Thermo-Dynamics of Plates and Shells

Foundations of Engineering Mechanics Series Editors: V.I. Babitsky, J. Wittenburg Jan Awrejcewicz . Vadim A. Krysko . ...

Author: Jan Awrejcewicz | Vadim Anatolevich Krys'ko | Anton V. Krys'ko

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Foundations of Engineering Mechanics Series Editors: V.I. Babitsky, J. Wittenburg

Jan Awrejcewicz . Vadim A. Krysko . Anton V. Krysko

Thermo-Dynamics of Plates and Shells

Series Editors: V.I. Babitsky Department of Mechanical Engineering Loughborough University Loughborough LE11 3TU, Leicestershire United Kingdom

J. Wittenburg Institut f u¨ r Technische Mechanik Universit¨at Karlsruhe (TH) Kaiserstraße 12 76128 Karlsruhe Germany

Authors: Jan Awrejcewicz Department of Automatics and Biomechanics Faculty of Mechanical Engineering Technical University of Lodz 1/15 Stefanowskiego St., 90-924 Lodz Poland

Anton V. Krysko Department of Mathematics Saratov State University 410054 Saratov, Russia

Vadim A. Krysko Department of Mathematics Saratov State University 410054 Saratov, Russia

ISSN print edition: 1612-1384 ISBN-10: 3-540-34261-3 ISBN-13: 978-3-540-34261-8

Springer Berlin Heidelberg New York

Library of Congress Control Number: 2006926221 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX package Typesetting:Data conversion by the author\and SPi using Springer LT Cover-Design: deblik, Berlin Printed on acid-free paper SPIN: 11603313 89/3100/SPI - 5 4 3 2 1 0

Preface

The present monograph is devoted to nonlinear dynamics of thin plates and shells with termosensitive excitation. Since the investigated mathematical models are of diﬀerent sizes (two- and three-dimensional diﬀerential equation) and diﬀerent types (diﬀerential equations of hyperbolic and parabolic types with respect to spatial coordinates), there is no hope to solve them analytically. On the other hand, the proposed mathematical models and the proposed methods of their solutions allow to achieve more accurate approximation to the real processes exhibited by dynamics of shell (plate) - type structures with thermosensitive excitation. Furthermore, in this monograph an emphasis is put into a rigorous mathematical treatment of the obtained diﬀerential equations, since it helps eﬃciently in further developing of various suitable numerical algorithms to solve the stated problems. It is well known that designing and constructing high technology electronic devices, industrial facilities, flying objects, embedded into a temperature field is of particular importance. Engineers working in various industrial branches, and particularly in civil, electronic and electrotechnic engineering are focused on a study of stress-strain states of plates and shells with various (sometimes hybrid types) damping along their contour, with both mechanical and temperature excitations, with a simultaneous account of heat sources influence and various temperature conditions. Very often heat processes decide on stability and durability of the mentioned objects. Since numerous empirical measurement of heat processes are rather expensive, therefore the advanced precise and economical numerical approaches are highly required. A brief monograph description follows. Chapter 1 of this monograph is devoted to a study of three-dimensional problems of theory of plates in a temperature field. First, a brief historical outline as well as a state-of-art of the mentioned problems is described in introductional section. In Section 1.2, the system of diﬀerential equations governing a broad class of problems in the coupled dynamic theory of thermoelasticity in three-dimensional formulation is derived. A diﬀerence variational approximation is given and the diﬀerence scheme error is derived. Also stability of an explicit diﬀerence scheme is rigorously studied. Section 1.3 includes a comparison of solving systems governed by either hyperbolic or elliptic equations through various iterative methods.

VI

Preface

In section 1.4 numerous results of solutions of broad class of elasticity and thermoelasticity problems including coupling of temperature and deformations, are illustrated and discussed. In Chapter 2, after a brief historical research review, the variational equations for shallow anisotropic shells embedded into a temperature are derived. Coupling conditions and stress-strain state of shallow shells are formulated. In section 2.2 universality and eﬃciency of finite diﬀerence method devoted to boundary value problems for elliptic equations if outlined. Diﬀerence schemes for series of multidimensional stationary heat transfer equations are proposed in both sections 2.2 and 2.3. In the last section 2.4, influence of heat sources on a shell stress-strain and its stability is studied. Chapter 3 is devoted to analysis of dynamical behaviour and stability of closed cylindrical shells subject to continuous thermal load. A brief historical background is followed by variational formulation of the coupled dynamical problem of thermoelasticity. Hybrid-type variational equations of thin conical composite orthotropic thermosensitive shells are derived, and a problem of their solution is rigorously discussed. Furthermore, a solution to the biharmonic equation in relation to forcing function, as well as reliability of the obtained results, are addressed. Dynamical stability loss and non-uniform thermal loading are also studied. Dynamical behaviour and stability of rectangular shells is addressed in Chapter 4. In section 4.1, the computational algorithm to analyse diﬀerential equations with the associated boundary conditions is derived. The associated finite diﬀerence equations are given, and reliability of the results are verified. Stationary state method to analyse statical and dynamical problems is illustrated in section 4.1.4. Various vibrational phenomena and stability loss are studied. Stability of thin shallow shells with both transversal and heat loads are examined in section 4.2. Section 4.3 is devoted to stability of thin conical shells subject to both longitudinal load and heat flow. Finally, dynamical stability of flexurable conical shells with convection is studied in section 4.4. In Chapter 5 dynamics and stability of flexurable sectorial shells with thermal loads are addressed. First, theory of flexurable sectorial shells is introduced. The fundamental relations are assumed, diﬀerential equations are derived and initial conditions are given. After introduction of a thermal field the numerical “set-up” technique is illustrated and discussed, and numerical results reliability is outlined. Then various examples of stability of sectorial shells with finite deflections are studied. In addition, chaotic dynamics of sectorial shells and its control is addressed. Chapter 6 is devoted to a study of coupled problems of thin shallow shells in temperature field within the Kirchhoﬀ-Love kinematic model. Fundamental assumptions and relations are introduced, and the diﬀerential equations are derived. The finite diﬀerence model of a solution to three dimensional heat conductivity equation is formulated. Numerical algorithm to solve the obtained equations is proposed, and then numerous examples of investigation of stability loss of shallow rectangular shells follow. Additional original method to solve a coupled thermoelastic problem is also proposed.

Preface

VII

In chapter 7 a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. The proposed method makes it possible to benefit from the essential advantages of both the direct method (universality, finitness of a computational process, exactness) and the iterational one (minimal amount of operational storage). Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in R3 is considered, where boundary value problems of the 1st, 2nd or 3rd order, or their combinations are taken into account. The governing diﬀerential equations are reduced to algebraic ones with the help of the finite element and the boundary element methods for diﬀerent meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a non-homogeneous shallow physically and geometrically non-linear shell subject to transversal uniformly distributed load. The partial diﬀerential equations are reduced to a system of non-linear algebraic equations with the error of O(h2x1 + h2x2 ). The linearization process is realized through either Newton method or diﬀerentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported. In the last Chapter 8, some rigorous mathematical treatments of a coupled thermomechanical problems are addressed. First, the suﬃcient conditions of existence, uniqueness and continuity dependence on initial data of the Cauchy problem solutions for diﬀerential-operational equation of hybrid type (a part of the equation is of hyperbolic type, and another part is of parabolic type) are given. It is shown that if the operational coeﬃcients are suitably chosen, the investigated equation can model a diﬀerential equations governing vibrations of a plate, i.e. the modified GermainLagrange equation of thermal conductivity (a parabolic equation). Second, a coupled thermo-mechanical of non-homogeneous shells with variable thickness and variable Young modulus (a so-called Timoshenko type model) is studied. The investigated problem is reduced to uniformly correct problem in the first form of a first order diﬀerence equation. Third, boundary conditions for a non-homogeneous first order operator – diﬀerential equation possessing a unique solution are derived. Two important theorems are formulated. Lodz, Saratov October 2003

J. Awrejcewicz V.A. Krysko A.V. Krysko

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

1

Three–Dimensional Problems of Theory of Plates in Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid . . . . . . . . . . . . 11 1.2.1 Variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Diﬀerence approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.4 Diﬀerence approximation Error . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.5 Diﬀerence approximation Stability . . . . . . . . . . . . . . . . . . . . . . 29 1.3 Methods of Solving Diﬀerence Equations . . . . . . . . . . . . . . . . . . . . . . 39 1.3.1 Dimensionless Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.3.2 Systems of Elliptic Diﬀerence Equations . . . . . . . . . . . . . . . . 41 1.3.3 Systems of Parabolic and Hyperbolic Diﬀerence Equations . 47 1.3.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.3.5 Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.3.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.4 Linear Problems in the Theory of Plates in 3D Space . . . . . . . . . . . . . 59 1.4.1 Static Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.4.2 Dynamic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1.4.3 Non-stationary temperature field . . . . . . . . . . . . . . . . . . . . . . . 82 1.4.4 Comparison of Solutions – non-isothermal Processes in Static Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.4.5 Inner heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 1.4.6 Deformation and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 114 1.5 3D Physically Non-Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 1.5.1 Diﬀerential equations and diﬀerence approximation . . . . . . . 130 1.5.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 1.5.3 Estimation of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 1.5.4 Temperature and Deformation Coupling . . . . . . . . . . . . . . . . . 136

2

Stability of Rectangular Shells within Temperature Field . . . . . . . . . . . 149 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 2.2 Flexible Anisotropic Shallow Shells in Temperature Fields . . . . . . . . 152

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2.2.1 Problem formulation and assumptions . . . . . . . . . . . . . . . . . . . 152 2.2.2 Fundamental relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2.2.3 Variational and diﬀerential equations . . . . . . . . . . . . . . . . . . . . 159 2.2.4 Boundary and compatibility conditions . . . . . . . . . . . . . . . . . . 167 2.2.5 Compatibility conditions for shallow shells equations . . . . . . 177 2.2.6 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 2.3 Solution of 3D Stationary Heat Transfer Equation . . . . . . . . . . . . . . . 186 2.3.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 2.3.2 Construction of diﬀerence schemes . . . . . . . . . . . . . . . . . . . . . 194 2.3.3 A priori convergence estimation . . . . . . . . . . . . . . . . . . . . . . . . 206 2.3.4 Algorithm of computation and compatibility conditions . . . . 209 2.3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 2.4 Algorithm for Diﬀerence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 2.4.1 Construction of diﬀerence equations . . . . . . . . . . . . . . . . . . . . 227 2.4.2 Stability problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 2.4.3 Reliability of obtained results . . . . . . . . . . . . . . . . . . . . . . . . . . 234 2.4.4 Transversal load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 2.4.5 Diﬀerent boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.5 Computations of Plates and Shells in a Temperature Field . . . . . . . . . 252 2.5.1 Stress-strain state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 2.5.2 Stress-strain state and shells stability . . . . . . . . . . . . . . . . . . . . 264 3

Dynamical Behaviour and Stability of Closed Cylindrical Shells . . . . . 267 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells . . . . . . . . . . . 276 3.2.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 3.2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3.2.3 Hybrid-Type Variational Equations . . . . . . . . . . . . . . . . . . . . . 283 3.2.4 Solution Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 3.2.5 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 3.3 Computational Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 3.3.1 Finite Diﬀerence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 3.3.2 Solution to Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . . 315 3.3.3 Reliability of the Obtained Results . . . . . . . . . . . . . . . . . . . . . . 320 3.3.4 Modified Relaxation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 328 3.4 Dynamical Stability Loss with Ununiform Force Excitation . . . . . . . 334 3.4.1 Criteria of Dynamical Stability Loss (A Review) . . . . . . . . . . 334 3.4.2 Nonuniform Impulse External Pressure . . . . . . . . . . . . . . . . . . 342 3.5 Dynamical Stability Loss and Non-uniform Thermal Load . . . . . . . . 366 3.5.1 Thermal Field Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 3.5.2 Influence of Time, Shell Geometry and Load . . . . . . . . . . . . . 373 3.5.3 Combined Static and Thermal Loads . . . . . . . . . . . . . . . . . . . . 382

Table of Contents

XI

4

Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 4.2.1 Diﬀerential Equations, Boundary and Initial Thermoelastic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 4.2.2 Finite Diﬀerence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 4.2.3 Reliability of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 4.2.4 Stationary State Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 4.3 Stability of Thin Shallow Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 4.3.1 Influence of Heat Stream Intensity . . . . . . . . . . . . . . . . . . . . . . 432 4.3.2 Shells with Transversal Load and Heat Flow . . . . . . . . . . . . . 436 4.3.3 Influence of Thermal and Mechanical Characteristics . . . . . . 446 4.4 Stability of Thin Conical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 4.4.1 Boundary Conditions and Surrounding Medium . . . . . . . . . . 458 4.4.2 Constant Compressing Load and Heat Flow . . . . . . . . . . . . . . 464 4.4.3 Harmonic Longitudinal Load and Heat Flow . . . . . . . . . . . . . 466 4.5 Stability of Flexurable Conical Shells with Convection . . . . . . . . . . . 479 4.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 4.5.2 Boundary and Thermal Fields Conditions . . . . . . . . . . . . . . . . 480 4.5.3 Critical Temperature Versus Heat Transfer Coeﬃcient . . . . . 483

5

Dynamical Behaviour and Stability of Flexurable Sectorial Shells . . . . 493 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 5.2 Flexurable Conical Sectorial Shells Computations . . . . . . . . . . . . . . . 498 5.2.1 Fundamental Relations, Diﬀerential Equations, Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 5.2.2 Thermal Field and Set-Up Method . . . . . . . . . . . . . . . . . . . . . . 509 5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 5.3 Stability of Sectorial Shells with Finite Deflections . . . . . . . . . . . . . . 520 5.3.1 Influence of the Sector’s Angle . . . . . . . . . . . . . . . . . . . . . . . . . 522 5.3.2 Set-Up Method and Determination of Critical Loads . . . . . . . 560 5.3.3 Heat Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 5.3.4 Local Surface Load With Infinite Duration . . . . . . . . . . . . . . . 603 5.4 Chaotic Dynamics of Sectorial Shells . . . . . . . . . . . . . . . . . . . . . . . . . . 614 5.4.1 Statement of the problem and computational algorithm . . . . . 614 5.4.2 Static problems and reliability of results . . . . . . . . . . . . . . . . . 617 5.4.3 Convergence of a finite diﬀerence method along spatial coordinates for non-stationary problems . . . . . . . . . . . . . . . . . 618 5.4.4 Investigation of chaotic vibrations of spherical sector-type shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 5.4.5 Transitions from harmonic to chaotic vibrations . . . . . . . . . . . 627 5.4.6 Control of chaotic vibrations of flexible spherical sector-type shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

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6

Coupled Problems of Thin Shallow Shells in a Temperature Field . . . . 633 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 6.2 Fundamental Assumptions and Relations . . . . . . . . . . . . . . . . . . . . . . . 634 6.3 Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 6.4 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 6.5 Solution to 3D heat conductivity equation . . . . . . . . . . . . . . . . . . . . . . 640 6.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 6.8 Additional Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

7

Novel Solution Method for a System of Linear Algebraic Equations . . 671 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 7.2 Elimination method for equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 7.3 Numerical solution of a three-dimensional equation of elliptic type . 684 7.4 Computation of geometrically non-linear non-homogenous shallow shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

8

Mathematical Approaches to Coupled Termomechanical Problems . . 705 8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 8.1.2 Basic assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 8.1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 8.2 On the Solution of a Coupled Thermo-mechanical Problem . . . . . . . 713 8.2.1 Introduction and Statement of the Problem . . . . . . . . . . . . . . . 713 8.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 8.3 On the Solvable Operators Generated by Uniformly Correct Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 8.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

In section 1.1 historical outline putting emphasis on not solved problems in threedimensional formulation of plates thermoelastic theory is given. Section 1.2 presents a system of diﬀerential equations describing a broad class of problems of the coupled dynamic theory of thermoelasticity in a complete, threedimensional formulation including material’s non-homogeneity. The investigated system of equations has been supplemented with an equation at singular points of the examined space (a cubicoid), such as ribs, corners and simple points where various boundary conditions meet. A diﬀerence approximation of the initial diﬀerential system has been formulated with the use of the variational-diﬀerence method (the method of integral identity). The margin of the diﬀerence scheme error has been estimated. A theorem concerning stability of an explicit diﬀerence scheme has been proven and the condition of stability that guarantees weak convergence of the difference scheme’s solution towards the solution of a diﬀerential system has been obtained. Section 1.3 contains a comparison of solving systems of hyperbolic equations (using an explicit diﬀerence scheme based on applying Runge-Kutta’s method with automatic choice of an integration step and Runge-Kutta’s method with a constant step). Additionally, the section presents a comparison of applied iterative methods of solving systems of elliptic equations (Seidel’s method, the upper relaxation, the explicit and implicit methods of variable directions, and the explicit method of variable directions with the so-called Chebyshev’s acceleration). Several model problems have been used to draw the comparisons and the most economical methods have been applied as far as accuracy of solutions and computation time are concerned. Algorithms of the described methods have been formulated and a package of programs for solving problems of statics, quasistatics and elasticity and thermoelasticity dynamics has been created. An optimum choice of a spatial mesh step and an integration step within a time interval has been made and legitimacy of the theoretically obtained (in the first section) stability condition has been numerically confirmed. Feasibility of the obtained results has also been proven by means of comparison with real processes. Section 1.4 presents numerous results of solutions to a broad class of elasticity and thermoelasticity problems within the range of static, quasistatic and dynamic problems. There is also an analysis of the influence of the temperature and deformations’ coupling’s eﬀect using some examples of thermal and mechanical impacts.

2

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Finally, section 1.5 contains formulation of the equations of coupled dynamic three-dimensional problems with physical non-linearities. Moreover, the finite difference methods, Runge-Kutta’s method and the method of additional loads have been combined to form a numerical algorithm of solutions. Convergence of an approximate solution to the real one (the one searched for) has been analysed. The results of problems concerning thermal and mechanical impacts beyond the elasticity fields have been presented and the eﬀects of the influence of reciprocal temperature and deformation fields’ coupling on the analysed processes have also been investigated in this chapter.

1.1 Introduction While designing and constructing electronic devices, industrial facilities, flying objects or technological instrumentation, the problems related to heat processes are particularly important. They appear due to the use of new materials, more complex loads aﬀecting every single element of analysed objects, and also due to an increase of permissible heat loads in devices of smaller and smaller dimensions. As it is generally known, heat processes determine stability of functioning and durability of analysed objects. On the other hand though, numerous empirical measurements of heat processes are extremely complex and expensive. Therefore, exact computational analyses (numerical, as well as analytical) ought to be conducted in order to obtain constructions of optimum characteristics. In fact, non-stationary temperature reactions in surrounding environment require more accurate calculations than classic modelling of thermomechanical phenomena. In 1845, Duhamel [188] was the first to formulate the theory of elasticity regarding thermal stresses. However it was not until 1956, that Biot [107] introduced a dissipation function into a thermal conduction equation to account for the heat caused by the material’s deformation. Thus, the problem of thermoelasticity and the variational principle of coupled theory of thermoplasticity were first formulated. Since then there has been a great interest in that sort of problems. Earlier works on the theory of thermoelasticity [188] presented a dominating view that a change of temperature within a time interval is small, and therefore it was possible to apply a simplified (quasistatic) method, that is to neglect inertial terms in equations of motion, without the risk of major errors. The next step, introduced by means of the theory of thermoelasticity to simplify the problem, was neglecting dilatation terms in heat conduction equations. Sometimes, when both of the above mentioned terms are neglected in diﬀerential equations [598], the solution of a static problem is found. It turns out though, that due to the significance of the problems such simplifications ought not to be made. Among such problems are: the problem of investigating stress waves in deformable bodies; the problems related to determining thermoelastic vibrations; the problems related to investigating stability of conservative elastic systems [119, 164, 267, 316, 356, 466]. In their works, Danilovskoya [160, 161, 162, 163, 164], Kartashova and Shefter [316] analysed the influence of inertial terms on bodies’ behaviour considering the inertia forces. They

1.1 Introduction

3

also proved that neglecting a dilatation term does not ensure qualitatively satisfactory results due to ineﬃcient examination of the coupling coeﬃcient’s influence on the phenomenon. All the factors mentioned above caused a growth of interest in complete (i.e. not simplified) problems which fruited in numerous analytical works. Works of Karlsoy and Eger [315], Lykov [451], Kovalenko [355] and Nowacki [512] contain analyses and generalisation of two, so far independent disciplines, i.e. the theory of elasticity and the theory of heat conduction, and also a definition of so called coupled problem. A full formulation of the principles of variational theories of thermoelasticity is to be found in works [107, 265]. Betti’s theorem on reciprocity of virtual works is discussed in monograph [516], and a generalisation of Maizel’s method may be found in work [453]. Formulation of flat and space problems of coupled quasistatic theory of thermoelasticity is described in the works of Podstrigach, Schvetz, and Nowacki [512, 516, 545, 546, 547, 548]. Nowacki’s monograph [513] introduces equations of the coupled theory of thermoelasticity into wave equations and a method of solving linear and non-linear variants of the problems listed above. Many popular methods of solving the equations of Galerkin’s [215] or Papkovich’s [528] classic theories of elasticity are generalized in Podstrigach’s or Nowacki’s works and applied into the theory of coupled thermoelasticity. The method of solving problems of the coupled theory of thermoelasticity in case of a boundless space was proposed by Zorski [727], who used Green’s function to solve a heat conduction equation and considered dilatation to be a heat source. Chadwick’s work [145] takes up generalized problems of solving boundary problems of the coupled theory of thermoelasticity with the use of integral methods, whereas Souler and Brul use the small parameter method [632]. The problems related to accuracy of formulated boundary problems of the coupled theory of thermoelasticity were described first in book [119], which investigates an initial boundary problem for an isotropic body, later extended also onto an anisotropic body in Ionescu work [277]. Numerous dynamical problems of mathematical physics apply various integral transformations, including Laplace’s transformation [294], the solution of which is related to the use of Fourier’s series. In their work, Kupradze and others [398] propose their theory of multidimensional singular integral equations that makes it possible to investigate the static and dynamic problems of stabilised continuous systems’ vibrations. Hybrid problems, investigated by Magnaradze [452], Kupradze and Burchuadze [397] may be solved with generalized integrals that correspond to diﬀerential equations with the use of harmonic and analytical functions. Defermos’ work [175] contains many theorems concerning basic problems of the theory of thermoelasticity, including their proofs. Work [101] investigates the so-called second and third boundary and initial boundary problems of the coupled theory of thermoelasticity with the use of the method of potential and Laplace’s transformation. Work [397] analyses four basic three-dimensional boundary problems of the theory of thermoelasticity in case of harmonic vibrations of a homogeneous isotropic medium with the following conditions set in its boundaries:

4

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1) displacement and distribution of temperature; 2) thermal stress and thermal flux; 3) displacement and thermal flux; 4) thermal stress and distribution of temperature. In addition, the authors formulate and prove many theorems concerning the existence and uniqueness of the above mentioned problems. The solutions to all of the four types of boundary conditions, presented in the form of generalized Fourier’s series, are to be found in Burchuladze’s work [135]. Fundamental results referring to the initial boundary problems of the theory of thermoelasticity have been obtained in the work of Kachnashviliev [294]. Nevertheless, fundamental solutions are still being perceived as classic. The conditions of smoothness appear to be too diﬃcult to achieve for solutions of a wave equation describing impact processes. Due to the fact that such solutions do not have derivatives of the first order, they need to be examined from a generalized perspective. Integral relations contain information about solutions and emphasise physical phenomena because information on solution’s smoothness is partially lost in diﬀerential equations. The generalized mathematical theory on diﬀerential equations of the coupled theory of thermoelasticity described by means of both hyperbolic and parabolic equation has been formulated relatively recently. The works of Ladyzhenskaya [405] and Ilyisn [276] that were published in early fifties, contain numerous vital results referring to the theory of boundary problems for one hyperbolic or parabolic equation of a general type. In order to prove the existence and uniqueness of a generalized equation, it is necessary to make an entirely new a priori estimation that would take into account the right parts of equations in the form of the weakest norm and thus would accurately emphasise the physical aspect of the problem. Qualitatively most adequate examinations of general solutions seem to be the ones that apply the finite diﬀerence method. The method definitely stands out among many other approximate methods. Owing to continuing research of Samarskiy, Gulin, Nikolaev [591, 593, 594, 595], a large number of problems concerning stability of diﬀerence schemes for all types of one-dimensional equations in mathematical physics have been solved. This also started the research on diﬀerence schemes in the theory of elasticity. Let us list only a few examples of important results obtained with the use of the theory of diﬀerence schemes. Work [419] describes an a priori estimation of a solution in spaces W22,2 , W22,1 made by means of energy inequalities for dynamic problems of the theory of thermoelasticity using Dirichlet’s homogeneous boundary conditions. The authors have also constructed and examined a non-overt diﬀerence scheme and proved its convergence. In his work [483], Moskalkov presents a method of constructing diﬀerence schemes for the coupled theory of thermoelasticity boundary problems that is also useful for the equations of variable or discontinuous coeﬃcients. Work [541] proposes a variational-diﬀerence formulation of the diﬀerence scheme of the coupled theory of thermoelasticity problems. Work [341] proves convergence of the diﬀerence solution towards the solution of a general hybrid problem for a hyperbolic equation with variable coeﬃcients. It also shows how to improve the accuracy of presently applied diﬀerence schemes. In works [419, 694], the relation between the smoothness of a solution to the coupled theory of thermoelasticity one-dimensional dynamic problems and the smoothness

1.1 Introduction

5

of input data is examined. Smoothness is examined with the use of terminology applied for Hilbert and Sobolev’s spaces. Two diﬀerence scheme families have been constructed and their stability and convergence have been studied. Works [419, 693] extend the investigated problems by taking into account two-dimensionality or many so-called layer problems. It is worth noticing that at present, many finite diﬀerential problems modelling the flat problem of the dynamic theory of elasticity and the theory of thermoelasticity have already been solved. A large number of schemes described by displacements of high accuracy, stability and short computation time have also been presented [79, 96, 97, 345, 484, 591, 592, 664]. Among the less thoroughly examined problems are the ones that refer to the diﬀerential method of solving initial-boundary problems of the three-dimensional theory of elasticity and the theory of thermoelasticity. A review work by Suslova [643] contains a broad bibliography of works on research focused on solving boundary problems of the three-dimensional theory of elasticity. It also lists several works concerning the theory of thermoelasticity [142, 293, 643]. In works [198, 199] Ermolenko describes constructing the solution of a hybrid problem for a cubicoid by cutting the finite space out and he proves stability and convergence of the cubic diﬀerence process by applying the transformation of Lam´e’s equations. He compares the result obtained in this way to the accurate one. In works [339, 340] Konovalov describes stability conditions for diﬀerence schemes for two-dimensional dynamic and static hybrid problems. The development of computational methods using computers and special algorithms has led to a sudden progress in the discussed field of science. A major contribution in the development of computational methods in the research on the dynamics of continuous media has been brought by the works of Godunov [224], Kukudzanov [393], Neuman [500], Rachmatulin [561, 562], Richtmyer [572], Wilkins [703] and Janenko [287]. Numerous examples of computations regarding the mechanics of a continuous medium are included in monographs [225, 287, 394, 573]. The problem of the coupled theory of thermoelasticity still remains a live issue due to its potential application and the numerical methods allow drawing a great deal of conclusions of a general nature. The examples of these may be the research and solutions of coupled thermoelasticity problems with the use of numerical methods for a number of particular issues: in work [546], Galerkin’s method is applied for solving a coupled problem in a finitely dimensional space with the use of a three-dimensional model; in work [616], the same method is applied to solve a two-dimensional problem; in work [430], a half-space finite diﬀerence method is applied for a one-dimensional problem, and in works [220, 721] – for a three-dimensional problem. In work [266], Huang and Shich compare solutions of free vibration problems regarding thermal processes in plates and spherical shells by applying dynamic and quasistatic theories. Non-stationary thermoelasticity problems for an infinite two-layered and initially heated plate consisting of various materials and thermally processed through interaction with fluids within Newton’s laws, have been examined in work [646]. Work [649] analyses stress-strain states of thick two-layered spheres with regard to axially symmetrical heat sources (the problem has been solved with

6

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

the use of the quasistatic theory). Work [648] investigates a system of coupled thermoelasticity diﬀerential equations with the use of a cylindrical coordinate system. Fourier’s method has been used to examine stress-strain states in a long circular cylinder with inserted rigid rings in work [504]. The finite diﬀerence method has been used to solve the problem of thermoelasticity for a rectangular orthotropic plate with regard to the dependence of its certain characteristics on temperature in work [641]. Work [663] investigates a non-stationary coupled thermoelasticity problem for an infinitely long, thick plate. The plate’s surfaces have been subjected to intensive heating and the coupling between the temperature field and the deformation has been analysed. The distribution of the temperature field in time has also been examined, as well as concentration of the stresses depending on the size of the stress field and the material’s thermodynamic properties. Dynamic loss of stability of thin plates has been analysed with the use of finite diﬀerence method in work [191], taking into account the eﬀect of reciprocal coupling of the temperature field and the deformation field. Work [324] presents a solution to the coupled thermoelasticity problem for a thin rectangular shell aﬀected by a three-dimensional temperature field. It also mathematically proves the convergence of the obtained approximate solution. All of the above mentioned works point out the diﬀerences which appear in solutions if the coupling of the deformation (strain) fields and the temperature fields are not taken into account. An increase of the coupling coeﬃcient leads to an increase of interactions, which consequently leads to damping of the produced thermoelastic waves. Works by Karnauchov [312] and Pobedria [541] are focused on the problem of coupling in the theory of thermoelasticity. The influence of coupling on the stressstrain state of elastic and elastoplastic constructions has been investigated in work [359]. Several works of Day [169, 170, 171, 172, 173, 174] are also worth attention since the author investigates the conditions of legitimacy of applying approximations of unbounded theory of thermoelasticity and also the conditions of applying the properties of the solutions of heat conductivity equations to the solutions of a onedimensional dynamic coupled thermoelasticity problem’s equations. Research on thermal processes with regard to finite velocity of heat transfer is another direction in the development of the theory of thermoelasticity, since an entire class of physical processes (highly intensive thermal processes, laser rays) should be presented from the perspective of generalized Fourier’s law [451]. Works [323, 429, 495, 496, 558, 627] have been dedicated to the research on dynamic processes in solid bodies with regard to the heat transfer finite speed. In the works of Engelbrecht and Ivanov [285], an analysis of one- and two-dimensional models of wave processes have been made. In Kolyano and Shter’s work [337], a variational principle of reciprocal coupling of thermoelasticity for non-homogeneous media has been investigated using a cantilever beam as an example. Coupling of the deformation field and the temperature field significantly aﬀects the solution’s character, especially in the problems of spreading impact fields in thermoelastic bodies. Therefore, the research on the dynamic coupling eﬀects occurring in thermoelastic bodies subjected to simultaneous thermal, impulse, impact and mechanic treatment is one

1.1 Introduction

7

of the most important issues these days. Danilovskaya [163, 164] was the first to examine the dynamic eﬀect in the “impact” problem along a half-space. The research was consequently carried on by Mura [489]. If the temperature on the surface of a body changes at a limited speed instead of sudden leaps, then the problem may be solved with a small parameter method [494]. In Pobrushin’s work [544], an analysis of some one-dimensional initial-boundary problems with thermal and mechanical impacts along the symmetry axis of an infinite rod has been made. The dynamic coupled thermoelasticity problem for a half-infinite plate at a simultaneous increase of temperature on its edge and with the use of Laplace’s integral transformation including the small parameter method has been solved in Sidlar’s work [617]. Dynamic behaviour of thin cylindrical shells subjected to impetuous thermal treatment has been investigated in work [632]. A coupled system of diﬀerential equations is derived with the use of Bubnov-Galerkin method and variational theorems, and also a simple-supported infinite cylindrical shell is investigated. Work [359] investigates dynamic thermoelastic processes during heat impacts in such construction elements as plates or spherical and cylindrical shells. The research has been conducted with the use of dynamic coupled thermoelastic equations and dynamic non-coupled equations of thermoelastoplasticity, and with the method of reduction to a series of non-coupled quasistatic problems, which in turn have been solved with KrylovBogolubov method. In Kuvyrkin’s work [402], a heat impact in the surface layer of a body limited by a curvilinear surface has been investigated. Shatalov’s work [608] shows that a decrease of equations’ couplings leads to a decrease of strain in the front of a thermoelastic wave. A method of expansion into power series in regard to a small parameter being the thermomechanical coupling has been applied in that case. Gayvas’ work [221] presents an analytical solution to a thermoelasticity problem for a plate with discontinuity caused by heat impact. The behaviours of plates subjected to steady mechanical load and rapid thermal transients on their both surfaces have been investigated in work [231]. Few of the solved problems that are related to impacts belong to the class of problems with aperiodic excitations. In this respect the theory of thermoelasticity seems to be a little underdeveloped and it faces some significant mathematical problems. Due to simultaneous mechanical and thermal impacts in constructions some small plastic deformations are ignored. The first work focused on investigation of elastoplastic stress states was published by Iliushin [272], and later by Rogoshinskov, who took non-uniformity of heating into account. Many works analyse also particular problems. Ionov’s works [278, 498] based on the theory of small elastoplastic deformations are among them. Work [148] describes a stress-strain state of an infinitely long cylindrical shell subjected to heating. In a series of works by Piskun [538, 539], cylindrical shells subjected to non-uniform heating and internal pressure have been examined. Work [307] contains some computations of thermoplastic deformations based on the variationaldiﬀerence method, and work [109] describes a stress-strain state of rotational shells in conditions of axially symmetrical heating. Monographs [609, 610] present a theory and computational methods concerning many problems of thermoplasticity at variable loads including also the history of loading (the objects of study included

8

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

cylinders, disks and low lift rotational shells). Work [126] applies Iliushin’s theory of plasticity to deal with heating an isotropic sphere with heat impacts of various shape and length (the problem was solved as a non-coupled one). Analytical description of thermoelastoplastic deformations is published in work [583]. In work [242], Birger’s method is applied to solve non-linear elasticity problems. Many interesting conclusions concerning dependence of physical and material parameters on temperature and work regime related to cooling shells and plates have been drawn in work [417]. Work [399] formulates a functional in order to find a variational solution to a plasticity theory problem at changing temperature for an elastoplastic material. Work [261] investigates the influence of the temperature load history, and work [150] analyses unique and continuous dependence on initial conditions in dynamic problems of non-linear thermoelasticity. A theory and a method of solving problems of thin-walled constructions heated by stationary and non-stationary heat sources are described in work [336], in which the dependence of physical and mechanical characteristics on temperature has been taken into account. A combination of the method applied for the theory of thermoelasticity with Vlasov’s variational method has been used to solve a three-dimensional problem of non-linear thermoelasticity in work [357]. It needs to be emphasised that coupling of the temperature and deformation fields (also in a quasistatic case) for problems of non-elastic material characteristics is taken into account only in selected works [180, 217, 259, 350, 584]. A recent Polish publication edited by Wo´zniak [708] contains a synthetic and abundant presentation of the level of modern knowledge of the theory of elastic plates and shells with specific reference to the contribution of Polish scientists in its development. In contrast to that approach this monograph puts more light to the contribution of scientists from the former eastern bloc into the development of the theory of plates in the temperature field. It is worth emphasising that names of the two first authors of this book are connected with a series of monographs on the theory of plates and shells published in Polish [37, 38, 39, 48, 50, 51, 53]. The latest theoretical achievements in non-classic analyses of the thermoelastic shell theory problems are described in monograph [39]. Numerous aspects of non-linear dynamics of shells and plates, including bifurcations, chaos and solitons, have been analysed in other works of the two first authors of this monograph [41, 45, 46, 47, 49, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 389, 390], which also seem to be worth recommendation for readers who wish to broaden their knowledge in the field of shells and plates. At this point, several conclusions need to be drawn. (i) All of the above mentioned works investigate classic initially-boundary problems, while a typical (combined) boundary conditions are the most important in the theory of elasticity and thermoelasticity. There is a noticeable lack of solutions of that type in both linear and non-linear problems. (ii) There is no evidence for stability of diﬀerence schemes of the coupled theory of thermoelasticity in three-dimensional formulation for a cubicoid. (iii) Complexity of a physically non-linear system of diﬀerential equations limits the number of examples of solutions to thermoelastoplastic problems to only a few.

1.1 Introduction

9

The authors of this chapter focused their attention on solving the following problems: 1) construct a system of diﬀerential equations of the coupled dynamic theory of thermoelasticity taking into account a three-dimensional model and singularities of all kinds; 2) apply the variational-diﬀerence method for solving the coupled thermoelasticity theory problems; 3) prove stability of the diﬀerence approximation for the examined class of problems; 4) solve a typical problems of the theory of elasticity and the theory of thermoelasticity; 5) formulate a method and solve physically non-linear, initially-boundary problems for a three-dimensional plate in the dynamic coupled approach, and examine the influence of temperature and deformation fields’ coupling. The following notation is used: xi , i = 1, ..., 3 W(x) t Q(x, t) hα n U(u1 , u2 , u3 ) T = T0 + θ T0 θ ατ λq λ c ei j σi j e

- coordinate of a point in space; - examined field; - time; - {x ∈ Ω(x), τ ∈ (τ0 , τ1 )}; lα - step in a mesh: hα = ; Nα - normal unit vector directed outside the field: ni, j+m = cos(ni, j+m , xi ); - displacement vector; - absolute temperature; - absolute temperature in a stress-free state; - temperature increase; - linear coeﬃcient of thermal expansion; - heat conduction coeﬃcient; - heat emission coeﬃcient; - thermal capacity; - strain tensor coeﬃcient; - stress tensor coeﬃcient; 3 - volumetric strains: e = eii ; i=1

λ, µ - Lam´e’s coeﬃcients: λ = E ρ ν P4 ∂Ωi 1 2 P(P , P , P3 ) f ( f1 , f2 , f3 )

- Young’s modulus; - material’s density; - Poisson’s ratio; - heat sources’ unit power; - plate’s wall; - volume (mass) force; - surface force;

E Eν ,µ= ; (1 + ν)(1 − 2ν) 2(1 + ν)

10

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

lα - plate’s dimension along xα axis; Nα - set of points of division towards xα axis; ω(ω1 × ω2 × ω3 ) - mesh surface: ω = {x(x1 , x2 , x3 ), xα ∈ ωα , α = 1, ..., 3}, ω ¯α = {xαiα , iα = 0, 1, ..., Nα−1 , Nα }, ωτ = ω1 × ω2 × ω3 × ω4 = ω × ω4 = {x(x1 , x2 , x3 , x4 ), xα ∈ ωα , α = 1, ..., 4}; 1 2 3 S (S , S , S ) - entropy vector; s - entropy flux; L2 (Ω) - Banach functional space of the following properties:

u2,Ω

⎛ ⎞ 12 ⎜⎜⎜ ⎟⎟⎟ 1 ⎜⎜⎜ ⎟ = ⎜⎜ |u|2 dx⎟⎟⎟⎟ ∼ u2,ω¯ = (u, u)ω¯2 , ⎝ ⎠

u x 2,Ω

Ω

⎛ ⎞ 12 ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ = ⎜⎜ u2x dx⎟⎟⎟⎟ ; ⎝ ⎠ Ω

W21 (Ω) - space of elements L2 (Ω) with generalized derivatives of the first order due to Ω and of the following properties: (u, υ)(1) 2,Ω =

(uυ + u x υ x )dx ∼ (u, υ)(1) 2,ω ¯ =

N

υ(x)u(x)h,

i=0

Ω

1

1 (1) 2 (1) 2 (u, (u, = u) ∼ u) ; u(1) 2,ω ¯ 2,Ω 2,Ω W21,0 (Ω) - Hilbert’s space composed of elements u(x, τ) belonging to space L2 (Qτ ), which have generalized derivatives of the first order due to Qτ of the following properties: (u, υ)(1,0) 2,Qτ =

(uυ + u x υ x )dxdτ, Qτ

12 (u, u)(1,0) ; u(1,0) 2,Qτ = 2,Qτ 2 β = 3(λ + µ)ατ , 3 υx = υ x¯ x =

υi+1 − υi , h

⎧ ⎪ ⎪ ⎨ hα , xα ∈ ωα α = ⎪ , ⎪ ⎩ h α , xα ∈ 0, lα 2 υ x¯ =

υi+1 − 2υi + υi+1 , h2

υi − υi−1 , h υ xy =

The following markings are applied:

υ0x =

h4 =

τ1 − τ0 , M

υi+1 − υi−1 , 2h

υi+1, j+1 − υi, j+1 − υi+1, j + υi j . hi h j

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

11

- free edge - simple support - clampededge - mechanicalimpact - thermalisolation - temperaturedistribution - thermalimpact

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid This chapter presents a variational method-based derivation of a system of coupled thermoelasticity diﬀerential equations for a three-dimensional plate, taking into account material’s non-homogeneity. The system includes equations within the plate’s field, at its edges, ribs in its corners and at simple contact points of numerous boundary conditions, which allows solving a substantial number of problems. A diﬀerence system is derived with the use of the variational-diﬀerence method by approximating the initial diﬀerential system with accuracy of such small values as O(h2 ). The obtained diﬀerence scheme’s stability theorem has been proven. 1.2.1 Variational equations We shall consider interaction between an elastic non-homogeneous body Ω and a medium that surrounds it in conditions in which thermal and mechanical processes are taken into account. Let us assume that at time instant τ = τ0 the body does not remain in the state of stress, i.e. the thermodynamic quantities that characterise the body such as absolute temperature T = T 0 , strain and stress tensor components and displacement vector components are equal to zero. Mechanical interaction makes displacement fields appear in the body. In every general case they accompany the change of the temperature field. Heating the body also causes perturbations in the investigated fields. Heat conductivity involves producing entropy, and strains cause a decrease of it, which in result leads to producing heat. Although thermoelastic damping is usually weak and for a short time interval it may be neglected (the noncoupled thermoelasticity theory), the relatively long-lasting processes require taking energy dissipation into account (the combined theory of thermoelasticity). Dissipation energy can be described by the following relation [63]: T 0 ∂S 2 1 dτ, (1.1) D= 2 λq ∂τ Ω

12

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

S =

cθ + βe, T0

(1.2)

where: S (S 1 , S 2 , S 3 ) is an entropy vector. Body Ω remains in motion, therefore according to Hamilton-Ostrogradski’s principle, integral τ1 (Π + K − A − B) dτ, (1.3) τ0

that describes the work utilised for the system’s movement within time interval (τ0 , τ1 ), assumes an extremal form in the movement’s trajectory: τ1 (Π + K − A − B) dτ = 0,

δ τ0

K=

ρ 2

3 Ω

2

ui dΩ,

(1.4)

(1.5)

i=1

where: K is the kinetic energy, Π = W + P denotes the elasticity potential, and in addition λ µei j ei j + e2 dΩ (1.6) W= 2 Ω

is the isothermal energy of strain. Moreover 1 c 2 1 4 θ dΩ + P θdΩ P0 = 2 T0 T0 Ω

(1.7)

Ω

denotes the thermal energy, B=

3 Ω

Pi ui dΩ

is the inertia forces’ work, whereas 3 i i f u d∂Ω + nθS n d∂Ω A= ∂Ω

i=1

(1.8)

i=1

(1.9)

∂Ω

denotes the external forces’ work. In spite of the fact that principle (1.3) does not take dissipation energy into account, it is essential to do it in energetic conditions of the coupled theory of thermoelasticity. That is why equation (1.4) takes the following form: τ1 (Π + K + D − A − B) dτ = 0.

δ τ0

(1.10)

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

13

Using Cauchy’s dependences ∂ui , ∂xi

eii =

ei j =

∂ui ∂u j + ∂x j ∂xi

(1.11)

and Duhamel-Neuman’s dependences σii = 2µei j + λe − βθ,

σi j = µei j ,

(1.12)

and expression (1.2), which is equivalent to the following dependence: θ=−

T0 div (S − βu) , c

(1.13)

we can transform expression (1.10). Thus we obtain a functional of the coupled thermoelasticity energy expressed by the displacement and entropy flux components. 1.2.2 Diﬀerential equations On basis of the energetic investigations discussed in 1.2.1 we shall construct a system of diﬀerential equations, the solution of which will be minimised by functional (1.10). A cubicoidal plate will serve as the object of investigation (Fig. 1.1). The edges and corners are characteristic for the surface of a cubicoid and they are sets of singular points. Also the points where the types of boundary conditions change and the points of application of concentrated central forces and heat sources

x2 l2

j W2 j W6 j W4

l1

j W5

jWm j W3

l3

j W1

x1

jWm

x3

Figure 1.1. The investigated cube-shaped plate.

14

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

can be called singular points. An analytical solution of the coupled thermoelasticity theory problems in the described field requires taking the field’s singularities into account. For instance, when applying the method of mesh it is necessary to thicken the mesh during the approach to the singular points. In this way the computation time needed to solve the problem will suddenly prolong. In order to avoid undesirable eﬀects it is necessary to create additional equations in the singular points, from now on called the consistency conditions, that will constitute a part of the diﬀerential equations which function as Euler’s system for functional (1.10), used by the authors of work [429]. In addition, surface integrals will be included to describe the whole of the additional conditions imposed on the plate at its edges, in its corners and places where boundary conditions meet. k ∂u ∂ui ∂uk ∂u j Dkk+m λ + (λ + 2µ) +λ − βθ nk,k+m d∂Ωk+m + R= ∂xi ∂xk ∂x j ∂xi ∂Ωk+m

k ∂ui ∂uk ∂u j ∂u λ + (λ + 2µ) +λ − βθ nk,k+m d∂Ωk+m + ∂xi ∂xk ∂x j ∂x j

Dkk+m ∂Ωk+m

∂uk ∂u j ∂u j µ + nk,k+m d∂Ωk+m + ∂x j ∂xk ∂xi

∂uk ∂ui ∂ui µ nk,k+m d∂Ωk+m + + ∂xi ∂xk ∂xi

∂uk ∂u j ∂ui µ + nk,k+m d∂Ωk+m + ∂x j ∂xk ∂x j

∂uk ∂ui ∂ui µ + nk,k+m d∂Ωk+m + ∂xi ∂xk ∂x j

j Dk+m ∂Ωk+m

Dik+m ∂Ωk+m

Dik+m ∂Ωk+m

Dik+m ∂Ωk+m

li Dik+m

i l j ∂ui ∂uk ∂ui ∂u ∂u j ∂ui i µ nk,k+m dxi + µ + D j+m + n j, j+m d∂x j + ∂xk ∂xi ∂xi ∂x j ∂xi ∂x j

0

0

lk Dii+m

i ∂ui ∂uk ∂u j ∂u λ + (λ + 2µ) +λ − βθ ni,i+m dxk + ∂xi ∂xk ∂x j ∂xk

0

Ai+m θ + Bi+m T i+m −

∂Ωi+m

Ci+m

0 Ai+m T i+m

2 λq ∂S i 1 ∂S i ∂S i ni,i+m + + Ci+m + S + ∂x j ∂xk 2 λ ∂τ

i

λq ∂S i ∂S i λq ∂S i ∂S i ni,i+m + Ci+m ni,i+m d∂Ωi+m + λ ∂τ ∂x j λ ∂τ ∂xk

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

∂Ωi+mn

li

15

2 λq ∂S i ∂S i ∂S i 1 0 ni,i+m + Ai+mn θ + Bi+mn T i+mn − Ai+mn T i+mn + Ci+mn + Si + ∂x j ∂xk 2 λ ∂τ λq ∂S i ∂S i λq ∂S i ∂S i 1 Ci+mn ni,i+m + Ci+mn ni,i+m ∂Ωi+mn + 2 λ ∂τ ∂x j λ ∂τ ∂xk

0 0 Ai+mn θ + Ai+mn θ + Bi+m T i+m + Bi+mn T i+mn − Ai+mn T i+m − Ai+mn T i+mn

0

∂S i ∂x j

λq ∂S i ∂S i λq ∂S i ∂S i Ci+mn ni,i+m + Ci+mn ni,i+m dxi , λ ∂τ ∂x j λ ∂τ ∂xk

+

∂S i + ∂xk

(1.14)

0 where: i → j → k, i, j, k = 1, ..., 3, m = 0, ..., 3, T i+mn (x, t), T i+mn (x, t) – are set respectively at the limit of the function of the heat flux and the medium’ temperature; j , Ai+m , Bi+mΛ , Ci+mΠ – are constants that assume values 0 or 1 depending on the Di+m type of the boundary conditions; indeces Λ and Π define parts of the plate’s wall (left or right), where the function is set. Additionally: l j lk li m=3 νd∂Ωi+m = ν dxk dx j 0 m=0 , ∂Ωi+m

0

0

l j lk/2 νd∂Ωi+mΠ =

m=3

i

∂Ωi+mΠ

0

0

l j

lk

νd∂Ωi+m = ∂Ωi+mn

ν dx j dxk l

ν dx j dxk 0

m=0

,

.

(1.15)

0 lk/2

By setting independent variations to the displacement vector u and to the entropy flux s and making integration by parts we obtain: c 2 ∂θ i 1 i θ dΩ = − θδS d∂Ωi+m + δS dΩ− δP = − δ 2 T0 ∂xi Ω

Ω

∂Ωi+m

βθδui d∂Ωi+m +

β Ω

∂Ωi+m

∂θ i δu dΩ + ∂xi

Ω

P4 i δS dΩ, T0

(1.16)

τ1 i 2 τ1 2 i 1 ∂u ∂ui i ∂u i δu dΩτ − ρ δK = ρ δ dΩdτ = ρ δu dΩdτ, 0 2 ∂τ ∂τ ∂τ2 τ0

Ω

Ω

j j Di+m fi+m δui d∂Ωi+m , ∂Ωi+m

Ω

(1.17)

δA =

τ0

(1.18)

16

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

δB =

Pi δui dΩ,

(1.19)

Ω

τ1 T0 T 0 ∂S δS dΩdτ, δD = S δS dΩ − λq λq ∂τ Ω

δW = − Ω

τ0

τ0

(1.20)

Ω

∂ ∂u i ∂ui i ∂ ∂ ∂u j i (λ + 2µ) µ δu + δu + µ δu + ∂x j ∂x j ∂xi ∂xi ∂x j ∂xi i

∂ ∂uk i ∂ ∂u j i ∂ ∂uk i ∂ ∂ui i µ δu + µ δu + λ δu + λ δu dΩ+ ∂xk ∂xk ∂xi ∂xk ∂x j ∂xi ∂xk ∂xi i j k ∂u ∂u ∂u ∂u j ∂ui i (λ + 2µ) δu + µ δui + +λ +λ +µ ∂xi ∂xi ∂xk ∂xi ∂x j ∂Ωi+m

∂ui ∂uk k δu d∂Ωi+m , +µ µ ∂xi ∂xk

∂uk ∂ui ∂u j Dk+m k λ + (λ + 2µ) +λ − βθ nk,k+m δuk dxi − ∂xi ∂xk ∂x j

li δR =

(1.21)

0 ∗ Dk k+m

∂Ωi+m

∂uk ∂ ∂ui ∂u j λ + (λ + 2µ) +λ − βθ nk,k+m δui d∂Ωk+m + ∂xi ∂xi ∂xk ∂x j

li Dik+m 0

∗ Di k+m ∂Ωk+m

∂uk ∂ui µ ni,i+m δui dx− +µ ∂xi ∂xk

∂ui ∂uk ∂ µ nk,k+m δui d∂Ωk+m + ...+ +µ ∂x j ∂xk ∂xi

0 Ai+m θ + Bi+m T i+m − Ai+m T i+mn δS i d∂Ωi+mn +

∂Ωi+m,n

lk

Bi+mn T i+mn −

0 Ai+mn T i+mn

δS dxk − i

∂Ωi+m

0

lk

∂Ωi+m

∗∗

∂

0 Bi+mn T i+mn − Ai+mn T i+mn δS i d∂Ωi+mn + ∂x j

B j+mn T j+mn − A j+mn T 0j+mn δS i dxk −

0

∂

B j+mn T j+mn − A j+mn T 0j+mn δS i d∂Ω j+m + ∂xk

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

∂Ωi+mn

∂Ωi+mn

λq ∂S i Ci+m ni,i+m δS i d∂Ωi+m + λ ∂τ

lk Ci+mn

17

λq ∂S i ni,i+m δS i dxk − λ ∂τ

0

lk λ q ∂2 S i λq ∂S i i i ni,i+m δS d∂Ωi+m + ... + Ci+mn ni,i+m δS + Ci+mn 0 λ ∂xk ∂τ λ ∂τ l ∗∗ C i+mn

λq ∂ ∂S i ni,i+m dxi . λ ∂x j ∂τ

(1.22)

0

Considering the fact that some integrals are equal to zero, expression δR can be reduced. The integrals marked with one star equal zero because integrands are stress derivatives, which in turn occur to be constant in relation to the variable, for which a derivative is calculated. The integrals marked with two stars are also equal to zero because integrands are constant derivatives in relation to the variable, according to which diﬀerentiation is made. Substituting expressions (1.16)–(1.22) into (1.10) and assuming ui , S i , θ as independent variables (their variations are arbitrary), we obtain the following system of diﬀerential equations: 3 3 m ∂θ ∂2 u s ∂ sm ∂u kαβ −β + Ps = ρ 2 , ∂xα ∂xβ ∂x s ∂τ α,β=1 m=1 ⎞ ⎛ 3 3 1 4 ∂ ⎜⎜⎜⎜ θ ∂u s ⎟⎟⎟⎟ ∂2 θ ⎟⎟ , ⎜⎜ + + P = β ∂τ ⎝ α s=1 ∂x s ⎠ ∂x2s T 0 s=1 ⎡ 3 3 ⎤ ⎢⎢⎢ ⎥⎥⎥ ∂um s sm s s ⎢ nα,i+m kαβ − βθn s,i+m ⎥⎥⎥⎦ + Ei+m u s = fi+m , Di+m ⎢⎢⎣ ∂xβ

(1.23)

(1.24)

(1.25)

α,β=1 m=1

Ci+m

λq ∂θ 0 ni,i+m + Ai+m θ − T i+m + Bi+m T i+m = 0, λ ∂xi ⎡ 3 3 ⎤ m ⎢⎢⎢ ⎥⎥⎥ ∂u s sm Di+m ⎢⎢⎣⎢ nα,i+m kαβ − βθn s,i+m ⎥⎥⎥⎦ + ∂xβ

(1.26)

α,β=1 m=1

⎡ 3 3 ⎤ ⎢⎢⎢ ⎥⎥⎥ ∂um s sm s s ⎢ nα, j+m kαβ − βθn s, j+m ⎥⎥⎥⎦ + Ei+m u s + E sj+m u s = fi+m, D j+m ⎢⎢⎣ j+m , ∂xβ

(1.27)

α,β=1 m=1

Ci+m

λq ∂θ λq ∂θ 0 ni,i+m + C j+m n j, j+m + Ai+m θ − T i+m + A j+m θ − T 0j+m = λ ∂xi λ ∂x j Bi+m T i+m + B j+m T j+m , ⎡ 3 3 ⎤ ⎢⎢⎢ ⎥⎥⎥ ∂um s sm ⎢ nα,i+m kαβ − βθn s,i+m ⎥⎥⎥⎦ + Di+m ⎢⎢⎣ ∂xβ α,β=1 m=1

(1.28)

18

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

⎡ 3 3 ⎤ m ⎢⎢⎢ ⎥⎥⎥ ∂u sm D sj+m ⎢⎢⎢⎣ nα, j+m kαβ − βθn s, j+m ⎥⎥⎥⎦ + ∂xβ α,β=1 m=1

⎡ 3 3 ⎤ ⎢⎢⎢ ⎥⎥⎥ ∂um s sm ⎢ nα,k+m kαβ − βθn s,k+m ⎥⎥⎥⎦ + Dk+m ⎢⎢⎣ ∂xβ α,β=1 m=1

s us Ei+m

s s + E sj+m u s + Ek+m u s = fi+m, j+m,k+m ,

(1.29)

λq ∂θ λq ∂θ λq ∂θ 0 ni,i+m + C j+m n j, j+m + Ck+m nk,k+m + Ai+m θ − T i+m + λ ∂xi λ ∂x j λ ∂xk

0 A j+m θ − T 0j+m + Ak+m θ − T k+m + Bi+m T i+m + B j+m T j+m + Bk+m T k+m = 0, (1.30) ⎡ 3 3 ⎤ m ⎢⎢⎢ ⎥⎥⎥ ∂u s sm ⎢⎢⎣⎢ Di+mn nα,i+m kαβ − βθn s,i+m ⎥⎥⎦⎥ + ∂xβ α,β=1 m=1 ⎡ 3 3 ⎤ ⎢⎢⎢ ⎥⎥⎥ ∂um s sm ⎢ nα,i+m kαβ − βθn s,i+m ⎥⎥⎦⎥ + Di+mn ⎢⎣⎢ ∂xβ α,β=1 m=1 Ci+m

s s s s u s + Ei+mn u s + Ek+m u s = fi+mπ , Ei+mπ λq ∂θ λq ∂θ 0 Ci+mn ni,i+m + Ai+mn (θ − T i+mn + Ci+mn )+ λ ∂xi λ ∂x j

0 Ai+mn θ − T i+mn + Bi+mn T i+mn + Bi+mn T i+mn = 0 .

(1.31)

(1.32)

The initial conditions are as follows [431]: ∂u s = q2s (x) , u s |τ=τ0 = q1s (x) , ∂τ τ=τ0 θ|τ=τ0 = q (x) , s = 1, 3, i −→ j −→ k, i, j, k = 1 . . . 3, ←

←

(1.33)

sm where: kαβ = µδαs δβm + (λ + µ)δαβ δ sm , δαβ denotes Kronecker’s symbol, α = λq /cρ, S S S S fi+m , ..., fi+m, j+m = fi+m + f j+m , f j+mΛ , are set functions corresponding to the surs face forces, whereas q(x), q1 (x), q2s (x) are set functions at an initial instant of time (coordinates). Thus obtained system contains: a) three expressions (1.23) describing dynamic behaviour of a three-dimensional plate including temperature stresses within the field, b) a generalized heat conduction equation (1.24), c) three equations (1.25) and (1.26) on the cubicoid’s walls, d) three equations (1.27) and (1.28) on the parallelepiped’s edges, e) three equations (1.29) and (1.30) in the parallelepiped’s corners, f) three equations (1.31) and (1.32) in the contact points of numerous boundary s s , Ak+m , E sj+m , Ai+m , Bi+m , Ci+m are equal conditions. Assuming that coeﬃcients Di+m to 0 or 1, we shall obtain boundary conditions well-known in the theory of elasticity and thermoelasticity [6, 198]:

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

19

1. Rigid fixing (the first boundary problem) j j a) Di+m = 0, Ei+m = 1, j = 1, ..., 3, or j j k i = Dii+m = Ei+m = 0, Ei+m = Ei+m = Dki+m = 1; b) Di+m

2. Jointedly supported edge (the third boundary problem) j j k i a) Dii+m = Ei+m = Ei+m = 1, Ei+m = Di+m = Dki+m = 0, or j j k i = Di+m = 1, Ei+m = Ei+m = Dki+m = 0; b) Dii+m = Ei+m

3. Free edge (the second boundary problem) j j Di+m = 1, Ei+m = 0, j = 1, ..., 3;

4. Temperature distribution (the first boundary problem) Ai+m = 1, Bi+m = Ci+m = 0; 5. Density of a heat flux’ normal component (the second boundary problem) Ai+m = 0, Ci+m = 1, Bi+m = 1; 6. Convectional heat transfer (the third boundary problem) Ci+m = Ai+m = Bk+m = 1 . Combined conditions on the plate’s surface will serve as an example. The boundary conditions related to the following walls of the cubicoid and corresponding equations (1.25), (1.26) will be considered: – wall ∂Ω1 is free and insulated from heat sources, thus (D11 = D31 = D21 = 1, E11 = E13 = E12 = 0, C1 = 1, A1 = B1 = 0): λ

∂u1 ∂u2 ∂u3 + (λ + 2µ) +λ − βθ = f11 (0, x2 , x3 , τ) , ∂x2 ∂x1 ∂x3 µ

∂u2 ∂u1 +µ = f12 (0, x2 , x3 , τ) , ∂x1 ∂x2

µ

∂u1 ∂u3 +µ = f13 (0, x2 , x3 , τ) , ∂x1 ∂x3 ∂θ = 0; ∂x1

– wall ∂Ω2 is fixed and aﬀected by heat impact (D12 = D32 = D22 = 1, E21 = E23 = E22 = 0, C2 = 1, A2 = 0, B2 = 1): u1 = 0,

20

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

u2 = 0, u3 = 0, ∂θ = T 2 (x1 , 0, x3 , τ); ∂x2 – wall ∂Ω3 is jointedly supported and the temperature distribution is as follows (D33 = E32 = E33 = 1, D23 = D13 = E31 = 0, C3 = 0, A3 = 1, B3 = 0): u1 = 0, u2 = 0, λ

∂u1 ∂u2 ∂u3 +λ + (λ + 2µ) − βθ = f33 (x1 , x2 , 0, τ) , ∂x1 ∂x2 ∂x3 θ = T 30 (x1 , x2 , 0, τ) .

Consistency conditions (1.27) and (1.28) in the investigated case are as follows: – at the edge (0, 0, x3 ): (λ + 2µ)

∂u1 ∂u2 ∂u3 +λ +λ − βθ + u1 = f11 (0, 0, x3 , τ) , ∂x1 ∂x2 ∂x3 µ

∂u1 ∂u2 +µ + u2 = f12 (0, 0, x3 , τ) , ∂x1 ∂x2

µ

∂u1 ∂u3 +µ + u3 = f13 (0, 0, x3 , τ) , ∂x1 ∂x3 ∂θ ∂θ + = T 2 (0, 0, x3 , τ) ; ∂x1 ∂x2

– at the edge (x1 , 0, 0):

u1 = 0, u2 = 0,

λ

∂u1 ∂u3 ∂u2 + (λ + 2µ) +λ − βθ + u3 = f33 (x1 , 0, x3 , τ) , ∂x1 ∂x3 ∂x2

∂θ + θ = T 2 (x1 , 0, 0, τ) + T 30 (x1 , 0, 0, τ) ; ∂x2 – at the edge (0, x2 , 0): (λ + 2µ)

∂u1 ∂u2 ∂u3 ∂u3 ∂u1 +λ +λ − βθ + µ +µ = ∂x1 ∂x2 ∂x3 ∂x1 ∂x3 f13 (0, x2 , 0, τ) + f13 (0, x2 , 0, τ) , µ

∂u2 ∂u1 +µ + u2 = f12 (0, x2 , 0, τ) , ∂x1 ∂x2

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

λ

21

∂u1 ∂u3 ∂u2 ∂u3 ∂u1 + (λ + 2µ) +λ − βθ + µ +µ = ∂x1 ∂x3 ∂x2 ∂x1 ∂x3 f33 (0, x2 , 0, τ) + f13 (0, x2 , 0, τ) , λq ∂θ + θ = T 30 (0, x2 , 0, τ) . λ ∂x1

Compatibility conditions (1.29), (1.30) for corner (0,0,0) in the investigated case are as follows: (λ + 2µ)

∂u2 ∂u3 ∂u1 +λ +λ − βθ + 2u1 = f11 (0, 0, 0, τ) , ∂x1 ∂x2 ∂x3 µ

µ

∂u1 ∂u2 +µ + 2u2 = f12 (0, 0, 0, τ) , ∂x1 ∂x2

∂u3 ∂u1 ∂u1 ∂u2 ∂u3 +µ +λ +λ + (λ + 2µ) + u3 = ∂x1 ∂x3 ∂x1 ∂x2 ∂x3 f33 (0, 0, 0, τ) + f13 (0, 0, 0, τ) ,

λq ∂θ λq ∂θ + + θ = T 2 (0, 0, 0, τ) + T 30 (0, 0, 0, τ) . λ ∂x1 λ ∂x2 1.2.3 Diﬀerence approximation The problems described by the system of equations (1.23)–(1.33) are going to be solved with the method of mesh. The method makes it possible to bring a system of partial diﬀerential equations to a system of algebraic equations. There are a lot of approximations of the same problem. Among them there is one that provides a required approximation order and is stable (the so-called convergent approximation). If it is possible, a diﬀerence scheme should model the primary diﬀerential problem. Variational-diﬀerence methods are the most appropriate for analysing the problems discussed in this chapter and they retain the properties of a diﬀerential system. In order to build a diﬀerence scheme we are going to use the method of integral identity [429], which is based on an assumption that the energy functional is expressed in the form of (1.10). The solutions of problems (1.23)–(1.33) are generalized when there are such functions u, θ ∈ W21,0 (Qτ ) that for arbitrary functions vi (x, τ) ∈ W21,0 (Qτ ) fulfil the following integral identity: I (u, θ) = I +

τ1 3 τ0

3

i=1

∂Ωi+m

Ω

i=1

1 4 P vi + P vi dΩdτ− T0 i

0 vi + T i+m vi = 0, fi+m vi + T i+m

(1.34)

22

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

τ1 ⎡⎢ 3 3 ⎢⎢⎢ 3 ∂θ ∂u s ∂v s ∂un ∂v s sm ⎢⎢⎣ + kαβ I= −β vs − ρ ∂xβ ∂xα ∂x s ∂τ ∂τ s=1 α,β=1 n=1 τ0

Ω

⎤ 3 ∂θ ∂v4 ∂v4 ∂u s 1 ∂v4 ⎥⎥⎥⎥ ⎥⎥ dΩdτ. +β − θ ∂x s ∂x s ∂τ ∂x s α ∂τ ⎦ s=1

(1.35)

The field of constant change of arguments Qτ is going to be replaced with a finite set of points (nodes) in the form of ωτ = ω × ω4 {xα = ihα , x4 = jh4 , i = 0, ..., Nα , j = 0, ..., M}. For every continuous function f (x, τ) set in field Qτ , functions fi jk (x, x4 ) = f (xi , x j , xk , x4 ) are going to be constructed and defined within ωτ . Identity (1.34) consists not only of equations (1.23), (1.24), but also of conditions (1.25)–(1.33). We are going to find out about it when making a transformation of dependence (1.34). The integral identity is approximated with a summing identity replacing the integrals with quadratic and derivative forms - diﬀerence quotients. Integrals (1.34) will be replaced with quadratic forms describing trapezoids, and integrals (1.35) will be replaced with linear combinations of various relations of left and right rectangles. Strictly speaking: ⎧ 1 τ ⎪ ⎪ 1 ⎪ ⎨ [A]dΩdτ ∼ [A] [A] hi h j hk h4 + h h h + h I1 = ⎪ i j k 4 ⎪ 24 ⎪ ⎩+ ωi ×+ ω j ×+ ωk ×ω¯ 4 ω+ ×ω+ ×+ ωk ×ω ¯4 τ0

Ω

i

[A] hi h j hk h4 +

ω+i ×+ ω j ×+ ωk ×ω ¯4

+

[A] hi h j hk h4 +

ω+i ×ω+j ×ω+k ×ω ¯4

[A] hi h j hk h4 +

I2 =

ρ τ0

Ω

τ1 τ0

Ω

I4 =

β Ω

[A] hi h j hk h4 ,

ω+i ×+ ω j ×ω+k ×ω ¯4

∂u s ∂ν s ∂u s ∂v s dΩdτ ∼ h1 h2 h3 h4 , ρ ∂τ ∂τ ∂τ ∂τ ω ×ω ×ω ×ω 1

I3 =

∂θ s ∂θ ∂v4 ∂un ∂v s −β v + , i −→ j −→ k, ← ← ∂xβ ∂xα ∂x s ∂x s ∂x s

τ1

τ0

[A] hi h j hk h4 +

+ ω ×+ ω ×ω+ ×ω i j k ¯4

+ ω ×ω+ ×ω+ ×ω i j k ¯4

sn A = kαβ

τ1

[A] hi h j hk h4 +

+ ω ×ω+ ×+ ω ×ω i k ¯4 j

j

2

3

4

1 ∂v4 1 ∂v4 θ dΩdτ ∼ θ h1 h2 h3 h4 , α ∂τ α ∂τ ω ×ω ×ω ×ω 1

2

3

4

∂u s ∂v4 ∂u s ∂v4 dΩdτ ∼ h1 h2 h3 h4 . β ∂x s ∂τ ∂x s ∂τ ω ×ω ×ω ×ω 1

2

3

(1.36)

4

The derivatives in formulas (1.36) are approximated with the following relations: ∂um ∂um ∼ ymxα on net + ωα , ∼ ymx¯β on net ω+β , ∂xα ∂xβ

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

∂v s ∂v s ∼ ηmxα on net + ωα , ∼ ηmx¯β on net ω+α , ∂xα ∂xα ∂θ ∂θ ∂u s ∼ y sxˆ4 , ∼ y4xˆα on net ωα , ∼ y4x¯α on net ω+α . ∂τ ∂xα ∂xα For instance:

11 k11

+ ω ×+ ω ×ω+ ×ω 1 2 3 ¯4

∂u1 ∂v1 h1 h2 h3 h¯ 4 ∼ ∂x1 ∂x1 +ω

1×

+ ω ×ω+ ×ω 2 3 ¯4

23

(1.37)

µy1x1 η1x1 h1 h2 h3 h¯ 4 .

An approximate solution of problem (1.23)–(1.33) will be such two mesh func¯ τ , which for arbitrary net functions ηi , i = 1, ..., 4, set in tions yi , i = 1, ..., 4, set on ω the same net field ω ¯ τ , fulfil the following identity: 3 3 3

I1h

+

s=1 α,β=1 m=1 3

I2h

+

I4h

+

I3h

+

3

¯ 1 ×ω ¯ 2 ×ω ¯ 3 ×ω ¯4 s=1 ω

1 4 s P η + P η h1 h2 h3 h4 + T0 s s

0 η s + T s+m η s h¯ i h j = 0 i, j = 1, 3, i j. f s+m η s + T s+m

¯ i ×ω ¯j s=1 ω

Assuming that mesh functions ηi , i = 1, ..., 4 are equal units in an arbitrary chosen single point, we obtain diﬀerence equations in that node, in which ηi does not equal 0. Passing through all the nodes in net ω ¯ τ , we obtain a net equivalent that approximates diﬀerential system (1.23)–(1.33). Eventually we obtain the following system of equations:

1 ! j (λ + 2µ) yix¯i + µyix¯ j + µyix¯k + λy x j + λy x¯j j + µy xj i + xi xj xk x xj x ¯ i i 2 "

+ Pi − βy4xˆi = ρyix¯4 x4 , (1.38) + µy x¯j i + λykxk + λykx¯k + µykxi + µykx¯i x¯ j xi x¯i xk x¯k ⎛ ⎞ 3 3 ⎟⎟⎟ 1 4 ⎜⎜⎜⎜ cρ 4 4 α y x¯α xα + P = ⎜⎜⎝ y + β y xˆα ⎟⎟⎟⎠ , (1.39) T0 λq α=1 α=1 x4

hi ! j i + µy x¯j i xi + λy x¯j j + Di+m ni,i+m (λ + 2µ) yix¯i xi + λy xˆj j + λykxˆk + λy x j x¯i xi xj x¯i xi 2 " #

j i yi = gii+m , µy x¯i xi + µyix¯ j + µyix¯k + Pi − βy4x¯i xi − βy4 + Ei+m xj xj xk "

hi ! j j Di+m ni,i+m µy x¯j i xi + µyixˆ j + µy x¯k + (λ + 2µ) y x¯j j + λyix¯i xi + x xj k 2 xj #

j j + µykx j + P j − βy4xˆ j + Ei+m y j = gi+m , λykxk + µyix j xj x¯i xi xk "

hi ! k k Di+m ni,i+m µykx¯i xi + µyixˆk + µy x¯ j + (λ + 2µ) y x¯j k + λyix¯i xi + xj xk 2 xk

24

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

µyixk

x¯i xi

#

j + λy xj j + µy xj k + Pk − βy4xˆk + Ei+m yk = gki+m , xk

(1.40)

xj

λq hi 4 0 ni,i+m y4x¯i xi + Ai+m y4 − T i+m + Bi+m T i+m + y x¯ j x j + y4x¯k xk + P4 = 0, (1.41) λ 2 $ % $ % j i i k 4 Di+m ni,i+m Hi (λ + 2µ) y x¯i xi + λy x¯i xi + λy xˆk − βy + Dij+m n j, j+m H j µyix¯ j x j + µy x¯j i xi + Ci+m

j !

Dii+m + Di+m H µyix¯k + λy x¯j j x j + µy x¯j i xi + λykxk + µykx¯i xi + xk x¯ j x j x¯i xi xk x¯i xi 2 % i 4 i i i i i Di+m − βy x¯i xi + Ei+m Hi y + E j+m H j y = gi+m, j+m , $ % $ % j ni,i+m Hi µy x¯j i xi + µyix¯ j x j + D jj+m n j, j+m H j (λ + 2µ) y x¯j j x j + λyix¯i xi + λykx¯k xk − βy4 + Di+m j + Dij+m Di+m

2

!

H µy x¯j k + λy x¯i xi xk

x¯ j x j

+ λykxˆk

x¯ j x j

"

+ µyix¯ j x j − βy4x¯ j x j + P j + xˆk

j j j Hi y j + E j+m H j y j = gi+m, Ei+m j+m , $ % $ % Dki+m ni,i+m Hi µykx¯i xi + µyixˆk + Dkj+m n j, j+m H j µykx¯ j x j + µy xˆj k +

Dki+m + Dkj+m 2

!

H (λ + 2µ) ykx¯k + λyix¯i xi + µyixk xk

k Ei+m Hi yk

xk

+

E kj+m H j yk

=

x¯i xi

(1.42)

"

+ λy x¯j j x j − βy4xˆk + Pk + xk

gki+m, j+m ,

Ci+m Hi ni,i+m y4x¯i xi + C j+m H j n j, j+m y4x¯ j x j + Bi+m Hi T i+m + B j+m H j T j+m +

Ci+m + C j+m

0 H y4x¯k xk + P4 = 0, (1.43) + A j+m H j y4 − T 0j+m + Ai+m Hi y4 − T i+m 2 $ % j i i Di+m ni,i+m H¯ i (λ + 2µ) y x¯i xi + λy x¯ j x j + λykx¯k xk − βy4 + $ % $ % Dij+m n j, j+m H¯ j µy x¯j i xi + µyix¯ j x j + Dik+m nk,k+m H¯ k µykx¯i xi + µyix¯k xk + Dii+m + Dij+m + Dik+m 3

!

H¯ λykx¯k xk

x¯i xi

+ µykx¯i xi

x¯k xk

+ λy x¯j j x j

x¯i xi

+ µy x¯j i xi

" x¯ j x j

+

i i H¯ i yi + E ij+m H¯ j yi + Ek+m H¯ k yi = gki+m, j+m,k+m , Ei+m $ % $ % j ni,i+m H¯ i µy x¯j i xi + µyix¯ j x j + D jj+m n j, j+m H¯ j (λ + 2µ) y x¯j j x j + λyix¯i xi + λykx¯k xk − βy4 + Di+m $ % j Dk+m nk,k+m H¯ k µy x¯j k xk + µykx¯ j x j + j j + D jj+m + Dk+m Di+m

3

!

H¯ λyix¯i xi

x¯ j x j

+ µyix¯ j x j

x¯i xi

+ λykx¯k xk

x¯i xi

+ µykx¯ j x j

j j j j H¯ i y j + E j+m H¯ j y j + Ek+m H¯ k y j = gi+m, Ei+m j+m,k+m , $ % $ % Dki+m ni,i+m H¯ i µykx¯i xi + µyix¯k xk + Dkj+m n j, j+m H¯ j µykx¯ j x j + µykx¯k xk +

" x¯k xk

+

(1.44)

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

$

25

%

Dkk+m nk,k+m H¯ k (λ + 2µ) ykx¯k xk + λyix¯i xi + λy x¯j j x j − βy4x¯k xk + Dki+m + Dkj+m + Dkk+m 3

!

H¯ λyix¯i xi

x¯k xk

+ µyix¯k xk

x¯i xi

+ λy x¯j j x j

x¯k xk

+ µy x¯j k xk

" x¯ j x j

+

k k Hi yk + E kj+m H j yk + Ek+m Hk yk = gki+m, j+m,k+m , Ei+m

λq 4 λq λq y H¯ i + C j+m n j, j+m H¯ j y4x¯ j x j + Ck+m nk,k+m H¯ k y4x¯k xk + λ x¯i xi λ λ

0 + Bi+m H¯ i T i+m + B j+m H¯ j T j+m + Bk+m H¯ k T k+m + Ai+m H¯ i y4 − T i+m

0 (1.45) A j+m H¯ j y4 − T 0j+m + Ak+m H¯ k y4 − T k+m + P4 H¯ = 0,

Ci+m ni,i+m

Hi = H¯ i =

hj hi h j hi , Hj = , H= , hi + h j hi + h j hi + h j

h j hk hi h j hi hk , H¯ j = , H¯ k = , hi h j + h j hk + hk hi hi h j + h j hk + hk hi hi h j + h j hk + hk hi H¯ =

hi h j hk , m = 0, ..., 3, i −→ j −→ k . ← ← hi h j + h j hk + hk hi

The diﬀerence derivatives in equations (1.40)–(1.45) have been notated in a twolevelled form because of their reciprocal dependence on the location of the node in which the equation has been written. The upper notation corresponds to such case of the node’s location, in which the direction of the coordinate axis coincides with possibility of notating the derivative within the field in the same direction (the right derivative). The lower notation corresponds to such a location, in which both the direction of the axis and the direction of the derivative notation are opposite each other (the left derivative). The note that belongs to edge (l1 , 0, 0), (l1 , 0, l3 ) will serve as an example of our investigation. Equations (1.42), (1.43) have the following form: $ % $ % D14 H1 (λ + 2µ) y1x¯1 + λy2x¯2 + λy2xˆ3 − βy4 − D12 H2 µy1x2 + µy2x¯1 + "

D14 + D12 ! 1 H µy x¯3 + λy2x2 + µy2x¯1 + λy3xˆ3 + µy3x¯1 − βy4x¯1 + P1 + x3 x¯1 x2 x1 xˆ3 2 E41 H1 y1 + E21 H2 y1 = g142 , $ % $ % D24 H1 µy1x2 + µy2x¯1 − D22 H2 (λ + 2µ) y2x¯2 + λy1x¯1 + λy3x3 − βy4 + "

D24 + D22 ! 2 H µy x¯3 + λy1x¯1 + λy3xˆ3 + µy1x2 + µy3x2 − βy4x2 + P2 + x3 x2 x2 x¯1 xˆ3 2 E42 H1 y2 + E22 H2 y2 = g242 , $ % $ % D3 + D32 !

H (λ + 2µ) y3x¯3 + D34 H1 µy1xˆ3 + µy3x¯1 − D32 H2 µy3x2 + µy2xˆ3 + 4 x3 2 "

λy1x¯1 + µy1xˆ3 + λy2x2 + µy3xˆ3 − βy4xˆ3 + P3 + xˆ3

x¯1

xˆ3

x2

26

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

E43 H1 y3 + E23 H2 y3 = g342 ,

(1.46)

λq 4 λq y − C2 H2 y4x2 + B4 H1 T 4 + B2 H2 T 2 + λ x¯1 λ

C4 + C2 P4 4 0 4 0 4 H y x¯3 x3 + = 0. A4 H1 y − T 4 + A2 H2 y − T 2 + 2 T0 C4 H1

The coeﬃcients that perform as net functions in equations (1.40)–(1.45), are selected in the following way: kh (x) = moreover:

k (x1 , x2 , x3 ) − k (x1 − h1 , x2 , x3 ) , 2

j j (x1 , x2 , x3 ) , = fi+m Ph (x) = P (x1 , x2 , x3 ) , gi+m

k k i i i + H j f j+m , gii+m j+mk+m = H¯ i fi+m + H¯ j f j+m + H¯ k fk+m . gki+m j+m = Hi fi+m

1.2.4 Diﬀerence approximation Error To start an investigation of the a priori characteristics in the form of a diﬀerence system it is necessary to begin with determining the order of the diﬀerence approximation error, since the diﬀerence scheme’s accuracy depends on it. Moreover, the function’s decomposition into Taylor’s series is used in this case. Let uih be a projection of solution ui , θ onto the mesh field ωτ , step h – a vector with norm |h| > 0, and let ui have a suﬃcient number of generalized derivatives. We shall investigate deviation Ψ = Lh uh − Ph − Lu − P, where Lh is a diﬀerence operator, and L is a diﬀerential operator. We say that Lh approximates L with order n on mesh ωτ , if |Ψ | < Mhn , where M − const > 0 does not depend on h. Let us make a separate estimation of the deviation of each of equations (1.36)–(1.38). We shall assume that index n, the smallest for all the equations, will be the system’s approximation order’s error. Let us analyse the error of equations (1.38) in detail:

Ψ 1 (ωτ ) = (λ + 2µ) uix¯i + µuix¯ j + µuix¯k + xi

xj

xk

1 ! j λu x j + λu x¯j j + µuixi + µuix¯i + λukxk + λukx¯k + xi xj xj xi x¯i x¯i 2 " i

∂ ∂u (λ + 2µ) + + Pi − βθ xˆi − ρuix¯4 x4 = µukxi + µukx¯i xk x¯k ∂xi ∂xi 2 h2i ∂2 ∂ ∂ui h j ∂2 ∂ui ∂ ∂ui ∂ui (λ + µ + µ + µ + + 2µ) 12 ∂xi2 ∂xi ∂x j ∂x j 12 ∂x2j ∂x j ∂xk ∂xk

h2k ∂2 ∂ui ∂ ∂u j h2i ∂2 ∂u j ∂ ∂ui µ + λ + λ + µ + 12 ∂xk2 ∂xk ∂xi ∂xi 12 ∂xi2 ∂xi ∂x j ∂xi

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

27

h2j ∂2 ∂ui ∂ ∂uk h2i ∂2 ∂uk ∂ ∂uk µ + λ + λ + µ + 2 2 12 ∂x j ∂xi ∂xi ∂xk 12 ∂xk ∂xk ∂xk ∂xi h2k ∂2 ∂uk ∂θ h2i ∂ ∂θ ∂2 ui i µ + P − β − β −ρ 2 − 2 12 ∂xk ∂xi ∂xi 2 ∂xi ∂xi ∂τ

h24 ∂2 ∂2 ui ρ + O h3 . 12 ∂τ2 ∂τ2

(1.47)

Taking (1.23) into account, we obtain Ψ 1 (ωτ ) = O(h2 ). It can be proven analogously that there is a second order approximation in equation (1.39): Ψ (ωτ ) = 2

3

θ x¯i xi + P − αθ xˆ4 − β 4

i=1

3 i=1

uixˆi x4

3 ⎡ 2 ⎢⎢⎢ ∂ θ h2i ∂2 ⎢⎣ 2 + = 12 ∂xi2 ∂xi i=1

⎛ 2 ⎞ ⎜⎜⎜ ∂ θ ⎟⎟⎟ ⎜⎝ 2 ⎟⎠ + ∂xi

⎤ h24 ∂2 ∂2 ui ⎥⎥⎥ ∂2 ui ∂θ h24 ∂ ∂θ ⎥⎦ + −β − β P −α − ∂τ 2 ∂τ ∂τ ∂xi ∂τ 12 ∂τ2 ∂xi ∂τ

O h2 + h24 . 4

(1.48)

Taking equations (1.24) into account, we obtain Ψ 2 (ωτ ) = O(h2 ). Equations (1.40) are of the same type for all nodes and they refer to any arbitrary wall of the cubicoid. Therefore, it seems useful to determine the error of the approximation of one of the wall’s equation. To illustrate that, for ∂Ω1 we obtain: Ψ 3 (0, x2 , x3 ) = (λ + 2µ) u1x¯1 + λu2xˆ2 + λu3xˆ3 + "

h1 ! 2 λu xˆ2 + µu2x1 + λu2xˆ2 + µu2x1 + µu1x¯2 + µu1x¯3 + P1 − βθ x1 −βθ−g11 = x1 xˆ2 x1 xˆ2 x2 x3 2 (λ + 2µ)

∂u1 ∂u1 h ∂ ∂u2 (λ + 2µ) + +λ + ∂x1 2 ∂xi ∂x1 ∂x2

∂u3 h ∂u3 ∂u3 h ∂ ∂u2 λ +λ + λ + 2 ∂x2 ∂x2 ∂x3 2 ∂x3 ∂x3

h ∂ ∂u2 ∂ ∂u2 ∂ ∂u1 ∂θ 1 − βθ − f11 + O h2 . λ + µ + µ +P −β 2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x3 ∂x3 ∂x1

(1.49)

Taking equations (1.25), (1.23) into account, we obtain Ψ 3 (0, x2 , x3 ) = O(h2 ). The last three equations of the investigated node on wall ∂Ω1 also contain second order approximation errors. We are going to prove that the edge equations contain approximation errors of the O(h2 ) order. Let us consider a deviation expression of one in 48 similar equations (1.42), (1.43): Ψ 4 (0, 0, x3 ) =

% % h1 $ 1 h2 $ (λ + 2µ) u1x1 + +λu2x2 + λu3xˆ3 − βθ + µu x2 + µu2x1 + h1 + h2 h1 + h2

28

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

"

h1 h2 ! 1 µu x¯3 + λu2x2 + µu2x1 + λu3xˆ3 + µu3x1 + P1 − βθ x1 − g112 = x3 x1 x2 x1 xˆ3 h1 + h2 1 1

∂u h1 ∂ ∂u ∂u2 h2 (λ + 2µ) (λ + 2µ) + + O h21 + λ + h1 + h2 ∂x1 2 ∂xi ∂x1 ∂x2 ⎤

⎥⎥ ∂u3 h23 ∂ ∂u3 h2 ∂ ∂u2 2 λ + O h2 + λ + λ − βθ⎥⎦⎥ + 2 ∂x2 ∂x2 ∂x3 12 ∂x3 ∂x3 1 2

∂u h2 ∂ ∂u ∂u2 h1 ∂ ∂u2 h1 2 µ + µ + O h2 + µ + µ + O h21 + h1 + h2 ∂x2 2 ∂x2 ∂x2 ∂x1 2 ∂x1 ∂x1 & ' & ' ∂ ∂u1 h1 h2 ∂ ∂u2 ∂ ∂u2 µ + O (h3 ) + λ + O h2 + µ + O h2 + h1 + h2 ∂x3 ∂x3 ∂x1 ∂x2 ∂x2 ∂x1

& ' & ' ∂ ∂u3 ∂θ ∂ ∂u3 1 λ + O h1 + µ + O h3 + P − β − O h1 − ∂x1 ∂x3 ∂x3 ∂x1 ∂x1

h21 h22 h2 ∂ ∂u2 ∂ ∂u2 h1 f11 − f21 = O h2 − λ − λ , h1 + h2 h1 + h2 2 (h1 + h2 ) ∂x2 ∂x2 2 (h1 + h2 ) ∂x1 ∂x1 h

h

h ∂ ∂u2 λ ∼ − λu2x2 , λu2x2 ∼ λu2x2 − λu2x2 = λy2x2 , (1.50) − x x2 2 2 ∂x2 ∂x2 2 2 3y2j − 4y2j−1 + y2j−2

. 2h According to equation (1.23) and boundary conditions (1.25) we obtain Ψ 4 (0, 0, x3 ) = O(h2 ). Let us investigate the approximation error of 32 equations in the field’s corners. The following investigates a deviation for one equation in corner (0, 0, 0): y2x2 =

Ψ 5 (0, 0, 0) =

% $ h2 h3 (λ + 2µ) u1x1 + λu2x2 + λu3x3 − βθ + h1 h2 + h2 h3 + h3 h1

$ % $ % h1 h3 h1 h2 µu1x2 + µu2x1 + µu1x3 + µu3x1 + h1 h2 + h2 h3 + h3 h1 h1 h2 + h2 h3 + h3 h1 !

"

h1 h2 h3 µu2x1 + λu2x2 + µu3x1 + λu1x3 + P1 − βθ x1 − g1123 = x2 x1 x3 x1 h1 h2 + h2 h3 + h3 h1

∂u1 h1 ∂ ∂u1 ∂u2 h2 h3 (λ + 2µ) (λ + 2µ) + + O h21 + λ + h1 h2 + h2 h3 + h3 h1 ∂x1 2 ∂x1 ∂x1 ∂x2

h2 ∂ ∂u2 ∂u3 h3 ∂ ∂u3 λ + O h22 + λ + λ + O h23 − βθ + 2 ∂x2 ∂x2 ∂x3 2 ∂x3 ∂x3 1

∂u h1 h2 h2 ∂ ∂u1 ∂u2 h1 ∂ ∂u2 µ + O h22 + O h21 + + + + h1 h2 + h2 h3 + h3 h1 ∂x2 2 ∂x2 ∂x2 ∂x1 2 ∂x2 ∂x1 1 1 3

∂u h1 h3 h3 ∂ ∂u ∂u h1 ∂ ∂u3 2 µ + O h3 + O h21 + + + + h1 h2 + h2 h3 + h3 h1 ∂x3 2 ∂x3 ∂x3 ∂x1 2 ∂x1 ∂x1

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

29

∂ ∂u2 ∂ ∂u2 ∂θ h1 h2 h3 ∂ ∂u3 ∂ ∂u1 µ +λ −β + + + P1 = h1 h2 + h2 h3 + h3 h1 ∂x2 ∂x1 ∂x3 ∂x1 ∂x1 ∂x2 ∂x1 ∂x3 ∂x1

O h2 −

h2 h23 h1 h22 ∂ ∂u3 ∂ ∂u2 λ − µ − 2 (h1 h2 + h2 h3 + h3 h1 ) ∂x3 ∂x3 2 (h1 h2 + h2 h3 + h3 h1 ) ∂x2 ∂x2 h3 h21 ∂ ∂u3 µ . 2 (h1 h2 + h2 h3 + h3 h1 ) ∂x1 ∂x1

(1.51)

Taking equations (1.23), (1.25), (1.51) into account, we obtain Ψ 5 (0, 0, 0) = O(h2 ). Every equation of system (1.38)–(1.45) belongs to one of the five investigated forms of equations. This being so, the approximation error does not exceed O(h2 ), and the general error’s order in each node of field ωτ is not smaller than the other. 1.2.5 Diﬀerence approximation Stability It may turn out that the knowledge of a diﬀerence scheme’s approximation error’s order is insuﬃcient to estimate the scheme’s quality. That is why, having determined the approximation order, it is necessary to analyse the scheme’s stability. The a priori estimation for yi is an essential part of the analysis of a diﬀerence scheme. If the system approximates the problem and it is stable, then its solution leads to the solution of a diﬀerential problem. The diﬀerence problem obtained with variational-diﬀerence methods are stable [384]. However, derivation of estimations imposes bounds upon the right parts and the coeﬃcients of equations and also upon the steps of the mesh field hi , which is particularly important while making calculations. Our reasoning will be based on mesh space W21 (ω), W21,0 (ωτ ) and the following energy estimation:

2 (1)2 2 2 2 2 (1.52) y(1,0) 2,ωτ ≤ C 1 g2,1∂Ω×ω4 + P2,1ωτ + C 2 q1 2,ω + q2,ω + q2 2,ω . While deriving the proof we are going to use the most popular net relations, such as: – diﬀerential transformation (a uni-dimensional case): (ω, v) x,i = ω x,i vi + ωi+1 v x,i = ω x,i vi + ωi+1 v x¯,i+1 , (ω, v) x¯,i = ω x¯,i vi + ωi−1 v x¯,i = ω x¯,i vi + ωi−1 v x,i−1 ;

(1.53)

– summation [429] (p.225) (a one-dimensional case): n

vi ω x¯,i h = −

i=m+1 n i=m+1

v x,i ωi h = −

n−1

v x,i ωi h + vn ωm − vm ωm ,

i=m n i=m+1

vi ω x¯,i h + vn ωn − vm+1 ωn ;

(1.54)

30

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

– multi-dimensional summation with zero boundary values of mesh functions

hω x,i vi = −

ω ¯

hωv x¯,i .

(1.55)

ω ¯

The one-dimensional relations have been shown due to the fact that the initial transformations are derived only for one variable. Additionally we apply the following equalities: 2hω x,i ωi = ω2i+1 − ω2i − h2 ω x,i , 2hω x¯,i ωi = ω2i − ω2i−1 + h2 ω x¯,i , 2h

N

ω x¯,i ωi = ω2N − ω20 + h

i=1

2h

N−1

(1.56)

N & ' h ω x¯,i 2 ,

(1.57)

i=1

ω x,i ωi = ω2N − ω20 − h

i=0

N−1 & ' h ω x,i 2 , i=0

2h4 y x¯4 (k) y (k) = y (k) − y (k − 1) + h24 y x¯4 (k) ,

(1.58)

2h4 y xi (k) y x¯4 xi (k) = y2x (k) − y2x (k − 1) + (δy x (k − 1))2 ,

(1.59)

2

2h4 y xi (k) y x4 xi (k) =

2

y2x

(k + 1) −

y2x

(k) − (δy x (k)) ,

δv (k) = v (k + 1) − v (k) ,

2

(1.60)

and also well known Cauchy’s algebraic inequalities: ⎛ ⎞ 12 ⎛ n ⎞ 12 n n ⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎜ αi j ai bi ≤ ⎜⎜⎜⎝ αi j ai a j ⎟⎟⎟⎠ ⎜⎜⎜⎝ αi j bi b j ⎟⎟⎟⎠ , i, j=1 i, j=0 i, j=0 1

1

|(u, v)ω¯ | ≤ (u, u)ω¯2 (v, v)ω¯2 = u2,ω v2,ω ,

(1.61)

in which ε is a small number |ab| ≤ εa2 +

1 2 b ∀ε > 0 . 4ε

(1.62)

The ideas that refer to the proof of the diﬀerence processes’ stability have been derived from a number of scientific works. Andreev and Samarskiy’s works present a priori estimations of approximate solutions of the linear elasticity problems in the static approach taking into account mixed boundary conditions for a two-dimensional case. Ladyzhenskaya’s work contains derivations of the first initially-boundary problem for a parabolic and a hyperbolic equation in a general form. Works [231, 241, 492] address extended research into hybrid types of problems. Treating those references as basis we are going to prove a theorem that refers to stability of approximate solutions to the coupled thermoelasticity problems for three-dimensional plates.

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

31

To make things simpler let us assume that hi = h. Relations (1.38)–(1.44) are universal because if we assume proper form of indeces it is possible to formulate a problem in a very general way. As the multitude of symbols makes it rather diﬃcult to comprehend the problem, we are going to investigate it within defined combined boundary conditions. The first and the second initially-boundary problems will serve as mechanical conditions. System (1.38)–(1.44) that describes thus formulated combination of conditions takes the following form:

1 ! 2 (λ + 2µ) y1x¯1 + λy1x¯2 + λy1x¯3 + λy x2 + µy2x1 + λy2x¯2 + x1 x2 x3 x x x¯1 1 2 2 "

µy2x¯1 + λy3x3 + µy3x1 + λy3x¯3 + µy3x¯1 − βy4xˆ1 + P1 = ρy1x¯4 x4 , (1.63) x¯2

x1

x3

x¯1

x¯3

1 ! 3 (λ + 2µ) y2x¯2 + λy2x¯1 + λy2x¯3 + λy x¯3 + µy2x¯2 + λy3x3 + x2 x1 x3 x ¯ x ¯ x2 2 3 2

" µy3x2 + λy1x¯1 + λy1x1 + µy1x2 + µy1x¯2 − βy4xˆ2 + P2 = ρy2x¯4 x4 ,

x3

(λ + 2µ) y3x¯3

µy1x3

x1

x¯2

x2

+ λy3x¯1

x1

+ λy3x¯2

+

x1

1 !

λy1x¯1

+ µy1x¯3 + λy1x1 +

x¯3 x¯1 x3 2

" + λy2x¯2 + λy2x2 + µy2x¯3 + µy2x¯3 − βy4xˆ3 + P3 = ρy3x¯4 x4 , (1.64) x¯3 x3 x¯2 x2

y4x¯1 x1 + y4x¯2 x2 + y4x¯3 x3 + P4 − β y1xˆ1 x1 + y2xˆ2 x4 + y3xˆ3 x4 = αy4x4

x3

x1

x2

inside plate ω:

h ! 3 µy x¯1 + µy3x¯2 + λy1x¯1 + µy1x3 + x1 x2 x3 x¯1 2 "

λy2x2 + µy2x3 + P3 − βy4x3 − βy4 + g33 = 0,

(λ + 2µ) y3x3 + λy2xˆ2 + λy1xˆ1 + x3

x2

h $

(λ + 2µ) y2x¯2 + µy2x¯1 + λy1x1 + λy3x3 + µy2x3 + µy3xˆ2 + x2 x1 x2 x2 2

%

µy1x2 + µy3x2 + P2 − βy4x2 + g23 = 0, x1 x3 !

h (λ + 2µ) y1x¯1 + µy1x¯2 + λy2x2 + λy3x3 + µy1x3 + µy3xˆ1 + x x x x1 1 2 1 2 "

µy2x1 + µy3x1 + P1 − βy4x1 + g1x3 = 0, x2

x3

(1.65)

λq 4 h

y x3 + y4x¯1 x1 + y4x¯2 x2 + P4 = g3 (1.66) λ 2 on the free wall ∂Ω3 aﬀected by a thermal impact. The other walls are fixed and their temperature distribution is as follows: y s = gis , s = 1, ..., 3 ,

(1.67)

32

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

y4 = gi , i = 1, 2, 3, 4, 5, 6 .

(1.68)

The consistency conditions at the edges (0, x2 , 0), (l1 , x2 , 0), (x1 , 0, 0), (x1 , l2 , 0) are of the same kind and they take the following form:

h ! 3 1 (λ + 2µ) y3x3 + λy2xˆ2 + λy1x1 + µy x1 + µy3x¯2 + λy1x1 + µy1x3 + λy2x2 + x x x x x3 1 2 3 1 2 2 " # 1 1 1 − βy4x3 + P3 − βy4 + y3 + g33 + g31 = 0, (1.69) x2 2 2 2

h !

1 (λ + 2µ) y2x¯2 + µy2x1 + λy1x1 + λy3x3 + µy2x3 + µy3xˆ2 + x2 x1 x2 x2 2 2 "#

1 1 1 µy1x2 + µy3x2 + P2 − βy4x2 + y2 + g23 + g21 = 0, x1 x3 2 2 2 !

1 h (λ + 2µ) y1x1 µy1x3 + µy3x1 + + µy1x¯2 + λy2x2 + λy3x3 + x1 x2 x1 x1 2 2

%# 1 1 1 µy2x1 + µy3x1 + P1 − βy4x1 + y1 + g13 + g11 = 0, x2 x3 2 2 2 1 4 h

1 1 1 y x3 + y4x¯2 x2 + P4 + y4 + g3 + g1 = 0 (1.70) 2 2 2 2 2

µy2x3

at the edge (0, x2 , 0). The consistency conditions in corners (0, 0, 0), (0, l2 , 0), (l1 , 0, 0), (l1 , l2 , 0) are also of the same kind and they take the following form:

h ! 3 1 (λ + 2µ) y3x3 + λy2x2 + λy1x1 + µy x1 + µy3x2 + λy1x1 + µy1x3 + λy2x2 + x1 x2 x3 x1 x3 3 3 " # 2 1 1 1 − βy4x3 + P3 − βy4 + y3 + g33 + g31 + g32 = 0, x2 3 3 3 3 !

1 h (λ + 2µ) y2x2 + µy2x1 + λy1x1 + λy3x3 + µy1x2 + µy2x3 + µy3x2 + x2 x1 x2 x2 x1 3 2 "# 2

1 1 1 µy3x2 + P2 − βy4x2 + y2 + g23 + g21 + g23 = 0, x3 3 3 3 3 !

1 h

(λ + 2µ) y1x1 + µy1x2 + λy2x2 + λy3x3 + µy2x1 + µy1x3 + µy3x1 + x x x x x2 1 2 1 1 3 3 "#

1 1 1 2 µy3x1 + P1 − βy4x1 + y1 + g13 + g11 + g12 = 0, (1.71) x3 3 3 3 3 h 2 1 1 1 1 λq 4 y + P4 + y4 + g3 + g1 + g2 = 0 (1.72) 3 λ x3 3 3 3 3 3 in corner (0, 0, 0). The initial conditions are as follows:

µy2x3

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

yi x =τ = qi1 , yix4 x =τ = qi2 , y4 x =τ = q . 4

0

4

0

4

0

33

(1.73)

To make it easier, we shall denote the coeﬃcients of the mechanical part of system (1.63)–(1.73) as ai, j,k , and the thermal part as bi, j,k . We shall assume that the net functions beyond the field’s boundaries are zero, i = 1, ..., N1 , j = 1, ..., N2 , k = 1, ..., N3 . THEOREM 1.1 Let the coeﬃcients of system (1.23)–(1.33) fulfil the following inequalities: 3 3 3 3 2 ξ2jl ≤ ai jk (x, τ) ξil ξ2jn ≤ µ2 ξin , (1.74) µ1 j=1 l=1

∂ai jk < µ3 , ∂τ

i=1 n=1

µ4 ξ2 ≤ bi jk (x, τ) ξi ξ j ≤ µ5 ξ2 ,

(1.75) (1.76)

where ξi, j is an arbitrary number and: ∀s, i P s (x, τ) ∈ L2,1 (Qτ ) , fis (x, τ) ∈ L2,1 (∂Ω × (τ0 , τ1 )) ,

(1.77)

q1s (x) ∈ W21 (Ω) , q2s (x) ∈ L2 (Ω) , q (x) ∈ L2 (Ω) ,

(1.78)

and if the following inequalities are fulfilled: h4 µ2 √ h4 3 = 1 − ε, 1 − 12µ5 2 = ε, ε ∈ (0, 1), √ µ1 h h

(1.79)

then diﬀerence scheme (1.38)–(1.45) uniquely determines mesh functions y s and their interpolations at hi → 0, which are weakly convergent in L2,1 (Qτ ) towards the generalized solution u s (x, τ) ∈ Ω21,0 (Qτ ), s = 1, ..., 4 of problem (1.23)–(1.33). In order to find a solution to system y s , s = 1, ..., 4 we are going to derive the energy estimations. Therefore, we multiply both sides of equations (1.63) by i h3 h4 [yix4 (m) + y x4 (m)], while both sides of equations (1.64) – by 2h1 h3 y4 (m + 1), s where y (m) denotes a net function on mh4 upper layer. The results are summed along all of the nodes ω × ω4 . In the same way we transform relations into boundary conditions. Both sides of equations (1.65), (1.67) are multiplied by h2 h4 [yix4 (x1 , x2 , 0, m) + yix4 (x1 , x2 , 0, m)], and both sides of equations (1.66), (1.68) – by 2h2 h4 y4 (x1 , x2 , 0, m + 1) and they are summed in field ω × ω2 × ω4 . Equations (1.69) are multiplied by h2 h4 [yix4 (0, x2 , 0, m)+yix4 (0, x2 , 0, m)], while equation (1.70) is multiplied by 2h2 h4 y4 (0, x2 , 0, m + 1) and summation is made on account of ω2 and ω4 . Both sides of equations (1.71) are multiplied by h2 h4 [yix4 (0, m) + yix4 (0, m)], and equations (1.72) are multiplied by 2h2 h4 y4 (0, m + 1) and summed on account of ω4 . The obtained relations are grouped and as a result we obtain the following expression (1.80): !

h4 h3 − ai jk y1x1 (m) y1x4 (m) + y1x4 (m) − ... ω1 ×ω2 ×ω3 ×ω4

x1

34

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1 !

× ai jk y3x¯1 y1x4 (m) + y1x¯4 (m) + ai jk y4x1 (m) y1x4 (m) + y1x¯4 (m) − x3 2 %

1 1 P y x4 (m) +y1x¯4 (m) + y1x¯4 x4 y1x4 (m) + y1x¯4 (m) + ... + $ 2h3 h4 −bi jk y1x¯1 x1 (m) y4 (m + 1) − ... − P4 y4 (m + 1) + bi jk y4x4 (m) y4 (m + 1) + % bi jk y1xˆ1 x4 y4 (m + 1) + ... + $

h4 h2 ai j0 y3x3 y3x4 (x1 , x2 , 0, m) + y3x¯4 (x1 , x2 , 0, m) + ...+ ω1 ×ω2 ×ω4

h !

y3x4 (x1 , x2 , 0, m) + y3x¯4 (x1 , x2 , 0, m) + ...+ ai j0 y3x¯1 x 1 2

ai j0 y4x3 (m) y3x¯4 (x1 , x2 , 0, m) + y3x4 (x1 , x2 , 0, m) +

P3 y3x¯4 (x1 , x2 , 0, m) + y3x4 (x1 , x2 , 0, m) −ai j0 y4 y3x4 (x1 , x2 , 0, m) + y3x¯4 (x1 , x2 , 0, m) +

g33 y3x4 (x1 , x2 , 0, m) + y3x¯4 (x1 , x2 , 0, m) + ... + 2bi j0 y4x3 y4 (x1 , x2 , 0, m + 1) + h

bi j0 2y4x¯4 x1 y4 (x1 , x2 , 0, m + 1) + ... + P4 2y4 (x1 , x2 , 0, m + 1) + ...+ 2 2g1 h4 h2 y4 (0, x2 , x3 , m + 1) + ω2 ×ω3 ×ω4

ω2 ×ω4

h4 h2

1$ a0 j0 y3x3 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) + ...+ 2

h

y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) − a0 j0 y2x3 x2 2

a0 j0 y4x3 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) + P3 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) −

a0 j0 y4 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) + y3 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) +

g33 y3x4 (0, x2 , 0, m) + y3x¯4 (0, x2 , 0, m) + ... + 2b0 j0 y4 (0, x2 , 0, m + 1) y4x3 + h

4 4 4 4 2b0 j0 y x¯1 x1 y (0, x2 , 0, m + 1) + ... + 2y y (0, x2 , 0, m + 1) + ... + 2 1

h

h4 a0 y3x3 y3x4 (0, m) + y3x¯4 (0, m) + ... + a0 y2x3 y3x4 (0, m) + y3x¯4 (0, m) + ... − x2 3 3 ω4

a0 y4x3 y3x4 (0, m) + y3x¯4 (0, m) +P3 y3x4 (0, m) + y3x¯4 (0, m) −a0 y4 y3x4 (0, m) + y3x¯4 (0, m) +

2y3 y3x4 (0, m) + y3x¯4 (0, m) + g33 y3x4 (0, m) + y3x¯4 (0, m) + ... + 2b0 y4 y4 (0, m + 1) + h 4 (1.80) b0 y x1 x1 2y4 (0, m + 1) + ... + 2y4 y4 (0, m + 1) = 0. 3

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

35

It seems necessary to emphasise that the obtained relation (1.80) represents the energy of thermoelasticity of a three-dimensional plate in the form of a mesh iteration. Let us make the following transformations:

h3 ai jk (m) y1x¯1 x1 (m) y1 (m + 1) − y1 (m − 1) = I1 (M) = − ω×ω4

=

ω×ω ¯ 4

h3 ai jk (m) y1x1 (m + 1)y1x1 (m) −

2

h

ω1 ×ω2 ×ω4

ω1 ×ω2 ×ω4

ai j0 (m) y1x1

ω×ω ¯ 4

y1x1

h3 ai jk (m) y1x1 (m)y1x1 (m − 1) −

(m + 1) − y1x1 (m − 1) ±

h2 yi (m) y1x1 (m + 1) − y1 (m − 1) = h3 ai jk (M + 1)y1x1 (M + 1) y1x1 (M) − ω

ω

I2 (M) = ω×ω ¯ 4

ω×ω ¯ 4

δai jk (m) 1 y x1 (m + 1) y1x1 (m) − h 4 ω×ω ¯ 4

h2 ai j0 y1x1 (m) y1 (m + 1) − y1 (m − 1) ±

ai jk (1)y1x1 (1) y1x1 (0) h3 − h4

ω1 ×ω2 ×ω4

2

(m)

h2 yi (m) y1 (m + 1) − y1 (m − 1) ,

ω1 ×ω2 ×ω4

h3 ai jk (1) y3x1 (1) δy1x3 (0) +

ai jk (m) y3x1

(m) δy1x1

(m) −

I3 (M) = −

ω1 ×ω2 ×ω4

ω×ω ¯ 4

ω×ω ¯ 4

ω1 ×ω2 ×ω4

h ai j0 y3x1 y1 (m + 1) − y1 (m − 1) ± 2

h3 h4 ai jk y4 (m) y1x4 x1 (m) + y1x¯4 x1 (m) ±

h3 ai jk y4 (m) y1x1 (m + 1) + y1x1 (m − 1) ,

ω×ω4

ω ¯

h3 ai jk (M) y3x1 (1) Mδy1x3 (M) +

ai jk h2 h4 y4 (m) y3x1 (1) y1x4 (m) +y1x¯4 (m) =

ωi ×ω j ×ω4

I3 (M) = −

h3 ai jk (m) y3 y1 (m + 1) − y1 (m − 1) ,

ω1 ×ω2 ×ω4

−

h3

h3 ai jk y4 (m) y3x3 (m + 1) + y3x3 (m − 1) +

h2 ai jk h4 y4 (m) y4 (m) y3x4 (m) + y3x¯4 (m) ±

ω1 ×ω2 ×ω4

h2 ai jk h4 y4 (m) y3x4 (m) + y3x¯4 (m) ,

36

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

I4 (M) = −

ω×ω4

I5 (M) =

(1.53)

I6 (M) =

(1.54)

I6 (M) =

3

2h4 h

ω×ω ¯ 4

ω×ω4

bi jk y4x1

ω

h4 h3 P y1x4 (m) + y1x¯4 (m) ,

h3 y1x¯4 (M + 1)2 + y1x4 (1)2 ,

(m) y4x1 (m + 1)±

2bi jk h4 h2 y4 (m)y4x3 (m) y4 (m + 1) =

ωi ×ω j ×ω4

ω×ω ¯ 4

2bi jk h4 h2 y4 (m)y4x1 (m) y4 (m + 1) ,

ωi ×ω j ×ω4

2h4 h3 bi jk y4x3 (m) y4x3 (m + 1)−

h3 bi jk −y4x3 (m)2 + y4x3 (m + 1)2 − δy4x3 (m)2 h4 − ωi ×ω j ×ω4

2bi jk h4 h2 y4x3 (m) y4 (m + 1) ,

I7 (M) = − I8 (M) =

ω×ω4

2h3 h4 P4 (m) y4 (m + 1),

ω×ω4

2h4 h3 bi jk y4x4 (m) y4 (m + 1) =

(1.59)

h2 bi jk (y4 (m + 1)2 − y4 (m)2 − δy4 (m)2 )h4 ,

ω×ω4

I9 (M) =

ω×ω4

ω

2bi jk y4x1 x4 (m) y4 (m + 1) = − (1.60)

2h3 y1x1 (M + 1) y4 (M + 1) bi jk −

I10 (M) = −

ω×ω4

ω×ω ¯ 4

3

h4 h

ai jk y4x¯2 x2

ω×ω4

ω

2bi jk h3 δy4 (m)y4x1 (m) +

2h3 y1x1 (0) y4 (0) bi jk ,

(m) y1x¯4 (m) + y1x4 (m)

h3 ai jk δy1x2 (m) + δy1x2 (m − 1) y2x2 (m)±

ωi ×ω j ×ω4

=

(1.54) (1.60)

ai jk h4 h2 y2x2 (m) y1x4 (m) + y1x¯4 (m) .

The other components of expression (1.80), which have already occurred in the previously examined solutions, appear during the process of summation on account of all nodes or they are similar to the already mentioned ones. Taking the signs into account, we estimate expression (1.80), and without introducing any additional theories we shall use the estimations published in works [285, 429]: ( ( (2 (( ( I1 (M)α > µ1 ((y1x (M + 1)(( − µ2 ((y1x1 (M + 1)(( ((δy1x1 (M)(( ≥ ( (2 (( ( √ h4 ( µ1 ((y1x (M + 1)(( − 2 3 ((y1x1 (M + 1)(( ((y1x¯1 (M + 1)(( , h inequality [285] is taken into account at this point:

1.2 Coupled 3D Thermoelasticity Problem for a Cubicoid

37

⎛ ⎞1 % ⎟⎟⎟ 2 h4 (( s (( ⎜⎜⎜ 3 $ s s (δy x (M)( = ⎜⎜⎜⎝h ≤ y x¯4 (i + 1, m + 1) − y x¯4 (i, m + 1) 2⎟⎟⎟⎠ h ω i

( √ h4 ( 2 3 ((y sx¯4 (M + 1)(( , h

( ( (( 1 ( ( ( ( (( (( 1 ( I1 (M)b ≤ µ2 (y x1 (1)( (y x1 (0)(( ≤ µ2 ((q x1 (( + h4 ((q x1 (( ((q x1 (( , ( ( ((y1 (m + 1)((( (((y1 (m)((( ≤ µ h ((y1 (m)(((2 , I1 (M)c ≤ µ3 h4 3 4 x x x ω4

(1.81)

ω4

( (( 1 (( ((P1 (m)((( (((y1 (m + 1)((( + (((y1 (m)((( ≤ 2P I4 (M) ≤ h4 x4 x¯4 2,1,ωτ max (y x¯4 (m)( , 1≤m≤M

ω4

( (2 ( (2 I5 (M) = ((y1x¯4 (M + 1)(( − ((y1x¯4 (1)(( , ( (2 ( (2 ( (2 I6 (M) = h4 bi jk ((y4x (m)(( + ((y4x (m + 1)(( − ((δy4x (m)(( , ω4

I6 (M)a ≥ µ4

( (2 ( (2 h4 ((y4x (m)(( + ((y4x (m + 1)(( ,

ω4

(2 µ5 (( 4 (δy (m)(( . 2 h In the investigated case, the following relation is applied: I6 (M)b ≤ 4 × 3

s 1 1 δy xi (x, m) = |δy s (i + 1, m) − δy s (i, m)| ≤ |δy s (i + 1, m) − δy s (i, m)| → h h (( s ((2 4 × 3 (δy x ( ≤ 2 δy s (m)2 , h I11 (M) =

3

α=1 ω4 ×ω j ×ωi

h4 h2 gαk yαx4 (x1 , x2 , 0, m) + yαx¯4 (x1 , x2 , 0, m) ,

I12 (M) = I7 (M) ≤ 2

ω4

ωi ×ω j

( (( ( ( ( h4 ((P4 (m)(( ((y4 (m + 1)(( ≤ ((P4 ((2,1,ω ,

I11 (M) ≤ 2 g (M)2,1,∂ωτ (1.61)

2h4 h2 gk y4 (x1 , x2 , 0, m) ,

τ

( ( max ((y x4 (m)(( ≤ ε g (M)22,1∂ω +

1≤m≤M

(1.62)

( (2 1 max ((y x¯4 (m)(( , 4ε 1≤m≤M ( (2 1 max ((y4 (m + 1)(( , I12 (M) ≤ ε g4 (M)22,1,∂ωτ + 4ε 1≤m≤M

38

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

⎛ ⎞ ⎜⎜⎜( ((2 (( 4 ((2 (( 4 ((⎟⎟⎟ ( 4 (δy (m)( − (y (0)(⎟⎟⎟⎠ . I8 (M) ≥ µ4 ⎜⎜⎝⎜(y (M + 1)( + ω4

Let us estimate the other components of formula (1.80): ⎛ ⎞ ⎟⎟⎟ ⎜⎜⎜ 3 3 1 3 1 3 3 1 h y x1 (m) δy x3 (m)⎟⎟⎟⎠ ≥ 0, |I2 (M)| ≥ µ1 ⎜⎜⎝⎜ h y x1 (1) δy x1 (0) + y x1 (M) δy x3 + 2 ω ¯ ω×ω ¯ 4 − |I3 (M)| ≤ −2µ2 (1.61)

( ((y4 (m)((( (((y3 (m + 1)((( , x3 ω4

⎞ ⎛ ⎜⎜⎜ ( (( (( 4 (( (( 1 (( (( 4 (( (( 1 ((⎟⎟⎟ ( 1 ⎜ −I9 (M) ≤ −2µ2 ⎜⎜⎝ (y x1 (m)( (δy (m)( + (y x1 (0)( (y (0)( − (y x1 (M + 1)(⎟⎟⎟⎠ , (1.61) ω4

⎛ ⎞ (( (( (( 4 (( (( 1 (( (( 4 ((⎟⎟⎟ (( 4 ⎜⎜⎜⎜ (( 1 (y (M + 1)( ≤ −2µ5 ⎜⎜⎝ (y x1 (m)( (δy (m)( + (y x1 (0)( (y (0)(⎟⎟⎟⎠ , ω4

1 2 3 δy x2 (m) y x2 (m) h ≥ 0. |I10 | ≥ 2µ1 ω×ω4 As a result, we obtain the following estimation (1.82): (( 1 ( ( (2 ( (2 ( (2 (2 (2 (y x¯4 (M+1)(( + ((y2x¯4 (M+1)(( + ((y3x¯4 (M+1)(( + µ1 ((y1x (M+1)(( + µ1 ((y2x (M+1)(( + ( (2 (( ( ( (( ( √ h4 ( µ1 ((y3x (M+1)(( − 2µ2 3 ((y1x (M+1)(( ((y1x¯4 (M+1)(( + ((y2x (M+1)(( ((y2x¯4 (M+1)(( + h ( (( (( 3 (( (2 ( ( ( (2 (( 3 (y (M+1)( (y (M+1)( + µ h ((y4 (m)(( + ((y4 (M+1)(( + µ ((y4 (M+1)(( + x

4

x¯4

4

x

x

ω4

( ( ((y1 (m)(((2 + (((y2 (m)(((2 + (((y3 (m)(((2 + ((δy4 (m)(((2 ≤µ (((q1 ((( + h (((q2 ((( q +µ h 2 4 1x 3 4 x x x x x ω4

ω4

( ( ( (( 1 (( ( ( ( ( ( ( ( (P (2,1ω max ((y1x¯4 (m)(( + ((P2 ((2,1,ω max ((y2x¯4 (m)(( + ((P3 ((2,1,ω max ((y3x¯4 (m)(( + τ

τ

1≤m≤M

τ

1≤m≤M

1≤m≤M

( (( 1 ((2 ( (2 ( (2 h ((δy4 (m)(((2 + (((P4 ((( (q2 (2,1ω + ((q22 ((2,1ω + ((q32 ((2,1ω + µ5 42 12 + 2,1ωτ τ τ τ h ω4 ( (( 4 ((2 ((δy4 (m)((( (((y3 (M+1)((( + (((y1 (M+1)((( + (((y2 (M+1)((( + (y (0)( + 2µ2 x3 x1 x2 ω4

2µ5

3 ( (( i (( ((δy4 (m)((( (((y1 (m)((( + (((y2 (m)((( + (((y3 (m)(((+ 2 (((qi ((( x1 x2 x3 i 2,1∂ω max (y x4 (m)( + ω4

τ

i=1

( ( 2 q4 2,1∂ωτ max ((y4 (m)(( . 1≤m≤M

1≤m≤M

(1.82)

We introduce similar terms and reinforce the inequality with relation (1.62) and impose condition (1.79) onto the steps of the net field ωτ . The result is as follows:

1.3 Methods of Solving Diﬀerence Equations

39

(( s (2 ( (2 ( (2 (y x¯4 (M + 1)(( + ((y sx (M + 1)(( + ((y4 (M + 1)(( + y s (M + 1)2 + (( 4 (2 ( (2 (y x4 (M + 1)(( + ((y4x (M + 1)(( ≤ ( (2 ( ( ( (2 C (τ, ε) P s 22,1ωτ + ((P4 ((2,1ω + g s 22,1∂ωτ + g4 22,1∂ωτ + g4 22,ω + ((g1s ((2,ω + ((g2s ((2,ω . τ (1.83) Inequality (1.83) occurs for every M and provides stability of the diﬀerence scheme (1.38)–(1.45). If the inequality is satisfied, the solutions (1.38)–(1.45) become stable and convergent towards the solution of problem (1.23)–(1.33). Further considerations are no diﬀerent from the classic ones, described in work [285], that is why we shall not quote them here. First, a uniform bound of norms in W21,0 (Qτ ) is derived from estimation (1.83), then weak compactness of continuous couplings us (1.23)–(1.24) in W21,0 (Qτ ) is derived to prove consequently that only if the subsequence weakly converges in W21,0 (Qτ ) upon a function, then the function is the generalized solution (1.23)–(1.33) that belongs to W22,1 (Qτ ).

1.3 Methods of Solving Diﬀerence Equations The theoretical foundations presented in the first section may appear useful in formulation of methods, algorithms or writing computation programs that would make it possible to automatically solve problems in the field of statics, quasistatics, dynamics and coupled thermoplasticity taking into account the broad class of typically classic and non-classic boundary conditions at a plate’s edges. The use of the variational-diﬀerence method for solving the system of equations (1.23)–(1.33) allows obtaining the system of diﬀerence equations (1.38)–(1.45). The system is characterised by the following specific features: a) it has a high order equal to the order of the mesh nodes multiplied by four; b) the matrix is dissected, which means that in every line there are several elements that are not equal to zero; c) the matrix’ non-zero elements are distributed according to a certain scheme - the matrix becomes a cellular matrix. Many diﬀerent methods may be applied to solve problems of such a type. The methods used in this chapter to solve diﬀerence systems have the best method characteristics for each of the specific problems (statics, quasistatics and dynamics). The term ‘method characteristic’ may be applied to numerical stability, accuracy, computation time economy or the amount of computer memory involved in the process of computation. This chapter describes the algorithms that carry optimum methods into eﬀect and verifies correctness of the results. Finally, it presents the investigation of the influence of continuity conditions on the solution’s behaviour. 1.3.1 Dimensionless Equations Many problems of mechanics that involve deformable bodies require bringing the equations to their dimensionless forms, which in consequence allows estimating

40

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

physical phenomena in many other similar objects. Dimensionless parameters in thermoelasticity problems ui , τ, xi , θ, are related to dimensional quantities through the folowing dependences [79, 217, 384]: xi = i xi , τ = ℵ=

τ32 ET 0 α2T , θ = θT 0 , ui = ui αT T 0 , β = , α (1 − 2ν)cρ

λq T 0 τT α2 ρ ET 0 αT = , P4 = P4 2 , Pi = Pi , f = f EαT T 0 , τM i ET 0 32 3

T i0 =

T 0i − T 0 T λq T 0 αi 1 2 , BI = , Ti = i , λ1 = , λ2 = , T0 λq i 2 3 ⎧ −2 ⎪ λ , i=1 ⎪ ⎪ ⎪ 3 3 ⎨ λ3 = , λ = ⎪ λ22 , i=2 , ⎪ ⎪ 1 ⎪ ⎩ 1, i=3

(1.84)

where ℵ is the system’s inertia coeﬃcient within the range of · 106 text− − − · 108 [217] at measured in centimetres. The system of diﬀerential equations for a homogeneous isotropic material of three-dimensional plate (1.23)–(1.24) takes the following form (the unknowns are dimensionless quantities): λ−2 λ2i ∂2 ui ∂2 u j ∂2 uk 1−ν k + + + 2 2 (1 + ν)(1 − 2ν) ∂xi 2(1 + ν) ∂x j 2(1 + ν) ∂xk2 2 j ∂u ∂θ 1 1 ∂2 uk ∂2 ui − + + Pi = λℵ 2 , 2(1 + ν)(1 − 2ν) ∂xi ∂x j ∂xk ∂xi 2(1 − 2ν) ∂xi ∂τ 2 1 2 2 2 3 2 2 2 ∂u ∂ u ∂ u ∂θ ∂ θ ∂θ ∂ θ + P4 = . (1.85) λ23 2 + λ−2 + 2 −β 2 2 ∂x1 ∂τ ∂x2 ∂τ ∂x3 ∂τ ∂τ ∂x1 ∂x2 ∂x3 The dimensionless conditions on the plate’s walls have the following form: j i (1 − 2ν) (1 + ν) i ∂u ∂u 1 ν ∂uk i θ ni,i+m + Ei+m fi+m , − + + i ui = Dii+m ∂xi 1 − ν ∂x j ∂xk 1 − 2ν 1−ν j i ∂u j j j 2 ∂u ni,i+m + Ei+m + λi j u j = 2λi (1 + ν) fi+m , Di+m ∂xi ∂x j k i ∂u k −2 ∂u k k ni,i+m + Ei+m + λk k uk = 2λ−1 Di+m k (1 + ν) fi+m , ∂xi ∂xk Ci+m

∂θ = BI(T i0 − θ)Ai+m + Bi+m T i , ∂xi

(1.86)

where α is a heat exchange coeﬃcient. The dimensionless coeﬃcients of the equations related with the parallel walls are equal.

1.3 Methods of Solving Diﬀerence Equations

41

1.3.2 Systems of Elliptic Diﬀerence Equations The problems concerning elliptic diﬀerence equations are solved with iterative methods. Certain interesting characteristics constituted of several unknowns [50, 52, 431] are the objects of study in this case. A Dirichlet model for Poisson equation will serve as a model problem in our considerations:

y x¯1 x1 + y x¯2 x2 = − fi j , fi j = f x1i , x2 j , yi j |γ = ϕi j .

(1.87)

The investigation will be carried on for the following case: a) yi j |γ1 +γ2 = ϕ0 , yi j |γ4 = ϕ1 , yi j |γ6 +γ7 = ϕ2 , 0 ≤ x1 ≤ m2 , x − m1 , 0 ≤ x2 ≤ l3 , m2 − m1 ⎧ ⎪ x2 − l1 ⎨ ϕ0 , γ3 yi j |γ8 = ϕ0 + (ϕ2 − ϕ1 ) , k=⎪ ⎩ϕ ,γ ; l2 − l 1 2 5 yi j |γ3 ,γ5 = k (ϕ1 − k)

(1.88)

b) on a square-shaped surface (Fig. 1.2b) 0 ≤ xi ≤ 1: yi j | x1 =0 = e3x2 , yi j | x2 =0 = cos 3x1 , yi j | x1 =1 = e3x2 cos 3, yi j | x2 =1 = e3 cos 3x1 , y = e3x2 cos 3x1 .

(1.89)

Seidel’s method is analysed and so is the method of upper relaxation, the explicit and implicit method of variable directions and the direct method of variable directions with Chebyshev’s acceleration. The methods are compared in consideration of the velocity of convergence and the problem of choosing parameters that makes it possible to accelerate the iterative processes is investigated. All the mentioned methods are characterised by simplicity of realisation and suﬃciently good convergence. Let us consider the following system of linear diﬀerence equations: Ay = f .

(1.90)

The equation that describes the method of upper relaxation takes the following form:

(k) (k−1) + 1 − ωopt y(k−1) + ωopt fi j , (1.91) y(k) i j = ωopt L1 yi j + L2 yi j ij where: A = E+D+F, L1 = E/D, L2 = F/D, whereas D, E and F denote respectively diagonal, upper and lower triangular components of the A passage. At ωopt = 1 method (1.91) transforms into Seidel’s method ωopt =

2 , ) 1 + 1 − λ21

(1.92)

42

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

a)

x1 m2

g4 g5

g3 m1

g6

g2 g1

g8

l1 b)

g7

l2

l3

x2

x1 l1

l2

x2

Figure 1.2. Dirichlet’s problem for Poisson’s equation: a) on a T-shaped surface, b) on a square-shaped surface.

where λ1 denotes the spectral radius of the A passage matrix corresponding to Seidel’s method. The theoretical value of λ1 (quoted from appropriate literary sources) in a rectangular field is equal to [431]: λ1 =

h21 2(h21

+

h22 )

cos

h22 πh2 πh1 cos + , 2 l2 l1 2(h1 + h22 )

cos

h21 h23 πh1 πh2 + 2 2 cos + l1 l2 h1 h2 + h22 h23 + h23 h21

yet, for a cubicoid it equals: λ1 =

h22 h23 h21 h22 + h22 h23 + h23 h21

(1.93)

1.3 Methods of Solving Diﬀerence Equations

h21 h22 h21 h22

+

h22 h23

+

h23 h21

cos

πh3 . l3

43

(1.94)

In order to determine ωopt within an arbitrary field, an approximate value of λ1 is used in equation (1.92). There is a fundamental relationship binding the spectral radius of Seidel’s matrix and the spectral radius of the m1 and ωopt upper relaxation matrix. (µ1 + ω − 1)2 = λ21 . (1.95) ω2 µ1 By substituting formula (1.95) into (1.92) is possible to estimate approximately = µ1Mm can be calculated according the value of ωopt . The approximate value of µm−1 1 to the following formula based on Seidel’s iterative process [52]: (( (( y(k+1) − y(k) ij ij (y(k+1) − y(k) ( ij ( (( = , (1.96) µm 1 = lim ( (k) k→∞ (y y(k) − y(k−1) − y(k−1) ( ij ij ij

where µm is a minimum number, for which the following condition is fulfilled: Mm µ1 (1.97) Mm−1 − 1 ≤ εµ , µ 1

where: ω0 , εm u are set quantities (we may also assume that ω0 equals one). It appears that the velocity of convergence may be significantly increased, if ω0 is chosen exact to one hundredth. It can be achieved by applying the equivalent rectangle principle that consists in searching for the spectral radius of Seidel’s method passage matrix for the field of an equivalent rectangle, according to formula (1.93). The equivalent rectangle is such a rectangle, the surface of which is equal to the investigated surface and which is constructed according to the following principles: its width is the measurement of the largest circle that can be drawn within the surface’s boundaries (the circle should not cross the boundary) and its length corresponds to the surface of the investigated field divided by the field’s width. The dependence of the number of iterations necessary for obtaining solution of the set accuracy εit = 10−6 on parameter ωopt can be found in Table 1.1. Table 1.1. The dependence of the number of iterations N on ω (the upper relaxation method). ω N

1 247

1.5847 100

1.6073 96

1.65 89

1.6752 94

The formulas that describe the overt method of variable directions have the following form [163]: ! " (k+ 12 ) (k+ 12 ) (k+ 1 ) + A2 y(k) yi j 2 = y(k) |∂r = ϕi j , i j − τk A1 yi j i j , yi j

44

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

! " (k+ 12 ) (k+ 12 ) (k+1) ∂r = ϕi j , y(k+1) = y − τ y + A y A , y(k+1) k 1 2 ij ij ij ij ij

(1.98)

where: A1 yi j = a1i j yi−1 j −

a0i j 2

yi j + a2i j yi j−1 , A2 yi j = a3i j yi+1 j −

a0i j 2

yi j + a4i j yi j+1 .

(1.99)

Parameter τk is chosen to be equal to 1/hav [170], and numerical experiments show that changes of τk have little influence on the velocity of convergence (Table 1.2). To increase the convergence velocity of the overt method of variable directions, applying Chebyshev acceleration of convergence [52] is recommended. In result, the iterative process takes the following form: (k+1) = y(k) − y(k) y(k+1) ij i j + αk (y i j ),

(1.100)

ij

where y(k+1) is calculated according to formulas (1.98), (1.99), αk are the coeﬃcients ij used for increasing the convergence velocity. In case of Chebyshev’s cyclic method, αk is determined according to formula [294, 318] −1

2k − 1 π αk = 2 M + m − (M − m) cos N

,

(1.101)

where m and M are respectively the minimum and the maximum proper values of the passage matrix in Chebyshev cyclic method, which is based not on formulas (1.98) and (1.99), but on the implicit method of variable directions. For the iterative process being discussed here, αk is expressed by means of M in the following way [52]:

M 2k−1 2 1 + cos 2N π .

(1.102) αk = 1 − M2 1 + cos 2k−1 2N π M is determined approximately with the use of Lusternik’s algorithm [316]. The iterative process is realised by making N1 iterations according to formulas (1.98) and (1.99) with (( ( ((y(k) − y(k−1) ((( i j i j ( ( , at λ(k) (1.103) λ(k) 1 = ( (k−1) 1 → M, when k → ∞ . (k−2) ( ((y − yi j (( ij

Table 1.2. The dependence of the number of iterations N on τ (the overt method of variable directions). τ N

15 77

16 75

20 64

26 70

32 81

1.3 Methods of Solving Diﬀerence Equations

45

(m) If λ(k) 1 begins to diﬀer slightly between subsequent iterations then λ1 may be assumed as approximate to M = λ(m) 1 . Next, αk is calculated for k = 1, ..., N and further operations are conducted with acceleration (the parameter changes periodically with the period equal to the number of parameters - N). The method is characterised by greater velocity of convergence than the overt method of variable directions and the above mentioned upper relaxation method. The results of the theoretical investigations of the convergence velocity [109, 431] have also been practically proven (Table 1.3).

Table 1.3. Comparison of iterative methods for a T -shaped field.

Seidel’s method Upper relaxation method Explicit method of variable directions Chebyshev’s acceleration method

Theoretical velocity of convergence π2 h2 2πh 2πh √ 2π h

N

εit

800 57 55 30

10−4 10−4 10−4 10−4

In order to choose the most eﬀective solution method, a comparison of a numerical solution and an exact solution (1.89) [50, 52] has been made on diﬀerent nets, with various parameters accelerating the iteration process, in relation to exact location of spectral characteristics of passage matrixes, with various approximation errors, and also in relation to the velocity of convergence (Table 1.4). Table 1.4. Comparison of iterative methods for a square-shaped field. Seidel’s method Scheme O(h2 ) O(h4 ) Parameter 1 1 N 205 196 εit 10−6 10−6 Error 7 · 10−3 6 · 10−4 Theoretical velocity of π2 h2 Λ0 2π2 h2 convergence

Upper relaxation Variable directions method method Implicit Explicit O(h2 ) O(h4 ) O(h2 ) O(h4 ) 1.8225 1.792 – 20 92 88 12 88 10−6 10−6 10−6 10−6 6 · 10−4 1 · 10−4 3 · 10−5 1 · 10−4 2πh

2.09πh

–

πh

This work does not include any iterative formulas of the implicit method of variable directions because of their complexity (see [431]). Numerical experiments have shown that among all the considered methods the most eﬀective one for a square is the upper relaxation method of an increased order of accuracy. The number of iterations necessary to achieve the set accuracy with the use of the implicit method of variable directions is smaller. However, it is not economical as far as such factors as machine computation time, sophistication of software and memory capacity (it

46

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

requires twice as much memory as the other methods) are concerned. The explicit method of variable directions works well in a T -shaped field (Table 1.3). The results obtained with the use of the upper (successive and block) relaxation method, the explicit and implicit methods of variable directions and the method of variable triangles with Chebyshev’s acceleration have been compared within a ring-shaped field [61] for the first and the third boundary condition in the polar coordinates system. Despite the fact that the theoretical and the numerical velocities of convergence in the method with Chebyshev’s acceleration are higher than in all the other methods, the method’s poor accuracy makes it practically useless (Table 1.5).

Table 1.5. Comparison of iterative methods for a ring-shaped field. εit

N

Error

10−3

7

0.01703

1 4 4 ln ln π2 εit η

10−3

15

0.00073

1 1 √ ln 4 η ε

Implicit method of variable directions Explicit method of variable directions

Number of iterations necessary to achieve set accuracy

Chebyshev’s acceleration method

2 ln ε √ √ 2 2 4η

10−3

37

0.06503

10−3

142

0.00015

ln

0.00015

* 1 ln ∆1 ln ε

Succesive upper relaxation method Block upper relaxation method

−3

10

47

* 1 ln ∆1 ε

The results of the investigation testing applicability of a number of methods have shown that the most useful method of analysing solutions for a cubicoid (also for static problems of thermoelasticity) is the upper relaxation method with error O(h2 ). An increased-order scheme within a rectangle-shaped field brings better results in case of a more complicated system of diﬀerential equations and a more complicated field, in which a slight diﬀerence in the number of iterations leads to significant complications of the diﬀerence scheme due to a larger number of approximation nodes. Another important problem is the choice of a digitisation step of the investigated field’s (cubicoid’s) mesh. Runge’s law [37], used so often, makes it possible to choose optimum steps h1 = h2 = h3 = 0.125. Further decrease of the net’s steps does not result in serious changes in obtained results.

1.3 Methods of Solving Diﬀerence Equations

47

1.3.3 Systems of Parabolic and Hyperbolic Diﬀerence Equations Works [97, 399, 419, 494] present a generalized method of constructing diﬀerence schemes for numerical solutions of non-stationary problems of heat conductivity and continuous media dynamics. A large number of publications devoted to that issue has been listed in book [399]. The system of diﬀerence equations (1.38)–(1.45) is presented in an explicit form - the most convenient for numerical integration. However, explicit schemes are not always stable. Conditions (1.74) are imposed on the diﬀerence scheme’s steps. It turns out that limitations concerning the practical use of explicit schemes may be very significant and that is the reason for applying implicit schemes. Implicit schemes are absolutely stable and the only limitation they impose on the time step is the condition of accuracy. Nevertheless, in contrast to explicit schemes, simple implicit schemes for multidimensional problems turn out to be uneconomical [97, 108]. That is why from now on we are going to consider only explicit schemes. We are going to present numerical experiments concerning a model problem applied for comparison of solutions with the use of explicit finite diﬀerence schemes (1.38)–(1.45) with accuracy O(h44 + h2 ), then with Runge-Kutta’s method with accuracy O(h44 + h2 ) and constant step and with Runge-Kutta’s method with accuracy O(h44 +h2 ) with automatic step choice. The model problem is going to be represented by Cauchy’s problem [51, 60] for the following system: d2 x1 = f1 (x1 , x2 ) , dτ2 d2 x2 = f2 (x1 , x2 ) , dτ2 x s (0) = x0s ,

(1.104)

describing movement of charged particles with initial velocities and motion start coordinates. Change of the variables allows decrease of the system’s order: dϕ s dx s = ϕ s (x1 , x2 ), = f s (x1 , x2 ), dτ dτ x s (0) = x0s , ϕ s (0) = x˙0s , s = 1, 2 .

(1.105)

Explicit diﬀerence scheme (1.38)–(1.45) takes the following form: yτs = ϕisj , y s (0) = x0si j , ϕτs = fisj , ϕ s (0) = x˙0si j , s = 1, 2 .

(1.106)

Formulas that describe Runge-Kutta’s method are commonly known [397, 474]. The most widely applied is the method with order O(h44 ). On one hand, the investigated methods reveal very good characteristics: in order to determine values in the next point they only require information about the previous point. This makes them

48

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

economical as far as the use of computer memory is concerned. On the other hand, the necessity to repeat calculation of the right sides at every integration step is a serious fault, since it takes significantly longer calculation time. The finite diﬀerence method (1.105) allows for computing the right sides only once but it considerably aﬀects accuracy of the results. The trajectory that corresponds to Runge-Kutta’s computation method is the closest to the real solution, which confirms theoretical considerations concerning the error’s order. Therefore, all further calculations have been made with the use of this method. It is necessary to point out that RungeKutta’s method has been “accepted” as stable. In case of the automatic integration step choice based on the approach described in publication [37], the calculation time increases two or four times depending on how far the first step is from the optimum step. That is the reason why such modification is not applied. Stability conditions that definitely impose limitations in regard to the integration step in time and to the integration step in space are presented in Section 2.2.3. They have been applied in order to obtain an optimum solution with the use of Runge-Kutta’s method. Numerical experiments show correctness of this approach. The following problems have been solved: a) thin plate’s vibrations, b) the problem of non-stationary heat transfer (problems no. 10 and 12 formulated in Section 2.3) with the use of various integration steps. Graphs presented in Figures 1.3 and 1.4 illustrate: a) deflections in the centre of the plate in time, b) temperature distribution in the centre of the plate in time. At steps h4 > h 4,opt and h4 > h

4,opt the process is unstable and it does not reflect the real physical phenomenon. Through decreasing the step and thus approaching the optimum value, that may be calculated from inequality (1.79) for h = 0.125, the processes start becoming stable, and at h4 ≤ h 4,opt and h4 ≤ h

4,opt they describe real vibrations and real changes of temperature. If the dynamic problems and the non-stationary heat conductivity problems were solved independently, then in every case the stability condition should be applied to quicken the calculation process, since the optimum steps h 4,opt > h

4,opt are diﬀerent from each other. All calculations in this work have been made with the following integration steps: h 4,opt = 0.05 and h

4,opt = 0.00125. In case of Runge-Kutta’s method, the theoretical basis for assuming stability conditions (1.79) is the fact that calculations are characterised with high accuracy [37] and are described by the following formula: yi+1 = yi + h f (x, y) ,

(1.107)

which refers to the finitely dimensional approximation (1.106) obtained in Section 2.1. The conducted consideration leads to a conclusion that Runge-Kutta’s method may be applied together with the conditions of the theorem presented in Section 2.1. 1.3.4 Algorithm Mathematical descriptions of all kinds of problems (static, quasistatic, dynamic and coupled) are characterised by many types of systems of diﬀerential equations (elliptic, parabolic, hyperbolic or parabolic-hyperbolic). Solutions to such equations can

1.3 Methods of Solving Diﬀerence Equations u

49

3

0.05

x2

x1 x3

Dt= Dt = 0.05 0.025

Dt= Dt = 0.2

0

4

6

8

t

10

k=25

-0.025

-0.05

Figure 1.3. The graph of the deflection in the plate’s centre in time (the problem of the nonstationary heat transfer during a thin plate’s vibrations).

be obtained by applying various methods or combinations of methods. For this reason, the algorithms that realise numerical calculations will be described separately. Algorithm 1. Solving static problems. Static problems in three-dimensional problems and stationary heat conduction problems are described mathematically by a system of equations or one elliptic equation formed as a result of “breaking” system (1.38)–(1.39) without taking

50

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 3

u ,µ 0.9

u

0.8

3

0.7 0.6 0.5

µ

0.4 0.3 0.2 0.1 0

0.01

0.02

0.03

0.04

t

-0.1 -0.2

h4'=0,1

-0.3

h4''=0,01

-0.4 -0.5 -0.6

Figure 1.4. Fig. 4. The temperature distribution in a plate’s centre in time (the problem of the non-stationary heat transfer during a thin plate’s vibrations).

inertia forces into account. In order to solve this problem we shall use the upper relaxation method. Each of the system’s four equations takes the following form:

Λy(i) = Λi xi , y xj j xi , ykxi xk , y4 , i → j → k,

(1.108)

where Λ denotes a diﬀerence analogue of Laplace’s diﬀerential operator. Formulation of boundary conditions at the field’s boundary coincides with conditions (1.40)–(1.45), thus the algorithms realising calculation methods are diﬀerent. The iteration formula, both for the system and one equation, can be notated in the following way: ai yis(k+1) = (1 − ωopt )ai yis(k) + ωopt

i−1 j=1

a j y s(k+1) − ωopt j

M

a j y s(k+1) + ωopt Λis , (1.109) j

j=1

where i denotes transitional numeration of the three-dimensional field of the vector’s elements, y s (y1s , ..., y sM ), M = N1 N2 N3 , k - number of iterations, ai - coeﬃcients at the unknowns. Parameter ωopt is chosen according to the relations made in Section 2.2. First, initial approximations are set in the entire field of the mesh and boundary conditions are set on its boundaries, where Dirichlet’s problem is considered. Then cyclic calculations are made according to formulas (1.96), (1.97), (1.95), (1.92),

1.3 Methods of Solving Diﬀerence Equations

51

(1.109). While solving Neuman’s problem or the third boundary problem in the boundary nodes for every iteration within the field, the calculations are made according to the following formulas (since it is impossible to present dimensionless conditions in a boundary node by means of an index, we shall use formulas for an edge and a corner as an example): −1 h2 h2 1−ν (1 − ν) h1 λ1 1 5 D1 + λ1 D1 · = h1 (1 − 2ν)(1 + ν) h2 2(1 + ν) h1 (1 − 2ν)(1 + ν) h1 1(k−1) v 2(k) 1(k) 1(k) 1 1(k) 3(k) D1 yi−N1 − − 3yi−N1 + 4yi−2N1 + y − y x3 − y 1 − ν x2 2 i h2 2(k−1) h2 1(k−1) h1 1(k) 1(k) D15 λ1 y1(k) y + + − 3y + 4y − y i−1 i+1 i+2 2h2 (1 + ν) λ 1 x1 2 i 1 4(k−1) 1 1 y , (1.110) f5 h1 + f1 h2 − 1 − 2ν i ⎞ ⎛ D22 ⎟⎟⎟ h1 ⎜⎜⎜ 1 (1 − ν) (1 − ν) 2(k) 2 + D5 · yi = ⎝⎜ ⎠⎟ h2 h1 2(1 + ν) (1 − 2ν)(1 + ν) h2 (1 − 2ν)(1 + ν) h2 2(k−1) h2 1(k) 2(k) 2(k) 2 2(k) 3(k−1) D5 yi+1 + ν − 3yi+1 + 4yi+2 + − y − y x3 y 1 − ν x1 2 i h1 2(k−1) h2 λ 1 2(k) 2(k) 2 2(k) −1 1(k) D y λ − h1 λ1 y x2 − − 3yi−N1 + 4yi−2N1 + y h1 2(1 + ν) 1 i−N1 1 2 i 1 y4 , f12 h2 − h1 f52 + 1 − 2ν −1 h2 3 2(k) h2 3 h1 3 h1 3(k−1) 1(k) y3(k) = D1 + D5 λ1 D1 yi−N1 + h1 λ−2 − y 3 y x3 − i h1 h2 h1 2 i % h1 3(k) h2 3(k−1) 3(k) 3 −1 2(k) 3y3(k) + 4y − i−N1 i−2N1 + h D5 yi+1 λ1 − h2 λ3 λ2 y x3 − 2 yi 2 %

3(k) 3 −1 −3y3(k) + 2 h2 f13 λ−1 3 − h1 f5 λ3 (1 + ν) , i+1 + 4yi+2

y1(k) i

= y4(k) i

1 $ 4(k) + B5 T 5 h2 + B1 T 1 h1 − BI C1 h2 y4(k−1) + +C5 h1 y4(k−1) yi−N1 + y4(k) + i i i+1 c1 + c5 h1

h2 4(k−1) 4(k) 4(k) 4 4 4 − 3yi+1 + 4yi+2 + (1.111) y y − 3yi−N1 + 4yi−2N1 . 2 i 2 i

The dimensionless relations of calculations in corner ∂Ω4 ∂Ω2 ∂Ω3 at h1 = h2 = h3 take the following form: 3(k−1) h1 1(k−1) $ 2(k) ν 1(k) 1 y1(k) = D − + y1+N1 N2 − y3(k) − y2 − y2(k−1) y 4 y1+N1 − 1 1 1 2(1 − ν) 2 1

52

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1 2(1 − 2ν)(1 + ν) 4(k−1) 1 (1 − ν) + h1 f1 + y + − 1−ν 1 − 2ν 1

3(k) $ % h1 1(k−1) 1(k−1) − 3y1(k) + 4y1(k) λ23 − + D13 y1(k) y1 1+N1 N2 − y1+N1 − y1 1+N 1+2N N N 1 2 1 2 2

h1 1(k−1) 2(k−1) − y2(k) λ−2 2(1 + ν)h f31 λ3 (1 − 2ν) + D12 y1(k) y 1 − 2 1+N1 − y1 2 1

% 1 − 2ν 1(k) 1 −1 1 1 1 (1 (1 D 3y1(k) + 4y f λ + − 2ν) D + D + 2 + ν) h , 2 2 1 4 2 3 2 3 2 (1 − ν)

1(k) h2 2(k−1) = D24 y2(k) − y1(k) − 3y2(k) + 4y2(k) y2(k) λ21 − − y1 1 1+N1 + y2 1 1+N 1+2N 1 1 2

1(k) % ν y 2 (1 + ν) h1 λ1 f42 (1 − 2ν) + D22 y2(k) + − y1(k) + y3(k) 2 1 1+N1 N2 − 2 (1 − ν) 1+N1 h2 1(k) (1 − 2v) (1 + ν) f22 + y3(k−1) + 4y1(k) − + y1 − 3y1(k) 1 2 3 2 h2 (1 − ν)

$ 1 y4(k) + 2 (1 + ν) + D23 y2(k) + y32 − y31 λ−2 2 − 1 1+N N 1 2 1 − 2ν % h2 2(k−1) 2(k) 2 −1 (1 (1 − 2ν) · − 3y2(k) + 4y f λ + 2 + ν) h y1 1 3 2 1+N 1+2N N N 1 2 1 2 2

% $ (1.112) 2 (1 − ν) D22 + (1 − 2ν) D24 + D23 , h2 3(k−1) 1(k) 1(k) = D34 y3(k) − − 3y3(k) + 4y3(k) − y3(k) y1 1 1+N1 + y1+N1 N2 − y1 1+N 1+2N 1 1 2

% $ 2(k) + y2(k) λ22 − 2 (1 + ν) h3 f43 (1 − 2ν) + D32 y3(k) 2 1+N1 N2 − y1 ν h1 3(k−1) 3(k) 3 3(k) (1 · − 3y3(k) + 4y − 2ν) + D y1 3 y1+N1 N2 + 2 3 2 2 (1 − v) h2 3(k−1)

1(k) 1(k) 2(k) 2(k) 3(k) 3(k) − 3y1+N1 N2 + 4y1+2N1 N2 + y1+N1 − y1 + y2 − y1 − y 2 1 $

%−1 1+ν 1 4(k−1) 3 (1 − 2ν) h1 f3 − y1 2 (1 − ν) 2(1 − ν)D33 + (1 − 2v) D32 + D34 , 1−ν 1 − 2ν C4 h 4(k−1) 4(k−1) 4(k) y4(k) = C4 y4(k) + − 3y4(k) + 4y + y1 1 1+N1 + B4 hT 4 − BIA4 y1 1+N 1+2N 1 1 2 C2 h 4(k−1) + B2 hT 2 − BIA2 y4(k−1) + − 3y4(k) + 4y4(k) + y1 C2 y4(k) 2 1 2 3 2 C3 h 4(k−1) 4(k−1) 4(k) 4(k) (C2 + C4 )−1 . + B hT − BIA y + − 3y + 4y C3 y1(k) y 3 3 3 1+N1 N2 1 1 1+N1 N2 1+2N1 N2 2 3y1(k) 1+N1

4y1(k) 1+2N1

%

Calculations are finished as soon as the following criterion is satisfied:

1.3 Methods of Solving Diﬀerence Equations

s(k) yi − yis(k+1) s(k) < εit . max i,s y

53

(1.113)

i

In case of a stationary heat transfer problem the applied algorithm is assumed for one equation in order to calculate y4 . When a static elasticity problem is solved the algorithm is applied in the system with unknowns y s , s = 1, ..., 3. For the problems that refer to thermal stress according to the algorithm, the temperature distribution is investigated first and then the algorithm is used again to determine the displacement distribution y s , s = 1, ..., 3. Algorithm 2. Solving dynamic problems, non-stationary heat transfer problems and coupled thermoelasticity problems. Similarly to the case of model problem (1.106)–(1.107), through changing the variables systems (1.38)–(1.45) lead to the system of seven equations. Apart from displacements y s , s = 1, ..., 3 and temperature y4 , the velocities of displacements are also unknown. At the first stage of solving the problem, the initial conditions are set for all unknown nodes. Then, depending on the type of the initial-boundary conditions, the known values of surface functions are set (in case of the first boundary problem). Next, a step in time is made using Runge-Kutta’s method formulas for every unknown node within the net’s field, and the analogous right sides of equations (1.38)–(1.39) are calculated. After completion of the calculations for all the field’s inner nodes, the analyses of the types of boundary problems are made at every time step. If the second or the third initial-boundary conditions are solved, then values y4 , that correspond to a given type of a problem, are calculated in the nodes according to relations (1.110), (1.111), (1.112). Calculations are made in the same manner within the whole time range. While solving uncoupled thermoelasticity problems, the discussed algorithm is applied to system (1.38)–(1.45), and the coeﬃcient placed at the dilatation term in the heat conductivity equation is assumed to be equal to zero. Algorithm 2 may be used to calculate the dynamic behaviour of plates aﬀected by all kinds of volume and surface forces, with zero coeﬃcients placed at the thermal gradient expression. The system of equations is reduced to six equations. The number of boundary conditions is also reduced. If algorithm 2 was to be applied solely to the heat conduction equation (1.39) with thermal boundary conditions (1.41), (1.43), (1.45), then the solution would be the distribution of the temperature field of a three-dimensional plate subjected to various types of thermal load, including inner heat sources. One particular feature (free form the influence of thermal loads) of solving dynamic problems for thick plates with the use of algorithm 2 is worth emphasising. The thicker the plate, the more unstable the solution is. An analysis of that process has shown that instability occurs in proximity of the field’s boundary. Since the boundary conditions are calculated according to formulas (1.110), (1.111), (1.112), i.e. while solving the third boundary problem, the instability penetrates into the field. The instability disappears if the free edge condition is replaced with another condition for any other type of support. The instability disappears also along with the decrease of the plate’s thickness (for λ1 = 1/4, 1/10, 1/50, 1/100 - the calculations

54

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

are stable; for λ1 = 1/2 - slight instability occurs; for λ1 = λ2 = λ3 = 1 - the calculations are unstable). Running the iteration process within the field’s boundaries prevents instability from occurring in case of thick plates. In case of thin plates though, the process still remains unstable (for λ1 = 1/4 - instability disappears after 16 time steps; for λ1 = 1/10, 1/100 - instability disappears after the first time step). Due to the conclusions drawn above, there may be two cases of algorithm 2, depending on the investigated plate’s thickness: a) the solutions for thin walls are obtained according to the described algorithm 2; b) the solutions for thick walls’ boundaries are obtained through multiple application of formulas (1.110), (1.111), (1.112). Case b) requires a more thorough study since it consists of the following stages: 1. Calculating y4i according to formulas (1.41), (1.43), (1.45). 2. Successive calculation of yi(m) , y j(m) , yk(m) is made according to formulas (1.110) for all nodes of the wall. 3. Successive calculation of yi(m) , y j(m) , yk(m) is made according to formulas (1.111) for all nodes of the edges. 4. Successive calculation of yi(m) , y j(m) , yk(m) is made according to formulas (1.112) for the corner. The initial values in the nodes are the values calculated in the previous time step. The formulas’ values of the field’s inner nodes do not change during the entire iteration process. Moreover, an assumption is made that the displacement has an index of an axis perpendicular to the investigated wall. 5. Assumption of condition (z) yi − y(z+1) i (z) < ε1it max i yi and return to stage 2 if the condition is not fulfilled. Algorithm 3. Solving quasistatic problems. Combinations of algorithms 1 and 2 presented above produce an algorithm of solutions to a quasistatic problem. In this case, a non-stationary heat conduction equation (1.39) is solved, and the plate’s stress-strain state is additionally determined at every time step, i.e. in system (1.38) inertial terms are assumed and a static problem is solved. Algorithm 2 for solving problems of non-stationary quasistatic of heat conductivity is interrupted at every time step in order to fulfil algorithm 1 for the mechanical part with the temperature gradient, then it is repeated to carry on the calculation. 1.3.5 Reliability The algorithms described in Section 2.4 have been written using FORTRAN software. The programs have been tested on model systems, thus proving their correctness. In case of statics, the following forms have been assumed for the investigated functions:

1.3 Methods of Solving Diﬀerence Equations

55

u1 = (x1 − 0.5)2 + (x2 − 0.5)2 x3 , u2 = (x2 − 0.5)2 + (x3 − 0.5)2 x1 ,

u3 = (x3 − 0.5)2 + (x1 − 0.5)2 x2 , θ = (x1 − 0.5)2 + (x2 − 0.5)2 x33 .

(1.114)

In case of dynamics, the functions have assumed the forms of:

u1 = (x1 − 0.5)2 + (x2 − 0.5)2 x3 τ2 , u2 = (x2 − 0.5)2 + (x3 − 0.5)2 x1 τ2 ,

u3 = (x3 − 0.5)2 + (x1 − 0.5)2 x2 τ2 , θ = (x1 − 0.5)2 + (x2 − 0.5)2 x33 τ. (1.115) There have been combined boundary conditions set upon the walls – free edge and fixing, moving support and free edge, fixing and moving support, temperature distribution and thermal insulation, thermal insulation and convectional heat circulation with surrounding medium, temperature distribution and heat circulation. The approximate solution has been compared to the exact one (reliability of calculations has been confirmed on basis of the results yielded by other authors). Consistency of the results has been achieved exact to 5-6 digits after comma (Tab. 1.6, Tab. 1.7). A comparison has been made in points (1/2, 1/2, 1/2), (7/8, 1/4, 1/2) with the following types of support: one free, thermally insulated wall and the others - fixed with set temperature distribution. The stress-strain state of an isotropic one-dimensional plate subjected to a uniformly distributed load has been investigated in statics. The results of the comparison (see Section 3.1, problem 2) are consistent with the results obtained by Kornishin [345]. As far as non-stationary problems of heat transfer are concerned, an investigation of the temperature field distribution subjected to a heat impact (∂θ/∂x1 = 13.3) onto one of a cubicoid’s walls (with the other walls thermally insulated: ∂θ/∂n = 0) has been conducted. The coordinates ‘temperature in the plate’s centre-time’ of the graph presented in figure 1.5a and the temperature distribution along the plate’s thickness at various time instants presented by the graph in figure 1.5b represent the results consistent with the ones obtained in work [198], in which the temperature field of an infinite plate aﬀected by heat impact has been investigated (the point curve represents Kovalenko’s results). Comparison of numerical and analytical solutions is also a vital confirmation of the reliability of calculations. Fourier’s method has been applied for a non-stationary problem of heat conductivity without the set temperature distribution at the field’s boundary and with a single, uniformly distributed inner heat source, in order to obtain a solution [49] of the following form: τ1 √ ∞ 10 √ −3π2 (τ −τ) 2 2 −14π2 τm 1 2 2 e dτ ≈ e ≈ 0.054 . θ (0.5, 0.5, 0.5, 0.01) = 14π2 m=1 m=1 τ0

The approximate numerical solution obtained in this point of the field equals y4 = 0.0544. The results of the thin plate’s dynamic behaviour obtained in this work have been compared to the results presented in work [261].

56

1 Three–Dimensional Problems of Theory of Plates in Temperature Field a)

µ

x1=1

100 90 80

x1=0.5

70 60

x1=0

x2

50 40

x1 x3

30 20 10 0

0.1

0.2

0.3

0.4

0.5

0.6

µ

b) t=0.6

120 110 100 90 80 70 60

t=0.1 t=0.05

50 40 30 20 10 0

Figure 1.5. The distribution of the temperature field aﬀected by a heat impact: a) in coordinates ‘temperature in the plate’s centre-time’, b) the temperature distribution along the plate’s thickness at various time instants.

1.3 Methods of Solving Diﬀerence Equations

57

An investigation of a plate with mechanical characteristics for aluminium: ν = to a uniformly distributed load P = 1, at initial 0.3, E = 7.2 · 108 g/cm2 , subjected conditions ui τ=τ = 0, ∂ui /∂ττ=τ = 0 has been conducted. 0 0 The following support conditions have been assumed: joint (D11 = E12 = E13 = 1, E31 = D23 = D33 = 0, D12 = E22 = E23 = 1, E51 = D25 = D35 = 0), free edge (D13 = D23 = D33 = 1, E3i = 0, i = 1, ..., 3, Di6 = 1, E6i = 0), null surface forces, and dimensionless parameters: λ1 = 1, λ2 = 50, λ3 = 1/50. The results obtained for a thin plate described according to Kirchhoﬀ-Love’s hypothesis with the use of combined Runge-Kutta’s method and the finite diﬀerence method have become the object of comparison [261]. Table 1.6. Comparison of exact and approximate solutions to static and stationary heat conduction problems.

u1 u2 u3 θ

Approximate solution Exact solution (1/2,1/2,1/2) (7/8,1/4,1/2) (1/2,1/2,1/2) (7/8,1/4,1/2) −0.35 · 10−5 0.1719 0 0.17188 −0.1 · 10−4 0.6249 · 10−1 0 0.625 · 10−1 −0.53 · 10−5 0.3151 · 10−1 0 0.3156 · 10−1 −6 −1 −0.408 · 10 0.4259 · 10 0 0.4297 · 10−1

Error (1/2,1/2,1/2) (7/8,1/4,1/2) 0.35 · 10−5 0.2 · 10−4 −4 0.1 · 10 0.1 · 10−4 −5 0.53 · 10 0.5 · 10−4 −6 0.41 · 10 0.4 · 10−4

Table 1.7. Comparison of exact and approximate solutions to dynamic and stationary heat conduction problems for τ = 0.1.

u1 u2 u3 θ

Approximate solution Exact solution (1/2,1/2,1/2) (7/8,1/4,1/2) (1/2,1/2,1/2) (7/8,1/4,1/2) 0.364 · 10−11 0.48 · 10−6 0 0 0.637 · 10−3 0.313 · 10−3 0 0.3125 · 10−3 −0.909 · 10−3 0.637 · 10−3 0 0.625 · 10−3 −5 −5 0.75 · 10 0.12 · 10 0 0

Error (1/2,1/2,1/2) (7/8,1/4,1/2) 0.36 · 10−11 0.48 · 10−6 0.641 · 10−11 0.5 · 10−6 0.91 · 10−11 0.1 · 10−5 −10 0.75 · 10 0.12 · 10−5

It is worth emphasising how precisely the curves ‘normal stress in the medium time’ overlap for the heat impact problem (which is described in Section 3.5 as Danilovskya’s problem). This phenomenon also confirms correct functioning of the program designed to solve dynamic problems of the theory of thermoelasticity. 1.3.6 Numerical Experiments Certain experiments concerning model problems of statics and dynamics (1.114)– (1.115) have been conducted in order to examine the influence of the compatibility conditions. While calculating the stress-strain state of a three-dimensional elastic plate (a cube) without taking mesh conditions into account, it can be observed that a decrease of the spatial mesh is accompanied by an increase of the calculation

58

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

time. Calculations that include the compatibility conditions in the singular points cause a decrease of the error’s value by 70% compared to exact values. The results presented in Table 1.8 suggest the necessity of taking the compatibility conditions into account. Such a conclusion has also been confirmed by calculations made for stationary heat transfer problems. In case of dynamic problems, considering the compatibility conditions in the calculations brings less satisfactory eﬀects - the error’s value at every time step is about 3%. This can be explained by the fact that the pattern of a diﬀerence scheme with the error of O(h2 ) does not include the singular points (edges, corners). Therefore, they appear to be extremely significant during the calculations of the approximate values within the singularity field; they occur in the calculations of approximate solutions on the walls in the zone adjoining to the boundary. However, the errors overlap with every new integration step, therefore during long-term analyses it is possible to recognise the results as correct without decreasing the spatial net and without taking the compatibility conditions into account.

Table 1.8. Comparison of results obtained with the compatibility conditions at decreasing steps of the net.

u1 u2 u3 Calculation time

Without compatibility 1/8 1/12 1/16 0.36920 0.25860 0.24808 0.33750 0.21093 0.19816 0.19015 0.18925 0.18585 15 min

1h

4h

With compatibility 1/8 0.24608 0.19530 0.18555

Exact solution

17 min

–

0.24609 0.19531 0.18655

Table 1.9. Relation of the number of iterations for plates of various thicknesses, including the compatibility conditions. Plate’s thickness λ1 = λ2 = λ3 = 1 λ1 = 1, λ2 = 1/10 λ1 = 1, λ2 = 1/50

Number of iterations 101 85 70

Error 10−3 10−3 10−3

Eventually, the influence of the compatibility conditions on the results of the analysis of plates of various thicknesses has been examined. Table 1.9 presents the results of numerical calculations, in which the dependence of the approximate solution’s accuracy (set in advance) on the plate’s thickness has been emphasised. The importance of the compatibility conditions decreases along with the decrease of the plate’s thickness.

1.4 Linear Problems in the Theory of Plates in 3D Space

59

1.4 Linear Problems in the Theory of Plates in 3D Space This section presents the examined results of linear problems obtained with the use of the methods described in the previous section. The problems become more complex as the physical models grow more sophisticated. 1.4.1 Static Problems The reaction of an elastic plate subjected to mechanical (surface and volume) excitations for the plate’s various relative thicknesses is investigated. Through reduction of inertial terms and temperature gradients we are going to examine the stress-strain states obtained exclusively on basis of mechanical relations. The system that describes the investigated group of problems is elliptic and it should be solved with the use of the iterative method of upper relaxation. 1. Let us examine a cube-shaped plate, the walls ∂Ω2 , ∂Ω3 , ∂Ω5 , ∂Ω6 of which are fixed (E2i = 1, Di2 = 0, E5i = 1, Di5 = 0, E3i = 1, Di3 = 0, E6i = 1, Di6 = 0, i = 1, ..., 3), and walls ∂Ω1 and ∂Ω4 are free (E1i = 0, Di1 = 1, E4i = 0, Di4 = 1, i = 1, ..., 3). Problem 1. The plate is aﬀected by surface forces f11 = f41 = 0.001, f12 = f13 = f42 = f43 = 0, and volume forces do not occur. Such a choice of mechanical reaction corresponds to a uniform distribution of an external transverse load, perpendicular to plate 0x2 x3 , with x1 = 0.5 (Fig. 1.6a) and x1 = 0.25 (Fig. 1.6b). Problem 2. The plate is aﬀected by volume force P1 = 0.001, P2 = P3 = 0, surface forces do not occur, which corresponds to a uniform inner load (Fig. 1.7.) Qualitative representations of the stress-strain states in both problems for thick plates are diﬀerent. The diﬀerence is that when the volume forces operate, displacement u1 within the entire volume is the same on plane x1 = 0.5 (Fig. 1.7a) and on plane x1 = 0.25 (Fig. 1.7b). When the surface forces operate though, displacement u1 near the active walls (Fig. 1.6b) is larger than in the plate’s centre (Fig. 1.6a). Along with the decrease of the plate’s thickness (the relative thickness is 0.5) one can observe similarity of qualitative representations of displacement distribution for problems 1 and 2 (Fig. 1.8 plane x2 = 0.5, x1 = 0.5). In the further stage of the experiment the plate’s thickness is reduced (Fig. 1.9 - the relative thickness is 0.02). The reaction to the surface load starts to coincide with the reaction to the volume force. Nevertheless, one can observe that the value of displacement u1 increases at equivalent loads along with the reduction of the plate’s size (Fig. 1.10 presents the relative thickness for corresponding curves) and it is identical along its thickness. In all the conducted experiments, displacements u2 and u3 are insignificantly small compared to displacement u1 . Solutions to problems 1 and 2 for plates of the relative thicknesses 0.1 and 0.02 have been compared to the results obtained by means of the method of mesh for a

60

1 Three–Dimensional Problems of Theory of Plates in Temperature Field b) x2

a) 1

u ·10

x1

-4

x3

4 3 2 1

1

x2

5 4 3 2 1

x3

u ·10

-4

x2

2

u ·10

x3

-4

4 3 2 1

x2

x3

3

u ·10

x2

2

u ·10

x2

-4

3

x3

x2

4 3 2 1

x3

u ·10

4 3 2 1

-4

-4

4 3 2 1

x3

Figure 1.6. The representation of the stress-strain state for problem 1 in case of: a) x1 = 0.5, b) x1 = 0.25.

plate described according to Kirchhoﬀ-Love’s displacement hypothesis [261] (Fig. 1.11), as well as with Kornishin’s results [345] obtained according to a diﬀerence scheme of an increased order of accuracy (see Tab. 1.10). The relative error for a 0.1-thick plate is between 5% in the centre and 10% at the plate’s edges. For a 0.02-thick plate it is between 3% and 11%, respectively. Diﬀerent relative errors along the middle plate and the increase of the errors as the thickness is reduced can be explained by means of Kirchhoﬀ-Love’s method,

1.4 Linear Problems in the Theory of Plates in 3D Space x2

a)

b)

x1 x3 1

8 7 6 5 4 3 2 1

u ·10

1

u ·10

-4

-4

8 7 6 5 4 3 2 1

x2

x2

x3

u2·10-4

2

x2

x3

3

u ·10

x3

u ·10

4 3 2 1

-4

x3

3

u ·10

-4

x3

4 3 2 1

x2

3 2 1

x2

61

x2

-4

4 3 2 1

x3

Figure 1.7. The representation of the stress-strain state for problem 2 in case of: a) x1 = 0.5, b) x1 = 0.25.

usually applied for analyses of thin plates. The problems regarding relative thickness larger than 0.125 cannot be solved with the use of this method. A following conclusion may be drawn from the already made investigation. When a plate’s thickness is reduced, the reaction to a uniformly distributed surface force coincides with the reaction to the volume force (λ2 = 1/8). It is also worth noticing that a 10−6 volume load (e.g. for aluminium alloys) leads to a situation in which the displacement is also slight (Fig. 1.12) and is equal

62

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

u ·10

-4

2

u ·10

16 14 12 10 8 6 4 2

-4

8 6 4 2

x3

x3

x1

x1 3

u ·10

1

u ·10

x3

-4

-4

x2

10 8 6 4 2

x1 x1

18 16 14 12 10 8 6 4 2

x3 2

u ·10

-4

10 8 6 4 2

x2

x2

x3

x3 u3·10-4 8 6 4 2

x2

x3

Figure 1.8. The representation of the displacement distribution for problems 1 and 2 on planes x2 = 0.5 and x1 = 0.5 (the relative thickness is 0.5).

to 10−7 ÷ 10−10 . It means that at small loads the volume force may be neglected, which has actually been carried out in work [595].

1.4 Linear Problems in the Theory of Plates in 3D Space

1

u ·10

2

u ·10

-2

63

-2

4 3 2 1

5 4 3 2 1

x3

x3

x1

x1

3

u ·10

-2

4 3 2 1

x3

x2

x1

x1 x3

1

6 5 4 3 2 1

u ·10

2

u ·10

-2

-2

4 3 2 1

x2

x2

x3

x3 3

u ·10

x2

-2

4 3 2 1

x3

Figure 1.9. The representation of the displacement distribution for problems 1 and 2 on planes x2 = 0.5 and x1 = 0.5 (the relative thickness is 0.02).

64

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

1

u ·10

-3

18 16 14 12 10 8 6 4 2

2

u ·10

6 4 2

x3

x3

3

u ·10

-3

x1

x1

-3

8 6 4 2

x2

x3

x1

x1

x3

2

u ·10 1

u ·10

-3

6 4 2

18 16 14 12 10 8 6 4 2

x3

-3

x3

x1

3

u ·10 x1

x3

-3

8 6 4 2

xi

Figure 1.10. The quantity of the relative thickness for corresponding curves at the transverse loading.

1.4 Linear Problems in the Theory of Plates in 3D Space

3

65

x2

-2

u ·10 5 4 3 2 1

x1

x3

x1 x3 1

3

u ·10

5 4 3 2 1

4 3 2 1

x3

-4

u ·10

-4

x1

x3

x1

Figure 1.11. Solution to problems 1 and 2 for plates of the relative thicknesses 0.1 and 0.02.

Table 1.10. Comparison of the results obtained by diﬀerent authors for a uniformly loaded plate (the value of displacement u3 in the plate’s centre is presented). Plate’s thickness Kirchhoﬀ-Love’s [242] hypothesis [261] The three-dimensional theory Relative [242] error [261]

λ2 = 1/50 0.063 0.077 0.068 8% 11%

λ2 = 1/10 0.089 0.085 0.095 5.5% 10.5%

λ1 = λ 2 = λ 3 = 1 – – 0.021 – –

66

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

u ·10

-8

36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

x1

x3

2

u ·10

-8

8 6 4 2

x3

x3

x1 1

u ·10

x1 3

u ·10

-3

8 6 4 2

-8

28 26 24 22 20 18 16 14 12 10 8 6 4 2

x1

x2

x3

x1

2

u ·10 8 6 4 2

x3

x1

x2

Figure 1.12. The quantity of displacements for a volumetric loading of 10−6 .

2. To illustrate the possibility of a three-dimensional modelling we are going to investigate solutions within the two-dimensional theories. Problem 3. Figure 1.13 presents displacements on plane x2 = 0.25 that have been obtained as a result of the operation of diﬀerent surface forces f11 = 0.01 and f41 = 0.001 (problem 1).

1.4 Linear Problems in the Theory of Plates in 3D Space a)

1

u ·10

b)

-2

10 9 8 7 6 5 4 3 2 1

2

u ·10

67

-4

6 4 2

x3

x1

3

u ·10

x3

-4

6 5 4 3 2 1

x1

2

u ·10

-3

4 3 2 1

x3

x3

x1

x1 x2 3

u ·10

-3

6 5 4 3 2 1

x3

x1 x3

3

u ·10

-4

18 16 14 12 10 8 6 4 2

x1

x3

x1

Figure 1.13. The representation of displacements caused during the operation of surface forces of various values ( f11 = 0.01 and f41 = 0.001): a) an increase of displacements at a decrease of the plate’s thickness; b and c) the asymmetrical distribution of displacements u2 and u3 in respect to each other.

In this case, similarly to the previous point, reducing the plate’s thickness causes a tendency to increase displacements (Fig. 1.13a). Due to the significant influence of the diﬀerent values of the surface forces that result from reducing the plate’s thickness, the displacements u2 and u3 are more asymmetrical to each other (Fig. 1.13b and 1.13c).

68

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Problem 4. Let us analyse a cube-shaped plate with fixed walls ∂Ω3 , ∂Ω6 (E3i = 1, E6i = 1, Di3 = 0, Di6 = 0, i = 1, ..., 3), and free walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 (E1i = E4i = E2i = E5i = 0, Di1 = Di4 == Di2 = Di5 = 1, i = 1, ..., 3), which is subjected to volume force P3 = 0.01 (the surface forces do not occur). Figure 1.14 presents displacement distribution on plane x3 = 0.5 for: a) fixed edges, b) free edges. Fixing of the edges decreases the displacements’ values within the entire investigated plane. It can be observed particularly near the edge and it corresponds to a physical interpretation of the phenomenon.

a)

b) 1

u ·10

1

u ·10

-3

4 3 2 1

3 2 1

x2

x2

x1

2

u ·10

x2

3

z3

-3 2

u ·10

4 3 2 1

-3

x2 x3

x3

x1

6 5 4 3 2 1

x1

-3

4 3 2 1

x2

x1

u ·10

x2

-3

3

u ·10

x2

-3

6 5 4 3 2 1

x3

Figure 1.14. The displacement distribution (planes x3 = 0.5 and x1 = 0.25) in case of: a) fixed edges, b) free edges.

1.4 Linear Problems in the Theory of Plates in 3D Space

69

Problem 5. Let us investigate a cube-shaped plate with free walls ∂Ω4 , ∂Ω5 , ∂Ω6 (E4i = E5i = E6i = 0, Di4 = Di5 = Di6 = 1, i = 1, ..., 3) and fixed walls ∂Ω1 , ∂Ω2 , ∂Ω3 (E1i = E3i = E2i = 1, Di1 = Di3 = Di2 = 0, i = 1, ..., 3), which is subjected to surface force f41 = 0.001, f42 = f43 = 0 (the volume force is not taken into account). Figure 1.15 presents displacement distribution on plane x3 = 7/8 for: a) fixed edges, b) free edges. Figure 1.16a illustrates displacement distribution in case of a)

1

u ·10

14 12 10 8 6 4 2

-4

x2 2

u ·10

x2

x1

3

u ·10

8 6 4 2

x1

x3

-4

x2

x1

-4

6 4 2

x2

x1

1

b)

16 14 12 10 8 6 4

u ·10

2

u ·10

x2

x1

3

8 6 4 2

x2

-4

u ·10

-4

8 6 4 2

x2

x1

-4

x1

Figure 1.15. The displacement distribution (plane x3 = 7/8) in case of: a) fixed edges, b) free edges.

70

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

fixed corner (0, 0, 0) (which means that ui (0, 0, 0) = 0), whereas Figure 1.16b illustrates an increased surface of fixing. The surface of fixing and the way the edge is fixed significantly influence the plate’s stress-strain state.

a) 1

u ·10

10 8 6 4 2

2

u ·10

-4

-4

8 6 4 2

x2 x2

x1

x1

3

u ·10

-4

x2

8 6 4 2

x2

b)

x1

1

u ·10

x3

-4

10 8 6 4 2

3

u ·10

x2

x1

x2 2

u ·10

x2

x1

-4

10 8 6 4 2

x1

-4

8 6 4 2

x1

Figure 1.16. The displacement distribution for problem 5 in case of: a) fixed corner (0, 0, 0), b) an increased fixing surface.

1.4 Linear Problems in the Theory of Plates in 3D Space

71

Problem 6. Thin plate λ1 = 1/50, λ2 = 1, λ3 = 50 with free walls ∂Ω1 , ∂Ω4 (E1i = E4i = 0, Di1 = Di4 = 1, i = 1, ..., 3) and fixed walls ∂Ω2 , ∂Ω3 , ∂O5 , ∂O6 (E2i = E3i = E5i = E6i = 1, Di2 =Di3 =Di5 = Di6 = 0, i = 1, ..., 3) are aﬀected by shearing forces f12 = −0.001, f42 = 0.001, f11 = f13 = f41 = f43 = 0 and volume forces, while other surface forces are equal to zero. The displacement distribution presented in Figure 1.17a on plane x3 = 0.5 and in Figure 1.17b on plane x1 = 0.5 is considerably diﬀerent from the displacement

a)

b)

u1·10-4

1

u ·10

4

-4

1

3 2

u2

1

u2

u1

2

-5

u ·10

u1

-1

2

-2

1

-3

x2

-4

-1

x1

-5 2

u ·10

-5 3

u ·10

2 1

x2 -1

-5

1

x1

x2

x1

u3·10-5

x2

1

x2

x1

x3

x1

Figure 1.17. The displacement distribution for problem 6 on plane: a) x3 = 0.5, b) x1 = 0.5.

72

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

distribution obtained at the transverse load (Fig. 1.10). The reaction to the shearing force is much weaker than the reaction to the transverse load of the same value. 3. Our further considerations are going to focus on a heated fixing-ring system, which can be static or dynamic. In this case we are going to present solutions to a number of static problems concerning the ring with fixing for which the following modelling conditions are set on the surface: ui (1, x2 , 7/8) = 0, ui (0, x2 , 7/8) = 0, ui (x1 , 1, 7/8) = 0, ui (x1 , 0, 7/8) = 0, i = 1, ..., 3 – will be called a ring with fixing in the first row of points, ui (1, x2 , 3/4) = 0, ui (0, x2 , 3/4) = 0, ui (x1 , 1, 3/4) = 0, ui (x1 , 0, 3/4) = 0, i = 1, ..., 3 – will be called a ring with fixing in the second row of points, ui (1, x2 , 5/8) = 0, ui (0, x2 , 5/8) = 0, ui (x1 , 1, 5/8) = 0, ui (x1 , 0, 5/8) = 0, i = 1, ..., 3 – will be called a ring with fixing in the third row of points.

Problem 7. We investigate a cubic plate with fixed walls ∂Ω3 , ∂Ω6 (E3i = E6i = 1, Di3 = Di6 = 0, i = 1, ..., 3) and free walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 (E ij = 0, Dij = 1, i = 1, ..., 3, j = 1, 2, 4, 5) subjected to volume force P3 = 0.01, while the surface forces are neglected. A ring is placed on the surface of the cube. The distribution of displacement is presented on planes a) x3 = 0.25 and b) x2 = 0.5 for the ring in the first row of points (Fig. 1.18), in the second row of points (Fig. 1.19) and in the third row of points (Fig. 1.20). The influence of the fixingring system’s reaction is clearly visible on plane x2 = 0.5. The stress-strain graphs are recognisably diﬀerent, especially for displacements u2 and u3 . The displacement distribution of the plate with the ring coincides with the displacement distribution of the plate without the ring on plate x3 = 0.25 (Fig. 1.14b). However, the values of displacements u1 , u2 , u3 tends to decrease as the ring changes its position along the axis towards plane x1 = 0.25, which also corresponds to the physics of the phenomenon.

1.4 Linear Problems in the Theory of Plates in 3D Space

a)

1

u ·10

73

-3 2

u ·10

4 3 2 1

x2

x1

3

u ·10

-3

4 3 2 1

x2

x1

-3

6 5 4 3 2

x2

1

x3

x2

x1

x1

b) 1

4 3 2 1

u ·10

2

u ·10

-3

-3

4 3 2 1

x3

x1

3

u ·10

x3

x1

-3

5 4 3 2 1

x3

x1

Figure 1.18. The displacement distribution for the ring in the first row of points on plane: a) x3 = 0.25, b) x2 = 0.5.

74

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

a)

1

3 2 1

u ·10

-3

2

u ·10

x2

x1 x2 3

u ·10

4 3 2 1

x1

-3

5 4 3 2 1

x2

x2

x1 x3

b)

-3

4 3 2 1

1

u ·10

x1

-3

2

u ·10

x3

4 3 2 1

x1 X3 3

u ·10

-3

x1

-3

6 5 4 3 2 1

x3

x1

Figure 1.19. The displacement distribution for the ring in the second row of points on plane: a) x3 = 0.25, b) x2 = 0.5.

1.4 Linear Problems in the Theory of Plates in 3D Space

a)

75

u ·10 1

-3

3 2 1

u ·10 2

x2

-3

x1

3 2 1

x2

x1

u ·10 3

-3

6 5 4 3 2 1

x2

x2

x1 x3

x1

b) u ·10 1

4 3 2 1

-3

u ·10 2

x2

-3

3 2 1

x1 x2

x1

u ·10 3

-3

5 4 3 2 1

x2

x1

Figure 1.20. The displacement distribution for the ring in the third row of points on plane: a) x3 = 0.25, b) x2 = 0.5.

76

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Problem 8. Figures 1.21 and 1.22 illustrate the solution of the previously formulated problem 7. This time with surface force f41 = 0.01 taken into account and the volume forces neglected.

a)

1

u ·10

-3

6 5 4 3 2 2

u ·10 x2

-3

3 2 1

x3 x2 3

x3

-3

u ·10 6

5 4 3 2 1

x2

x2

x3 x3

b)

1

u ·10

x1

-3

2

u ·10

6 5 4 3 2

-3

3 2 1

x2

x2

x3

x3

3

u ·10

-3

4 3 2 1

x2

x3

Figure 1.21. The displacement distribution taking the surface force into account on plane x1 = 0: a) without a ring, b) with a ring in the third row of points.

1.4 Linear Problems in the Theory of Plates in 3D Space a)

1

u ·10

77

-3

7 6 5 4 3

2

x ·10

-3

3 2 1

x3 x3

x1

x1

3

u ·10

-3

x2

5 4 3 2 1

x3

x3

x1

x1

b)

1

u ·10

-3

2

u ·10

6 5 4 3 2 1

-3

4 3 2 1

x3

x3

x1

x1

3

u ·10

-3

4 3 2 1

x3

x1

Figure 1.22. The displacement distribution taking the surface force into account on plane x2 = 0.5: a) without a ring, b) with a ring in the third row of points.

78

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Figure 1.21 presents a displacement graph on plane x1 = 0 and Figure 1.22 – on plane x2 = 0.5: a) without a ring, b) with a ring in the third row of points. The ring in case b) aﬀects the character of the stress-strain state in all the investigated planes, in which the values of displacements change in the ring’s proximity. It is clearly visible in the place where the ring is located (Fig. 1.21b). Problem 9. Figure 1.23 shows plane x2 = 7/8 with the graphs of displacement distribution of the plate with a ring (problem 8) aﬀected by a pair of surface forces f11 = 0.001 and f41 = 0.001. The solutions to problems 8 and 9 are used in further investigations to compare the results.

1

u ·10

-4

9 8 7 6 5 4 3 2

x2

x1

2

u ·10

-4

4 3 2 1

x2

x1

x2 3

u ·10

x1

x3

x2

-4

5 4 3 2 1

x1

Figure 1.23. The displacement graphs for a plate with a ring aﬀected by a pair of surface forces.

1.4 Linear Problems in the Theory of Plates in 3D Space

79

Comparison of the stress-strain state of a cube with a ring aﬀected by surface and volume forces may lead to a conclusion that the ring’s presence significantly influences the values of displacements. Moreover, for less accurate calculations, the surface force may be equivalently replaced with the volume force, and vice versa (the relative calculation error decreases along with reducing the plate’s thickness). Three-dimensional problems make it possible to investigate stress-strain states for such types of fixings and external forces that cannot be precisely modelled by means of the two-dimensional approach and the use of the two-dimensional theory may result in major errors in consequence. 1.4.2 Dynamic problems The process of solving dynamic problems allows investigating various kinds of changes that occur in time in the examined object. The following section presents the results of a number of problems regarding the influence of mechanical loads on plates in its dynamic aspect (the hyperbolic system of equations (1.23) does not include only the temperature gradient). Runge-Kutta’s method is applied to find the solution. A plate of variable relative thickness subjected to surface and volume forces is the object of the following analysis. Problem 10. The object of investigation is a plate with free edges ∂Ω3 , ∂Ω6 (E3i = E6i = 0, Di3 = Di6 = 1, i = 1, ..., 3) and jointedly supported walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 (Ei1 = 0, D1i = 1, Ei2 = Ei3 = 1, D2i = D3i = 0, i = 1, 2, 4, 5) aﬀected by volume force P3 = 0.01 that causes free vibration. The initial conditions (1.33) are assumed as zero. Figure 1.24 illustrates movement of points (0.5, 0.5, 0.5) with respect to axis x3 , that is displacement u3 . For the sake of experiment, the plate’s thickness has been reduced. This has led to an increase of the vibration’s frequency and amplitude. Changing the dimensionless coeﬃcient at the inertial terms also causes changes of the plate’s vibrations’ amplitude and frequency. As the inertia coeﬃcient ℵ decreases, the vibrations’ amplitude and frequency increase. The broken line in Figure 1.24 marks the solution of the analysed problem (the relative thickness is 0.1) with coeﬃcient ℵ = 0.01 and the full line marks the solution with coeﬃcient ℵ = 1. The explanation for it may be a decrease of the characteristic mechanical time τ M = l3 /C, which is used to express coeﬃcient ℵ = τ2M , c = ((λ + 2µ)/ρ)0.5 .

80

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 3

u (0.5;0.5;0.5) 0.2 x2 =0.01

0.015 x3

0.02

x1

0.1

0.05

0

4

8

12

16

20

32

0

2

4

-0.05

-0.1

Figure 1.24. The change of displacement u3 of a plate with free edges and jointedly supported walls.

Problem 11. The object of investigation is a cube-shaped plate with a fixing-ring system in the second row of points, aﬀected by a pair of surface forces f11 = 0.001 and f41 = 0.001. Walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 are free (E1i = E2i = E4i = E5i = 0, Di1 = Di2 = Di4 = Di5 = 1, i = 1, ..., 3), and walls ∂Ω3 , ∂Ω6 are fixed (E3i = E6i = 1, Di3 = Di6 = 0, i = 1, ..., 3). Null boundary conditions are assumed for calculations. The problem has been solved using a static formulation (problem 9). On plane x2 = 7/8 in Figures 1.25 and 1.26 one can observe changes that occur in the plate’s stress-strain state at diﬀerent time instants: Fig. 1.25a τ = 0.005, Fig. 1.25b τ = 0.01, Fig. 1.26a τ = 0.0125 and Fig. 1.26b τ = 0.0175. At the initial stage of the analysed process the changes of the ring’s behaviour become clearly visible. Besides, in course of time the stress-strain state approaches the static solution (Fig. 1.26 and Fig. 1.23). The investigated time intervals are small (dτ = 0.1 ÷ 0.5).

1.4 Linear Problems in the Theory of Plates in 3D Space

a)

1

12 10 8 6 4 2

u ·10

81

-5

2

u ·10

-5

4 3 2 1

x3

x1 3

u ·10

x3

x1

-5

8 6 4 2

x3

x1

b) 1

u ·10

-5

10 8 6 4 2

2

u ·10

x1

x3

3

u ·10

x3

4 3 2 1

x1

-5

x2

6 4 2

x3

-5

x1

x1

Figure 1.25. The displacement distribution of a cube-shaped plate with a fixing-ring system in the second row of points at various time instants: a) τ = 0.005, b) τ = 0.01.

82

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

a)

1

u ·10

-5

12 10 8 6

2

u ·10

-5

6 4 2

x3

x1

x3 3

u ·10

x1

-5

3 2 1

x3

x2

x1 x3

b)

1

u ·10

x1

-5

10 8 6 4 2

2

u ·10

-5

6 4 2

x3

x1

x3 3

u ·10

x1

-5

8 6 4 2

x3

x1

Figure 1.26. The displacement distribution of a cube-shaped plate with a fixing-ring system in the second row of points at various time instants: a) τ = 0.0125, b) τ = 0.0175.

1.4.3 Non-stationary temperature field One of the most important practical subjects of investigation is the influence of the occurrence of heat sources and heat fluxes inside an elastic body. Both continuous and discrete systems can become heat sources. This section analyses the temperature distribution modelled by general heat conductivity equation (1.24) (the term that corresponds to dilatation is assumed as equal to zero). A parabolic equation is solved by means of Runge-Kutta’s method. A three-dimensional body’s temperature field with and without a heat source is the object of the following investigation.

1.4 Linear Problems in the Theory of Plates in 3D Space

83

Problem 12. The temperature distribution in a cubicoid-shaped field is the object of research. The initial-boundary conditions are following: thermal insulation (Ci = 1, Ai = 0, Bi = 0, i = 2, 3) on the entire surface of the plate except for the heat impact (B1 = 1, T 1 = 133); in the centre of wall ∂Ω1 (x1 = 1, 0 ≤ x2 ≤ 1, 0.5 ≤ x3 ≤ 1, θ|τ0 = 0).

a)

µ·10 ·10

x2

3 2 1

x1

x3

x1

x3

x1

x3

µ·10 ·10 3 2 1

x2

x1

µ·10 ·10

x3

x1

µ·10 ·10

4 3 2 1

x2

x1

x1

4 3 2 1

3 2 1

x2

Figure 1.27. The temperature distribution in a cubicoid (problem 12) at time instant: a) τ = 0.002, b) τ = 0.04, c) τ = 0.07.

84

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

b)

µ·10 ·10 3 2 1

x2 x1

x3

x3

x1

µ·10 ·10 4

x1

x3

3 2 1

x1

x3

µ·10 ·10 3 2 1

x2

x1

3 2 1

µ·10 ·10 5 4 3 2 1

x1

µ·10 ·10

x2

x1

x2

Figure 1.27. cont.

Figure 1.27a illustrates the temperature distribution at τ = 0.002, Fig. 1.27b at τ = 0.04, and Fig. 1.27c at τ = 0.07. The heat transfer is easily observable from the side of the active wall and such is the temperature increase within the entire plate in course of time.

1.4 Linear Problems in the Theory of Plates in 3D Space

85

µ·10

c)

4 3 2 1

x2

X1

X3

x3

x1

X1

X3

µ·10 7 6 5 4 3 2 1

X1

X3

X2

X1

µ·10 6 5 4

µ·10

2 1

X1

X2

3 2 1

X1

X2

Figure 1.27. cont.

86

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Problem 13. Let us assume that the plate’s centre (0.5, 0.5, 0.5) contains a heat source of power P4 = 300 dimensionless units, which corresponds to the power of 150 cal/(cm3 · s) = 636 W/cm3 .

µ·10 ·10 µ·10 ·10

6 5 4 3 2 1

x2 5 4 3 2 1

x3

x1

x3 x1

x1

x2

µ·10 ·10 µ·10 ·10

7 6 5 4 3 2 1

x3

x2

7 6 5 4 3 2

x1

x3

x1 x1

x2

µ·10 ·10 7

x2

6 5 4 3 2 1

x3

µ·10 ·10 8 7 6 5 4 3

x1

x1

x3

1

x1

x2

Figure 1.28. The temperature distribution in a cubicoid (problem 13) at time instant: a) τ = 0.002, b) τ = 0.04, c) τ = 0.07.

The previous problem’s heat conductivity equation (1.24) did not take the investigated field’s inner heat sources into account. Figure 1.28a–c illustrates the temperature distribution at the same time instants as in problem 12 (on planes

1.4 Linear Problems in the Theory of Plates in 3D Space

87

x2 = x3 = 0.5). The influence of heat sources at initial time instants is characterised by an increase of temperature in the place where the heat source is located (τ = 0.002). Then, the source’s reaction starts to exceed the external heat impact’s reaction (τ = 0.1). The highest temperature can be observed in the proximity of the active wall. Solving the non-stationary three-dimensional equation of heat conductivity demonstrates the temperature field’s non-linearity along the plate’s thickness. 1.4.4 Comparison of Solutions – non-isothermal Processes in Static Problems The results of the investigations of the plate’s stress-strain state have so far been obtained without taking temperature into account. The same temperature has been assumed in every point of the field, and even the occurrence of deformations has never changed it. In reality, the plate’s deformations cause changes of temperature and the temperature’s change causes deformations of the plate due to the material’s thermal expansion. Introducing heat loading into the processes analysed in Sections 3.1 and 3.2 will additionally expand and complicate them. The mathematical model that describes those processes is diﬀerent from the previously discussed models because it requires solving the system of equations (1.23) (with or without the inertial terms) and the equation of heat conductivity (1.24) (stationary or non-stationary) simultaneously. The following thermal conditions have been assumed for all of the problems presented in this section: temperature distribution (Bi = 0, Ai = 1, Ci = 0) on the 0 = 0 and constant temperature distribution within the plate’s entire surface at T i+m 0 ring, T i+m = 1. Problem 14. The object of analysis is a cube-shaped plate with fixed walls ∂Ω3 , ∂Ω6 , edges (E3i = E6i = 1, Di3 = Di6 = 0, i = 1, ..., 3) and free walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 (E1i = E2i = E4i = E5i = 0, Di1 = Di2 = Di4 = Di5 = 1, i = 1, ..., 3). In such a case a stationary problem is solved, which means that in system (1.23), (1.24) time-dependent inertial terms and temperature derivatives are neglected. Next, the problems of stationary heat conductivity and static elasticity are successively solved. The iterative upper relaxation method is used to solve the problem. Figure 1.29 illustrates the influence of thermal excitations on plane x2 = 7/8 (Fig. 1.29a), the influence of displacement u1 on plane x2 = 0.5 (Fig. 1.29b–c), the influence of displacement u2 (Fig. 1.29d) and the influence of displacement u3 (Fig. 1.29e). The volume and surface forces are neglected. The fixing-thermal ring system is placed in the first row (see sections 3.1, 3.3 of this chapter). Let us pay attention to the fact that the scale of the graphs illustrating displacement u3 is two times smaller than that of the graphs showing displacements u1 and u2 . The changes of displacement u3 are most visible because the ring is placed perpendicularly to axis x3 . Besides, plane x2 = 7/8, which is parallel to axis x3 , is investigated and it is the plane and along which displacement u3 occurs. Due to the fact that wall ∂Ω3 and the

88

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

-1 µ·10 ·10

8 7 6 5 4 3 2 1

1

u ·10-3

x3

x1

x3

x3

x3

4 3 2 1 -1 x 1 -2 -3 -4 -5

1

u ·10-3

5 4 3 2 1

x2 -1

-2 -3

x2

x2

x1

x3

x1

x3

x1

2

u ·10-3 4 3 2 1

x3 -1

-2 -3 -4 -5

x1

3

u ·10-3

-6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48

Figure 1.29. The analysis of a cube-shaped plate (problem 14): a) the influence of thermal excitations on plane x2 = 7/8, b–c) the influence of displacement u1 on plane x2 = 0.5, d) the influence of displacement u2 , e) the influence of displacement u3 .

1.4 Linear Problems in the Theory of Plates in 3D Space

89

ring are rigidly fixed in the first row of the points in the plane’s proximity, displacement u3 that disappears during stresses may freely spread out only in the opposite direction to axis x3 (which Fig. 1.29d illustrates). The influence of the fixing-ring system located on the ring’s plane (x1 , x2 , 7/8) makes displacements occur only inside the plate. The temperature’s influence is symmetrical to plane x1 = 0.5, and as a result, the plate’s reaction to plane x1 = 0.5 is also symmetrical (Fig. 1.29e and 1.29c). The stress-strain state changes and if the influence of mechanical forces is taken into account next to the temperature, then the displacement distribution starts to become asymmetrical.

Problem 15. The plate with parameters described in problem 14 is furthermore aﬀected by surface forces f11 = f41 = 0.001, f12 = f13 = f41 = f43 = 0 and volume forces P3 = 4 · 10−7 (Fig. 1.30). Symmetry of the distribution of displacements u1 and u2 disappears because the surface forces aﬀect the parallel walls ∂Ω1 and ∂Ω4 in one direction along axis x1 . Thus mechanical loads strengthen the temperature-caused deformations along axis x1 and lessen them along axis x2 , which is particularly visible in case of displacement u1 . The value of the volume force is low – about 10−5 .

Problem 16. The plate investigated in problem 15 is encircled with a ring in the second row of points (Fig. 1.31). In this case, the first row of points becomes free and displacement distribution changes. While being heated, the body expands in various directions starting from the fixing-ring. Displacement u3 reveals the most relevant changes since its positive values appear between the ring and the rigid fixing of wall ∂Ω3 . Problem 17. In this very interesting case the thermal ring and the fixing-ring are located next to each other along axis x3 . The thermal ring is in the first row and the fixing-ring is in the second row. Figures 1.32a (without surface forces) and 1.32b (with surface forces f11 = f41 = 0.001, f12 = f13 = f41 = f43 = 0) show plane x2 = 7/8 on which the fixing-ring ”dashes” the expansion incited by the thermal impact. It is worth noticing that the graphic representations of all displacement distributions in this problem are in identical scale.

90

1 Three–Dimensional Problems of Theory of Plates in Temperature Field -1 µ·10 ·10

1

u ·10-3

9 8 7 6 5 4 3 2 1

x3

-1 x 1 -2 -3 -4 -5

x1

x3

4 3 2 1

2

x2

x3

5 4 3 2 1

x2

x1

x3

x1

x3 -1

-2 -3 -4 -5

x3

u ·10-3

x1

x1 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

Figure 1.30. The analysis of a cube-shaped plate as in problem 14, additionally considering surface and volume forces.

1.4 Linear Problems in the Theory of Plates in 3D Space -1 µ·10 ·10

91

1

u ·10-3

10 9 8 7 6 5 4 3 2 1

4 3 2

x3

x3

-1 x1 -2 -3 -4 -5

x1 3

u ·10-3

x2

x2

x1

x3

26 24 22 20 18 16 14 12 10 8 6 4 2

x3

x1

2

u ·10-3 6 5 4 3 2 1

x3

x3

x1

x1 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -24 -26 -28 -40 -42 -44 -46 -48 -50

Figure 1.31. The analysis of a cube-shaped plate as in problem 15 with an additional ring in the second row of points.

92

1 Three–Dimensional Problems of Theory of Plates in Temperature Field -1 µ·10 ·10

u1·10-2 9 6 5 4 3 2 1

9 8 7 6 5 4 3 2 1

x3

x3

x1

x1

1

8 u ·10 7 6 5 4 3 2 1

-2

x2

x3

x2

x2

x1

x3

x1

x3

3

u ·10-2

x3

4 3 2 1

x1

7 6 5 4 3 2 1

x3

u2·10-2

x1

3

u ·10-2 4 3 2 1

x2

x1

Figure 1.32. The distribution of the changes of temperature and displacements in case when the thermal ring and the fixing-ring are place one after another along axis x3 : a) without the surface forces, b) with the surface forces.

1.4 Linear Problems in the Theory of Plates in 3D Space -1 µ·10 ·10

93

1

u ·10-2

9 8 7 6 5 4 3 2

-1 -2 -3 -4 -5 -6 -7 -8 -9

x3

x3

x1

x2

6 5 4 3 2 1

x2

x1

3

x3

x1

x3

u ·10-2

x1

5 4 3 2 1

u2·10-2 6 5 4 3 2 1

x3

x1

x2

3

u ·10-2 x1

3 2 1

x3

Figure 1.32. cont.

x1

94

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Problem 18. In this problem the boundary temperature conditions are slightly altered and plate (∂θ/∂n = 0) is thermally insulated by heating it with the fixingthermal ring in the first row of points.

-1

·10 9 8 7 6 5 4 3 2 1

x1

x3

-u ·10-2 1

4 3 2 1

x3

x2

-1 x1 -2 -3 -4 -5

x2

-u ·10-2 3

x3

-u ·10-2

x1

x1

x3

3 2 1

2

4 3 2 1

x3 -1

x3

x1

x1

-2 -3 -4

Figure 1.33. The displacement fields’ distribution in a thermally insulated plate and heated by a fixing-thermal ring in the first row of points.

Figure 1.33 illustrates non-uniform and asymmetrical displacement fields’ distributions at symmetrical temperature distribution inside the plate, which are caused by surface forces ( f11 = f41 = 0.001, f12 = f13 = f42 = f43 = 0) operating in one direction.

1.4 Linear Problems in the Theory of Plates in 3D Space

95

Problem 19. A plate with the same type of fixing as previously, described according to a quasistatic problem is heated by the ring in the first row of points (the boundary conditions of problem 15). Null initial conditions are assumed.

1

-1 µ·10 ·10

8 7 6 5 4 3 2 1

u ·10-3

5 4 3 2 1

x3

-1 x1 -2 -3 -4 -5 -6

x1

x3

u1·10-3

x2

4 3 2 1

x3 x3 -1 -2 -3

6 5 4 3 2 1 -1 x3 -2 -3 -4 -5 -5

x2

x1

x1

x3

x1 3

u ·10-3 6 4 2 2

u ·10-3

x3

x1

-2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

x1

Figure 1.34. A plate with the same type of fixing as in problem 15 for null initial conditions at: a) τ = 0.005, b) τ = 0.0175, c) τ = 0.03625, d) τ = 0.1125.

96

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

-1 µ·10 ·10

u ·10-3

9 8 7 6 5 4 3 2 1

x1

x2

u3·10-3

x2

x1

x3

x3

8 6 4 2

x1

x3

2

u ·10-3 5 4 3 2 1

x3

-1 x 1 -2 -3 -4 -5 -6 -7

x3

x3

5 4 3 2 1

x1

Figure 1.34. cont.

x1 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44

1.4 Linear Problems in the Theory of Plates in 3D Space -1 µ·10 ·10

97

1

u ·10-3

9 8 7 6 5 4 3 2 1

3 2 1

x2

x2

-1x 1 -2 -3 -4

x1

3

x2

x1

x3

u ·10-3

x2

x3

8 6 4 2

x1

x3

2

6 5 4 3 2 1

x3

u ·10-3

x1

Figure 1.34. cont.

x1 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46

98

1 Three–Dimensional Problems of Theory of Plates in Temperature Field -1 µ·10 ·10

1

u ·10-3

9 8 7 6 5 4 3 2 1

x3

x2

x2

x1

x3 2

x3

x1

x1

x3

6 5 4 3 2 1

4 3 2 1

u ·10

x3

3

u ·10-3

8 6 4 2

x1 x3

-3

x1

-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

x1

Figure 1.34. cont.

In such a case ∂2 ui /∂τ2 = 0 is assumed in the system of equations (1.23)–(1.33), and the value of dilatation in the heat conductivity equation is assumed to be zero. Two methods are combined during calculations: the heat conductivity equation is solved with the use of Runge-Kutta’s method, and the upper relaxation method is applied to construct a graph illustrating the stress-strain state at every time step. Figure 1.34a–d presents the distribution of the temperature and displacements changes in time.

1.4 Linear Problems in the Theory of Plates in 3D Space

99

Problem 20. Problem 16 is investigated again, however its formulation has been changed. Figure 1.35a–d presents the plate’s stress-strain state distribution in time.

-1 µ·10 ·10

9 8 7 6

1

u ·10-3

5 4 3 2 1

x1

x3

x1

x3

x2

x3

3

u ·10-3 24 22 20 18 16 14 12 10 8 6 4 2

x2

x1

x3

x1

2

u ·10-3 4 3 2 1

x3

4 3 2

x3 x1

-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

x1

Figure 1.35. The distribution of a plate’s stress-strain state at: a) τ = 0.005, b) τ = 0.0175, c) τ = 0.03625, d) τ = 0.625.

100

1 Three–Dimensional Problems of Theory of Plates in Temperature Field -1 µ·10 ·10

1

u ·10-3

9 8 7 6 5 4 3 2 1

4 3 2 1 -1x1 -2 -3

x3

x1

x3

3

u ·10-3 x2

x2

x1

x3

28 26 24 22 20 18 16 14 12 10 8 6 4 2

x3

x1

2

u ·10-3 4 3 2 1

x3 -1

-2 -3 -4

x3

x1

x1 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

Figure 1.35. cont.

The following conclusion may be drawn for problems 19 and 20: in course of time the stress-strain state “grows stable” and approaches the distribution obtained with taking static problems into account. For instance: at τ = 0.0365 the results

1.4 Linear Problems in the Theory of Plates in 3D Space

101

-1 µ·10 ·10

1

u ·10-3

9 8 7 6 5 4 3 2 1

3 2 1 -1 x 1 -2 -3

x3

x1

x3

3

u ·10-3 x2

20

x2

18 16 14

x3

x1

x3

12 10

x1

8 6 4

2

u ·10-3 4 3 2 1

x3

-2 -3 -4

2

x1

x3

x1

-6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

Figure 1.35. cont.

coincide with convergence between 1% and 12% (Fig. 1.30 and 1.35a, and Fig. 1.31 and 1.35d for problem 20). Further calculations in time do not bring any changes to

102

1 Three–Dimensional Problems of Theory of Plates in Temperature Field u ·10-1 u1·10-3 9 8 7 6 5 4 3 2 1

5 4 3 2 1

x3

-1x 1 -2 -3

x1

x3

u ·10-3 3

x2

x3

22

x2

x1

x3

20 18 16 14

x1

12 10 8 6 4 2

u2·10-3 6 5 4 3 2 1

x3 -1

-2

x3

x1

-2 x 1 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

Figure 1.35. cont.

the stress-strain state. Therefore, observation of the moments of direct influence of the heat impact (dτ = 0.01) becomes most interesting in this case. The thermal ring

1.4 Linear Problems in the Theory of Plates in 3D Space

103

with heat distribution constant in time is one of the forms of the heat impact [101]. The problems concerning stationary thermal boundary conditions may be analysed within quasistatic considerations. Problem 21. Let us analyse two last cases of solving problems with a movable thermal ring. Initially, the fixing-ring is moved towards the second row of points. The thermal ring is moved towards the first row and then to the second row of points at τ = 0.0125. 1

u ·10-3

6 5 4 3 2 1

x3

9 8 7 6 5 4 3 2 1

x3

-1 x1 -2 -3 -4 -5 -6 -7 -8 -9

-1 µ·10 ·10

x2

x2

x1 2

u ·10-2

7 6 5 4 3 2 1

x3

x3

x1

x3

3

u ·10-2 x1

x3

x1

3 2 1

x1

Figure 1.36. The distribution of a plate’s stress-strain state with a movable thermal ring (the static approach) at: a) τ = 0.05, b) τ = 0.01, c) τ = 0.0125, d) τ = 0.01625, e) τ = 0.0175, f) τ = 0.03.

104

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

u ·10-2 6 5 4 3 2 1 -1 x1 -2 -3 -4 -5 -6 -7 -8

x3

-1 µ·10 ·10

8 7 6 5 4 3 2 1

x3

x2

x1 x3 2

u ·10-2

x3

7 6 5 4 3 2 1

x2

x1

x1

x3

3

u ·10-2 x1

x3

3 2 1

x1

Figure 1.36. cont.

Figure 1.36a–f illustrates all changes of the stress-strain state related to the temperature field’s change. Until the thermal ring is displaced at τ = 0.0125, from the very first time instant the graph of the stress-strain state approaches the solution obtained in the analogous problem 17 (Fig. 1.32b) in the static approach. As the thermal ring is displaced, the stress-strain state also approaches the solution obtained in the analogous problem (Fig. 1.31), but at τ = 0.0125 (the moment when the thermal ring is displaced) the representation of the temperature distribution does not resemble the previously investigated ones. A similar conclusion may be drawn for the displacement distribution. It is also important to notice that the scale of the graphs at time instant τ = 0.0125 suddenly changes by one order of magnitude. With the quasistatic approach it has been possible to discover the stress-strain state - an achievement impossible to make with the use of any static methods. By integrating the system of equations (1.23) at the same time with the equation of heat

1.4 Linear Problems in the Theory of Plates in 3D Space

105

1

u ·10-3

-1 µ·10 ·10

5 4 3 2 1

10 9 8 7 6 5 4 3 2 1

-1 x 1 -2 -3 -4

x3

x3

x1 3

u ·10-3 x2

x1

x3

20 18 16 14 12 10 8 6 4 2

x2

x3

x1

3

u ·10-3 4 3 2 1 -1 x3 -2 -3 -4

x3

x1

-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

x1

Figure 1.36. cont.

conductivity (1.24), assuming null dilatation and using Runge-Kutta’s method only, we obtain a complete dynamic problem of the theory of thermoelasticity.

106

1 Three–Dimensional Problems of Theory of Plates in Temperature Field u ·10-1

u ·10-3 1

9 8 7 6

4 3 2 1

5 4 3 2 1

x3

x3

-1 x 1 -2 -3 -4

x1

x2

x3

x3

x1

u ·10-3 2

4 3 2 1

x3 -1

-2 -3 -4

3

26 24 22 20 18 16 14 12 10 8 6 4 2

x2

x1

u ·10-3

x3

x1

x1

Figure 1.36. cont.

-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40

1.4 Linear Problems in the Theory of Plates in 3D Space

107

-1 µ·10 ·10

11 10 9 8 7 6 5 4 3 2 1

1

u ·10-3 4 3 2 1 -1 x 1 -2 -3

x3

x3

x1 3

u ·10-3 x2

x3

24 22 20 18 16 14 12 10 8 6 4 2

x2

x1

x3

x1

2

u ·10-3 4 3 2 1

x3 -1

-2 -3 -4 -5

x3

x1

Figure 1.36. cont.

x1 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

108

1 Three–Dimensional Problems of Theory of Plates in Temperature Field -1 µ·10 ·10

1

u ·10-3

9 8 7 6 5 4 3 2 1

3 2 1

x3

x1

-1 -2 -3

x3

x1 3

u ·10-3

x2

x2

x1

x3

x3

x1

24 22 20 18 16 14 12 10 8 6 4 2

x3 2

u ·10

-3

4 3 2 1

x3

x1

Figure 1.36. cont.

x1 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42

1.4 Linear Problems in the Theory of Plates in 3D Space

109

Problem 22. A problem formulated in the same way as problem 21 with a movable thermal ring is solved according to the dynamic approach with supplementary mechanical initial conditions (1.33). An assumption that qS1 , qS2 , q, s = 1, ..., 3 are equal to zero has been made in this problem. Figure 1.37a–f presents temperature and displacement graphs plotted on plane x2 = 7/8 at the same time instant as in case of the quasistatic analysis. Comparison of those results with the results obtained in an analogous quasistatic problem (Fig. 1.36a–f) leads to a conclusion that the temperature distributions coincide, the scale of the displacement graphs at τ = 0.0125 changes by one order of magnitude, but the similarity of the stress-strain state remains unchanged (it does not “keep up” with assuming the form it has in statics and quasistatics). That is why the time of calculation is prolonged until τ = 0.055 (Fig. 1.37f), which still seems to be not enough to reach stability of the desired stress-strain state portrait in statics (Fig. 1.31). Time-variable heat impacts belong to the most interesting phenomena investigated in the theory of thermoelasticity. They should be analysed with methods which are most related to real dynamic processes. The principle condition for obtaining reliable results is taking dynamic eﬀects into account. Finally, it is important to notice that the parts of the thermal ring that are perpendicular to plane x2 = 7/8, as well as that part of the ring that belongs to wall ∂Ω2 , parallel to plane x2 = 7/8, reveal their influence upon the investigated plane x2 = 7/8. 1.4.5 Inner heat sources It seems trivial to claim that including heat sources into investigations makes the mathematical relations concerning solving problems more complicated. In fact, the right sides of heat conductivity equations solved through the analytical approach equal zero in very few problems of the theory of elasticity. Mostly axially symmetrical or two-dimensional problems belong to this group. The methodology presented in this chapter enables scientists to solve problems of the theory of thermoelasticity with stationary or movable heat sources or heat fluxes in three-dimensional plates. If the influence of heat sources causes a change of temperature, then the plate undergoes deformations even without mechanical excitations. Dynamic problem (1.23)– (1.33) must be formulated and Runge-Kutta’s method must be applied to find the solution in this case.

110

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

u ·10-2

5 4 3 2 1

-1 µ·10 ·10

x3

9 8 7 6 5 4 3 2 1

x2

x1

x3

x2

x2

X1 x1

x3

3

u ·10

x1 2

4 3 2 1

-1 x1 -2 -3 -4 -5

u ·10

-2

5 3 2 1

-3

x3

x3

x1

x1

1

u ·10-2

4 3 2 1

x3

-1 x1 -2 -3 -4 -5

µ·10 ·10-1 10 9 8 7 6 5 4 3 2 1

x2

x1

x3

x3

x1

x2

x1

x3

3

u ·10-2 3 2 1

2

u ·10-2

x3

x1

3 2 1

x3

x1

Figure 1.37. The distribution of a plate’s stress-strain state with a movable thermal ring (the dynamic approach) at: a) τ = 0.005, b) τ = 0.01, c) τ = 0.0125, d) τ = 0.01625, e) τ = 0.03, f) τ = 0.055.

1.4 Linear Problems in the Theory of Plates in 3D Space µ·10 ·10

-1

1

u ·10-2

9 8 7 6 5 4 3 2 1

3 2 1

x3

x3

x1

x2

x1

x2

x1

x3

x1

x3 3

u ·10-2

3 2 1

2

4 3 2 1

u ·10-2

x1

x3

x3

x1

1

u ·10-3

µ·10 ·10

3 2 1

-1

x3

9 8 7 6 5 4 3 2 1

-1 x 1 -2

x2

x3

x3

x1

2

x3

x1

x1

x3

3

u ·10-3 14 12 10 8 6 4 2

u ·10-3 3 2 1

x2

x3

x1

Figure 1.37. cont.

-2 x1 -4 -6 -8 -10 -12 -14 -16 -18

111

112

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

u ·10-3 3 2 1 -1 µ·10 ·10

x3

-1 x1 -2

9 8 7 6 5 4 3 2 1

x2

x2

x1

x3

x3

x1

x3

x1 3

u ·10-3

14 12 10 8 6 4 2

2

u ·10-3 3 2 1

x3

x3

-2 x1 -4 -6 -8 -10 -12 -14 -16

x1

1

u ·10-3 µ·10 ·10

3 2 1

-1

-1 x1 -2 -3

x3

9 8 7 6 5 4 3 2 1

x2

x3

x3

x1

u ·10-3

-1 x3 -2 -3 -4

x1

x1

x3 3

u ·10

-3

16 14 12 10 8 6 4 2

2

4 3 2 1

x2

x3

x1

Figure 1.37. cont.

-2 x1 -4 -6 -8 -10 -12 -14

1.4 Linear Problems in the Theory of Plates in 3D Space

113

Problem 23. Let us investigate a cube-shaped plate with free walls ∂Ω1 , ∂Ω2 , ∂Ω4 , ∂Ω5 (E1i = E2i = E4i = E5i = 0, Di1 = Di2 = Di4 = Di5 = 1, i = 1, ..., 3) and rigidly fixed walls ∂Ω3 , ∂Ω6 (E3i = E6i = 1, Di3 = Di6 = 0, i = 1, ..., 3). The temperature distribution on the entire surface of the plate is equal to (Ci = Bi = 0, Ai = 1). On plane (x3 = 0.5, 0.125 ≤ x1 ≤ 0.5, 0.125 ≤ x2 ≤ 0.875) inside the plate there is a uniformly distributed heat source of dimensionless unite power.

-4 µ·10 ·10

6 5 4 3 2 1

1

u ·10-3

x3

5 4 3 2 1

x1

x2

x3

x3

x2

x1

x3

x1

x1

3

u ·10-3 5 4 3 2 1

u2·10-3 4 3 2 1

x3

x3

x1

x1

Figure 1.38. The distribution of the changes of temperature and displacements (plane x2 = 0.5) for the plate investigated in problem 23.

Figure 1.38 presents temperature and displacement distribution on plane x2 = 0.5. The displacement graphs clearly illustrate the plate’s expansion in all directions away from the heat source: symmetrically towards x3 and asymmetrically towards x1 and x2 . The behaviour of displacement u1 appears to be the most interesting: the layers located closer to the heat source’s plane move towards x1 , whereas the further layers move in the opposite direction to axis x1 . As a result, an increase of

114

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

temperature causes the strongest stresses between those layers. Further numerical calculations are possible when the heat source’s surface and volume are reduced or enlarged. Problem 24. A volumetric heat source move inside a plate (the source’s form is presented in Figure 1.29 on plane x2 = 7/8, but the problem is not described analytically). The following conditions are assumed on the plate’s boundary: null temperature distribution, surface forces f11 = f41 = 0.001, f12 = f13 = f42 = f43 = 0, and volume force P3 = 4 · 10−6 , P1 = P2 = 0. The boundary conditions are the same as in problem 16. Figure 1.39a–c illustrates temperature distribution and displacement graphs. Initially (τ = 0.00875), the plate reveals stronger reaction to mechanical excitations. Figures 1.39b and 1.39c display asymmetrical range of displacement u1 , caused by the surface forces. However, already at τ = 0.03 all three displacements increase by one order of magnitude due to the reaction to heat impact which absorbs the reaction incited by surface forces. Symmetry of the heat source makes the displacements symmetrical to axis (1/2, 1/2, 1/3) grow in importance. 1.4.6 Deformation and Temperature Heat impacts have been investigated in works [98, 101, 200, 350]. The results obtained by Danilovskaya [163, 164] have become classic and hence often referred to. She has pointed out a possibility of occurrence of compressive and tensile stresses in an infinitely elastic space during heating, which can be observed only through a complete dynamic formulation of the problem. This section contains an analysis of Danilovskaya’s problem for a three-dimensional plate. System of equations (1.23)–(1.33) has been solved with the use of Runge-Kutta’s method Problem 25. The object of investigation is a plate with fixed walls ∂Ω2 , ∂Ω3 , ∂Ω5 , ∂Ω6 , and two opposite walls Ω1 and Ω4 . Wall Ω1 is aﬀected by heat impact ∂θ/∂x1 = 13.3. The other walls are thermally insulated (∂θ/∂n = 0). The initial conditions are null (1.33). Comparison of the results described in work [98] with the results obtained by the authors of this work (Fig. 1.40) allows investigating the changes of the normal stress in point (0.5, 0.5, 0.5) (point c in the graph) in time. Figure 1.40 presents illustration of the results of calculations made for several sections of the plate in points: a) (1/8, x2 , x3 ), b) (1/4, x2 , x3 ), c) (1/2, x2 , x3 ), d) (7/8, x2 , x3 ). A configuration of curves a, b, c, d may be calculated with the use of analytical relations [98]. The compressive stresses increase until τ = x1 /a, in which a = 1/(1 + ν)(1 − 2ν) is the coeﬃcient at the main derivatives in the movement equations, x1 is the first coordinate of the examined object. The values of the characteristic time interval obtained with this formula coincide with the obtained approximate values τa ≈ 0.065, τb ≈ 0.13, τc ≈ 0.26, td ≈ 0.42.

1.4 Linear Problems in the Theory of Plates in 3D Space µ·10 ·10

5 4 3 2 1

115

-1

1

u ·10-5

x3

6 5 4 3 2 1

x1

x2

x2 x3

x1

x3 2

u ·10

4 3 2 1

u3·10-5

x1

x3

x1

-5

5 4 3 2 1

x3

x3

x1

x1

1

u ·10-5 5 4 3 2 1

-1 µ·10 ·10

4 3 2 1

x1

x3 x1

x3

x2

x3 2

u ·10-5

x2

x1

x3

3

u ·10-5 4 3 2 1

x1

x1

x3

4 3 2 1

x3

x1

Figure 1.39. The temperature distribution and the displacement graphs in case of a volumetric (movable) heat source at: a) τ = 0.00875, b) τ = 0.01525, c) τ = 0.03.

116

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

U ·10-4

5 4 3 2 1

4 3 2 1

µ·10-1 X3

X1

X3

X1

x2

x2

x1

x3

3

U ·10-4 4 3 2 1

x1

x3

2

U ·10-4

X3

X1

3 2 1

X3

X1

Figure 1.39. cont. ¾x x

1 1

a

x2

20

b c

x1

x3 0.1

0.2

t

t

0.4

0.5

-20 d

x2

x3

x1

Figure 1.40. The stress distribution for the plate investigated in problem 25.

1.4 Linear Problems in the Theory of Plates in 3D Space

117

The scatter of values between the maximum compressive stress and the maximum tensile stress for every section is identical, S is the coeﬃcient at the temperature derivatives of the movement equations and T 0 = 1. The results presented in Figure 1.40 coincide with the classic ones until the time instant in which the stress for the half-infinite space starts to converge towards zero. In case of a thick plate, the normal stresses regain their compressive nature in course of time, yet it is more intense than at the beginning. It can be best observed for section c). The full curve illustrates the results of calculations without taking deformation and temperature fields’ coupling (β = 0) into account. The broken curve represents the results including the coupling, which means that the generalized heat conductivity equation (1.24) has been included with all terms in the calculations. Coupling coeﬃcient β = 0.03 [595] for aluminium alloys may reach arbitrarily high values. The coupling eﬀect can be obtained at τ = 0.3125 – 10%. Problem 26. In order to investigate this problem, the distribution of normal stress along the axis of plane (x1 , 0.5, 0.5) has been presented in several time instants in Figure 1.41a. The same refers to Figure 1.41b, although the considerations include the second boundary conditions: identical heat impact, but the plate’s walls ∂Ωi , i = 2, ..., 6 have null temperature distribution. The stresses in this problem have very large quantities and they undergo significant changes. The thermal reaction of all six walls and adequately quicker stabilisation of the temperature distribution can serve as an explanation of this phenomenon. The coupling eﬀect occurs at τ = 0.3125 – 12%. Problem 27. The plate examined in the previous problem is aﬀected by simultaneous thermal and mechanical impacts, which cause a displacement on wall ∂Ω1 , which consequently approaches a stationary state of the following value: ⎧ ⎪ ⎨ 0, τ ≤ τ∗ 1 u x =l = u0 f (τ) , f (τ) = ⎪ ⎩ 1, τ > τ∗ , 1 1 where: u0 = 0.01, and f (τ) is Heaviside’s function. The full line in Figure 1.42a illustrates the normal stress distribution. Figure 1.42b presents the static stress on axis (x1 , 0.5, 0.5). Problem 28. This case refers to the following numerical experiment concerning the plate described in problem 27: sudden thermal impact has been neglected, i.e. the analysis takes only the mechanical impact into account.

118

1 Three–Dimensional Problems of Theory of Plates in Temperature Field a)

¾x x

1 1

0.5 t=0.03 0

-0.5

-1

x1

t=0.1 t=0.2

t=0.3

-1.5 t=0.5

b)

¾x x

1 1

0.5

0

-0.5

0.03 0.1

x1

0.2 0.4 0.3

-1

Figure 1.41. The normal stress distribution along the axis of plane (x1 , 0.5, 0.5): a) the first thermal boundary conditions, b) the second thermal boundary conditions.

1.4 Linear Problems in the Theory of Plates in 3D Space

119

a)

¾x x

1 1

x2

3

2

x1

x3

1

0 t=0.01

x1

t=0.03

-1

-2 x2 t=0.0125

-3

-4

x3

x1

t=0.05

-5

-6

-7

t=0.03

Figure 1.42. The stress distribution in time for the plate investigated in problem 26 regarding additional heat and mechanical impacts: a) the normal stress distribution, b) stresses tangential to axis (x1 , 0.5, 0.5).

120

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

b)

¾x x

1 1

1

0

t=0.001 t=0.03

-1 t=0.05 -2

-3

t=0.0125

-4

-5

-6

-7

t=0.03 t=0.04

Figure 1.42. cont.

x1

1.4 Linear Problems in the Theory of Plates in 3D Space

121

The results of calculation have been illustrated with the broken line in Figure 1.43a–b. The dash-dot curve illustrates the results obtained taking the temperature and the deformation fields’ couplings into account. The diﬀerence in case of this eﬀect does not exceed 1% at τ = 0.05. Figure 1.43a presents the behaviour of the normal stress in the plate’s centre within a time interval. The mechanical impact weakens the influence of the thermal impact. The impact wave becomes smooth (Fig. 1.43a–b). a)

¾x x

1 1

2 1 0.02

0.03

0.04

0.05

-1

t

-2

b)

¾x x

1 1

0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.4 0.45 0.5 0.55 0.6 0.65

-0.5

t

-1

Figure 1.43. The stress distribution in time for the plate investigated in problem 27: a) the normal stress in the plate’s centre, b) the tangential stresses.

Problem 29. A mechanical impact of value u0 = −0.01 increases the influence of the heat impact investigated in this section. Figure 1.44a–d illustrates the development of the heat impact process on plane x2 = 0.5. Displacements u1 and u3 in the investigated plane are characterised by significant changes. Displacements u2 and u3 are symmetrical to each other. In order to observe this, it is necessary to present the distribution of displacement u3 on plane x2 = 0.5, and displacement u2 on plane x3 = 0.5 (Fig. 1.44d). Similarly to the previously investigated thermoelasticity problems, a strong influence of the mechanical impact on the stress-strain state reveals first (Fig. 1.44a–b), and then thermal eﬀects occur (Fig. 1.44c–d). In Figure 1.45, the distribution of the normal

122

1 Three–Dimensional Problems of Theory of Plates in Temperature Field a)

1

-u ·10-2 4 3 2

0 µ·10 ·10

x3

4 3 2 1

x3

x1

x2

x1

x2

2

u ·10-3 4 3 2 1

x3

x1

x1

x3 3

u ·10-3 4 3 2 1

x1

x3

x3

x1

1

-u ·10-2 4 3 2 1

b)

6 5 4 3 2 1

x3

µ

x1

3

u ·10-3

x3

x1

x2

x3

x2

x1

x3

x3

4 3 2 1

x1

x1

Figure 1.44. The development of the heat impact process on plane x2 = 0.5 at: a) τ = 0.0125, b) τ = 0.1725, c) τ = 0.3325, d) τ = 0.5.

1.4 Linear Problems in the Theory of Plates in 3D Space µ 9 8 7 6 5 4 3 2 1

c)

1

u ·10-2 5 4 3 2 1

x2

x2

x1

x3

3

x1

x3

u ·10-3

4 3 2 1

x1

x3

u2·10-3 5 4

-1 x3 -2 -3 -4

3 2 1

x1

2

u ·10-2 d)

x1

x3

x1

x3

4 3 2 1

1

u ·10-2

-1 x3 -2 -3 -4 -5 -6 -7

x1

4 3 2

x3

x1

µ 10 9 8 7 6 5 4 3 2 1

x2

x3

x1

x1

x3

3

x3

x1

u2·10-3 3 2 1

x3

x2

x3

x1

Figure 1.44. cont.

u ·10-3 4 3 2 1 -1 x1 -2 -3 -4 -5 -6 -7

123

124

1 Three–Dimensional Problems of Theory of Plates in Temperature Field ¾x x

1 1

0.5

t=0.2

0.375 0.25 0.125 0 -0.125

t=0.1 t=0.4 X1 t=0.01 t=0.03 t=0.05

-0.25 -0.375 0.5

t=0.5

-0.625 -0.75 -0.875 -1 -1.125 -1.25

Figure 1.45. The distribution of the normal stress along axis (x1 , 0.5, 0.5) for the plate investigated in problem 29.

stress along axis (x1 , 0.5, 0.5) is demonstrated by a thermoelastic wave generated by the active wall towards a parallel wall. The reciprocal coupling eﬀect in the plate’s centre equals 1% at τ = 0.5. Problem 30. A complete three-dimensional description makes it possible to spot and observe heat impacts operating along a part of the plate’s wall. Let us investigate the influence of the heat impact formulated according to problem 29, yet limited to only a half of wall ∂Ω1 (x1 = 1, 0 ≤ x1 ≤ 1, 0.5 ≤ x3 ≤ 1), while the field of the mechanical impact’s operation is limited to a quarter of wall ∂Ω1 (0.5 ≤ x1 ≤ 1). Figure 1.46a–c presents the plate’s reaction to impacts at various time instants on plane x2 = 0.5. Reducing the active impact surface leads to substantial reduction of the plate’s stresses. It is particularly visible in Figure 1.43b, in which the full line

1.4 Linear Problems in the Theory of Plates in 3D Space

125

1

-u ·10-2

a)

3 2 1

x3

x1

µ 2 1

x3

2

u ·10-3

x1

x3

4 3 2 1 -1 x1

2

u ·10-3

x3

4 3 2 1

x2

-1 x1

x3

b)

x2

x1

x1

x3

µ·10 ·10 4 3 2 1

1

-u ·10-2 3 2 1

x1

x3 x3 x2

x1

x2 3

u ·10-3 x1

x3

x3

4 3 2 1

x1

2

u ·10-3 4 3 2 1 -1 x3 -2 -3 -4 -5

x3

x1

-1 x1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -111 -1 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21

Figure 1.46. The plate’s reaction to impacts (plane x2 = 0.5) at various time instants: a) τ = 0.1725, b) τ = 0.3325, c) τ = 0.5.

126

1 Three–Dimensional Problems of Theory of Plates in Temperature Field c) 3

u ·10-3 µ·10

4 4 3 2 1

x3

2

x3 -2

x1

x1

-4

x2

x2

-6 -8 -10

x1

x3

x3 1

x1

-12

-2

-14

-u ·10

-16

3 2 1

x3

-18

x1

-20 2

u ·10-3

-22

4 3 2 1

x3

-24 -26

x1

-28 -30 -32 -34

Figure 1.46. The plate’s reaction to impacts (plane x2 = 0.5) at various time instants: a) τ = 0.1725, b) τ = 0.3325, c) τ = 0.5.

marks the stress in the centre of the plate aﬀected by thermal or mechanical impacts on the entire surface of wall ∂Ω1 , while the broken line marks the same impacts along a part of wall ∂Ω1 . The coupling eﬀect for problem 30 is weaker than it is in problems 25 and 26. Its value at τ = 0.4 equals 1%. The thermoelastic wave (Fig. 1.47) is clearly visible and in course of time its front moves away from the active wall towards the opposite wall. The graph of the normal stress with respect to axis (x1 , 0.5, 0.5) takes the general value for every investigated case. Problem 31. In order to analyse the occurrence of the reciprocal coupling eﬀect caused by inner heat conductivity processes incited by a heat source, the following investigation is conducted: a cube-shaped plate with fixed walls ∂Ω2 , ∂Ω3 , ∂Ω5 , ∂Ω6 and free walls ∂Ω1 , ∂Ω4 is aﬀected by a uniformly distributed heat source (P4 = 1) located on plane x1 = 0.5; the temperature distribution equals zero (T i0 = 0). The energy source is non-stationary, it operates within time interval dτ = 0 ÷ 0.4 dimensionless units and then it is switched oﬀ.

1.4 Linear Problems in the Theory of Plates in 3D Space

127

¾x x

1 1

t=0.2 t=0.3

0.05

0

0.1

t=0.4 t=0.1 t=0.01 x t=0.02 1 t=0.03

0.2 0.3

-0.25

t=0.5 t=0.6

0.4

-0.5 0.5 0.6

Figure 1.47. The graph of the normal stress in respect to axis (x1 , 0.5, 0.5).

The graph in Figure 1.48a presents normal stress σ x1 x1 on straight line (x1 , 0.5, 0.5) at various instants of time. The full curves represent the source’s operation at τ = 0.4. The source creates a tension-related heat wave that spreads from the source to the plate’s edge, then it reflects and returns to the source’s centre. At the same time the heat source is switched oﬀ and the compressive stresses turn into tensile. Initially, until τ = 0.5, the distribution of the normal stress is not identical to the one at τ = 0.4 and its tension configuration is displaced into the positive field. However, already at τ = 0.6 the normal stresses have positive values only – it is characteristic for displacements that approach the initial state of equilibrium. The strongest stresses occur on the edges of the investigated plate. Figure 1.48b illustrates changes of temperature in the plate’s centre within a time interval. Figure 1.48c illustrates changes of the normal stress both in the plate’s centre and in point (1/8, 1/2, 1/2) within a time interval. The broken curve marks the results obtained taking the reciprocal coupling into account. The quantities of the reciprocal influence exerted on the investigated time interval diﬀer from each other. Table 1.11 presents the relative percentage diﬀerence in calculations in which the reciprocal coupling has and has not been taken into account. At the initial stage of heating, the coupling does not occur until the heat regime become stable (unchangeable). After the heat source is switched oﬀ, the meaning of the coupling

128

1 Three–Dimensional Problems of Theory of Plates in Temperature Field a)

¾x x

t=0.72

1 1

0.5

t=0.7

0.4 0.3 0.2 0.1 t=0.5 x1 t=0.21

0 -0.1

t=0.1

-0.2 -0.3

t=0.5 t=0.2

-0.4

t=0.42 t=0.4

x2

x2

x1 x3

x1 x3

b) µ·10-1 0.1 0.1

µ

¾x x

1 1

0.2

0.3

0.4

0.5

0.6

t

0%

0%

0%

5%

0.7%

0.3%

0.1%

0%

0%

0.3%

6%

0.7%

0.4%

0.05%

Figure 1.48. The analyses of the occurrence of the reciprocal coupling eﬀect (problem 31): a) the graph of the normal stress on the straight (x1 , 0.5, 0.5) at various time instants, b) the temperature change in the plate’s centre in time, c) the changes of the normal stress in time in the plate’s centre and in point (1/8, 1/2, 1/2).

1.5 3D Physically Non-Linear Problems

129

c)

¾x x

1 1

0.8

0.6

(1/8,1/2,1/2)

0.4

0.2

0.1

0.2

0.3

(1/2,1/2,1/2)

0.5

t

0.6

-0.2

-0.4

Figure 1.48. cont.

Table 1.11. The relative diﬀerence in calculations with and without considering the reciprocal coupling. τ θ σ x1 x1

0.1 0% 0%

0.2 0% 0%

0.3 0% 0.3%

0.4 5% 6%

0.5 0.7% 0.7%

0.6 0.3% 0.4%

0.7 0.1% 0.05%

eﬀect significantly increases since it reaches 6% during dτ = 0.01 for the normal stress and 5% for the temperature. In course of time the heat regime becomes stable again and the eﬀect disappears. The process described here confirms the fact that couplings must be investigated and taken into account especially at clearly distinguished non-stationary heat regimes.

1.5 3D Physically Non-Linear Problems Relations between couplings and deformations are non-linear if the body’s load exceeds the yield point. Variety of diﬀerent kinds of elastoplastic bodies leads to creating a number of various mathematical models that describe physical states of the

130

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

investigated bodies. There are several theories of plasticity that describe physical relations and precisely explain physical phenomena that characterise elastoplastic deformations. Therefore it is extremely important to answer the question about usefulness and applicability of those theories. Surprisingly enough, the problem may be solved only for one class of loading, namely the simple loading. The theory of small elastoplastic deformations [89, 171, 357] proves to be handy while describing phenomena of that kind. Its simplicity of relations and possibility to utilise general solving methods have made the theory of small elastoplastic deformations widely applied. This chapter uses small elastoplastic deformations to present a method of solving coupled problems of heat and mechanical loading of a three-dimensional plate. 1.5.1 Diﬀerential equations and diﬀerence approximation Let us investigate the stress-strain state at an initial time instant of an isotropic homogeneous body (plate) that is non-uniformly heated and aﬀected by surface and volume forces. The principal conditions of the applied theory have been presented in work [360]. The theory is based on the hypothesis of proportionality of the stress tensor and the strain tensor’s components:

(1.116) ei j − δi j e = ψ σi j − δi j σ , ψ=

3εi . 2σi

(1.117)

For small elastoplastic deformations the following function relation is eﬀective: σi = f (εi ),

(1.118)

which is similar to the relations that between stresses and deformations during tension σ = f (e) , (1.119) i.e.: the volumetric strain is elastic σ = ke. Moreover, during uniformly distributed heating we obtain σ e = + αT θ. k The components of the strain tensor take the form of superposition of the strain (p) components’ elasticity e(e) i j and plasticity ei j , i.e.: (p)

ei j = e(e) i j + ei j ,

(1.120)

where: e(e) i j is determined according to Duhamel-Neuman law, considering the fact that the coeﬃcients depend on temperature. Equation (1.116) is true, if the body’s loading is simple. In this case, without taking temperature relations into account,

1.5 3D Physically Non-Linear Problems

131

the external loadings increase until they are proportionally imposed to one general parameter [171]: (1.121) Pi = λ (τ) P0i , i 0i = λ(τ) f s+m , i, s = 1...3, m = 0, 3 . f s+m

(1.122)

During non-uniform heating in thermal quasistatic regime (slow gradual increase of temperature) the temperature loadings change in proportion to parameter [357] 0 = χ(τ)T 00 . (1.123) T i+m The external forces increase together with the increase of the temperature (exact to reliable multipliers) to such extent that the stress intensity increases along with the increase of the strain intensity (1.118), i.e.: Pi = BT γ , where: A=

0i f s+m

T 00γ

i f s+m = AT γ ,

,

B=

P0i T 00γ

,

(1.124)

(1.125)

and γ < 1 is a material constant. In case of a non-stationary heat regime, the simple stress occurs when 0 = T 00 eατ . T i+m

(1.126)

The external forces change according to formula (1.124), but the material characteristics do not depend on temperature. Otherwise, the stress is simple [145] if we investigate an incompressible material in plastic and elastic zones, where: e = 2τθ, ν = 0.5, k → ∞, E = 3G,

(1.127)

σi j = f (θ, εi ) .

(1.128)

The inner forces and the thermal deformations increase proportionally to some parameters and fulfil conditions (1.121), (1.122) and equation ατ (θ)θ = χ(τ)αT (T 0 )T 0 . In a general case, when all thermal and physical characteristics of the material depend on temperature, we obtain [198] T αT (T ) =

αT (ξ)dξ. T0

Function f (εi ) is determined from an experiment. Functional relations for f (εi ) that require meticulous calculations have been collected and analytically described in Krysko’s work [172]. Function f (θ, εi ) is also experimentally determined at various temperatures and presented in space εi , σi , T for certain surfaces (see [483]). In an elastic deformation field σi ≤ 1/ (3σ0 (T ))0.5 where σ0 (T ) is the flow interval, the surface turns non-linear and the equation takes the following form:

132

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

σi = 2µ (T ) ξi ,

(1.129)

where: µ(T ) denotes the time-dependable elasticity modulus. We may assume that at relatively high temperatures there a slight change of αT , which consequently leads to the assumption that αT = const. Therefore, a heat impact at such temperatures may result in plastic deformations, i.e.: E(θ), ν(θ). The equations of the theory of small elastoplastic deformations take the following form [356] σi j = 2 f (εi , T ) ei j +

3K − f (εi , T ) eδi j − 3KαT θδi j , 3

(1.130)

and the material characteristics depend on temperature: σi j = 2 f (εi ) ei j +

3K − f (εi ) eδi j − 3KαT θδi j , 3

(1.131)

or the material characteristics do not depend on temperature: + 3εi j εi j , εi = 2 eδi j . (1.132) 3 The system of diﬀerential equations of the non-linear theory of elasticity differs from system (1.23)–(1.33), the non-linearity of which results from additional expressions that characterise plastic deformations ε(p) i j [484], included within volumetric forces Pi : 2 j ∂2 ui ∂2 u j ∂2 uk ∂ u ∂2 uk (λ + 2µ) 2 + λ 2 + λ 2 + µ − + ∂xi ∂x j ∂xi ∂xk ∂xi ∂x j ∂xk εi j = ei j −

3Kατ

∂2 ui ∂θ − Pi − Pi∗ = ρ 2 , ∂xi ∂τ

3 ∂2 θ ∂ ∂ui 1 ∂θ 1 4 − P , −β = 2 ∂τ ∂x α ∂τ β ∂x i α α=1

(1.133) (1.134)

i ∂u 1 ∂u j G− + + 2Ψ ∂x j ∂xi k ∂u 1 ∂ui 1 ∂ui ∂u j ∂uk ∂ 1 ∂ui ∂ 2G − + G− . − ( + + + ∂xi Ψ ∂xi 3 ∂xi ∂x j ∂xk ∂xk 2Ψ ∂xi ∂xk (1.135) The boundary mechanical conditions are analogous to (1.25), (1.27), (1.29), (1.31) considering iP i i∗ = f s+m + f s+m , (1.136) f s+m where:

∂ P = ∂x j i∗

1.5 3D Physically Non-Linear Problems

where:

133

1 ∂ui 1 ∂ui ∂u j ∂uk ni,s+m + = 2G − − + + Ψ ∂xi 3 ∂xi ∂x j ∂xk i k ∂u ∂u 1 ∂u j 1 ∂ui G− n j,s+m + G − nk,s+m , + + 2Ψ ∂x j ∂xi 2Ψ ∂xi ∂xk i∗ f s+m

i −→ j −→ k, m = 0, 3 . ←

←

(1.137)

The thermal boundary conditions and the initial conditions are identical with those presented in Section 1.1 (1.26), (1.28), (1.30), (1.32), (1.33). Diﬀerence approximation (1.133)–(1.137) is assumed analogously to approximation (1.38)–(1.45), excluding the approximations of the right sides of the equai i∗ are approximated in a usual way, while functions Pi∗ , f s+m tions. Functions Pi , f s+m take the following form: 1 1

Pi∗ ∼ 2Gi − yix¯i xi − yix¯i xi + y xj i x j + ykxi xk + Ψi 3 ⎛ ⎞ ⎜⎜⎜ Ψ ⎟ 1

j k ⎜⎝2G xi + x2i ⎟⎟⎟⎠ yi0,xi − yi0,xi + y0,x + y 0,xk + j 3 Ψi ⎞ ⎛ ⎜⎜ Ψ x j ⎟⎟ i 1 i j ⎟⎟⎠ y + y j + ⎜ ⎜ y x¯ j x j + y xi x j + ⎝G x j + Gi − 0,x 0,x j i 2Ψi 2Ψi2 ⎞ ⎛

⎜⎜ Ψ xk ⎟⎟⎟ i 1 k Gi − yix¯k xk + ykxk xi + ⎜⎝⎜G xk + (1.138) ⎠⎟ y0,xk + y0,xi , 2 2Ψi 2Ψi 1 1

j i∗ k yi0,xi − yi0,xi + y0,x ∼ 2Gi − + y f s+m 0,xk ni,s+m + j Ψi 3

% 1 $ i j k i Gi − y0,x j + y0,x + y + y n n . (1.139) j,s+m k,s+m 0,xi 0,xk i 2Ψi 1.5.2 Algorithm The non-linear system of diﬀerential equations (1.133)–(1.137) does not have an analytical solution, therefore numerical methods that are variants of the successive approximations method [64, 65, 172] are applied. Solving the theory of plasticity problems usually leads to solving a number of linear problems that can be interpreted as the theory of elasticity problems (the method of elastic solutions). This work focuses on the use of the variational method of elastic solutions – the method of additional loading. i∗ , that occur as a result of taking additional terms in If quantities Pi∗ and f s+m (1.133)–(1.137) into account, are assumed to be known, then the following system of reciprocally coupled theory of thermoelasticity is obtained. The investigated system is integrated in consideration of time by means of Runge-Kutta’s method. Initially,

134

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

s∗ at τ = τ0 P s∗ and f s+m = 0 are assumed. Next, functions u s are corrected at every time step with the use of the additional loading method, i.e.: successive approximations of the solution of a properly chosen deformation graph are made according to the following scheme: 1) at the first approximation, solution u s(0) = u s is assumed at previous time step; 2) the obtained displacements u s(0) make it possible to find (0) (0) deformation ε(0) i j (1.11) and strain intensity εi (1.132); 3) the stress intensity σi is determined according to the assumed diagram of deformations [259] or on basis of an appropriate thermomechanical surface [489] from charts or according to a graph, (0) s(0) , Ψ (1) the then Ψ (1) = 1.5ε(0) i /σi is calculated; 4) with known displacements u s∗ can be calculated according to relation (1.135), (1.137); 5) values of P s∗ and fi+m Seidel’s iterative method (1.84) is applied for system (1.133)–(1.137) with unknown right sides without inertial terms. The condition of completing the iterative process is the following estimation: y s(k) − y s(k−1) j j < ε1it , max 1≤ j≤M y s(k) j

where: ε1it is the set value, M = N1 N2 N3 . The result is approximation u s(1) = u s(k) (1) (1) (1) (2) and other values ε(1) = 1.5ε(1) i j , εi , σi , Ψ i /σi , etc. The solving process with the use of the additional loading method should be carried on until the diﬀerence between approximation u s(n) and the preceding approximation u s(n−1) is suﬃciently small and equal to the set value ε2 . Thus obtained corrected solution u s(n) serves as basis for making the next time step with the use of Runge-Kutta’s method, which is next substituted into system (1.133)–(1.137). The combination of the successive approximations method and Runge-Kutta’s method has been applied in work [466] in order to investigate the dynamic behaviour of thin plates and shells (described by means of Kirchhoﬀ’s hypothesis) aﬀected by local mechanical loading. The diﬀerence between the approaches applied in work [466] and the above-described modification lies in using a combination of Seidel’s method, the additional loading method and Runge-Kutta’s method. 1.5.3 Estimation of Convergence The previous section presents the approach that utilises a combination of three numerical methods: solving a linear hyperbolic-parabolic system of diﬀerential equations by means of Runge-Kutta’s method; solving a non-linear elliptic system by means of the additional loading method; solving Lam´e’s linear elliptic diﬀerence system by means of Seidel’s method. The notion of convergence of the general approximate approach to the solution of a non-linear problem should be applied to investigate the convergence of one method on basis of the solution obtained with the use of another method, starting from the inner process. Let us verify the conditions imposed on the coeﬃcients and the right sides of the diﬀerence equations system that approximates the diﬀerential system (1.133)–(1.137). The convergence of the external overt process is provided by the conditions of the theorem quoted in the first chapter. The coeﬃcients of system (1.133)–(1.137)

1.5 3D Physically Non-Linear Problems

135

are assumed to be positive constants and they are bounded throughout the entire calculation process. Such a measure is made due to the fact that the plate is initially investigated as isotropic and homogeneous, therefore according to the additional loading method the following conditions are fulfilled (1.74), (1.75). Conditions (1.77) that refer to an elastoplastic case take the following form: sp s s∗ = fi+m + fi+m ∈ L2 (∂Ω × (τ0 , τ1 )). P sp = P s + P s∗ ∈ L2 (Qτ ) , fi+m

(1.140)

s ∈ They are fulfilled in consideration of the following: a) P s ∈ L2 (Qτ ), fi+m L2 (∂Ω × (τ0 , τ1 )) according to the condition of the theorem quoted in section 1 s∗ – and they do not change throughout the entire calculation process; b) P s∗ and fi+m superpositions of derivatives after the spatial variables of the solution obtained at the previous integration stage, i.e.: functions u s |τ=τ=0 ∈ W21,0 (Qτ ). The compactness of inclusion of W21 (Qτ ) within L2 (Ω) in [285] is also fulfilled for the solution field (a cubicoid), and besides W21,0 (Qτ ) is a Hilbert’s space, which means that it fulfils such an equality that if a, b ∈ W21,0 (Qτ ), then α(a + b) = αa + αb and P s∗ ∈ L2 (Qτ ), s∗ ∈ L2 (∂Ω × (τ01 , τk1 )) respectively. fi+m Boundaries on the right sides of the initial conditions (1.33) at the first time step depend on the plate’s stress at the initial time instant and they coincide with the conditions determined by formula (1.75). The solutions obtained with the use of the fulfilled approximations inside the field are assumed to be the initial conditions at the successive stages of calculations. For the convergence of the external process, the following conditions must be fulfilled:

q1s = u s | τ = τ0l+1 = u s(n) ∈ L2 (Ω) , ∂u s ∂u s τ=τ0 = ∈ W21 (Ω) , q2s = ∂τ ∂τ τ=τkl l+1 q2s ∈ W21 (Ω) ,

u s |τ=τl ∈ W21,0 (Qτ ) ,

(1.141)

q1s ∈ L2 (Ω) according to definition, which is conditioned by relation θ|τk1 = θ|τ0l+1 of

class W21,0 (Qτ ). According to the convergence theorem, which has been proven for solution u s(m) = u s of an elliptic diﬀerence scheme and published in works [429, 430], the inner process is stable and it converges towards the solution of Lam´e’s diﬀerential system. In this case, the following conditions are additionally imposed on the surface and the volume forces: P1s = P s + P s∗ − 3KαT

∂θ ∈ L2 (Ω) , ∂x s

s s s∗ = f1i=m + f1i=m − 3KαT θ ∈ L2 (∂Ω), f1i=m

(1.142) (1.143)

which are consistent with relation (1.140) and concern the solution of an equation of heat conductivity θ in class W21,0 (Qτ ). As conditions (1.143) are fulfilled, the convergence of the external process towards the desired solution and at the same time the convergence of the inner process of solving Lam´e’s diﬀerence system occur.

136

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

The proof of Seidel’s method’s convergence is presented in work [431]. It shows that self-coupling and the transition operator’s positivity are suﬃcient conditions for convergence. Work [431] also discusses certain properties of the operator that may be utilised in some problems of the theory of elasticity. It is important to mention that further stages of investigation utilise the simple loading, thus the external mechanical stresses and the inner volumetric stresses remain constant in time, i.e.: they fulfil conditions (1.121) and (1.122) when λ(τ) = 1. Moreover, the thermal conditions that describe insulation and heat impact occur on a part of one wall, which is provided by conditions (1.128) at αT (θ) = const. The investigated type of loading renders it possible to use the relations between the theory of small elastoplastic deformations and the solutions obtained by means of the additional loading method. The convergence of the additional loading method has been discussed in work [171], and the practical application of the presented iterative process has confirmed its good convergence. 1.5.4 Temperature and Deformation Coupling Investigation of reciprocally coupled temperature and deformation fields beyond the range of elasticity for three-dimensional plates has not yet been discussed in literature. The method, the numerical algorithm and the computation program presented in this work may be applied in order to conduct such an analysis in Section 3.6 for a physically non-linear plate. Problem 32. Let us analyse a thick plate described as a physically non-linear body, which is rigidly fixed along walls ∂Ω2 , ∂Ω3 , ∂Ω4 , ∂Ω5 , ∂Ω6 . Initial displacements u1 = 0.01, u2 = u3 = 0 are assumed on a quarter of wall ∂Ω1 , (x1 = 1, 0.5 ≤ x2 ≤ 1, 0.5 ≤ x3 ≤ 1) and the remaining part of wall ∂Ω1 is rigidly fixed. The problem is solved in the static approach, which means that inertial terms are neglected in system (1.133). The algorithm described in 4.2 is reduced due to excluding Runge-Kutta’s method: the additional loading method is combined with Seidel’s method. The following graph of aluminium strain is used: εi , σi = σ s 1 − exp − εs ε s = 0.98 · 10−3 , σ s = 3e s .

(1.144)

The results are presented in Figure 1.49a on plane x2 = 0.5. Table 1.12 shows the qualitative representation of the stress-strain state in both linear and non-linear approaches. Figure 1.49b presents the plastic deformation zone (the shaded area). Almost entire area is included within the plastic part of the deformation diagram, and only the angular area located away from the active area wall remains elastic. The plasticity field is asymmetrical, which corresponds to the asymmetrically operating loading.

1.5 3D Physically Non-Linear Problems

137

2

u ·10-3

a)

4 3 2 1

1

-u ·10-2 5 4 3 2 1

x3

x3

x1

x1

3

u ·10-3

x2

3 2 1

x3

x1

x3

x1

b) x2

x1 x3

Figure 1.49. The analysis of the thick plate described in problem 32: a) the displacement distribution on plane x2 = 0.5, b) the representation of the plastic deformation zone (the shadowed area).

Problem 33. We shall determine the stress-strain state of a plate aﬀected by a mechanical impact (see: problem 28) on a quarter of wall ∂Ω1 and a heat impact (∂θ/∂x1 = 13.3) imposed along a half of wall ∂Ω1 . Location of the coordinates system and the initial conditions (1.33) are identical to those in problem 31. Deformation graph (1.144) for an aluminium plate without considering the dependence on temperature is assumed. Such a step is fully justifiable, since at the

138

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

Table 1.12. Comparison of the values of displacements caused by heat impacts in linear and physically-non-linear approaches at τ = 0.3225.

u1 u2 u3

(1/4, 1/2, 1/2) (1/2, 1/2, 1/2) (3/4, 1/2, 1/2) Elastic Nonelastic Elastic Nonelastic Elastic Nonelastic -0.001610 -0.002562 -0.008710 -0.008879 -0.01510 -0.009788 -0.000001 -0.000036 -0.000193 -0.000465 -0.00165 -0.001215 -0.000892 -0.001508 -0.003450 -0.005207 -0.01630 -0.006735

initial deformation stages, which are not the object of our interest now, the temperature distribution is arbitrarily small and in any point it does not exceed the investigated field of certain temperature corresponding to aluminium flow range (T ≈ 300◦ ). Figure 1.50a–d presents the distribution of temperature and displacements u1 , u2 and u3 on plane x2 = 0.5 at various time instants τ = 0.0225, 0.1, 0.175, 0.3325. The results compared with the ones obtained in analogous problem 31 (Fig. 1.46c, 1.47, 1.48a) in the static approach reveal aluminium’s soft, non-elastic reaction. Some static approach impact-related phenomena can be clearly observed, the waves of thermoelastoplasticity do not disappear though (Fig. 1.51). Table 1.12 shows comparison of the displacement values for an elastic and a non-elastic problem in the central point (1/2, 1/2, 1/2), in a point close to the active wall ∂Ω1 (3/4, 1/2, 1/2), and in a point located away from the active wall (1/4, 1/2, 1/2). It can be observed that the points located further from the active wall and the central point are the places where the material’s reaction increases in case of the non-linear problem. Dislocations are much stronger in the points located in proximity of the active wall where the problem of elasticity is investigated. The distribution of the normal stress is non-uniform along the plate’s thickness. The displacement distribution and the dynamic behaviour in the less loaded area resemble the normal stress distribution for a linear problem (Fig. 1.48b–c). The stresses increase in the area close to the active wall, but the thermoplastic wave does not appear. The coupling eﬀect in the investigated time interval has not been detected. The coupling coeﬃcient β = 0.03. Problem 34. The boundary and the initial conditions in this problem are identical with those in problem 33. A steel plate’s reaction to mechanical and thermal (∂θ/∂x1 = 200) impacts are investigated. The deformation graph includes the temperature phenomena. Based on Table 1.13 [489], an instantaneous thermomechanical steel surface has been assumed. An increase of the heat impact does not bring any qualitative changes into the obtained results (Fig. 1.52a). Problem 35. The assumptions made in problem 34 remain the same. The heat impact’s force changes (∂θ/∂x1 = 850). The value of the temperature field suddenly increases (Fig. 1.52b, τ = 0.2), the values of displacements correspond to the physically non-linear problem’s values

1.5 3D Physically Non-Linear Problems a)

0 µ·10 ·10

4 3 2 1

1

-u ·10-2 4 3 2 1

x1

x3

x3 x2

139

x1

x2

2

u ·10-3 x1

x3

x3

4 3 2 1

x1

x3

x1

b)

1

-u ·10-2 µ·10 ·10

4 3 2 1

0

5 4 3 2 1

x1

x3 x1

x3

x2

x2

2

u ·10-3 3 2 1

x3

x1

x1

x3 3

u ·10 x1

x3

x3

-3

4 3 2 1

x1

Figure 1.50. The distribution of temperature and displacements u1 , u2 and u3 (plane x2 = 0.5) at various time instants: a) τ = 0.0225, b) τ = 0.1, c) τ = 0.175, d) τ = 0.3325.

and are placed within naturally physical intervals. In contrast to the investigated problems 33 and 34 with low temperatures in which displacement u1 is much bigger, there are also significant changes of the value of displacement u3 in the direction of both impacts’ operation in this problem. It is caused by an immense influence of

140

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

-u ·10-2 c)

4 3 2 1

0 µ·10 ·10

4 3 2 1

x3

x3

x1

x1

x2

x2

2

u ·10-3 4 3 2 1

x3

x1

x1

x3 3

u ·10-3

x3

x1

4 3 2 1

x3

x1

1

-u ·10-2 d)

3 2 1

x3

x1

0 µ·10 ·10

5 4 3 2 1

x1

x3

x3 x2

x3

x2

x1

x3

x1

Figure 1.50. cont.

3

u ·10-3 3 2 1

x1

1.5 3D Physically Non-Linear Problems

141

¾x x

1 1

0.25

t=0.35 t=0.3

0.125

t=0.2

(1/4,1/2,1/2)

t=0.1 x1 t=0.01 t=0.02 t=0.03

-0.125 -0.25

¾x x

1 1

0.5 0.25 0.1

0.2

0.3

0.4

t

(1/2,1/2,1/2)

-0.25 -0.5

(3/4,1/2,1/2)

Figure 1.51. The impact-related phenomena in the elastic approach (problem 33). Table 1.13. Values of instantaneous thermomechanical surface for steel. εi ∗ 102 0 0.1 0.2 0.4 0.6 0.8 1.0 1.2 3728

0 0 1950 3900 5735 6295 6690 6945 7060 8820

100 0 1925 3850 5630 6175 6500 6690 6790 8550

200 0 1880 3760 5520 5935 6180 6340 6420 8160

σi , N/cm2 300 400 0 0 1800 1700 3600 3390 5175 5000 5650 5375 5925 5585 6075 5725 6150 5795 7700 7335

500 0 1560 3083 4580 4910 5020 5080 5112 6520

600 0 1400 2760 3885 4130 4295 4380 4410 5730

700 0 1200 2340 3225 3465 3625 3705 4410 4730

temperature which absorbs the reaction to mechanical operation and at the same time expands the plate’s material in all directions starting from the middle of wall ∂Ω1 (x1 = 1, 0 ≤ x2 ≤ 1, 0.5 ≤ x3 ≤ 1). Due to the fact that wall ∂Ω3 is rigidly fixed the expansion proceeds much easier opposite axis x3 inside the plate. The broken line in Figure 1.53 marks the normal stress distribution along axis (x1 , 0.5, 0.5) at

142

1 Three–Dimensional Problems of Theory of Plates in Temperature Field x2

µ·10 ·10

x2

0

4 3 2 1

x3

x1

x3

x1

x3

1

-u ·10

-2

3 2 1

x1

x3 2

u ·10

x1

-3

4 3 2 1

x3

3

u ·10-3

x1

3 2 1

x3

x1

1

-u ·10-2 3 2 1

x3

µ·10 ·10

x1

2

3 2 1

2

u ·10-2 4 3 2 1

x1

x3

x1

x3 3

u ·10

-3

4 3 2 1

x3

x2

x2

x1 x3

x1

x3

x1

Figure 1.52. The displacement distribution for problem 34 at an increased thermal impact: a) τ = 0.1, b) τ = 0.2.

1.5 3D Physically Non-Linear Problems

143

¾x x

1 1

1.5

t=0.5

1.0

t=0.4

0.5

t=0.3 t=0.2 t=0.01 X1 t=0.02 t=0.03 t=0.1

0 t=0.2 -0.5

-1.0

-1.5

-2.0

-2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

t

-1.0 (3/4,1/2,1/2)

-2.0

(1/2,1/2,1/2)

¾x x

1 1

Figure 1.53. The normal stress distribution along axis (x1 , 0.5, 0.5) for problem 34 (the broken line) and for problem 35 (the full line).

144

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

various time instants for problem 34, whereas the full line - for problem 35. In both cases the stresses are identical. The thermoelastoplasticity wave does not occur within the investigated time interval. This problem’s normal stress does not exceed the stress in problem 34, which corresponds to a sudden increase of the temperature field. The coupling eﬀect (β = 0.01) is not observed. Based on the physical investigation equivalent, the assumption has been made of Duhamel-Neuman principle, that can be applied in case of small deformations and when the following condition is met: θ − T 0 1 . (1.145) T0 The additional loading method is also based on Duhamel-Neuman principle and on condition (1.145). Non-linear equation of heat conductivity should be applied at higher relative temperatures (problems 33 and 35). However the calculations can also be used as sort of an approximation of solution. Problem 36. The displacement distribution graphs change, if the impacts - not the temperature - are assumed to be the reason for the occurrence of the body’s plastic deformation. Work [356] discusses theses and describes experiments that confirm the fact that plastic deformations may occur caused by thermal impacts at arbitrarily low temperatures, which makes it possible to formulate a physically non-linear description of a model. The thermal impact in further experiments takes the form of ∂θ/∂x1 = 0.1. The boundary and the initial conditions are identical as those set in problems 34 and 35. The analytic form of the deformation diagram for aluminium (1.144) is used. The results of calculations are presented in Figure 1.54a–c on plane x2 = 0.5. Although the temperature distributions have the same configuration as in the previously investigated problems, they are significantly smaller. The displacement distribution at various time instants (Fig. 1.54a–c) indicates small displacements u3 in comparison to displacements u3 investigated in problems 34 and 35 at high temperature and at an arbitrarily rapid change in time (until τ = 0.1 value u3 decreases from 0 to −0.85 · 10−2 , and then increases up to −0.25 · 10−2 at τ = 0.43 in point (7/8, 0.5, 0.5)). The graph in Figure 1.55a illustrates the normal stress along axis (x1 , 0.5, 0.5) compared to the analogous graph plotted in the similar problem 33. A sudden increase of the stress can be explained by the fact that the most remote points of the field are neglected in this problem and the thermoelastoplastic wave starts to appear in point (3/4, 0.5, 0.5) at τ = 0.45. The comparison of figures illustrating the normal stress in the plate’s centre and in point (3/4, 0.5, 0.5) in time (Fig. 1.51, 1.55a–b) also shows diﬀerence between the investigated phenomena due to comparable mechanical and thermal eﬀects that occur in the last problem. Problem 36 has been solved with the use of the coupled approach with coupling parameter β = 0.03, without taking the fields of temperature and deformation into account. The diﬀerences between the temperature distributions at τ = 0.45 are 1.5% in the plate’s centre and 0.5% at

1.5 3D Physically Non-Linear Problems

145

1

-u ·10-2 a)

4 3 2 1

-2 µ·10 ·10

5 4 3 2 1

x3

x1

x1

x3

3

-u ·10-2

2

u ·10-3 3 2 1

4 3 2 1

x3

x1

x1

x3

x2

x2

x1

x3

x1

x3 1

b)

-u ·10-2 4 3 2 1

-2 µ·10 ·10

5 4 3 2 1

x3

x3

x1

x1

x2

x2

2

u ·10-3 4 3 2 1

x3

x3

x1

x1

x3 3

u ·10-4 x1

x3

3 2 1

x1

Figure 1.54. The distribution of temperature and displacements (plane x2 = 0.5) for the plate investigated in problem 36 at time instant: a) τ = 0.1, b) τ = 0.3, c) τ = 0.43.

146

1 Three–Dimensional Problems of Theory of Plates in Temperature Field 1

-u ·10-2

c)

4 3 2 1

-1 µ·10 ·10

x3

4 3 2 1

x3

x1

3

x1

2

·10

-4

3 2 1

-3

x3

x1

3 2 1

x3

·10

x1 x2

x3

x2

x1

x3

x1

Figure 1.54. cont.

the active wall. The coupling eﬀect for displacements is 1% in the plate’s centre at τ = 0.45. The eﬀect disappears towards the edges like in the elastic problem. The changes of displacement u3 in time in points (7/8, 0.5, 0.5) and (7/8, 0.5, 7/8) are illustrated in Figure 1.56a for problem 33 and in Figure 1.56b for problem 36. Displacements u1 and u2 increase uniformly in time like in the problem with high temperature, and the order of the values of displacements u1 and u2 is the same as in problem 35 (u1 is 10−1 , whereas u2 is 10−2 ). In course of time, the coupled problem displays the tendency to relatively increase the temperature field and to reduce the deformations, i.e.: the plate’s reaction is damped. Comparison of the results of the temperature and the deformation fields’ coupling’s influence in the linear and the physically non-linear approaches, and also the solutions of problems with various boundary conditions in the linear approach, leads to various conclusions as for the eﬀects of the coupling’s influence. The type of thermal boundary conditions (within the investigated class of problems) gives evidence of little influence on the coupling eﬀect (Fig. 1.41a – 12%, Fig. 1.41b – 10%, problems 25 and 26 at τ = 0.4). The character of the problem’s formulation also weakly depends on the coupling eﬀect (in problem 33’s non-linear

1.5 3D Physically Non-Linear Problems a)

147 0.47 0.45 0.4

¾x x

1 1

0.3

0.1 0.2

0.05

0.1 0.3 0.1

0

x1

-0.05

b)

¾x x

1 1

0.1

0.05

0

0.1

0.2

0.3

0.4

t

0.5

Figure 1.55. The graph of the normal stress: a) along axis (x1 , 0.5, 0.5) - σ x1 x2 (x1), b) σ x1 x1 (τ).

approach it is not visible within the investigated time interval, whereas in problem 30 with the linear approach there is a small coupling eﬀect – 1% in the plate’s centre). The type of mechanical boundary conditions exerts the greatest influence on the coupling eﬀect (in problem 30 with rigidly fixed walls – 1%, in problem 25 with free walls – 12%–10%). The results of the calculations confirm the theoretical assumptions that the larger the fixing surface and the thermal insulation surface are, the more the body’s inner state reflects the converse adiabatic state. An isoentropic process occurs in the form of the converse adiabatic process. If a body is thermally insulated but it can be affected by a force (a mechanical impact in the form of Heaviside’s function: problems 29, 30, 33–36), then it may also participate in the converse adiabatic process and it

148

1 Three–Dimensional Problems of Theory of Plates in Temperature Field

a) 3

u ·10-3 4 2 0

0.2

0.3

0.4

0.5

t

0.5

t

-2 (7/8,0.5,7/8)

-4 -6

(7/8,0.5,0.5)

-8

b)

3

u ·10-3 4 2 0

(7/8,0.5,7/8)

0.1

0.2

0.3

0.4

-2 -4

(7/8,0.5,7/8)

-6 -8

Figure 1.56. The changes of displacement u3 in time in points (7/8, 0.5, 0.5) and (7/8, 0.5, 7/8): a) problem 33, b) problem 36.

is characterised by constant entropy. In a general case, investigation is carried out into the processes of heat conduction and into the inner irreversible processes characterised by the increasing entropy and the increasing entropy flux, which serves as the basis for the theory of the reciprocal coupling of the temperature and the deformation fields. The observation of such processes may help explain the increase of the eﬀect of reciprocal coupling at the mechanical (the second and the third boundary problem) and the thermal (the first and the third boundary problem) “liberation” of the plate’s edges.

2 Stability of Rectangular Shells within Temperature Field

A brief historical research review is given in section 2.1. In section 2.2 variational equations in a hybrid form in curvilinear coordinates are derived for shallow anisotropic shells, as well as the variational and diﬀerential equations in rectangular coordinates for shallow homogeneous anisotropic shell within temperature field are reported. Compatibility relations of boundary conditions for homogeneous anisotropic rectangular shallow shells in a corner point and in the points, where the boundary conditions are changed, are derived. Coupling conditions for isotropic homogeneous shallow shells are given. Finally, the problem of stress-strain state of shallow shells in temperature field is formulated. In Section 2.3 universality and eﬃciency of the finite diﬀerence method devoted to boundary value problems for elliptic equations is discussed and illustrated. It is shown that for multi-dimensional stationary heat transfer problems an application of a 4th order finite diﬀerence method is suﬃcient. In particular, it is outlined that upper relaxation method possesses a relatively high convergence velocity, is simple in realization, and requires small amount of storage memory. Diﬀerence schemes with approximation error o(|h|4 ) for series of multi-dimensional stationary heat transfer equations governing temperature field distribution in isotropic, orthotropic and anisotropic homogeneous and non-homogeneous media are constructed. Theorem on convergence of the proposed diﬀerence scheme to solution of an initial diﬀerential system with velocity of o(|h|4 ) is formulated and proved. In addition, algorithm of 3D heat transfer stationary equation is proposed. Compatibility conditions for diﬀerence boundary value problems with approximation error p(|h|4 ) are obtained. Eﬃciency of the algorithm is illustrated using model problem with various boundary conditions. In section 2.4 the diﬀerence equations approximating the system of nonlinear diﬀerential equations of shallow shell with approximation error o(|h|4 ) are constructed. It is illustrated by comparing computational results of shells with approximation o(|h|4 ) and o(|h|2 ), that the approximation o(|h|4 ) is more eﬃcient. Intervals of iterational parameter variations applied in the nonlinear relaxation method, are defined experimentally. Owing to computations of flexible anisotropic homogeneous shells with planes of stiﬀ symmetry orthogonal to axis z, fibres orientation in stiﬀ symmetry plane have essential influence on shell stability. Among other results, it is detected that a change of fixation type along shell contour essentially influences both a value of critical loads and shell stress-strain state. Owing to increase

150

2 Stability of Rectangular Shells within Temperature Field

of geometrical parameters k1 , k2 , the shell becomes more sensitive to fixation type along its contour side. In the section 2.5 we show that heat sources occurrence influences a shell stressstrain state and its stability. An essential influence of temperature field type defined by the corresponding boundary conditions on the stress-strain shell state is illustrated. It is also demonstrated and discussed, how both fixation type along shell contour side and transversal load action modify shell stress-strain state and its stability within a temperature field.

2.1 Introduction Shells are members of many structures and machines in many timeline branches of technics. A wide spectrum of shells application is motivated by design of simultaneously strength and light constructions. It is clear that to achieve this requirement a real stress-strain state estimation is highly required. This is a reason for development of precise and economical computational techniques devoted to analysis of various constructions including shells and plates being their members. Nowadays an investigation of plates and shells in condition of high temperatures is very challenging, since the thermal stresses can lead to stability loss or collapse of constructions. Thermal stresses have been investigated for a long time. Duhamel (1837-1838) and Neuman (1841) derived equations governing thermoelastic stresses behaviour. In 1879 Hopkinson, and in 1900 Aliband constructed equations of thermoelastic equilibrium state in the form used nowadays. Famous scientists of last century have been involved in creation of mathematical fundamentals of heat transfer theory like Ostrogradskiy, Kelvin, Duhamel, Kirchhoﬀ, Maxwell, Stokes, Lam´e, Bossinesque, Rayleigh, Lamb, and others. Many fundamental results of heat transfer theory are included in monographs [292, 451]. In the book [678] practical methods devoted to solutions of heat transfer problems are reported. Some achievements of nonlinear heat transfer problems up to 1975 are given in the monograph [358]. It includes approximate analytical and numerical methods of nonlinear problems of energy pumping or investigation of physical media possessing mathematical model analogous to heat transfer processes. However, only 2D (two dimensional) heat transfer problems are solved and the finite diﬀerence method of higher order is not applied. Various aspect of heat transfer theory are discussed in monographs [193, 401]. It is worth noticing that in this period an essential contribution to theory of thermal stresses, and in particular into solution of stationary problems have been carried out by Central an Eastern Europe scientists like Galerkin, Dinnik, Lebedev, Maslov, Muschelishvili, Papkovitch, and others. Lebedev’s monograph [420] gives state-of-art of earlier period of thermal stresses problems development. Long time ago Maizel [453] applied Betty’s theorem to investigate heat transfer phenomena. Further development in this field can be traced through monographs [342, 471, 515].

2.1 Introduction

151

Analysis of 3D (three dimensional stationary) heat transfer equation is complicated. Majority of authors introduces series of simplified hypotheses of this equation. Problem devoted to thermomechanical shell state is often reduced to consideration of the corresponded 2D problem via representation of a being sought quantities in the form of series with respect to powers of the coordinate x3 . Temperature representation as a series of infinite length is proposed first by Malkin [456]. Analogous approach to solve the problems of theory of elasticity is proposed by Lurie [446]. In the reference [460], assuming a linear temperature distribution along the plate thickness, the corresponding heat transfer equations are obtained through temperature averaging along thickness. This method has been further extended into the case of unstationary regime for thin shells. Applying similar like assumptions with respect to temperature distribution along thickness, the analogous equations are obtained using the variational principle by Bolotin [124]. Polynomial representation of temperature distribution along plate and shell thickness is also applied by Danilovskaya [162]. Monograph [555] is devoted to fundamental problems of heat transfer and thermal elasticity of thin shells. It includes methods of solutions for statical, quasistatical and dynamical problems of thermoelasticity of plates and shells for diﬀerent cases of heating. Dynamical problems of linear theory of shells are considered in reference [552]. Computations of statical problems of theory of isotropic plates and shells with linear temperature distribution along thickness accounting of temperature dependence of linear expansion coeﬃcient and Young modulus are carried out in references [374, 535]. Axially symmetric geometrical nonlinear problems using both Bubnov and Ritz methods with a linear temperature distribution along a thickness are studied in references [247, 306]. Let us briefly analyse geometrically nonlinear problems of statics of plates and shells. Equations of plates and shell finite deflections behaviour using Kirchhoﬀ-Love hypothesis are derived in references [211, 308] and they are generalized into the case of curvilinear surfaces in references [461, 682]. Finite diﬀerence method with approximation O(h2 ) has been first applied to analyse geometrically nonlinear plates in reference [296]. Then a wide class of flexible plates problems are solved in references [345, 346, 347]. In the cited Kornishin’s works a higher order finite diﬀerence method to solve equations governing behaviour of flexible plates is used for the first time. Mathematical problems devoted to solvability of boundary value problems of nonlinear plates and shells theory have been initiated in references [187, 205, 329, 482, 688, 718]. Nowadays engineers working in various industrial branches, and particularly in civil, and electronic and electrotechnic engineering are focused on analysis of stressstrain states of plates and shells with various (sometimes hybrid types) clamping

152

2 Stability of Rectangular Shells within Temperature Field

along their contours with both mechanical and temperature excitations and accounting influence of heat sources and various temperature conditions. Both actions of transversal load and temperature field on the flexible rectangular shells are rarely investigated. Among others, the following unsolved questions are addressed in this chapter: (i) temperature field investigation directly from three dimensional stationary heat transfer equations; (ii) influence of various shell contour clamping; (iii) influence of heat sources and temperature conditions on stress-strain shell and its stability; (iv) derivations of compatibility conditions for orthotropic shallow shells.

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields 2.2.1 Problem formulation and assumptions Consider a shell with its middle surface bounded by a closed curve Γ. Let us attach to its middle surface an orthogonal system of curvilinear coordinates α, β (note that the coordinates do not need to overlap with main shell curvatures). It is assumed that Lam´e parameters A, B and curvature radii R 1 , R 12 , R 2 of the middle surface are continuous together with their first derivatives with respect to directions α, β. Contrary to main radiuses R1 , R2 (solid curves) notation, radiuses of a curvature in directions α, β are denoted by dashed curves. It is assumed that the function H(α, β) does not have first order discontinuities, and Hmax ≡ H0 is essentially smaller than the smallest main curvature radius Rmin . Furthermore, it is assumed that H0 /Rmin can be neglected in comparison to 1 (shells which satisfy this assumption are called thin shells). Let us denote displacements in direction α, β, γ by u, v, w, respectively. Since a shell has low stiﬀness in direction γ, deflection w is of the same order as H, and hence the inequality w/H 1 is not satisfied. Owing to this observation, a geometrically nonlinear theory is introduced, and instead of nonlinear terms w/H, derivatives of deflection with respect to coordinates appear. Therefore, the relation w ∼ H defines shell stiﬀness in various directions in an indirect way. All displacement components are taken significantly less than a characteristic shell dimension of middle surface. Let us introduce the initial deflection w0 (α, β). This function gives imperfections of initial form of the middle surface in prebuckling state. It is assumed that it is continuous together with its first and second derivatives with respect to α and β, and it is of thickness order. Deformations ε11 , ε12 , ε22 in middle surface are assumed to be small in comparison to one. However, it does not mean that a coupling between stresses and deformations should be linear. Variations of curvatures are characterized by the parameters κ11 , κ12 , κ22 . In this chapter technical theory of shells is used assuming that an influence of displacements u, v on the curvature variation parameters and first derivatives with

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

153

respect to deflection stress function with the multiplier AB/R1 R2 can be neglected in comparison to high order derivatives. It is assumed that shell material is non-homogeneous and anisotropic. The introduced theory is based on normal hypothesis. In typical formulation of straight normals, it is assumed that a length of normal elements is conserved. It means that deformations εγγ are neglected in comparison to one. In the introduced theory εγγ is approximately defined through condition of in-plane strain state δγγ = 0. A shell can be loaded through distributed (along its edges) perpendicular Q0n , normal T n0 and tangential T 0 forces in middle plane, and the bending moment Mn0 . Continuously distributed surface load Z is assumed to be normal to middle surface. In the case H const tangential pressure components normal to surfaces 1 ∂H γ = ±H(α, β) are neglected. In addition, derivatives A1 ∂H ∂d , B ∂β are assumed to be small in comparison to one. Volume forces in middle surface with potential U and temperature field T depending on three coordinates are accounted. 2.2.2 Fundamental relations Owing to straight normals hypothesis, shell deformation in its arbitrary point has the form ([555], p.73): eαα = ε11 + γκ11 ,

eββ = ε22 + γκ22 ,

eαβ = ε12 + 2γκ12 .

(2.1)

Deformations in middle surface [306, 685] read ε11 =

ε12

2 2 1 ∂A w 1 1 ∂w1 1 ∂u 1 1 ∂w + v− + − , A ∂α AB ∂β R1 2 A ∂α 2 A ∂α

2 2 1 ∂B w 1 1 ∂w1 1 ∂v 1 1 ∂w0 + u− + − , ε22 = B ∂β AB ∂α R2 2 B ∂β 2 B ∂β A ∂ u B ∂ v 2w 1 ∂w1 1 ∂w1 1 ∂w0 1 ∂w0 · − · , = + + + B ∂β A A ∂α B R12 A ∂α B ∂β A ∂α B ∂β

where: w1 = w + w0 . Parameters of middle surface curvature variation have te form [306]: 1 ∂A 1 ∂w 1 ∂ 1 ∂w − ≡ −w

αα , κ11 = − A ∂α A ∂α AB ∂β B ∂β 1 ∂B 1 ∂w 1 ∂ 1 ∂w − ≡ −w

ββ , κ22 = − B ∂β B ∂β AB ∂α A ∂α 1 ∂A 1 ∂w 1 ∂ 1 ∂w + ≡ −w

αβ . κ12 = − A ∂α B ∂β AB ∂β A ∂α

(2.2)

(2.3)

Note that these expressions are approximated even within a linear theory. In exact formulas also terms u/R1 , v/R1 are accounted. Neglection of these terms belongs

154

2 Stability of Rectangular Shells within Temperature Field

to one of the technical shell theory assumptions. In relations (2.3) nonlinear terms do not appear, since angles of rotation are neglected. Relation between stresses and deformations is presented in the form of generalized Hook’s law [14] eαα = a11 σαα + a12 σββ + a13 σγγ + a16 σαβ + αT11 T, eββ = a12 σαα + a22 σββ + a23 σγγ + a26 σαβ + αT22 T, eγγ = a13 σαα + a23 σββ + a33 σγγ + a36 σαβ + αT33 T, eαβ = a16 σαα + a26 σββ + a36 σγγ + a66 σαβ + αT12 T, eαγ = a45 σβγ + a55 σαγ + αT13 T,

eβγ = a44 σβγ + a45 σαγ + αT23 T,

(2.4)

where: αTii , αTij are temperature coeﬃcients of linear extension and shears of an anisotropic body. Solving first four equations of (2.4) with respect to σαα , σββ , σγγ , σαβ , one gets σαα = c11 eαα + c12 eββ + c13 eγγ + c16 eαβ − βo11 T, σββ = c12 eαα + c22 eββ + c23 eγγ + c26 eαβ − βo22 T, σγγ = c13 eαα + c23 eββ + c33 eγγ + c36 eαβ − βo33 T, σαβ = c16 eαα + c26 eββ + c36 εγγ + c66 eαβ − βo12 T, where:

c11 = m−1 a22 a33 a66 − a22 a236 − a223 a66 + 2a23 a26 a36 − a226 a33 ,

c22 = m−1 a11 a33 a66 − a11 a236 − a213 a66 + 2a13 a16 a36 − a216 a33 ,

c33 = m−1 a11 a22 a66 − a11 a226 − a212 a66 + 2a12 a16 a26 − a216 a22 ,

c66 = m−1 a11 a22 a33 − a11 a223 − a212 a33 + 2a12 a13 a23 − a213 a22 ,

c12 = m−1 a12 a236 − a12 a33 a66 + a13 a23 a66 − a13 a26 a36 + a16 a26 a33 − a16 a23 a36 ) , c13 = m−1 (a12 a23 a66 − a12 a26 a36 − a13 a22 a66 + a13 a226 + a16 a22 a36 − a16 a26 a23 , c26 = m−1 (a11 a23 a36 + a11 a33 a26 − a13 a12 a36 + a213 a26 + a16 a12 a33 − a16 a13 a23 , c16 = m−1 (a12 a26 a33 − a12 a23 a36 + a13 a22 a36 − a13 a23 a26 − a16 a22 a33 + a16 a223 ,

c36 = m−1 a11 a26 a23 − a11 a22 a36 + a212 a23 +

(2.5)

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

155

a16 a13 a22 − a12 a13 a26 − a16 a12 a23 ) , c23 = m−1 (a11 a36 a26 − a11 a23 a66 + a13 a12 a66 − a13 a16 a26 − a16 a12 a36 + a216 a23 , a11 a12 a13 a16 a a a a m = 12 22 23 26 , a13 a23 a33 a36 a16 a26 a36 a66 βo11 = αT11 c11 + αT22 c12 + αT33 c13 + αT12 c16 , βo22 = αT11 c12 + αT22 c22 + αT33 c23 + αT12 c26 , βo33 = αT11 c13 + αT22 c23 + αT33 c33 + αT12 c36 , βo12 = αT11 c16 + αT22 c26 + αT33 c36 + αT12 c66 .

(2.6)

Note that in relation (2.4) ai j = ai j (α, β, γ). In theory of anisotropic shells a transformation of material coeﬃcients into a new coordinate system α , β , γ is required, assuming that their values are known in another coordinates system α, β, γ. Let a structure of material in each point body point has only one plane of elastic symmetry, parallel to the middle surface, which coincides with the shifted surfaces αβ and α β . Let the coordinates α, β, γ and α , β , γ be identical after rotation in amount of angle ϕ around general axis γ = γ . The following transformation formulas are obtained [424]: a 11 = a11 cos4 ϕ + (2a12 + a66 ) sin2 ϕ cos2 ϕ + a22 sin4 ϕ+ (a16 cos2 ϕ + a26 sin2 ϕ)sin2ϕ, a 22 = a11 sin4 ϕ + (2a12 + a66 ) sin2 ϕ cos2 ϕ + a22 cos4 ϕ− (a16 sin2 ϕ + a26 cos2 ϕ) sin 2ϕ, a 12 = a12 + (a11 + a22 − 2a12 − a66 ) sin2 ϕ cos2 ϕ+ 1 (a16 − a26 ) cos 2ϕsin2ϕ, 2 a 66 = a66 + 4(a11 + a22 − 2a12 − a66 ) sin2 ϕ cos2 ϕ−

a 16

2(a16 − a26 ) cos 2ϕsin2ϕ, 1 2 2 = a22 sin ϕ − a11 cos ϕ + a12 + a66 cos 2ϕ sin 2ϕ+ 2

a16 cos2 ϕ(cos2 ϕ − 3 sin2 ϕ) + a26 sin2 ϕ(3 cos2 ϕ − sin2 ϕ), 1

2 2 a26 = a22 cos ϕ − a11 sin ϕ − a12 + a66 cos 2ϕ sin 2ϕ+ 2 a16 sin2 ϕ(3 cos2 ϕ − sin2 ϕ) + a26 cos2 ϕ(cos2 ϕ − 3 sin2 ϕ),

156

2 Stability of Rectangular Shells within Temperature Field

a 13 = a13 cos2 ϕ + a36 sin ϕ cos ϕ + a23 sin2 ϕ, a 23 = a13 sin2 ϕ − a36 sin ϕ cos ϕ + a23 cos2 ϕ, a 36 = (a23 − a13 ) sin 2ϕ + a36 cos 2ϕ,

a 33 = a33 .

(2.7)

In a particular case, if a body is orthotropic and old axes α, β, γ are the main axes of elasticity, i.e. they are orthogonal to planes of elastic symmetry, then formulas describing elastic constants (2.7) are more simplified owing to introduction of the technical elasticity constants Ei , Gi j , νi j [424]: a 11 =

1 cos4 ϕ 2ν12 sin4 ϕ sin2 ϕ cos2 ϕ + + − , E1 G12 E1 E2

1 sin4 ϕ 2ν12 cos4 ϕ sin2 ϕ cos2 ϕ + + − , E1 G12 E1 E2 1 1 2ν12 1 ν12 a 12 = sin2 ϕ cos2 ϕ − + + − , E1 E2 E1 G12 E1 1 1 2ν12 1 1

a66 = 4 sin2 ϕ cos2 ϕ + + + − , E1 E2 E1 G12 G12 2 1 sin ϕ cos2 ϕ 2ν12

+ cos 2ϕ sin ϕcosϕ, − − a16 = 2 E2 E1 G12 E1 2 1 cos ϕ sin2 ϕ 2ν12

− cos 2ϕ sin ϕ cos ϕ, − − a26 = 2 E2 E1 G12 E1 ν23 ν23 ν13 ν13 2 2

2 2 =− sin ϕ + cos ϕ , a23 = − cos ϕ + sin ϕ , E2 E1 E2 E1 1 ν13 ν23 a 33 = sin 2ϕ. (2.8) , a 36 = − E3 E1 E2 a 22 =

a 13

Owing to σγγ = 0, the third equation of (2.5) yields σαα = B11 eαα + B12 eββ + B16 eαβ − β11 T, σββ = B12 eαα + B22 eββ + B26 eαβ − β22 T, σαβ = B16 eαα + B26 eββ + B66 eαβ − β12 T,

(2.9)

where: B11 = c11 −

c213 , c33

B26 = c26 −

B12 = c12 −

c23 c36 , c33

c13 c29 , c33

B22 = c22 −

c223 , c33

B16 = c16 −

c13 c36 , c33

B66 = c66 −

c236 , c33

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

β11 = βo11 −

c13 o β , c33 33

β22 = βo22 −

c23 o β , c33 33

β12 = βo12 −

157

c36 o β . c33 33

Substituting (2.1) into (2.9) gives σαα = B11 ε11 + B12 ε22 + B16 ε12 + γ(B11 κ11 + B12 κ22 + 2B16 κ12 ) − β11 T, σββ = B12 ε11 + B22 ε22 + B26 ε12 + γ(B12 κ11 + B22 κ22 + 2B26 κ12 ) − β22 T, σαβ = B16 ε11 + B26 ε22 + B66 ε12 + γ(B16 κ11 + B26 κ22 + 2B66 κ12 ) − β12 T.

(2.10)

Integrating stresses with respect to γ, the following middle surface forces are obtained T 11 = D11,0 ε11 + D12,0 ε22 + D16,0 ε12 + D11,1 κ11 + D12,1 κ22 + 2D16,1 κ12 − t11,0 , T 22 = D12,0 ε11 + D22,0 ε22 + D26,0 ε12 + D12,1 κ11 + D22,1 κ22 + 2D26,1 κ12 − t22,0 , T 12 = D16,0 ε11 + D26,0 ε22 + D66,0 ε12 + D16,1 κ11 + D26,1 κ22 + 2D66,1 κ12 − t12,0 . (2.11) Multiplying stresses by γ and integrating over shell thickness, the following banding and rotation moments are obtained M11 = D11,1 ε11 + D12,1 ε22 + D16,1 ε12 + D11,2 κ11 + D12,2 κ22 + 2D16,2 κ12 − t11,1 , M22 = D12,1 ε11 + D22,1 ε22 + D26,1 ε12 + D12,2 κ11 + D12,1 κ22 + 2D26,2 κ12 − t22,1 , M12 = D16,1 ε11 + D26,1 ε22 + D66,1 ε12 + D16,2 κ11 + D26,2 κ22 + 2D66,2 κ12 − t12,1 . (2.12) Positive direction of forces and moments are shown in monograph [686]. Coeﬃcients occurred in relations (2.11), (2.12) are the functions of α, β, and they are coupled through integrals H Di j,k (α, β) =

Bi j γk dγ,

i, j = 1, 2, 6,

k = 0, 1, 2.

(2.13)

−H

Temperature components of forces and moments read H ti j,k =

βi j T γk dγ,

i, j = 1, 2,

k = 0, 1.

(2.14)

−H

Formulas (2.11) yield middle surface deformations ε11 = A11 T 11 + A12 T 22 + A16 T 12 − d11 κ11 − d12 κ22 − d16 κ12 + T 1 , ε22 = A12 T 11 + A22 T 22 + A26 T 12 − d21 κ11 − d22 κ22 − d26 κ12 + T 2 , ε12 = A16 T 11 + A26 T 22 + A66 T 12 − d61 κ11 − d62 κ22 − d66 κ12 + T 12 , where coeﬃcients Ai j and di j have the form

(2.15)

158

2 Stability of Rectangular Shells within Temperature Field 2 A11 = m−1 1 (D22,0 D66,0 − D26,0 ),

A12 = m−1 1 (D16,0 D26,0 − D12,0 D66,0 ),

A16 = m−1 1 (D12,0 D26,0 − D16,0 D22,0 ),

2 A22 = m−1 1 (D11,0 D66,0 − D16,0 ),

A26 = m−1 1 (D12,0 D16,0 − D11,0 D26,0 ),

2 A66 = m−1 1 (D11,0 D22,0 − D12,0 ),

d11 = A11 D11,1 + A12 D12,1 + A16 D16,1 , d12 = A11 D12,1 + A12 D22,1 + A16 D16,1 , & ' d16 = 2 A11 D16,1 + A12 D26,1 + A16 D66,1 , d21 = A12 D11,1 + A22 D12,1 + A26 D16,1 , d22 = A12 D12,1 + A22 D22,1 + A26 D26,1 , & ' d26 = 2 A12 D16,1 + A22 D26,1 + A26 D66,1 , d61 = A16 D11,1 + A26 D12,1 + A66 D16,1 , d62 = A16 D12,1 + A26 D22,1 + A66 D26,1 , & ' d66 = 2 A16 D16,1 + A26 D26,1 + A66 D66,1 , T 1 = A11 t11,0 + A12 t22,0 + A16 t12,0 , T 2 = A12 t11,0 + A22 t22,0 + A26 t12,0 , T 12 = A16 t11,0 + A26 t22,0 + A66 t12,0 , D11,0 D12,0 D16,0 m1 = D12,0 D22,0 D26,0 . D16,0 D26,0 D66,0

(2.16)

Owing to the following relations, the stress function in the middle surface [306]

T 11 = Fββ + U,

T 22 = Fαα + U,

T 12 = −Fαβ ,

(2.17)

are introduced. Function U represents a potential of the volume forces having the following projections X=−

1 ∂ U, A ∂α

Y=−

1 ∂ U, B ∂β

into the axes α, β. Recall known operators in curvilinear coordinates [306]: ∂ A ∂ 1 ∂ B ∂ + , ∆= AB ∂α A ∂α ∂β B ∂β ∂ 1 ∂ ∂ 1 ∂ ∂ 1 A ∂ 1 ∂ 1 B ∂ + + + . ∆k = AB ∂α R 2 A ∂α ∂α R12 ∂β ∂β R12 ∂α ∂β R 1 B ∂β

(2.18)

(2.19)

(2.20)

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

159

2.2.3 Variational and diﬀerential equations Owing to a principle of virtual displacements, the variation of full energy δ of a deformed shell is equal to zero in its equilibrium state δ≡ δV + δu +δc = 0,

(2.21)

where: δV - external forces work variation; δu - potential energy variation, occurred in result of bending deformation; δc - energy variation yielded by the deformation in middle surface. Variation of external forces work reads [306]

(X · u + Y · v + Z · w) − δ T no un + T lo ul − Mno w n + Qon w dl , (2.22) δV = −δ where: un , ul - normal and tangent edge displacement components in the middle surface; w u - derivative with respect to deflection in normal direction to shell edge. It is assumed that the loads T no and T lo do not appear on edge intervals, where w and w n are not defined. Potential energy variation (deflection [306]) has the form (M11 δκ11 + M22 δκ22 + 2M12 δκ12 ) dS , δu = (2.23) and variation of middle surface deformation energy is given by (T 11 δε11 + T 22 δε22 + T 12 δε12 ) dS = δc ≡ δ

(T 11 ε11 + T 22 ε22 + T 12 ε12 ) dS− (ε11 δT 11 + ε22 δT 22 + ε12 δT 12 ) dS . (2.24)

This transformation allows for obtaining variational hybrid type equation (both deflection and stress function are variated). Substituting (2.22), (2.23) and (2.24) into (2.21) we get (T 11 ε11 + T 22 ε22 + T 12 ε12 − X · u − Y · v − Z · w) dS + δ≡ δ (M11 δκ11 + M22 δκ22 + 2M12 δκ12 − ε11 δT 11 − ε22 δT 22 − ε12 δT 12 ) dS − δ

T no un + T lo ul − Mno w n + Qon w dl = 0.

(2.25)

Consider the expression (2.25) as the sum of two terms δ= δ1 +δ2 = 0, where:

(2.26)

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2 Stability of Rectangular Shells within Temperature Field

δ1 = δ

o o (T 11 ε11 + T 22 ε22 + T 12 ε12 − Xu − Yv) dS − T n un + T l ul dl ,

δ2 =

(M11 δκ11 + M22 δκ22 + 2M12 δκ12 − ε11 δT 11 −

ε22 δT 22 − ε12 δT 12 ) dS − δ

ZwdS −

&

Mno w n

−

' Qon w dl

.

(2.27)

Substituting expression (2.2) instead of deformations into δ1 and integrating by parts we get [306] 1 ∂ 1 ∂ 2 ∂B (BT 11 ) − T 22 + δ1 = −δ A T 12 + X udS − AB ∂α ∂α A ∂β 1 ∂ 2 1 ∂ ∂A (AT 22 ) − T 11 + δ B T 12 + Y vdS + AB ∂β ∂β B ∂α ⎤ ⎡ ⎡ ⎧ ⎪ ⎢⎢⎢ 1 ∂w 2 2 ∂w0 1 ∂w ⎥⎥⎥ ⎢⎢⎢ 1 ∂w 2 ⎪ 1 ⎨ ⎥ ⎢ ⎢ ∂ + + T ⎢ ⎥ + T 22 ⎢⎣ ⎪ ⎪ ⎩ 11 ⎣ A ∂α 2 A ∂α A ∂α ⎦ B ∂β

1 ∂w 1 ∂w0 1 ∂w 1 ∂w 1 ∂w 1 ∂w0 2 + 2T 12 + + B ∂β B ∂β A ∂α B ∂β A ∂α B ∂β T 11 1 ∂w 1 ∂w0 T 12 T 22 dS − δ −2 + wdS + B ∂β A ∂α R 1 R12 R2 $

% ' & δ T 11 − T no un + T 12 − T l0 ul dl.

(2.28)

∂F ∂F , RAB are neThe forces are governed by formulas (2.17), the terms RAB 1 R2 ∂α 1 R2 ∂β AB glected in comparison to higher derivatives of F (except of the multiplier l1 l2 ) and the following boundary conditions are attached

T 11 = T no ,

T 12 = T lo .

(2.29)

One may observe that first and second integrals over space and contour in δ1 are equal to zero. Note that two first underintegral expressions in (2.28) represent multiplication of left hand sides of the equilibrium equations in a middle surface within the technical theory of displacements. In these equations the transversal forces do not appear, owing to neglection of terms u/R1 and v/R1 standing in expression describing a curvature [306]. Recall that the neglection of first derivatives of F in comparison with the multiplier AB/R1 R2 is the second assumption of the technical shell theory. If at least one of main curvature radiuses is infinite, than an error does not appear. Applying integrations by parts to the first term (2.28), and the Gauss formula to the fourth term, the following is obtained [306]:

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

1 δ1 = − δ 2

161

1 1 2∆k F + L2 (w2 , F) − Φ(w2 ) + 2 + U wdS + R1 R2 1 ∂w ∂w δ T 11 + T 12 wdl, (2.30) 2 ∂n ∂l

where: w2 = w + 2w0 ,

Φ(ϕ) = X

1 ∂ϕ 1 ∂ϕ +Y − U∆ϕ, A ∂α B ∂β

L2 (ϕ, ψ) = ϕ

αα ψ

ββ − 2ϕ

αβ ψ

αβ + ϕ

ββ ψ

αα . Integral along a contour in (2.30) is equal to zero, since in the case of clamping w = 0, and on the free side T n = T e = 0 owing to (2.29). It should be emphasized that the physical material parameters do not appear in (2.30). Let us substitute into relations for moments (2.12) the deformation values in the middle surface (2.15), expressed via stress function and curvatures, in the form M11 = D11 T 11 + D12 T 22 + D16 T 12 − d11 κ11 − d12 κ22 − d16 κ12 + T 1∗ , M22 = D21 T 11 + D22 T 22 + D26 T 12 − d21 κ11 − d22 κ22 − d26 κ12 + T 2∗ , ∗ M12 = D16 T 11 + D62 T 22 + D66 T !2 − d61 κ11 − d62 κ22 − d66 κ12 + T 12 ,

where: D11 = D11,1 A11 + D12,1 A12 + D16,1 A16 , D12 = D11,1 A12 + D12,1 A22 + D16,1 A26 , D16 = D11,1 A16 + D12,1 A26 + D16,1 A66 , D21 = D12,1 A11 + D22,1 A12 + D26,1 A16 , D22 = D12,1 A12 + D22,1 A22 + D26,1 A26 , D26 = D12,1 A16 + D22,1 A26 + D26,1 A66 , d11 = D11,1 d11 + D12,1 d21 + D16,1 d61 − D11,2 , d12 = D11,1 d12 + D12,1 d22 + D16,1 d62 − D12,2 , d16 = D11,1 d16 + D12,1 d26 + D16,1 d66 − 2D16,2 , d21 = D12,1 d11 + D22,1 d21 + D26,1 d61 − D12,2 , d22 = D12,1 d12 + D22,1 d22 + D26,1 d62 + D22,2 , d26 = D12,1 d16 + D22,1 d26 + D26,1 d66 − D26,2 , T 1∗ = D11,1 T 1 + D12,1 T 2 + D16,1 T 12 − t11,1 , T 2∗ = D12,1 T 1 + D22,1 T 2 + D26,1 T 12 − t22,1 , ∗ = D16,1 T 1 + D26,1 T 2 + D66,1 T 12 − t12,1 , T 12

D61 = D16,1 A11 + D26,1 A12 + D66,1 A16 ,

(2.31)

162

2 Stability of Rectangular Shells within Temperature Field

D62 = D16,1 A12 + D26,1 A22 + D66,1 A26 , D66 = D16,1 A16 + D26,1 A26 + D66,1 A66 , d61 = D16,1 d11 + D26,1 d21 + D66,1 d61 − D16,2 , d62 = D16,1 d12 + D26,1 d22 + D66,1 d62 − D26,2 , d66 = D16,1 d16 + D26,1 d26 + D66,1 d66 − 2D66,2 .

(2.32)

Moments (2.31) and deformations in the middle surface (2.15) are substituted into δ2 in the form (2.27) to yield [R(F, w) − R(w, w) − R(F, F)] dS − δ δ2 = ZwdS + δ

∂w Mno ∂n

−

Qon w

dl +

N T − M T dS ,

(2.33)

where: R(F, w) = (D11 T 11 + D12 T 22 + D16 T 12 )δk11 + (D21 T 11 + D22 T 22 + D26 T 12 )δk22 + 2(D61 T 11 − D62 T 22 + D66 T 22 )δk12 − (d11 k11 + d12 k22 + d16 k12 )δT 11 − (d21 k11 + d22 k22 + d26 k12 )δT 22 − (d61 k11 + d62 k22 + d66 k12 )δT 12 , R(w, w) = (d11 k11 + d12 k22 + d16 k12 )δk11 + (d21 k11 + d22 k22 + d26 k12 )δk22 + 2(d61 k11 + d62 k22 + d66 k12 )δk12 , R(F, F) = (A11 T 11 + A12 T 22 + A16 T 12 )δT 11 + (A12 T 11 + A22 T 22 + A26 T 12 )δT 22 + (A16 T 11 + A26 T 22 + A66 T 12 )δT 12 ,

(2.34)

M = T 1 δT 11 + T 2 δT 22 + T 12 δT 12 , T

∗ N T = T 1∗ δk11 + T 2∗ δk22 + 2T 12 δk12 ,

(2.35)

Owing to substitution of δ1 and δ2 into (2.26), the being sought variational equation is obtained. Observe that operation of function variations having physical parameters is not used during variations of the functions. The being sought variational equation is of a hybrid type, since both varied and being sought functions w and F appear independently. The obtained variational equation generalizes equations obtained in references [212, 686]. In the case of homogeneous and physically linear shells, the hybrid types variational equations are reported by Alumiae [13]. In the case of isotropic non-homogeneous isotropic shells with physical and geometrical non-linearities the hybrid type variational equation has been obtained by Kantor [306], whereas for the case of elastic-plastic material including loading and second order plastic deformations it has been derived by Krys’ko [369]. Variational background of theory of thin plates and shells is given in references [212, 533].

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

163

It is worth noticing that the obtained variational equation can be applied during calculations of composite shells, since the first order discontinuities of Young modulus and Poisson’s coeﬃcients do not violate existence of integrals Di j,k . In the case of rectangular coordinates x, y, z the Lam´e coeﬃcients A = B = 1. For a given homogeneous anisotropic shell material with heat and elastic symmetry plane, perpendicular to middle shell surface and orthogonal to axis z, the coeﬃcients in (2.4) are constant. Owing to the Kirchhoﬀ-Love hypothesis and relation ezz = 0, the generalized Hook’s law can be rewritten in the form e xx = a11 σ xx + a12 σyy + a16 σ xy + αT11 T (x, y, z), eyy = a12 σ xx + a22 σyy + a26 σ xy + αT22 T (x, y, z), e xy = a16 σ xx + a26 σyy + a66 σ xy + αT12 T (x, y, z).

(2.36)

Solving (2.36) with respect to the stresses σ xx , σyy , σ xy , the following equations are obtained σ xx = B11 e xx + B12 eyy + B16 e xy − β11 T, σyy = B12 e xx + B22 eyy + B26 e xy − β22 T, σ xy = B16 e xx + B26 eyy + B66 e xy − β12 T, where:

2 B11 = m−1 2 (a22 a66 − a26 ),

(2.37)

B12 = m−1 2 (a26 a16 − a12 a66 ),

B16 = m−1 2 (a12 a26 − a22 a16 ),

2 B22 = m−1 2 (a11 a66 − a16 ),

B26 = m−1 2 (a12 a16 − a11 a26 ),

2 B66 = m−1 2 (a11 a22 − a12 ),

β11 = αT11 B11 + αT22 T B12 + αT12 B16 , β12 = αT11 B16 + αT22 B26 + αT12 B66 , β22 = αT11 B12 + αT22 B22 + αT12 B26 , a11 a12 a16 m2 = a12 a22 a26 . a16 a26 a66 Owing to hypothesis of straight normals, the relations similar to (2.1) occur, namely e xx = ε11 + zκ11 , eyy = ε22 + zκ22 , e xy = ε12 + 2zκ12 , where: ε11 =

2 ∂u 1 ∂w − κ1 w + , ∂x 2 ∂x ε12 =

ε22 =

2 ∂v 1 ∂w − κ2 w + , ∂y 2 ∂y

∂u ∂v ∂w ∂w + + . ∂y ∂x ∂x ∂y

Substituting these relations into multiplied by z equation (2.37), and into (2.36), and integrating them with respect to z from −H to H, the following relations governing deformations, stresses and moments in the shell middle surface are obtained

164

2 Stability of Rectangular Shells within Temperature Field

1

a11 T 11 + a12 T 22 + a16 T 12 + αT11 T N , 2H 1

= a12 T 11 + a22 T 22 + a26 T 12 + αT22 T N , 2H 1

= a16 T 11 + a26 T 22 + a66 T 12 + αT12 T N , 2H

ε11 = ε22 ε12 H M11 ≡

σ xx zdz =

2H 3 (B11 κ11 + B12 κ22 + 2B16 κ12 ) − β11 T M , 3

σyy zdz =

2H 3 (B12 κ11 + B22 κ22 + 2B26 κ12 ) − β22 T M , 3

σ xy zdz =

2H 3 (B16 κ11 + B26 κ22 + 2B66 κ12 ) − β12 T M , 3

−H

H M22 ≡ −H

H M12 ≡ −H

where: κ11 = − H T 11 ≡ −H

∂2 w , ∂x2

κ22 = −

∂2 F σ xx dz = 2 , ∂y

∂2 w , ∂y2

κ12 = − H

T 22 ≡

σ xy dx = − −H

H TN ≡

∂2 F , ∂x2

∂2 F , ∂x∂y H

T (x, y, z)dz, −H

σyy dz = −H

H T 12 ≡

∂2 w , ∂x∂y

TM ≡

zT (x, y, z)dz. −H

The obtained relations are substituted into (2.26) instead of deformations, stresses and moments, and the following variational equation in rectangular coordinates system for homogeneous anisotropic shell subjected to an action of transversal load q and temperature field T (x, y, z) is obtained ∂4 w ∂4 w ∂4 w (2H)3 B11 4 + 2(B12 + 2B66 ) 2 2 + B22 4 + δ= 12 ∂x ∂x ∂y ∂y ∂4 w ∂4 w ∂2 T M ∂2 T M ∂2 T M − + β11 4B16 3 + 4B26 + β22 + 2β12 3 2 2 ∂x∂y ∂x ∂y ∂x∂y ∂x ∂y ∂2 F ∂2 F ∂ 2 w ∂2 F ∂ 2 w ∂2 F ∂ 2 w ∂2 F − q δwdxdy− k1 2 − k2 2 − 2 2 − 2 + 2 ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂y ∂x2

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

∂4 F 1 ∂4 F ∂4 F ∂4 F a11 4 + (2a12 + a66 ) 2 2 + a22 4 − 2a16 − 2H ∂y ∂x ∂y ∂x ∂x∂y3 2 2 2 2 2 ∂ w∂ w ∂4 F T ∂ TN T ∂ TN T ∂ TN + 2H − α12 − 2a26 3 + α11 2 + α22 ∂x∂y ∂x ∂y ∂y ∂x2 ∂x2 ∂y2 ⎞⎤ 2 2 2 ∂w ∂2 w ∂2 w ⎟⎟⎟⎟⎥⎥⎥⎥ ∂ F ∂u ∂2 F ∂v + − + k1 2 2 ⎟⎠⎥⎦ δFdxdy + δ ∂x∂y ∂y ∂x ∂y2 ∂x ∂x2 ∂y l2 ∂ (2H)3 ∂2 F ∂u ∂v ∂2 w ∂2 w dxdy + − B11 2 + B12 2 + ∂x∂y ∂y ∂x 12 ∂x ∂x ∂y

0

∂ w ∂2 w ∂2 w ∂2 w ∂ δw + B11 2 + B12 2 + 2B16 δw− ∂x∂y ∂x∂y ∂x ∂x ∂y ∂ ∂2 w ∂2 w ∂2 w ∂ B16 2 + B26 2 + 2B66 δw + β11 T M δw− 2 ∂y ∂x∂y ∂x ∂x ∂y l1 ∂T M ∂T M ∂w ∂2 F ∂w ∂2 F β11 δw − 2β12 δw + δw δw − dy+ ∂x ∂y ∂x ∂y2 ∂y ∂x∂y x=0 2

2B16

l1

(2H)3 ∂2 w ∂2 w ∂2 w ∂ − B12 2 + B22 2 + 2B26 δw+ 12 ∂y ∂x∂y ∂x ∂y

0

∂ ∂2 w ∂2 w ∂2 w ∂ ∂2 w δw − 2 B16 2 + B12 2 + B22 2 + 2B26 ∂x∂y ∂y ∂x ∂x ∂y ∂x ∂2 w ∂2 w ∂T M ∂ δw − β22 δw + β22 T M δw− B26 2 + 2B66 ∂x∂y ∂y ∂y ∂y l2 ∂w ∂2 F ∂2 F ∂w ∂T M 2β12 δw + δw δw − dx+ ∂x ∂y ∂x2 ∂x∂y ∂x y=0 l1 l2 (2H)3 ∂2 w ∂2 w ∂2 w + B16 2 + B26 2 + 2B66 δw + 2β12 T M δw 6 ∂x∂y ∂x ∂y x=0 y=0

l2 0

1 ∂ ∂2 F ∂2 F ∂2 F T a12 2 + a22 2 − a26 + α22 T N δF− 2H ∂x ∂x∂y ∂y ∂x

∂ ∂ ∂2 F ∂2 F ∂2 F ∂2 F T + α δF − a16 2 + + a − a T 22 26 22 N 2 2 ∂x∂y ∂x ∂y ∂y ∂x ∂y 2 1 ∂w ∂ ∂2 F ∂2 F ∂w + αT12 T N δF + δF + k2 δF− a26 2 − a66 ∂x∂y 2 ∂y ∂x ∂x ∂x

a12

165

166

2 Stability of Rectangular Shells within Temperature Field

∂2 w ∂w ∂ δF k2 w δF + 2 ∂x ∂y ∂x

l1

l1 dy +

x=0

0

∂ F ∂ F ∂2 F ∂2 F ∂ T T +α T N δF − a16 2 +a26 −a66 +α T N δF− a16 ∂x∂y 11 ∂x ∂x∂y ∂x∂y 12 ∂y 2 ∂ ∂2 F ∂2 F ∂2 F 1 ∂w ∂ T a11 2 + a12 2 − a16 + α11 T N δF + δF+ ∂x∂y ∂y 2 ∂x ∂y ∂y ∂x l2 ∂2 w ∂w ∂w ∂ 1 ∂w ∂2 F k1 δF − k1 w δF − k1 w δF + 2 δF a16 2 + dx + ∂y ∂y ∂y 2H ∂x ∂y ∂y y=0 l1 l2 ∂w ∂w ∂2 F ∂2 F = 0. + αT12 T n δF − δF (2.38) a26 2 − α66 ∂x∂y ∂x ∂y ∂x x=0 y=0 2

1 ∂ ∂2 F ∂2 F a11 2 + a12 2 − 2H ∂y ∂y ∂x

2

Comparing the coeﬃcients by δw and δF for x, y ∈ G {0 < x < l1 , 0 < y < l2 } in the variational equations (2.38), the following system of nonlinear diﬀerential equations is yielded ∂4 w ∂4 w (2H)3 ∂4 w B11 4 + 2(B16 + 2B66 ) 2 2 + B22 4 + −∇2κ F − L(w, F) − q + 12 ∂x ∂x ∂y ∂y ∂4 w ∂4 w − Ψ1 (x, y) = 0, 4B16 3 + 4B26 ∂x ∂y ∂x∂y3 1 1 ∂4 F ∂4 F ∂4 F 2 a22 4 + a11 4 + (2a12 + a66 ) 2 2 − ∇κ w + L(w, w) + 2 2H ∂x ∂y ∂x ∂y ∂4 F ∂4 F − 2a26 3 + Ψ2 (x, y) = 0, (2.39) 2a16 3 ∂x∂y ∂x ∂y where:

H ∂2 T ∂2 T ∂2 T Ψ1 (x, y) = dz, z β11 2 + β22 2 + 2β12 ∂x∂y ∂x ∂y −H

H Ψ2 (x, y) = −H

∂2 T αT11 2 ∂y

+

∂2 T αT22 2 ∂x

−

αT12

∂2 T dz, ∂x∂y

∂ 2 w ∂2 F ∂ 2 w ∂2 F ∂ 2 w ∂2 F , + − 2 ∂x∂y ∂x∂y ∂x2 ∂y2 ∂y2 ∂x2 2 2 ∂w 1 ∂ 2 w ∂2 w L(w, w) = 2 2 − , 2 ∂x∂y ∂x ∂y

L(w, F) =

∇2κ F = κ1

∂2 F ∂2 F + κ , 2 ∂y2 ∂x2

∇2κ w = κ1

∂2 w ∂2 w + κ . 2 ∂y2 ∂x2

(2.40)

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

167

2.2.4 Boundary and compatibility conditions Owing to various types of supports in real shell structures, a rich set of mathematical models of boundary value problems appear. A choice of boundary conditions has essential influence on the choice of initial diﬀerential equations (either in hybrid form or with respect to displacements). First, some of the boundary conditions for diﬀerential equations (2.39) are briefly recalled. 1. Free support of shell edges

2. Sliding clamping a)

b)

w = M11 = T 11 = T 12 = 0

for x = 0, l1 ;

w = M22 = T 22 = T 12 = 0

for y = 0, l2 .

w=

∂w = 0, ∂x

T 11 = ε22 = 0,

for x = 0, l1 ;

w=

∂w = 0, ∂y

T 22 = ε11 = 0,

for y = 0, l2 .

w=

∂w = 0, ∂x

T 11 = T 12 = 0,

for x = 0, l1 ;

w=

∂w = 0, ∂y

T 11 = T 12 = 0,

for y = 0, l2 .

(2.41)

(2.42)

(2.43)

3. Free support on flexible non-extended (non-compressed) in tangential plane ribs w = M11 = T 11 = ε22 = 0 w = M22 = T 22 = ε11 = 0

for x = 0, l1 ; for y = 0, l2 .

(2.44)

4. Hybrid type boundary condition for for

x = 0, l1 x = 0, l2

- free support; - sliding clamping.

(2.45)

Note that various combination of boundary conditions (2.41)–(2.44) are possible not only along whole contour, but also along each of shell sides. In reference [592] on an example of bi-harmonic equation governing behaviour of thin homogeneous isotropic plate for small deflections, coupling conditions for fourth order equations and all possible boundary conditions are formulated, and the variational technique is applied. A similar like approach to the system of nonlinear diﬀerential equations (2.39) of flexible shallow homogeneous anisotropic shells is applied now. So far, in order to get the variational equation (2.38), a fixation type of a shell boundary has been not accounted. Accounting of one the fixation ways for x = y = 0, it is assumed that its energy is

168

2 Stability of Rectangular Shells within Temperature Field

1 = + 2

⎫ 2 2 2 2 2 l2 ⎧ ⎪ ⎪ ⎪ ⎪ ∂ w ∂w ∂w ⎨ 2⎬ + C1 + α1 + β1 w ⎪ B1 ⎪ ⎪ ⎪ 2 ⎩ ⎭ ∂x∂y ∂x ∂y 0

⎫ 2 2 2 2 2 l1 ⎧ ⎪ ⎪ ⎪ ⎪ ∂ w ∂ w ∂w ⎨ 2⎬ + C2 + α2 + β2 w ⎪ B2 ⎪ ⎪ ⎪ 2 ⎩ ⎭ ∂x∂y ∂y ∂x

1 2

dy+ x=0

dx+

y=0

0

1 2

⎫ 2 2 2 l2 ⎧ ⎪ ⎪ ⎪ ∂F ∂ F ∂F ∂F ⎪ ⎨ ⎬ + K1 + 2L1 A ⎪ ⎪ ⎪ 2 ⎩ 1 ∂y ⎭ ∂y ∂x ⎪ ∂y 0

1 2

dy+ x=0

⎫ 2 2 2 l1 ⎧ ⎪ ⎪ ⎪ ∂F ∂ F ∂F ∂F ⎪ ⎨ ⎬ + K2 + 2L2 A ⎪ ⎪ ⎪ 2 ⎩ 2 ∂x ⎭ ∂x ∂y ⎪ ∂x 0

dx,

(2.46)

y=0

where the coeﬃcients Bi , Ci , αi , βi , Ai , Ki , Li (i = 1, 2) have fully defined values depending on boundary conditions type, and corresponds to variational equation (2.38) δ= 0. Owing to an account of external forces work on fixation curve, the functional corresponding to the problem (2.39) with non-homogeneous boundary conditions takes the form (2.47) ˜ = +1 , where: l2 1 = −

∂w M1o (y) ∂x

+

M2o (x) M1o (y),

dy − x=0

0

e1

l2

Qo1 (y)w

∂w + Qo2 (x)w ∂y

y=0

∂F + qo1 (y)F ∂x

0

e1 dx −

mo1 (y) mo2 (x)

0 0 o o o o o o M2 (x), Q1 (y), Q2 (x), m1 (y), m2 (x), q1 (y), qo2 (x)

∂F + qo2 (x)F ∂y

dy− x=0

dx, y=0

are external forces acting on the boundary, and is defined with the help of (2.46). Let us compute first variation of the functional (2.47) with respect to w and compare it to zero. Applying variational equation (2.38) and carrying out integration by parts, only terms related to the edge x = 0 are given (2H)3 12

l2

∂ ∂2 w ∂2 w ∂2 w B11 2 + B12 2 + 2B16 δw− ∂x ∂x∂y ∂x ∂y

0

∂ w ∂2 w ∂2 w ∂ ∂2 w ∂2 w ∂ δw + 2 B + B + 2B + B + 12 16 16 26 ∂x∂y ∂x ∂y ∂x2 ∂y2 ∂x2 ∂y2 2

B11

l2 ∂2 w ∂T M ∂ ∂T M δw δw + β11 T M δw − 2β12 δw+ β11 dy − 2B66 ∂x∂y ∂x ∂x ∂y x=0 0

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

l2

∂2 F ∂w ∂w ∂2 F δw − δw ∂x∂y ∂y ∂x ∂y2

β1 wδw + α1

∂ ∂w δw ∂x ∂x

dy − x=0

B1

169

∂4 w ∂3 w ∂ δw+ δw − C1 4 ∂y ∂x∂y2 ∂x

0

l2 x=0

−M1o (y)

dy +

∂ δw − Qo1 (y)δw ∂x

0

dy. x=0

∂ ∂x δw

and δw, the being sought boundary conComparing the terms standing by ditions for x = 0 are obtained (2H)3 ∂w ∂2 w ∂2 w ∂2 w − B11 2 + B12 2 + 2B16 + α1 ∂x 12 ∂x∂y ∂x ∂y ∂3 w C1 + β11 T M = M1o (y), ∂x∂y2 ∂2 w ∂2 w ∂2 w ∂ (2H)3 ∂ B11 2 + B12 2 + 2B16 + β11 T M + ∂x 12 ∂x ∂x∂y ∂x ∂y ∂ (2H)3 ∂2 w ∂2 w ∂2 w B16 2 + B26 2 + 2B66 + β12 T M = 2 ∂y 12 ∂x∂y ∂x ∂y 2 2 2 ∂2 F ∂w ∂ ∂ F ∂w + 2 . (2.48) −β1 w − 2 B1 2 + Qo1 (y) − ∂x∂y ∂y ∂x ∂y ∂y Furthermore, relations for moments in the middle surface in the case of orthotropic material are as follows (2H)3 ∂2 w ∂2 w ∂2 w B11 2 + B12 2 + 2B16 − β11 T M , M11 = − 12 ∂x∂y ∂x ∂y (2H)3 ∂2 w ∂2 w ∂2 w B12 2 + B22 2 + 2B26 − β22 T M , M22 = − 12 ∂x∂y ∂x ∂y (2H)3 ∂2 w ∂2 w ∂2 w B16 2 + B26 2 + 2B66 − β12 T M , (2.49) M12 = − 12 ∂x∂y ∂x ∂y where:

H TM =

zT (x, y, z)dz. −H

Then boundary conditions (2.48) take the form ∂ ∂w ∂2 w + M11 = C1 + M1o (y), α1 ∂x ∂y ∂x∂y ∂ ∂2 w ∂2 F ∂w ∂2 F ∂w ∂ ∂2 M11 + 2 M12 = β1 w + 2 B1 2 + − 2 − Qo1 (y). (2.50) ∂x ∂y ∂x∂y ∂y ∂y ∂y ∂y ∂x

170

2 Stability of Rectangular Shells within Temperature Field

Let us compute first variation of the functional˜ with respect to F and let derive the terms with respect to the edge x = 0: ⎫ 2 l2 ⎧ ⎪ ⎪ ⎪ ⎪ 1 ∂w ∂ ∂ ∂w ⎬ ⎨ ∂w ∂2 w δF + κ2 δF − κ2 w δF ⎪ δF + δ˜ = ⎪ ⎪ 2 ⎭ ⎩ ∂x ∂y 2 ∂y ∂x ∂x ∂x ⎪ 0

1 2H

l2

dy+ x=0

∂ ∂2 F ∂2 F ∂2 F a12 2 + a22 2 − a26 + αT22 T N δF− ∂x ∂x∂y ∂y ∂x

0

∂ ∂2 F ∂2 F ∂2 F T + α22 T N δF− a12 2 + a22 2 − a26 ∂x∂y ∂x ∂y ∂x ∂ ∂ ∂2 F ∂2 F ∂2 F T a16 2 + a26 2 − a66 + a12 T N δF dy+ ∂y ∂x∂y ∂y ∂y ∂x x=0 l2 −A1

∂2 F ∂2 F ∂4 F ∂F ∂ + K δF− δF + L1 − L 1 1 2 4 ∂x∂y ∂y ∂x ∂y ∂y

0

mo1 (y)

∂ δF − qo1 (y)δF ∂x

dy + . . . = 0. x=0

∂ δF and δF to zero, the being sought Comparing coeﬃcients standing by ∂x boundary conditions for x = 0 for F are obtained

2 1 ∂2 F ∂2 F ∂2 F 1 ∂w a12 2 + a22 2 − a26 + αT22 T N − − κ2 w − 2 ∂y 2H ∂x∂y ∂y ∂x L1 κ2

∂F = mo1 (y), ∂y

1 ∂ ∂w ∂w ∂2 w ∂2 F ∂2 F ∂2 F T + a + α + + a − a T 12 22 26 22 N − ∂x ∂x ∂y2 2H ∂x ∂x∂y ∂y2 ∂x2 ∂2 F ∂2 F ∂2 F ∂2 F ∂2 F ∂ a16 2 + a26 2 − a66 + αT12 T N − A1 2 − L1 + ∂y ∂x∂y ∂x∂y ∂y ∂x ∂y ∂2 ∂2 F K = qo1 (y). (2.51) 1 ∂y2 ∂y2

Owing to relations for deformation in middle surface ∂2 F ∂2 F ∂2 F 1 a11 2 + a12 2 − a16 + αT11 T N , ε11 = 2H ∂x∂y ∂y ∂x ∂2 F ∂2 F ∂2 F 1 T a12 2 + a22 2 − a26 + α22 T N , ε22 = 2H ∂x∂y ∂y ∂x

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

ε12 =

171

1 ∂2 F ∂2 F ∂2 F a16 2 + a26 2 − a66 + αT12 T N , 2H ∂x∂y ∂y ∂x

the boundary conditions (2.51) are transformed to the form 2 1 ∂w ∂F = mo1 (y), − κ2 w − ε22 − L1 2 ∂y ∂y ∂2 ∂ ∂w ∂w ∂2 w ∂ ∂2 F ∂2 F ∂2 F + + K1 2 = q◦1 (y). (2.52) + ε22 − ε12 − A1 2 − L1 κ2 ∂x ∂x ∂y2 ∂x ∂y ∂x∂y ∂y2 ∂y ∂y The most important particular cases of boundary conditions (2.50)–(2.52) for the edge x = 0 are as follows: (a) α1 = C1 = 0, β1 = ∞, K1 = A1 = L1 = 0; this case corresponds to ball type unmovable support w = 0, M11 = M1o (y), ε22 = mo1 (y),

κ2

∂ ∂ ∂w + ε22 − ε12 = qo1 (y). ∂x ∂x ∂y

(2.53)

(b) α1 = β1 = ∞, A1 = ∞, L1 = 0; this case corresponds to sliding clamping ∂w = 0, ∂x

w=

ε22 = mo1 (y),

∂2 F = 0. ∂y2

(2.54)

(c) α1 = C1 = 0, β1 = ∞, A1 = ∞, L1 = 0; in this case the edge is supported on flexible in tangential plane unstretched rib w = 0,

M11 = M1o (y),

∂2 F = 0, ∂y2

ε22 = mo1 (y).

(2.55)

(d) α1 = C1 = 0, β1 = ∞, L1 = A1 = ∞; in this case the edge is freely supported w = 0,

M11 = M1o (y),

∂2 F = 0, ∂y2

∂2 F = 0. ∂x∂y

(2.56)

The relations (2.53)–(2.56) yield the homogeneous boundary conditions for M1o (y) ≡ 0, Qo1 (y) ≡ 0, mo1 (y) ≡ 0, qo1 (y) ≡ 0. Our next aim is focused on obtaining the compatibility conditions in the cusps of an anisotropic shell clamped on sides x = 0, y = 0. First, variation of the functional (2.47) with respect to w is obtained. Introducing typical transformations and equaling to zero the variational terms related to the cusp corner point x = 0, y = 0, one gets

172

2 Stability of Rectangular Shells within Temperature Field

(2H)3 2 12

l1 l2 ∂2 w ∂2 w ∂2 w + B16 2 + B26 2 + 2B66 δw ∂x∂y ∂x ∂y x=0 y=0

l ∂2 w ∂ ∂ ∂2 w ∂2 w ∂ 1 2 y=0 − B1 2 δw + B1 δw− 2 {β12 T M δw}lx=0 δw − C 1 ∂y ∂y2 ∂x∂y ∂x ∂y ∂y B2

∂2 w ∂ ∂ ∂2 w ∂62w ∂ δw + B2 δw = 0. δw − C2 2 ∂x ∂x2 ∂x∂y ∂y ∂x ∂x

(2.57)

∂ ∂ Comparing to zero the coeﬃcients standing by δw, ∂x δw, ∂y δw, the compatibility conditions x = y = 0 for the function w are obtained ∂2 w ∂2 w ∂2 w (2H)3 2 B16 2 + B26 2 + 2B66 + 2β12 T M + 12 ∂x∂y ∂x ∂y ∂ ∂2 w ∂2 w ∂ B1 2 + B2 2 = 0, ∂y ∂x ∂y ∂x

C1

∂2 w ∂2 w + B2 2 = 0, ∂x∂y ∂x

B1

∂2 w ∂2 w = 0. + C 2 ∂x∂y ∂y2

(2.58)

The obtained compatibility conditions (2.58) correspond to anisotropic shell governed by nonlinear equations (2.39) with boundary conditions (2.50). Note that ∂ ∂ δw, ∂y δw are diﬀerthe compatibility conditions (2.58) are required only if δw, ∂x ent from zero. For Bi = Ci = 0 (i = 1, 2) only one compatibility condition exists M12 = 0. Problems devoted to compatibility conditions in the corners during searching for solutions of boundary value problems of elasticity are addressed in monograph [592]. Now we are going to derive continuation conditions in anisotropic shell corner for the function F. Let us compute first variation of the functional (2.47) with respect to F. Owing only to the terms related to the corner point x = y = 0 and comparing ∂ ∂ δF, ∂y δF, the following compatibility successively to zero coeﬃcients by δF, ∂x conditions hold ∂w ∂w ∂2 F ∂2 F ∂2 F ∂F 1 a16 2 + a26 2 − a66 + αT12 T N − + L2 + 2H ∂x∂y ∂x ∂y ∂y ∂y ∂x ∂ ∂ ∂F ∂F ∂2 F ∂F ∂2 F + L1 − K1 2 + A 2 − K2 2 = 0, A1 ∂y ∂x ∂y ∂x ∂x ∂y ∂x K1

∂2 F = 0, ∂y2

K2 =

∂2 F = 0. ∂x2

(2.59)

Using relations for deformations in middle surface, the following compatibility conditions for the function F in the point x = y = 0 are obtained ∂w ∂w ∂F ∂F ∂F ∂F 1 ε12 − + A1 + L1 + A2 + L2 − 2H ∂x ∂y ∂y ∂x ∂y ∂y

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

173

∂ ∂ ∂2 F ∂2 F K2 2 − K1 2 = 0, ∂x ∂y ∂x ∂y K1

∂2 F = 0, ∂y2

K2

∂2 F = 0. ∂x2

(2.60)

Owing to the special case of boundary conditions, the following compatibility conditions for the function F in the anisotropic shell corners are obtained. For K1 = Ai = Li = 0 (i = 1, 2), what corresponds to the boundary condition (2.53), (2.54), one gets ∂w ∂w = 0. (2.61) ε12 − ∂x ∂y For Ai = ∞, Li = 0 (i = 1, 2), we obtain ∂F ∂F = = 0. ∂y ∂x

(2.62)

This compatibility condition corresponds to the boundary condition (2.55). For Ai = L1 = ∞ (i = 1, 2) we get ∂F ∂F = = 0. ∂y ∂x

(2.63)

This compatibility condition corresponds to the boundary condition (2.56). One may conclude, owing to (2.62), (2.63), that for the boundary condition (2.55), (2.56) the function F is constant in the corner points. Applying compatibility conditions (2.62), (2.63) into boundary conditions (2.55), (2.56), correspondingly, the function F can be defined in the considered space boundary. Hence, the boundary conditions (2.55) for the functions F read 1 ∂2 F ∂2 F a22 2 − a26 + αT22 T N = mo1 (y) for x = 0, F = C0 , 2H ∂x∂y ∂x ∂2 F ∂2 F 1 T a11 2 − a16 + α11 T N = mo2 (x) for y = 0. (2.64) F = C0 , 2H ∂x∂y ∂y Proceeding in a similar way, the function F can be defined for the boundary condition (2.56), which takes the following form ∂2 F = 0, ∂x∂y

∂2 F =0 ∂y2

for x = 0,

∂2 F = 0, ∂x∂y

∂2 F =0 ∂x2

for y = 0.

(2.65)

Using compatibility condition (2.63) and boundary conditions (2.65), the function F = 0 can be defined on the space boundary. The boundary conditions (2.65) for F read

174

2 Stability of Rectangular Shells within Temperature Field

∂F =0 ∂x ∂F =0 ∂y

F = 0, F = 0,

for x = 0, for y = 0.

(2.66)

In what follows, the compatibility conditions satisfied by function w and F in the point y = yc lying on a rectangular side (i.e. for 0 ≤ y ≤ yc one type boundary condition is given, whereas for yc ≤ y ≤ l2 another type of boundary condition is given) are derived. The functional, corresponding to this problem, has the form 1 ˜ = + 2

⎫ 2 2 2 2 2 yl ⎧ ⎪ ⎪ ⎪ ⎪ ∂w ∂ w ∂w ⎨ 2⎬ + C1 + α1 + β1 w ⎪ B1 ⎪ ⎪ ⎪ 2 ⎩ ⎭ ∂x∂y ∂x ∂y 0

1 2

⎫ 2 2 2 2 2 l2 ⎧ ⎪ ⎪ ⎪ ⎪ ∂w ∂ w ∂w ⎨ 2⎬ + C + α + β w B2 ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎩ ⎭ ∂x∂y ∂x ∂y2 yl

yl

∂w Mio (y) ∂x

+

dy − x=0

0

1 2

l2

Qo1 (y)w

∂w M2o (y) ∂x

+

⎫ 2 2 2 l2 ⎧ ⎪ ⎪ ⎪ ∂F ∂ F ∂F ∂F ⎪ ⎨ ⎬ + K2 + 2L2 A ⎪ ⎪ ⎪ ⎩ 2 ∂y ⎭ ∂y ∂x ⎪ ∂y2 yl

∂F mo1 (y) ∂x

0

+

dy+ x=0

dy− x=0

l2

∂F mo2 (y)

dy −

qo1 (y)F x=0

dy+ x=0

yl

⎫ 2 2 2 yc ⎧ ⎪ ⎪ ⎪ ∂F ∂ F ∂F ∂F ⎪ ⎨ ⎬ + K1 + 2L1 A ⎪ ⎪ ⎪ ⎩ 1 ∂y ⎭ ∂y ∂x ⎪ ∂y2

yc

dy− x=0

Q02 (y)w

0

1 2

dy+ x=0

∂x

yc

+

qo2 (y)F

dy.

(2.67)

x=0

Let us derive first variation of the functional (2.67) with respect to w and compare it to zero. Integrating by parts and remaining only terms related to edge x = 0, one gets l2 −

∂2 w ∂2 w ∂2 w ∂ (2H)3 B11 2 + B12 2 + 2B16 + β11 T M δw+ ∂x 12 ∂x∂y ∂x ∂y

0

∂ (2H)3 ∂2 w ∂2 w ∂2 w B11 2 + B12 2 + 2B16 + β11 T M δw− 12 ∂x∂y ∂x ∂x ∂y ∂2 w ∂2 w ∂2 w ∂ (2H)3 B16 2 + B26 2 + 2B66 + β12 T M δw− 2 ∂y 12 ∂x∂y ∂x ∂y

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

yl

∂w ∂2 F ∂2 F ∂w δw + δw ∂x∂y ∂y ∂x ∂y2

dy + x=0

∂w ∂ β1 wδw + α1 δw ∂x ∂x l2

∂4 w ∂3 w ∂ δw+ δw − C1 4 ∂y ∂x∂y2 ∂x

0

yl dy + x=0

−M1o (y)

∂ δw − Qo1 (y)δw ∂x

0

l2

−M2o (y) yl

B1

∂ δw − Qo2 (y)δw ∂x

175

dy + x=0

B2 yl

dy+ x=0

∂4 w ∂3 w ∂ δw+ δw − C 2 ∂y4 ∂x∂y2 ∂x

∂w ∂ β2 wδw + α2 δw ∂x ∂x

dy = 0. x=0

∂ δw and δw, the boundary Now, comparing to zero the coeﬃcients standing by ∂x conditions for x = 0 for the function w are obtained ∂ ∂w ∂2 w + M11 − C1 = M1o (y), α1 ∂x ∂y ∂x∂y ∂2 F ∂w ∂2 F ∂w ∂ ∂2 ∂ 2 w ∂ M11 + 2 M12 = β1 w + B1 2 − 2 − Qo1 (y). + 2 ∂x ∂y ∂x∂y ∂y ∂y ∂y ∂y ∂x

Recall that this boundary condition holds for x = 0 for w (0 ≤ y ≤ yc ). Similarly, one obtains ∂ ∂w ∂2 w + M11 − C2 = M2o (y), α2 ∂x ∂y ∂x∂y ∂2 F ∂w ∂2 F ∂w ∂ ∂2 ∂2 w ∂ M11 + 2 M12 = β2 w + 2 B2 2 + − 2 − Qo2 (y). ∂x ∂y ∂x∂y ∂y ∂y ∂y ∂y ∂x for x = 0, yl ≤ y ≤ l2 . We are going to calculate the first functional variation (2.67) with respect to F and compare it to zero. Integrating by parts and carrying out some transformations, the following boundary conditions for the function F for x = 0, 0 ≤ y ≤ yc are obtained ∂ ∂2 ∂ 2 F ∂2 F ∂2 F ∂w ∂2 w ∂w ∂ ε22 − ε12 + K1 2 + k2 + 2 = qo1 (y), − A 1 2 − L1 2 ∂x ∂y ∂x∂y ∂x ∂y ∂y ∂y ∂y ∂x 2 ∂F 1 ∂w = mo1 (y). −ε22 − k2 w + − L1 2 ∂y ∂y

(2.68)

Similarly, for x = 0, yl ≤ y ≤ l2 , one gets ∂ ∂ ∂2 ∂ 2 F ∂2 F ∂2 F ∂w ∂2 w ∂w ε22 − ε12 + K2 2 + κ2 + = qo2 (y), − A 2 2 − L2 2 ∂x ∂y ∂x∂y ∂x ∂y ∂x ∂y ∂y ∂y

176

2 Stability of Rectangular Shells within Temperature Field

−ε22 − κ2 w +

2 ∂F 1 ∂w = mo2 (y). − L2 2 ∂y ∂y

(2.69)

For example, the boundary conditions (2.68), (2.69) have the form α1 = C1 = 0, β1 = ∞, A1 = ∞, L1 = 0 w = 0, M11 = M1o (y), ∂2 F = 0, ∂y2

for x = 0, 0 ≤ y ≤ yc ,

ε22 = mo1 (y).

(2.70)

α2 = β2 = ∞, L2 = A2 = ∞, ∂w = 0, w = 0, ∂x ∂2 F ∂2 F = 0, = 0. ∂x∂y ∂y2

for x = 0, 0 ≤ y ≤ l2 ,

(2.71)

i.e. in the point y = yc a change of boundary conditions occurs. Comparing to zero the terms of functional (2.67) variation with respect to w and the point y = yc , the following equations are obtained (B1 − B2 )

∂2 w = 0, ∂y2

(C1 − C2 )

∂2 w = 0, ∂x∂y

(B1 − B2 )

∂3 w = 0. ∂y3

(2.72)

For the function F in the point y = yc the compatibility conditions for x = 0 are as follows ∂F ∂F ∂3 F + (L2 − L1 ) + (K1 − K2 ) 3 = 0, (A1 − A2 ) ∂y ∂x ∂y (K2 − K1 )

∂2 F = 0. ∂y2

(2.73)

Owing to compatibility conditions (2.72), (2.73), for the same boundary conditions to the left and to the right of the point yc , there are not any compatibility conditions in the point yc , whereas for diﬀerent boundary conditions the compatibility conditions should be satisfied. These conditions will be given for the boundary ∂2 w = 0 in the point y = yc ; conditions (2.70), (2.71). Since C1 = 0, C2 0, then ∂x∂y ∂F and since α1 = 0, α2 = ∞, then ∂x = 0 in the point y = yc , x = 0. In other words, for the boundary conditions (2.70), (2.71) given to left and to right from the point y = yc , in the point yc the following compatibility conditions ∂2 w = 0, ∂x∂y should be satisfied.

∂F = 0. ∂x

(2.74)

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

177

2.2.5 Compatibility conditions for shallow shells equations Let us consider a rectangular isotropic shell occupying the space G+ {0 ≤ x ≤ l1 , 0 ≤ l ≤ l2 } in the plane oxy. Energy of this shell, without account of boundary conditions, has the form [212]: ⎡ ⎛ ⎞⎤ l1l2 ⎧ ⎪ ⎜⎜⎜ ∂2 w 2 ∂2 w ∂2 w ⎟⎟⎟⎥⎥⎥ ⎪ ⎨ D ⎢⎢⎢⎢ ∂2 w ∂2 w ⎜ ⎟⎟⎥⎥ − + 2(1 − ν) + − = ⎢ ⎜⎝ ⎪ ⎪ ⎩ 2 ⎣ ∂x2 ∂x∂y ∂y2 ∂x2 ∂y2 ⎠⎦ +

0 0

⎡ ⎞⎤ ⎛ 2 ⎜⎜⎜ ∂2 F 2 ∂2 F ∂2 F ⎟⎟⎟⎥⎥⎥ ∂2 F 1 ⎢⎢⎢⎢ ∂2 F ∂2 F + 2 + 2(1 + ν) ⎜⎜⎝ − 2 2 ⎟⎟⎠⎥⎥⎦ − κ1 2 w− ⎢⎣ 2 2EH ∂x ∂x∂y ∂y ∂x ∂y ∂y ⎡ ⎤ 2 2 ∂2 F 1 ⎢⎢ ∂2 F ∂w ∂2 F ∂w ∂2 F ∂w ∂w ⎥⎥⎥⎥ κ2 w 2 + ⎢⎢⎢⎣ 2 + 2 −2 ⎥− 2 ∂x ∂y ∂x∂y ∂x ∂y ⎦ ∂x ∂y ∂x l1l2 Φ(x, y)dxdy,

q(x, y)w} dxdy = 0 0

where Φ(x, y) denotes the underintegral expression. Let the considered shell (with respect to edge x = 0) is stiﬄy linked with a supporting rod. The rod possesses bending stiﬀness coeﬃcient B+ , and rotational stiﬀness coeﬃcient C + . Energy of the shell stiﬀened by a rib along the edge x = 0 is given by the equation 1 ¯ + =+ + 2

2 2 2 ⎫ l2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ + ∂2 w ⎬ + ∂ w + C B ⎪ ⎪ ⎪ ⎩ ⎭ ∂x∂y ⎪ ∂y2 0

1 2

dy+ x=0

⎫ 2 l2 ⎧ ⎪ ⎪ 2 ⎪ ⎪ ⎨ + ∂F +∂ F + ∂F ∂F ⎬ + K1 2 + 2L A ⎪ ⎪ ⎪ ⎩ ⎭ ∂y ∂y ∂x ⎪ ∂y 0

dy. x=0

Consider one more shell occupying the space G− {−l1 ≤ x ≤ 0, 0 ≤ y ≤ l2 } and also stiﬀened along x = 0 by a rod with stiﬀnesses B− , C − . Its energy reads −

0 l2

=

Φ(x, y)dxdy, −l1 0

whereas the energy accounting supporting rod has the form 1 ¯ − =− + 2

2 2 ⎤ l2 ⎡⎢ 2 2 ⎥⎥⎥ ⎢⎢⎢ − ∂ w − ∂ w ⎥⎥ dy+ + C ⎢⎣ B ∂x∂y ⎦ x=0 ∂y2 0

178

2 Stability of Rectangular Shells within Temperature Field

⎤ 2 2 l2 ⎡⎢ 2 ⎥⎥ ⎢⎢⎢ − ∂F − ∂ F − ∂F ∂F ⎥ ⎣⎢A ∂y + K ∂y2 + 2L ∂y ∂x ⎥⎥⎦ dy, x=0

1 2

0

˜ ¯ + + ¯ − + 1 = 2

⎫ 2 l2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ∂w 2⎬ + βw α ⎪ ⎪ ⎪ ⎪ ⎩ ∂x ⎭ 0

dy.

(2.75)

x=0

In the above α denotes joint stiﬀness coeﬃcient; β is the support stiﬀness co˜ F). We are eﬃcient under the joint. Let us construct the functional J(w, F) = (w, focused on searching its first variation with respect to w, which is then compared to zero. Obtained relation is further transformed through integration by parts to yield

l2 δw J =

D 0

l1 l2 2 ∂2 w ∂ ∂ w ∂2 w ∂ ∂2 w δw +ν 2 dy − D +ν 2 + ∂x ∂x2 ∂x2 ∂y ∂x ∂y x=0 0

l1 l1 l2 2 ∂ F ∂w ∂2 F ∂w ∂2 w ∂ δw dy + − δw 2 (1 − ν) dy+ x=0 ∂y ∂x∂y ∂y2 ∂x ∂x∂y ∂y x=0 0

l2 α

+∂

3 w + ∂ w ∂ δw + β1 wδw B δw − C ∂y4 ∂x∂y2 ∂x 4

0

∂ ∂w ∂w ∂ δw δw dy+ − − ∂x x=+0 ∂x x=−0 ∂x ∂x x=+0 x=−0

l2 D 0

l2 0

+ x=0

0 ∂2 w ∂2 w ∂ δw +ν 2 dy− ∂x2 ∂y ∂x x=−l1

0 ∂2 w ∂ ∂ ∂2 w ∂2 w δw dy+ D + ν 2 + 2 (1 − ν) 2 ∂x ∂x ∂y ∂x∂y ∂y

−l1

l2 0

l2 0

0 ∂2 F ∂w ∂2 F ∂w − δw dy+ ∂y2 ∂x ∂x∂y ∂y x=−l1

−∂

3 w − ∂ w ∂ δw B δw − C ∂y4 ∂x∂y2 ∂x 4

dy + . . . x=0

where terms not related to the joint are denoted by dots. Owing to (2.75), the function w(x, y) should be continuous for x = 0, and hence also its variation δw should be continuous. Introducing the notation

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

179

∂2 w ∂2 w , + ν ∂x2 ∂y2 2 ∂w ∂2 w ∂ ∂ ∂2 w (1 − ν)D , − 2 + ν Q1 (x, y) = − D ∂x ∂y ∂x∂y ∂x2 ∂y2 M1 (x, y) = −D

(2.76)

one obtains l2 δw J =

∂ ∂ M1 δw − Q1 δw| x=+0 − M1 δw + Q1 δw| x=−0 + ∂x ∂x x=+0 x=−0

0

∂2 F ∂w ∂2 F ∂w − δw − 2 x=−0 ∂y ∂x ∂x∂y ∂y 2 ∂2 +∂ w B δw + 2 2 x=+0 ∂y ∂y 2 ∂ ∂ + ∂ w C δw − ∂y ∂x∂y ∂x x=+0 α

∂2 F ∂w ∂2 F ∂w − δw + 2 x=+0 ∂y ∂x ∂x∂y ∂y 2 ∂2 −∂ w B δw − 2 2 x=−0 ∂y ∂y 2 ∂ ∂ − ∂ w C δw + ∂y ∂x∂y ∂x x=−0

∂ ∂w ∂w ∂ δw δw + − − ∂x x=+0 ∂x x=−0 ∂x ∂x x=+0 x=−0 / βwδw| x=0 dy + . . . = 0.

Comparing successively to zero the coeﬃcients standing by δw, ∂ ∂x δw x=−0 , the following relations are obtained [592]: ' ∂2 w ∂2 & [Q1 ]| x=0 = βw + 2 B+ + B− − ∂y ∂y2 x=0

∂ ∂x δw x=+0 ,

∂2 F ∂w ∂2 F ∂w + , ∂y2 ∂x x=0 ∂x∂y ∂y x=0 2 ∂w ∂ + ∂ w C α , = −M1 + ∂x x=0 ∂y ∂x∂y x=+0

∂w α ∂x

2 ∂ − ∂ w C . = −M1 − ∂y ∂x∂y x=0 x=−0

Conjugation conditions for the function w(x, y) can be given in more symmetric form, if instead of two last relations their half-sum and diﬀerence are taken. Owing to attachment of continuity condition of w for x = 0, one obtains that [w] = 0, 2 2 ∂w ∂ w 1 ∂w ∂2 w ∂2 w = D +ν 2 +D +ν 2 + α ∂x 2 ∂x2 ∂y x=+0 ∂x2 ∂y x=−0

180

2 Stability of Rectangular Shells within Temperature Field

2 2 ∂ ∂ + ∂ w − ∂ w C C , − ∂y ∂x∂y x=+0 ∂y ∂x∂y x=−0 2 2 2 ∂ ∂w ∂2 w ∂ + ∂ w − ∂ w C C = − , D + ν + ∂y ∂x∂y x=+0 ∂y ∂x∂y x=−0 ∂x2 ∂y2 ∂ ∂2 w ∂2 w ∂ ∂2 w D = + ν 2 + 2 (1 − ν)D ∂x ∂x2 ∂y ∂x∂y ∂y ' ∂2 w ∂2 F ∂w ∂2 F ∂w ∂2 & − (2.77) −βw − 2 B+ + B− + ∂x∂y ∂y ∂y ∂y2 ∂y2 ∂x for x = 0. Owing to notation (2.76), most interesting particular cases of conditions (2.77) are further analysed (a) α = ∞, β = B± + C ± = 0 ([592]): ∂w [w] = = [M1 ] = [Q1 ] = 0 ∂x

for x = 0.

(2.78)

Note that these conditions exhibit stiﬀ coupling between shells. They can be rewritten also to the form 2 ∂w ∂ ∂2 w ∂w = D 2 = D 2 =0 for x = 0. [w] = ∂x ∂x ∂x ∂x (b) α = β = 0, B+ = B− = B, C + = C − = C ([592]): ∂2 w ∂ [w] = 0, C M1 − = 0, ∂y ∂x∂y x=+0 ∂2 w ∂ ∂2 ∂2 w ∂2 F ∂w C . M1 + = 0, [Q1 ] = 2 2 B 2 − 2 ∂y ∂x∂y x=−0 ∂y ∂y ∂y ∂x Note that now we deal with two same shells coupled through an ideal joint. Shell edges, in the joint neighbourhood, are strengthened by ribs, which are exhibited to both bending and rotation. (c) α = ∞, β = 0, B+ = B− = B, C + = C − = C ([592]): 2 ∂w ∂2 w ∂ w ∂ ∂2 [w] = = 0, [M1 ] = 2 C , [Q1 ] = 2 2 B 2 . ∂x ∂y ∂x∂y ∂y ply These conditions correspond to whole shell reinforced by a rib. (d) α = β = C + = C − = 0, B+ = B− = B ([592]): [w] = 0, M1 | x=±0 = 0, 2 ∂ w ∂2 F ∂w ∂ [Q1 ] = 2 2 B 2 − 2 for x = 0. ∂y ∂y ∂y ∂x Observe that these conditions diﬀer from the (b) case. Namely, in this case shells can be considered not as stiﬄy coupled with reinforced rods, but us lying on them. 2

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

181

(e) α = ∞, β = C + = C − = 0, B+ = B− = B ([592]): 2 2 ∂w ∂ w ∂w ∂2 = D 2 = 0, [Q1 ] = 2 2 B 2 . [w] = ∂x ∂x ∂y ∂y In this case the rod plays a role of support for stiﬄy coupled shells (whole shell). In what follows we are going to derive conditions for F along linking line x = 0 of two shells. Again, we start with computation of first variation of the functional J with respect to F. Integrating by parts and remaining the terms related to the rod x = 0, one gets l2 δF J = −

1 EH

0

l1 2 l2 ∂2 F ∂ ∂ F ∂2 F ∂ 1 δF −ν 2 dy + + EH ∂x ∂x2 ∂y ∂x ∂x2 x=0 0

⎫l1 ⎛ 2 ⎞ l1 l2 ⎧ ⎪ ⎪ ⎪ ⎪ ∂2 F 1 ∂w ⎟⎟⎟⎟ ∂ ⎬ ⎨⎜⎜⎜⎜ (2 + ν) 2 δF dy + ⎪ dy+ ⎟⎠ δF ⎪ ⎜⎝−k2 w + ⎪ ⎪ ⎭ ⎩ 2 ∂y ∂x ∂y x=0 x=0

0

l2 0

l1 l2 2 2 ∂w ∂w ∂2 w +∂ F + ∂ F + δF dy+ k2 δF −A dy + − L ∂x ∂x ∂y2 ∂x∂y ∂y2 x=0 x=0 0

l2

+ ∂F

∂ δF L ∂y ∂x

0

ν

l2

+∂

F K δF ∂y4

dy + x=0

0 ∂2 F ∂ δF + ∂y2 ∂x x=−l1

4

0

l1

1 EH

l2 dy − x=0

1 EH

∂2 F − ∂x2

0

0 ∂ ∂2 F ∂2 F δF + (2 + ν) dy+ ∂x ∂x2 ∂y2 x=−l1

0

⎫0 2⎞ 0 l2 ⎪ ⎪ ⎪ ⎪ ∂w ∂w ∂2 w 1 ∂w ⎟⎟⎟⎟ ∂ ⎬ ⎨⎜⎜⎜⎜ + k2 δF dy+ ⎟ δF ⎪ dy + ⎜−k w + ⎪ ⎪ ⎭ ⎩⎝ 2 2 ∂y ⎠∂x ⎪ ∂x ∂x ∂y2 x=−l1

l2⎧⎛ 0

x=−l1

l2

∂2 F ∂2 F δF −A1 2 − L− ∂x∂y ∂y

0

l2 0

K−

0

l2 dy + x=0

∂4 F δF ∂y4

− ∂F

∂ δF L ∂y ∂x

0

dy+ x=0

dy + . . . , x=0

where dots denote terms not referred to the rod x = 0. Assuming F as continuous function, also δF is continuous one. ∂ ∂ δF x=+0 , ∂x δF x=−0 to zero, the followComparing the terms standing by δF, ∂x ing equations are obtained

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2 Stability of Rectangular Shells within Temperature Field

∂w ∂2 w 1 ∂ ∂2 F ∂2 + ∂w + − + (2 + ν) 2 − k2 + EH ∂x ∂x2 ∂x ∂x ∂y2 x=+0 ∂y x=+0 ∂w ∂2 w ∂2 F 1 ∂ ∂2 F − ∂w + + (2 + ν) 2 + k2 + EH ∂x ∂x2 ∂x ∂x ∂y2 x=−0 ∂y x=−0 2 2 4 +∂ F + ∂ F + ∂ F −A −L + K + ∂x∂y x=+0 ∂y2 ∂y4 x=+0 2 2 4 −∂ F − ∂ F − ∂ F −A −L + K = 0, ∂x∂y x=−0 ∂y2 ∂y4 x=−0 1 EH 1 − EH

∂2 F ∂2 F −ν ∂x2 ∂y2

+ x=+0

⎧ 2 ⎫ ⎪ ⎪ ⎪ 1 ∂w ⎪ ⎨ + ⎬ w− k ⎪ ⎪ 2 ⎪ ⎩ ⎭ 2 ∂y ⎪

⎧ 2 ⎫ ⎪ ⎪ ⎪ ∂2 F ∂2 F 1 ∂w ⎪ ⎨ − ⎬ −ν 2 + ⎪ −k2 w + ⎪ ⎪ 2 ⎩ ⎭ 2 ∂y ⎪ ∂x ∂y x=−0

+ L = 0, ∂y x=+0

+ ∂F

x=+0

+ L x=−0

− ∂F

∂y

= 0.

(2.79)

x=−0

Assuming that k2+ = k2− = k2 (i.e. both shells have the same curvature k2 ) the first relation can be rewritten to the following one 4 ∂w ∂w ∂w ∂2 F 1 ∂2 F +∂ F − ν − κ − + K − − 2 1 EH ∂x2 ∂x x=0 ∂x ∂y x=0 ∂y2 x=0 ∂y4 2 2 4 2 2 +∂ F + ∂ F −∂ F −∂ F − ∂ F A −L + K −A −L = 0. ∂x∂y x=+0 ∂x∂y x=−0 ∂y2 ∂y4 ∂y2 Two last relations in (2.79) are combined, and for k2+ = k2− = k2 one gets ⎡ 2 ⎤ 1 ∂2 F ∂2 F 1 ⎢⎢⎢⎢ ∂w ⎥⎥⎥⎥ − ν 2 + k2 [w]| x=0 − ⎣⎢ ⎥ = EH ∂x2 2 ∂y ⎦ ∂y x=0 x=0 − ∂F − L − L , ∂y x=+0 ∂y x=−0 2 ∂ F 1 ∂2 F 1 ∂2 F ∂2 F − ν + − ν + EH ∂x2 ∂y2 x=+0 EH ∂x2 ∂y2 x=−0 ⎧ ⎧ 2 ⎫ 2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂w ⎪ 1 ∂w ⎪ ⎨ ⎬ ⎬ ⎨ +⎪ + k2 w − k w− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ 2 2 ∂y 2 ∂y ⎪ x=+0 x=−0 ∂F ∂F L+ + L− = 0. ∂y x=+0 ∂y x=−0

+ ∂F

(2.80)

Computing various particular cases of conditions (2.79), (2.80), the following coupling conditions for the function F are obtained

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

183

(a) K + = K − = A+ = A− = L+ = L− = 0 2 ∂ ∂ F 1 ∂2 F ∂w + k + (2 + ν) = 0, 2 EH ∂x ∂x2 ∂x x=±0 ∂y2 1 ∂2 F ∂2 F − ν = 0. EH ∂x2 ∂y2 x=±0 (b) A+ = A− = ∞, L+ = L− = 0 ∂2 F = 0, ∂y2 x=±0

∂2 F ∂2 F −ν 2 2 ∂x ∂y

(c) A+ = A− = ∞, L+ = L− = ∞ ∂2 F = 0, ∂y2 x=±0

= 0. x=±0

∂2 F = 0. ∂x∂y x=±0

Observe that during derivation of coupling conditions for shallow shells, an action of external forces is no accounted, and the obtained coupling conditions are homogeneous ones. Account of external forces will give non-homogeneous coupling conditions. 2.2.6 Temperature field In order to study a stress-strain state and stability loss of thin plates working in conditions of non-uniform heating, a temperature field should be defined. Both theoretical background and methods devoted to solution of the heat transfer problems of thin walled elements are required. In majority of references various methods are used to reduce 3D heat transfer equation into 2D one for plates and shells, since then the problem is essentially simplified. In this section we are going to solve directly 3D heat transfer equation to define temperature filed occurred in plates and shells. For shells made from isotropic material the stationary heat transfer equation follows W0 ∂T ∂2 T =− , (2.81) ∆T + 2 + 2k ∂z λ ∂z where k = 12 (k1 + k2 ) - average curvature of the shell surface; λ - heat transfer coeﬃcient of an isotropic body; ∆ - Laplace operator (2.19). In the case of a plate, the equation (2.81) reads W0 ∂2 T ∂2 T ∂2 T + 2 + 2 =− . 2 λ ∂x ∂y ∂z

(2.82)

It should be emphasized that many technical materials have a heat transfer coeﬃcient depended on heat stream direction. Recall that since the equations (2.81),

184

2 Stability of Rectangular Shells within Temperature Field

(2.82) are obtained for scalar heat transfer coeﬃcient, they are not valid for such materials. For an anisotropic material, the heat transfer coeﬃcient represents the second order tensor. In this case ∂T ∂T ∂T + λ xy + λ xz i+ div(λgradT ) = div λ xx ∂x ∂y ∂z ∂T ∂T ∂T ∂T ∂T ∂T + λyy + λyz + λzy + λzz j + λzx k = −W0 , (2.83) λyx ∂x ∂y ∂z ∂x ∂y ∂z where: i, j, k are eigenvectors of a rectangular coordinates. If components of a heat transfer tensor do not depend on coordinates, the expression (2.83) is reduced to the form div(λgradT ) = λ xx (λ xy + λyx )

∂2 T ∂2 T ∂2 T + λyy 2 + λzz 2 + 2 ∂x ∂y ∂z

∂2 T ∂2 T ∂2 T + (λ xz + λzx ) + (λzy + λyz ) = −W0 . ∂x∂y ∂x∂z ∂z∂y

(2.84)

In order to solve the stationary heat transfer equations, the boundary conditions should be attached. First (I), second (II), third (III) and fourth (IV) boundary conditions are mainly applied [315]: First order boundary conditions. A temperature distribution on the body surface S as the coordinates function is applied T s = g(x, y, z),

x, y, z ∈ S .

(2.85)

Heating and cooling processes of the body for a given temperature variation on its boundary or for intensive heat exchange on its surface, when the surface temperature, are representive examples. However, these conditions are rather rarely to be met in practice, and (2.85) are applied for purely mathematical purposes and errors estimating only. Second order boundary conditions. Heat stream distribution is assumed on a body surface as the function of coordinates W s = θ(x, y, z),

x, y, z ∈ S .

Owing to the Fourier rule, the condition (2.86) reads ∂T = θ(x, y, z), x, y, z ∈ S . −λ ∂n S

(2.86)

(2.87)

In the particular case, when a density of a heat stream on the body surface is constant, then W s = W0 = const. Such heat transfer conditions can be realized during bodies heating through high temperature sources, and when heat transfer occurs through radiation within the

2.2 Flexible Anisotropic Shallow Shells in Temperature Fields

185

Stefan-Boltzman rule, assuming that a body temperature is significantly less than a temperature of a radiating surface. In the case ∂T = 0, Ws = − ∂n S a so called heat isolation takes place. Third order boundary condition. On the space body boundaries a dependence of heat stream density, caused by heat transfer between surface body temperature T, and surrounding medium temperature T 0 , is given. In the case of a body cooling (T S > T 0 ), one gets W s = ξ(T S − T 0 ),

(2.88)

where ξ is proportionality coeﬃcient, known as heat transfer coeﬃcient and measured as W/(m2 grad). Equivalently ∂T = ξ(T S − T 0 ). (2.89) ∂n The equation (2.89) is the analytical expression governing third order boundary condition, which is widely used in heat transfer investigation in solid bodies surrounded by a fluid stream. Observe that third order boundary conditions yields also first and second order boundary conditions as its particular case. If ξ/a → ∞ (ξ → ∞ for λ = const or λ → 0 for ξ = const), then the first order boundary conditions are yielded 1 ∂T = 0, T S − T 0 = lim ξ/λ→∞ ξ/λ ∂n S −λ

i.e. T S = T 0 . If ξ → 0, than a particular case of the second order boundary conditions is obtained ∂T = 0. −λ ∂n s Fourth order boundary conditions. They govern heat transfer between a body surface and surrounding medium (convection between a body and a fluid) or between two contacting solid bodies, where a temperature of contacting surfaces is the same T 1S = T 2S , −λ1

∂T 1 ∂n

= −λ2 S

(2.90) ∂T 2 ∂n

.

(2.91)

S

Equation (2.90) exhibits continuity condition of the temperature filed, whereas equation (2.91) governs energy conservation rule on contacting bodies surface.

186

2 Stability of Rectangular Shells within Temperature Field

2.3 Solution of 3D Stationary Heat Transfer Equation 2.3.1 The method In order to solve a stationary heat transfer equation, the finite diﬀerence method is applied. Process of solution of PDEs through finite diﬀerence method includes two fundamental steps: (i) a transformation of diﬀerential equations and boundary conditions into diﬀerence (mesh) system of equations; (ii) solution of the obtained diﬀerence equations. On the first step, a question how to estimate an accuracy of applied approximation appears. In this chapter the 4th order approximation is applied. Owing to such mesh, the diﬀerence equations order is decreased, which plays an important role during solution to 3D equation. The second step is focused on the following problem: which method (direct or iterational) should be used to solve the system of diﬀerence equations. Here an iterational method is applied, since the system of diﬀerence equation is of high order, and a direct approach can not be used. Owing to occurrence of various iterational methods, an important question arises: how to choose the most suitable solution method for a given problem. In practice, a method requiring less computation time with simultaneous conservation of a given accuracy, occupying less amount of computer memory, and simple in realization is recommended. Owing to experimental comparison of various methods, the mostly suitable one (with respect to arithmetic calculus members or required computer time to solve the whole problem) is chosen (see for example [367]). Let us reconsider briefly the problem of eﬃciency of various iterational methods in application to Dirichlet problem for Laplace and Poisson’s equations in the following spaces: square and T-shape space (Figure 2.1). Iterational methods of successive upper Seidel’s relaxation, triangle method, triangle method with Chebyshev’s acceleration, implicit method of variable directions are considered. Two ways of partial derivatives approximation are studied: fifth and ninth order schemes, with the corresponding approximation o(h2 ) and o(h4 ). A comparison of the mentioned methods with respect to convergence velocity, computer memory amount and to their complexity is carried out, and the problem of parameters accelerating iterational methods convergence is also addressed. For the Laplace equation ∆T = 0, the Dirichlet problem with boundary conditions on the unit square sides ⎧ 3x1 ⎪ e , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ cos 3x2 , T |Γ = ⎪ ⎪ ⎪ e3x1 cos 3, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e3 cos 3x , 2

x2 = 0, 0 ≤ x1 ≤ 1 x1 = 0, 0 ≤ x2 ≤ 1 x2 = 1, 0 ≤ x1 ≤ 1

.

(2.92)

x1 = 1, 0 ≤ x2 ≤ 1

is solved. Recall that the exact solution to this problem is T (x1 , x2 ) = e3x1 cos 3x2 .

2.3 Solution of 3D Stationary Heat Transfer Equation x2

x2

1.0

b

Γ4 Γ5

Γ3

(1)

x2

Γ2

Γ6 Γ7

Γ1

1.0

0

187

Γ8

x1

(1) x1

0

(2)

x1

a

x1

Figure 2.1. Square and T-shirt spaces.

For the Poisson equation ∆T = − f , the Dirichlet problem within the T-shape space and with respect to the boundary Γ is solved (Γ is composed of straight intervals ΓK ). The boundary conditions have the form T |Γ1 ,Γ2 = ϕ0 ,

T |Γ4 = ϕ1 ,

T |Γ3 ,Γ5 = K + (ϕ1 − K)

x2 − x2(1)

, (1)

x2(2) − x2

T |Γ8 = ϕ0 + ϕ2

T |Γ6 ,Γ7 = ϕ2 , K=

x1 − x1(1) x2(2) − x2(1)

,

ϕ0 for Γ3 ϕ2 for Γ5 (2.93)

where: ϕ0 , ϕ1 , ϕ2 are constant quantities. Let us derive a system of diﬀerence equations, approximating the equation ∆T = − f with the boundary conditions T |Γ = ϕ(x, y) in the form AT = F,

(2.94)

where: A = K + D + N; D - diagonal; K - upper triangle; N - lower triangle matrices. The method of top relaxation applied to (2.94) yields [463]: $ % (2.95) DT (n) = BT (n−1) − ω KT (n) + (D + N)T (n−1) − F . For ω = 1 this method is transformed to Seidel one. Owing to five-points approximation, the diﬀerence equations have the form

(1) (n) (2) (n) (3) (n−1) (4) (n−1) (n−1) , (2.96) T i(n) j = ω ai j T i−1 j +ai j T i j−1 +ai j T i+1 j +ai j T i j+1 + f + (1 − ω)T i j h2

h2

(3) (2) (4) 2 1 where: a(1) i j = ai j = 2(h21 +h22 ) , ai j = ai j = 2(h21 +h22 ) , h1 , h2 - are mesh cell length in direction of x1 and x2 , correspondingly. For nine-points approximation, one gets

(1) (n) (n) (n) (3) (n−1) (4) (n−1) (5) (n−1) T i(n) j = ω ai j T i−1 j + ai j T i j−1 + ai j T i+1 j + ai j T i j+1 + ai j T i−1 j+1 +

188

2 Stability of Rectangular Shells within Temperature Field

(n) (7) (n) (8) (n−1) (n−1) a(6) , i j T i−1 j−1 + ai j T i+1 j−1 + ai j T i+1 j+1 + f + (1 − ω)T i j where: (3) a(1) i j = ai j =

1 5h22 − h21 , 10 h21 + h22

(4) a(2) i j = ai j =

(2.97)

1 5h21 − h22 , 10 h21 + h22

1 . 20 Iterational multiplier is obtained through well known formula [463]: (6) (7) (8) a(5) i j = ai j = ai j = ai j =

ωopt =

2 , ) 1 + 1 − λ21

(2.98)

where: λ1 is a spectral matrix radius, corresponding to Seidel method. Recall that λ is known for both five- and nine-points approximation, i.e. λ(5) 1 = λ(9) 1

h22 2(h21 + h22 )

cos

h21 πh1 πh2 + , cos a b 2(h21 + h22 )

⎛ ⎞ πh2 5h22 − h21 1 ⎜⎜⎜ πh1 5h21 − h22 πh2 ⎟⎟⎟ πh1 ⎟⎠ , cos + 2 + 2 = ⎜⎝cos cos cos 5 a b a b h1 + h22 h1 + h22

(2.99)

(2.100)

where: a, b are length of rectangular sides (0 ≤ x1 ≤ a, 0 ≤ x2 ≤ b). In order to get ωopt for an arbitrary space, and approximation value λ1 is used in (2.98). There exist a relation [463], which couples spectral radiuses of both matrix and iteration process (2.95) via µ1 , λ1 and ω of the form (µ1 + ω − 1)2 = λ21 . ω2 µ1

(2.101)

Therefore, one may define an approximated relaxation multiplier ωopt via the formula 2 . (2.102) ωm+1 = + 1+

1−

µ(m) 1 +ωm −1 2 µ(m) 1 ωm

The approximated value µ(m) 1 is yielded by the Lusternik method [463]: (( ( (m+1) (m+1) (m) ( Ti j − T i(m) ( ( i j T − T j ( ( , ≈ = lim µ(m) 1 m→∞ ( (T (m) − T (m−1) (( i j T (m) − T (m−1) ij ij where: m denotes the minimal number, for which the inequality holds (m) µ1 (m−1) − 1 ≤ εµ , µ1

(2.103)

(2.104)

2.3 Solution of 3D Stationary Heat Transfer Equation

189

and εµ is a small given quantity, ω0 is a given value (for example, ω0 = 1). Carrying is defined. However, the velocity out m1 iterations through Seidel method, µ(m−1) 1 convergence essentially increases, if one takes ω0 either as 0.1 or as 0.2. The latter observation can be achieved applying the so called equivalent rectangular method. Owing to this method, the spectral radius of a matrix in Seidel method is sought for the equivalent rectangular, i.e. with respect to surface equal to given space. It can be obtained through diameter of the largest circle, which can be drawn within the space boundaries (a circle should not either intersect a boundary curve nor include an external space); its length is yielded by dividing the space area via width. Besides, ωopt can be taken as the following asymptotic value ωopt

2πh1 h2 =2− ) h21 + h22

+

1 1 + a2 b2

(2.105)

(for five-points scheme) 0 ωopt = 2 −

6πh1 h22

1 1 , + (h21 + 10h22 )(h21 + h22 )(20h22 − h21 ) a2 b2 h21 + 25h22

(2.106)

(for nine-points scheme), where: h1 , h2 , a, b are the sam as in (2.99). It is worth noticing that the iteration number will be smaller, if ωopt is given either directly through the formulas (2.105), (2.106) or through equivalent rectangular method. It follows form (2.104) that iteration number h depends on εµ . Numerical experiment shows, that for a square the optimal interval for (i) εµ is [0.001 ÷ 0.005] for h1 = h2 = 0.625 for five-points approximation; (ii) εµ ∈ [0.005 ÷ 0.01] for nine-points approximation; (iii) εµ ∈ [0.001; 0.01] for five-points approximation of T-shape space. Notice that εν decreases with a step decrease. Among all explicit methods of variable directions, good estimation of convergence speed and economical realization of spatial step exhibit triangle methods, since the triangular matrices are simply transformed. In addition, each spatial step is realized via explicit formulas. For example, if the initial matrix (2.94) is divided into two triangular matrices 1 (1) (2) (A1 T )i j = ai j T i−1 j − T i j + ai j T i j−1 , 2 1 (4) (A2 T )i j = a(3) (2.107) i j T i+1 j − T i j + ai j T i j+1 , 2 then the corresponding explicit method of variable directions is yielded by formulas ! 1 1 1 (n) (n+ 1 ) (n+ 12 ) (2) (n+ 2 ) (3) (n) + τ a(1) Ti j 2 = T τ ij i j T i−1 j + ai j T i j−1 + ai j T i+1 j + 1+ 2

190

2 Stability of Rectangular Shells within Temperature Field (n) a(4) i j T i j+1

1 − T i(n) 2 j

2(n+1) = T ij (n+ 12 ) a(2) i j T i j−1

1 1+

,

i = 1, 2, . . . , N1 − 1, j = 1, 2, . . . , N2 − 1,

1

(n+ 12 )

Ti j

τ 2

1 (n+ 1 ) − Ti j 2 2

(2.108)

! 1 (1) (n+ 2 ) (4) 2(n+1) 2(n+1) + τ a(3) i j T i+1 j + ai j T i−1 j + ai j T i j+1 +

,

i = N1 − 1, . . . , 1; j = N2 − 1, . . . , 1.

(2.109)

Numerical experiment shows, that multiplier τ increasing a convergence in (2.108), (2.109) is equal to 2 . τcp = h1 + h2 In order to increase a speed of triangular method convergence the Chebyshew convergence acceleration is applied. Iterational process is formulated in the following manner: iterational formulas of triangular method (2.108), (2.109) are supplemented by the following formula

(n+1) 2 = T i(n) − T i(n) (2.110) T i(n+1) j j + λn T i j j , where: λn are certain parameters improving the iterational process convergence αn =

−1 2n − 1 M 2n − 1 M 1 + cos π 1− 1 + cos π , n = 1, . . . , N. (2.111) 2 2N 2 2N

Quantity M (maximal matrix eigenvalue of iterational process) is usually found approximately. For this aim, for instance, the Lusternik algorithm (2.103) can be used. Owing to n1 iterations through formulas (2.108), (2.109), i.e. without Chebyshew acceleration, the following formula is obtained λ1(n1 )

⎞⎛ ⎞ ⎛ ⎟ ⎜⎜ ⎟−1 ⎜⎜⎜ ⎟ ⎟ ⎜ (n1 +1) (n1 ) ⎟ (n1 ) (n1 −1) ⎟ ⎟ ⎜ ⎜ = ⎜⎜⎝ T i j − T i j ⎟⎟⎠ ⎜⎜⎝ T i j − T i j ⎟⎟⎟⎠ , ij

(2.112)

ij

which tends to λ1 = M for n1 → ∞. If it changes from iteration to iteration slightly, this quantity can be taken as approximating value λ1 , i.e. as M. The λn is calculated, and iterations are realized through Chebyshew acceleration. Further, let us focus on consideration of the Pisman-Pakford method. In this approach a transition from one iteration to another one is realized via two steps [594]:

1 (n+ 12 ) (n) (n+ 12 ) T + A T A , T T (n+ 2 ) = T (n) − τ(1) Γ = ϕ, 1 2 n

1 1 (2.113) A1 T (n+ 2 ) + A2 T (n+1) , T (n+1) Γ = ϕ, T (n+1) = T (n+ 2 ) − τ(2) n (2) where: τ(1) n and τn are the parameters accelerating an iterational process. The first formula in (2.113) is implicit one with respect to a horizontal direction, whereas the second one is implicit with respect to vertical direction. The following formulas hold

2.3 Solution of 3D Stationary Heat Transfer Equation

−Aα T = Λα T = T xα xα ,

191

α = 1, 2,

1 (0) (3) (−A1 T )i j = a(1) i j T i−1 j − ai j T i j + ai j T i+1 j , 2 1 (0) (4) (−A2 T )i j = a(2) i j T i j−1 − ai j T i j + ai j T i j+1 , 2 where: a(k) i j are defined in (2.97). In the case of high order accuracy scheme A = A1 + A2 −

h21 + h22 A1 A2 . 12

Fundamental idea of the method of variable directions consists of reduction of transition from one to another iteration yielding a solution along rows and columns of one-dimensional problems, which are solved via iterational method. Algorithm of solution of algebraic equations (2.113) using this method is reduced to successive solution along rows of the following equations 1

1

(n+ 2 ) = Fn , T (n+ 2 ) = τ(1) n A1 T

where: (n) Fn = T (n) − τ(1) n A2 T ,

(2.114)

1 T (n+ 2 ) = ϕ, Γ

and along columns of the following equations (n+1) T (n+1) + τ(2) = Fn+ 12 , n A2 T

where:

1

(2.115)

1

(n+ 2 ) Fn+ 12 = T (n+ 2 ) − τ(2) . n A1 T

For the scheme of higher order approximation formulas analogous to (2.114), (2.115), can be rewritten to the following form [591]: ⎞ ⎞ ⎛ ⎛ ⎜⎜⎜ ⎜⎜⎜ (1) h21 ⎟⎟⎟ ⎟⎟⎟ (n+ 1 ) (2.116) ⎝⎜E + ⎝⎜τn − ⎠⎟ A1 ⎠⎟ T 2 = Φn , 12 where:

⎞ ⎞ ⎛ ⎛ h2 + h22 ⎜⎜⎜ ⎜⎜⎜ (1) h22 ⎟⎟⎟ ⎟⎟⎟ (n) 1 A2 ϕ, Φn = ⎝⎜E − ⎜⎝τn + ⎟⎠ A2 ⎟⎠ T , T (n+ 2 ) = ϕ − 1 Γ 12 12 ⎞ ⎞ ⎛ ⎛ ⎜⎜⎜ ⎜⎜⎜ (2) h22 ⎟⎟⎟ ⎟⎟⎟ (n+1) = Φn+ 12 , ⎝⎜E + ⎝⎜τn − ⎠⎟ A2 ⎟⎠ T 12 ⎞ ⎞ ⎛ ⎛ h21 ⎟⎟⎟ ⎟⎟⎟ (n+ 1 ) ⎜⎜ ⎜⎜ (n+ 12 ) 2 , ⎟ ⎟ Φn+ 12 = ⎜⎜⎝E − ⎜⎜⎝τ(2) + T A T Γ = ϕ. ⎠ 1⎠ n 12

(2.117)

Acceleration of iterational process convergence is achieved via the appropriate (2) choice of the parameters τ(1) n and τn . Following [591], a computation of optimal

192

2 Stability of Rectangular Shells within Temperature Field

(2) parameters τ(1) n and τn for the problem (2.93), (2.115) is reduced to the following formulas: S ωn + τ S ωn − τ , τ(2) , (2.118) τ(1) n = n = 1 + ωn p 1 − ωn p

where: S =τ+ τ= ωn =

1− p ∆

,

p=

∆1 − ∆2 + (∆1 + ∆2 )p

κ−ξ , κ+ξ ,

ξ=

(1 + 2θ)(1 + θσ ) , 2θσ/2 (1 + θ1−σ + θ1+σ )

σ=

ν≈

2∆1 ∆2

1 4 4 ln ln , π2 ε η

κ= 0

(∆1 − δ1 )∆2 (∆2 + δ1 )∆1

,

(∆1 − δ1 )(∆2 − δ2 ) (∆1 + δ2 )(∆2 + δ1 )

2n − 1 , 2n 1 2 1 θ= η 1 + η2 , 16 2

,

n = 1, 2, . . . , ν, η=

1−ξ . 1+ξ

In the above δ1 , δ2 are the minimal eigenvalues of the operators A1 and A2 , respectively; A1 , A2 are the maximal eigenvalues of these operators. They are assumed to be known. Finally, ε is the required accuracy of the iterational process. For a scheme of higher order accuracy (2.116), (2.117), computations of iterational parameters can be carried out through formulas (2.118), substituting δα , ∆α by δ˜ α , ∆˜ α . The latter are coupled by relations δ˜ α =

δα , 1 − κα δα

∆˜ α =

∆α 1 − κα ∆α

,

(2.119)

where:

h2α . 12 For a rectangular space, a higher order accuracy scheme with an optimal choice of parameter series is realized through formulas (2.118), (2.119). Results are given in Table 2.1. They show, that the method is fastly convergent, since after 12 iteration the required accuracy 10−6 is achieved. However, owing to comparison to explicit methods, i.e. top relaxation, triangular with Chebyshev’s acceleration, machine time required for one iteration computation using the scheme (2.116)–(2.119) is two times larger. Besides, for the scheme (2.116)–(2.119), the memory volume required for storα = 1, 2,

(n+ 1 )

κα =

2 is equal to 2N, where N denotes number of points in the space. age of T i(n) j , Ti j In all considered iterational methods the computation is continued until the following inequality is achieved T (n) − T (n−1) ij ij < εum , (2.120) maxi, j (n) T ij

where: εum is given small quantity.

2.3 Solution of 3D Stationary Heat Transfer Equation

193

In Table 2.2 the discussed methods for five-points approximation in application to the problem (2.93) are compared. In Table 2.3 dependencies of iteration number for upper relaxation and Seidel methods using five- and nine-points approximations reported. Nine-points approximation has higher convergence velocity than five-points one. Besides, nine-points approximation decreases an order of diﬀerence equations, and hence a shorter machine time is required to achieve a given accuracy in comparison to five-points scheme. Table 2.1. Comparison of computational results using various computational schemes and applying formulas (2.147) and (2.119) (h = h1 = h2 = 0.0625, εum = 10−6 ; squared space). Method Scheme

Variable directions Seidel Upper relaxation triangular implicit fivenine- fiveninefiveninepoints points points points points points

Iterational (2) parameters 1 1 1.65 1.65 20 τ(1) n , τn Memory storage N N N N N 2N Iterations number 247 225 89 49 80 12 Error in point (0.5; 0.5) 0.001 0.0006 0.0005 0.00009 0.0001 0.00003 Convergence π2 h2 speed 1.2π2 h2 2πh 2.09πh wπh 2

Table 2.2. Computational eﬃciency of various methods applied to problem (2.93) (εum = 10−4 , h1 = h2 = h = 0.25, T-shape space). Method Seidel Upper relaxation Triangular With Chebyshev’s acceleration

Convergence Iterations Iterational velocity number Storage parameters π2 h2 2

300

N

ω=1

2πh

46

N

ω0 = 1.5

2πh

55

N

τcp

√ 2π h

30

N

λn

In Table 2.1 Seidel, upper relaxation and variable directions methods with fiveand nine-points approximation are compared on example of the problem (2.92) for a squared space.

194

2 Stability of Rectangular Shells within Temperature Field

Table 2.3. Dependence of iteration number for upper relaxation and Seidel’s methods using five (n(5) ) and nine (n(9) ) points approximations (h1 = h2 = h = 0.0625, εum = 10−6 , squared space). ω 1 1.58 n(9) 225 68 n(5) 247 100

1.61 60 96

1.65 49 89

1.675 50 94

Owing to Table 2.1, for the squared space the method of variable directions with the scheme of higher order accuracy is mostly eﬀective. Then upper relaxation with nine-points approximation follows. The latter one can be applied for an arbitrary space, since an optimal iterational parameter is computed in the computational process. Since a priori knowledge of maximal eigenvalue is not required, this method seems to be the most universal. Besides, the method of over relaxation is distinguished by its simplicity and requires a minimal memory storage (one working field N). To conclude, on a basis of carried out numerical experiments top relaxation method is the mostly economical one, and hence it is further applied to solve the 3D heat transfer equation. 2.3.2 Construction of diﬀerence schemes As it has been mentioned already in section 2.1.1, diﬀerence scheme of fourth order accuracy allows to take more larger mesh in comparison to the schemes o(h2 ). This yields essential decrease of an order of diﬀerence equations system, which is essentially important for solutions of multidimensional problems. In the reference [592] the schemes at 4th and 6th accuracy order for the 2D Poisson equation are constructed, as well as the 4th order scheme for the equation ∂2 T ∂2 T ∂2 T + 2a + = − f (x1 , x2 ), ∂x1 ∂x2 ∂x22 ∂x12 where |a| < 1 is constant number, is given. In what follows the 4th order accuracy diﬀerential scheme for the multidimensional equation n n ∂2 T ∂2 T LT ≡ + 2 K = − f (x), i j ∂xi ∂x j ∂xi2 i=1 i, j=1 x = (x1 , x2 , . . . , xn ) ∈ Gn {0 ≤ xα ≤ lα , α = 1, 2, . . . , n} .

(2.121)

is constructed. Consider the diﬀerence operator [592] Λv ≡

n i=1

v xi xi +

n

$ % Ki j v xi x j + v xi x j .

(2.122)

i, j=1

Diﬀerence operators in (2.122) are splitted into series with power h (h - mesh step) using the Taylor series

2.3 Solution of 3D Stationary Heat Transfer Equation

∂2 v h2 ∂4 v + + o(h4 ), ∂xi2 12 ∂xi4

v xi xi = v xi x j =

195

∂2 v h ∂3 v h2 ∂4 v h2 ∂4 v − + + + ∂xi ∂x j 2 ∂xi2 ∂x j 6 ∂xi3 ∂x j 6 ∂xi ∂x3j h ∂3 v h2 ∂4 v − + o(h3 ), 2 ∂xi ∂x2j 4 ∂xi2 ∂x2j

v xi x j =

∂2 v h ∂3 v h ∂3 v h2 ∂4 v + − + + 2 3 ∂xi ∂x j 2 ∂xi ∂x j 2 ∂xi ∂x j 6 ∂xi3 ∂x j h2 ∂4 v h2 ∂4 v − + o(h3 ). 6 ∂xi ∂x3j 4 ∂xi2 ∂x2j

Substituting these series in (2.122), one gets ⎞ ⎡ n ⎛ 2 n ⎜⎜⎜ ∂ v h2 ∂4 v ⎟⎟⎟ ⎢⎢⎢ ∂2 v h2 ∂4 v ⎜ ⎟ ⎢ Λv ≡ K + + + 2 ⎝ 2+ ⎠ ⎣ i j 12 ∂xi4 ∂xi ∂x j 3 ∂xi3 ∂x j ∂xi i=1 i, j=1 ⎤ n n h2 ∂4 v ⎥⎥⎥⎥ h2 ∂4 v ∂2 v ∂2 v 4 − ) = + 2 K + + o(h ⎥ i j 3 ∂xi ∂x3j 2 ∂xi2 ∂x2j ⎦ ∂xi ∂x j ∂xi2 i=1 i=1 i j

⎛ n ⎡ ⎤⎞ n 4 4 ⎢⎢⎢ ∂4 v ⎥⎥⎟⎟⎟ v v h2 ⎜⎜⎜⎜ ∂4 v ∂ ∂ ⎜⎜⎝ + Ki j ⎢⎢⎣4 3 +4 − 6 2 2 ⎥⎥⎥⎦⎟⎟⎟⎠ + o(h4 ) = 4 3 12 i=1 ∂xi i, j=1 ∂xi ∂x j ∂xi ∂x j ∂xi ∂x j Lv +

n ∂4 v h2 2 h2

L v− + o(h4 ) = 1 + 3Ki j + 2Ki2j 12 6 i=1 ∂xi2 ∂x2j i j

−f −

n ∂4 v h2

h2 Lf − + o(h4 ). 1 + 3Ki j + 2Ki2j 12 6 i=1 ∂xi2 ∂x2j i j

In what follows, the diﬀerence equation Λ v ≡

n i=1

v xi xi +

n $

Ki j v xi x j + v xi x j +

i=1 i j

h2

2 1 + 3Ki j + 2Ki j v xi xi x j x j = −ϕ , 6 where ϕ = f +

h2 L f, 12

(2.123)

(2.124)

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2 Stability of Rectangular Shells within Temperature Field

approximates the equation (2.121) with error o(h4 ). If ki j ≡ 0 for i j, the the equation (2.121) is transformed into the multidimensional Poisson’s equation, and the diﬀerence equation n n h2 h2

(2.125) v xi xi + vx x x x = − f + L f Λv≡ 6 i, j=1 i i j j 12 i=1 approximates multidimensional Poisson’s equation. This diﬀerence equation (2.125) coincides on the squared mesh with a diﬀerence scheme given in reference [592] for n-dimensional Poisson’s equation, i.e. it will approximate the equation (2.82). Observe that the general type equation n

Kii

i=1

∂2 T ∂x2i

+2

n i, j=1

Ki j

∂2 T = − f (x), ∂xi x j

(2.126)

where ki j is constant, is transformed to the form(2.121) via the following variables transformation 3 xi = Kii xi . Therefore, the diﬀerence equation (2.124) is approximating the equation (2.84) with the error o(h4 ) after the variables transformation 3 3 3 x1 = K11 x1 , x2 = K22 x2 , x3 = K33 x3 . Let us construct a diﬀerence scheme for the equation LT ≡

n

Li T = − f (x),

i=1

x = (x1 , x2 , . . . , xn ) ∈ Gn {0 ≤ xα ≤ lα , α = 1, 2, . . . , n}, Li T =

∂2 T ∂T + Ki (x) . ∂xi ∂xi2

(2.127)

It is assumed that ki (x), f (x) ∈ C (4) (Gn ), T (x) ∈ C (6) (Gn ). Consider the diﬀerence operator Λv =

n

◦

Λi v,

◦

where Λi v = v xi xi + ai x x◦i , ai = Ki (x).

i=1

Applying the Taylor series with respect to step h of the form v xi xi =

∂2 v h2 ∂4 v + + o(h4 ), ∂xi2 12 ∂xi4

v x◦i =

∂v h2 ∂3 v + + o(h4 ). ∂xi 6 ∂xi3

(2.128)

2.3 Solution of 3D Stationary Heat Transfer Equation

Then

⎤ n n ⎡ h2 ⎢⎢⎢ ∂4 v ∂2 v ∂v ∂3 v ⎥⎥⎥ ⎢⎣ 4 + 2ai 3 ⎥⎦ + o(h4 ) = Λv = + ai + 2 ∂x 12 ∂x ∂xi ∂xi i i i=1 i=1 ⎤ n ⎡ ∂3 v ⎥⎥⎥ h2 ⎢⎢⎢ ∂4 v ⎢⎣ + 2ai (x) 3 ⎥⎦ + o(h4 ). Lv + 12 i=1 ∂xi4 ∂xi

Let us calculate

⎞⎛ n ⎞ ⎛ n ⎜⎜⎜ ∂2 ∂ ⎟⎟⎟⎟ ⎜⎜⎜⎜ ∂2 v ∂v ⎟⎟⎟⎟ ⎜ L v = ⎜⎝ + ai (x) + ai (x) ⎟⎠ = ⎟⎜ ∂x1 ⎠ ⎝ i=1 ∂xi2 ∂xi ∂xi2 i=1 2

n ⎛ 4 ⎜⎜⎜ ∂ v ∂2 ai ∂v ∂3 v ∂2 v ∂ai ∂2 v ⎜⎝ 4 + 2 +2 + ai 3 + a2i 2 + 2 ∂xi ∂xi ∂xi ∂xi ∂xi ∂xi ∂xi i=1 ⎞ ⎛ ⎞ n n ⎜⎜⎜ ∂4 v ∂ai ∂v ∂3 v ⎟⎟⎟ ∂3 v ⎟⎟⎟ ⎜⎝ 4 + 2ai 3 ⎟⎠ + + ai 3 ⎠⎟ + Li L j v = ai ∂xi ∂xi ∂xi ∂xi ∂xi i=1 i=1 i j

⎞ ⎤ n ⎡⎛ 2 n ⎢⎢⎢⎜⎜⎜ ∂ ai ∂ai ⎟⎟⎟ ∂v ∂a1 ∂2 v ⎥⎥⎥ 2 ⎢⎣⎜⎝ 2 + ai ⎟⎠ ⎥ + ai + 2 Li L j v. + ⎦ ∂xi ∂xi ∂xi ∂xi2 ∂xi i=1 i=1 i j h2 2 12 L v

Now, the expression is added and removed from Λv: ⎡ ⎤ ⎢⎢ n ⎛⎛ ⎥⎥⎥ ⎞ 2 ⎞ n 2 ⎢ 2 ⎢ ⎥⎥⎥ ⎟ ∂ ∂a ∂ai ⎟⎟⎟ ∂v v h ⎢⎢⎢ ⎜⎜⎜⎜⎜⎜ ∂ ai ⎟ i ⎟⎟⎠ + ⎥⎥⎥ ⎟⎠ ⎢⎢⎢ ⎜⎝⎜⎝ 2 + ai + a2i + 2 L L v i j ⎥⎥⎥ 12 ⎢⎢⎣ i=1 ∂xi ∂xi ∂xi ∂xi ∂xi2 ⎦ i=1 i j

Therefore one gets ⎞ ⎤ ⎛ n ⎡ h2 ⎢⎢⎢ 2 ∂ai ∂2 v ⎜⎜⎜ ∂2 ai ∂ai ⎟⎟⎟ ∂v ⎥⎥⎥ h2 2 ⎢⎣ ai + 2 ⎟⎠ ⎥⎦ − + ⎜⎝ 2 + ai Λv = Lv + L v − 12 12 i=1 ∂xi ∂xi2 ∂xi ∂xi ∂xi n n ⎡ h2 ⎢⎢⎢ 2 h2 ∂ai ∂2 v h2 4 ⎢ Li L j v + o(h ) = − f L f − + ⎣ ai + 2 12 i=1 12 12 i=1 ∂xi ∂xi2 i j

⎛ 2 ⎞ ⎤ n ⎜⎜⎜ ∂ ai ∂a ⎟ ∂v ⎥⎥⎥ h2 ⎜⎝ 2 + ai i ⎟⎟⎟⎠ ⎥⎦ − Li L j v + o(h4 ). ∂xi ∂xi 12 i=1 ∂xi i j

The following approximation is introduced ⎞ ⎞⎛ ⎛ 2 ⎜⎜⎜ ∂ ∂ ⎟⎟⎟ ⎜⎜⎜⎜ ∂2 v ∂v ⎟⎟⎟⎟ ◦ ◦ ⎟⎠ ⎜ Li L j v = ⎜⎝ 2 + ai + aj ⎟ = Λi Λ j v + o(h2 ), ∂xi ⎝ ∂x2j ∂x j ⎠ ∂xi

197

198

2 Stability of Rectangular Shells within Temperature Field

and hence the diﬀerence equation ⎞ ⎤ ⎛ 2 n ⎡ ⎜⎜⎜ ∂ ai h2 ⎢⎢⎢ 2 ∂ai ∂ai ⎟⎟⎟ ⎥⎥⎥

⎟⎠ v ◦ ⎥⎦ + Λ v ≡ Λv + v xi xi + ⎜⎝ 2 + ai ⎣⎢ ai + 2 12 i=1 ∂xi ∂xi xi ∂xi n i=1 i j

h2 Λi Λ j v = − f + L f 12 ◦

◦

(2.129)

approximates equation (2.127) with the error o(h4 ). The diﬀerence equation (2.129) can be rewritten to the form ⎤ ⎫ ⎡ n ⎧ ⎪ ⎢⎢⎢ ∂ai ∂2 ai ⎥⎥⎥ ⎪ ∂ai h2 2 ⎨ ⎬

Λv≡ ⎪ ⎩ 1 + 12 ai + 2 ∂xi v xi xi + ⎣⎢ai + ai ∂xi + ∂x2 ⎥⎦ v x◦i ⎪ ⎭+ i i=1 n h2 h2 ◦ ◦ Λi Λ j v = − f + L f , 12 i=1 12

(2.130)

i j

where:

◦

◦

Λi Λ j v = v xi xi x j x j + ai v x j x j x◦i + ⎛ 2 ⎞ ⎜⎜⎜ ∂ a j ∂a j ⎟⎟⎟ ∂a j ⎜⎝ 2 + ai + ai a j v x◦i x◦j . ⎠⎟ v ◦ + 2 ∂xi x j ∂xi ∂xi ◦

Note that the form of Λi Λ j v follows from approximation Li L j v with the error o(h2 ) ⎞ ⎞⎛ ⎛ 2 ⎜⎜⎜ ∂ ∂ ⎟⎟⎟ ⎜⎜⎜⎜ ∂2 v ∂v ⎟⎟⎟⎟ ∂3 v ∂4 v ⎟⎠ ⎜⎝ 2 + a j Li L j v = ⎜⎝ 2 + ai + ⎟⎠ = 2 2 + ai ∂xi ∂x j ∂x j ∂xi ∂xi ∂x j ∂xi ∂x2j ∂4 v ∂v ∂ ∂v ∂3 v ∂3 v ∂2 a + a a = + a + a + j i j i j ∂x j ∂xi ∂x j ∂xi2 ∂xi2 ∂x2j ∂xi ∂x2j ∂xi2 ∂x j ∂a j ∂v ∂a j ∂2 v ∂2 a j ∂v ∂2 v + 2 + a a + ai = i j ∂xi ∂xi ∂x j ∂xi ∂x j ∂xi ∂x j ∂xi2 ∂x j ⎞ ⎛ 2 ⎜⎜⎜ ∂ a j ∂a j ⎟⎟⎟ ⎟⎠ v ◦ + v xi xi x j x j + ai v x j x j x◦i + a j v xi xi x◦j + ⎝⎜ 2 + ai ∂xi x j ∂xi ∂a j 2 + ai a j v x◦i x◦j + o(h2 ). ∂xi If ai (x) = ai = const, then the scheme (2.130) is given by n n h2 2 h2 ◦ ◦ Λv≡ 1 + ai v xi xi + ai v x◦i + Λi Λ j v = 12 12 i=1 i=1

i j

2.3 Solution of 3D Stationary Heat Transfer Equation

− f+ where:

◦

199

h2 Lf , 12

(2.131)

◦

Λi Λ j v = v xi xi x j x j + ai v x j x j x◦i + a j v xi xi x◦j + ai a j v x◦i x◦j . Let us construct a diﬀerence scheme for the equation (2.81), which is rewritten in the abbreviated form LT ≡ A1

LT ≡

3

∂2 T ∂2 ∂2 T ∂T + A2 2 + A3 2 + a3 = − f (x), 2 ∂x3 ∂x1 ∂x2 ∂x3

Ai Li T + a3 L3 T = − f,

x ∈ G3 {0 ≤ xα ≤ lα , α = 1, 2, 3} .

(2.132)

i=1

Consider the diﬀerence operator Λv = A1 v x1 x1 + A2 v x2 x2 + A3 v x3 x3 + a3 v x◦3 . First, we apply the series development with respect to h of diﬀerence operators (2.128) ⎛ ⎞⎤ 3 ⎡ h23 ∂3 v ⎢⎢⎢ ∂2 v h2i ∂4 v ⎜⎜⎜ ∂v ⎟⎟⎥⎥ 4 4 ⎢⎣Ai 2 + + o(hi ) + a3 ⎜⎝ + + o(h3 )⎟⎟⎠⎥⎥⎦ = Λv = 4 3 12 ∂xi ∂x3 6 ∂x3 ∂xi i=1 3

(Ai Li v) + a3 L3 v +

i=1

Lv +

3 i=1

Ai

⎤ 3 ⎡ h2 ⎢⎢⎢ h21 2 ⎥ ⎢⎣Ai Li v + 3 a3 L3 L3 v + o(|h|4 )⎥⎥⎥⎦ = 12 6 i=1

h2 h2i 2 Li v + 3 2a3 L3 L3 v + o(|h|4 ), 12 12

|h|4 = h21 + h42 + . . . + h4n .

Second, observe that v is a solution of equation (2.132) Lv ≡ A1 L1 v + A2 L2 v + A3 L3 v + a3 L3 v = − f. Then, one finds

h23 h2i 2 12 Ai Li v, 12 a3 L3 L3 v,

h21 h2 h2 h2 h2 A1 L12 v = − 1 A2 L1 L2 v − 1 A3 L1 L3 v − 1 a3 L1 L3 v − 1 L1 f, 12 12 12 12 12 h22 h2 h2 h2 h2 A2 L22 v = − 2 A1 L2 L1 v − 2 A3 L2 L3 v − 2 L2 L3 v − 2 L2 f, 12 12 12 12 12 h23 h2 h2 h2 h2 A3 L32 v = − 3 A1 L3 L1 v − 3 A2 L3 L2 v − 3 a3 L3 L3 v − 3 L3 f, 12 12 12 12 12 h2 a3 h2 a3 h2 a3 h2 a3 h23 a3 L3 L3 v = − 3 A1 L3 L1 v − 3 A2 L3 L2 v − 3 a3 L3 L3 v − 3 L3 f. 12 12 A3 12 A3 12 A3 12 A3

200

2 Stability of Rectangular Shells within Temperature Field

The found relations are substituted to diﬀerence operator ⎛ 2 h2 A3 + h23 A1 ⎜⎜ h A2 + h22 A1 L1 L2 v + 1 L1 L3 v+ Λv = Lv − ⎜⎝⎜ 1 12 12 h2 A3 + h23 A1 h22 A3 + h23 A2 a3 (h22 A3 + h23 A2 ) L2 L3 v + a3 1 L3 L1 v + L3 L2 v+ 12 12A3 12A3 ⎞ h23 h23 h23 a3 h21 h22 ⎟⎟ 2 a3 L3 v + L1 f + L2 f + L3 f + L3 f ⎟⎠⎟ + o(|h|4 ). 12A3 12 12 12 12 A3 Requiring to be approximated by diﬀerence equation (2.132) with error of o(|h|4 ), it is suﬃcient to take it in the form ⎞ ⎛ h2 a2 ⎟⎟ ⎜⎜ Λ v ≡ A1 v x1 x1 + A2 v x2 x2 + ⎜⎜⎝A3 + 3 3 ⎟⎟⎠ v x3 x3 + a3 v x◦3 + 12 A3 h2 A3 + h23 A1 h2 A3 + h23 A2 h21 A2 + h22 A1 v x1 x1 x2 x2 + 1 v x1 x1 x3 x3 + 2 v x2 x2 x3 x3 + 12 12 12 h21 A3 + h23 A1 h2 A3 + h23 A2 v x1 x1 x◦3 + a3 2 v x2 x2 x◦3 = 12A3 12A3 ⎞ ⎛ h23 ∂2 f h23 ∂ f ⎟⎟⎟ h22 ∂2 f h21 ∂2 f ⎜⎜⎜ − ⎝⎜ f + + + + ⎠⎟ . 12 ∂x12 12 ∂x22 12 ∂x32 12A3 ∂x3 a3

(2.133)

If h1 = h2 = h3 = h, then the diﬀerence equation (2.133) takes the form ⎞ ⎛ ⎜⎜ h2 a23 ⎟⎟⎟ ⎟⎠ v x3 x3 + a3 v x◦ + Λ v ≡ A1 v x1 x1 + A2 v x2 x2 + ⎜⎜⎝A3 + 3 12 A3 h2 h2 h2 (A2 + A1 )v x1 x1 x2 x2 + (A3 + A1 )v x1 x1 x3 x3 + (A2 + A3 )v x2 x2 x3 x3 + 12 12 12 2 $ % a3 h (A3 + A1 )v x1 x1 x◦3 + (A3 + A2 )v x2 x2 x◦3 = 12A3 ⎡ ⎞⎤ ⎛ 3 ⎢⎢⎢ a3 ∂ f ⎟⎟⎟⎟⎥⎥⎥⎥ h2 ⎜⎜⎜⎜ ∂2 f ⎢ ⎟⎟⎥⎥ . ⎜⎜ + − ⎢⎢⎣ f + (2.134) 12 ⎝ i=1 ∂a21 A3 ∂x3 ⎠⎦ Let the following equation is given ⎞ n ⎛ ⎜⎜⎜ ∂2 T ∂T ⎟⎟⎟ ⎟⎠ = − f (x), LT ≡ ⎝⎜A1 2 + ai ∂x1 ∂xi i=1 x ∈ Gn {0 ≤ xα ≤ lα , α = 1, 2, . . . , n} . Proceeding in analogical way, one find that the diﬀerence equation

(2.135)

2.3 Solution of 3D Stationary Heat Transfer Equation

Λ v ≡

⎞ n ⎡⎛ h2 a2 ⎟ ⎢⎢⎢⎜⎜⎜ ⎢⎣⎜⎝Ai + i i ⎟⎟⎟⎠ v xi xi 12 Ai i=1

201

⎡ ⎤ n ⎢ h2 A + h2 A ⎥⎥⎥ ⎢⎢⎢ i j j i v xi xi x j x j + + ai v x◦i ⎥⎦ + ⎢⎣ 12 i, j=1 i j

a1

h2j Ai + h2i A j 12Ai

⎤ ⎥⎥ h2i v x j x j x◦i + a j v xi xi x◦j + ai a j v x◦i x◦j ⎥⎥⎥⎦ = 12A j 12Ai ⎞ ⎛ ⎛ ⎞ n ⎜⎜⎜ h2i ai ∂ f ⎟⎟⎟⎟⎟⎟ ⎜⎜⎜ h2i ∂2 f ⎜ ⎜⎝ + − ⎜⎝ f + ⎠⎟⎟⎟⎠ 2 12 12 A ∂x ∂x i i i i=1 h2i A j + h2j Ai

(2.136)

approximates (2.135) with error of o(|h|4 ). Let us construct a diﬀerence scheme for equation with variable coeﬃcients ⎞ n ⎛ ⎜⎜⎜ ∂2 T ∂T ⎟⎟⎟ LT ≡ ⎝⎜Ai 2 + ai (x) ⎠⎟ = − f (x), ∂xi ∂xi i=1 LT ≡

n

Ai Li + ai (x)Li T

i=1

x=

(x1j , . . . , xnj )

∈ Gn {0 ≤ xα ≤ lα , α = 1, . . . , n} .

(2.137)

Consider the diﬀerence operator

Λv ≡ i = 1n Ai v xi xi + ai (x)v x◦i , ei h . x = (x1j , . . . , xnj ) ∈ Gn xij = jhi , i = 1, . . . , n, j = 0, 1, . . . , N1 , Ni = hi Applying series development of diﬀerence operators with respect to hi , we get ⎞ ⎞⎤ ⎛ n ⎡ ⎛ 2 ⎢⎢⎢ ⎜⎜⎜ ∂ v h2i ∂4 v ⎟⎟⎟ ⎜⎜⎜ ∂v h2i ∂3 v ⎟⎟⎟⎥⎥⎥ ⎢⎣Ai ⎜⎝ 2 + ⎟ ⎟⎥ + o(|h|4 ) = ⎜ Λv ≡ (x) + + a ⎝ i 4⎠ 3 ⎠⎦ 12 ∂x 6 ∂x ∂x ∂x i i i i i=1 Lv +

⎤ n ⎡ h2i ∂3 v ⎥⎥⎥ ⎢⎢⎢ h2i ∂4 v ⎢⎣Ai ⎥ + o(|h|4 ). + 2a (x) i 4 3⎦ 12 12 ∂x ∂x i i i=1

The following diﬀerence operator is obtained ⎞ n ⎛ h2i ⎜⎜⎜ h2i 2 ⎟⎟ 4 Λv = − f + ⎝⎜Ai Li v + 2ai (x) Li Li v⎟⎟⎠ + o(|h| ). 12 12 i=1 Ai

n n h2 h2i 2 h2i

h2i

Li v = − Li a j L j v − i Li f, A j Li L j v − 12 12 12 12 j=1 j=1 ji

202

2 Stability of Rectangular Shells within Temperature Field

⎤ n ⎡ 2

⎥⎥

h2i h2i 2 ⎢⎢⎢ hi ⎢⎣ Ai Li v + ai Li Li v = − A j Li L j v + Li a j L j v ⎥⎥⎦ − 12 12 12 j=1 ji

⎛ ⎞ h2i ⎜⎜⎜ ∂2 ai ∂v ∂ai ∂2 v ⎟⎟⎟ h2i ⎜⎝ 2 ⎟⎠ − Li f, +2 12 ∂xi ∂xi ∂xi ∂xi2 12 ai

n

h2i h2i ai

Li L i v = − A j Li L j v + Li a j L j v − 12 12 Ai j=1 ji

⎛ ⎞ h2i ai ⎜⎜⎜ ∂ai ∂v ∂2 v ⎟⎟ h2 ai ⎜⎝ + ai 2 ⎟⎟⎠ − i Li f. 12 Ai ∂xi ∂xi 12 Ai ∂xi One may conclude that the equation (2.137) will be approximated by a diﬀerence equation with error o(|h|4 ), if it is taken in the form ⎛ ⎛ ⎞⎤ ⎞⎤ ⎫ ⎡ n ⎧⎡ ⎪ h2i ⎜⎜⎜ a2i h2i ⎜⎜⎜ ∂2 ai ai ∂ai ⎟⎟⎟⎥⎥⎥ ⎪ ⎢⎢⎢ ∂ai ⎟⎟⎟⎥⎥⎥ ⎨⎢⎢⎢ ⎬

Λ v≡ ⎪ ⎩⎣⎢Ai + 12 ⎝⎜ Ai +2 ∂xi ⎠⎟⎦⎥ v xi xi + ⎣⎢ai + 12 ⎝⎜ ∂x2 + Ai ∂xi ⎟⎠⎥⎦ v x◦i ⎪ ⎭+ i i=1 ⎤ ⎡ 2 ⎧⎡ n h2i ⎪ ⎢⎢⎢ ∂ a j ai ∂ak ⎥⎥⎥ ai ⎨⎢⎢⎢ ⎥⎦ v ◦ + ⎪⎢⎣A j v xi xi x j x j + a j v xi xi x◦j + A j v x j x j x◦i + ⎢⎣ 2 + 12 ⎩ Ai Ai ∂xi xk ∂xi i, j=1 i j

⎛ ⎛ ⎞⎞ n ⎜⎜⎜ h2i ⎜⎜⎜ ∂2 f ⎟⎟⎟⎟⎟⎟ ∂a j ai a ∂ f i ⎜⎝ 2 + ⎟⎠⎟⎟ . + a j v x◦i x◦j = − ⎜⎜⎝ f + 2 ∂xi Ai 12 ∂xi Ai ∂xi ⎠ i=1

(2.138)

Finally, diﬀerence schemes with errors o(h4 ) are constructed for the equation n ∂ ∂T LT ≡ = − f (x), K(x) ∂xi ∂xi i=1 x = (x1 , . . . , xn ) ∈ Gn {0 ≤ xα ≤ lα , α = 1, . . . , n} ,

(2.139)

where LT ≡

n i=1

Li T, Li T =

∂ ∂T , K(x) = K(x1 , x2 , . . . , xn ). K(x) ∂xi ∂xi

This equation governs stationary temperature distribution of a non-homogeneous medium. 1 , and consider the diﬀerence operator Denote p(x) = k(x) Λv =

n 1 i=1

a

.

v xi xi

2.3 Solution of 3D Stationary Heat Transfer Equation

203

Developing the diﬀerence operator into the series with respect to h [592], one gets ∂ 1 ∂v h2 1 v xi = + Li (pLi v) + o(h4 ), a ∂xi p ∂xi 12 and the diﬀerence operator reads Λv ≡

n 1 i=1

a

=

v xi xi

n

Li v +

i=1

n h2 Li (pLi v) + o(h4 ). 12 i=1

(2.140)

Using the fact, that v is solution of equation (2.140), one obtains ⎛ n ⎞ ⎜⎜⎜ ⎟⎟⎟ Lv = − f, pLv = −p f, Li p ⎜⎜⎝ Li v⎟⎟⎠ = −Li p f, i=1

Li (pLi v) = −Li

n

pL j v − Li p f,

j=1 i j n

Li (pLi v) = −

i=1

n

Li (pL j v) −

i, j=1

n

Li p f.

i=1

Substituting the obtained relation into the diﬀerence operator (2.140), one gets ⎛ ⎞ ⎜⎜ ⎟⎟⎟ n n n n ⎜ ⎟⎟⎟ 2 ⎜ 1 h ⎜⎜⎜⎜ Λv = v xi = Li v − Li (pL j v) + Li p f ⎟⎟⎟⎟⎟ + o(h4 ) = ⎜⎜⎜ a 12 ⎜⎜⎝i, j=1 ⎟⎟⎠ xi i=1 i=1 i=1 i j

−

n n h2 Li (pL j v) − f − Li p f + o(h4 ). 12 i, j=1 i=1 i j

It is suﬃcient for diﬀerence equation to approximate the equation (2.139) with the error o(h4 ), if it is taken in the form n n 1 h2 h2

vx + (2.141) Λi (pΛ j v) = − f + L(p f ) , Λv≡ a i xi 12 i, j=1 12 i=1 i j

where: Λi v =

1 a v xi xi ,

and the coeﬃcient

⎛ ⎞ 1 ⎜⎜ 1 1

4 1 ⎟⎟⎟⎟ ⎜ ⎜ + + ⎟. ai = pi−1 + 4pi− 12 + pi = ⎜⎝ 6 6 ki−1 ki− 12 ki ⎠ Note that the coeﬃcients ai are obtained from the following condition

(2.142)

204

2 Stability of Rectangular Shells within Temperature Field

1 Ψ = Λi v − Li v = v xi a

xi

∂ 1 ∂v = o(h2 ). − ∂xi p ∂xi

In reference [592] it is illustrated, that it takes place when the following condition is satisfied

1 1 1 1 1 1 1 1 + = + o(h2 ), − = + o(h2 ). 2 a(x + h) a(x) p(x) h a(x + h) a(x) p This condition is satisfied, if ai are taken in the form (2.142). Consequently, the diﬀerence equation (2.141) approximates the initial equation (2.139) with the error of o(h4 ), where ai are given by (2.142). Let us now construct a diﬀerence system with an error o(h4 ) for the equation with variable coeﬃcients Lu ≡

n

Ai (x)

i=1

∂2 u = − f (x), ∂xi2

(2.143)

x = (x1 , . . . , xn ) ∈ Gn {0 ≤ xα ≤ lα , α = 1, . . . , n} , n n Λv = Ai (x)Λi v = Ai (x)v xi xi . i=1

Λi v = v xi xi =

i=1

∂ v + L2 v + o(h4 ), ∂xi2 12 i 2

h2i

where Li v =

∂2 v , ∂xi2

where Ai (x), f (x) are suﬃciently smooth functions. Developing the diﬀerence operator into the series with respect to h of of the form Λv =

n

Ai (x)Li v +

i=1

n h2i Ai (x)Li2 v + o(h4 ), 12 i=1

(2.144)

and using the fact that v is a solution to equation (2.143), the following relations are obtained Lv =

n

Ai (x)Li v = − f (x),

i=1

Li v = −

n Aj 1 L j v − f (x), A A i i j=1 i j

and therefore Li2 v

=−

n j=1 i j

Aj f . Li L j v − Li Ai Ai

Substituting the obtained result into the diﬀerence operator (2.144), one gets

2.3 Solution of 3D Stationary Heat Transfer Equation

Λv =

n

205

Ai (x)Λi v =

i=1 n

Ai (x)Li v −

i=1

n i=1

⎡ ⎤ ⎢⎢⎢ ⎥⎥⎥⎥ n ⎢⎢⎢ Aj f ⎥⎥⎥⎥ Ai ⎢⎢⎢⎢⎢ Li L j v + Li ⎥ + o(|h|4 ) = 12 ⎢⎢⎣ j=1 Ai Ai ⎥⎥⎥⎥⎦ h2i

i j

n n n h2i h2i Aj f − f (x) − Ai Li Ai − Li L j v + o(|h|4 ). 12 Ai 12 j=1 Ai i=1 i=1 i j

The following diﬀerence equation

Λv≡

n j=1

n h2i Aj Ai Λi Ai (x)Λi v + Λ jv = 12 Ai i, j=1 i j

⎛ ⎞ n ⎜⎜⎜ h2i f ⎟⎟⎟⎟ Ai Li − ⎜⎜⎝ f (x) + ⎟ 12 Ai ⎠ i=1 approximates (2.143) with the error of o(|h|4 ). This diﬀerence equation is transformed to the following form ⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎛ ⎞ n n n ⎜⎜⎜⎜ ⎜⎜⎜ h2i h2i A j ⎟⎟⎟⎟ f ⎟⎟⎟⎟

Ai Li Ai (x)Λi ⎜⎜⎜⎜E + Λ j ⎟⎟⎟⎟ v = − ⎜⎜⎝ f (x) + Λv≡ ⎟ . (2.145) 12 j=1 Ai ⎟⎟⎠ 12 Ai ⎠ ⎜⎜⎝ i=1 i=1 i j

If Ai (x) = Ai = const, then from (2.145) the following diﬀerence equation is obtained ⎛ ⎞ n n n ⎜⎜⎜ ⎟⎟ h2i h2i

A j Λi Λ j v = − ⎜⎜⎝ f (x) + Li f ⎟⎟⎟⎠ , Ai Λi v + Λv≡ (2.146) 12 12 i=1 i, j=1 i=1 i j

which for n = 3 approximates the equation (2.147) with the error of o(|h|4 ): LT ≡

3 i=1

Ai

∂2 T = − f (x). ∂xi2

(2.147)

In what follows, the Dirichlet problem for this equation in the space G3 {0 ≤ xα ≤ lα }, {α = 1, 2, 3} with the boundary Γ T |Γ = g(x), is analysed.

x ∈ Γ,

(2.148)

206

2 Stability of Rectangular Shells within Temperature Field

2.3.3 A priori convergence estimation In order to prove a convergence of diﬀerence scheme (2.146) a background of difference scheme theory [592] is briefly described. Let ω is the finite set of nodes (mesh) in a certain bounded space of n-dimensional Euclidean space and pεω is the mesh point. Let us consider the equation B(P, Q)v(Q) + F(P), P ∈ ω (2.149) A(P)v(P) = Q∈Ξ (P)

for the function v(P) given on the mesh ω. Let A(P) and B(P, Q) are the equation coeﬃcients; F(P) (denotes the right hand side of equation) are given mesh functions; Ξ (P) is the set of nodes of mesh ω except of the node R (or a neighbourhood of a node P). A pattern of mesh equation (2.149) in the node P consists of the node P and its neighbourhood Ξ (P). It is assumed that the coeﬃcients A(P) and B(P, Q) satisfy the conditions A(P) > 0, B(P, Q) ≥ 0 for P ∈ ω and Q ∈ Ξ (P), B(P, Q) ≥ 0. D(P) = A(P) −

(2.150)

Q∈Ξ (P)

The point P is said to be a boundary node of the mesh ω, if in this point the function v(P) value is given, i.e. v(P) = g(P)

for P ∈ Γ,

(2.151)

where Γ is a set of boundary nodes. Comparing (2.151) with (2.149) one may conclude, that on the boundary Γ the following formal relations can be introduced A(P) ≡ 1,

B(P, Q) = 0,

F(P) = g(P).

Owing to the notation Zv(P) = A(P)v(P) −

B(P, Q)v(Q),

Q∈Ξ (P)

the equation (2.149) takes the form Zv(P) = F(P).

(2.152)

THEOREM 2.1 If D(x) > 0 on ω, then the problem (2.149)–(2.152) with ϕ(x) = 0 can be a priori estimated via the following inequality (( ( F(x) (( ( , vl ≤ ((( D(x) (l where vl = max x∈ω |v(x)| .

2.3 Solution of 3D Stationary Heat Transfer Equation

207

THEOREM 2.2 Let v(x) is the solution to the problem (2.149)–(2.152), and v(x) is the solution obtained via change in (2.152), (2.151) the functions F(x), ϕ(x) by the functions F(x), g(x), and there exist at least one node x0 of the mesh ω where D(x0 ) > 0. Then, if the following conditions are satisfied |F(x)| ≤ F(x), x ∈ ω; |g(x)| ≤ g(x), x ∈ Γ, the following inequality |v(x)| ≤ v(x) holds on ω. Proof of these two theorems is given in reference [592]. Consider now the diﬀerence Dirichlet problem approximating (2.149), (2.150) with the error of o(|h|4 ) with a help of diﬀerence scheme (2.126), which can be presented in the following form Λ v ≡

3 i=1

Ai v xi xi +

3 h2i A j v xi xi x j x j = −ϕ(x), x ∈ ω 12 i, j=1

(2.153)

i j

v = g(x) for x ∈ Γ,

ϕ(x) = f +

3 h2i ∂2 f , 12 ∂xi2 i=1

where ω is the mesh space G3 {0 ≤ xα ≤ lα , α = 1, 2, 3}. THEOREM 2.3 Diﬀerence scheme (2.153) is uniformly convergent with the velocity o(|h|4 ) (it possesses fourth accuracy order), if a solution to initial diﬀerential problem (2.148) T (x) ∈ C (6) (G3 ), function f (x) ∈ C (4) (G3 ), and the following conditions are satisfied 4A2 A1 A3 4A3 A2 A1 4A1 A2 A3 − 2 − 2 ≥ 0, − 2 − 2 ≥ 0, − 2 − 2 ≥ 0. 2 2 h1 h2 h3 h2 h1 h3 h23 h2 h1

(2.154)

Proof. For the error τ = v − T the following problem is considered Λ τ = −Ψ, x ∈ ω;

τ = 0, x ∈ Γ,

(2.155)

where 3

Ψ = Λ v + ϕ = o(|h|4 ) for x ∈ ω, if v ∈ C (6) (G ), f ∈ C (4) (G3 ). Let us check the condition (2.150). For this purpose the scheme (2.153) is rewritten in the form (2.149): B(P, Q)v(Q) + F(P). A(P)v(P) = Q∈Ξ (P)

Scheme (2.153) has the following form

Av = B1 v(+11 ) + v(−11 ) + B2 v(+12 ) + v(−12 ) + B3 v(+13 ) + v(−13 ) +

208

2 Stability of Rectangular Shells within Temperature Field

B4 v(+11 ,+12 ) + v(+11 ,−12 ) + v(−11 ,+12 ) + v(−11 ,+12 ) +

B5 v(+11 ,+13 ) + v(+11 ,−13 ) + v(−11 ,+13 ) + v(−11 ,−13 ) +

B6 v(+12 ,+13 ) + v(+13 ,−13 ) + v(−12 ,+13 ) + v(−12 ,−13 ) + ϕ,

(2.156)

where A and Bi are expressed via coeﬃcients of the equation (2.147) and hi : ⎡ ⎡ ⎤ ⎤ 1 ⎢⎢⎢ 4A1 A2 A3 ⎥⎥⎥ 4 ⎢⎢⎢ A1 A2 A3 ⎥⎥⎥ A = ⎣⎢ 3 + 2 + 2 ⎦⎥ > 0, B1 = ⎣⎢ 2 − 2 − 2 ⎦⎥ , 3 h1 6 h1 h2 h3 h2 h3 ⎡ ⎡ ⎤ ⎤ 1 ⎢⎢⎢ 4A2 A1 A3 ⎥⎥⎥ 1 ⎢⎢⎢ 4A3 A1 A2 ⎥⎥⎥ B2 = ⎣⎢ 2 − 2 − 2 ⎦⎥ , B3 = ⎣⎢ 2 − 2 − 2 ⎦⎥ , 6 h2 6 h3 h1 h3 h1 h2 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 1 ⎢⎢⎢ A1 A2 ⎥⎥⎥ 1 ⎢⎢⎢ A1 A3 ⎥⎥⎥ 1 ⎢⎢⎢ A2 A3 ⎥⎥⎥ ⎢⎣ 2 + 2 ⎥⎦ , B5 = ⎢⎣ 2 + 2 ⎥⎦ , B6 = ⎢⎣ 2 + 2 ⎥⎦ . B4 = 12 h1 12 h1 12 h2 h2 h3 h3 The following relation holds ⎛ ⎛ ⎞ ⎞ 4 ⎜⎜ A1 A2 A3 ⎟⎟ 1 ⎜⎜ 4A1 A2 A3 ⎟⎟ D(P) = A(P) − B(P, Q) = ⎜⎜⎝ 2 + 2 + 2 ⎟⎟⎠ − ⎜⎜⎝ 2 − 2 − 2 ⎟⎟⎠ − 3 h1 3 h1 h2 h3 h2 h3 Q∈Ξ (P) ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 1 ⎜⎜⎜ 4A2 A1 A3 ⎟⎟⎟ 1 ⎜⎜⎜ 4A3 A1 A2 ⎟⎟⎟ 1 ⎜⎜⎜ A1 A2 ⎟⎟⎟ ⎜⎝ 2 − 2 − 2 ⎟⎠ − ⎜⎝ 2 − 2 − 2 ⎟⎠ − ⎜⎝ 2 + 2 ⎟⎠ − 3 h2 3 h3 3 h1 h1 h3 h1 h2 h2 ⎛ ⎛ ⎞ ⎞ 1 ⎜⎜⎜ A1 A3 ⎟⎟⎟ 1 ⎜⎜⎜ A2 A3 ⎟⎟⎟ ⎜⎝ + 2 ⎟⎠ − ⎜⎝ 2 + 2 ⎟⎠ = 0, 3 h21 3 h2 h3 h3 for the boundary points D(P) = 1 > 0. In other words, conditions (2.150) are satisfied if 4A2 A1 A3 4A3 A1 A2 4A1 A2 A3 − 2 − 2 ≥ 0, − 2 − 2 ≥ 0, − 2 − 2 ≥ 0. h21 h2 h3 h22 h1 h3 h23 h1 h2 In order to estimate a solution to the problem (2.155) the following majorant function is constructed ⎛ ⎞ l2 x2 ⎟ l2 x2 x2 C0 ⎜⎜⎜ l12 ⎜⎝ + 2 + 3 − 1 − 2 + 3 ⎟⎟⎟⎠ . τ(x) = 6 A1 A2 A3 A1 A2 A3 Observe that Λ τ =

3 i=1

Ai τ xi xi +

3 h2i A j τ xi xi x j x j = −C0 . 12 i, j=1 i j

For the function τ(x) in the space G3 {0 ≤ xα ≤ lα , α = 1, 2, 3} the following estimation holds

2.3 Solution of 3D Stationary Heat Transfer Equation

209

⎛ ⎞ l32 ⎟⎟⎟ l22 C0 ⎜⎜⎜ l12 0 ≤ τ(x) ≤ + ⎟⎠ . ⎝⎜ + 6 A1 A2 A3 In order to solve the problem (2.155), assuming C0 = max x∈ω |Ψ (x)| = ||Ψ ||e and applying Theorem 3, the following estimation is obtained ⎛ ⎞ l2 ⎟⎟ l2 1 ⎜⎜ l2 max x∈ω |τ(x)| = max x∈ω |v − T | ≤ τ(x) ≤ ⎜⎜⎝ 1 + 2 + 3 ⎟⎟⎠ max x∈ω |Ψ (x)| 6 A1 A2 A3 under condition (2.154), i.e. ⎛ ⎞ l32 ⎟⎟⎟ l22 1 ⎜⎜⎜ l12 ⎜ + ⎟⎠ Ψ l . τl ≤ ⎝ + 6 A1 A2 A3 It means, that a solution of diﬀerence problem (2.153) is uniformly convergent with velocity o(|h|4 ) to solution of diﬀerential problem, what finishes the proof. Remark. For a mesh h1 = h2 = h h3 the condition (2.154) is satisfied, if h2 4A3 A3 ≤ 32 ≤ , 4A2 − A1 h A1 + A2

h2 A3 ≤ 32 . 4A1 − A2 h

For a squared mesh h1 = h2 = h3 = h the condition (2.154) is satisfied, if 4A1 ≥ A2 + A3 , 4A2 ≥ A1 + A3 , 4A3 ≥ A1 + A2 . 2.3.4 Algorithm of computation and compatibility conditions In this section, a stationary three dimensional heat transfer equation for shallow shells [555] is analysed LT ≡ a1

∂T ∂2 T ∂2 T ∂2 T + a2 2 + a3 2 + k = − f (x1 , x2 , x3 ), 2 ∂x3 ∂x1 ∂x2 ∂x3

where: k=

(2.157)

∂l12 l22 l1 k1 + k , λ = , f = f (l1 x1 , l2 x2 , 2Hx3 ), 2 l2 λ2 (2H)2 2 2H 1 2H T, ki = 2 ki , (i = 1, 2), T = α11 l1 li

xi = li xi , (i = 1, 2),

x3 = 2hx3 ,

2 l2 1 2 a1 = 2 , a2 = 1, a3 = λ1 = . 2H λ

Equation (2.157) is already in non-dimensional form, and bars over nondimensional parameters are omitted. Applying approximation (2.133), where a3 = k, Ai = ai , the diﬀerence equation approximating (2.157) with the error o(|h|4 ) has the form

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2 Stability of Rectangular Shells within Temperature Field

ΛT ≡

3 i=1

3 k2 h23 a j h2i Λi Λ j T + kΛ˜ 3 T + Λi T + Λ3 T + 12 12a3 i, j=1 i j

⎛ 2 ⎞ h22 a3 + h23 a2 ⎜⎜⎜ h1 a3 + h23 a1 ⎟⎟ ˜ ˜ k ⎝⎜ Λ3 Λ! T + Λ3 Λ2 T ⎟⎟⎠ = −ϕ, 12a3 12a3 where: ϕ = f (x) +

(2.158)

3 kh23 ∂ f h2i ∂2 f + , 12 ∂xi2 12a3 ∂x3 i=1

and Λi T , Λ˜ 3 T (i = 1, 2, 3) are approximated by central diﬀerences with the error of o(|h|2 ). Substituting in (2.158) the diﬀerence operators by central diﬀerences relaxations, the following diﬀerence equation is obtained

AT i jk = B1 T i+1 jk + T i−1 jk + B2 T i j+1k + T i j−1k + B3 T i jk+1 + T i jk−1 +

B5 T i+1 jk+1 + T i+1 jk−1 + T i−1 jk+1 + T i−1 jk−1 + B6 T i j+1k+1 + T ik+1k−1 +

T i j−1k+1 + T i j−1k−1 + B4 T i+1 j+1k + T i+1 j−1k + T i−1 j+1k + T i−1 j−1k +

B7 T i jk+1 + T i jk−1 + B8 T i+1 jk+1 + T i−1 jk+1 − T i+1 jk−1 − T i−1 jk−1 +

B9 T i j+1k−1 + T i j−1k+1 − T i j+1k−1 − T i j−1k−1 + ϕ, (2.159) where:

⎛ ⎞ 4 ⎜⎜⎜ a1 a2 a3 ⎟⎟⎟ A = ⎜⎝ 2 + 2 + 2 ⎟⎠ + 3 h1 h2 h3 ⎛ ⎞ a1 1 ⎜⎜ a12 a13 ⎟⎟ B1 = 2 − ⎜⎜⎝ 2 2 + 2 2 ⎟⎟⎠ , a12 h1 6 h1 h2 h1 h3 ⎛ ⎞ a23 ⎟⎟ a2 1 ⎜⎜ a12 B2 = 2 − ⎜⎜⎝ 2 2 + 2 2 ⎟⎟⎠ , a13 h2 6 h1 h2 h2 h3 ⎛ ⎞ a23 ⎟⎟ a3 1 ⎜⎜ a13 k2 B3 = 2 − ⎜⎜⎝ 2 2 + 2 2 ⎟⎟⎠ + , 12a3 h3 6 h1 h3 h2 h3 B4 =

k2 , 6a3 = a1 h22 + a2 h21 , = a1 h23 + a3 h21 , a23 = a2 h23 + a3 h22 ,

a12 a13 a23 k a13 , B5 = , B6 = , B8 = , 2 2 2 2 2 2 2h3 12h21 a3 12h1 h2 12h1 h3 12h2 h3 ⎛ ⎞ a23 k ⎜⎜⎜ a23 ⎟⎟⎟ k a13 B7 = . (2.160) ⎝⎜1 − 2 − 2 ⎟⎠ , B9 = 2h3 2h3 12h22 a3 6h1 a3 6h2 a3

We are going to find approximation of boundary conditions and compatibility conditions for the equation (2.157) with the error o(|h|4 ). The boundary condition takes the form

2.3 Solution of 3D Stationary Heat Transfer Equation

kα −kα

∂T = κ−α − g−α (xβ , xγ ), ∂xα

211

xα = 0,

∂T = κ+α − g+α (xβ , xγ ), xα = 1, α, β, γ = 1, 2, 3; α β γ, ∂xα

(2.161)

in the space G3 {0 ≤ xα ≤ 1, α = 1, 2, 3}. One may get from the boundary condition (2.161) the first, second and third order boundary conditions by giving the coeﬃcients kα , κα , κ+α values of 0 or 1. In the case kα = 0, i.e. for the first boundary value problem, the boundary conditions are approximated exactly. Consider approximation of second order boundary conditions and compatibility conditions, i.e. when κ−α = κ+α = 0 ∂T = −g−α (xβ , xγ ), xα = 0; ∂xα

−

∂T = −g+α (xβ , xγ ), xα = 1. ∂xα

(2.162)

Introducing notations ⎧ 2 ⎪ v x , xα = 0 ⎪ ⎪ ⎪ ⎨ hα α v , xα 0; 1 α = 1, 2, 3 , Λα v = ⎪ x α xα ⎪ ⎪ ⎪ ⎩ − 2 v xα , xα = 1 hα

(2.163)

the boundary conditions for x1 = 0 take the form ∂T = −g−1 (x2 , x3 ). ∂x1

(2.164)

Let us develop T (x) into Taylor series in neighbourhood of the node x1 = 0. Then the approximation of the boundary condition (2.164) with the error o(h2 ) has the form (recall, that T (x) is solution of equation (2.157)) 2a1 2a1 T x1 + a2 T x2 x2 + a3 T x3 x3 + kT x◦3 = − f (x) + g−1 (x2 , x3 ) . (2.165) h1 h1 Assume that x1 = 0, x2 = 0, x3 0, 1. Similarly to the previous case one may observe that the expression (2.166) approximates a compatibility condition along the rib x1 = x2 = 0 with the second order accuracy 2a2 2a1 2a2 2a1 (2.166) Tx + T x + a3 T x3 x3 + kT x◦3 = − f (x) + g−1 + g−2 . h1 1 h2 2 h1 h2 Diﬀerence equation (2.158) approximates the equation (2.157) with the error o(|h|4 ), if the point (x1 , x2 , x3 ) ∈ ω. Let us construct a diﬀerence scheme, approximating the problem (2.157) with boundary conditions (2.161) with the error amount of o(|h|4 ). A diﬀerence scheme is sought in the form analogous to (2.158)

Λ T ≡ a1 Λ1 T + a2 Λ2 T + a3 Λ3 T +

a2 h21 + a1 h22 Λ1 Λ2 T + 12

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2 Stability of Rectangular Shells within Temperature Field

a3 h21 + a1 h23 a3 h22 + a2 h23 k2 h23 Λ1 Λ3 T + Λ2 Λ3 T + Λ3 T + 12 12 12a3 ⎛ 2 ⎞ h22 a3 + a2 h23 ⎜⎜⎜ h1 a3 + a1 h23 ⎟⎟ ˜ ˜ Λ3 Λ1 T + Λ3 Λ2 T ⎟⎟⎠ = −Φ . k ⎝⎜ 12a3 12a3

(2.167)

Right hand side Φ for x ∈ ω should be defined in order to obtain error of approximation o(|h|4 ). If for x ∈ ω one assumes Φ = Φ, then the equation (2.167) coincides with the equation (2.158), and hence it approximates (2.157) with the error of o(|h|4 ). Finally, we need to give Φ on a boundary of the space G3 {0 ≤ xα ≤ 1, (α = 1, 2, 3)}. Consider first the equation (2.167) for x1 = 0, x2 0, 1, x3 0, 1. Owing to (2.163), we rewrite the equation (2.167), multiplying it by h1 /2, in the form a1 T x1 +

a2 h21 + a1 h22 h1 h1 h1 a2 T x2 x2 + a3 T x3 x3 + kT x◦3 + T x1 x2 x2 + 2 2 2 12a1

a3 h21 + a1 h23 a3 h22 + a2 h23 h1 k2 h23 h1 T x2 x2 x3 x3 + Tx x + T x1 x3 x3 + 12a1 12 2 12a3 2 3 3 ⎛ ⎞ a3 h22 + a2 h23 ⎟⎟ h1 kh1 ⎜⎜⎜ a3 h21 + a2 h23 T x◦3 x1 x1 + T x◦3 x2 x2 ⎟⎟⎠ + Φ = 0. ⎝⎜ 2 12a3 12a3 2 In order to compute an error of approximation Ψ of the equation, T is developed into Taylor series ⎛ ⎜⎜ ∂T h1 ∂2 T h21 ∂3 T h31 ∂4 T + + + + o(h41 )+ Ψ (0, x2 , x3 ) = ⎜⎝⎜ ∂x1 2 ∂x12 6 ∂x13 24 ∂x14 ⎛ ⎛ ⎞ ⎞ ⎟⎟⎟ h1 ⎜⎜⎜ ∂2 T h23 ∂4 T ⎟⎟ h1 ⎜⎜⎜ ∂2 T h22 ∂4 T 4 4 a2 ⎜⎝ 2 + + o(h2 )⎟⎠ + a3 ⎜⎝ 2 + + o(h3 )⎟⎟⎠ + 4 4 2 2 ∂x2 12 ∂x2 ∂x3 12 ∂x3 ⎛ ⎛ a3 h21 + a1 h23 ⎜⎜⎜ ∂ ∂2 T a2 h21 + a1 h22 ⎜⎜⎜ ∂ ∂2 T h1 ∂2 ∂2 T 2 ⎜⎝ ⎜⎝ + + o(h ) + + 1 12 ∂x1 ∂x22 2 ∂x12 ∂x22 12 ∂x1 ∂x32 ⎛ ⎞ ⎞ ⎟⎟ a3 h22 + a2 h23 h1 ⎜⎜⎜ ∂2 ∂2 T ⎟⎟ h1 ∂2 ∂2 T 2 ⎟ 2 ⎟ ⎜ ⎟ ⎟ + o(h ) + o(h ) + ⎝ 2 2 1 ⎠ 2 ⎠+ 2 2 2 ∂x1 ∂x3 12 2 ∂x2 ∂x3 ⎛ ⎞ h2 h1 ⎜⎜⎜ ∂2 T h23 ∂4 T ⎟⎟ 4 ⎟ ⎜⎝ 2 + ⎟ + o(h ) k2 3 3 ⎠+ 4 12a3 2 ∂x3 12 ∂x3 ⎡ ⎛ ⎞ ⎟⎟ h1 ⎢⎢ a3 h21 + a1 h23 ⎜⎜⎜ ∂ ∂2 T h3 ∂2 ∂2 T 2 ⎟ ⎜⎝ ⎟ + + o(h ) k ⎢⎢⎣ 3 ⎠ + 2 2 2 2 12a3 ∂x3 ∂x1 2 ∂x3 ∂x1 ⎛ ⎞⎤ a3 h22 + a2 h23 ⎜⎜⎜ ∂ ∂2 T h3 ∂2 ∂2 T ⎟⎟⎥⎥⎥ h1

2 ⎟ ⎜⎝ ⎟⎥ Φ. + + o(h ) 3 ⎠⎦ + 2 2 2 12a3 ∂x3 ∂x2 2 ∂x3 ∂x2 2

2.3 Solution of 3D Stationary Heat Transfer Equation

213

Since T is a solution of the equation (2.157) with attached boundary conditions (2.161), then in order to achieve Ψ by approximation for x1 = 0 with error o(|h|4 ), it is suﬃcient to take ⎛ h2 ∂2 f h2 ∂2 f h2 ∂2 f h2 ∂ f 2 ⎜⎜⎜ ⎜⎝g−1 + 1 + Φ = f + 1 2 + 2 2 + 3 2 + 12 ∂x1 12 ∂x2 12 ∂x3 h1 6 ∂x1 ⎞ a1 h22 − a2 h21 ∂2 g−1 a1 h23 − a3 h21 ∂2 g−1 ⎟⎟⎟ ⎟⎠ = + 12a1 12a1 ∂x22 ∂x32

⎛ ⎞ a1 h22 + a2 h21 ∂2 g−1 a1 h23 − h21 a3 ∂2 g−1 ⎟⎟⎟ h21 ∂ f 2 ⎜⎜⎜ ⎜⎝g−1 + ⎟⎠ . + + ϕ + h1 6 ∂x1 12a1 12a1 ∂x22 ∂x32

(2.168)

Approximation of a boundary condition with the error of o(|h|4 ) for x1 = 0 has the following form 2 2 a2 h21 + a1 h22 a1 T x1 + a2 T x2 x2 + a3 T x3 x3 + T x1 x2 x2 + kT x◦3 + h1 h1 12a1 a3 h22 + a2 h23 h2 2 a3 h21 + a1 h23 T x2 x2 x3 x3 + k2 3 T x3 x3 + T x1 x3 x3 + h1 12a1 12 12a3 ⎛ ⎞ 2 2 2 2 a3 h2 + a2 h3 ⎜⎜ a3 h1 + a1 h3 ⎟⎟ k ⎜⎜⎝ T x◦3 x1 x1 + T x◦3 x2 x2 ⎟⎟⎠ = −Φ , 12a3 12a3

(2.169)

where Φ is defined by (2.168). Let us define an approximation of a compatibility condition along the rib x1 = 0, x2 = 0, x3 0, 1. Owing to (2.163), the equation (2.167) takes the form a3 h21 + a1 h22 2 2 2 2 a1 T x1 + a2 T x2 + a3 T x3 x3 + kT x◦3 + Tx x + h1 h2 12 h1 h2 1 2 a3 h21 + a1 h23 2 a3 h22 + a2 h23 2 k2 h23 T x1 x3 x3 + T x3 x3 x2 + Tx x + 12 h1 12 h2 12a3 3 3 k

h21 a3 + a1 h23 2 h2 a3 + a2 h23 2 T x1 x◦3 + k 2 T ◦ + Φ = 0. 12a3 h1 12a3 h2 x2 x3

Error of approximation of this equation reads ⎛ ⎞ ⎟ 2 ⎜⎜⎜ ∂T h1 ∂2 T h21 ∂3 T h31 ∂4 T 4 ⎟ ⎜ Ψ (0, 0, x3 ) = a1 ⎝ + + + + o(h1 )⎟⎟⎠ + h1 ∂x1 2 ∂x12 6 ∂x13 24 ∂x14 ⎛ h2 ∂2 T h22 ∂3 T h32 ∂4 T 2 ⎜⎜⎜ ∂T a2 ⎜⎝ + + + + o(h4 )+ h2 ∂x2 2 ∂x22 6 ∂x23 24 ∂x24 ⎞⎛ 2 ⎛ 2 2 ⎞ ⎞ ⎛ h2 ∂ 3 T ⎟⎟ ⎜⎜ ∂ T h2 ∂4 T ⎜⎜⎜ k h3 ⎟⎟ ⎟⎟ ⎜⎜ ∂T ⎜⎝ + a3 ⎟⎟⎠ ⎜⎜⎝ 2 + 3 4 + o(h43 )⎟⎟⎠ + k ⎜⎜⎝ + 3 3 + o(h43 )⎟⎟⎠ + 12a3 ∂x3 6 ∂x3 ∂x3 12 ∂x3

214

2 Stability of Rectangular Shells within Temperature Field

⎛ a2 h21 + a1 h22 2 2 ⎜⎜⎜ ∂ ∂T ∂ ∂T h2 ∂2 ∂T ⎜⎝σ + (1 − σ) + + 12 h1 h2 ∂x2 ∂x1 ∂x1 ∂x2 2 ∂x22 ∂x1 ⎞ h22 ∂3 ∂T h21 ∂3 ∂T ⎟ h1 h2 ∂4 T h1 ∂2 ∂T 3 ⎟ + + + + o(|h| )⎟⎟⎠ + 2 ∂x12 ∂x2 6 ∂x23 ∂x1 6 ∂x13 ∂x2 4 ∂x12 ∂x22 ⎛ ⎛ ⎞ a3 h21 + a1 h23 ⎜⎜⎜ ∂2 ∂T ⎟⎟⎟ a3 h22 + a2 h23 ⎜⎜⎜ ∂2 ∂T h1 ∂2 ∂2 T 2 ⎜⎝ 2 ⎜⎝ 2 + + o(|h| )⎟⎠ + + 6h1 2 ∂x32 ∂x12 6h2 ∂x3 ∂x1 ∂x3 ∂x2 ⎛ ⎞ h21 a3 + a1 h23 ⎜⎜⎜ ∂ ∂T ⎟⎟ h2 ∂2 ∂2 T h1 ∂ ∂2 T 2 ⎟ ⎜ ⎟ + o(|h| ) + + + k ⎝ ⎠ 2 ∂x32 ∂x22 6h1 a3 ∂x3 ∂x1 2 ∂x3 ∂x12 ⎞ h22 a3 + a2 h23 ∂ ∂T h21 ∂ ∂2 T h31 ∂ ∂4 T ⎟⎟ 4 ⎟ ⎟ + + o(h ) + + k 1 ⎠ 6 ∂x3 ∂x13 24 ∂x3 ∂x14 6h2 a3 ∂x3 ∂x2 ⎞ ⎟ h2 ∂ ∂2 T h22 ∂ ∂3 T h32 ∂ ∂4 T 4 ⎟ + + + o(h2 )⎟⎠⎟ + Φ . 2 ∂x3 ∂x22 6 ∂x3 ∂x23 24 ∂x3 ∂x24 Using (2.157) and boundary conditions (2.161), the expression for error of approximation takes the form ⎛ 2 2 h2 ∂2 h2 ∂2 ⎜⎜ h ∂ 2 ∂T 2 ∂T + a2 + LT + ⎜⎜⎝ 1 2 + 2 2 + 3 2 + Ψ (0, 0, x3 ) = a1 h1 ∂x1 h2 ∂x2 12 ∂x1 12 ∂x2 12 ∂x3 ⎞ h23 ∂ ⎟⎟⎟ a1 h22 − a2 h21 ∂2 ∂T h2 ∂ h ∂ ⎟⎠ LT + 1 k LT + + LT + 2 12a3 ∂x3 3 ∂x1 6h1 3 ∂x2 ∂x2 ∂x1 a1 h23 − a3 h21 ∂2 ∂T a2 h23 − a3 h2 ∂2 ∂T a2 h21 − a1 h22 ∂2 ∂T + + + 2 2 6h1 6h2 6h2 ∂x3 ∂x1 ∂x3 ∂x2 ∂x12 ∂x2 h1 (a2 h21 + a1 h22 ) ∂3 ∂T h2 (a2 h21 + a1 h22 ) ∂3 ∂T + + 18h1 18h2 ∂x23 ∂x1 ∂x13 ∂x2 a2 h21 + a1 h22 ∂ ∂T ∂ ∂T σ + Φ + o(|h|4 ) = + (1 − σ) 3h1 h2 ∂x2 ∂x1 ∂x1 ∂x2 −f − ⎛ 2 ⎜⎜⎜ ⎜⎝g−1 + h1 ⎛ 2 ⎜⎜⎜ ⎜⎝g−2 + h2

h23 ∂2 f h23 ∂ f h22 ∂2 f h21 ∂2 f − − − k − 12 ∂x12 12 ∂x22 12 ∂x32 12a3 ∂x3

⎞ a1 h22 − a2 h21 ∂2 g−1 a1 h23 − a3 h21 ∂2 g−1 ⎟⎟⎟ h21 ∂ f ⎟⎠ − + + 6 ∂x1 12a1 12a1 ∂x22 ∂x32 ⎞ a2 h21 − a1 h22 ∂2 g−2 a2 h23 − a3 h22 ∂2 g−2 ⎟⎟⎟ h22 ∂ f ⎟⎠ − + + 6 ∂x2 12a2 12a2 ∂x12 ∂x32 ⎛ 4 ⎜⎜⎜ a2 h21 + a1 h22 ∂g−1 h22 (a2 h21 + a1 h22 ) ∂3 g−1 ⎜⎝ σ + + h1 h2 12a1 ∂x2 72a1 ∂x23

2.3 Solution of 3D Stationary Heat Transfer Equation

215

⎞ a2 h21 + a1 h22 ∂g−2 h21 (a2 h21 + a1 h22 ) ∂3 g−2 ⎟⎟⎟

4 (1 − σ) + ⎠⎟ + Φ + o(|h| ). 12a2 ∂x1 72a2 ∂x13 Therefore, to achieve error of compatibility condition approximation of amount of o(h41 + h42 + h43 ) for x1 = x2 = 0, it is suﬃcient to take Φ = f +

h23 ∂2 f h23 ∂ f h21 ∂2 f h22 ∂2 f + + + k + 12 ∂x12 12 ∂x22 12 ∂x32 12a3 ∂x3

⎡ 2 ⎢⎢⎢ ⎢⎣g−1 + h1 ⎡ 2 ⎢⎢⎢ ⎢⎣g−2 + h2

⎤ a1 h22 − a2 h21 ∂2 g−1 a1 h23 − a3 h21 ∂2 g−1 ⎥⎥⎥ h21 ∂ f ⎥⎦ + + + 6 ∂x1 12a1 12a1 ∂x22 ∂x32 ⎤ a2 h21 − a1 h22 ∂2 g−1 a2 h23 − a3 h22 ∂2 g−2 ⎥⎥⎥ h22 ∂ f ⎥⎦ + + + 6 ∂x2 12a2 12a2 ∂x12 ∂x32 ⎡ 4 ⎢⎢⎢ a2 h21 + a1 h22 ∂g−1 h22 (a2 h21 + a1 h22 ) ∂3 g−1 σ + + ⎣⎢ h1 h2 12a1 ∂x2 72a1 ∂x23 ⎤ a2 h21 + a1 h22 ∂g−2 h21 (a2 h21 + a1 h22 ) ∂3 g−2 ⎥⎥⎥ ⎥⎦ . (1 − σ) + 12a2 ∂x1 72a2 ∂x13

(2.170)

In what follows, approximation of compatibility condition for x1 = x2 = 0 with the error of o(h41 + h42 + h43 ) has the following form a2 h21 + a1 h22 2 2 a1 T x1 + a2 T x2 + a3 T x3 x3 + T x1 x2 + h1 h2 3h1 h2 a3 h21 + a1 h23 a3 h22 + a2 h23 k2 h23 T x1 x3 x3 + T x2 x3 x3 + kT x◦3 + Tx x + 6h1 6h2 12a3 3 3 k

h21 a3 + a1 h23 h2 a3 + a2 h23 T x1 x◦3 + k 2 T x2 x◦3 = −Φ , 6a3 h1 6a3 h2

(2.171)

where Φ is defined in (2.170). One may obtain in an analogous way also compatibility conditions for x1 = x3 = 0 and x2 = x3 = 0. 2.3.5 Problems In order to verify eﬃciency of the constructed approximation using the diﬀerence scheme (2.134), a model-type problem for the equation (2.157) with attached all possible boundary conditions will be solved in the space G {0 ≤ x ≤ 1 , 0 ≤ y ≤ 1, −0.5 ≤ z ≤ 0.5}. Below some of the boundary conditions are given. $ % 1. T | x=0;1 = 0.25 + (y − 0.5)2 2Hz4 , x, y , $ % (2.172) T |z=±0.5 = ± (x − 0.5)2 + (y − 0.5)2 H0.25 .

216

2 Stability of Rectangular Shells within Temperature Field

2.

$ % = −0.75 + (y − 0.5)2 2Hz3 , x, y , x=0;1 $ % T |z=±0.5 = ± (x − 0.5)2 + (y − 0.5)2 H0.25 .

∂T +T ∂x

3.

T |z=±0.5

∂T = −2Hz3 , ∂x x=0;1

4.

∂T = −2Hz3 , x, y , ∂x x=0;1 $ % = ± (x − 0.5)2 + (y − 0.5)2 H0.25 .

(2.173)

(2.174)

x, y ,

$ % = (x − 0.5)2 + (y − 0.5)2 (0.75 + H0.25), z=0.5 $ % T |z=−0.5 = − (x − 0.5)2 + (y − 0.5)2 H0.25 . (2.175)

∂T + 2HT ∂z

∂T = −2Hz3 , x, y , ∂x x=0.1 $ % = − (x − 0.5)2 + (y − 0.5)2 H0.25;

5.

T | x=−0.5 $ % ∂T = (x − 0.5)2 + (y − 0.5)2 1.5H . ∂z z=0.5

(2.176)

In all problems the following function serves as the right hand sides of equation (2.157) 3

3 2 2 f = − 8Hz + z (x − 0.5) + (y − 0.5) . (2.177) H The function

% $ T = (x − 0.5)2 + (y − 0.5)2 2Hz3

(2.178)

is the exact solution of the mentioned boundary value problems. They are solved for 2H = 0.1, 0.02, 0.01. For the mesh steps h1 = h2 = h3 = 18 the obtained solutions coincide with exact solution (2.178). It means that the proposed diﬀerence scheme (2.134) with the error o(|h|4 ) gives a good approximation to a being sought solution. On example of this model problem, an investigation of error computation of the functions Ψ1 (x, y), Ψ2 (x, y), δ1 (y), δ2 (x) in dependence of 2H is carried out. The given function are transformed to non-dimensional form using the following relations 2 l1 1 2H T , x = l, x, y = l2 y, z = 2Hz, λ = , T= T l2 α11 l1

2.3 Solution of 3D Stationary Heat Transfer Equation

217

αTij βi j , α = , (i, j = 1, 2), ij B11 αT11 αT11

βi j =

H ∂2 T ∂2 T ∂2 T B11 (2H)4 dz = Ψ (x, y) = z β11 2 + β22 2 + 2β12 × ∂x∂y ∂x ∂y l12 l22 −H

0.5 −0.5

⎞ ⎛ ⎜⎜⎜ ∂2 T ∂2 T ∂2 T ⎟⎟⎟ B11 (2H)4 −2 ⎟ z ⎝⎜β11 λ + β + 2β λ Ψ1 (x, y), dz = ⎠ 22 12 ∂x∂y l12 l22 ∂x2 ∂y2

⎞ 0.5⎛ 2 (2H)2 ⎜⎜⎜ ∂2 T ∂2 T ⎟⎟⎟ (2H)2 T −2 ∂ T T ⎟ Ψ2 (x, y) = 2 2 ⎜⎝ 2 + α22 λ −α λ = Ψ 2 (x, y). dz ⎠ 12 ∂x∂y l1 l2 l12 l22 ∂y ∂x2 −0.5

H δ1 (y) = 12β11 −H

H δ2 (x) = 12β22 −H

σ1 (y) =

σ2 (x) =

αT22

αT11

(2H)4 B11 zT dz = 12β11 l12

0.5 zT dz =

−0.5

(2H)4 B1 zT dz = 12β22 λ−2 l22

(2H)4 B11 δ1 (y), l12

0.5 zT dz =

−0.5

1 2H

1 2H

H −H

H −H

2H T dz = l1

2H T dz = l2

2

0.5 α22 −0.5

2 λ

−2

0.5

−0.5

(2H)4 B11 δ2 (x), l22

2H T dz = l1

2H T dz = l2

2

2

σ1 (y),

λ−2 σ2 (x).

The bars over non-dimensional quantities are omitted. In the case of boundary value problem (2.172) for equation (2.157) for 2H = 0.1, 2H = 0.02 the maximal error for the function Ψ1 (x, y) is in amount of 0.4% (0.5%). For the function Ψ2 (x, y), having its exact value equal to zero, the maximal absolute error is equal to 10−14 (10−15 ) for 2H = 0.1 (2H = 0.01). For boundary value problem (2.173)–(2.176) and equation (2.157) for 2H = 0.1, the maximal error of Ψ1 (x, y) achieved 0.1%, whereas for 2H = 0.01 it achieved 0.2%. The investigations have shown that the diﬀerence scheme (2.134) approximating (2.157) with various boundary conditions yields good approximation either for temperature field T (x, y, z) and the function Ψ1 (x, y), Ψ2 (x, y), δ1 (y), δ2 (x), σ1 (y), σ2 (x). The following boundary value problems are considered for the equation (2.157): 1.

$ % ∂T = 0 x, y , T |z=±0.5 = ± (x − 0.5)2 + (y − 0.5)2 H · 0.25 . (2.179) ∂x x=0.1 Heat source function f is defined via (2.177).

218

2.

3.

2 Stability of Rectangular Shells within Temperature Field

∂T ∂T = −2Hz3 , x, y ; = 0, ∂x x=0.1 ∂z z=−0.5 $ % T |z=0.5 = (x − 0.5)2 + (y − 0.5)2 H · 0.25,

(2.180)

f is defined in the form of (2.177). ∂T ∂T = 0 x, y , = 0, ∂x x=0.1 ∂z z=−0.5 $ % T |z=0.5 = (x − 0.5)2 + (y + 0.5)2 H · 0.25, f = 0.

(2.181)

In figures 2.2–2.9 temperature fields in cross sections y = 0.25 and y = 0.5, obtained during solutions of the boundary value problems (2.172) for f , (2.177), (2.172) for f = 0, (2.179), (2.177), (2.179) for f = 0, (2.180), (2.181) are reported. In all mentioned figures by (a) curves characterizing behaviour of temperature function T (z) in the cross section y = 0.25: x = 0 (curve 1), x = 0.125 (curve 2), x = 0.25 (curve 3) are denoted. Comparison of the Figures 2.2 and 2.3 shows, that for f = 0 temperature field inside G3 {0 < x < 1, 0 < y < 1, −0.5 < z < 0.5} is almost linear with respect to z (see curves 2, 3, 2 , 3 in Figure 2.3a), although on the space boundary for x = 0 temperature field is given in the form of z3 , i.e. inside of the space without the heat sources ( f = 0) and with boundary conditions (2.172) the temperature field is linear. Lack of heat sources ( f = 0) is exhibited by Ψ1 (x, y) behaviour. In Figure 2.4a the function Φ1 (x, y) is presented, corresponding to boundary value problem (2.172) for f = 0, and also the curves of Ψ1 (x, y) function in the cross sections y = 0.5 (curve 1), y = 0.25 (curve 2) y = 0.125 (curve 3) are reported. Curve 4 corresponds to the function Ψ1 (x, y) = −0.00065 being a solution of the problem (2.157), (2.172) for f (2.177). Owing to comparison of the curves 1,2,3 and 3, it is observed that for f = 0 the function Ψ1 (x, y) changes qualitatively and becomes nonlinear with respect to x and y. For the problem (2.157), (2.172) for f (2.177), the temperature field is nonlinear (Fig. 2.2) and Ψ1 (x, y) = const, whereas for f = 0 a temperature field is close to linear one (Fig. 2.3) inside of the space with respect to z, and Ψ1 (x, y) in nonlinear (Fig. 2.4a). Temperature field, corresponding to the problem (2.157), (2.179) for f (2.177), is reported in Fig. 2.5. Comparing the results in Fig. 2.5 and Fig. 2.2, one may conclude that heat isolation of edge shell surface does not influence a qualitative behaviour of temperature field with respect to z (it remains nonlinear along z). However, Ψ1 (x, y) becomes nonlinear with respect to x, y (see Fig. 2.4b) contrary to Ψ1 (x, y) = const for the problems (2.172), (2.157) for f (2.177). Temperature field of the problem (2.157), (2.179) for f = 0 is reported in Fig. 2.6. It is linear with respect to z inside of the space. Temperature field for the problem (2.180), (2.157) for f (2.177) is shown in Fig. 2.7. It is nonlinear with respect to z. Both Ψ1 (x, y) and Ψ2 (x, y) are nonlinear with respect to x, y (see Fig. 2.8).

2.3 Solution of 3D Stationary Heat Transfer Equation

4·10-4 T(Z) 2 1 3

Z 0.5

0 1` 2` 3`

a) -4 4·10 T(X;0.25;Z)

-0.5

0.5

Z

1 X -4 3·10 T(X;0.25;Z)

-0.5

0

0.5

X

Z

1

b) Figure 2.2. Temperature field distributions (see text for more details).

219

220

2 Stability of Rectangular Shells within Temperature Field

-4

4·10 T(Z)

1 2

3 -0.5 3`

Z 0.5

0 2`

3` 4·10

-4

a)

T(X;0.5;Z) 3·10-4

-0.5

0 0.5 Z 1 X

-4

4·10

-0.5

T(X;0.25;Z)

0 0.5 Z 1 X

b)

Figure 2.3. Temperature field distributions (see text for more details).

2.3 Solution of 3D Stationary Heat Transfer Equation y1(x) -4

6·10

3 3·10

1/8

x

0

y1(x,y) 6.5·10

-4

4

-4

1

2

-3·10-4

0.5

1/8

7/8

7/8

x

y

a) 2·10

-3

y1(x)

0

y1(x,y) -2·10 1/8

2

-3

0 1/8

7/8

7/8

x

x 0.5 1

3

1

1

y

b) Figure 2.4. Function Ψ1 (x, y) and its cross sections (see text for more details).

221

222

2 Stability of Rectangular Shells within Temperature Field -4

4·10 T(Z) 2 1 3

-0.5 3`

Z 0.5

0

2` 1`

a) 4·10-4

-0.5

T(X;0.25;Z)

0

0.5 X

Z

1

3·10-4 T(X;0.5;Z)

-0.5

0 0.5

X

Z

1

b) Figure 2.5. Temperature field distributions (see text for more details).

2.3 Solution of 3D Stationary Heat Transfer Equation 4·10-4 T(Z) 3

-0.5

2

1

Z 0.5

0 3` 2` 1`

a) -0.5

3.3·10-4

T(X;0.25;Z)

0

0.5

Z

1 X T(X;0.5;Z) -0.5

-4

3·10

0

0.5

Z

1 X

b)

Figure 2.6. Temperature field distributions (see text for more details).

223

224

2 Stability of Rectangular Shells within Temperature Field -3

1

2

2·10 T(Z)

1` 2`

3 3` 0.5

0

-0.5

a)

Z

T(X;0.5;Z)

1·10-3 1.5·10-3 0.5

Z

1 X

T(X;0.5;Z)

0.75·10-3 1.2·10

-3

0.5 Z X

1

b) Figure 2.7. Temperature field distributions (see text for more details).

2.3 Solution of 3D Stationary Heat Transfer Equation -3

4·10

1

(x)

225

2 3

-3

2·10

(x,y)

x 0.5

0

1

1 -3

-2·10

1/8

0 1/8

-3

-4·10

7/8

7/8

y x

a) 2

1/8

(x,y)

1/8

-3

7/8

7/8

2·10

(x)

2

2

y

x

3 0 1

0.5

x

-3

-2·10

b) Figure 2.8. Function Ψ1 (x, y) (a) and Ψ2 (x, y) (b) and their cross sections (see text for more details).

226

2 Stability of Rectangular Shells within Temperature Field 4·10-4 T(Z)

1

1`

2 2`

3

3` 0

0.5

a)

Z

T(X;0.25;Z) 3.3·10-4 4·10

-4

0.5

X

Z

1

T(X;0.5;Z)

-4

2.5·10

3·10

-4

0.5 Z 1 X

b) Figure 2.9. Temperature field distributions (see text for more details).

2.4 Algorithm for Diﬀerence Equations

227

In Fig. 2.9 the temperature field corresponding to the problem (2.157), (2.181) for f = 0 is given. Owing to this figure, during heat shell isolation from edge surfaces and the surface z = −0.5 and with a lack of heat sources ( f = 0) the temperature field in linear with respect to z inside the space. Owing to analysis of boundary problems (2.172), (2.179)–(2.181) for the equation (2.157), a lack of heat sources ( f = 0) is associated with almost linear temperature field with respect to z. Functions Ψ1 (x, y) and Ψ2 (x, y) are nonlinear with respect to x, y. An occurrence of heat sources of the type (2.177) is associated with nonlinear (with respect to z) temperature field occurrence.

2.4 Algorithm for Diﬀerence Equations 2.4.1 Construction of diﬀerence equations We begin with transformation of the system (2.39) into non-dimensional form using the following relations w = 2Hw,

F = B11 (2H)3 F,

q=

Bi j = Bi j B−1 11 , ai j = ai j B11 , αi j = βi j = βi j (B11 α11 )−1 , i, j = 1, 2, λ =

B11 (2H)4 q, l12 l22

αi j , i, j = 1, 2, α11

l1 2H , k1 = 2 ki , i = 1, 2. l2 li

(2.182)

The system (2.39) possesses the following non-dimensional form (bars are omitted): ∂ 2 w ∂2 F ∂2 F ∂2 F ∂ 2 w ∂2 F ∂ 2 w ∂2 F − q+ + 2 −k1 2 − k2 2 − 2 2 − 2 ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂y ∂x2 ∂4 w 1 −2 ∂4 w ∂4 w λ + λ2 B22 4 + 2(B12 + 2B66 ) 2 2 + 4 12 ∂x ∂y ∂x ∂y ∂4 w ∂4 w − Ψ1 (x, y) = 0, 2B16 λ−1 3 + 2B26 λ ∂x ∂y ∂x∂y3 2 2 ∂w ∂2 w ∂2 w ∂ 2 w ∂2 w ∂4 F + a22 λ−2 4 + k1 2 + k2 2 + 2 2 − ∂x∂y ∂y ∂x ∂x ∂y ∂x a11 λ2

∂4 F ∂4 F ∂4 F + (2a + a ) − 2a λ − 12 66 16 ∂y4 ∂x2 ∂y2 ∂x∂y3 2a26 λ−1

where x, y ∈ G {0 ≤ x, y ≤ 1},

∂4 F + Ψ2 (x, y) = 0, ∂x3 ∂y

(2.183)

228

2 Stability of Rectangular Shells within Temperature Field

0.5

λ−2 β11

Ψ1 (x, y) = −0.5

0.5 Ψ2 (x, y) = −0.5

∂2 T ∂2 T ∂2 T −1 zdz, + β + 2λ β 22 12 ∂x∂y ∂x2 ∂y2

2 2 ∂2 T −2 ∂ T −1 ∂ T dz. + α λ − α λ 22 12 ∂x∂y ∂y2 ∂x2

Boundary conditions for the system (2.183) have the following generalized form ∂w T T li w, , Mnn , εnn , T n , T 12 , M , N = 0, (2.184) ∂n where:

0.5 M = T

0.5 zT (x, y, z)dz,

−0.5

N = T

T (x, y, z)dz. −0.5

Recall that in (2.183) and (2.184) bars are omitted. The system (2.183) is rewritten into the form ∂2 w ∂2 F ∂2 w ∂2 F ∂4 w ∂4 w ∂4 w + k + − A x 4 + A xy 2 2 + Ay 4 = k1 + 2 2 ∂x ∂x ∂y ∂y ∂x ∂y2 ∂y2 ∂x2 ∂2 F ∂2 w ∂4 w ∂4 w + q + A16 3 + A26 + Ψ1 (x, y), ∂x∂y ∂x∂y ∂x ∂y ∂x∂y3 2 2 ∂w ∂4 F ∂4 F ∂4 F ∂2 w ∂2 w a x 4 + a xy 2 2 + ay 4 = − k1 + 2 − ∂x∂y ∂x ∂x ∂y ∂y ∂x ∂y2 2

k2 where: Ax =

∂4 F ∂2 w ∂4 F + 2a16 λ + 2a26 λ−1 3 − Ψ2 (x, y), 2 3 ∂x ∂x∂y ∂x ∂y

(2.185)

1 −2 1 2 1 (2B12 + 4B66 ) , λ , Ay = λ B22 , A xy = 12 12 12 a x = a22 λ−2 , ay = a11 λ2 , a xy = (2a12 + a66 ) , A16 = −2B16 λ−1 , A26 = −2B26 λ.

The partial derivatives occurred in (2.185) are approximated by diﬀerence relations with the error o(|h|4 ) applying Taylor series development with respect to h powers (h is the mesh step of the space Gh {0 ≤ xi , y j ≤ 1, xi = ih, y j = jh, i, j = 0, 1, . . . , N; N = 1h ). h2 ∂4 u ∂2 u ≈ u − + o(h4 ) = xx 12 ∂x4 ∂x2

% 1 $ −ui+2 j + 16ui+1 j − 30ui j + 16ui−1 j − ui−2 j + o(h4 ), 2 12h

2.4 Algorithm for Diﬀerence Equations

229

∂4 u h2 ∂6 u 1 $ ≈ u − + o(h4 ) = 4 −ui j+3 + 12ui j+2 − 39ui j+1 + yyyy 4 6 6 ∂y ∂y 6h % 56ui j − 39ui j−1 + 12ui j−2 − ui j−3 + o(h4 ). The diﬀerence relations approximating derivatives with o(h4 ) error are denoted by

∂k l ∂xl ∂yk−l

ij

∂k u ∂xk

ij

= lhxk ui j o(h4 ),

= lhxl yk−l ui j + o(h4 ), k = 2, 4; l = 1, 2, 3,

where:

1 $ 72ui j + 20 ui+1 j+1 + ui+1 j−1 + ui−1 j+1 + ui−1 j−1 − 4 12h

38 ui+1 j + ui j+1 + ui−1 j + ui j−1 + 2 ui+2 j + ui−2 j + ui j+2 + ui j−2 −

ui+1 j+2 + ui+1 j−2 + ui+2 j+1 + ui−2 j+1 + ui−1 j−2 + % ui−2 j−1 + ui−1 j+2 + ui+2 j−1 , l x2 y2 ui j =

1 $ ui+1 j−2 − ui−1 j−2 + ui−1 j+2 + ui−2 j+1 − ui+1 j+2 + 24h2

% ui+2 j−1 − ui−2 j−1 + 10 ui+1 j+1 − ui−1 j+1 − ui+1 j−1 + ui−1 j−1

ehxy ui j =

Substituting the obtained diﬀerence relations into the system (2.185), the following system of nonlinear algebraic equation is obtained (its order is equal to nodes number in space Gn )

1 A x 56 + A xy 36 + Ay 56 wi j = A x l˜hx4 wi j + Ay l˜yh4 wi j + A xy l˜hx2 y2 wi j + 2 $ % % $ 1 k2 12h2 + ly2 wi j lhx2 Fi j + k1 12h2 + l x2 wi j ly2 Fi j − 24 1 h h l wi j l xy Fi j + (q + Ψ1 (x, y) 6h4 , 2 yx

1 (a x 56 + a xy 36 + ay 56)Fi j = a x l˜x4 Fi j + ay l˜y4 Fi j + a xy l˜x2 y2 Fi j + 2 2 %

$ 2 1 1 h (2.186) l xy wi j − k1 12h2 + l x2 wi j ly2 wi j − k2 l x2 wi j − Ψ2 (x, y)6h4 , 96 24 2 where: l˜x4 u, l˜x2 y2 u, l˜y4 u means that in this diﬀerence relation there is a lack of a term in central point i, j. Denoting by f1 (w, F), ϕ1 (w, F) the right hand sides of the system (2.186), depending on the function w, F values in mesh nodes, the system (2.186) can be recast into the form

230

2 Stability of Rectangular Shells within Temperature Field

Awi j = f1 (w, F) + (q + Ψ1 )6h4 , aFi j = ϕ1 (w, F) − Ψ2 6h4 ,

(2.187)

where: A = A x 56 + A xy 36 + Ay 56, a = a x 56 + a xy 36 + ay 56. Owing to diﬀerence relations of the operators lhxk u, for the system (2.185), approximation 25-points pattern is applied and two series out-contour nodes are required. The values of w and F in out-contour nodes are defined via boundary and compatibility conditions. The boundary value condition (2.53)–(2.56) accounting of compatibility conditions for the functions w and F (2.57)–(2.61) for x = 0 have the following non-dimensional form w = 0,

∂F ∂w = 0, F = 0, = 0, ∂x ∂x

∂2 w w = 0, = 12β11 ∂x2

0.5

∂F = 0, ∂x

(2.189)

T dz, F = 0,

(2.190)

zT dz, F = 0,

−0.5

∂w ∂2 F = 0, a22 2 = −α22 w = 0, ∂x ∂x

(2.188)

0.5

−0.5

∂2 w = 12β11 w = 0, ∂x2

0.5

−0.5

∂2 F zT dz, F = 0, a22 2 = −α22 ∂x

0.5 T dz.

(2.191)

−0.5

The derivatives, occurred in boundary conditions (2.188)–(2.191) are approximated by an error o(h4 ). The following two types of approximations are used to define w and F values of the out-contour points 2 ∂u 1

= −ui+2 j + 16ui+1 j − 30ui j + 16ui−1 j − ui−2 j + o(h4 ), 2 2 ∂x i j 12h

∂2 u ∂x2

= ij

∂u ∂x

∂u ∂x

= ij

1

− 20u + 6u + 4u − u 11u + o(h4 ), i+1 j i j i−1 j i−2 j i−3 j 12h2 = ij

1

−ui+2 j + 8ui+1 j − 8ui−1 j + ui−2 j + o(h4 ), 12h

1

3ui+1 j + 10ui j − 18ui−1 j + 6ui−2 j − ui−3 j + o(h4 ), 12h

where a point with index (i, j) lies on a space boundary, and (i+1, j), (i+2, j) denote out-contour nodes. For example, boundary value conditions (2.191) have the following diﬀerence representation for x = 0

2.4 Algorithm for Diﬀerence Equations

231

wi j = 0, 0.5 1

zT dz = δ1 (y j ), wi+2 j + 16wi+1 j + 16wi−1 j − wi−2 j = 12β11 12h2 −0.5

0.5 1

zT dz = δ1 (y j ). 11wi+1 j + 6wi−1 j + 4wi−2 j − wi−3 j = 12β11 12h2 −0.5

Observe that to find w values in out-contour nodes, there are two equations with two unknowns wi+2, j and wi+1, j . The second equation yields wi+1, j , assuming wi j = 0 on the boundary x = 0: % 1 $ −6wi−1 j − 4wi−2 j + wi−3 j + 12h2 δ1 (y j ) , 11 % 1 $ = 80wi−1 j − 75wi−2 j + 16wi−3 j + 204h2 δ1 (y j ) , 11 0.5 δ1 (y j ) = 12β11 zT (x, y, z)dz, x = 0.

wi+1 j = wi+2 j

−0.5

The boundary condition w = 0,

∂w = 0, ∂x

can be approximated analogously 1

18wi−1 j − 6wi−2 j + wi−3 j , 3 1

wi+2 j = 120wi−1 j − 45wi−2 j + δwi−3 j . 3 These boundary conditions can be rewritten in the following generalized form wi+1 j =

wi+1 j = wi+2 j =

(c) 2 a(c) 2 wi−1 j + a3 wi−2 j + wi−3 j + c12h f (y) ,

1

a(c) 1

(c) (c) 2 a(c) 4 wi−1 j + a5 wi−2 j + a6 wi−3 j + c204h f (y) ,

1

a(c) 1

(2.192)

where the coeﬃcients di(c) and c take the fully defined values depended on boundary condition type (Table 2.4). Similarly, boundary conditions with respect to function F are defined by (2.192), where F is given for (2.190), (2.191) in the following way α22 f = σ1 (y) = − a22

0.5 T (x, y, z)dz, x = 0, −0.5

232

2 Stability of Rectangular Shells within Temperature Field Table 2.4. Coeﬃcients a(c) i and c with the corresponding boundary conditions. a(c) 1 11 3

∂2 u ∂n2 ∂u ∂n

a(c) 2 -6 18

a(c) 3 -4 -6

a(c) 4 80 120

1 f = σ2 (x) = − λ−2 a11

a(c) 5 -75 -45

a(c) 6 16 8

c 1 0

f (y) δ1 (y)

0.5 T (x, y, z)dz, y = 0.

−0.5

It is worth noticing that various boundary conditions (2.188)–(2.191) are obtained when in initial data the coeﬃcients a(c) i , c, f are variated. This observation exhibits universality of the finite diﬀerence method in comparison with other methods, where owing to variation of boundary condition type the computation algorithm must be changed. 2.4.2 Stability problems In order to investigate a stability of shallow shells within a stationary temperature field, a method of direct solution of nonlinear equations governing their stress-strain state is applied. Critical loads (pure temperature, force, force and temperature) are defined via limiting points of the problem (load-deflection diagram). Both top and larger critical loads are defined. Process of solution of a statical problem of thermoelasticity contains of two parts. The first one includes temperature field T (x, y, z) definition, and it is reduced to solution of a boundary value problem (2.81) (see the Section 2.4). In result, the h temperature field T in 3D mesh space G3 {0 ≤ xi ≤ 1 , 0 ≤ y j ≤ 1, −0.5 ≤ zk ≤ 0.5} is obtained. Next, the following quantities are defined 0.5 ∂2 T ∂2 T ∂2 T zdz, 0 < x1 , y j < 1, Ψ1 (xi , yi ) = β11 λ−2 2 + β22 2 + 2β12 λ−1 ∂x∂y ∂x ∂y −0.5

0.5 Ψ2 (xi , y j ) = −0.5

α22 λ−2

2 ∂2 T ∂ 2 T −1 ∂ T dz, 0 < xi , y j < 1, + − α λ 12 ∂x∂y ∂x2 ∂y2 0.5

δ1 (y j ) = 12β11

zT (x, y, z)dz, x = 0, 0 ≤ y j ≤ 1,

−0.5

12β22 −2 δ2 (xi ) = λ B22

0.5

−0.5

zT (x, y, z)dz, y = 0, 0 ≤ xi ≤ 1,

2.4 Algorithm for Diﬀerence Equations

σ1 (y j ) = −

α22 a22

0.5 T (x, y, z)dz, x = 0, 0 ≤ y j ≤ 1, −0.5

1 σ1 (xi ) = − λ−2 a11

233

0.5 T (x, y, z)dz, y = 0, 0 ≤ xi ≤ 1,

(2.193)

−0.5

in nodes of mesh space Gh 0 ≤ x1 ≤ 1, 0 ≤ y j ≤ 1 with application of Simpson’s formula. Now second part begins, which is focused on solution to nonlinear algebraic equations system (2.186), corresponding to the problem (2.183) with boundary equation (2.188)–(2.191). Nonlinear system of algebraic equations, obtained through finite diﬀerence method with approximation o(h4 ) is solved via nonlinear relaxation method. The system (2.187) is cast in the form wi j =

1 6h4

f1 (w, F) + K q0 + Ψ1 (xi , y j ) , A A

Fi j =

1 6h4 ϕ1 (w, F) − KΨ2 (xi , y j ), a a

(2.194)

where: Kq0 = q. In addition, equations approximating boundary conditions (2.188)–(2.191) are attached, in the following form wi+1 j = wi+2 j =

(c) 2 w + a w + w + cK12h δ (y ) , a(c) i−1 j i−2 j i−3 j 1 j 2 3

1

a(c) 1

(c) (c) 2 a(c) 4 wi−1 j + a5 wi−2 j + a6 wi−3 j + cK204h δ1 (y j ) ,

1

a(c) 1

wi j+1 =

1

(c) 2 w + a w + w + cK12h δ (x ) , a(c) i j−1 i j−2 i j−3 2 i 2 3

a(c) 1 1 (c) (c) 2 wi j+1 = (c) a4 wi j−1 + a(c) w + a w + cK204h δ (x ) , i j−2 i j−3 2 i 5 6 a1 1

Fi+1 j = (c) a2(c) Fi−1 j + a3(c) Fi−2 j + Fi−3 j + cK12h2 σ1 (y j ) , a1 1

Fi+2 j = (c) a4(c) Fi−1 j + a5(c) Fi−2 j + a6(c) Fi−3 j + cK204h2 σ1 (y1 ) , a1 1

2 Fi j+1 = (c) a2(c) Fi j−1 + a(c) 3 F i j−2 + F i j−3 + cK12h σ2 (xi ) , a1 1

Fi j+2 = (c) a4(c) Fi j−1 + a5(c) Fi j−2 + a6(c) Fi j−3 + cK204h2 σ2 (xi ) , (2.195) a1

234

2 Stability of Rectangular Shells within Temperature Field

where the coeﬃcients c and c can take the value of 1 or 0. The values c = 0, c = 0 correspond to boundary conditions (2.188); c = 1, c = 0 correspond to (2.189); c = 0, c = 1 – (2.190); c = c = 1 – (2.191). Owing to the boundary conditions form (2.195), either boundary value problems of (2.188)–(2.191) type are solved or their combination on diﬀerent or the same shell side(s) can be used. In this case the corresponding compatibility conditions in the points of a boundary condition change are added. The values i = j = 0 correspond to the central shell point x = y = 0.5, where the computational process starts. In the equation corresponding to this point (n−1) + hw (n - step number and contrary to all other equations, the value w(n) 00 = w00 with respect to w; hw - deflection step in the centre point) is given. Then the quantity K is defined, which is used in all other remaining equations of the system (2.194) solvable with respect to wi j , Fi j . Index i = w correspond to boundary nodes along straight line x = 0; j = N boundary nodes along contour y = 0; i = N + 1, N + 2 - out contour nodes; j = N + 2, N + 2 - out-contour nodes. Parameter q0 should be given, and it characterizes a load amount in the quantity defined by the critical load K(q0 +Ψ1 (x, y)). For q0 = 0 purely temperature-like problem is obtained. For q0 > 0 one deals with force and temperature loads, and for δ1 = δ2 = Ψ1 (x, y) = Ψ2 (x, y) = σ1 = σ2 = 0 purely force load occurs. Note that a computational process is carried out through iterational scheme until the values of wi j , Fi j on the previous and next iteration coincide within the assumed accuracy. Then a next step with respect to w is realized, i.e. the value = w(n) w(n+1) 00 0 + hw , in the central point is computed, an a calculus is repeated. In result of solution computation within M steps with respect to w, the dependence load-deflection is obtained, and upper and lower critical loads are obtained. 2.4.3 Reliability of obtained results The problem (2.183), (2.184) is going to be solved applying finite diﬀerence method with approximation error o(h4 ). It is well known that occurrence of high order of algebraic equations belongs to main drawback of finite diﬀerence method o(h2 ). However, this problem can be omitted, if higher order approximations are used. The latter allow to take less density mesh and to get a solution with assumed accuracy of 1.5 – 2 times faster then applying approximation o(h2 ). Let us cut the space G {0 ≤ x ≤ 1, 0 ≤ y ≤ 1} by squared mesh with a step h and let substitute all partial derivatives in (2.183), (2.188), (2.191) by the corresponding diﬀerence relation with errors o(h2 ) and o(h4 ). Owing to a change of diﬀerential problem into diﬀerence problem, a system of nonlinear algebraic equations is obtained, which has the following generalized form fi (x1 , x2 , . . . , xn ) = 0,

i = 1, 2, . . . , n,

(2.196)

where unknown variables x1 , x2 , . . . , xn represent unknown values of the functions wi j , Fi j in the mesh nodes. It is assumed that (2.196) includes also boundary conditions.

2.4 Algorithm for Diﬀerence Equations

235

To solve the system of nonlinear algebraic equations (2.196) and iterational method of nonlinear relaxation is applied, which is governed by the following equation:

(k+1) (k) , xi , . . . , xn(k) , i = 1, 2, . . . , n . (2.197) xi(k+1) = xi(k) + ω fi xi(k+1) , . . . , xi−1 It is well recognized that for a given constraint into the initial nonlinear system (2.196) the iterational process (2.197) is convergent only if accelerating parameter ω ∈ (0, 1]. Experiment shows, that for both approximation (i.e. o(h2 ) and o(h4 )) the optimal interval of ω is the following one [0.25; 0.75] for the problem associated with boundary condition (2.188), and the interval [0.1; 0.5] for the problem associated with the boundary condition (2.191), where ω essentially depends on k1 and k2 . Owing to increase of k1 and k2 , an optimal parameter ω decreases. The Runge principle is used for the step h estimation, whereas for the diﬀerence scheme o(h2 ) for the space G {0 ≤ x ≤ 1, 0 ≤ y ≤ 1} the step h = 1/8 is used (for approximation o(h4 ) the step h = 1/16 is applied). Results for diﬀerent steps are reported in Table 2.5 for the point x = y = 0.5. Also results of the Bubnov-Galerkin method are attached. A comparison of computation computer time are given for the boundary conditions (2.188), (2.191) in Tables 2.6, 2.7, correspondingly. As it is seen in Table 2.6, computation time for the problem (2.183), (2.188) with approximation o(h4 ) is less comparing with o(h2 ) of amount of 1.5–2 times, and for the problem (2.183)– (2.191) (see Table 2.7) the computational time amount of o(h4 ) scheme is two time less than that of o(h2 ) approximation. Hence, for rectangular shell computations the approximation o(h4 ) is recommended (see also [344]).

Table 2.5. Comparison of the finite diﬀerence methods with approximations o(h4 ) and o(h2 ) and the Bubnov-Galerkin approach. k1 = k2 = 10, o(h4 ) k1 = k2 = 10, o(h2 ) h q F h q F 1/6 38.8 0.32 1/8 38.06 0.32 1/12 38.8 0.32 1/16 38.12 0.32

Bubnov-Galerkin method q F 38.8 0.32

Table 2.6. Computational time intervals distribution for the boundary condition (2.188). wchw = 0.1 0.1–3.5 0.1–5.0 0.1–7.5

k1 = k2 10 18 24

(2.188) o(h4 ), 9 15 38

o(h2 ), 11 18 71

236

2 Stability of Rectangular Shells within Temperature Field

Table 2.7. Computational time intervals distribution for the boundary condition (2.191). wchw = 0.1 0.1–2.0 0.1–3.0 0.1–2.5

k1 = k2 0 24 36

(2.191) o(h4 ), 3 8 13

o(h2 ), 6 12 33

2.4.4 Transversal load Owing to stability computations of flexible anisotropic shells during rotation of the coordinate system of amount of angle ϕ, the relations (2.8)–(2.10) are used to compute coeﬃcients of equation (2.39). For a shell with geometrical parameters k1 = k2 = 24, subjected to an action of uniformly distributed load, the boundary value problems (2.54), (2.55) are solved. In Fig. 2.10 dependencies “deflection-load” in the shell centre q (w(0.5; 0.5)), obtained for the (2.54) problem (solid curve), (2.55) (dashed curve) for ϕ = 0◦ , 30◦ , 45◦ , 60◦ , 90◦ are reported. Curves, corresponding to the mentioned ϕ values, are denoted by digits 1, 2, 3, 4, 5. For ϕ = 30◦ and 90◦ , ϕ = 45◦ and 60◦ the curves load-deflection coincide (in Fig. 2.10 curve 3 coincides with curve 4, curve 2 coincides with curve 5) for the problem (2.54), (2.55). Increasing value from 0◦ to 53◦ , a value of top critical load increases for the problem (2.54), (2.55), whereas increasing ϕ form 53◦ to 90◦ , it decreases. Shells with k1 = k2 = 24 with boundary (2.55) loose their stability for smaller amount of top critical load than those with boundary condition (2.54). Here the upper critical load dependence on angle ϕ is displayed. Solid curve corresponds to problem (2.54), whereas dashed curve corresponds to problem (2.55). Maximal top critical load value corresponds to the angle ϕ = 53◦ . Development of shells stress-strain state, using an example of shell k1 = k2 = 24, in dependence of boundary conditions and angle ϕ can be traced in Fig. 2.11a for the points x = y, and in Fig. 2.11b for the points y = 0.5 for boundary conditions (2.54), (2.55). Diagrams of fundamental functions w, w

xx , F, F

xx corresponding to the problem (2.54) for w(0.5; 0.5) = 1.5 are represented by solid curves. Similarly, diagrams of these functions for w(0.5; 0.5) = 1.5 corresponding to problem (2.55) are marked by dashed curves. Solution of the problems (2.55), (2.54) for angles ϕ = 0◦ , 30◦ , 45◦ are marked by digits 1, 2, 3, correspondingly. Comparing solutions with the same boundary conditions but for diﬀerent angle ϕ, one may conclude that diagrams of the functions w, w

xx related to diﬀerent angle ϕ coincide, but the diagrams of the function F, F

xx are diﬀerent. In the latter case, larger absolute value of ϕ corresponds to larger value of F, F

xx in the shell centre. Owing to analysis of shell solutions with boundary conditions (2.54), (2.55) for the same values of angle ϕ, the diagrams of the functions w

xx essentially diﬀer either along x = y or y = 0.5. In the latter case, a diﬀerence along y = 0.5 is essential not only in the shell centre, but in its contour also. Diagrams F, F

xx for the problems (2.54), (2.55) computed in the shell centre and corresponding to the problem (2.55) are in absolute value larger of amount of 25% for the function F, and on amount of 10% for the function

237 W

X

1 0

0

100

200 100

200

300 q

3,4

300

q

1

1

2,5

2,5

30

3,4

60

2

90

j

3

Y`

0

Y

5

2

j

X`

4

2.4 Algorithm for Diﬀerence Equations

Figure 2.10. “Load-deflection” dependence in the shell centre.

F

xx than in the problem (2.54). The following data are used during computations: kg 5 kg 5 kg G = 0.42 · 105 cm 2 , ν2 = 0.07, E 1 = 2.1 · 10 cm2 , E 2 = 1.6 · 10 cm2 (glass-plastic material). Note that reported calculations can be used for optimal construction design in the sense of maximal critical load in dependence on angle ϕ.

238

2 Stability of Rectangular Shells within Temperature Field -30 1.2

1,2,3

-12

F

2

3 1.5

-15

W 1,2,3

3

0,6

x=y 0.5

0

3

2

1

1,2,3

1,2,3

3 2

1

1

1

0

W``xx a)

x=y 0.5

0

F``xx

-30 1.2

F

2

1,2,3 3

-15

1.5 1,2,3

-6

2

1,2,3 1,2,3

0,6

1

1 2

3 2 3

1

1 2

x 0.5 0

x 0.5

F``xx

W``xx b)

Figure 2.11. Fundamental functions w, w

xx , F, F xx for x = y (a) and y = 0.5 (b) in the shell centre (boundary conditions (2.54), (2.55)).

2.4.5 Diﬀerent boundary conditions Consider isotropic plates and shells (ν = 0.3) for diﬀerent boundary conditions (2.188), (2.191) along their sides under action of uniformly distributed transversal load. Digit 9 denotes ball type conditions (2.191), digit 5 denotes clamping edge of

2.4 Algorithm for Diﬀerence Equations

239

the type (2.188), digits 1–4, 6–8 denote their combinations along a shell side. The mentioned notation is used in Figures 2.12, 2.13. In Figures 2.12a, 2.13a the graphs “load-deflection” in the plate centre (0.5; 0.5) for all clamping types, denoted by digits 1–3, are reported. The curve number is the same as boundary type number. Comparing the curves in Fig. 2.12a one may conclude that ball type fixation on the corner points (curve 4) practically does not influence the plate stress-strain state (curves 4 and 5 coincide). This conclusion is supported by results reported in Figure 2.13b, where the diagrams of fundamental functions w, w

xx , F, F

xx in the point (0.5; 0.5) along the diagonal x = y are given. Now the curves are denoted in the same way as in Fig. 2.12. Decreasing length of ball supported contour, in the neighbourhood of corner points (curves 3, 4) the graph “load-deflection” approaches curve 5, corresponding to clamped contour. In Fig. 2.12b the graphs for clamping 4 and 9 diﬀer in shell centre and in neighbourhood of the contour. Owing to decrease of clamped contour length in direction 1–4, the w

xx value in the plates centre increases with respect to absolute value and achieves maximum for fully clamped contour (curve 5). Comparing the curves in Fig. 2.13a, it is seen, that clamping of corner points has influence on the stress-strain plate state (≈ 20%, compare Figures band 9). Owing to decrease of the length of clamped contour, in the neighbourhood of corner points (curves 6, 7), the graph “load-deflection” tends to curve 9, i.e. it describes the plate supported by balls along whole contour. Increasing length of clamped contour along a side in vicinity of corner points on amount of 50% (curve 7), a plate strength is increased on amount of 50% comparing with ball supported plate. In Fig. 2.13b, curves of the fundamental functions w, w

xx , F, F

xx in the point w(0.5; 0.5) = 1.5 along the diagonal x = y are shown. A curvature changes its sign. Curvature values w

xx depends on the length of clamped contour along a side. A plate with large interval of camped contour (curve 8) is associated with large curvature w

xx absolute value in the plate centre. Absolute value of the function F decreases with increase of clamped contour length along a shell side. Function F

xx changes its sign. Increasing length of shell clamped contour, the graphs F

xx approach curve 5, governing behaviour of F

xx for a plate with clamped contour. In Fig. 2.14 the graphs “load-deflection” in the central shell point for boundaries denoted by digits 1-5, 9 are reported. Solid curves correspond to a shell with geometrical parameters k1 = k2 = 24, whereas dashed curves correspond to k1 = k2 = 28. Let us denote shell side lengths by l0 , a length clamped interval along a side by l2 , a length of ball-supported part along a side by l1 = l0 − l2 . Owing to behaviour of curves reported in Fig. 2.14, clamping in the middle of shell side essentially influences the upper critical load, since it is decreased on amount of 30% in comparison with ball-supported shell (compare curves 2 and 9, 3 and 9). Further increase of clamped contour length l2 ≥ 34 l0 in the middle of a shell side does not influence a value of the upper critical load (curves 4 and 5 fully coincide; curve 3 diﬀers from 5 amount of 1%). For a shell with geometrical parameters k1 = k2 = 18 (dashed curves) an increase of clamped contour length in the middle of a side l2 ≥ 14 l0 does not influence stress-strain shell state, governed by “load-deflection” dependence in

1.5 b)

xy 0.5

5,4 1

F 0

45

90

135

q

0.5

9 1 2 3 4 5

a)

1.0

9

1

2

4,5

3

W

1.5

0

-0.1

-0.2

-0.3

0

0.6

1.2

1.8 W

2

1

9

9

3

5,4,3

2

x=y 0.5

2

9

4,5

5,4

3

3

2

9

1

1

2

F``xx

4,5

0

1.5

3

6

W``xx

0

-6

-12

2 Stability of Rectangular Shells within Temperature Field

-18

240

Figure 2.12. “Load-deflection” dependence in the shell centre (a) and fundamental function for x = y and y = 0.5 (b) (see the text for more details).

1.5

0

1.5

241

F 0

45

90

135

q

0.5

9 6 7 8 5

a)

1

9

6

5

7

8

W

1.5

0

-0.1

-0.2

-0.3

0

1.2

0.6

6

1.8 W

9

7

6

7

9

5,8

5,8

b)

x=y 0.5

x=y 0.5

5,8

7

5

7

6

9

F``xx

3

W``xx 6

9

6

8

0

-6

-12

-18

2.4 Algorithm for Diﬀerence Equations

Figure 2.13. “Load-deflection” dependence in the shell centre (a) and fundamental function for x = y and y = 0.5 (b) (see the text for more details).

242

2 Stability of Rectangular Shells within Temperature Field

the central point (0.5; 0.5) (curves 3, 4, 5 coincide with each other). For l2 ≤ 14 l0 a shell looses its stability, and a value of upper critical load is decreased on amount of 15% in comparison to the ball-supported shell (compare curves 1 and 9). In Fig. 2.14–2.18 graphs of fundamental functions w, w

xx , F, F

xx for the shell k1 = k2 = 24, corresponding to boundary condition 1-4 along lines x = y and y = 5 are shown. In Figures 2.19, 2.20 graphs of fundamental function w, w

xx , F, F

xx for the shell k1 = k2 = 24 corresponding to clamped contour 5 and ball-supported shell 9 are shown. Curves on Figures 2.15–2.20 are denoted by the same number as a type of fixation. Index identifies a deflection in the shell centre with a corresponding curve. For example, 52 corresponds to a shell with clamped contour, and deflection value in the central point w = 1.7. In Fig. 2.21 graphs “load-deflection” in the central shell point for fixations denoted in the figure by points 5-9 are shown. Solid curves correspond to a shell with geometric parameters k1 = k2 = 24, whereas dashed zones correspond to parameters k1 = k2 = 18. Owing to analysis of Figure 2.21, clamping of corner points have an important influence on upper critical load value, decreasing it in comparison with ball supported shell on 13b (compare curves 6 and 9). An increase of clamped contour length l2 , beginning from corner points, up to 34 l0 decreases upper critical load value up to 40% for the shell k1 = k2 = 24 (compare the curves 7, 8, 9). Further increase of the clamped contour length l2 ≥ 34 l0 does not influence upper critical load value. Curves 5 and 8 up to deflection w(0.5; 0.5) = 1.7, coincide. In Fig. 2.21a the dependencies of top (curve ) and below (curve 1) critical loads on clamped contour length in the shell middle side are reported. A development of shell stress-strain state with k1 = k2 = 24 dependence on fixation type along a shell side can be traced in Figures 2.22–2.24, where graphs of fundamental functions w, w

xx , F, F

xx , corresponding to fixation 6-8 along both diagonals x = y and y = 0.5 are reported. Again, curves notation is linked with their fixation types and indeces notation is similar to the previous discussed case. In Figure 2.25 curves of equal deflections for the shell k1 = k2 = 24 for all fixation types 1-9 for w(0.5; 0.5) = 1.5 are shown. Figure 2.25 displays, how a shape of deformable shell surface is changed owing to fixation contour type. The curves diﬀer from those occurred in problems 1, 2, 5, 6, 8, 9, and the same observation holds for deflection level values. For the problems 4, 5, the curves of equal deflection coincide either in a shape or in a magnitude. For the problems 6, 7, 9, the curves are closed in shape, but are diﬀerent in magnitude. Comparing curves 4 an 5 in Fig. 2.14, 9 and 6 in Fig. 2.19 one may conclude that fixation type of corner points essentially influences a value of top critical shell load (≈ 15%), whereas a ball-type support of the shell corner points has no influence on stress-strain shell state (curves 4 and 5 coincide). In Figure 2.26, the dependencies “load-deflection” in the central point q(w(0.5; 0.5)) for the shell with k1 = k2 = 36 for all considered type of fixation along a shell side 1-9 are reported. The curves, similarly to those in Figs. 2.14, 2.21, have the same number as fixation type. It should be emphasized that the shell k1 = k2 = 36 is

4 5 l0 0

100

200

300

q

1.1

2

1

l0

5,4,3

9

9 1 2 3

3,4,5

2.2

1

2

3.3

9

4.4

243

W

2.4 Algorithm for Diﬀerence Equations

Figure 2.14. Load-deflection function in the shell centre for the boundary condition 1-5, 9 (solid curves - k1 = k2 = 24; dashed curves k1 = k2 = 28; see text for more details).

more sensitive to variation of fixation along a shell side, than a shell with k1 = k2 = 24.

244

2 Stability of Rectangular Shells within Temperature Field 1.0 -50 F

-10 13

13 25

0

W

13

12

12 11

12

0.5 -25

11

x=y 11

W``xx

11

x=y

00

0.5

13

12

0.5

F``xx

a) 1.0 -50

-10

F

13

0.5 -25

13 11 0

x

12

11

00

0.5

-5

11

11

12

12

13

12

13 25

0

x

0

0.5 11

W``xx

5

25

13 10

11(W=0.9), 12(W=1.6), 13(W=2.5) b)

F``xx

15

Figure 2.15. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 1-4 for x = y (a) and y = 0.5 (b) (see text for more details).

A ball-type support in the middle of a shell side yields an increase of top critical load already for l1 = 14 l0 in comparison to fully clamped shell in amount of 5%. Further increase of the ball-type support length in the middle shell side increases shell stability, and for l1 = 34 l0 an increase of top critical load (in comparison to clamped shell) is in amount of 60%. The ball-type clamping in contour points for l1 = 14 l0 practically does not influence a shell stress-strain state. Curves 4 and 5 coincide, and for l1 = 12 l0 the value of upper critical load increases on amount of 1% in comparison to the clamped contour 5. Further increase of ball-type support

2.4 Algorithm for Diﬀerence Equations 0.6

0.4

23

23 22

W

22

24

F

-50

24

25

245

24

22

-25 0.2

23 24

-8

23

21

-4

22 21

21 0

x=y

21 0

0.5

x=y

0

0.5

F``xx

0

W``xx a) 0.6

W

23

22,24 22

-9 23 24

21

-25

21

22

22 21 0

0.3

23 24

23

-50

24

3

F

21

x

0

0.5

0

x

0

0.5

25

W``xx

21 9

50

21(W=1), 22(W=1.7), 23(W=2.2), 24(W=2.7) b)

23 24

F``xx

18

Figure 2.16. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 1-4 for x = y (a) and y = 0.5 (b) (see text for more details).

interval l1 ≥ 12 l0 increases the value of the top critical shell load, and for l1 = 34 l0 and it is in amount of 10% in comparison to clamped contour.

246

2 Stability of Rectangular Shells within Temperature Field -50 0.7

34 33 25

32

33

34

33

31 0

x=y

31

31

32

32

31

x=y

0

0.5

0

0.5

34

32,34

-25

W

-7

33

F

F``xx

0

W``xx a)

-50 0.7

34 33

F 33

33,,34 2.5

W

33

34

31

31

31

32 0

-25

32

32

-7

34 32

31

x

0 0

0.5

x

0

0.5

25

W``xx

7

50

31(W=0.9), 32(W=1.8), 33(W=2.2), 34(W=2.6) b)

F``xx

14

Figure 2.17. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 1-4 for x = y (a) and y = 0.5 (b) (see text for more details).

In Figs. 2.27a, b the graphs dependence of top critical load on fixation type for shells k1 = k2 = 36 (curve 1), k1 = k2 = 24 (curve 3) are reported. Figure 2.27a corresponds to fixation 1-5, 9 (a length of ball-type supports l1 is variated beginning from corner points), whereas Fig. 2.27b corresponds to fixation 5-9 (a length of ball-type support l1 is variated beginning from the middle of a side). Owing to behaviour of curves 1 and 2 in Fig. 2.27a, an increase of ball-support length up to l1 ≤ 12 l0 beginning from corner points, slightly influences a shell stress-strain

2.4 Algorithm for Diﬀerence Equations

247

-50

-10

44 43 25

W

43

44

42

42 41

0

x=y

0.5

F

41

W``xx

44 -5

42

43,44

41

41

0

0.5

42,43

43

-25

x=y

0

0

0.5

F``xx

a)

-50

-10

44 43 25

W

43

-25

42

44

0.5

42 41 0

42

F

43,44

0

0.5

0

-5

42

41

x

43 44

41

41

x

0

0.5

-25 5

W``xx

-50 10

41(W=0.9), 42(W=1.8), 43(W=2.2), 44(W=2.6) b)

15

F``xx

Figure 2.18. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 1-4 for x = y (a) and y = 0.5 (b) (see text for more details).

state. For l1 ≥ 34 l0 an upper critical load value increases in comparison with fully clamped shell, and for ball-supported shell an increase of upper critical load value is in amount of 68% in comparison to clamped shell (for k1 = k2 = 24). For the shell k1 = k2 = 36 its stability increases for l1 ≥ 14 l0 , and for ball-supported shell an

248

2 Stability of Rectangular Shells within Temperature Field 0.6 -60

54

W

-9

53,54

-45

53 3

F

53

0.3

53 54

52

51

1.5

52

52 51

51

0

x=y

0

0

0.5

0

0.5

-3

51

-15

x=y

-6

54

52

-30

W``xx

W``xx

3

a) -50

3

W

53 52 51 0

0.4

53

54

F

53,54 -8

52 52

-25

52

0.2

51

51

x

0 0

0.5

54

51

x

0

0.5

25 54

W``xx

8

50

51(W=1), 52(W=1.7), 53(W=2.2), 54(W=2.6) b)

53 54

F``xx

16

Figure 2.19. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 5, 9 for x = y (a) and y = 0.5 (b) (see text for more details).

increase of upper critical load is on amount of 70% comparing with a clamped shell. In Fig. 2.27b it is seen, that an increase of a length of ball-support in the middle of a side with l1 = 14 l0 (for shell k1 = k2 = 36), l1 = 12 l0 (for k1 = k2 = 24), l1 = 34 l0 (for k1 = k2 = 18) influences the value of upper critical shell load. Therefore, ball-support in the middle of a side more essentially influences on shell stability,

2.4 Algorithm for Diﬀerence Equations

249

-80 1.4

6

-14

F

-60

W

4

92

93

94

93

91

91

-40

94

92

0.7

-7

94

93

93

-20

2 92 91 0

x=y

92

94 91

x=y

00

0.5

W``xx

F``xx

0.5

0

20

a)

1.4

6

W

93

94

-14

F 92

92 93 94 91

93

0.7 -30

3 92 91 0

-7

91

94 93

91

92

94

x

00

0.5

x 0.5

F``xx

0

91(W=0.8), 92(W=1.3), 93(W=3.5), 94(W=5.8)

W``xx b)

Figure 2.20. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 with boundary conditions 5, 9 for x = y (a) and y = 0.5 (b) (see text for more details).

than ball-type support on a shell side ends. In addition, owing to increase of the parameters k1 , k2 , the value of l1 decreases, for which the value of top critical shell load begins to increase.

W 0

100

200

300

q

5

8

5,8

10

6

6

9

9

7

5 6 7 8 9

5

20

5

8

30

6

8

40

0

150

300

q

1

0.5

1

2

7

1.0

50

2 Stability of Rectangular Shells within Temperature Field

2

250

Figure 2.21. Load-deflection function for the boundary condition 5-9 and the dependence qcr (R1 /R2 ) for k1 = k2 = 24 (solid curves) and k1 = k2 = 18 (dashed curves).

2.4 Algorithm for Diﬀerence Equations 75

-75

W 64

1.2

64

50

F

-12 62

63

61 0.6

62 61 0

-6

63 64

62 61

x=y

0 0

x=y 0.5

W``xx

0

0.5

F``xx

25

6

-75

W

1.2

5.0

-50

64

63

62

0.6

63

62 61

64

-25

63 62 61

F

61

64 63

0

61

64

-25

25

2.5

62

63

-50

63

7.5

251

64

62

x

61

0 0

0.5

x 0.5

F``xx

0

61(W=0.9), 62(W=1.4), 63(W=3.6), 64(W=5.8) 25

W``xx

Figure 2.22. Fundamental functions w, w

xx , F, F xx for the boundary conditions 6-8 for x = y and y = 0.5 (see text for more details).

In Figure 2.27c, the curves of top critical load value dependence on magnitudes of geometrical shell parameters k1 , k2 , for the following fixation types: 1, 6, 7, 9 (a curves number corresponds to fixation case) are shown. The results displayed in Figure 2.27c show that the shell with k1 = k2 = 28 possesses the same value of top

252

2 Stability of Rectangular Shells within Temperature Field -60

4.5

W

1.0 -45

74

74 73

3.0

71 0

72

x=y

00

0.5

-5 71

71

71

x=y

73

72 74

-15

72

72

73 74

-30 0.5

73 1.5

-10

F

0

0.5

15

W``xx

F``xx

-75

5

74 5.0

W

74

-50

73

1.0

F 72

73 2.5

72

72

-25

0.5

-10

72 73

73

74

-5

74

71

71 71 0

x

0.5

00

W``xx

x 0.5

F``xx

0

25

71(W=0.9), 72(W=1.6), 73(W=3.0), 74(W=4.7)

Figure 2.23. Fundamental functions w, w

xx , F, F xx for the boundary conditions 6-8 for x = y and y = 0.5 (see text for more details).

critical load for fixation 1 and 6, whereas the shell with k1 = k2 = 32 - for fixations 1 and 9.

2.5 Computations of Plates and Shells in a Temperature Field 2.5.1 Stress-strain state In this section, investigation of influence of geometric parameters k1 , k2 on the temperature field T (x, y, z) and functions Ψ1 (x, y), Ψ2 (x, y) is carried out. Stationary 3D

2.5 Computations of Plates and Shells in a Temperature Field

253

-60

84 0.8

30

W

81

0

x=y

81

x=y

0

0.5

-4

81

81

82 84

84 82,84

0.4

82

-8

83

82

83

-30

83

83

F

0

0.5

W``xx

F``xx

4

-60

84

30

W

0

F

-30

83

84 83 82 81

0.8

0.4

82

82

83

84 83

82

-4 81

81

x

81 00

0.5

W``xx

30

-8

84

x

0

0.5

F``xx

4

81(W=0.9), 82(W=1.4), 83(W=2.2), 84(W=3.0)

Figure 2.24. Fundamental functions w, w

xx , F, F xx for the boundary conditions 6-8 for x = y and y = 0.5 (see text for more details).

heat transfer equation for shallow shells (2.157) is solved through diﬀerence scheme (2.133) with an error o(|h|4 ) and with occurrence of the form 2k ∂T ∂z , where either 1 k = 2 (k1 + k2 ) or k = 0. As numerical parameters for k1 , k2 ≤ 24 on the temperature field is weak (less than 1%), however, an increase of these parameters increases this influence. The functions Ψ1 (x, y), Ψ2 (x, y) also diﬀer slightly for variations of k1 , k2 ≤ 24. The graphs “load-deflection” obtained accounting or not the term 2k ∂T ∂z , practically coincide.

254

2 Stability of Rectangular Shells within Temperature Field 9

1

2 0.3

6

8

0.4

6

8 0.6 7 1.1 43 1.

6 0.3 0 0.9 34 1.

6 0.3 0 0.9 34 1.

1.5

1.5

1.5

3

7

8

5

0.4

1.5

6 0.9 35 1.

0 0.6 9 1.0 40 1.

6 0.3 1 0.9 34 1.

1.5

3

1.5

4

5

6

4

1.3

1.3

1.5

09

1.0

7 0.5 3 1.0 38 1.

1.5

1.5

Figure 2.25. Curve of equal deflections for the shell k1 = k2 = 24 for the boundary conditions 1-9 and w(0.5; 0.5) = 1.5.

Consider a plate under action of uniformly distributed transversal load and temperature field. In Fig. 2.28a, the curves “load-deflection” in the centre of the plate clamped along contour (2.190) and subjected to an action of purely force load (curve 1), load and temperature field (2.179) (curve 2), load and temperature field (2.181) for f (2.177) (curve 3), load and temperature field (2.181) for f = 0 (curve 4),

2.5 Computations of Plates and Shells in a Temperature Field 840

255

q 1

6

630

9 8

7

420 2 4

8

5

210

W 1.5

3

4.5

6

Figure 2.26. Load-deflection function in the shell (k1 = k2 = 36) centre for the boundary condition 1-9.

and also for ball-supported plates along contour (2.189) under action of load and temperature field (2.181) for f (2.177) (curve 5), load and temperature field (2.181) for f = 0 (curve 6), and load (curve 7) are reported. Comparing the curves 2, 3, 4, with 1, one observes that for clamped plate along its contour, the temperature field

256

2 Stability of Rectangular Shells within Temperature Field

830

800

q`kpu

q`kpu 1

1

415

400 2 2

3

l1

l1 0

0.5

1

0

0.5

l1

l1

a) 850

1

b) q`kpu

1

9

425

6

7

0

18

36 Kx

c) Figure 2.27. Top critical load for the boundary conditions 1-5, 9 (a), 5-9 (b), and qncr (k x ) (c) for diﬀerent boundary conditions (see text for more details).

2.5 Computations of Plates and Shells in a Temperature Field

257

(2.179) (corresponding to heat isolation of plate sides surfaces) yields an increase of shell strengthness on amount of 7%, and temperature field (2.181) for f = 0 corresponding to heat isolations of plate sides surfaces and the surface z = −0.5 (without heat sources, the plate strength is decreased on amount of 8% in comparison with pure force). The temperature field (2.177), decreases strength of a shell with clamped contour in amount of 40% in comparison with an action of purely forcing load. Owing to the curves 3 and 4, an occurrence of heat sources of the form (2.177) for the temperature field (2.181) decreases shell strength in amount of 35%. Occurrence of heat sources (2.177), for the temperature field (2.181), increases strength of a ball-supported plate along surface in amount of 40% (see curves 5 and 6). An action of temperature field (2.181) without heat sources ( f = 0) on a ballsupported plate is rather weak, since curves 6 and 7 coincide. In Figure 2.28b graphs of functions w

xx , F

xx corresponding to w(0.5; 0.5) = 1.5 along the axis y = 0.5 are displayed. Curves have the same notations as in Figure 2.28a. An action of temperature moment on ball-supported plate contour, corresponding to temperature field (2.181) for f (2.177) is presented also in Figure 2.28b (see curves 5-7). All results discussed in the above are obtained for q0 = 0.01. The results given in Figure 2.28a exhibit an influence on fixation type on plate thermal-stress state (compare curves 3 and 5, 4 and 6). For the plate, being under action of temperature filed (2.181) for f (2.177), fixation of the contour (2.190) increases plate strength ability in amount of 55% in comparison to ball-support (2.189). On the other hand, for the plate being under action of temperature field (2.181) for f = 0, fixation of the contour (2.190) increases plate strength in amount of 230%, whereas purely forcing load action increases plate strength in amount of 260% in comparison to ball-support (curves 1, 7). Let us analyse now shells k1 = k2 = 18 and k1 = k2 = 24, ball-supported along their contours (2.189), and subjected to an action of uniformly distributed load (q0 = 0.01) and temperature field, corresponding to the problem (2.181) for f (2.177), (2.181) for f = 0, and also to the problem (2.179) for f (2.177). In Figure 2.29 graphs “load-deflection” in the centre shell point q(w(0.5; 0.5)), being under action of purely forcing load (curve 1), load and temperature field (2.179) (curve 4), load and temperature field (2.181) for f (2.177) (curve 2), load and temperature field (2.181) for f = 0 (curve 3), are reported. The curves related to shell k1 = k2 = 24 (k1 = k2 = 18) are solid (dashed). Owing to results shown in Figure 2.29, a temperature field (2.179) weakly influences upper critical load value in comparison to an action of only forcing load, decreasing it in amount of 1.5%. For the shell k1 = k2 = 24, the temperature field (2.181) for (2.177), decreases the value of upper critical load in amount of 13%, whereas the value of lower critical load is increased in amount of 13% in comparison to an action of only forcing load. The temperature field (2.181) with a lack of heat sources ( f = 0) decreases upper critical load value in amount of 8%, whereas lower critical load value is not changed comparing to an action of purely forcing load. In what follows, comparing the curves 2, 3, one may conclude that occurrence of heat sources f (2.177) for the temperature field (2.181) decreases shell k1 = k2 = 24 stability: top (low) critical load value decreases on

258

2 Stability of Rectangular Shells within Temperature Field 170 q

2 1 4

85

3

5

W

0

1

a) 80

6,7

1.5

W``xx 1,2

60

40

F``xx

1,2

4

2

20 6,7

5

x=y 0.5

0

0

6,7

-20

-2 5

-4

b)

-6

Figure 2.28. Load-deflection graphs (a) and the functions w

xx , F xx (b) for diﬀerent plate excitations (see text for more details).

2.5 Computations of Plates and Shells in a Temperature Field

259

amount of 5% (14%). A similar behaviour occurs for the shell k1 = k2 = 18. Graphs of the fundamental function w, w

xx , F, F

xx along the axis y = 0.5 for the shell k1 = k2 = 24 under an action of the temperature field (2.181) for f (2.177) and for f = 0, and forcing load, are shown in Figure 2.30a. Description of the curves is the same as in Figure 2.29. Index associated with a number denotes deflection value in the central point corresponding to this curve, for instance, 21 (w = 1), 22 (w = 2), 23 (w = 3). Consider now the shell k1 = k2 = 18, ball-supported along its contour (2.189), within a temperature field. In Figure 2.31, the dependencies “load-deflection” in the central shell point (0.5; 0.5) under an action of temperature field and forcing load are reported. Curve 1 corresponds to the shell under an action of only forcing load, curve 2 corresponds to the shell under actions of forcing load and temperature field (2.178), curve 3 - load and temperature field (2.179), curves 4, 5 - load and temperature field (2.180). The corresponding temperature field are shown in Figures 2, 5, 7. The curves 2, 3, 5 are obtained for the value of q0 = 0.01, curve 4 corresponds to q0 = 0.02. Comparing the curves given in Figure 2.31a, one may conclude that heat isolation of the shell surfaces sides slightly influences thermo-stress state (curves 2 and 3 graphically coincide). Heat isolation of upper stability (upper (lower) critical load value increases in amount of 10% (50%) - see curves 4 and 2). An increase of the load amount from 0.01 to 0.02 yields a decrease of upper (lower) critical load in amount of 7% (20%) (compare Figures 2.4 and 2.5). Comparing the curves 2-5 with the curve 1, corresponding to the shell subjected to pure load action, one may conclude that the thermal field (2.178), (2.179) increases the top critical load in amount of 7%, whereas the temperature filed (2.180) causes an increase of the top critical load in amount of 16% in comparison to the action of one load. A development of stress strain state of ball-supported shell can be traced in Figures 2.30b, 2.31b, where the functions w

xx , F

xx along the axis y = 0.5 are displayed. The curves in these figures have the same number as the curves “load-deflection” in Figure 2.31. The centre deflection values, for which graphs are constructed, are reported in the figure and denoted by a corresponding index associated with a number. Comparing the curves 21 with 3, 22 with 32 , one may trace an influence of shell sides heat isolation on its thermo-stress state. Influence of curvature increases with respect to absolute value in the shell centre in amount of 60%, whereas in zone of fixed contour increases in amount of a few times. Comparing the curves 4i and 5i (i = 1-5) the conclusions follow: increasing the load amount from 0.01 to 0.02, a curvature value decreases in a zone of fixed contour. A reason is that a temperature torque decreases on the stability boundary. The function F

xx values also are decreased in a zone of fixed contour, since the temperature stress decreases on the space boundary. Consider now the shell k1 = k2 = 24, clamped along its contour (2.188), subjected to an action of a load and temperature field (2.180) shown in Figure 2.7. The curves “load-deflection” in the central shell point q(w(0.05; 0.05)) are displayed in Figure 2.32a. The curves 3, 4 correspond to a shell being within the temperature

2.2 0

45

90

4

1.1

1

3

3

2

1

4

Kx=Ky=18

2

Kx=Ky=24

3.3

4

1,3

W

4.4

2 Stability of Rectangular Shells within Temperature Field

135 q

260

Figure 2.29. Load-deflection curves diﬀerently excited (see text for more details).

2.5 Computations of Plates and Shells in a Temperature Field

261

-9

32,42 33 4

3

23

23

22

33

W

32,42

33,43

x

42 32

21

0.5

0

21

23

W``xx 0.8 F 33,43

32

42

0.4 31,41

W``xx 2

2

-45

F``xx

21 X 0.5

18

a)

F``xx

25

W``xx

31

0

0.5

23

-30

9

22

q=0.01 x

0

-15

30

23

0

15

0

0.5

41

31

x

0

31,41

21

22

22 31,41

-30

43

23

0 -3

22 21

q=0.01

32 33

F``xx

x

34

0

0.5 33

-6

23

-25

21

-9

21(W=0.5), 22(W=1.2), 23(W=2) b)

32 34

31

-50

31(W=0.5), 32(W=1.2), 33(W=2.0), 34(W=3.5)

Figure 2.30. Fundamental functions w, w

xx , F, F xx for the shell k1 = k2 = 24 (y = 0.5) with diﬀerent excitations (see text for more details).

262

2 Stability of Rectangular Shells within Temperature Field

180

q Kx=Ky=18 4

1

5

2,3

90

0

1.5

3.0

4.5

a) 60

50

W``xx 41

42 43

51

0.5

45 44

45

44

43

0

55

F``xx 55

54

52 5 3

0

52

42

41

q=0.01 53

F``xx 0

x

0

W``xx

q=0.01

51

-50

54

-10

51(W=0.5), 52(W=1.3), 53(W=2.6), 54(W=4.2), 55(W=4.5)

-60

-12 41(W=0.5), 42(W=1.3), 43(W=2.6), 44(W=4.2), 45(W=4.5)

b)

Figure 2.31. Load-deflection graphs (a) and the functions w

xx , F xx (b) for diﬀerent shell excitation (see text for more details).

field (2.180). This field is characterized in the following way: heat isolation of upper = 0, whereas its lower surface is attacked by the temperature shell surface ∂T $ ∂z z=−0.5 % 2 T |z=0.5 = (x − 0.5) + (y − 0.5)2 0.25H. The curves 1, 2 correspond to the heat isolated shell from below, i.e. the temperature field satisfies the following boundary conditions $ % ∂T = 0, T |z=−0.5 = − (x − 0.5)2 + (y − 0.5)2 0.25H, ∂x z=0.5 ∂T = 0 x, y . (2.198) ∂x x=0;1

2.5 Computations of Plates and Shells in a Temperature Field 200

q

263

2 1

3

100

4

W 3.2

0 a) 200

q 1

2

100

3,4,5

W 0

b)

3.2

Figure 2.32. Load-deflection graphs for diﬀerent excitations (see text for more details).

The source f is governed by the equation (2.177). Curves 1, 3 are obtained for q0 = 0.0; curve 4 - for q0 = 0.004, curve 2 - for q0 = 0.008. Comparing the curves 1 and 3 one may conclude that an action of temperature field (2.198) increases a value of upper critical shell load in comparison to temperature field (2.180). Decreasing the load amount from 0.01 to 0.008 an action of temperature field (2.198) increases

264

2 Stability of Rectangular Shells within Temperature Field

yielding the shell stability increase (compare the curves 1 and 2), and a value of the upper critical load increases in amount of 5.5%. Decreasing the load amount from 0.01 to 0.004, and action of temperature field (2.180) increases yielding a shell stability decrease (curve 4 lies below curve 3). It should be emphasized that the temperature field (2.198) is negative. Curve 2 (curve 1) corresponds to an amount of absolute temperature value T max = 125◦ (90◦ ). The temperature field (2.180) is positive. Hence, these two temperature fields act on the shell in opposite manner. 2.5.2 Stress-strain state and shells stability The stress with geometrical parameters k1 = k2 = 24 for diﬀerent boundary conditions along their sides and being subjected to an action of uniformly distributed transversal load and temperature field are investigated. Notation with respect to fixation type and the corresponding curve number introduced in Section 2.5.1 is applied. In Figure 2.32b graphs “load-deflection” in shell centre (0.5; 0.5) for the boundary conditions denoted by digits 1-5, 9 for q0 = 0.01 are reported. The curves, corresponding to a shell under action of temperature fields (2.180) and (2.198) are shown in Figures 2.22b and 2.33a, correspondingly. Recall that two applied temperature fields have the same absolute magnitude but diﬀerent sign. Temperature field (2.198) is negative. Comparing the curves in Figure 2.32b one may observe that increasing a lengths of ball-supported contour is a vicinity of contour points up to 1 2 l0 , the graphs “load-deflection” approaches curve 4, corresponding to the clamped contour. In Figure 2.32b the curves 3-5 coincide. A variation of the clamped contour length l2 from 12 l0 to 0 in the middle of a side increases shell stability (curves 1, 2 lie over curve 5) in comparison to clamped shell. Analysing the results reported in Figure 2.33a one may conclude that increasing a length of ball-supported contour in a vicinity of corner points up to l1 ≤ 14 l0 , the “load-deflection” graph is shifted into curve 5 corresponding to clamped contour (curves 4, 5 coincide). Decreasing a length of clamped part of the contour l2 from 34 l0 to 0 in the middle of a side, decreases shell stability in comparison to shell clamped along the whole contour (curves 1-3 lie below curve 5). A value of top critical load for the shell corresponding to fixation 1 is decreased on amount of 10% in comparison to shell 5, clamped along the whole contour. In other words, increasing a length of clamped contour part in the middle of temperature field (2.180) side decreases shell stability, whereas the field (2.198) increases shell stability. Graphs “load-deflection” in the shell k1 = k2 = 24 centre (0.5; 0.5) for boundary conditions 1, 2 under actions of purely forcing load (curves 1, 2), of temperature field (2.180) and load (q0 = 0.01) (curves 1 , 2

) are displayed in Figure 2.33b. Digits 1, 2 denote fixation way, whereas dashing corresponds to temperature field. Comparing 1, 1 , and 1

one may conclude that mostly stable shell under an action of only forcing load (fixation 1), the temperature field (2.180) decreases a value of top critical load in amount of 14%, whereas the field (2.198) - in amount of 19% in comparison with load action. For the fixation type, the top critical load value is the largest one for the field (2.198) action, i.e. the field (2.198) increases the value of top critical load in amount of 1.3%, whereas the field (2.180) decreases a value

2.5 Computations of Plates and Shells in a Temperature Field

265

of the upper critical load in amount of 23% in comparison to an action of only one load (compare 2, 2 and 2

).

q

4,5

3

1

2

9

9 1 2 3 4 5 W

0

0.8

1.6 a)

2.4

200 q

1

Kx=Ky=24

2

2``

3.2

1` 2`

1``

110

W 0

0.8

1.6 b)

2.4

Figure 2.33. Load-deflection curves for diﬀerent shell boundary conditions.

3.2

266

2 Stability of Rectangular Shells within Temperature Field

Recall that computations are carried out without an influence material properties variations with respect to temperature. For a series of design materials, in particular for steel with small cole amount, or steel the Young modulus is changed in amount of less than 5% on the considered temperature intervals.

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells with Continuous Thermal Load

A historical background of dynamical behaviour and stability of shells thermally loaded is given in section 3.1. Dynamical problems of thermoelastic thin thermosensitive cylindrical shells are studied in section 3.2. Namely, after a general introduction, the variational formulation of the coupled dynamical problem of thermoelasticity is addressed. Next, the hybrid-type variational equations of thin conical composite orthotropic thermosensitive shells are derived. The problem of solution existence is rigorously discussed, and then a classification of thermoelastic problems is given. Computational algorithms are illustrated and discussed in section 3.3. A solution to the biharmonic equation in relation to forcing function, as well as the reliability of the obtained results, are studied. The modified relaxation method is described in section 3.3.4. Section 3.4 is devoted to analysis of dynamical stability loss with uniform force excitation. Criteria of dynamical stability loss are first reviewed, and then many examples are studied in more detail. Dynamical stability loss and non-uniform thermal load is addressed in section 3.4. Thermal field computation influence of time, shell geometry and load, as well as combined static and thermal loads action is studied in some detail.

3.1 Introduction Owing to the development of modern technics and technology, an investigation of dynamical behaviour in diﬀerent constructions with thermal and force excitations becomes very important. A full description of these problems can be given in a frame of is possible thanks to the dynamical theory of thermoelasticity [514, 515]. Nowadays the thermoelastic theory is well established and clearly separated from other trends. It includes the following behaviour: the heat transfer (stationary and unstationary) between a body and an external environment; thermoelastic stresses caused by the temperature gradient; the dynamical eﬀect accompanying sudden unstationary heating processes, in particular, thermoelastic vibrations of thin walled constructions caused by a heat impact; thermomechanical eﬀect caused by the interaction between the deformation fields and the temperature. Fundamental results of the thermoelasticity theory are obtained in the quasistatical conditions. The inertial therms are not included in the governing equations

268

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

and a linking term in the heat transfer equation as well as the material characteristics do not depend on the temperature. The first step of a solution to the quasistatic problem of thermoelasticity and the problem of temperature stresses is focused on the definition of the temperature field by the methods of the heat transfer theory. A systematical approach to these problems is given in the monographs of Lykov [451], Kozdoba [358], Carslow and Eger [143], Podstrigatch and Koliano [553], Belaev and Riadno [92], and others. In a general case, a heat transfer equation is a nonstationary and three dimensional one. In order to simplify a mathematical statement of this complex problem, the three dimensional heat transfer equation is reduced to the lower dimension equation using the physical and mathematical properties of the problem. In particular, in the case of thin bodies, diﬀerent dependencies of the temperature versus a normal coordinate are used (as the series of normalized Legendre polynomial or the orthogonal functions), the asymptotic method, the method of development on the eigenfuctions, and so on [487, 549, 554, 726]. In some works the three dimensional heat transfer equation has been used directly [671]. To the first thermoelastic problems belong these of a heat impact on the surface of a halfspace investigated by Danilovskaya [161]. She has explained the singularities of the dynamical heat stresses propagations. The fundamentals of the modern plates and shells theory have been established in the works of Ambartsumian, Bolotin, Bubnov, Vlasov, Volmir, Vorovitch, Goldenveizer, Dinnik, Germain, Iliuˇsin, Kane, Karman, Kiltchevskij, Kirchhoﬀ, Labenson, Lurie, Love, Muscheliˇsvili, Muˇstari, Galimov, Novoˇzilov, Obraztsov, Pankovitch, Pogrelov, Rabotnov, Rˇzamitsin, Timoshenko, Filonenko-Boroditch, Tchernych and others. The fundamental steps and directions in the development of the plates and shells theory are given in the works of Ainola and Nigul [9], Alfutov [11], Alumiae [12], Ambartsumian [14], Bolotin [120, 121], Burmistrov [136], Valisvili [675], Vol’mir [684, 685, 686], Vorovitch [689, 690], Galimov [218], Goldenblat [227], Goldenveizer [228], Grigoliuk, Kabanov, Srebovskij [238, 240], Guz’ and Babitch [256], Darevskij [165], Dzanelidze [190], Iliuˇsin and Pobiedria [270], Kantor [303], Karmiˇsin, Skurlatov, Startsev, Feldstein [310], Kaiuk [297], Kornisin [345], Kossovitch [351], Krysko [369], Morozov [481], Mushtari and Galimov [492], Novozilov [507, 508, 509, 510], Ogibalov [518], Ogibalov and Koltunov [520], Pelech [533], Satchenkov [601], Srubshtchik [633], Timoshenko [655, 656, 657], Filin [207], Vol’mir [684], Darevskij [165], Karmishin et al. [309], Satchenkov [600], Ramm [566], Wunderlich [710], Zienkiewicz [724], Thompson, Hunt [654], Hermann [258], Yamaki [711], Hinton, Owen, Zienkiewicz [260], Kleiber [328], Borkowski [125], K¨onig [352], Leissa [426, 427], Galdersmith [214], Powell [559], Sare, Massonet [596], Ashton, Whitney [36], Aalami, Williams [1], Voyiadjis, Karamanlidis [692], Lekhsritskii [428] and others.

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269

The researchers attention is focused on the problems of stability investigations and the definition of stress deformation states of plates and shells. A deep investigations of those processes is caused by the industrial needs. In the recent time, vibrations and stability investigations in a frame of the elastic theory are carried out by Andreev, Lebiediev and Obodan [28], Baˇzenov [75], ˇ Zigalko [728], Kolometz [334], Krysko [373, 379], Kossovitch [351], Makarenko [454], Pertzev and Platonov [534], Satchenkov [600], Kr¨atzig, Onate [361], Crisfield [159], Wrigpers, Wagner [709], Gorman [233, 234], Singh, Dey [622], Sakata, Takahaski [590], Soedel [629], Bogdanovich [112], Desturgnder, Salann [177], Hjelmstadt [262], Gould [235], Laguese, Lions [407], Jawao [288], Reddy [569], Krissen, Skalond [368], Yi-Yuan Ya [713], Sawczuk, Sokol-Supel [603], Gilgert, Hackl [223], Voyiadjis, Karamandlidis [692], Reismann [570], Valid [674], Waszczyszyn et al. [701], Elishakoﬀ et al. [195] and others. Dynamical stability of shells can be investigated using the following properties: – Type of a shell (cylindrical, spherical, conical); – Physical-mechanical material properties (elastic, elastic-plastic, plastic); – Load properties (axial load, external pressure, concentrated load, matching of diﬀerent loads); – Time dependence of a load (impulse load, rectangular impulse load, cyclic load and others). It should be noted that we still have not got a definition of dynamical stability including all of the mentioned properties. The description called the ”dynamical stability” is related to many problems of different physical behaviour. According to Simitses [620] classification, we can formulate three classes of problems related to the dynamical stability (unstability). 1. Unstability, occurring as a result of periodic excitations causing a parametric resonance (for certain combinations of shell’s free vibrations’ frequencies and excitation frequencies). The problem of stability investigation is reduced to that of the Mathieu-Hill problem [114, 146]. 2. Unstability occurring due to the occurrence of nonconservative external forces (mainly aerodynamical), which is characterized by self-excited vibrations (flatter). The stability investigation is reduced to the explanation of self-excitations. The problem of post-critical design behaviour is reduced to the analysis of a limit cycle [73, 121, 184, 684]. 3. Unstability caused by a load being an arbitrary aperiodic function of time. In this case, the definition of dynamical stability loses its clear meaning. This situation occurs during a load-carrying ability in conditions of the explosion type load. Estimation of the load-carrying ability is rather complicated in this case because of a sudden change of equilibrium states. Very small perturbations of external conditions (load, boundary conditions, end so on) may lead to a qualitative change of the initial state. A proper approach, including all singularities of those processes, is possible only when the geometrical nonlinear shells theory is

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

applied. The first work devoted to the shells’ stability loss in the geometrically nonlinear frame of investigations has been carried out by Grigoliuk [239]. The criterion choice in the problems of dynamical stability loss has been discussed, among others, in the following references [132, 165, 310, 375, 387, 404, 454, 491, 612, 684]. The investigation problem of thin shells with the nonuniform load’s reactions has a crucial meaning as far as application is concerned. In a frame of the linear theory those problems have been considered in the works of Agafonov [3], Lyapunov, Roˇzikova [448], Yao [654], Keer et al. [320], Reisman, Pawlik [571], and others. Behaviour of closed cylindrical shells with a nonuniform external pressure, including nonlinearities, still needs further investigation. We have to mention the work of Kormiˇsin et al. [310], devoted to the experimental-theoretical investigation of interaction processes between thin walled constructions and strength impact waves in a gas, and also the work of Solonenko [631]. In the latter one, on the basis of the equations of nonlinear theory of shells, a stress-deformation state is analysed, when the external pressure is applied. This pressure acts in a part of the surface of diﬀerent shapes and sizes. Among others, it has been shown that the largest dynamical eﬀect is concentrated in the zone located along the generating line. The problems of cylindrical shells’ dynamical stability have been considered by Makarenko [454, 455]. In his works, the influence of nonuniformity and a load speed on the critical load values and the post-critical deformation states have been considered. The deflection and force functions have been presented in the longitudinal direction by half-sinusoids. In the circle direction the Bubnov-Galerkin method is applied. The equation related to the force function components has been solved due to the transition to the Cauchy problem and integration of the initial vector using the Newton’s method has been carried out. In order to describe physical nonlinearities, the theory of small elastic deformation has been used. In particular, a weak influence of the plastic material properties on the deformation properties and critical time have been shown. We have to mention a group of works oriented on the dynamical stability of cylindrical shells with a nonuniform load carried out by Kolomoietz and Krysko[334, 378, 379]. In these works, by means of Bubnov-Galerkin method in higher approximations many problems of cylindrical shells’ dynamical stability with the rectangular type loads have been solved. Among others, the results of the critical load’s dependence on the excitation time and on the angle of the load’s action have been reported. Among the experimental works devoted to the dynamical shell buckling, the following ones should be mentioned: Bivin and Naida [111], Tchuiko [653], Skurlatov [625], Bushtyrkov and Naida [139], Andrieev et al. [20, 22, 23], Karolev [313] and others. In the work of Karmishin et al. [310] an experimental investigation of shells buckling is carried out. For small values of the axial compressing force, in comparison to the statical critical force, the buckling is characterized by an area including the loaded part. A short-wave component is added to the fundamental half-wave.

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271

The experiments have shown that the unloaded part is practically undeformed. If the longitudinal force is close to the critical one, then the buckling form approximates that of the statical stability loss. In this case, the critical impulse value is small and it plays a role of the exciting factor acting on the potential threshold, separating the old and new equilibrium positions. The results of instability areas’ experimental investigations of smooth cylindrical shells with an impulse external load in combination with the external (or internal) transversal statical pressure are given in the work of Manevitch at al. [457]. It has been shown that the external statical pressure in the area of values higher than 0, 6q (q - the statical pressure of buckling) has essential influence on the stability loss. In this case, the stability loss is initiated by a sudden jump and then large deformations occur. The internal pressure has stabilizing influence of the shell and leads to a significant increase of the critical impulse. According to the authors’ conclusions a quasi linear dependence of the critical impulse on the internal pressure is observed. On the other hand, in the work of Baskakov et al.[86], where the experimental results of investigation of the influence of the static internal pressure and load velocity on smooth cylindrical shells with an impulse type external load stability are outlined, the following conclusions have been obtained: 1. The internal pressure essentially increases a magnitude of the critical load of shells’ stability loss. This dependence has a nonlinear character, and the dynamical overload coeﬃcient increases to the moment when the internal pressure causes the shell’s material strength loss. 2. The shells with the internal pressure have got relatively small final deformations and not always lose their load-carrying abilities. The results of the experimental investigations of the influence of geometrical shells’ parameters and of the boundary conditions on the magnitude of dynamical critical load with an external impulse-type pressure are given in the work of Andrieev et al. [21]. The results of the research show a weak sensitivity of the shells with average length (L/R = 1, . . . , 4) on the boundary condition with that type of the load. The stability loss of the short shells (L/R ≤ 0.5) is characterized by an increase of the critical impulse value. With the increase of the R/h parameter the dynamical critical load is decreased. A dependence of the critical impulse on the model’s radius for the fixed R/h value is linear. There are only a few works devoted to the experimental investigation of the dynamical stability loss with an ununiform load. We mention only, where the problem of stability estimation of a cylindrical shell with the impulse-type pressure is investigated. An influence of the load zone on the dynamical critical load value is analysed beginning with an impulse type to quasistatic type loads. In real constructions, applying enough careful experiments we need to take into account the initial deformations of the analysed shells, the occurrence of the load eccentricity, additional vibrations and other deviations and excitations. The influence of those factors during the analytical and experimental investigations leads to an extremely complicated but valid problem [36, 131, 196, 326, 434, 435, 440, 441, 467].

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Analytical solutions to the dynamical stability problems obtained on the basis of a simplified approach can be found in the works Antsiferov, Pavlenko [34], Karmiˇshin, Feldˇstein [310], Lindberg, Florence [433]. From the computational point of view, the buckling problem of rotational shells with an axiallysymetric load is eﬀectively solved using many algorithms of one dimensional discretization [138]. High dimensional problem include many diﬃculties from the point of view of the computation time length [76]. The investigation of the thermoelastic processes in the thin-wolled constructions (plates, shells) has some singularities. A proper choice of the computational model leads to eﬀective results. However, it is impossible to give general recommendations to this problem. It can only be mentioned that the model should be possibly simple, but including principal properties of a real system. According to the work of Pertsev and Paltonov [534] the qualitative analysis of the wave processes in a construction plays an important role in the choice of a proper model. This analysis sometimes allows for an a priori judgement, concerning the question of which processes (and at what time) play a significant role. It also makes it possible to establish an appropriate physical and mathematical model. This remark is valid for the heat transfer, the elastic as well as the thermoelastic problems. In the latter case, that analysis has a particular role because processes with diﬀerent time scales are considered. The fundamental mathematical models of a shell deformation with a local dynamical heating are considered in the works of Medvedenko, Obodan [470], Andreev et al. [29]. In these works special time scales are introduced. A defined mathematical model of thermoelastic deformation corresponds to each of them. In particular, it has been shown that in the first time moments, after applying a heat flow, the shell can be treated as a three dimensional body, where a thermo-stress wave is propagated. Further, after heating along the thickness, a thin-walled property is observed, and a deformation has a static character. What should be noted here is the theoretical-experimental investigation carried out by Andrieev and Obodan [27]. Depending on the load parameters the resonance phenomena and the dynamic stability loss occurring in a time similar to the heating time along the thickness. In the case of a statical load of the thin-walled construction, the phenomenon of the statical stability loss caused by the heat impulse is investigated. In the work of McQuillen, Brull [469] it has been shown that in a frame of the halfly linked theory of thermoelasticity (in the equation of the heat transfer a mechanical linking term is omitted) dynamical eﬀects are dominating when the shell’s wall becomes thin enough, i.e. the heating time has a value of the lowest vibration period order. In the work of Podstrigatch and Shvetz [555] the thin-walled shell conditions from the point of view of the heat transfer are given in the following form h 1, l

kh 1,

h2 1, aT a

where: a - coeﬃcient of temperature conductivity; T a - time of an observable temperature increase; k - curvature of the shell.

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273

In the work of Bolotin [123], where the thermoelastic equations are established in the theory of plates and shells together with the Kirchhoﬀ-Love hypothesis about a normal element, the analogical one is introduced in the heat transfer equation. It is assumed that the temperature along the thickness is linearly distributed. A wrong assumption of this hypothesis has been shown by Guliaev et al. [251] and Guliaev, Tchibiriakov [252]. The heat transfer equation (in those works), has been reduced to the second order equation using the Legandre polynomials. It has been shown that (as a result of the heat impulse) in the beginning a rapid change of temperature occurs. It causes internal stresses which lead to the stability loss and then a heat stabilization occurs. The problem has been considered as the geometrically linear one. In the work of Kovalenko [354] it has been also shown that when a rapid unstationary space temperature field is imposed, then a purely heat deformation along the thickness of a thin-walled shell or plate essentially diﬀers from the linear one. Therefore, a hypothesis about the unchangeability of a normal element (in a general case) does not correspond to the hypothesis of the linear change of heat stresses along the thickness. However, the application of general purely heated deformation reduces the thermoelastic problem with a volume temperature field to the two-dimensional problem of the isothermic theory of plates and shells. The problem of vibration excitation of the thin-walled elements of constructions with a heat impulse load is considered in works of Boley [114], Boley, Barber [116], Kraus [365], Sinitsin [624], and others. In these references, the fundamental properties of the phenomena with a heat impact are described and illustrated. Among others, it has been noted, that the influence of the inertial terms during the investigation of the temperature stresses in the thin-walled elements of constructions leads to the solutions, where the vibrations appear. The intervals of parameters, where the unstationary behaviour of the construction occurs, are given. The singularities of the dynamical behaviour when a cylindrical anisotropical shell is heated are discussed by Shvetz and Flatchok [615]. The investigation of thermostability of plates and shells is described in the monograph by Ogibalov and Gribanov [519]. The correct formulations of the mechanical thermostability problems are given. Furthermore, the methods and their solutions and some examples of stability calculations of plates and shells in a frame of the physical nonlinear theory are discussed. All the above mentioned works have been carried out using the assumption, that the temperature increase is small and that all materials’ characteristics have been independent of the temperature. As it is known in many practical problems, those assumptions can not be made. A lack of those limitations does not change the assumption about small deformations but only leads to the occurrence of the variable coeﬃcients in the thermoelasticity equations. In the work of Kovalenko [354] the thermoelasticity theory has been formulated without those limitations and in a frame of the small deformations including the dependence of the elastic and heat material properties on the temperature. Computations of statical problems of shells and plates with a linear temperature distribution along the thickness, including the temperature change of both the

274

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

elasticity modules and the linear heat expansion coeﬃcient, have been carried out in the works of Burak, Ogirko [134], Petrov [535], Karpov and Filatov [314], Nerubailo and Ivanov [499], Kamiya and Fukui [299], and others. Attention has been paid to the investigation of the design behaviour influence on the statical load’s joint action and the temperature field (Butenko and Chalilov [140], Lebedev [422], Lykianenko and Makarenkov [449], Mishulin and Sinitsin [475] and others). In the work of Stroud and Mayers [639] the dynamical thermoelastic behaviour of the rectangular plate made from the temperature depending material with an arbitrary temperature action, is analysed. It has been shown that only a full dependence of the material versus the temperature leads to correct results. A review of material stresses of diﬀerent materials with properties depending on the temperature can be found in the work of Noda [505]. Dynamic stability of thin-walled slowed shells with a temperature impact taking into account the temperature dependence of their fundamental characteristics has been analysed by Krysko and Fedorov [373, 375, 376]. In these works it has been assumed that the heat flow has been uniformly distributed along the shell. This assumption reduced the problem to the one dimensional heat transfer equation. In order to solve the thermoelastic equations the method of finite diﬀerences and the Runge-Kutta method have been used. The authors have concluded that the temperature dependencies of all material characteristic are needed for a proper investigation of dynamic instability eﬀects with a heat impact. The experimental data of dynamical stability loss with a local external heat impact with an internal pressure are very rare. We mention the work of Kostoglotov et al. [352], where the experimental investigations of smooth alloy and with the internal pressure, have been carried out using a laser generator. The experimental investigations of the plates and shells instability with a local heat impact and diﬀerent external loads have been described in references [23, 30, 31, 532]. All of the experiments have been carried out using the technique called “impulse” and described in reference [24]. A local impulse type heating has been reached using an optical quantum generator. The occurrence of the wave-forms and the buckling occurrence of the cylindrical shells have been analysed in relation to the power of the heating flow [30]. It has been concluded that the influence of the heat impulse initiates a buckled form localized in the area of the heat action. In the work [23] a circle plate with a local heat impact has been analysed. The temperature distribution after the heat impact and the wave occurrence along the plate radius have been analysed, too. In the work [378] the phenomenon of the stability loss of cylindrical shells with a local unstationary heating and with a uniform pressure or an axial compression is described and illustrated. As a result of the investigations, three types of the stability loss occurs: local (temperature), general (corresponding to the load type) and general with a time delay. The latter one is linked with the heat distribution and with the nonmonotonic dependencies of the critical loads on the load exponent variations of the unsymetric deformations. In the work [532] the influence of the heat impact on

3.1 Introduction

275

the cylindrical shells with a combination of diﬀerent statical loads (an axial compression and an external pressure) has been investigated. Also, the most dangerous external loads combinations have been outlined which essentially decrease the carrying abilities of a load. Some of the works are devoted to the approximate methods to solutions with the heating load excitations [118]. The variational methods play an important role in solving these problems. The variational principle allows for a compact form formulation of the problem, because it includes the fundamental equations and the boundary conditions in a general form. It allows (sometimes) for a low and high limits estimation of the variational integral. It describes the quantity being more suitable in direct practical applications. It should be noted, that not all diﬀerential treatment of the problem has its variational formulation in the classical meaning, i.e. in the sense of an integral which should approach a maximum or a minimum [474]. Gurtin [253] has developed a variational principle where the initial conditions are included in the functional. A general variational approach applied for an orbitrary nonlinear system has been developed by Tonti [659]. The problems of the variational equations have been considered in the references of Balabuch and Shapovalov [82], Bugrij [133], Gribanov and Panitchkin [237], Grigorienko et al. [246], Flatchok [209], and others. Among the works devoted to the qualitative investigations of the operator equations (the theorems of existence, the estimation of the solutions’ number, the existence of nontrivial solutions, the applications of diﬀerent schemes of the approximated solutions, the analysis of the equations with the parameters, and so on), we mention the research of Vainberg [672, 673], Vishik [679, 680], Dubinskij [186], Krasnoselskij [362], Krasnoselskij and Zabreiko [363], Koshelov [349], Ladyzenskaya [405, 406], Michlin [474], Nikolskij [503], Sobolev [628]. The qualitative investigations of nonlinear behaviour of plates and shells and their statical properties have been carried out by many authors. In the monograph by Vorovitch [689] a wide spectrum of possible approaches to many diﬀerent problems of this type is given. Essentially smaller amount of works is devoted to the problems of dynamical behaviour of plates and shells. Among others, the works of Vorovitch [690], Morozov [481] and Lions [438] are remarkable. The geometrically nonlinear linked problems of slope shells’ thermoelasticity have been considered by Kiritchenko and Krysko [324]. The up-to-date dynamic stability problems of cylindrical shells with a nonuniform thermal load lead to the following conclusions and observations. 1. The problem of geometrical parameters of cylindrical shells’ influence, their boundary conditions, the load parameters causing dynamical instability with an ununiform external pressure in a frame of the geometrical nonlinear theory is not suﬃciently investigated.

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

2. Further research presents the problem of dynamic stability loss of a thin-walled cylindrical shell in the condition of a combined nonuniform thermal load including a priori given static load. 3. An adequate choice and a theoretical analysis of the calculation model during a numerical solution to the problem of dynamical stability loss of a geometrical nonlinear and a thermal sensitive cylindrical shell in the condition of an ununiform thermal load still need further investigations and clarifications.

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells 3.2.1 General Introduction Let us consider an elastic body as a continuum, with temperature T , entropy s (thermodynamics parameters), stresses σi j and deformations εi j . The entropy and the temperature, the corresponding parameters of stresses and deformations are the selfcoupled state parameters [555]. In order to establish the relations between the state parameters we need to formulate an expression for free energy ψ as a function of the deformation components εi j and the temperature T . We assume that the body in the initial undeformed state has the temperature T 0 = const. We assume that the temperature increase (T − T 0 ) yields a purely ther4T mal expansion T α∗ dT (α∗ - true coeﬃcient of the linear thermal expression) with 0 a magnitude of one order smaller than εi j . This assumption is not in opposition to the fundamental hypotheses of the linear elasticity theory of deformations. It allows to omit a limitation governed by the inequality (T − T 0 )/T 0 1. However, mechanical and thermal nonuniformities caused by high temperature influence should be accounted. Therefore, developing the function of free energy into the Taylor series, second εi j powers remain. The function ψ and the state equations have the following form [354] λ ψ = ε2kk + µεi j εi j − (3λ + 2µ) αT (T − T 0 ) εkk − 2

T

T dT

T0

Cε=0 dT , T

(3.1)

T0

1 ∂λ 2 ∂ψ ∂µ =− ε − εi j εi j + ∂T 2 ∂T kk ∂T T 6 Cε=0 ∂ 5 (3λ + 2µ) αT (T − T 0 ) εkk + dT, ∂T T s=−

(3.2)

T0

σi j =

5 6 ∂ψ = 2µεi j + λεkk − (3λ + 2µ) αT (T − T 0 ) δi j . ∂εi j

(3.3)

In the above, αT denotes the average coeﬃcient of a linear thermal expansion in the temperature interval (T 0 , T ), defined by the relation

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

277

T

1 αT = T − T0

α∗ dT ,

(3.4)

T0

where Cε=0 is the thermal capacity of an undeformed state, λ, µ are the Lam´e coeﬃcients for the isothermical deformation. They are linked with the isothermical elasticity modulus E and the Poisson’s coeﬃcient ν by the relations λ=

νE , (1 + ν) (1 − 2ν)

µ=

E . 2 (1 + ν)

(3.5)

−u = (u , u , u ) and deformations have the The relations between displacements → 1 2 3 following form [676] 1

(3.6) εi j = ui, j + u j,i + um,i um, j . 2 Below, a general thermoelastic problem is formulated. We need to determine the stress components σi j , the deformations εi j , the dis−u and the temperature T (when the mechanical and thermal excitations placements → are given) satisfying the following governing equations 5&

' 6 d2 ui δim + ui,m σκm ,κ + P¯ i = ρ0 2 , dt

(3.7)

where P¯ i is the component of the external load, ρ0 is the material’s density, and κ is the physical-geometrical parameter. The relations between stresses and deformations are given by (3.3), and between deformations and displacements are defined via the equation (3.6). The heat transfer equation reads T

' ds & = λT T ,i , j + r, dt

(3.8)

(λT is the heat transfer coeﬃcient; r - the power of the heat sources) for the given initial and boundary conditions. Using the relations (3.2), (3.3), the equations (3.7), (3.8) are presented in the following form ⎧ ⎤⎫ ⎡ ⎤ ⎡ ⎪ T ⎥⎥⎥⎪ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ '⎢⎢⎢ d2 ui ⎢⎢ ⎥⎥ ⎨& ⎬ ¯ ⎥⎥⎥⎥⎪ ⎢ ⎥ (3λ + u + − + 2µ) α dT + Pi = ρ0 2 , (3.9) λε δ δ 2µε ⎢ ⎥ ⎢ ⎪ ⎪ im i,m ⎢ κm ⎢ kk ∗ κm ⎥ ⎥ ⎪ ⎪ ⎥ ⎢ ⎥ ⎢ ⎪ dt ⎦⎪ ⎣ ⎦ ⎣ ⎪ ⎪ ⎩ ⎭ T0

⎧ ⎪ ⎪ ⎪ ⎪ ∂2 ⎨ 1 ∂ 2 λ 2 ∂2 µ ε − ε ε + T⎪ − i j i j kk ⎪ 2 ⎪ ∂T 2 ∂T 2 ⎪ ⎩ 2 ∂T

,κ

⎫ ⎡ ⎤ ⎪ T ⎢⎢⎢ ⎥⎥⎥ ⎪ ⎪ Cε=0 ⎪ ⎢⎢⎢ ⎥⎥⎥ ⎬ dT − ⎢⎢⎢(3λ + 2µ) α∗ dT ⎥⎥⎥εkk + ⎪ ⎪ ⎪ T ⎣ ⎦ ⎪ ⎭ dt T0

⎧ ⎤ ⎫ ⎡ ⎪ T ⎥⎥⎥ ⎪ ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ∂µ ∂λ ∂µ ∂λ ⎥⎥⎥ ⎪ ⎢ ⎨ ⎬ dεi j ⎢ ⎢⎢⎢ εkk − 3 (3λ ε + 2 = T⎪ + α dT − + 2µ) α 2 δ ⎥ ⎪ i j ∗ ∗ i j ⎥ ⎪ ⎥ ⎢ ⎪ ⎪ ∂T ∂T ⎦ ⎪ ⎣ ∂T ⎪ ⎪ ⎩ ∂T ⎭ dt T0

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

&

λT T ,i

'

,j

+ r.

(3.10)

The initial conditions are defined by a distribution of components of the displacei ment vector ui , their velocities du dt and the temperature T in the whole volume V of the elastic body of the form

− x , (3.11) ui |t=0 = u0i → dui du0i → −x , |t=0 = dt dt

− x . T| =T → t=0

0

(3.12) (3.13)

The boundary conditions on the surface Ω of the elastic body are composed of mechanical and thermal conditions. The mechanical boundary conditions are given either for the displacements

− x,t , (3.14) u = uˆ → i

or for the stresses

i

− ˆi → x,t . σi j n j = σ

(3.15)

The thermal boundary conditions are expressed by the ones given below: – temperature distribution along the body’s surface

−

− T → x , t = Tˆ → x,t ,

(3.16)

– intensity of the heat flow qT through the body’s surface

− ∂T , (3.17) x , t = −λT qT → ∂n – a surrounding medium temperature and a rule of convection between the body surface and the medium ∂T = α (T − υ) , (3.18) −λ ∂n where α is the thermal coeﬃcient. 3.2.2 Variational Formulation Many works have been devoted to the variational formulation of the coupled dynamical problem of thermoelasticity. Among others, we have to mention the references [82, 133, 209, 555]. A review of diﬀerent formulations of the thermoelasticity problems is described by Shatchnev in the book of Nowacki [515]. We have to mention the works of Biot [108], where only a quasistatical case is considered, and the work of Nickell and Sackman [501], where an approach, introduced by Gurtin [253], is applied. A case, in which a temperature increase is not small, has been considered by Lebon, Lambermont [423]. The method of limited variations and a series of simplified hypotheses have been used. In the work of Belli, Morosi [94] a critical review of the exciting approaches to the variational formulations in the case of

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

279

linked thermoelastic problem is given. Additionally, the variational formulation of that problem for small thermal excitations using a standard method (see Vainberg [673]) is outlined. Owing to that approach we consider a possibility of variational formulation of the linked dynamical thermoelastic problem when the thermal excitations are not small. In a general case, a variational formulation to the problem N(u) = f,

u ∈ D(N),

(3.19)

where N(u) is the nonlinear operator, defined on D(N). It depends on the existing character of the bilinear form in relation to which operator N(u) should satisfy the following condition [661] 8 7 8 & ' 7

(3.20) Nu ϕ, χ = Nu χ, ϕ , ∀ϕ, χ ∈ D Nu , where Nu is the linear part of the Gateaux derivative of the Nu operator, defined by the expression N (u + εϕ) − N (u) . (3.21) Nu ϕ = lim ε→0 ε The condition (3.20) defines the Nu symmetry in relation to the chosen bilinear form. As it has been shown in the works of Gurtin [253] and Tonti [659] the existence of the variational formulation is guaranteed when a bilinear form is taken as the following operator convolution tK u (tK − t) υ (t) dt = υ (t) ∗ u (t) ,

u (t) ∗ υ (t) =

(3.22)

0

dγ du ∗ υ (t) = u ∗ + u (tK ) υ (0) − u (0) γ (tK ) , (3.23) dt dt where tK denotes the end of the considered time interval. For the considered initialboundary thermoelasticity problem (3.9), (3.10), (3.11)–(3.18) we take the bilinear form in the following way u, υ =

−

− u→ x,t ∗ υ → x , t dV.

(3.24)

V

The Gateaux derivative can be presented in the following symbolic form Au BT

, (3.25) NuT = Cu DT where the operators Au , BT , Cu , DT satisfy the homogeneous initial and boundary conditions. The symmetry condition (3.21) is equivalent to the satisfaction of the following three conditions (3.26) Au χ, ϕ = Au ϕ, χ ,

280

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

BT χ, ϕ = Cu ϕ, χ , DT χ, ϕ = DT ϕ, χ ,

∀ϕ, χ ∈ D

&

(3.27)

' NuT .

(3.28)

The operators Au , BT , Cu and DT have the following form:

− → 9& ' Au → u , T −u 1 = δim + ui,m (2µε1κm + λε1kk δκm ) + ⎛ ⎞ ⎡ ⎤ T ⎜⎜⎜ ⎟⎟⎟ ⎢⎢⎢ ⎥⎥⎥ ⎜ ⎟ ⎢ ⎥ u1i,m ⎜⎜⎜⎜⎜2µεκm + ⎢⎢⎢⎢⎢λεkk − (3λ + 2µ) α∗ dT ⎥⎥⎥⎥⎥ δκm ⎟⎟⎟⎟⎟ + ⎝ ⎠ ⎣ ⎦ T0

,κ

$

− % d u1i d2 u1i ρ0 2 = A1 → u , T u1,i + ρ0 2 , ,j dt dt

→ & ' ∂µ − BT u , T T 1 = δim + ui,m 2 εκm T 1 + ∂T 2

(3.29)

⎡ T ⎢⎢⎢ ∂λ ∂α∗ ⎢⎢⎢ ∂λ dT · T 1 − 3 + ⎢⎢⎢ εkk T 1 − (3λ + 2µ) ∂T ∂T ⎣ ∂T T0

⎤ ⎫ T ⎥⎥⎥ ⎪ ⎪

− ⎪ ∂µ ⎥⎥⎥ ⎪ ⎬ T1 2 α∗ dT ⎥⎥⎥ δκm ⎪ = B1 → u , T T1, ⎪ ∂T ⎦ ⎪ ⎪ ⎭ T0

(3.30)

,κ

− → ∂ λ ∂2 µ Cu → u , T −u 1 = T − 2 εkk ε1kk − 2 2 εi j ε1i j + ∂T ∂T ⎫ ⎡ ⎤ ⎪ T ⎢ ⎥⎥⎥ ⎪ ⎪ dεi j ∂µ ∂λ ∂2 ⎢⎢⎢⎢⎢ ⎬ dT ⎥⎥⎥⎥ ε ⎪ (3λ − T 2 ε ε − α dT + δ + 2µ) ⎢ ∗ 1i j 1kk i j ⎥⎥⎦ 1kk ⎪ ⎪ ⎪ dt ∂T ∂T dt ∂T 2 ⎢⎢⎣ ⎪ ⎭ 2

T0

T ∂µ ∂µ ∂λ ∂λ εkk − 3 + 2 T 2 εi j + α∗ dT − ∂T ∂T ∂T ∂T

T0

− →

− du1,i dε1i j 6 (3λ + 2µ) α∗ δi j = C1 → , u , T −u 1,i + C2 → u,T dt dt

→ ∂3 µ 1 ∂3 λ 2 − DT u , T = T − ε T − εi j εi j T 1 + 1 kk 2 ∂T 3 ∂T 3 ⎤ ⎡ T ⎥ ⎢ ∂2 ⎢⎢⎢⎢⎢ ∂α∗ ⎥⎥⎥⎥⎥ (3λ dT + 2µ) ⎥⎥⎥ εkk T 1 + ⎢ ∂T ∂T 2 ⎢⎢⎣ ⎦ T0

(3.31)

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

281

⎡ ⎤ T ⎥⎥⎥ ⎢ 2 ⎢ ⎢ ∂µ ∂Cε=0 T 1 dT ∂ ⎢⎢⎢ ∂λ ⎥⎥⎥ + 2 + α dT T + 3 ε ⎢ ⎥ ∗ ⎥⎥⎦ kk 1 ∂T ∂T T dt ∂T 2 ⎢⎢⎣ ∂T T0

⎧ ⎫ ⎡ ⎤ ⎪ ⎪ T ⎢⎢⎢ 2 ⎥⎥⎥ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ∂ µ Cε=0 ⎪ ⎢⎢⎢ ∂ ⎥⎥⎥ ⎨ 1∂λ ⎬ dT 1 T⎪ + ε − ε ε + ⎢⎢⎢ 2 (3λ + 2µ) α∗ dT ⎥⎥⎥ εkk + − ⎪ ⎪ ⎪ 2 kk 2 ij ij ⎪ ⎪ 2 T ∂T ∂T ∂T ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ dt T0

⎧ ⎫ ⎡ ⎤ ⎪ ⎪ T ⎢⎢⎢ 2 ⎥⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 µ ⎢⎢⎢ ∂ ⎥⎥⎥ Cε=0 ⎪ ⎨ 1 ∂2 λ ⎬ dT (3λ T1 ⎪ − ε − ε ε + α dT + 2µ) − + ⎢ ⎥ ⎪ kk i j i j ∗ ⎢ ⎥ ⎪ 2 2 2 ⎢ ⎥ ⎪ ⎪ T ⎪ ∂T ⎣ ∂T ⎦ ⎪ ⎪ ⎩ 2 ∂T ⎭ dt T0

T 2 ∂2 µ ∂µ ∂ λ ∂λ ∂α∗ T 2 2 εi j T 1 + +2 dT + εkk T 1 − 3 2 ∂T ∂T ∂T ∂T ∂T T0

3

∂2 λ ∂2 µ +2 2 2 ∂T ∂T

α∗ dT + (3λ + 2µ)

∂α∗ + ∂T

T0

dεi j ∂µ ∂µ ∂λ ∂λ +2 α∗ T 1 δ i j − T 1 2 εi j + εkk − 3 ∂T ∂T dt ∂T ∂T ⎤ ⎫ T ⎥⎥⎥ ⎪ ⎪ ⎪ ' ∂µ ∂λ ⎥ ⎪ ⎬ dεi j & +2 − λT T 1,i , j − 3 α∗ dT − (3λ + 2µ) α∗ ⎥⎥⎥⎥⎥ δi j ⎪ ⎪ ∂T ∂T dt ⎦ ⎪ ⎪ ⎭

T

T0

→

→ dT 1 & ' ∂λT ∂λT − − T 1 T ,i = D1 u , T T 1 + D2 u , T − λT T 1,i , j − T 1 T,i . ∂T dt ∂T ,j ,j

(3.32)

In the above the following notations is used ε1i j = u1 j,i + u1i, j + u1m,i um, j + u1m, j um,i . We prove that for the operator Au the condition (3.25) is satisfied Au ϕ, χ =

&

A1u ϕ,i

V

'

, j ∗ χdV +

ρ0 V

A1u ϕ,i ∗ χdΩ −

Ω

d2 ϕ ∗ χdV = dt2 ρ0

A1u ϕ,i ∗ χ, j dV + V

V

dϕ dχ ∗ + dt dt

dϕ (tK ) dϕ (0) · χ (0) − ρ0 · χ (tK ) dV = dt dt dϕ dχ ∗ dV = − A1u χ, j ∗ ϕdΩ+ − A1u χ, j ∗ ϕ,i dV + ρ0 dt dt ρ0

V

V

Ω

(3.33)

282

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

A1u χ, j

,i

∗ ϕdV +

V

ρ0

d2 χ ∗ ϕdV+ dt2

V

dχ dχ (tK ) − ϕ (0) · ρ0 dV = ϕ (tK ) · ρ0 dt dt V

&

A1u χ,i

'

,j

∗ ϕdV +

ρ0

V

d2 χ ∗ ϕdV = Au χ, ϕ. dt2

(3.34)

V

The integrals acting on the surface Ω, and those including the functions in the initial and end time moments are equal to zero in accordance with the Gateaux derivative definition. We check the (3.27) condition (i.e. the symmetry condition) of the operator DT χ, ϕ =

D1 χ ∗ ϕdV +

V

∂λT χT ,i ∂T

,j

D2 V

dϕ ∗ χdV − dt

dϕ ∗ χdV − dt

V

λ T ϕ, j

V

D2

,i

Ω

λ T ϕ, j

,i

,j

∗ ϕdV−

χ∗

λT χ,i ∗ ϕdΩ− Ω

∂λT χT ,i ∗ ϕ, j dV = ∂T

V

dD2 ϕdV − dt

dϕ ∗ χdV − dt

V

V

D2

'

D2 χ (tK ) ϕ (0) −

λT Y, j ∗ χ,i dΩ+ Ω

∂λT ϕT ,i ∂T

V

D1 ϕ ∗ χdV + V

∂λT χT ,i ∗ ϕdΩ − ∂T

∗ χdV +

5

dD2 ϕdV+ dt

V

χ

∂λT χT ,i ∗ ϕdΩ+ ∂T

Ω

D1 ϕ ∗ χdV + V

λT χ,i

V

V

λT χ,i ∗ ϕ, j dV −

D1 ϕ ∗ χdV +

V

D2 χ (0) ϕ (tK )] dV +

&

V

∗ ϕdV =

V

V

V

dχ ∗ ϕdV + D2 dt

∗ χdV − V

χ∗ V

∂λT ϕT ,i ∂T

,j

∗ χdV−

dD2 ϕdV+ dt

,j

∗ χdV.

2 This expression implies that symmetry is achieved only if dD dt = 0 and operator D2 has the form

→ ∂2 µ 1 ∂2 λ − ε − εi j εi j + D2 u , T = T − KK 2 ∂T 2 ∂T 2

(3.35) ∂λT ∂T

= 0. The

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

⎫ ⎡ ⎤ ⎪ T ⎢⎢⎢ 2 ⎥⎥⎥ ⎪ ⎪ Cε=0 ⎪ ⎢⎢⎢ ∂ ⎥⎥⎥ ⎬ (3λ α dT + + 2µ) ε . ⎢⎢⎢ 2 ⎥⎥⎥ KK ⎪ ∗ ⎪ T ⎪ ⎣ ∂T ⎦ ⎪ ⎭ T

283

(3.36)

0

It means that D2 is time independent in the case of unstationary temperature field when the simultaneous conditions given below are satisfied for an arbitrary material: ∂µ ∂α∗ ∂λ = 0, = 0, = 0, Cε=0 = const. (3.37) T ∂T ∂T Therefore, the symmetry of the DT operator is achieved only in the case, when the fundamental material characteristics do not depend on the temperature (and, consequently, on time). It means that λ, µ, αK , Cε=0 , λT should be constants. We check the condition (3.26) for the mixed terms of the Gateaux derivative to the operator of the linked dynamical thermoelastic problem with constant coefficients λ, . . . , λT . As it has been shown in the reference [615], the given symmetry condition is satisfied, when as the mechanical variable we take a distribution of the velocity vector, and not of the displacement vector. In the conditions of nonlinear dependence between the displacement and the deformations (3.6) this choice can dε i not be realized, because during the search for dti j in the expression both ui and du dt appear. Thus, it has been proved that a variational formulation of the operator related to the coupled dynamical thermoelastic problem is only possible, when the fundamental material characteristics λ, . . . and λT do not depend on the temperature (on time) and therefore the relation between the displacements and the deformations is linear. The fact that the variational formulation of that problem is only possible in the case of a linear thermoelasticity and small thermal excitations has a physical meaning. It should be noted that with a change of the diﬀerential operator caused by introduction of the integral multiplier [659], the variational formulation of the linked dynamical thermoelastic problem for the case in which the thermic excitation is not small, the characteristics λ, . . . , λT depend on the temperature (time) and the link between the deformations is nonlinear. However, this variational formulation leads to complex integro-diﬀerential equations. They are not applicable to the works, where the problem is formulated using only the diﬀerential equations. 3.2.3 Hybrid-Type Variational Equations Consider a composite thin shell made from an arbitrary number of orthotropic or/and design-orthotropic composites (see Fig. 3.1). The coordinate surface z = 0 can be applied to an arbitrary s-th composite. It can overlap with one of the composite surface contacts, the shell’s boundary surfaces or it can be located inside of the composite. The coordinate surface z = 0 is related to the curvilinear coordinate system α, β. The coordinate lines α and β overlap with the main curvatures of the given coordinate system. The external normal is oriented to the centre of the shell’s curvature. In order to describe the shell’s thermoelastic dynamical behaviour we introduce the following assumptions and hypotheses:

284

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.1. A thin shell composed of arbitrary number of orthotropic composites.

1. The hypothesis about the undeformated normals, given for the whole shell’s pack globally [14]. 2. The fundamental assumptions of the nonlinear technical theory of conical shells [228]. 3. The Duhamel-Neuman hypothesis for the orthotropic material [513]. It has also been assumed that the fundamental material characteristics depend on the temperature. In this case, from the generalized Hook’s law, we can obtain the relations between the components of the stressed and the deformable state for an orthotropic and a design-orthotropic composite

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

T σα = B11 ε1 + B12 ε2 + z (B11 κ1 + B12 κ2 ) − B11

T α1 dT − B12

T0

T0

T

T

σβ = B21 ε1 + B22 ε2 + z (B21 κ1 + B22 κ2 ) − B21

α1 dT − B22 T0

B11 = B12 = B21 =

E1 (T ) , 1 − ν1 ν2

B22 =

α2 dT ,

(3.38)

α2 dT ,

(3.39)

T0

ταβ = B66 εαβ + zB66 κ12 , where:

285

(3.40)

E2 (T ) , 1 − ν1 ν2

ν2 E1 (T ) ν1 E2 (T ) = , 1 − ν1 ν2 1 − ν1 ν2

B66 = G12 .

The curvatures κ(∗) are defined via relations (3.42). E1 , E2 denote the elasticity modulus; G12 is the shear modulus; ν1 , ν2 are the Poisson’s coeﬃcients; α1 , α2 are the coeﬃcients of the linear thermal excitation of the orthotropic composite (the shell’s layer). For the design-orthotropic shell’s layer νi = νγi , where γi is the coefficient of filling of the layer [290]. The relations between the deformations and the displacements of the coordinate surface and the expressions for the curvatures have the following form ε1 =

2 ∂u 1 ∂w − k1 w + , ∂α 2 ∂α

2 ∂υ 1 ∂w − k2 w + ε2 = , ∂β 2 ∂β ε12 = κ1 =

∂2 w , ∂α2

∂u ∂υ ∂w ∂w + + · , ∂β ∂α ∂α ∂β κ2 =

∂2 w , ∂β2

κ12 = 2

(3.41) ∂2 w . ∂αδβ

(3.42)

The u, υ, w denote the longitudinal, circle and transversal displacements of the coordinate surface z = 0, correspondingly. The κ1 , κ2 denote the main curvatures of the coordinate surface. Integrating (3.38)–(3.40) along the shell’s thickness, we get the expressions for the internal forces T 1 , T 2 , s12 . Multiplying (3.38)–(3.40) by z and integrating along the shell’s thickness we obtain the moments M1 , M2 , H12 . The final expressions have the form T 1 = C11 ε1 + C12 ε2 + K11 κ1 + K12 κ2 − C1T , T 2 = C21 ε1 + C22 ε2 + K21 κ1 + K22 κ2 − C2T , s12 = C66 ε12 + K66 κ12 = s21 ,

286

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

M1 = D11 κ1 + D12 κ2 + K11 ε1 + K12 ε2 − K1T ,

(3.43)

M2 = D21 κ1 + D22 κ2 + K21 ε1 + K22 ε2 − K2T , H12 = D66 K12 + K66 ε12 = H21 , where Ci j =

δ −∆ m+n s s=1 δ

s=1 δ

Di j =

CiT =

δ −∆ m+n s s=1 δ

s−1 −∆

Bisj zdz,

s−1 −∆

δ −∆ m+n s s=1 δ

Bisj dz,

s−1 −∆

δ −∆ m+n s

Ki j =

(3.44)

Bisj z2 dz,

s−1 −∆

⎛ s ⎞ ⎜⎜⎜T ⎟⎟⎟ ⎜⎜ ⎟⎟ s ⎜ s ⎜ Bii ⎜⎜ βi (T ) dT ⎟⎟⎟⎟ dz, ⎜⎝ ⎟⎠ T 0s

⎛ s ⎞ δ −∆ ⎜⎜⎜T ⎟⎟⎟ m+n s ⎜ ⎟⎟ ⎜ Biis z ⎜⎜⎜⎜ βis (T ) dT ⎟⎟⎟⎟ dz, KiT = ⎟⎠ ⎝⎜ s=1 δ

s−1 −∆

β1s = α1s + ν2s α2s , i = 1, 2,

T 0s

β2s = α1s ν1s + α2s ,

j = 1, 2,

i = j = 6.

During considerations of the conical shell’s equations in the hybrid form the following functions are introduced T1 =

∂2 F , ∂β2

T2 =

∂2 F , ∂α2

s12 = s21 = s = −

∂2 F . ∂α∂β

(3.45)

In this case the relations between deformations and stresses have the following form 2 2 ∂ F ∂ F ∂2 w ∂2 w + A + d + C + C + d , ε1 = A11 1T 12 2T 11 12 ∂β2 ∂α2 ∂α2 ∂β2 2 2 ∂ F ∂ F ∂2 w ∂2 w + A + d + C + C + d , ε2 = A12 1T 22 2T 21 22 ∂β2 ∂α2 ∂α2 ∂β2 ε12 = A66

∂2 F ∂2 w + 2d66 , ∂α∂β ∂α∂β

where A11 = C22 /Ω,

A12 = −C12 /Ω,

A22 = C11 /Ω,

(3.46)

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells 2 Ω = C11C22 − C12 ,

A66 =

1 , C66

d66 =

287

K66 , C66

C22 K11 − C12 K12 C22 K12 − C12 K22 , d12 = , Ω Ω C11 K12 − C12 K11 C11 K22 − C12 K12 , d22 = . d21 = Ω Ω The corresponding relations between the moments and the forces have the form 2 2 ∂ F ∂ F + C1T + d21 + C2T + M1 = d11 ∂β2 ∂α2 d11 =

&

' ∂2 w & ∗ ' ∂2 w + D12 − D12 − K1T , 2 ∂α ∂β2 2 2 ∂ F ∂ F M2 = d12 + d + C + C 1T 22 2T + ∂β2 ∂α2 &

where:

D∗11 − D11

(3.47)

' ∂2 w & ∗ ' ∂2 w + D − D − K2T , 22 22 ∂α2 ∂β2

∂2 w ∂2 F + 2 D∗66 − D66 , H = −d66 ∂α∂β ∂α∂β

D∗12 − D12

D∗11 = K11 d11 + K12 d21 ,

D∗22 = K12 d12 + K22 d22 ,

D∗12 = K11 d12 + K12 d22 = K12 d11 + K22 d21 ,

D∗66 = K6 d66 .

For the variational formulation we use the principle of virtual displacements [676] tK [δK − δV + δA] dt = 0, (3.48) 0

where: V - energy of an elastic body’s deformation; K - kinetic energy of an elastic body; A - work of external forces. It should be noted that this variational method is also valid for the problems of the initial deformations and the thermal stresses [676]. The expression for the deformation energy of the s-th shell’s layer has the following form ⎞ ⎡ ⎛ ⎟⎟⎟ T s ⎢⎢⎢ ⎜⎜⎜ 1 ⎟⎟ ⎢⎢⎢⎢ s ⎜⎜⎜⎜ s s V = ⎢⎢⎢σα ⎜⎜⎜ε1 + zκ1 − α1 dT ⎟⎟⎟⎟⎟ + 2 ⎠ ⎣ ⎝ s V

T0

⎞ ⎤ ⎛ ⎟⎟⎟ ⎥⎥⎥ ⎜⎜⎜ T s ⎟⎟⎟ ⎥⎥ ⎜⎜ s⎜ s s ⎟ ⎜ σβ ⎜⎜ε1 + zκ2 − α2 dT ⎟⎟ + ταβ (ε12 + zκ12 )⎥⎥⎥⎥ dV . ⎟⎠ ⎥⎦ ⎜⎝ T 0s

(3.49)

288

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Integrating along the thickness and summing all the shell’s layers, after some transformations, the expression for energy of the shell’s deformation has the following form 1 1 Fββ uα − k1 w + w2α + Fαα υβ − k2 w + w2β − V= 2 2 Ω

1$ 2 + Fαβ uβ + υα + wα wβ − A11 Fββ + 2A12 Fββ Fαα + A22 Fαα 2 & & & ' ' ' 2 − D11 − D∗11 w2αα − 2 D12 − D∗12 wαα wββ − D22 − D∗22 w2ββ − A66 Fαβ

% $ 4 D66 − D∗66 w2αβ − d11 Fββ wαα + d21 Fαα wαα + d12 Fββ wββ + % d22 Fαα wββ − 2d66 wαβ Fαβ − [(d11C1T + C2T d21 − K1T ) wαα + (d12C1T + d22C2T − K2T ) wββ + (A11C1T + A12C2T ) Fββ + (A12C1T + A22C2T ) Fαα ]} dαdβ.

(3.50)

The relations (3.41)–(3.47) are used in order to get the expression (3.50). The kinetic energy of a multilayer orthotropic shell without a rotational inertia and without inertial eﬀect in the tangentional direction is written as 1 K= 2 where m∗ =

m+n s=1

m∗ Ω

2 ∂w dαdβ , ∂t

(3.51)

ρ s (δ s − δ s−1 ) denotes the mass of the shell element.

A variation of the external forces’ work can be presented in the form [676] ∂δw + F¯ α δu − z qδwdαdβ − δA = − ∂α Ω

Ω1

∂δw ¯ ¯ Fβ δυ − z + Fn δw dΩ, ∂β

(3.52)

where F¯ α , F¯ β , F¯ n are the components of the external force, given on the part of the surface the surface Ω1 and q denotes 4 4 4 load. Introducing 4 the following 4 notations ¯ α = F¯ α zdz, T¯ β = F¯ β dz, M ¯ β = F¯ β zdz, T¯ n = F¯ n dz, we T¯ α = F¯ α dz, M substitute the derivatives of the variations δwα and δwβ by the following expressions δwα = µ1 δwµ − µ2 δwt ,

δwβ = µ2 δwµ − µ1 δwt ,

(3.53)

− −u ,→ where → t denote the unit vectors of the orthogonal coordinates attached to the boundary contour. The µ1 , µ2 denote a normal vector projection onto the coordinates, associated with the coordinate lines of the deformed surface. A variation of the work is defined as follows

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

δA = −

qδwdαdβ −

289

:

T¯ α δu + T¯ β δυ+

Ω

$ %

¯ α − µ1 M ¯ β δw − µ1 M ¯ α − µ2 M ¯ β δwµ ds. T¯ n − µ2 M t

(3.54)

The energy variation of the whole shell can be expressed by the partial variation δV = δVu + δVυ + δVF + δVw ,

(3.55)

where the partial variation element has the following form [290] ⎡⎢ ∂V1 ∂V1 ∂V1 ∂V1 ⎢⎢⎢ ∂V1 − + + + δV f = ⎣⎢ ∂ f − ∂ f ∂ fβ β ∂ fαα αα ∂ fαβ αβ α α ⎤ : ⎥⎥⎥ ∂V1 ∂V1 ∂V1 ⎥⎥⎦ δ f dαdβ + µ1 + µ2 − µ1 − ∂ fα ∂ fβ ∂ fαα α ββ ⎫ ⎪ ∂V1 ∂V1 ∂V1 ⎪ 1 1 ⎬ − µ1 − µ2 µ2 δf+ ⎪ ⎭ ∂ fββ β 2 ∂ fαβ β 2 ∂ fαβ αβ ⎪ 1 ∂V1 ∂V1 ∂V1 1 ∂V1 δ fβ ds. + µ2 δ fα + µ2 + µ1 µ1 ∂ fαβ 2 ∂ fαβ ∂ fββ 2 ∂ fαβ

∂V1 ∂ fββ

(3.56)

In the above, V1 denotes the subintegral expression of (3.50). Taking into account (3.53), the partial variations in the explicit form is presented :

(3.57) δVu = µ1 Fββ − µ2 Fαβ δuds, δVυ = δVF =

−

:

−µ1 Fαβ + µ2 Fαα δυds,

A12 Fββ + A22 Fαα

αα

+

Ω

(3.58)

1

A66 Fαβ + αβ 2

1

A11 Fββ + A12 Fαα + A66 Fαβ + d21 wαα + d22 wββ + k2 − ββ αβ αα 2

d66 wαβ + d11 wαα + d12 wββ + k1 w − d66 wαβ + αβ

2

ββ

αβ

1

wαα wββ − 2wαβ wαβ + wββ wαα + (A12C1T + A22C2T )αα + (A11C1T + A12C2T )ββ δFdαdβ,

δVw = Ω

1

− d11 Fββ + d21 Fαα + k2 F

αα

(3.59)

− d66 Fαβ + αβ

290

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

− d66 Fαβ + ββ αβ $

$& % % & ∗ ' ' ∗ ∗ D11 − D11 wαα + D12 − D12 wββ + 2 D66 − D66 wαβ + αα αβ $

$& % % & ∗ ' ' ∗ ∗ D12 − D12 wαα + D22 − D22 wββ + 2 D66 − D66 wαβ + ββ αβ

Fββ wαα − 2Fαβ wαβ + Fαα wββ + (d11C1T + d21C2T − K1T )αα +

d12 Fββ + d22 Fαα + k1 F

(d12C1T + d22C2T − K2T )ββ δwdαdβ+ : $

µ1 Fββ wα − Fαβ wβ + M1α + Hβ +

µ2 Fαα wβ − Fαβ wα + M2β + Hα δwµ −

µ21 M1 + 2µ1 µ2 H + µ22 M2 δwµ + % $

% µ1 µ2 (M1 − M2 ) − H µ21 − µ22 δw ds.

(3.60)

t

The kinetic energy variation of the shell has the following form [684] ∂2 w δK = − m∗ 2 δwdαdβ . ∂t

(3.61)

Ω

Substituting (3.54), (3.57)–(3.61) into (3.48) we get the equilibrium equation, the equilibrium of deformation continuity and the boundary condition for a dynamical problem of the elastic thermo-sensitive thin-walled shell with the orthotropic and design-orthotropic layers. – The equilibrium equation

d11 Fββ + d21 Fαα + k2 F

αα

+ d66 Fαβ

αβ

$&

'

+ d12 Fββ + d22 Fαα + k1 F + ββ

&

'

%

+ D∗11 − D11 wαα + D∗12 − D12 wββ + αα $& $

% % & ' ' 2 D∗66 − D66 wαβ + D∗12 − D12 wαα + D∗22 − D22 wββ + αβ ββ

$

% 2 D∗66 − D66 wαβ + Fββ wαα − 2Fαβ wαβ + Fαα wββ − d66 Fαβ

αβ

αβ

(K1T − d11C1T − d21C2T )αα− (K2T − d12C1T − d22C2T )ββ + q − m∗

∂2 w = 0. (3.62) ∂t2

– Equation of continuity deformation

1

A66 Fαβ + A11 Fββ + A12 Fαα + αβ ββ 2

+ d21 wαα + d22 wββ + k2 w − d66 wαβ +

A12 Fββ + A22 Fαα 1

A66 Fαβ αβ 2

αα

+

αβ

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

d11 wαα + d12 wββ + k1 w − d66 wαβ

αβ

291

+

1

wαα wββ − 2wαβ wαβ + wββ wαα + (A12C1T + A22C2T )αα + 2 (A11C1T + A12C2T )ββ = 0 .

(3.63)

– Boundary conditions

$

¯ α − µ1 M ¯ β − µ1 Fββ wα − Fαβ wβ + M1α + Hβ + T¯ n − µ2 M t

µ2 Fαα wβ − Fαβ wα + M2β + Hα −

% µ1 µ2 (M1 − M2 )t + Ht µ21 − µ22 = 0,

¯ α + µ2 M ¯ β − µ21 M12 + 2µ1 µ2 H + µ22 M22 = 0, µ1 M

T¯ β − −µ1 Fαβ + µ2 Fαα = 0,

T¯ α − µ1 Fββ − µ2 Fαβ = 0 .

(3.64) (3.65) (3.66) (3.67)

The kinematic boundary conditions read [290] u = uK ,

υ = υK ,

w = wK ,

∂w = γK , ∂µ

(3.68)

where uK , υK , wK , γK are the given contour displacements the angular displacement −u . of the normal → Consider more detailly, the boundary conditions (3.64)–(3.67). We introduce the system of the orthogonal coordinates αK , βK , z on the shell contour (see Fig. 3.2). −˜ −˜ → −µ˜ ,→ Let the unit vectors → t , l of the coordinates define the right hand coordinate −r˜ be the radius-vector of the sursystem. Let ds be the element of the contour arc; → −˜ → −˜ → −˜ → face; and let l 1 , l 2 , l n be the unit vectors of the deformable surface. Considering the deformations as small and taking the quantities ds, ti , ui as the same for both the deformable and the undeformable surfaces [492], we get → −˜ → − t = t,

→ −µ˜ = µ,

−˜ → −˜ → −˜ → −˜ → −˜ → − −˜ → −r˜ β = → l 1 t1 + l 2 t2 = l 2 µ1 − l 1 µ2 , t ≈ t =→ r s = −r˜α α s + → β s → −˜ → −˜ → −˜ → −˜ − − → → −µ˜ = µ = → t × l n = l 1 t2 − l 2 t1 = l 1 µ1 + l 2 µ2 . → −˜ → −˜ → − → − → − In the above t 1 , t 2 are the vector t projections on the directions l 1 , l 2 −µ2 =

dα = t1 = − sin α0 , ds

t2 = cos α0 =

dβ = µ1 , ds

292

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.2. An orthogonal coordinate system on the shell contour.

−˜ −u˜ and → where α0 denotes the angle between two unit vectors → l 1 . If the limiting cross section overlaps with the α curve, then α0 =

π , 2

→ −˜ t1 = − l 1 ,

−˜ → −µ˜ = → l 2,

T¯ β = Fαα , ¯ β − M2 = 0, M

µ2 = 1,

µ1 = 0,

T¯ α = −Fα,β ,

¯ α − Fαα wββ − Fαβ wα + M2β + 2Hα = 0. T¯ n + M α

(3.69)

If the limiting cross section overlaps with the β curve, then α0 = 0,

¯ α − M1 = 0, M

µ2 = 0,

µ1 = 1,

→ −˜ → l 1 = −µ˜ ,

→ −˜ → −˜ l2= t,

T¯ β + Fαβ = 0, T¯ α + Fββ = 0,

¯ β − Fββ wα − Fαβ wβ + M1α + 2Hβ = 0. T¯ n + M β

(3.70)

If the shell does not have the boundaries, the limiting conditions are substituted by the periodicity conditions along the two coordinates. If the shell is closed along one of the coordinates, then the periodicity condition is used only along one of the coordinates. Considering shells with initial imperfections the equations’ formulations are analogical to the ones described earlier. Suppose that a shell has the initial buckling w0 (α, β). We assume that the amplitude of those displacements does not exceed

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

293

the shell’s thickness. We also take it to guarantee that the occurring form imperfections resulting from those displacements have a character of sloped parts. Then, the expressions governing the deformations have the following form [685] 1 1 ε1 = uα − k1 w + w2α − k1 w0 + w20α , 2 2 1 1 ε2 = υβ − k2 w + w2β − k2 w0 + w20β , 2 2 ε12 = uβ + υα + wα wβ − w0α w0β .

(3.71)

The quantity w − w0 should be used in order to define full deformations. Below, we give the final expressions for the equilibrium equation and the deformation continuity. – The equilibrium equation

d11 Fββ + d21 Fαα + k2 F

αα

− d66 Fαβ

αβ

+ d12 Fββ + d22 Fαα + k1 F − ββ

$&

%

& ' ' + D∗11 − D11 (w − w0 )αα + D∗12 − D12 (w − w0 )ββ + αα $

% ∗ 2 D66 − D66 (w − w0 )αβ + αβ % $& & ' ' D∗12 − D12 (w − w0 )αα + D∗22 − D22 (w − w0 )ββ + ββ

% $

2 D∗66 − D66 (w − w0 )αβ + Fββ wαα − 2Fαβ wαβ + Fαα wββ −

d66 Fαβ

αβ

αβ

(K1T − d11C1T − d21C2T )αα− (K2T − d12C1T − d22C2T )ββ+ q − m∗

∂2 w = 0. (3.72) ∂t2

– The equation of deformation continuities

A12 Fββ + A22 Fαα

αα

+

1

A66 Fαβ + A11 Fββ + A12 Fαα + αβ ββ 2

$ % 1

A66 Fαβ + d21 (w − w0 )αα + d22 (w − w0 )ββ + k2 (w − w0 ) − αβ αα 2 $ % % $ d66 (w − w0 )αβ + d11 (w − w0 )αα + d12 (w − w0 )ββ + k1 (w − w0 ) + αβ

ββ

$ % d66 (w − w0 )αβ

1

+ wαα wββ − 2wαβ wαβ + wββ wαα − αβ 2 1

w0αα w0ββ − 2w0αβ w0αβ + w0ββ w0αα + 2 (A12C1T + A22C2T )αα + (A11C1T + A12C2T )ββ = 0.

(3.73)

294

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

In the above, in both (3.72) and (3.73) w denotes a full deflection. The boundary conditions are identical to the boundary conditions (3.64)–(3.67). Consider a one-layer homogeneous shell. In this case we have ∆=

h , 2

δi = 1, B12 = B21 =

δ1 = h, G=

νE , 1 − ν2

E1 = E2 = E,

E , 2(1 + ν)

B11 =

E , 2(1 + ν)

B66 =

ν1 = ν2 = ν, E = B22 , 1 − ν2 β1 = α∗ (1 + ν) = β2 .

We introduce the following variables [519] h

B (α, β) =

1 1 − ν2

2 E (T ) dz, − h2 h

1 A (α, β) = 1 − ν2

2 E (T ) zdz, − h2 h

D (α, β) = h

1 NT (α, β) = 1−ν

2 − h2

1 1 − ν2

2 E (T ) z2 dz, − h2

⎛ T ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ E (T ) ⎜⎜⎜ α∗ (T ) dT ⎟⎟⎟⎟⎟ dz = C1T = C2T , ⎝ ⎠ T0

⎛ T ⎞ h 2 ⎜⎜⎜ ⎟⎟⎟ 1 ⎜⎜ ⎟ ⎜ (T ) (α, (T ) E z ⎜⎜⎜ α∗ MT β) = dT ⎟⎟⎟⎟⎟ dz = K1T = K2T , 1−ν ⎝ ⎠ − h2

T0

and we express the other variables and the expression using the functions C11 = C22 = B,

C12 = νB = C21 ,

K11 = K22 = A, K12 = νA = K21 , 1−ν 1−ν B, K66 = a, C66 = 2 2 D11 = D22 = D, A11 = A22 =

D12 = D21 = νD,

1 , B(1 − ν2 )

A12 = A21

1−ν D, 2 1 1 , =− 1−νB

D66 =

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

295

A 2 1 , d12 = d21 = 0, A66 = , B 1−νB A A2 νA2 1 − ν A2 , D∗12 = D∗21 = , D∗66 = . (3.74) d66 = , D∗11 = D∗22 = B B B 2 B The relations between the forces, moments and deformations have the form d11 = d22 =

T 1 = B (ε1 + νε2 ) + A (κ1 + νκ2 ) − NT , T 2 = B (νε1 + ε2 ) + A (νκ1 + κ2 ) − NT , 1−ν (Bε12 + Aκ12 ) = s21 = s, s12 = 2 M1 = D (κ1 + νκ2 ) + A (ε1 + νε2 ) − MT , M2 = D (νκ1 + κ2 ) + A (νε1 + ε2 ) − MT , 1−ν (Dκ12 + Aε12 ) = H21 = H. (3.75) H12 = 2 The relations between the deformations and the forces and between the curvatures and moments are as follows $ % A 1 ε1 = & ' Fββ − νFαα + (1 − ν) NT + wαα , 2 B B 1−ν $ % A 1 ' −νFββ + Fαα + (1 − ν) NT + wββ , 2 B B 1−ν 1 2 2A · Fαβ + wαβ , ε12 = − 1−ν B B

2 A A A − D wαα + νwββ + NT − MT , M1 = Fββ + B B B

2 A A A − D νwαα + wββ + NT − MT , M2 = Fαα + B B B 2 A A H = − Fαβ + (1 − ν) − D wαβ . B B

ε2 =

&

(3.76)

– The equilibrium equation to the problem of the initial imperfections has the form $ 2 2 % A A − D ∇2 ∇2 (w − w0 ) + 2 − D ∇2 (w − w0 ) + α B B α $ 2 % A2 A − D ∇2 (w − w0 ) + ∇2 − D ∇2 (w − w0 ) − 2 β B B α 2 A A (1 − ν) L − D, w − w0 + L , F + L (w, F) − B B A ∂2 w ∇2 MT − ∇2 NT + q − m 2 + k2 Fαα + k1 Fββ = 0. (3.77) B ∂t

296

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

– The equation of deformation continuity to the problem of the initial imperfection has the form

1 1 1 2 2 2 2 2 1 ∇ ∇ F+2 ∇2 F− ∇ F +2 ∇ F +∇ α β B B α B β B A

1 , F + L , w − w0 · 1 − ν 2 + B B $ %

1 − ν2 k2 (w − w0 )αα + k1 (w − w0 )ββ + N 1

T 1 − ν2 [L (w, w) − L (w0 , w0 )] + (1 − ν) ∇2 = 0. 2 B

(1 − ν) L

(3.78)

In (3.77) and (3.78) the following operators are introduced ∇2 ∇2 f = fαααα + 2 fααββ + fββββ , ∇2 f = fαε + fββ , L ( f, g) = fαα gββ − 2 fαβ gαβ + fββ gαα . It should be noted that in the case of dynamical problems (3.77), (3.78) a simplification is not achieved by taking the coordinate surface in order to get A = 0. This is caused by an occurrence of the unstationary temperature fields, and it is impossible to choose the constant in the surface. For a stationary temperature field the equations (3.77), (3.78) (using the simplification A = 0) are identical to the equations given in [519]. To conclude a generalization of the thermoelastic equations for dynamical problems of a thin-walled thermosensitive conical shell has been developed. For the temperature stresses determination we need to define the temperature field. The heat transfer equation can be obtained from the equation of (3.10) type, neglecting the time dependent terms. The heat transfer equation for a thin-walled multilayer orthotropic conical shell has the form [486] s s ∂T s ∂ s ∂T s ∂ s ∂T s ∂ s ∂T s ∂T λT α + λT β + λT z + 2kλTs z = Cε=0 . (3.79) ∂α ∂α ∂β ∂β ∂z ∂z ∂z ∂t Above λTs α , λTs β , λTs z are the heat transfer coeﬃcients, and Cε=0 is the thermal capacity for a constant deformation of the s-th layer orthotropic material. On the shell’s boundary surfaces the boundary conditions (3.16)–(3.18) should be satisfied, whereas between the layers we have the following continuity conditions of the heat flows and the temperature: λTs−1 z

∂T s−1 ∂T s = λTs z , ∂z ∂z T s−1 = T s .

(3.80) (3.81)

In some works [519, 549] it has been noted that for thin shells the temperature distribution along the thickness can be obtained with a high accuracy using the heat transfer equation for a flat wall.

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

297

3.2.4 Solution Existence The problems of solution existence in mechanics and physics are related to the qualitative investigations of the operator equations. A suitability of the theory which reduces that problem to the mathematical scheme of the physical world behaviour independently of the experiments should be verified [206]. On one hand, it corresponds to the theory of multidimensional singular potentials and singular integral equations. On the other hand, it is related to the theory of generalized solutions to the diﬀerential equations (the methods of Hilbert’s spaces, the variational methods) [396]. In this chapter, the latter approach is used, which is more powerful and which includes the case of the variable coeﬃcients and variable boundary conditions. It includes the following fundamental steps: – a derivation of the a priori estimation; – an application of those estimations. A choice of the functional spaces, where a solution is located, plays an essential role to getting and applying a priori estimation of the compactness method [438]. The obtained (in this work) results are generalizations of the known ones [438, 481, 690] to the case of the thermosensitive conical shell’s vibrations. A special attention has been paid to the formulation and the theorem’s proof (on the existence) for the outlined problem in the case of a suﬃcient regular surface using the Dirichlet boundary conditions. This approach is supported by an application of the Green’s function method. Consider the bounded space Ω0 in Rz (Ω0 is the vibrated shell). We are looking for a function pair w, F defined in Ω0 ×]0, tK [, satisfying the equations (3.62), (3.63) and governing dynamical behaviour of a conical shell with the material characteristics depending on the temperature, and with the boundary conditions ∂w = 0, ∂n ∂F = 0, F, ∂n

w,

,

(3.82)

∂w = w 0 . ∂t t=0

(3.83)

on

and with the following initial conditions w|t=0 = w0 ,

We assume that the temperature field is stationary. In this case we can simplify the equations (3.62), (3.63), when the coordinate surface is chosen to get A = 0. Denoting the surface by z0 and introducing the notations α = x, β = y, we get the following equation: h

A (x, y) =

1 1 − ν2

2 E (x, y, z) (z − z0 )dz = 0. − h2

(3.84)

298

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

The quantities B and NT in the formula (3.73) are unchanged, whereas D and MT have the following form: h

1 D = D (x, y) = 1 − ν2 h

MT = MT (x, y) =

1 1 − ν2

2 − h2

2 E (T ) (z − z0 )2 dz, − h2

⎛ T ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ E (T ) (z − z0 ) ⎜⎜⎜ α∗ (T ) dT ⎟⎟⎟⎟⎟dz. ⎝ ⎠

(3.85)

T0

It should be noted that in the considered problem the temperature is supposed to be known. Therefore the functions B, D, NT are also known. We rewrite the equations (3.62), (3.63) taking into account the above assumption:

(3.86) mw

+ ∆21D + ν∆22D w − [w, F] − {k, F} + ∇2 MT − q = 0, N

1

T ∆21H − ν∆22H F + 1 − ν2 [w, w] + 1 − ν2 {k, w} (1 − ν) ∇2 = 0. (3.87) 2 B Above, the notation given in [536] is used ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 G + 2 G 2 , ∆21G ≡ 2 G 2 + 2 ∂x∂y ∂x∂y ∂x ∂x ∂y ∂y

∂2 f ∂2 f + k , [u, υ] = L(u, υ), 2 ∂y2 ∂x2 ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 G + 2 G 2 . ≡ 2 G 2 −2 ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x {k, f } ≡ k1

∆22G

We introduce the space H02 (Ω0 ) [438] ∂υ = 0 on Γ H02 (Ω0 ) = υ| υ ∈ H02 (Ω0 ) , υ = 0, ∂n

(3.88)

(3.89)

Therefore, H02 (Ω0 ) creates the Sobolev space [628] of functions taking zero values on the space boundary together with their derivatives. Denote by Q the cylinder in R2 × Rt : Q = Ω0 ×]0, tK [, where tK is finite, and by its side’s limit: = Γ×]0, tK [. p By L (0, tK ; Ω0 ) [438] we denote the space of the functions t - f (t) :]0, tK [→ Ω0 , which are measured, having the values in Ω0 and satisfying the condition ⎛ tK ⎞ 1p ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟ p ⎜⎜⎝ f Ω0 dt⎟⎟⎟⎟⎠ = f L p (0,tK ;Ω0 ) < ∞. 0

(3.90)

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

299

If p = ∞, then the form (3.90) is changed by the norm [438] sup ess f (t)Ω = f L∞ (0,tK ;Ω0 ) ,

t ∈ ]0, tK ]

(3.91)

and we have L p (0, tK ; L p (Ω)) = L p (Q). THEOREM 3.1 Let the q, ∇2 MT , ∇2 ( NBT ), k1 , k2 , w0 , w 0 be given, and q ∈ L2 (Q) ,

(3.92)

∇2 MT ∈ L2 (Ω) , N T ∇2 ∈ L1 (Ω) , B

(3.93)

k1 , k2 ∈ L2 (Ω) , w0 ∈

H02

(Ω) .

w 0 ∈ L2 (Ω)

(3.94) (3.95) (3.96) (3.97)

Then, there exist w and F, which satisfy (3.86), (3.87), (3.82), (3.83), and

(3.98) w ∈ L∞ 0, tK ; H02 (Ω) ,

w ∈ L∞ 0, tK ; L2 (Ω)

F ∈ L∞ 0, tK ; H02 (Ω) .

(3.99) (3.100)

Comment 3.1 From (3.98), (3.100) and the definition (3.88) it results that [ω, F] ∈ L∞ (0,K , L1 (Ω)) and the equation (3.86) leads to the inclusion ω

∈ L∞ (0, tK ; H −2 (Ω)) that the conditions (3.83) have sense. In the work [438] some properties of [u, υ] are proved, which are used in our further considerations. Lemma 3.1 The transformation u, υ → [u, υ] is the bilinear transformation H02 (Ω)× H02 (Ω) → H −2 (Ω). Corollary 3.1 The form u, υ, f → ([u, υ], f ) is the continuous trilinear form in H02 (Ω). Lemma 3.2 The trilinear form u, υ, f → ([u, υ], f ) is symmetric in H02 (Ω). In the analogical way it can be proved that the same properties are applied to the form k, u, υ → ({k, u}, υ). Lemma 3.3 The trilinear form k, u, υ → ({k, u}, υ) is the symmetric trilinear form in H02 (Ω). Lemma 3.4 The operators ∆21D + ν∆22D and ∆21H + ν∆22H are symmetric, positive defined operators [481]. Lemma 3.5 The operators ∆21D + ν∆22D and ∆21H + ν∆22H are strong elliptic operators [679, 680].

300

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Corollary 3.2 The operator ∆21D + ν∆22D has in Ω the whole system of vectors [680]. From the given properties the following lemma is concluded. Lemma 3.6 A strongly elliptic operator ∆21H + ν∆22H transforms the space H02 (Ω) into the conjugated space H −2 (Ω) heomorphicly [184]. In particular, it means that there exists the operator G F , inversed to ∆21H + ν∆22H , which is the continuous operator form H −2 (Ω) → H02 (Ω). It means that there exists the “Green’s operator” inversed to ∆21H + ν∆22H in Ω with the Dirichlet conditions. Proof I. The construction of the approximate solution. Let υ1 , . . . , υm be the basis defined by the functions in D(Ω) (g ∈ D(Ω) - the space of the functions of the class C ∞ in Ω, having the compact carrier in Ω), having the following properties: a) υi ∈ H02 (Ω) ∀i; b) ∀m υ1 , . . . , υm are linearly independent; c) the linear combinations υi are densed in H02 (Ω). Let wm (t) satisfy the following conditions wm (t) ∈ [υ1 , ..., υm ] ,

i.e. wm (t) =

m

gim (t) υi ,

(3.101)

i=1

w

m (t) , υ j + ∆21D + ν∆22D wm (t) , υ j − [Fm (t) , wm (t)] , υ j −

(3.102) {k, Fm (t)} , υ j + ∇2 MT , υ j − q, υ j = 0 , 1 ≤ j ≤ m, wm (0) = w0m ∈ [υ1 , ..., υm ] , w0m → w0 in H02 (Ω) ,

(3.103)

w m

(3.104)

(0) =

w 0m

∈ [υ1 , ..., υm ] ,

w 0m

→

w 0

in L (Ω) . 2

Define Fm (t) by the relation

1 ∆21H − ν∆22H Fm (t) = − (1 − ν) [wm (t) , wm (t)] − 2 N

T 1 − ν2 {k, wm (t)} − (1 − ν) ∇2 (3.105) = 0, Fm (t) ∈ H02 (Ω) . B Observe that Fm (t) do not need to belong to [υ1 , . . . , υm ]. In accordance with the Lemma 3.6, Fm (t) can be presented in the following form 1

Fm (t) = G F − 1 − ν2 [wm (t) , wm (t)] − 2 N

T . (3.106) 1 − ν2 {k, wm (t)} − (1 − ν) ∇2 B Substituting these relations to (3.102), we can get the system of ordinary diﬀerential equations in relation to wm (t). Therefore, we can believe on the existence of wm (t) according to the solvability of the ordinary diﬀerential equations, and, on the existence of Fm (t) in a certain interval [0, tm ], tm > 0.

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

301

II. A priori estimation. Multiplying (3.102) by g m and making a sum in relation to j, we get '

&

wm (t) , w m (t) + ∆21D + ν∆22D wm (t) , w m (t) − ' & ' & [Fm (t) , wm (t)] , w m (t) − {k, Fm (t)} , w m (t) + &

' (3.107) ∇2 MT , w m − q, w m = 0. However, according to the Lemmas 3.2 and 3.3, we have ' &9 / ' & {k, Fm (t)} , w m (t) = k, w m (t) , Fm (t) , ' &5 6 ' & [Fm (t) , wm (t)] , w m (t) = wm (t) , w m (t) , Fm (t) = 1d [wm (t) , wm (t)] , Fm (t) . 2 dt

(3.108)

According to (3.105), the equation (3.108) is transformed to the following one 1 d

2 2 (t) (t) − ν∆ , F − F − ∆ m m 1H 2H 1 − ν2 dt d 1 d 2 NT (3.109) ∇ , Fm (t) . {k, wm (t)} , Fm (t) − dt 1 + ν dt B Substituting (3.108), (3.109) and (3.107), we obtain '

&

wm (t) , w m (t) + ∆21D + ν∆22D wm (t) , w m (t) + &9 / ' 1 d

2 ∆1D + ν∆22D Fm (t) , Fm (t) + k, w m (t) , Fm (t) − 2 1 − ν dt & / '

' &9 (3.110) k, w m (t) , Fm (t) + ∇2 MT , w m (t) − q, w m (t) = 0. Furthermore, using the relations, and according to the operators ∆21H + ν∆22H and ∆21D + ν∆22D properties, we get

1 d

d ∆21H − ν∆22H Fm (t) , Fm (t) ≡ ∆21H − ν∆22H Fm , Fm , dt 2 dt 1 d

∆21D + ν∆22D wm (t) , wm (t) , ∆21D + ν∆22D wm (t) , w m (t) ≡ 2 dt &

' 1d &

' 1 d & '2 (3.111) wm (t) , w m (t) = w (t) , w m (t) = w (t) , 2 dt m 2 dt m the expression (3.110) is transformed to the following form 1 d 2

2 wm (t) + ∆1D + ν∆22D wm (t) , wm (t) + 2 dt &

'

1 ∆21H − ν∆22H Fm (t) , Fm (t) = q, w m (t) − ∇2 MT , w m (t) , (3.112) 2 1−ν

302

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

and it implies that 1 2

2 wm (t) + ∆1D + ν∆22D wm (t) , wm (t) + 2 & 1

2 2 ∆1H − ν∆2H Fm (t) , Fm (t) = q (σ) , w m (σ) dσ− 1 − ν2 t

0

t

2

1 ∇2 MT , w m (σ) dσ + w 0m + ∆21D + ν∆22D w0m , w0m + 2

0

1

2 2 (0) (0) − ν∆ , F . ∆ F m m 1H 2H 1 − ν2 According to (3.103), (3.104) we have 2

2 w0m + ∆1D + ν∆22D w0m , w0m ≤ const.,

(3.113)

and taking into account (3.106) we get 1

Fm (0) = G F − 1 − ν2 [wm0 , wm0 ] − 2 N T . (3.114) 1 − ν2 {k, wm0 } − (1 − ν) ∇2 B However, [w0m , w0m ] belongs to the bounded set in L1 (Ω), and correspondingly, in H −2 (Ω). Therefore, Fm (0) belongs to the bounded set in H02 (Ω) and

∆21H − ν∆22H Fm (0) , Fm (0) ≤ const.

Furthermore, using the property of positively defined operators ∆21D + ν∆22D and ∆21H +ν∆22H , the Cauchy inequality for ε, and using the Gronwille’s lemma [437], it can be shown that tm = tK and

wm , Fm are bounded in L∞ 0, tK ; H02 (Ω) , (3.115)

w m are bounded in L∞ 0, tK ; L2 (Ω) .

(3.116)

III. Limiting transformations. According to (3.71), (3.72) we can define the series wµ , Fµ , in the following manner: wµ → w is weak in L∞ (0, tK ; H02 (Ω)), Fµ → F is weak in L∞ (0, tK ; H02 (Ω)), w µ → w is weak in L∞ (0, tK ; H02 (Ω)),

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

wµ → w is strong in L2 (Q) (according to Sobolev theorem).

303

(3.117)

Let the functions ϕ j , 1 ≤ j ≤ j0 belong to 1 ([0, tK ]), ϕ (tK ) = 0 and ψ =

j0

ϕ j ⊗ υ j,

(3.118)

j=1

Owing to (3.102) and for m = µ > j0 we have −

tK

w µ ψ

dt +

0

tK

tK

∆21D

+

ν∆22D

wµ , ψ dt −

0

(q, ψ) dt −

k, Fµ , ψ dt =

0

% Fµ , wµ , ψ dt−

0

tK

tK $

0

tK

∇2 MT , ψ dt + w µ (0) , ψ (0) . (3.119)

0

However, according to the Lemma 3.2, we have tK $

%

Fµ , wµ , ψ dt =

0

tK $

% ψ, Fµ , wµ dt,

0

[ψ, Fµ ] → [ψ, F] is weak in L2 (Q), and because wµ → w is strong in L2 (Q), then we get tK $

%

tK

Fµ , wµ , ψ dt →

0

&5

6

'

tK ([w, F] , ψ) dt,

ψ, F , w dt =

0

0

and (3.119) in the limit achieves the expression tK

&

'

w , ψ dt +

−

tK

0

tK

+

ν∆22D

0

tK ({k, F} , ψ) dt =

0

∆21D

(q, ψ) dt − 0

w, ψ dt − ([w, F] , ψ) dt− tK

0

tK

∇2 MT , ψ dt + w 0 , ψ (0) ,

(3.120)

0

which is true for all ψ of the (3.118) type. Using the limiting transition we deduce that (3.120) is satisfied for all ψ ∈ L2 (0, tK H 2 (Ω)), and ψ ∈ L2 (0, tK ; L2 (Ω)). Therefore it has been shown that w, F satisfy (3.86) and w (0) = w 0 . In order to get (3.87) we can directly achieve the limit in (3.105) (for m = µ). Taking into account that [wµ , wµ ] → [w, w] for instance, in D (Q) (a space of distribution). If ϕ ∈ D(Q), then

304

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

tK $ 0

%

wµ , wµ , ϕ dt =

tK $

% wµ , ϕ , wµ dt

0

and it is possible to achieve the limit, as in the above. A proof of existence and uniqueness theorems for the boundary value problems of the heat transfer is given in reference [405]. 3.2.5 Classification The given above mathematical model of dynamical problem of the temperature stresses of thin thermosensitive shells is constructed under many assumptions and hypotheses, and it includes diﬀerent types of possible deviations. We consider now a class of problems, which can be solved in a frame of the given mathematical model using the approach in which “...the model accuracy should not extend the intervals of deviation of the initial data, and no one of the quantities should be calculated more accurately than it is required from the point of view of the considered problem” [508]. Using the Kirchhoﬀ-Love hypothesis in order to reduce the three dimensional boundary value problem, the deviation Rh is assumed. Using this estimation and taking into account the computational accuracy (5%) we assume that only relatively 1 ) are considered [534]. long shells ( Rh ≤ 20 The theory of conical shells, used for the fundamental motion equations’ derivation, can be also applied to the shells with zero Gauss’ curvature and to the shells with a large changes exponent [14]. A phenomenological approach assuming a speed of the heat distribution as infinitely large while considering the heat transfer equations has been used. This assumption is verified by the computations of the temperature fields in diﬀerent bodies in usual conditions occured in practice. As it has been shown in reference [444], for the thermoelasticity problems, already for temperature higher than 293K (a room temperature) we can limit ourselves to the consideration of a “usual” heat transfer equation. In the given model a coupling mechanical term in equation is not included. A quantitative influence of the omitted term in the considered problems is small, although in some cases it can help to illustrate and describe some of the qualitative eﬀects (for instance, damping and dispersion of the elastic waves) [119, 555]. The inertial parameter certainly belongs to one of the fundamental parameters characterizing the mathematical model in both heat transfer and motion equations. Its role suﬃciently depends on the influence of a load duration. It has been shown that for a purely elastic problems considered in reference [17], a loading time duration is considerably smaller in comparison to the mechanical time characteristics of the shell. The load can be considered as the impulse-type load. In the case of the length comparable to the dynamical one (quasi-impulse), and in the case of longer time duration - to the quasistatical one. The analogical approach is applied also for the heat transfer problems [119]. Therefore, we can assume that the role of inertia in

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

305

the temperature stresses’ problems really depends on the relations between the time durations of the mechanical and heat changes characteristics, and on the duration of the heating loads. Consider a class of problems in which the characteristics time of mechanical and heat behaviour are closely related and are comparable to the time duration of the thermal load. In this case we need to consider the unstationary processes of heat distribution along the shell as well as the dynamical behaviour of the shell. We introduce the following quantities: τmech - characteristic mechanical time; τheat - characteristic heat time; (τimp )mech - time of forcing action; (τimp )heat - time of heat action. Therefore, the latter condition is presented in the following form

≈ τimp . (3.121) τmech ≈ τheat ≈ τimp mech

heat

As the characteristic mechanical time we take that equal to the 0.25 of the maximal period of free vibrations of the homogeneous cylindrical shell [17]: + + ) ' ρ R 4 & π T mech 1 l = · · 3 1 − ν2 , = (3.122) τmech = 4 2ωmin k1 R E h where: ωmin - the minimal frequency of free vibrations of a cylindrical shell; k1 the coeﬃcient characterizing the influence of the boundary conditions (k1 = 1 for a rolling support; k2 = 1.5 - for a stiﬄy supported shell; k1 = 1.25 for the unsymmetrical boundary conditions). The minimal frequency of free vibrations of the cylindrical shell can be found from the following expression [301] ⎧ ⎫ 2 ⎤ 12 ⎪

⎪ 0 ⎡⎢ mπR 4 ⎪ ⎪ h2 ⎪ ⎪ 4 2 ⎥ ⎪ ⎪ ⎥ ⎢ + n − 1 n ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎪ 12(1−ν2 )R2 ⎨ 1 E ⎢⎢⎢⎢ l ⎬ ⎥ ⎢ ωmin = min ⎪ , (3.123) ⎪ ⎥ ⎢

⎪ ⎪ ⎥ ⎢ 2 ' & ⎪ ⎪ n,m ⎪ ⎥ ⎢ R ρ mπR ⎪ 2 2 ⎦ ⎣ ⎪ ⎪ + n + 1 n ⎪ ⎪ l ⎩ ⎭ where n denotes the waves’ number in the circled direction, and m denotes the halfwaves’ number in the longitudinal direction. The time corresponding to the time of heat distribution along the shell serve as the characteristic heat time τheat = k2

b2 , a

b = (l, R, h) ,

(3.124)

where a denotes the temperature transfer coeﬃcient, and k2 denotes the coeﬃcient characterizing the order of the shell’s overheating, which lies in the interval from 0.08 to 1.0. 1 are considered and in order to Assuming that the thin-walled shells with Rh ≤ 20 satisfy the condition (3.121), we need to take into account the process of heat distribution along the shell’s thickness. For example, for a shell made from aluminium

306

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

and magnesium with the following physical characteristics [392]: E = 7100MPa, 2 ρ = 2700 mkg3 , ν = 0.3, a = 5 · 10−5 ms and the geometrical characteristics Rh = 250, l −3 R = 4, h = 10 m the characteristic mechanical time τmech and the characteristic heat time τheat are equal to (k1 = 1, k2 = 0.1) τmech ≈ 2 · 10−3 s,

(τheat )h ≈ 2 · 10−3 s,

l2 l2 R2 = k2 2 · = 16 (τheat )R = a a R l2 R2 h2 = 106 (τheat )h >> (τheat )h . k2 2 · 2 · (3.125) a R h The condition (3.121) (in this case) can be presented in the following form + + )

' ρ R 4 & h2

l 3 1 − ν2 ≈ k2 ≈ τimp ≈ τimp . (3.126) mech heat 2k1 E h a (τheat )l = k2

A choice of the k2 coeﬃcient is motivated by the following model problem. Consider an infinite plate made from a material with constant characteristics and having zero initial temperature. In the initial time the external side of the plate is influenced by the action of higher temperature T , whereas the internal side is heat isolated. As the characteristic heat time we can take τheat (after that the heat achieves the shell’s internal side). According to the references [119] this time is given by: τheat ≈ 0.0885

h2 , a

k2 = 0.0885.

(3.127)

We have to add that in this time moment the temperature moment MT achieves its maximum (maximal gradient of temperature). During the estimation of free vibration frequencies of the heated shell a minimal frequency of the cylindrical shell (ununiformely heated along its thickness) is obtained: 1 T max + T min 2 , (3.128) ωT = ωmin 1 + nE 2 where: E = E(z) = E0 (1 + nE T ), T max and T min are characterized by the temperature change along the thickness. It results from (3.128) that free vibration frequency of the heated shell is smaller than that of a cold shell (3.123). Therefore, in the case of dynamical behaviour of thin thermosensitive elastic shells, for a simultaneous inclusion of inertial eﬀect in the motion and heat transfer equations a special attention should be paid to the heat distribution processes only along the shell’s thickness. The heat distribution in the average surface can be neglected. During the analysis of the thermoelastic processes some limitations are given to the temperature changes. As the criterion of the limiting state when a thermal load is applied, the following stresses condition can be used σ (T ) ≤ σ s (T ) ,

(3.129)

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

307

where σ(T ) denotes the acting stresses, and the σ s (T ) denotes the elasticity border for a given temperature. In the process of heating of the load the deformation continuity conditions can be changed. Therefore, only a certain part of the whole heat deformation causes the occurrence of the thermal stresses [661] Fσ (T ) ≤ σ s (T ) .

(3.130)

In the above, F denotes the function defining the continuity deformations condition depended on the boundary conditions and on the time heating excitation duration. The F function is equal to the relation between the thermal stresses and the maximal possible stresses. It is changed in the interval from 0 to 1 [661]. The theoretical and experimental investigations [93, 436] proved that with a lack of structural and phase changes, heat capacity for a constant pressure C p can be treated as linearly coupled via the temperature. In this case, the fundamental material characteristics also linearly depends on the temperature: E = E0 (1 + nE T ) , ν = ν0 (1 + nν T ) ,

α∗ = α∗0 (1 + nα T ) ,

λT = λT 0 (1 + nλ T ) ,

C p = Cε=0 (1 + nc T ) .

(3.131)

Above ne , nα , nν , nλ , nc are the corresponding temperature coeﬃcients from the interval (1...10) × 10−4 , E0 , α∗0 , ν0 , λT 0 are the elasticity modulus, the coeﬃcient of a linear heat expansion, the Poisson’s coeﬃcient, the heat transfer coeﬃcient for the initial temperature, respectively. In further considerations we neglect the Poisson’s coeﬃcient’s dependence on the temperature. The criterion (3.130) (in its most generalized case) can be presented in the following form [661]: ⎧ ⎪ T 2f l T ⎪ ⎪ ⎪ T ⎪ ) (1 , C T ≤ 0.5 + n k ⎪ 3 ε=0 c ⎪ ⎪ T T n1 E0 (1 + nE T ) ⎨ α∗0 (1 + nα T ) dT ≤ ⎪ (3.132) F ⎪ ⎪ 1 − RT ν T 4f l T ⎪ ⎪ ⎪ T0 ⎪ ⎪ ⎩ k4Cε=0 (1 + nc T ) T 3 , T > 0.6 fl

where RT possesses the following values: RT = 1 - for one basic deformation continuity; RT = 1 - for two basic deformations’ continuity; RT = 2 - for three basic deformation’s continuity; T f l - the material flow temperature; k3 , k4 - material constants. It should also be noted that the heat capacity for higher temperature is constant [644]. The Machutov’s relation [635] can be used for the elasticity threshold σ s0 against the temperature estimation. In this case the criterion (3.130) has the following form

308

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

E0 (1 + nE T ) F 1 − RT ν

T T0

1 1 − , α∗0 (1 + nα T ) dT ≤ σ s0 exp κ T T0

(3.133)

where: σ s0 denotes the elasticity material threshold for T 0 temperature; κ is a material constant. In the temperature range TTf l < 0.6 both formulas (3.132), (3.133) are in good agreement (an error does not exceed 5%). To conclude, knowing the function F for each problem we can define a maximal allowed temperature increase (∆T )max according to the formula (3.132), (3.133). On the other hand, thermal sensitivity governed by (3.131) should be taken into account only if its magnitude increases the allowed error (in our case 5%). The above given condition bounds the temperature increment from below and can be presented in the following way 1 (∆T )δ > δ (3.134) + signnK · T 0 , |nK | where δ denotes the allowed error of 5%, nK = (nE , nα , nc , nλ ) and T 0 is the initial temperature. Depending on the relations between (∆T )max and (∆T )δ the problem can be classified in the following manner A. (∆T )δ < (∆T )max . 1) (∆T )δ < ∆T < (∆T )max - a thermoelastic problem with inclusion of the material characteristics versus temperature; 2) ∆T < (∆T )δ - a thermoelastic problem without inclusion of the material characteristics versus temperature; 3) ∆T > (∆T )max - a thermoplastic problem with inclusion of the material characteristics versus temperature. B. (∆T )δ > (∆T )max . 1) ∆T < (∆T )max - a thermoelastic problem without inclusion of the material characteristics versus temperature; 2) (∆T )max < ∆T < (∆T )δ - a thermoplastic problem without inclusion of the material characteristics versus temperature; 3) ∆T > (∆T )δ - a thermoplastic problem with inclusion of the material characteristics versus temperature. As an example, in a frame of the given classification, consider a body with two basic deformations continuity (RT = 2) made from the material DI6T with the following material characteristics [635]: E0 = 7.2 · 1010 Pa,

α∗0 = 24.3 · 10−6

1 , K

ν = 0.35,

κ = 301.04K,

3.2 Thermoelastic Thin Thermosensitive Cylindrical Shells

T 0 = 293K,

T pl = 933K,

nα = 5 · 10−4 ,

σ s0 = 340MPa,

nc = 3.9 · 10−4 ,

309

nE = 5.38 · 10−4 ,

nλ = 4.5 · 10−4 .

In the Fig. 3.3 the dependence of a relative temperature increment ∆T T versus the magnitude of the F functions according to (3.133) for the considered material DI6T is presented.

Figure 3.3. The relative temperature increase versus F function.

The low boundaries of the temperature increment using the temperature dependencies of diﬀerent considered materials read 5

(∆T )δ

5

(∆T )δ

6 nE

6 nc

= 79K, = 142K,

5

(∆T )δ

5

6

(∆T )δ

nα

6 nλ

= 115K, = 126K.

It means that for the considered material DI6T the temperature dependence in the thermoelastic problems should be considered in the following cases: – – – –

for the elasticity modulus E for F < 0.527; for the linear coeﬃcient of the thermal expansion α∗ for F < 0.329; for the thermal capacity C p for F < 0.248; for the heat transfer coeﬃcient λT for F < 0.291.

To conclude, the described approach separated the class of problems concerning thermoelastic thin cylindrical shells with a simultaneous inclusion of both inertial terms (in the shell’s motion equation and in the heat transfer equation) and material characteristics depending on the temperature.

310

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

3.3 Computational Algorithms 3.3.1 Finite Diﬀerence Equations In order to solve the dynamical problem of a thermosensitive thin homogeneous shell with geometrical nonlinearities we apply the method of finite diﬀerence. The fundamental equations (3.77)–(3.80) are presented in the following non-dimensional form 2 $ % A˜ R A˜ 2 1 2 2 2 ˜ ˜ ( ) ( ) ∇ + − D ∇ − D w ˜ − w ˜ + 2 w ˜ − w ˜ ∇ & ' 0 0 l B˜ 1 − ν2 B˜ x˜ x˜ 2 $ % ˜ l A˜ 2 2 2 A ˜ ˜ 2 − D ∇ (w˜ − w˜ 0 ) + ∇ − D ∇2 (w˜ − w˜ 0 ) − y˜ R B˜ B˜ y˜ 2

A˜ ˜ A˜ ˜T− ˜ ˜ w˜ − 1 ∇2 M (1 − ν) L − D, w˜ − w˜ 0 + L , F + L F, ˜ ˜ 1−ν B B 2 2 2 A˜ τ M ∂ w˜ 1 ∇2 + ky F˜ x˜ x˜ + k x F˜ y˜y˜ = 0, (3.135) NT + q˜ − 1−ν τT ∂t˜2 B˜ 1 2 2˜ R 1 2 ˜ l 1 2 ˜ ∇ ∇ F+2 ∇ F +2 ∇ F + x˜ y˜ l B˜ x˜ R B˜ y B˜ 1 1 2˜ A˜ ∇ F − (1 + ν) L , F˜ + L , w˜ − w˜ 0 + ky (w˜ − w˜ 0 ) x˜ x˜ + ∇2 ˜ ˜ B B B˜ ˜ 1 2 NT ˜ w) ˜ − L (w˜ 0 − w˜ 0 )] + ∇ = 0, (3.136) k x (w˜ − w˜ 0 )y˜y˜ + [L (w, 2 B˜ ∂T˜ ∂ ∂T˜ = λT T˜ . (3.137) ∂t˜ ∂˜z ∂˜z The operators ∇2 , ∇2 ∇2 have the following form R l w˜ x˜ x˜ + w˜ y˜y˜ , l R 2 R 2 l 2 2 ∇ ∇ w˜ = w˜ x˜ x˜ x˜ x˜ + 2w˜ x˜ x˜y˜y˜ + w˜ y˜y˜y˜y˜ , l R ∇2 w˜ =

whereas non-dimensional parameters read ˜ x = l x˜, y = R˜y, z = h˜z, w = hw, ˜ F = E0 h3 F, t=

Cε=0 h2 E0 h4 h2 ˜ R2 T , q = 2 2 q, t˜, T = ˜ q1 = 2 q, ˜ λT 0 lRα∗0 l R h

3.3 Computational Algorithms

qT = B=

hλT 0 lR q˜ T , τ M = lRα∗0 h

+

311

ρ h2Cε=0 l R , τT = , k x = , ky = , E0 λT 0 h h

E0 h ˜ E0 h3 ˜ E0 h2 ˜ E0 h3 ˜ B, D = D, A = A, N NT , = T (1 − ν) lR 1 − ν2 1 − ν2 1 − ν2 1

E0 h4 ˜ MT , B˜ = MT = (1 − ν) lR 1

2 A˜ = − 12

2

1 2

˜ ˜ E T d˜z, D = E T˜ z˜2 d˜z,

− 12

− 12

⎛ ⎞ 1 2 ⎜⎜⎜T˜ ⎟⎟⎟

⎜ ⎟ E T˜ z˜d˜z, N˜ T = E T˜ ⎜⎜⎜⎜⎜ α∗ T˜ dT˜ ⎟⎟⎟⎟⎟ d˜z, ⎝ ⎠ − 12

1

2 ˜T = M − 12

T0

⎛ T˜ ⎞

⎜⎜⎜⎜

⎟⎟⎟⎟

E (T ) E T˜ z˜ ⎜⎜⎜⎜⎜ α∗ T˜ dT˜ ⎟⎟⎟⎟⎟ d˜z, E T˜ = , E0 ⎝ ⎠ T0

α∗ (T ) lRα∗0 w0 . , T˜ 0 = T 0 , w˜ 0 = α∗ T˜ = 2 α∗0 h h The numerical investigation of the problem is carried out with the following boundary conditions for the equations (3.135), (3.136) w˜ = 0, w˜ x˜ x˜ = 0, F˜ = 0, F˜ x˜ x˜ = 0, x˜ = 0, x˜ = 1.0,

(3.138)

w˜ = 0, w˜ x˜ = 0, F˜ = 0, F˜ x˜ x˜ = 0, x˜ = 0, x˜ = 1.0,

(3.139)

whereas for the equation (3.137) it reads 1 1 ∂T˜ = 0, z˜ = , T˜ = T˜ 0 , z = − , 2 ∂˜z 2

∂T˜ 1 1 ∂T˜ = q˜ T 0 , z = − , = 0 z˜ = . λT T˜ ∂˜z 2 ∂˜z 2 The initial condition for the equation (3.135) have the following form w˜ = 0,

∂w˜ = 0, t˜ = 0, ∂t˜

(3.140) (3.141)

(3.142)

and for the equation (3.137) T˜ = 0, t˜ = 0.

(3.143)

Solving the problems (3.135), (3.136), (3.138), (3.142) and (3.135), (3.136), (3.139), (3.142) a symmetry related to the line y˜ = 0 and y˜ = π is used. On the symmetry lines the following symmetry conditions are taken: w˜ y˜ = 0, w˜ y˜y˜y˜ = 0, F˜ y˜ = 0, F˜ y˜y˜y˜ = 0.

(3.144)

312

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

We change the considered area by the arguments 9 / D0 = 0 ( x˜, y˜ , z˜) , 0 ≤ x˜ ≤ 1, 0 ≤ y˜ ≤ π, t˜ ≥ 0 to the area of their discrete values in the nodes of the mesh ωhx hyτ = ωhx ωhy ωτ = {(xk , yl , tn ) , xk = kh x } , yl = lhy , tn = nτ, k = 0, N, l = 0, M, n = 0, 1, 2, . . . , with the constant steps h x , hy , τ (see Fig. 3.4).

Figure 3.4. The mesh used in the finite diﬀerence method.

Instead of the functions w, ˜ F˜ we consider the mesh functions wnkl , Fkln given in the mesh nodes (xk , yl , tn ). Analogically, during solutions to the problems (3.137), (3.140), (3.143) and (3.137), (3.141), (3.143) the area of the continuous argument variations 1 1 ˜ D0T = z˜, − ≤ z˜ ≤ , t ≥ 0 2 2 is substituted by the area of discrete values in the nodes of diﬀerence mesh ωhz τ = ωhz ωτ = {(zm , tn ) , zm = mhz , tn = nτ, m = 0, K, n = 0, 1, 2, ...

3.3 Computational Algorithms

313

with the constant step hz . Instead of the function T˜ we consider its mesh analog T˜ mn given in the mesh nodes (zm , tn ). The nonlinear heat transfer equation is solved for each point of the computational area, where the heat excitation is diﬀerent from zero. Changing the partial derivatives, occurring in (3.135)–(3.144), by the centraldiﬀerence relations [594], we obtain the following finite diﬀerence equations A2 − D Λ4 (w − w0 ) + B l A2 R A2 − D [Λ2 (w − w0 )] x◦ + 2 − D ◦ [Λ2 (w − w0 )]y◦ + 2 ◦ l B R B y x 2 2 A A − D Λ2 (w − w0 ) − (1 − ν) L1 − D, w − w0 + Λ2 B B A A 1 1 Λ2 MT − Λ2 NT + L1 , F + L1 (F, w) − B 1−ν 1−ν B ky F x¯ x + k x Fy¯y ,

τM τT

2

1 wt¯t = q − 1 − ν2

(3.145)

1 R 1 l 1 (Λ2 F) x◦ + 2 (Λ2 F)y◦ + Λ4 F + 2 B l B x◦ R B y◦ A 1 1 Λ2 F − (1 + ν) L1 , F + L1 , w − w0 + Λ2 B B B 1 [L1 (w, w) − L1 (w0 , w0 )] + 2 N T Λ2 = 0, B ◦ T t = λT (T ) T z ◦ ,

ky (w − w0 ) x¯ x + k x (w − w0 )y¯y +

z

where for the diﬀerence derivatives the following notations are used

R2 n l2 n n + 2 w + , w w kl kl x¯ x x¯ x x¯ x¯yy l2 R2 kl y¯y¯yy

Λ4 wnkl =

R n l n Λ2 wnkl = , wkl + w x¯ x l R kl y¯y

L1 wnkl , Fkln = wnkl Fkln − 2 wnkl x◦ y◦ Fkln x◦ y◦ + wnkl Fkln , x¯ x

wnkl

◦

x

wnkl

=

y¯ y

x¯ x

wn − 2wnkl + wnk−1l = k+1l , h2x − wnk−1l , A = Ankl , B = Bnkl , 2h x

wnk+1l

y¯ y

x¯ x

(3.146) (3.147)

314

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

D = Dnkl , T = T mn , w0 = w0kl , MT = (MT )nkl , NT = (NT )nkl . The boundary and initial conditions (3.138)–(3.144) are expressed by the diﬀerence analogies of the form:

= 0, k = 0, k = N; (3.148) wnkl = wnkl = Fkln = Fkln x¯ x

wnkl = wnkl

x¯ x

= Fkln = Fkln = 0, k = 0, k = N; x¯ x & ' T mn = T 0 , m = 0, T mn z◦ = 0, m = K; & ' & '& ' λT T mn T mn z◦ = qT 0 , m = 0, T mn z◦ = 0, m = K,

w

kl = 0, wnkl ◦ = 0, n = 0; ◦

x

t

T mn = 0, n = 0; ! "

" wnkl y◦ = wnkl ◦ = Fkln y◦ = Fkln ◦ = 0. !

y¯ y y

y¯ y y

(3.149) (3.150) (3.151) (3.152) (3.153) (3.154)

Two series of the past contour points are introduced in the space whx hy (see Fig. 3.4) and two post contour points in the contour whz . The values of the mesh functions in those points are defined using the boundary conditions (3.148)–(3.151), (3.154). The obtained equations together with the boundary conditions (3.152)–(3.154) are solved using the following steps. 1. In each time step the temperature field for each point of the computational space whx hz is found, where the thermal excitation is diﬀerent from zero. Further, using a numerical integration along the shell thickness by the Simpson’s method the functions Ankl are defined. Those functions, as well as the temperature field, can be found in the whole time interval independently of the solution to the motion equation. However, in order to improve the economy of the computer memory and to increase the algorithm’s eﬀectiveness the calculations of the mentioned functions and the temperature field have been carried out using the described algorithm. 2. Using found in the previous step wnkl values and taking into account the heat functions (see section 1.1.4) the right hand side of the system of algebraic equations in relation to the function Fkln is formulated. 3. The obtained values of Fkln are substituted to the right hand side of the equation (3.145), and the wn+1 kl is obtained. Then the process is repeated. The theoretical basis of the described algorithm is given and the existence of the solution is proved (see Chapter 3.2.4).

3.3 Computational Algorithms

315

3.3.2 Solution to Biharmonic Equation A fundamental diﬃculty while solving the equations (3.145)–(3.147) occurs during the F function estimation. The diﬃculty of finding a solution to the equation (3.146) is caused by a high order of the initial equation (3.155). One of the possible approaches to solve the problem is focused on the reduction of the problem to the lower order equations (possibly, of the second order). In a general case formulation and realization of the diﬀerence schemes to the second order equations is much easier [330]. The method of splitting the biharmonic operator into two second order equations has been used for the first time by Marcus in 1925 [604]. For the problems, which can not be splitted into two independent elliptical problems for the second order equations (for instance, the Dirichlet problem), the iterational method of introducing a small perturbation parameter into the boundary condition and of solving the splitted problems on each iteration, has been applied. Such an approach has been presented in the references [77, 527], and others. In this work, the boundary conditions of the form (3.148), (3.149) and (3.154) give a possibility to apply a splitting method directly. Using the biharmonic operator Λ4 and the ϕ for the other terms (the values of these terms are taken from a previous three steps) the equation (3.146) is presented in the following form Λ4 F = Λ2 (Λ2 F) = −ϕ,

(3.155)

F = 0, F x¯ x = Λ2 F = 0, k = 0, k = N,

Fy◦ = 0, Fy¯y y◦ = (Λ2 F)y◦ = 0, l = 0, l = M .

(3.156) (3.157)

We get the following equations in the splitted form Λ2 Φ =

and Λ2 F =

R2 R Φ x¯ x + Φy¯y = Λ2x Φ + Λ2y Φ = − ϕ = −ϕ1 , l l2

(3.158)

Φ = 0, k = 0, k = N,

(3.159)

Φy◦ = 0, l = 0, l = M,

(3.160)

R2 R F x¯ x + Fy¯y = Λ2x F + Λ2y F = − Φ = −Φ1 , 2 l l F = 0, k = 0, k = N, Fy◦ = 0, l = 0, l = M.

(3.161) (3.162) (3.163)

Therefore, a solution to the biharmonic equation (3.155) has been reduced to the sequent solution of equations (3.158) and (3.161). They are solved using the Fourier series approach [594]. We consider the application of this method to our problem. We consider the being sought functions Φnkl = Φ (k, l), Fkln = F (k, l) and the given function ϕnkl = ϕ (k, l) for the fixed k, 0 ≤ k ≤ N as the mesh functions with the l argument. Developing the given functions into the sums in relation to the operator Λ2y eigenfunctions, we get

316

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Φ (k, l) =

M i2 =0

F (k, l) =

M i2 =0

Fi2 (k) µ(2) i2 (l),

Φi2 (k) µ(2) i2 (l),

ϕ (k, l) =

M i2 =0

ϕi2 (k) µ(2) i2 (l),

(3.164)

⎧+ ⎪ ⎪ 1 i2 πl ⎪ ⎪ ⎪ cos , i2 = 0, M ⎪ ⎪ ⎨ π M (l) = µ(2) + ⎪ i2 ⎪ ⎪ ⎪ i2 πl 2 ⎪ ⎪ ⎪ cos , 1 ≤ i2 ≤ M - 1 ⎩ π M denotes the eigenfunction of the Λ2y operator

where:

(2) (2) Λ2y µ(2) i2 + λi2 µi2 = 0, 1 ≤ l ≤ M − 1,

µ(2) i2

◦

y

= 0, l = 0, l = M,

(3.165)

which corresponds to the eigenvalue λ(2) i2 =

4 i2 π , i2 = 0, 1, ..., M. sin 2M h2y

The Fourier coeﬃcient ϕi2 (k) for each k, 1 ≤ k ≤ N − 1 is calculated using the formulas ϕi2 (k) =

M−1

% $ (2) (2) (l) (0) (k, (M) (k, hy ϕ (k, l) µi(2) + 0.5h + ϕ M) µ . ϕ 0) µ y i i 2 2 2

l=1

Substituting (3.164) to (3.158)–(3.163) we get ϕi2 (k) =

M

ρl ϕ1 (k, l) cos

l=0

i2 πl , M

0 ≤ i2 ≤ M, 1 ≤ k ≤ N − 1, h2x l2 h2 l2 −υi2 (k − 1) + 2 + 2 υi2 (k) − υi2 (k + 1) = x 2 ϕi2 (k) , R R 1 ≤ k ≤ N, υi2 (0) = 0, 2 2 h l h2x l2 (N) (N 1 + x 2 λ(2) υ ϕi (N) , − υ − 1) = i i 2 2 2R i2 2R2 2 Φ (k, l) =

M i2 πl 2 , ρl υi2 (k) cos M i =0 M 2

0 ≤ l ≤ M, 1 ≤ k ≤ N,

(3.166)

(3.167)

3.3 Computational Algorithms

ρl =

l = 0, l = M , 1≤l≤ M−1

0.5, 1,

Φi2 (k) =

M

ρl Φ1 (k, l) cos

l=0

317

(3.168)

i2 πl , M

0 ≤ i2 ≤ M, 1 ≤ k ≤ N − 1 h2x l2 (2) h2 l2 −Pi2 (k − 1) + 2 + 2 λi2 Pi2 (k) − Pi2 (k + 1) = x 2 ψi2 (k) , R R

(3.169)

a ≤ l ≤ N − 1, Pi2 (0) = 0, h2 l2 h2x l2 1 + x 2 λ(2) ψi (N) , i2 Pi2 (N) − Pi2 (N − 1) = 2R 2R2 2 0 ≤ i2 ≤ M, F (k, l) =

(3.170)

M i2 πl 2 , ρi Pi (k) cos M i =0 2 2 M 2

1 ≤ k ≤ N − 1, 0 ≤ l ≤ M.

(3.171)

In order to solve three point boundary value problems (3.167), (3.170) the multigrid method is used. The sums (3.166), (3.168), (3.169), (3.171) are obtained using the algorithm of the discrete Fourier transformations (for the nodes’ numbers, being a power of two). However, in order to have a more flexible possibility of diﬀerent type node’s choice a usual summation is used. A sequential solution to the equation (3.158) and (3.161) leads to the essential reduction of the required operations. It allows to get the Φ1 (k, l) values in (3.168) as the series related to the eigenfunctions of the diﬀerence operator Λ2y . Only the components of the υi2 development are used while finding the being sought F(k, l). It should be noted [594] that during the method of one dimensional Fourier development the eigenfunctions of the diﬀerence operator Λ2x are not used. We only need to separate the variables of Λ2x , and, therefore, it can be substituted by a more generalized operator. In order to solve the equations (3.155) the method of matrices for the five points equations [594] is used. We give the matrix structure and the coeﬃcient values used for calculations. The equation (3.155) can be presented in the following way − −u = → fh Ah→ h where

(3.172)

→ −u = {F ...F ...F ...F ...F ...F }T , h 11 M1 1i Mi 1N MN → − f h = {ϕ11 ...ϕm1 ...ϕ1i ...ϕ Mi ...ϕ1N ...ϕ MN }T .

The Ah matrix with the dimension (N × M) × (N × M) has the following form

318

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

(( (( A1 − A2 (( A2 (( (( A3 ( . I ( Ah = 2 2 ((( h x hy (( . (( (( (( ((

A3 A1 A2 .

A4 A2 A1 . A3 . .

A3 A2 . A2 .

A3 . A1 . A3

. A2 . A2 A3

. A3 . A1 A2 2A3

(( (( (( (( (( . . (( (( , ( . . ((( A2 A3 ((( A1 + A3 A2 (( ( 2A2 A1 (

and the matrices A1 , A2 , A3 can be presented as follows (( (( a1 2a2 2a1 (( a2 a1 + a3 a2 (( a1 a2 a3 ( a3 a2 . . . . . A1 = ((( . (( a a a a 3 2 1 2 (( a3 a2 a1 + a3 (( ( 2a3 2a2

(( (( (( (( ( . ((( , a3 ((( a2 ((( a1 (

⎛ ⎞ 2 h2 ⎜⎜⎜ R2 h2y ⎟⎟ l x a1 = 2 ⎜⎜⎝3 2 2 + 3 2 2 + 4⎟⎟⎟⎠ , l hx R hy ⎛ 2 2 ⎞ ⎜⎜⎜ l h x ⎟⎟ h2x l2 ⎜ a3 = 2 2 , a2 = −4 ⎝ 2 2 + 1⎟⎟⎠ , hy R R hy with dimension M × M,

(( (( b1 (( b2 (( ( . A2 = ((( (( . (( (( (

2b2 b1 . b2 .

b2 . b1 . b2

(( (( (( ( . (( (( , ( . ((( b2 ((( b1 (

. b2 . b1 2b2 ⎛ ⎞ ⎜⎜⎜ R2 h2y ⎟⎟ ⎜ b1 = −4 ⎜⎝ 2 2 + 1⎟⎟⎟⎠ , b2 = 2, l hx with dimension M × M,

(( (( (( C1 (( (( C2 (( (( (( . (( (( (( , . A3 = (( (( (( . (( (( C1 (( (( ( C1 (

2

C1 =

R2 hy , l2 h2x

3.3 Computational Algorithms

319

with dimension M × M. In order to solve the equations system (3.158), (3.161) also the matrix method is used for the three point equations [594]. We give the matrices structure and the coeﬃcients’ values necessary to carry out the computations in this case. The equation (3.158) and (3.161) can be presented in the following way: → − −r , Bh V h = → h

(3.173)

where the Bh matrix with the dimension (N × M) × (N × M) has the form (( (( (( B1 B2 (( (( B2 B1 B2 (( (( (( . . . . . . (( I (( (( (, B2 B1 B2 Bh = h x hy (( . . . . . . ((( (( ( B2 B1 B2 ((( (( ( 2B2 B1 ( and the matrices B1 and B2 with the dimension M × M can be presented in the form (( (( b1 (( b2 (( ( . B1 = ((( (( . (( (( (

2b2 b1 . b2 .

b2 . b1 . b2

. b2 . b1 2b2

(( (( (( ( . (( (( , ( . ((( b2 ((( b1 (

R hy l hx l hx , b2 = + , l h x R hy R hy (( (( (( d1 (( (( d1 (( (( (( . (( (( l hy (( , d1 = . B2 = (( . R hx (( (( . (( ( d1 ((( (( ( d1 (

b1 = −2

A comparison of three algorithms of the solution to the biharmonic equation (3.155) has proved that the most eﬀective was the algorithm which used the splitting of the diﬀerence operator into two equations (3.158), (3.161) with the sequential solution to each of those problems using the method of one dimensional Fourier series’ development (see Table 3.1).

320

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells Table 3.1. Eﬃciency of the used algorithms. Method Five point matrices Splitting + three point matrices Splitting + one dimensional Fourier series development

Estimation of the Estimation of the remembered information value artihmetic operations number ∼ 3M 2 (N + 1)

∼ 2M 3 N

∼ M 2 (N + 1)

∼ M3 N

∼ N(3M + 5)

∼ M2 N

3.3.3 Reliability of the Obtained Results On the stability of the obtained diﬀerence schemes A fundamental problem of the finite diﬀerence method accuracy is reduced to the analysis of a deviation of the approximation and stability of the used schemes. The application of the centre-diﬀerence relations for the motion equations of shells and boundary conditions leads to the reduction of the diﬀerence scheme (it has the second order approximation in relation to h x , hy , τ). The diﬀerence scheme of the heat transfer equation has the first order approximation in relation to τ and the second order approximation in relation to hz . To the most complicated analyses belong the one related to the diﬀerence scheme stability and an investigation of a continuous solution dependence on the initial input data. It should be noted that the stability corresponds to the internal scheme property, dependent neither on the approximation nor on the diﬀerence scheme link with the diﬀerential equations [591]. Therefore, the stability condition should be formulated as a certain relation between the diﬀerence operators. In the theory of the diﬀerence scheme it has been shown that approximation and stability secure the convergence of the diﬀerence scheme [572, 591]. We investigate the stability of the diﬀerence equations system (3.145)–(3.147) with the boundary conditions (3.148)–(3.154). The stability of the diﬀerence equation (1.13) can be proved using its representation in the splitted form of (3.158)– (3.163). The stability of each of the equations (3.158) and (3.161) can be proved using the maximum principle [592]. The stability of equation (3.145) is not a trivial problem because in this case the variable coeﬃcients appear, and the equation is a nonlinear one. In order to estimate the stability condition for this case, we use the approach given in the work [572]. A local stability of the linearized equation is investigated, obtained from the initial nonlinear equations, for which a well-known theory of equations with constant coeﬃcients is used. Because the analysed equation belongs to the parabolic type (the infinite velocity of excitation distribution), the approach described in the references

3.3 Computational Algorithms

321

[572, 637] is used. According to it, the operator stability is investigated via the stability of its main part (i. e. the terms, including the higher order derivatives). Thus, we investigate the stability of the following linear equation

τM τT

2

A2 Λ4 u = f . ut¯t + D − B

(3.174)

We also investigate a stability in relation to the perturbation of the initial data ( f = 0). The stability investigation of (3.174) include two diﬀerent approaches: – spectral stability; – general stability theory of the three-layer scheme. Owing to the spectral stability method [572], a solution to the diﬀerence equation (3.174) is sought in the form of harmonics dependent on two real parameters: unkl = λn (α, β) ei(αk+βl) .

(3.175)

A solution to the diﬀerence problem is stable when the spectrum is located in the unit circle |λ| ≤ 1. Substituting (3.175) to (3.174) we get the characteristic equation which leads to the stability condition [439]. In the case of h x = hy , it can be expressed in the following form ⎧ 2 ⎫ 12 √ τM ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 τM 2 1 ⎬ ⎨ τT

⎪ = h. τ < min ⎪ 2 ⎪ ⎪ A ⎪ ⎪ 4 2 τT x ⎭ ⎩ D−

(3.176)

B

In order to use a general stability theory of the diﬀerence schemes [585] we reduce the equation (3.174) to the canonical form by its multiplication of τ2 : τ2

τM τT

2

A2 Λ4 u = 0. ut¯t + τ2 D − B

(3.177)

2

2 It is easy to check that B1 = 0, R = ττMT E, A1Λ = τ2 D − AB Λ4 (using the notations of the reference [591]) and the stability theorem can be applied. The stability conditions of the scheme (3.177) are represented by the following inequalities 2 τM 1 τ2 . B1 ≥ 0, R > A1Λ or Λ4 < 4 4ρ τT

(3.178)

Therefore, in order to estimate the stability we need to estimate a norm of the diﬀerence operator Λ4 . Using the results of the works [77, 681] it can be shown that for the operator norm the following estimation is valid: ⎞ ⎛

⎜⎜⎜ 16 16 ⎟⎟⎟ 2 2 (3.179) Λ4 ≤ 2∆ = 2 ∆1 + ∆2 = 2 ⎝⎜ 4 + 4 ⎠⎟ . h x hy

322

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

As a result, in the case of h x = hy for τ the estimation, similar to that of (3.176) is valid. The empirical formula [74] is used for a final choice of time step limitations of the nonlinear diﬀerence scheme (3.145). The time step for the nonlinear equation is recommended to be three or four times smaller than the limiting step defined from the stability condition of the linearized equation, which is well verified by the numerical experiments [181, 691]. As an example of the approach described above for the stability estimation we consider a numerical solution of the problem related to a loaded cylindrical shell ( Rl = 2.2, ky = 112.5) by a step type load q1 = 0.8 with infinite time duration and uniformly distributed along the rectangular shape α x = 0.2, αy = 0.1 (see Fig. 3.10). As the diﬀerence mesh the nodes along the longtidunal coordinate and 33 nodes along the circle coordinate are used. Taking into account the estimation (3.176) and the empirical rule, the time step in this case should not exceed 2 · 10−3 . In the Table 3.2 the results of the deflection function’s values, the load function F, the M x , My moment and the forces N x , Ny for the diﬀerent time moments for the following steps: 0.00025, 0.0005, 0.001, 0.002 are given. Table 3.2. Computational results. Coordinate

Variation

W/h tmax

F tmax

Mx tmax

My tmax

Nx tmax

Ny tmax

Circle

65

14.07 0.2335

2.177 0.227

302.4 0.267

610.1 0.251

58.14 0.2445

67.24 0.261

N = 11

49

14.29 0.2335

2.229 0.237

309.2 0.2585

616.2 0.2445

57.13 0.2485

71.77 0.250

τ = 0.0005

33

14.95 0.248

2.437 0.238

357.2 0.242

562.4 0.317

56.10 0.326

73.64 0.2515

17

13.63 0.159

7.528 0.390

380.1 0.318

539.3 0.350

44.62 0.147

168.5 0.3385

longitudinal

21

14.76 0.2495

2.665 0.239

343.3 0.249

573.0 0.317

56.49 0.321

71.46 0.2385

M = 33

6

15.66 0.247

2.864 0.274

332.4 0.2395

556.1 0.243

52.90 0.2505

76.76 0.2255

Time

0.00025

14.95 0.248

2.430 0.238

357.2 0.242

562.1 0.317

56.08 0.326

73.72 0.2515

N = 11

0.001

13.63 0.159

2.430 0.238

357.2 0.242

562.1 0.317

56.08 0.326

73.25 0.251

M = 33

0.002

13.63 0.159

2.447 0.236

355.4 0.240

565.3 0.314

56.19 0.326

72.59 0.254

τ = 0.0005

The calculation results yield the 2 · 10−3 step as a suitable one from the stability point of view of the diﬀerence scheme, as well as from the point of view of achieving the required accuracy. Therefore, the above described approach for the stability estimation of the nonlinear diﬀerence equation (3.145) creates a good approximation for the time step choice of the diﬀerence scheme. The final choice of the time step has been realized using the computational experiments. A stability of the diﬀerence scheme (3.147) has been carried out in reference [593]. It has been shown that the diﬀerence scheme is stable, when the following

3.3 Computational Algorithms

323

inequality is satisfied: h2z . (3.180) 2maxλT (T ) Because we consider the processes for which λT (T ) = λT0 (1 + nλ ∆T ), the maximal value of λT (T ) does not exceed 2λT0 . The estimation (3.180) serves for hz estimation, i.e. the nodes’ number along the thickness: τ≤

K<

1 √ , 2 τ

(3.181)

where τ denotes the step time, taken from the stability condition of the diﬀerence schema for equation (3.145). Convergence of the numerical solution and modelled problems. An exact investigation of the convergence of the numerical solution to the nonlinear problems is extremely diﬃcult. Therefore an accuracy estimation of the applied algorithm is given by the solutions to the modelled problems and by the results of the numerical experiment. The latter one is realized by computations for diﬀerent values of the initial problems, therefore a solution should also not depend on them. Practically, because of the finite dimensions of the computational boxes, this condition is formulated as a requirement of a weak solution dependence on the schemes parameters [310]. In the Table 3.2 the results of the solution to the above mentioned problem for diﬀerent parameters of the diﬀerence scheme are given. It can be easily concluded that a solution converges in relation to the space and time coordinates. As a modelled problem to the heat transfer equation we again consider the problem of the temperature field definition of an infinite plate with constant characteristics and with an initial temperature. The surface z = − 12 for t = 0 is attacked by the temperature T 0 , whereas the z = 12 surface is ideally heat isolated [118]. For the difference scheme of equation (3.147) the 11 nodes along the thickness have been used and the time step τ = 4 · 10−3 (λT = const = 1.0) has been applied. In the Fig. 3.5 a comparison of the temperature distribution along the thickness for diﬀerent time instants, obtained using numerical and analytical methods, are given. Good results coincidence is observed. In order to estimate the accuracy of the whole algorithm, the numerical and analytical solutions of the modelled problem are compared. In order to get the analytical solution the testing method of nonlinear unstationary termomechanics problem, described in [506], is used. Consider the temperature field T = T (x, y, z, t) = f1 (x) f2 (y) f3 (z) f4 (t) , 2 1 1 z− , f4 (t) = te−t , f1 (x) = 1.0, f3 = 2 2

324

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.5. The relative temperature distribution along the shell thickness z.

f2 (y) =

cos y, 0 ≤ y ≤ π2 . 0, π2 ≤ y ≤ π

(3.182)

We assume that the temperature material characteristics are known: E = E (T ) = 1 + nE T , α∗ = α∗ (T ) = 1 + nα T , λT = λT (T ) = 1 + nλ T .

(3.183)

Therefore, knowing (3.182), (3.183) we can define all functions A, B, D, MT , NT , in relation to the formula (3.147). From the boundary condition (3.141) we can define a value of the decreasing heat flow ∂T 1 cos yte−t z=− 1 = −λT · cos yte−t . (3.184) = λT z − qT (x, y, t) = λT 1 2 ∂z z=− 2 2 Therefore, the modelled problem for a nonlinear heat transfer equation is formulated. The second boundary condition corresponds to the heat isolation condition. The initial condition corresponds to a zero value initial temperature. We consider now the shell motion equations (3.135) and (3.136).The deflection function w and the forcing F have the forms w = w (x, y, t) = Aw sin ωt sin αx cos βy,

(3.185)

F = F (x, y, t) = AF sin ωt sin αx cos βy.

(3.186)

The given functions satisfy the boundary conditions (3.138) and (3.144) (α = π, β is integer).

3.3 Computational Algorithms

325

We substitute (3.185) and (3.186) to the equation (3.136) and as a result we get an equation which allows to obtain the AF coeﬃcient via the Aw one (because this equation is very complex it is not reported here). Furthermore, substituting (3.185) and (3.186) into equation (3.135), we get an equation for the load q = qmod (again it is not given here because of its complexity). Finally, substituting (3.185) into the initial conditions (3.142) we get w = 0, ∂w = Awω sin πx cos βy, t = 0. (3.187) ∂t Therefore, the process of an appropriate modelling problem of temperature stresses of a thin cylindrical shell including temperature dependence of the fundamental material characteristics (3.183) with boundary conditions (3.138) and (3.144), the initial conditions (3.187) with a force excitation action qmod and the heat flow qT (3.184) (the inertial surface is thermoisolated) has been defined and solved. In addition, the given problem has been solved numerically for the following geometrical characteristics (h = 10−3 , Rh = 200, Rl = 2.5) and with the following parameters of the diﬀerence scheme N = 11, M = 33, K = 11, τ = 0.001 by using the algorithm given in section 3.4.1 and comparing the obtained solution with the analytical ones (3.182), (3.185), (3.186) (for Aw = 1, β = 3, ω = 1). The results show that the numerical solution to the given modelled problem diﬀers from the analytical one no more than by 5%, in both the temperature field determination as well as the deflection and forcing functions. A comparison with other results. In the reference [379] a comparison of the results obtained using the proposed algorithm with the results obtained using the higher approximation method of Bubnov-Galerkin is carried out. As an example a cylindrical shell’s behaviour with a rectangular impulse action of the external, finite and infinite time duration load is analysed. Using the higher order approximations Bubnov-Galerkin method for the being sought solution is presented in the following form (w, F) =

M

N

Ai j (t) , Bi j (t) sin iπx cos jy .

(3.188)

i=1 j=1

Applying the Bubnov-Galerkin procedure to each of the equations (3.135) and (3.136) we get (from the linear equation in relation to the forcing function) an explicitly expressed relation for the coeﬃcients Bυz through the Ai j coeﬃcient. Substituting those coeﬃcients into the equation related to the deflection function, we get the following system of diﬀerential equations in relation to the unknown function Ai j (t) (the integrals of the Bubnov-Galerkin method have been calculated analytically). Each equation of this system represents the second order ordinary diﬀerential

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

equation. The obtained system is reduced to the dynamical (autonomous) system of diﬀerential equations, which is solved using the Runge-Kutta method. A convergence investigation of the methods with the following geometrical parameters Rl = 2.2, ky = 112.5 and boundary conditions related to the rolling support, with a rectangular impulse of the finite time duration uniformly distributed has been carried out. For a given loading type a being sought solution in the longtidunal direction is well approximated by a half-sinusoid. We can take N = 1 in the development (3.188), and in the finite diﬀerence method only 11 nodes of the longitudinal coordinate have been used. As a convergence condition of the methods we have used an overlapping of the previous approximation with the next one in relation to the functions w and F. In order to get a solution with the accuracy of 3% in (3.188) it is eﬃcient to take M = 17, whereas in the method of finite diﬀerences it is suﬃcient to take 33 nodes in the circle direction. In the case of the infinite length impulse in time the given approximations secure higher order solution accuracy.

Figure 3.6. Time histories of the relative shell deflections.

The integration step is taken numerically from the stability condition equal to a 0.0004 in the case of the Runge-Kutta method, whereas in the case of time diﬀerences - 0.001. In the Fig. 3.6 the dependencies wh (0.5; 0) − t for diﬀerent load intensities for T imp = 0.2, α = 0.5, α x = 1.0 are given. The curves 1-4 are obtained using the Bubnov-Galerkin method, and the curves 5-8 - the finite diﬀerence method. The curves 1 and 5 correspond to q1 = 0.1, the curves 2 and 6 - q1 = 0.15, the curves 3 and 7 - q1 = 0.17, the curves 4 and 8 q1 = 0.20. Observe that the solutions obtained using diﬀerent methods are close to each other.

3.3 Computational Algorithms

327

Comparison with experimental data. In order to verify the algorithm a comparison of the obtained numerical results with the experimental data, reported in the references [21], has been carried out. In this reference the problem of a cylindrical shell stability loaded by the impulse pressure has been considered. In order to realize the external dynamical pressure on the shell side surface, electromagnetic forces have been applied. A metalic shell has been situated in the impulse magnetic field, which has been realized by a condensator and an induction coil. In the metal of the shell some eddy points are induced, whose interactions with the coil magnetic fields lead to the occurrence of the forces which deformated the shell. The nonuniformity of the dynamical forces distribution has been achieved by a current intensity control. For the material having a low electrical conductivity a layer of copper has been covered to realize the required excitations.

Figure 3.7. The relative critical load of the cylindrical shell versus the angle ϕ.

In Fig. 3.7 the dependence between the relative critical load versus the load attacking angle for a shell made from rustless steel 12XI8H9 with the following geometrical sizes R = 25mm, l = 12mm, h = 0.2mm has been given. An impulse duration achieved 280µs, which corresponds to a quasistatical load. As the relative critical load value the relation of a critical pressure with nonuniform load to the critical pressure value with an uniform load obtained experimentally has been taken. As the critical experiment loads, the loads yielding the final deformations equal to the shell’s thickness have been taken. After each loading the shell’s profile has been registered. The registered deformation form of the surface has been used to define the number and amplitude values. The results of the investigations have been compared with the results of the numerical solutions using the algorithm developed

328

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

earlier. They were also compared with numerical solution, obtained for a statical influence [27] and recalculated for a given shell geometry with the help of the similarity relations [25]. The experimental results and the calculation results are in good agreement and they prove “a resonance” character of the critical load dependence versus the loading angle. A comparison of the wave-forms along the circle coordinate during the numerical calculations has shown a qualitative agreement with the profilegrams obtained during the laboratory experiment. Particularly, the maximum deflection occurs on the edge of the load zone. 3.3.4 Modified Relaxation Method The solutions of the stationary problems defining an equilibrium state are often found using a transitional process approach. A stationary state is a limit of the transitional approach. It is sometimes easier to achieve a direct numerical calculation of the equilibrium state instead of solving the nonlinear algebraic equations. In the literature, this approach is also called the method of dynamical relaxation, or the Richardson second order method. The fundaments of the method are given in reference [168]. Fedos’ev [202] was one of the first researchers who has applied this method to the stability problems of nonlinear deformable systems. This method has been widely used and developed among others by Krysko [369]. In the reference [364], a theoretical introduction to the nonlinear problems of plates equilibrium of the arbitrary contour from the point of view of Liapunov asymptotical stability theory has been considered. In the works [650, 651] the method convergence using initial equations of the diﬀerential form in a frame of the functional analysis is discussed. A wide review of possible variants of the dynamical relaxation method can be found in the work [317]. When the equations of that method are presented in the diﬀerence form, it can be shown, that the method is related to a class of iterational methods, the theory of which has been developed by Samarskiy [591, 592]. In this case the method corresponds to the relaxation iterational method. In the relaxation method a solution to the equations system in the form of partial derivatives can be reduced to the solution to the Cauchy problem for the diﬀerential equations. We briefly elaborate on the advantages of this method. From the mathematical point of view, the relaxation method can be considered as the iterational method of solutions to the nonlinear algebraic equations system, where each step in time creates a new approximation to an exact solution of the problem. The relaxation method, as the iterational one, possesses a high order accuracy. In addition, it does not possess the main drawback of the iterational methods, namely a high sensitivity to the initial approximation. This is related to the physical meaning of the equations, describing the plates and shells vibrations in the viscous medium. The additional advantage of the relaxation method is a very simple technique to obtain the multiple solutions of the static problems. The Cauchy problem essentially

3.3 Computational Algorithms

329

depends on the initial conditions. Providing various conditions, we can get various solutions to the problem. Besides, solving the homogeneous equations’ system using traditional methods (in order to get nontrivial solutions) we need to include the initial deviation: small transversal load, small curvature, or a small initial deflection. The introduction of those initial deformations always influences the solution. Solving the same problems using the relaxation method the initial deformations are expressed by the non-homogeneous initial conditions, whereas a change of the initial conditions does not influence the obtained statical solutions to the problem. One more advantage of the relaxation method is expressed by its simple computer realization. It happens so, since there exist many eﬀective algorithms to solve the Cauchy problem. A disadvantage is related to the fact that using this method only stable solutions are found. Therefore, the problem of an optimal continuum damping coeﬃcient choice is opened. We consider the method to find unstable solutions using an example of the nonlinear equation f (x) = 0 . (3.189) For this equation we can formulate two diﬀerent diﬀerential equation of the state stationary method (see also the monograph [43]) c ( x¨ + ε x˙) = f (x) ,

(3.190)

c ( x¨ + ε x˙) = − f (x) .

(3.191)

or In order to get all solutions to the equation (3.189) we need to solve both of the mentioned equations. Solving the equation (3.190) we get a half of the possible solutions of (3.189), stable in relation to equation (3.190). The unstable to that equations solutions are stable for the diﬀerential equation (3.191). Therefore, solving the equation (3.191), we can find the second part of the roots to the equation (3.189). To conclude, we can obtain all solutions to the equation (3.189). In practice, the algorithm can be realized in the following manner. Giving arbitrarily the initial approximation x0 and solving the diﬀerential equation (3.190), we get the solution x1 . Further, given an initial value x0 = x1 + δ and solving the equation (3.191) we can get the solution x2 . Then, again solving the solution (3.191) we can get the solution x2 . Then, again solving the equation (3.190) we get the third root of equation (3.189), an so on. As an example we consider the equation: x4 − 12x3 + 47x2 − 60x = 0 .

(3.192)

Knowing four real roots x1 = 0, x2 = 3, x3 = 4, x4 = 5, the relaxation method (RM) and the Newton method (NM) are applied for their estimation. In the Table 3.3 the obtained solutions of equations and the number of iterations required to achieve

330

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells Table 3.3. Numerical eﬃciency of Newton and relaxation methods Initial approximation

10

4.5

3.5

2.5

Newton’s method (NM)

5

3

5

5

Relaxation method (RM)

5

4

3

0

Number of iterations (MN)

10

9

12

7

Number of iterations (MR)

60

87

54

68

them are given. As it is clearly seen from the table, the relaxation method always finds, contrary to the Newton method, the stable root of the equation. Consider one more example for the roots determination using the relaxation method of the equation: arctg(x) = 0 , (3.193) having the root x = 0. It is known that for this equation the Newton’s method is converged to the root only if the initial approximation |x| < 1.39 holds. The relaxation method converges to the equation root for arbitrary initial approximations. Taking as the initial value x = 2, one gets the solution after 33 iterations. The results of solutions to the equations (3.192), (3.193) prove a high accuracy of the relaxation method. Therefore, with a very high probability we can believe in the obtained results related to plate and shells investigation with the longitudinal and transversal loads. The equations (3.190) and (3.191) can be presented in the form of one equation (3.190), but the mass coeﬃcient c can get both positive and negative values. We consider some characteristic properties of the modified relaxation method. Using this method, a term including damping of the surrounding continuum into the initial equations of motion is added. The equation (3.145) can be presented in the following form 2 2 ∂w τM ∂ w + A1 (w, F) = f , +ε (3.194) τT ∂t ∂t2 where ε is the coeﬃcient, including damping of the surrounding medium; A1 - the operator of the statical equation, including functions w and F; f - the function, characterizing the statical, force and temperature load. We rewrite the equation (3.194) into the diﬀerence form:

τM τT

2

wn+1 − wn−1 wn+1 − 2wn + wn−1 + A1 wn = f n . +ε 2 2τ τ

(3.195)

According to the reference [594], we reduce it to the canonical form of the three layered iterational scheme: B1 yK+1 = αK+1 (B1 − τK+1 A1 ) yK + (1 − αK+1 ) B1 yK−1 + αK+1 τK+1 f , where:

(3.196)

3.3 Computational Algorithms

B1 = 1 +

τT τM

2

ετ , 2

! τ2 1 +

αK+1 =

τ 2 "

ετ 2

1+

331

2

τ 2 ,

ετ 2

T

τM

T

τM

, y K = wn . 2 In order to estimate the iterational parameters τK+1 , αK+1 and the coeﬃcient ε and τ the following approach is used. During a solution to the nonlinear equations (operator A1 - nonlinear) the parameters of the corresponding linear equations [226] can serve as helpful ones (the analogical approach is applied for a stability investigation of the diﬀerence schemes). In this case, the speed of the stationary three-layered iteration process (3.196) convergence will be the highest, if we take τK+1 and αK+1 as the iterational parameters [594] of the form τK+1 =

2 , γ1 + γ2 √ 1− ξ ρ1 = √ , 1+ ξ

τK+1 = τ0 = αK+1 = α = 1 + ρ21 ,

ξ=

γ1 , γ2

(3.197)

where γ1 , γ2 - the constants of the energetic equivalence of the operator A1Λ : γ1 B1 ≤ A1Λ ≤ γ2 B1 ,

(3.198)

where A1Λ denotes a linear part of the A1 operator. The (3.198) inequality is true, since A and B operators are self-conjugated and positively defined in the real Hilbert space. As the A1Λ we take the bi-harmonic operator A2 Λ4 yK = DΛ4 yK . (3.199) A1Λ yK = D − B In the above A = 0, since we have the stationary temperature field. Therefore, in order to get the iteration parameters (3.197) we need to estimate the eigenvalues of the diﬀerence operator (because 1 + ετ 2 = const, E - the unit matrix). In order to estimate the maximum eigenvalue of this operator we use the Gershgorin’s theorem [77] n s , (3.200) = max |γ | ≤ γ 2

2max

i

ij

j=1

where si j are the elements of the matrix of the bi-harmonic operator. Using the matrix representation (3.172), we get for l = R, h x = hy : ⎛ 1 ⎞ ⎜⎜⎜2 ⎟⎟⎟ ⎟ 64 64 64 ⎜⎜⎜⎜ 2 ⎟ , γ2 < 4 · maxD = 4 · max ⎜⎜⎜ (1 + nE T ) z dz⎟⎟⎟⎟⎟ ≤ hx hx ⎜⎝ ⎟⎠ 12h4x − 12

(3.201)

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

where the last inequality is true since nE < 0. In order to estimate the lower boundary of the operator eigenvalue, the results of the reference [77] are applied

γ1 = δ21 + δ22 D,

δ1 =

πh x 4 sin2 , 2l x h2x

δ2 =

πhy 4 sin2 , 2ly h2y

(3.202)

where: δ1 , δ2 are the minimal eigenvalues of the Poisson’s operator Λ2 in the rectangular l x × ly . Using estimation for those eigenvalues [652] we get δ1 >

8 , l2x

δ2 >

8 . ly2

(3.203)

It yields for our case (l x = 1.0, ly = π) 1

2 γ1 ≥ 64 min D = 64

(1 + nE T max ) z2 dz =

64 (1 + nE T max ) . 12

(3.204)

− 12

Taking into account the estimations (3.201) and (3.204), from (3.196) the values ε and τ are obtained, corresponding to higher convergence of the iteration process using (3.197) (T max = 0, ττMT = 1 - elasticity problem): √ 3 2 2 h , ≈ τ= 2 x (γ1 + γ2 ) 12 ⎛ ⎞ ⎟ 2h2x 2 2 8 ⎜⎜⎜ 2⎟ ⎜ ε= − 1 ≈ √ ⎝αK+1 = , ξ = h x ⎟⎟⎠ . 2 τ αK+1 − 1 1 + h 3 y

(3.205)

Note that a step of the iteration process in the case of an elasticity problem overlaps with the step obtained from the stability condition of the linear problem (3.176). During the nonlinear problem considerations, as it has been shown by experiments of a wide class of computations, the iterational process stability is secured by decreasing of the step, defined by the formulas (3.205) in order of 1.2 to 2 times. A correction of the ε parameter for the nonlinear problems has been carried out using the numerical experiment in order to secure a vibrating process and to realize damping quickly enough. As an example of the relaxation method application the problem of deformation of the critical cylindrical load with the parameters Rl = 2.2, ky = 112.5 is investigated. During a solution to the problem the following diﬀerence scheme has been used: N = 11, M = 33, τ = 0.001. The shell has been loaded by a constant band pressure q1 = 0.1 with the wideness of 0.9 value. In the Fig. 3.8 the deflection dependencies of the loading zone versus time are given. The solid curve corresponds to the value ε = 0.0 (without damping), the dashed one with two dots - ε = 10, the dashed one with one dot - ε = 20 and the dashed line - ε = 40.

3.3 Computational Algorithms

333

Figure 3.8. The relative deflection time history in the loading center for diﬀerent damping values.

Figure 3.9. The relative deflection time history subjected to a continuous load action.

It should be noted that the described relaxation method allows to solve the dynamical problems with an account of the stress-deformation state in a frame of one algorithm. Solving the statical problem we can change the parameters of the diﬀerence scheme (decrease a time step and take ε = 0), and prolong a solution of the obtained dynamical problem (being in a frame of one mathematical model). From

334

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

the physical point of view it can be explained in the following manner. Namely, a statical load can be treated as a slowly applied dynamical load. In the Fig. 3.9 a solution to the dynamical behaviour of a cylindrical shell with a band load action q1 = 0.05, αy = 0.3 and with an infinite time duration with initially statical internal pressure q1stat = 0.06, with the following geometrical parameters: l R = 2.2, ky = 112.5, and the parameters ε = 20, N = 11, M = 33, τust = τ = 0.001, is presented. As the finishing computation process condition, a relative diﬀerence between the displacements values corresponding to the time moments of the neighbourhood maximum has been taken (for example 5% of the initially given values). In order to estimate this value and according to the recommendation of the work [310] we can take 0(h2 ), where h denotes the mesh step.

3.4 Dynamical Stability Loss with Ununiform Force Excitation 3.4.1 Criteria of Dynamical Stability Loss (A Review) A key point of dynamical stability problems lies in the choice of stability criterion. It is extremely important during the stability investigation of geometrical nonlinear shell governing equations, because a wrong choice of the criterion can delete the advantages of the nonlinear theory. In its most general form, a stability is defined as the system property related to a small deviation from the initial motion or equilibrium state. The dynamical criterion has been used by Lagrange during the investigation of the conservative system with a finite degrees-of-freedom system. An exact mathematical description of this criterion for a certain class of the system has been given by Liapunov [447]. The main idea of this criterion is: the initial motion form (or an equilibrium form) is stable, if small perturbations cause small deviation of those forms, which can be realized as arbitrary small for decreasing the perturbations. A system is called unstable, if arbitrary small perturbations cause finite deviations from the initial form. General concepts and methods used for the critical loads estimation applied to elastic systems can be classified (on the basis of Simitses work [621]) into the following groups: 1. The approach associated directly to the motion equations [132]. The motion equations are solved numerically for diﬀerent load parameters, and therefore, the system reaction is obtained. The load parameter, for which the largest time deviation occurs, is called a critical one. 2. The approach, using the phase energetic space [239, 264, 686]. The critical conditions are related to the phase space system characteristics and the suﬃcient stability (or instability) conditions are estimated (low or top stability boundaries). 3. The energetic approach [263, 619]. This approach is applicable only to the conservative system. Using it, one can estimate lower and top boundaries of the critical boundaries of the dynamical stability loss.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

335

The criterion, given by Budianskij and Roth [132] has been mainly applied to the statical load. It exhibits a stability of the limiting point type or a bifurcation with an unstable post-critical branch, for systems, which during a statical load exhibit a buckling form with stable post-critical branches (plates, columns). It is rather difficult using this criterion, to estimate a critical load [621]. Close to this criterion is the approach presented by Vol’mir [684], where as the dynamical criterion of unstability a rapid increase of the deflection caused by a small change of the load is taken. The given criterion is realized on the basis of the maximal displacement parameter against the parameter of the load’s amplitude (obtained using diﬀerent curves of the dynamical displacements). However, in a frame of this approach there exists non-uniqueness of the buckling moment identification (the beginning of a rapid deflection increase, the bifurcation point of deflection, the average meaning of the strong buckling interval, and so on). In the work of Stevens, Fulton [636] the load, defined by an intersection point of two tangents to the maximum deflection parameter versus load parameter, is taken as a critical one. The authors believe that this definition of the critical load has the advantage that with a strong change of the maximal deflection parameter the results are in good agreement with those obtained for the determination of the given dependence inflaction point. For a larger load concentration the results are closer to the real ones. A clearer character of the maximum deflection parameter versus the load parameter can be obtained using the logarithmic coordinates (see reference [310]). In this case, the discussed dependence is composed of two linear parts. Mente [472] suggested considering a maximal circled deflectional deformation as a reaction of the cylindrical shell. According to the author, the latter one is the most sensitive shell’s parameter, when the unsymmetric pressure is applied. As the stability boundary the author suggest to take a deflection point of dependence of the maximal deflection deformation versus the load parameter. In the work of Krysko and Fedorov [376] the criterion related to the stressdeformation state has been given. The dependencies of the deflection versus time are established for diﬀerent load parameters. The first maximum of this dependence serves to define the membrane stress. The load, for which a change of the given stress sign occurs, is called the critical one. In the work [404] another criterion is introduced. According to it, a critical dynamical load is defined by a condition in which a new equilibrium state occurs for a zero velocity. This criterion includes the following steps. In the beginning of the load’s action the inertial forces exceed the external load, then, after a transition through zero and after a change of the sign, they begin to approach the external load. The acceleration also changes its sign. It means that in a certain time moment the shell’s velocity becomes equal to zero, and therefore a rapid increase of deflection occurs. The moment when the shell velocity approaches zero is taken as the critical one. In the work [454] as an unstable motion of the construction the motion where an increase of the shell’s deflection corresponding to the reaction decrease is

336

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

recognized. On the unstable solution branches (in the statical problems) a displacement increase causes the decrease of the load and the membrane forces. As the instability criterion a sign change of the displacement derivatives and relations, which characterizes a decrease of the construction resistance in global, is taken. It is noted that this criterion is not related only to constraints on the material characteristics (elasticy or plasticity). In the work of [612] an interesting behaviour is reported. The deflection diagrams versus time for diﬀerent values of load parameters are considered. The time, necessary to achieve a maximal deflection amplitude, increases with the load’s increase. But, beginning from a certain load value, the described deflection value is achieved earlier in time. This load value is defined as the critical one. In some works the approach related to the transformation of a dynamical problem to the quasistatical one is presented. The former critical forces for the averaged shell’s surface are defined (including the inertial forces) from the dynamical relations. The buckling process is analysed from the point of view of statics [684]. This technique of “freezing in time” is usually used in practice for complex designs. This approach (in its general form) has not been proved yet. There exists a big diﬀerence in time intervals of the characteristics time load and the buckling time [76]. This approach has been used by Ekkes [72], who considered a conical spherical arc with uniformly distributed rectangular pressure infinite impulses. The dynamical stability criterion has been given by Darevskij [165], where two essential conditions are required. 1. When the time of the load’s increase from zero to the required value tends to infinity, a statical criterion can be used. 2. The criterion should have its meaning, when the load is acting on the finite time interval. The latter requirement is extremely important, because a usual approach to the stability problem of the elastic system with a dynamical load is related to its behaviour in time, which tends to infinite excitation. By excitation the author means a change of the initial conditions for a certain time value (a change of displacements’ and velocities’ field), which is diﬀerent from the shell’s displacement as a stiﬀ body. The introduced criterion is formulated in the following way: “...if there exists a time moment, when an elastic body can be perturbated without a change of all its velocities and accelerations (this perturbation is exhibited only by a change of displacement), then in this time instant a deformation of the elastic body is unstable. If not (when in the considered time instant we can realize the described perturbation) a deformation is refered to as a stable one”. Srubshtchik [634] has considered the changes of stable and unstable forms of the elastic shells by the investigation of statically unstable equilibrium states. The author defines a critical statical load. It corresponds to the smallest load, for which a transition from a fundamental equilibrium to an unstable post-critical state (with a potential energy equal to the potential energy of the initial time) is possible. In the work of Krysko [387] the dynamical criterion is introduced. Small vibrations of the shell’s constructions are analysed in the neighbourhood of a

3.4 Dynamical Stability Loss with Ununiform Force Excitation

337

configuration defined by the solution to the nonlinear equations for given time instants and load values. The initial shell configuration is defined by the solution to geometrically nonlinear motion equations of the conical shells theory. A solution to the obtained linearized system of diﬀerential equations with variable coeﬃcients is found using the Bubnov-Galerkin method. The being sought function has the form δw (x, y, t) = δA (t) Wi j (x, y) , δF (x, y, t) = δB (t) Fi j (x, y) . After transformations we get an ordinary linear homogeneous diﬀerential second order equation (in relation to time) of the form: δA¨ (t) + λδA (t) = 0, where λ denotes the second power of the free vibration frequency of the perturbated motion. For each time moment we have an uncountable set of the eigenfrequencies with a discrete spectrum. The integrals of that equation have the form:

√ δA (t) = C1 cos t λ + ϕ , λ > 0, δA (t) = C2 t + C3 , λ = 0, δA (t) = C4 et

√

−λ

+ C5 e−t

√ −λ

, λ < 0.

¯ are the constants defined by the initial conditions. The formulas Here Ci (i = 1.5) imply that if all of them are positive, then the form of motion is stable. Therefore, stability or instability of the shell’s state is obtained on each time step for a given load by the consideration of the corresponding spectrum of small shell’s vibration. As far as the estimation of the load-carrying design abilities is concerned the problem of dynamical instability can be generalized and considered as the problem of the fundamental parameters constants controlling its behaviour [621, 684]. The integral deviation characteristics serve as a control parameter in the case of a dynamical load. In the case of an impulse load, the time of the plastic deformations occurrence can be used. Kantor [303], during considerations of the spherical shell’s nonlinear vibrations took the value of k = 2 f /h as the stability criterion where f is the height of the shell. In the work of Tchuiko [653], as the dynamical instability criterion the occurrence of plastic deformations is taken. In some works [22, 457] it is assumed that a shell loses its stability, if a deflection reaches a value equal to the shell’s thickness. However, the latter criterion can not be applied in the case of an unsymmetric load [455]. In the work [310] it has been shown that a choice of the threshold deflection value should be realized in relation to the shell’s geometry, because it essentially depends on the geometrical parameters (in particular on R(h)). A value of the threshold deflection does not need to be equal to the shell’s thickness. The following approach is presented, when the threshold is defined by a certain deformation of the deflection fmax = max f (t). The fmax parameter is the functional defined on the set of functions

338

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

characterizing the load P = P(t). Choosing a certain class of loads, for example of the form

Pmax 1 − tt0 , t ≤ t0 P (t) = 0, t > t0 , it can be characterized by a finite number of parameters. In this case, we have Pmax and t0 or their combinations I = 12 Pmax t0 (a maximal load and an impulse). Then fmax will be the function of those parameters ( fmax = ϕ(Pmax , I)). Realizing the experiment series, when Pmax , I are varied, we can get the ϕ function as the family of the curves ϕ(Pmax , I) = C. Then, the area of the surface Pmax − I, covered by a family of lines for which C < fmax is safety (the other one is unsafety). This approach has been applied to the problems of the impact waves action on a construction. It should be noted that a critical value Pdyn is the functional on a set of all possible functions P(t). It means that Pdyn , defined by the numerical calculations, characterizes a shell only for a given loading type, but not in general. In some investigations of dynamical stability the load P(t) = Ctα (α ≥ 1) [455, 502] has been taken into account, although it is clear that an infinitely increasing load causes instability and construction damage. As it has been shown in the reference [310] a maximal deflection (and the corresponding deformation) plays a key role during a consideration of the carrying-load abilities. Therefore, the load considerations can be regarded as an illustration to the dynamical unstable behaviour [418]. In a general case, the stability problem with a combined load has been first solved by Papkovitch [529]. It has been shown that in the case of a statical combined loads a boundary line, separating the stable and unstable zones, can not be oriented by its convexity part into the unstable zone. The same property is also valid for a combined dynamical load [310]. In the reference [532] it has been shown that during a joint action of a statical load and a heat impact the above property is also valid. In order to define a load-carrying ability of thin shells with a combined load stability zone for the fundamental parameters of the load should be determined. None of the described above criteria can be treated as universal. Each of them possesses advantages and disadvantages. One of them is applied only to a special type of shells, the second one is diﬃcult to apply in order to solve the problem in a geometrically nonlinear formulation with an arbitrary load action. The third one defines a critical load only approximately. We consider an application of some dynamical criteria of the stability loss on the example of a cylindrical shell with the following geometrical parameters l/R = 2.2, ky = 112.5, with a rolling support and with an infinitely long external dynamical pressure uniformly distributed dy = 0.9 (see Fig. 3.10) In the Fig. 3.11 the dependencies of a relative deflection in the point (0.5, 0.0) versus time for diﬀerent load amplitudes are given. It should be noted that increasing the amplitude load a character of the given dependencies w/h(t) is significantly changed. A remarkable change occurs for q1 ∈ (0.125, 0.130). For the load smaller than q1 = 0.130 the dependence has a vibrating character close to the harmonic one.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

339

Figure 3.10. The cylindrical shell subjected to the uniform load action.

When q1 = 0.125 is exceeded, the vibration frequency change, and an essential increase of the deflection amplitude is observed. An analogical character is observed for the dependencies M x − t, My − t.

Figure 3.11. The relative deflections in the shell point (0.5; 0.0) for the diﬀerent values of the load amplitudes.

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

A given change in the cylindrical shell behaviour can be explained

using the following qualitative approach. Consider a typical dependence q1 wh , characterizing the instability of a singular point type with a statical load of the cylindrical shell (Fig. 3.12).

Figure 3.12. A typical dependents q1 (w/h).

For the loads, which do not exceed the q1cr value the vibrations close to the configuration defined by the load q1bcr (before critical) occur. Increasing the q1cr value a jump onto a new stable branch occurs with vibration about a new equilibrium configuration defined by q1acr (after critical) load. The given scheme fully agrees with the results given in reference [17]. In the latter work it is shown that for an infinite impulse pressure duration the critical dynamical load approaches the critical statical load. We compare the critical load values found using diﬀerent criteria. According to the Vol’mir [684] we build the dependence max wh − q1 (see Fig. 3.13). In what follows the critical load lies in the interval (0.125, 0.130). There exists an uncertainty during the critical load choice, because in the case of a non-uniform load (even for a low load level) the shell form is similar to the higher load value. Using the logarithmic coordinates, according to the recommendations given in reference [310], the critical load value belongs to the range given above (see Fig. 3.14) (using also the criterion given in [376]). According to the Shian-Soong-Roth criterion [508] the critical load corresponds to the value of q1 = 0.14 (see Fig. 3.11). We also note a disadvantage of applying the criterion wh < 1 for a given type of the load. During a solution to the problem with the help of the geometrical linear approach, a sudden deflection change is not observed. The only deflection increases proportional to the load is observed. Therefore, the dynamical stability loss criteria, given in the references [310, 376, 508, 684], are in good agreement with each other.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

341

Figure 3.13. The maximal relative shell deflection versus the load q1 .

Figure 3.14. The maximal relative shell deflection versus the load q1 in the logarithm scale.

In this work, a critical load is defined using the Vol’mir’s criterion [684]. One may (with suﬃcient accuracy) define a critical dynamical load for the nonlinear case. This criterion is very suitable for numerical calculations. It is analogical to the static criterion given by K´arm´an and Tsien [308]. During the investigation of w/h(t) dependence, the maximal deflections have been found in the zones lying in the neighbourhood of the loading zones. It is important for the computation time decrease. The critical load has been defined with the accuracy of 5%:

342

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

q1bkr ≤ q1kr ≤ q1acr , q1acr − q1bcr ≤ 0.05 . q1bcr As the characteristic properties of the stability loss, essential frequency changes in the dependencies wh − t, M x − t, My − t and a sign changes in the dependencies N x (t), Ny (t) are observed. In addition, also a limitation given by the plasticity condition has been taken into account. 3.4.2 Nonuniform Impulse External Pressure In this item we use the nonlinear geometrical approach to analyse the stress-strain state and the stability of a closed cylindrical shell of a circle cross section subjected to the impulse of the external pressure.

Figure 3.15. The relative deflection time history.

The stress-strain state and the dynamical critical loads are defined on the basis of the solution to the initial-boundary problems (3.145)–(3.154) using an algorithm of the finite diﬀerence method. The numerical investigation of convergence (see Table 3.2) has shown that to get critical loads it is suﬃcient to use the following diﬀerence mesh - 11 nodes along the longitudinal coordinate (on the half shell length), 33 nodes on the circled coordinate and the time step 0.001. During the investigation of stress-strain state we should increase the nodes number along the circled coordinate to 49.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

343

Figure 3.16. The stress function versus time.

Figure 3.17. The moment M x versus time.

Consider a shell with a rolling support and the following geometrical parameters: Rl = 2.2, ky = 112.5. The shell is subjected to an impulse of the external pressure with the infinite length uniformly distributed along the rectangular area in

344

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.18. The moment My versus time.

Figure 3.19. The force N x versus time.

the shell centre (see Fig. 3.10) ⎧ ⎪ ⎨ q0 , (x, y) ∈ s , t ∈ [0, ∞[ q (x, y, t) = ⎪ ⎩ 0, (x, y) ∈ s, t ∈ [0, ∞[ ,

(3.206)

where s = x0 ≤ x ≤ x0 + α x ∩ −αy ≤ y ≤ αy . Because for a given loading type the load is jumply varied in the longitudinal and circled directions, than the analyses of deflection and the stress-strain state in the mentioned directions are important.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

345

Figure 3.20. The force Ny versus time.

In the Figs. 3.15–3.29 the characteristic dependence of the relative deflection the forces function F, the moments M x , My and the forces N x , Ny against time, the longitudinal and circled coordinates for two values of the load: q1bcr = 0.4 and q1acr = 0.8 for the following loads parameters α x = 0.2, x0 = 0.4, αy = 0.1, are given. The load q1bcr = 0.4 corresponding to the pre-critical load, whereas the load q1acr = 0.8 - to the post-critical load. The function values in relation to the time are given in the points corresponding to the maximum values, whereas the function values in relation to the longitudinal and circle coordinates are given in the moments of achieving maximal values of the being sought functions and in the cross sections, where those maximums are achieved. The solid curve denotes function corresponding to the maximum for q1acr = 0.8, whereas the dashed curve denotes the function corresponding to the maximum for q1bcr = 0.4. The dashed curve with points denotes the function for q1acr = 0.8 corresponding to the achieved maximum for the pre-critical loads. The dashed curve with two points denote the analogical function with an action of the pre-critical load q1bcr = 0.4. Similarly to the case of a band loading, we have to mention an essential diﬀerence in a cylindrical shell behaviour before and after the initial loads for a given type of the load. It is expressed by the amplitude increase and the frequency change for the functions wh , M x , My , N x , Ny . The deflection form has a local dent character with boundaries and it is called the “wave-shaped” form. The wave-shapes of the shell in both longitudinal and circled directions for pre- and post-critical loads are similar to each other. On the other hand, the function change, characterizing the shell stress-deformation state in the circle direction essentially diﬀers from pre and post critical loads. The function wh , M x , My , N x , Ny distribution in relation t0 the longitudinal coordinate only slightly diﬀers for pre and post critical loads. w h,

346

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.21. The relative deflection distribution along the coordinate y for various loads and their durations.

To conclude, we have to mention that the investigated functions are localized in the loading zone and in the zones of the loading jumps.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

347

Figure 3.22. The stress function distribution along the coordinate y for diﬀerent load values and their action durations.

348

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.23. The distribution of the M x moment along the coordinate y for diﬀerent load values and their action durations.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

349

Figure 3.24. The moment My distribution along the variable y for diﬀerent loads and their action durations.

350

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.25. The force N x distribution along the variable y for diﬀerent loads and their action durations.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

351

Figure 3.26. The force Ny distribution along the variable y for diﬀerent loads and their action durations.

352

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.27. The stress function and relative deflection with distribution along x coordinate.

Figure 3.28. The moment M x and My distribution along x coordinate.

Figure 3.29. The forces N x and Ny distributions along x coordinate.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

353

Impulse-type load of a finite duration in time. A linear problem of unsymmetric deviation analysis of rollingly supported shell for the impulse loads is discussed in the reference by Yao [712]. An analytical solution of a given problem has been obtained by the development into two dimensional Fourier series and by application of the Laplace transformations. In this work, a comparison of the numerical and analytical (given in [712]) results is carried out. It shows a good agreement.

Figure 3.30. The shell vibrations and the unit load.

In the Figs. 3.30–3.32 the calculation results for the following shell geometry 1 5 = 2.2, ky = 112.5 and the load parameters α x = 18 , αy = 16 , x0 = 16 are given when the infinite duration impulses and the finite increasing time duration impulses are applied, respectively. It should be noted that the shell behaviour for the given loads essentially diﬀers from the behaviour of one mass system [712]. It is known that when an infinitely long in time impulse is applied to one mass system, it oscillates around the zero equilibrium position with a maximum amplitude equal to twiced displacement corresponding to the statical load. The results obtained in this work using a few degree of freedom systems show (see Fig. 3.30) that the vibrations occur around the certain (diﬀerent from zero) equilibrium. An application of the finite length impulse (see Fig. 3.31) indicates an essential diﬀerence in comparison to the one mass behaviour (in spite of that vibrations occur around the zero equilibrium position). When an impulse with a finite increasing time is applied, the maximum deflection amplitude decreases in comparison to the amplitude with an action of the impulse with a sudden increasing time (see Fig. 3.32). The obtained results indicate the necessity of a few degree of freedom approximation of the shell behaviour. l R

354

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.31. The shell vibrations and the rectangular load.

Figure 3.32. The shell vibrations and the combination of triangle and rectangular loads.

We consider now a behaviour of the ball-type supported shell with the essential dynamical loads which may lead to the stability loss. The investigated shell is loaded by band pressure with the α x = 1.0, αy = 0.5 (see Fig. 3.10). In the Fig. 3.33 a dependence of the critical dynamical load in relation to the impulse duration in the case of the pressure impulse stage ( Rl = 2.2, ky = 112.5) is given. It should be noted that with the impulse duration greater than 0.2 the

3.4 Dynamical Stability Loss with Ununiform Force Excitation

355

Figure 3.33. The critical load depends on the impulse duration.

Figure 3.34. The critical load impulse dependence on its duration.

magnitudes of the dynamical critical loads overlap for both the infinite length and the finite length impulses. In the Fig. 3.34 the latter dependence is build in the coordinates the critical load impulse versus its time duration (I = q1 T imp ). For the impulse duration smaller than 0.05 the shell’s behaviour is defined by the impulse magnitude. It is interesting to compare the considered range of the impulse duration with the characteristic time of the shell behaviour, which is equal to 1/4 of the free vibration

356

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.35. Typical loads: (a) rectangular-type; (b) triangle-type; (c) exponential-type.

Figure 3.36. The dependence q(I) in the logarithmic scale for diﬀerent loads.

period (the time of increase to the maximum value). Using the estimation of the shell characteristic time, given in Chapter 3.2.4, we have for a given geometry: T ≈ 0.0606 . 4 Then, using the approach of Anderson and Lindberg [17] the load with impulse duration smaller than τmech is called the impulse load (the impulse character is important), the load with impulse duration compared with the shell characteristic time is called the quasi-impulse load (the impulse, its amplitude, and time dependence is important), the load with impulse duration greater than the characteristic time τmech is called the quasistatical one (the load magnitude is important). The results of these approaches are in a full agreement with the results given in Figs. 3.33, 3.34. Consider now the applied load form influence on the stability loss curve for the exponential, the triangle and the rectangular type load in relations to time (see Fig. 3.35). τmech =

3.4 Dynamical Stability Loss with Ununiform Force Excitation

357

Figure 3.37. The functions q(I) in the logarithmic scale for various shell ky parameters.

Figure 3.38. The functions q(I) in the logarithmic scale for various shell l/R parameters.

In the Fig. 3.36 the critical stability loss curves are presented for the given loading forms for the following geometrical shell parameters: Rl = 2.2, ky = 112.5. The results are presented in a double logarithmic scale. The largest diﬀerence between the curves has achieved 12% and 6% for the triangle and the exponential type impulses; correspondingly, in relation to the curve corresponding to the rectangular type impulse. That diﬀerence does not belong to the essential ones, and therefore,

358

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.39. The dependence of the critical load value versus T 0 .

the loading form influence can be considered as the second order for the considered range of the load. In Figs. 3.37, 3.38 the critical curves for the diﬀerent values of the geometrical shell parameters are presented. As it has been concluded from the given results, a change of l/R and ky essentially causes a change of the asymptotes of the load and the impulse. As it has been already mentioned, in the case of a linear problem the impulse action yields the largest shell’s reaction. A similar behaviour is observed in the case of the nonlinear problems and loads yielding the stability loss. In the Fig. 3.39 the dependence of the critical dynamical load versus the infinite duration impulse is presented. It is clear that an infinite time of the impulse corresponds to the statical load action. In accordance with the result of Kabanov and Michajlov [291] as well as Kolometz [334] the statical load value (αy = 0.5) is equal to 0.176. Therefore, a change of the finite velocity impulse by the rapid increase improves the shell stability. Impulse type loads with infinite duration. We consider an influence of the geometrical loads parameters (x0 , α x , αy ) on the critical dynamical load magnitude for ball-type supported cylindrical shells loaded by the external load of the step impulse form with the infinite length duration in time ( Rl = 2.2, ky = 112.5). In the Fig. 3.40 the dependence of the critical load in relation to the position of the area load x0 (α x = 0.5, αy = 0.5) is reported. It follows that the smallest critical value corresponds to the loading area, located in the shell centre.

3.4 Dynamical Stability Loss with Ununiform Force Excitation

359

Figure 3.40. The critical load value dependence on its position.

Figure 3.41. The critical load versus the longitudinal dimension of the load surface.

The dependence of the critical dynamical load versus the longitudinal shell size α x of the load area (αy = 0.5) is presented in Fig. 3.41. A decrease of α x leads to the monotonic increase of the critical load value. One half-wave occurs along the shell length. The smallest critical load corresponds to the whole shell length (α x = 1.0). The critical dynamical load dependence versus the circle size αy of the loading area (α x = 1.0) has the resonance character (see Fig. 3.42). The extremum points correspond to the multiple of the waves length for post critical shell deflection of the shell loading size analogically to the statical load [28]. In order to illustrate the latter conclusion, the characteristics diagrams of the wave occurrence in the shell central cross section (x = 0.5) are presented for diﬀerent load area dimensions in the circle direction (α x = 1.0) (see Fig. 3.43). In the Fig. 3.42 the dependence of the critical dynamical load versus the circle size αy for the surface load is also presented (α x = 1.0). It should be noted that the initial curves for αy = 0.5 and α x = 0.5 are similar to each other, which allows to

360

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.42. The critical load dependence on the load dimension characterized via the circle sector αy .

analyse only a basic stability curve (for instance for α x = 1.0). All other values are defined by calculations using the similarity coeﬃcient. Analysis of the dependencies presented in the Figs. 3.40–3.42 show that the smallest critical dynamical load corresponds to the surface load covering the whole shell length in the longitudinal direction and equal to the length of the wave of a fully loaded shell (see Fig. 3.44). Consider now an influence of the shell geometrical parameters and the shell boundary conditions on the magnitude of the critical load of the cylindrical shell with the rectangular type load of the external pressure of infinite duration and uniformly distributed along αy (α x = 1.0). In the Fig. 3.45 the dependence of the critical dynamical load in relation to the relative length Rl (ky = 112.5, αy = 0.5) for diﬀerent boundary conditions is reported. The boundary conditions correspond to the ball type support (the solid curve), clamping (the dashed curve) and the hybrid-type support (dashed curve with the points). The latter one corresponds to the case when one edge has the ball-type support, and the second edge is clamped. The given dependence q1 ( Rl ) is constructed in the non-dimensional form (curves 1), as well as in the dimensional form (curves 2). As it has been seen in the Fig. 3.45 the boundary conditions influence visibly the

3.4 Dynamical Stability Loss with Ununiform Force Excitation

361

Figure 3.43. Buckling shell forms for diﬀerent loads.

dynamical load for relatively short shells Rl < 1.0. In the Fig. 3.46 the dependence of the relative deflection in the shell centre versus time ( Rl = 1.0, ky = 112.5, α x = 1.0, αy = 0.5, q1 = 0.150) for a given boundary condition is reported, which confirms the earlier formulated conclusions. It should be noted that a diﬀerence in the shell behaviour for diﬀerent boundary conditions in the point situated in the shell central cross section occurs after some time. The characteristics area in Fig. 3.45 in the neighbourhood of Rl = 1.8 is caused by a loading character (band-type load) and a multiple of the wave length for post

362

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.44. The critical shell load versus α x and αy .

critical shell deflection in relation to the range load. It is exhibited by wave-shapes characters in relation to the circle coordinate for Rl = 1.0, 1.9, 2.2 and 4.0. In order to estimate the wave length for post-critical load we can apply the circle with an uniform statical load caused by the external pressure of the cylindrical shell [238]. + 1 l 1 √4 1 h 2 , ω = 34 & 3− n = ' · πR R , 2 ω χ 12 1 − ν

3.4 Dynamical Stability Loss with Ununiform Force Excitation

363

Figure 3.45. The critical shell load versus the relative length l/R.

+ ) & ' l R 4 2 . χ = 12 1 − ν πR h

(3.207)

364

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.46. Time histories of the shell centre deflection.

Figure 3.47. The critical load versus αy for diﬀerent l/R.

The maximum values of the stability curve q1 ( Rl ) correspond to the wave length of the post critical deflection defined by the formula (3.207) and to the band load length (2αy = 1.0). It is achieved for Rl = 1.9, which is in a full agreement with the character of the curve presented in Fig. 3.45. In the Fig. 3.47 the dependencies of the critical dynamical load versus the circle size αy of the loading area (α x = 1.0) for the rollingly supported cylindrical shell (ky = 112.5) for diﬀerent values of the relative length Rl are given. The given

3.4 Dynamical Stability Loss with Ununiform Force Excitation

365

Figure 3.48. The critical shell load value versus the relative curvature ky .

dependencies have the resonance character, the maximum and minimum points estimation corresponds to the values obtained from the relation (3.207).

366

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

In the Fig. 3.48 the dependence of the critical dynamical load versus the relative curvature ky ( Rl ) for the rollingly supported cylindrical shell is given. The dependence is constructed for the non-dimensional form (curve 1), the dimensional form (curve 2) and for the band type external pressure (α x = 1.0, αy = 0.5) of the infinite length in time. Here also the wave-shapes form in the characteristics areas of the stable curves (ky = 50, 112.5, 170 and 300) are presented. It is seen that with an increase of the ky parameter the value of the critical load decreases.

3.5 Dynamical Stability Loss and Non-uniform Thermal Load The analysis of thin-walled structures with non-stationary dynamical thermal loads belongs to one of the most important problems during the design of the air planes, and also in some branches of chemical and nuclear technics [119, 237, 297, 469, 486, 515, 555]. Owing to experimental and theoretical investigations of cylindrical shells with non-uniform heating in the condition of heat impact, the thermal stresses can lead to the buckling of shells [352, 519]. In the last years, a special attention has been paid to the investigations of the construction behaviour in the condition of combined thermal load [31, 118, 422, 532]. As a result of heating, the mechanical and stiﬀness characteristics change, the non-uniform thermal field occurs, and also diﬀerent time durations of heating and elastic processes should be taken into account [26, 30, 134, 299, 373, 505]. In this section a dynamical behaviour of a cylindrical shell in the condition of non-uniform thermal load with inclusion of thermal changes of the material characteristics is investigated. The influence of the inertial parameter, local and gradient character of the temperature field thermosensitivity eﬀect are analysed. All of the calculations are carried out in a frame of halfly coupled theory of dynamical thermoelasticity without any account of the thermomechanical interaction of the heat transfer equation. 3.5.1 Thermal Field Computation We discuss some calculation peculiarities with an action of the heat load on the example of the heat impact action on a cylindrical shell ( Rl = 2.2, ky = 112.5). We assume that in the initial time the external surface of the cylindrical shell undergoes a temperature increase. The temperature distribution on the shell’s surface is governed by the equation T0 (3.208) T = T (x, y, t) =

4 . 1 + by In the above, T 0 denotes the initial temperature increase, b denotes the coeﬃcient of the local temperature distribution on the shell surface. The internal shell surface is thermally isolated. It is assumed that the shell edges are supported on cold diafragrams. As it has been shown (in section 3.1 during the considerations of dynamical

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

367

Figure 3.49. The relative temperature versus time.

Figure 3.50. The relative temperature distribution along the shell thickness for diﬀerent time instants.

thermoelastic problems of thin shells) the unstationary processes of the heat transfer along the shell thickness play a crucial role. A change of the thermal field configuration along the shell surface can be neglected in this case. Since the halfly coupled thermoelastic theory is considered, the temperature field and all the functions depending on the temperature are treated as known. However, in the case of nonlinear

368

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

problems of unstationary heat transfer, owing to a diﬃculty to obtain an analytical solution , the solutions of the temperature problem on each calculation step seem to be more eﬀective. In the Fig. 3.49 the relative temperature versus time is presented in the points of external (curve 1) and internal (curve 2) shell surface. In the Fig. 3.50 the temperature distribution along the shell thickness for diﬀerent time instants with an action of the heat impact (T 0 = 200, b = 3.0) on the thermosensitive cylindrical shell Rl = 2.2, ky = 112.5 is illustrated. Observe that the temperature distribution along the thickness has essentially nonlinear character up to the thickness overheating (τ = 1.0). Therefore, the hypothesis of the linear temperature distribution along the thickness can lead to wrong results. It is known that mechanical and heating material properties are changed with temperature, and that is why this behaviour should be included in the calculations. In this work the investigation of the influence of the temperature material dependence on the stress-stain shell state ( Rl = 200, b = 112.5) made from material AMg6 with an action of heating impact (T 0 = 200, b = 3.00) is carried out. The given dependencies E(T ), α∗ (T ) and λT (T ) for the considered material in the considered temperature range can be approximated by the following linear function: E (T ) = 1 − 1.10−3 T , α∗ (T ) = 1 + 7.810−4 T, λT (T ) = 1 + 4.810−4 T.

(3.209)

Table 3.4. Maximal values of the functions w, f , M x , Ny , ei for various E(T ), αT (T ), λT (T ) Variants of temperature dependence

w

f

−1.53 0.601 E(T ), αT (T ), λT (T ) 100% 100%

Mx 10.5 100%

Ny

ei

87.6 0.00672 100% 100%

E(T ), αT (T ), λT (0)

98

96

99

99

98

E(T ), αT (0), λT (T )

93

98

94

92

93

E(T ), αT (0), λT (0)

92

94

93

91

91

E(0), αT (T ), λT (T )

140

107

110

125

98

E(0), αT (T ), λT (0)

138

104

108

124

96

E(0), αT (0), λT (T )

130

106

104

117

91

E(0), αT (0), λT (0)

129

102

103

115

89

In the Table 3.4, the maximal values of the relative deflection, forces functions, moment M x , force Ny and the deformation intensivity ei for the mean shell surface (x = 0.5, y = 0) for diﬀerent temperature material characteristics are given. The other variants are reported in events related to this case. As it has been seen form

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

369

the Table 3.4, the largest amount for the characteristics wh , F, M x , Ny and ei for a given material is introduced by the linear thermal expansion coeﬃcient and the elasticity modulus against the temperature.

Figure 3.51. Time histories of the A coeﬃcient.

Figure 3.52. Time histories of the B coeﬃcient.

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3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

A consideration of the problem of the quasistatic temperature stresses, where the design inertia is neglected and the time is introduced as the parameter of the known stationary temperature distribution, belongs to one of the most important steps during the analysis of processes with the heating load.

Figure 3.53. Time histories of the D coeﬃcient.

Figure 3.54. Time histories of the thermal moment MT .

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

371

Figure 3.55. Time histories of the force NT .

Figure 3.56. Time histories of ∇2 MT .

In the Figs. 3.51–3.57 the characteristic coeﬃcients A, B, C, the temperatural moment MT and force NT , and their Laplace operators are presented in the point x = 0.5, y = 0 (for the case, when the material characteristics depend on the temperature (curve 1) and when they are constant (curve 2) during an action of the

372

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.57. Time histories of NT .

Figure 3.58. Time histories of the relative deflection w/h.

heating impact (T 0 = 200, b = 3.0)). The variation of those functions, as well as their space derivatives describe “slow” (quasistatical) shell behaviour. In section 3.1 the importance of full inclusion of all stiﬀness coeﬃcients during considerations of dynamical thermoelasticity problems (particularly, an account of A coeﬃcient in the governing equations) was pointed out.

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

373

Figure 3.59. Time history of the moment My .

In the Figs. 3.58–3.59 the relative deflection and the moment My versus time dependencies are given in the point x = 0.5, y = 0.0 for A = 0 (the solid curve) and A 0 (the dashed curve) under an action of the heat impact (T 0 = 200, b = 3.0). When A coeﬃcient is neglected, then the error of the moment My may achieve 20%. 3.5.2 Influence of Time, Shell Geometry and Load As the parameter representing inertia the coeﬃcient standing by the second time derivative of the normal deflection in the motion equation, equal to the ratio of the heating time τT to the characteristics time τm is taken. Since the inertial parameter is very large, the inertial forces vanish and the problem becomes a quasistatical one. When this parameter becomes very small, the inertial forces play a crucial role and the buckling does not occur [555]. In the Figs. 3.60–3.62 the dependencies of the relative deflection moment My , the force N x (correspondingly) calculated in the shell’s centre (x = 0.5, y = 0, Rl = 2.2, ky = 112.5) in the case of the heating impact (T 0 = 200, b = 3.0) for the four values of the inertial parameter: 0.01 (solid curve); 0.1 (the dashed curve); 1.0 (the dashed curve with a dot); 10.0 (the dashed curve with two dots) have been reported. Dynamical solution oscillates around the corresponding quasistatical solution. The oscillation period is inversly proportional to the inertial parameter. It can be shown, using the formula for the minimal frequency, obtained on the basis of conical shells theory with the inclusion of only normal component of the inertial forces [310] and the definition of the τT and τ M parameters, that

374

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.60. Time histories of the relative deflection for diﬀerent values of the inertial coefficient.

Figure 3.61. Time history of the moment My for diﬀerent values of the inertial coeﬃcient.

ωmin

τT = A0 = πk1 τM

+

R 1 τT , 34 & ' h 3 1 − ν2 τ M

(3.210)

where k1 denotes the coeﬃcient representing influence of the boundary conditions. In the Figs. 3.63–3.65 the relative deflection, the moment My and the force N x in the shell centre x = 0.5, y = 0, Rl = 2.2, ky = 112.5 are presented for diﬀerent

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

375

Figure 3.62. Time histories of N x for diﬀerent values of the inertial coeﬃcient.

Figure 3.63. Time histories of the relative deflection for diﬀerent values of thermal load durations.

values of the heating load duration: 1.0 (the solid curve); 0.03 (the dashed curve); 0.06 (the dashed curve with a dot); 0.09 (the dashed curve with two dots). In accordance with the earlier remarks, after finishing of the heat loading, the shell motion occurs in the neighbourhood of a certain equilibrium defined by the level of stationary temperature force NT . From this point of view, we can define a critical time of the heating load action.

376

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.64. Time histories of the My moment for diﬀerent thermal load time durations of t.

Figure 3.65. Time histories of the N x force for diﬀerent thermal load time durations of t.

It has been shown that more rapid loading increase leads to the enlargement of the shell dynamical deviation. In the Figs. 3.66–3.68 the dependencies of the relative deflection, the moment My and the Ny in the shell centre (x = 0.5, y = 0, Rl = 2.2, ky = 112.5) are presented for diﬀerent values of the heating load: 0.0 (the solid curve); 0.1 (the dashed curve); 0.2 (the dashed curve with one dot); 0.3 (the dashed curve with two dots).

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

377

Figure 3.66. Time histories of the relative deflection for diﬀerent values of thermal load increase.

Figure 3.67. Time histories of the moment My for diﬀerent values of thermal load increase.

Now some peculiarities of the thermosensitive cylindrical shell behaviour with the heating load and with rather complicated time dependence are illustrated. Assume that for the shell x = 0.5, y = 0, Rl = 2.2, ky = 112.5 the following heat flow is applied [568] q (x, y, t) = q1 (x) q2 (y) q3 (t) ,

378

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.68. Time histories of the force Ny for diﬀerent values of thermal load increase.

Figure 3.69. The maximal relative deflection versus t p = ε.

q1 (x) = 1.0,

0 < x < 1,

q2 (y) =

q0 , 1 + (by)4

⎧ 1 2t ⎪ ⎪ ⎪ , 0 ≤ t ≤ εt p ⎪ ⎪ ⎪ εt 2 ⎪ p ⎪ ⎪ ⎪ ⎨ 2t 1 q3 (t) = ⎪ ⎪ 2− , εt p < t ≤ εt p ⎪ ⎪ ⎪ εt p 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, εt p < t ≤ t p , ε ≤ 1.

(3.211)

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

379

Figure 3.70. The maximal shell deflection versus t p and ε.

Figure 3.71. The maximal shell deflection versus the local parameter b.

Observe that the given rule of the heat flow distribution is often used to model the influence of optical quantum generators. A frequency of the heat impulses occurrence plays a key role in the shell deviation for a given load. The largest deflection is achieved when the impulses period (t p ) overlaps with the largest period of free oscillations t p = ε = T osc = 0.24 (see Fig. 3.69).

380

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.72. The deflection shaper for diﬀerent b values.

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

381

Figure 3.73. The maximal relative deflections versus the parameter l/R.

Figure 3.74. The maximal relative deflections versus the parameter ky .

The decrease of the separated impulse ε duration for a given time t p (t p = 0.24), as well as the increase of t p for a fixed length of the impulse ε (ε = 0.24) yields the decrease of the shell reaction (see Fig. 3.70).

382

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

We investigate the influence of the order of heating non-uniformity on the cylindrical shell behaviour ( Rl = 2.2, ky = 112.5) in the condition of the heating impact (T 0 = 100). In the Fig. 3.71 the dependence of the maximum deflection versus the parameter of locality b is reported. There exists the resonance value of the local parameter (b = 3.2), which fully agrees with the results obtained for the non-uniform force load (see section 3.3). In the Fig. 3.72 the characteristic wave-forms for diﬀerent b values are presented. For small b values the area of large deflection is located in the reversed side to the load in the shape of a local indentation. With the increase of b the waveshapes forms are changed, and the local warp for large parameter values (b > 3.0) occur. The investigation of reaction dependence versus the geometrical shell parameters with the heat impact action (T 0 = 200, b = 3.0) has shown that relative maximal deflection increases with the increase of the relative length Rl for the fixed curvature parameter ky (ky = 112.5, see Fig. 3.73), as well as with the increase of ky for the fixed relative length parameter Rl ( Rl = 2.2, see Fig. 3.74). The dependencies exhibit oscillating character yielded by the deflection forms interactions, the nonuniformity load (b = 3.0) and by the eigenforms of deflection under an action of the uniform load. 3.5.3 Combined Static and Thermal Loads One of very frequently met loadings of the thin-walled structures is a combination of statical and dynamical loads. The investigation of cylindrical shell stability has been carried out by rather small amount of researchers [86, 343, 457, 625]. It should be noted that an essential influence on the shell buckling have the material properties, forms and loading conditions [326]. In this section sensitivity of the cylindrical shell to the initial imperfections, as well as stability of the initial statical loading (external and internal pressure, extension and compression) are considered. The stability zones are outlined for the mentioned loads combination. Sensitivity to the initial imperfections. Following the reference [472], we consider sensitivity to the initial imperfections of the form: (3.212) w0 = a0 sin πx cos ny . A change of dynamical behaviour caused by the amplitude a0 variations and the harmonics number n, have been analysed for the case of a loaded cylindrical shell ( Rl = 2.3, ky = 112.5) with the impulse of external dynamical pressure with the infinite time duration uniformly distributed along the band (α x = 1.0, αy = 0.3). In the Fig. 3.75 the dependence of the maximal radial dynamical displacement in relation to the value of an ideal shell displacement (a0 = 0) versus the amplitude of initial imperfection, related to the shell thickness, are presented. The solid curve

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

383

Figure 3.75. The relative initial imperfection amplitude versus the relative deflection for diﬀerent load values.

Figure 3.76. The relative initial imperfection amplitude versus the values.

li (li )a0 =0

for diﬀerent load

load corresponds to the pre-critical load level (q1 = 0.05), whereas the dashed curve (q1 = 0.1) and the dashed curve with a dot (q1 = 0.15) correspond to post critical load level.

384

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.77. The relative shell deflection versus the initial harmonic imperfection number.

Figure 3.78. The parameter

(li ) (li )a0 =0

versus the initial harmonic imperfection number.

The analogical dependence for the maximal deformation intensity for the averaged surface is reported in Fig. 3.76. From the comparison of results presented in Fig. 3.75 and 3.76 one concludes that the initial imperfections with the amplitudes over 1/10 of the shell thickness can essentially influence the shell behaviour. The initial imperfections have more important influence on the dynamical shell behaviour in the pre-critical area than on its behaviour in the post-critical zone. It

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

385

should be noted that the deflection function is more sensitive to the change of initial imperfections amplitude, than on the deformations intensity. The shell sensitivity to the form and the harmonics number n of the initial imperfections are shown in the Fig. 3.77 and 3.78 (displacements and deformations intensity on the middle surface, respectively). The maximal dynamical displacements and the deformations intensity for the imperfections amplitude equal to 1/10 of the shell thickness are related to the corresponding displacement and the deformation intensity for a perfect shell. They are given in relation to the harmonics number along a circle for: q1 = 0.05 (the dashed curve with a dot). It is seen that the shell dynamical behaviour with respect to geometric parameters ( Rl = 2.2, ky = 112.5) and a banding load (α x = 1.0, αy = 0.3) is most sensitive for the sixth and the seventh harmonics. The biggest sensitivity to the harmonic number of the initial imperfection is achieved in the region of critical load (q1 = 0.1). Note that the dynamical displacements are more sensitive to the imperfection that the deformations intensity. In the Fig. 3.77 the harmonics number in the case of the opposed imperfection orientations (a1 = −0.1) is given. An essential strengthening eﬀect occurred. Therefore, the initial imperfections, considered in the above and corresponding to the fundamental forms of stability loss of the shell (3.210) lead to an essential change of the shell behaviour. Combined load. Consider a dynamical stability loss of the cylindrical shell ( Rl = 2.2, ky = 112.5) with the initial imperfections (a0 = 0.05, n = 6) under an action of stepped pressure impulse of the infinite duration in time and uniformly distributed on the band (α x = 1.0, αy = 0.3). A solution to the problem of the combined static-dynamical load is found applying the algorithm described in section 3.2.4.

Figure 3.79. The stability threshold of the cylindrical shell initially compressed or extended via the force N.

386

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.80. The stability threshold of the cylindrical shell initially compressed or extended statically.

In the Fig. 3.79 a stability loss threshold for a cylindrical shell in the case of the impulse load on the initially compressed or stretched shell is presented. Observe that a given threshold for the introduced shell parameters and the load is practically a straight line for both compression and stretching processes. The analogical stability loss threshold is presented in Fig. 3.80 for the case of impulse load action on the initially loaded by uniform static external and internal pressure on the cylindrical shell. In order to estimate the stability loss, and the stability loss threshold for the mentioned cases the following approximation can be used R s = Rd = I,

(3.213)

where: R s is the relative statical load and Rd is the relative dynamical load. In the above, q s , qd denote the critical values of static (axial compression, external pressure) and dynamical load with the combined loading; q scr , qdcr are the same for separated load actions. Owing to the results presented in Fig. 3.79 and 3.80, both an internal pressure and axial stretching have a stabilizing influence on the shell behaviour. Their action leads to increase of the critical impulse value in comparison with the action of only one (separated) load. The reason is because during an occurrence of the initial internal pressure (axial stretching) the stretching ring type stresses appear, which oppose to the non-axial symmetric buckling. Besides, owing to the internal pressure (axial stretching) the amplitudes of the initial imperfections decrease. This observation is illustrated in Fig. 3.81. In Fig. 3.81 the results related to the eigen-vibrations of the cylindrical shell initially loaded by the static pressure are presented. The solid curve corresponds to the external statical pressure (q s = −0.05). The oscillations occur in the neighbourhood of the equilibrium state corresponding to the zone of initial imperfections (a0 = 0.05, n = 6).

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

387

Figure 3.81. Time histories of the relative deflections of the shell initially loaded statically.

Figure 3.82. The shell deflection yielded by an action of combined (static or dynamic) loads.

In the case of the external pressure for the point corresponded to the initial shell deflection the oscillations amplitude exceeds approximately two times the amplitude of that point for the internal pressure. The wave-shapes peculiarities are observed in the case of a combined loading (see Fig. 3.82). For a statical load (for instance, the external pressure) in the shell

388

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.83. Stability zones of the shell freely supported for the combined load.

with initial imperfections (a0 = 0.05, n = 6) one half-wave along the shell length and six waves in the circle direction occur (see Fig. 3.82a). The action of purely impulse load on the same shell yields only one local indentation, which corresponds to the loading zone (α x = 1.0, αy = 0.3) with five waves in the circle direction (see Fig. 3.82b). In the area of larger statical load values (q s = 0.15), the stability loss form with the combined load analogical to that of a purely dynamical load appears (the occurrence of the local indentation). However, the waves number in the circle direction (n = 6) corresponds to the waves number of the cylindrical shell stability loss for the purely statical load (see Fig. 3.82c). For small levels of the initial statical load (q s = 0.05) and with the internal pressure (q s = −0.05) the wraping form is similar to that of the purely impulse load action with the same waves numbers in the circle direction (see Fig. 3.82d). In the Fig. 3.83 the stable zone for a rollingly supported shell in the case of three load combinations: axial compression (stretching), external (internal) pressure and the impulse non-homogeneous load (αy = 0.3) are presented. The values of statical critical loading for a given zone of the internal imperfections (a0 = 0.05, n = 6) have been taken using a linear interpolation and extrapolation. The wall of stability prism in the area of positive values of statical loads (axial compression and external pressure) presents practically a flat surface. It is identical with the corresponding

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

389

Figure 3.84. The shell stability threshold under the heat impact.

Figure 3.85. Time histories of the relative shell centre deflection for diﬀerent load values.

stability curves build for q1 = 0.05 and for fixed values of both external pressure and axial compression. The stability zone of initially loaded (static external pressure) imperfect shell ( Rl = 2.2, ky = 112.5, a0 = 0.05, n = 6) in the case of heating impact action (b = 3.0) is presented in the Fig. 3.84. As in the case of force loading action, the stability boundary (for the given shell’s parameters and the load) has practically the form of a straight line and for its estimation the approximation (3.211) holds ( Rl = 2.2, ky = 112.5, a0 = 0.05, n = 6).

390

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.86. Time histories of the moment M x for diﬀerent shell loading.

Figure 3.87. Time histories of the moment My for diﬀerent shell loading.

Consider the case of a combined thermal-forced load. Assume that for the cylindrical thermo-sensitive shell initially loaded and statically (external pressure P = 0.05 and the compression force N = 10.0) simultaneously the impulse of external dynamical pressure (q1 = 0.05) uniformly distributed and the heating impact (T 0 = 200, b = 0.3) with finite length in time (T imp = 0.10) are applied. In the Figs. 3.85–3.89 the variations in time of the relative deflection, moments M x and My , forces N x and Ny (for x = 0.5, y = 0.0) are presented. The solid curve corresponds to the thermal force load, whereas the dashed curve corresponds to the

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

391

Figure 3.88. Time histories of the force N x for diﬀerent shell loading.

Figure 3.89. Time histories of the force Ny for diﬀerent shell loading.

purely forced load, and the dashed curve with one dot corresponds to the purely heating load. In the Figs. 3.90–3.92 the variations of deflection, forcing functions, moment My and the forces Ny along the longitudinal coordinate (y = 0.0) in relation to their maximal values are presented. Similarly, in the Figs. 3.93–3.95 the same variables are presented for the circle coordinate (x = 0.5) for diﬀerent input, in the instants of approaching a maximal

392

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.90. The relative shell deflection along x coordinate for diﬀerent shell loads.

Figure 3.91. The moment My distribution along x coordinate for diﬀerent shell loads.

deflection value in the case of thermal forced load action. The interaction of heating and forcing dynamical load yielded the shell complex motion. The heating and the forcing load act in the opposite manner. It yields the decrease of the absolute deflection values, moments and forces in the case of the thermal-forced load action.

3.5 Dynamical Stability Loss and Non-uniform Thermal Load

Figure 3.92. The N x force distribution along x coordinate for diﬀerent shell loads.

Figure 3.93. The relative shell deflection along x coordinate for diﬀerent shell loads.

393

394

3 Dynamical Behaviour and Stability of Closed Cylindrical Shells

Figure 3.94. The moment My distribution along the coordinate y.

Figure 3.95. The force N x distribution along the coordinate y.

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

First (section 4.1), a historical review devoted to dynamical behaviour and stability of rectangular shells is given. In section 4.2 computational algorithm to analysed diﬀerential equations with the associated boundary and initial thermoelastic conditions is proposed. Then finite diﬀerence equations are derived, and reliability of the results is verified. Application of stationary state method to statical and dynamical problems is addressed in section 4.2.4 including numerous graphs of various critical dependencies of cylindrical and spherical shells. Various vibrational processes and stability loss phenomena are illustrated and discussed. Stability of thin shallow shells with both transversal and heat loads are addressed in section 4.3. First, influence of heat stream intensity is analysed, and then stability of shells with signchangeable transversal load subject to stream action is studied. Influence of thermal and mechanical shallow shells characteristics on shell stressstrain state is investigated in section 4.4.3. Section 4.5 deals with stability of thin conical shells subject to both longitudinal load and heat flow. First, influence of boundary conditions and surrounding medium is studied. Then an influence of constant compressing load and heat stream is examined. Further analysis includes harmonic longitudinal load and heat stream action. Many practical conclusions are reported. Dynamical stability of flexurable conical shell with convection is addressed in section 4.5. After formulation of the problem, boundary and thermal fields conditions are given. Dynamical stability is analysed including influence of many factors. The study is associated with many graphs and applicationally oriented useful conclusions.

4.1 Introduction Historical Review The most developed investigations are concerned with the stress strain state of shells action in a stationary temperature field. As it has been pointed out by Kovalenko [354], the main attention has been paid to the problem of so called non-coupled thermal elasticity, where a heat transfer and deformations are considered independently (in the heat transfer equations a coupling term describing a relation between the temperature and deformation fields is neglected).

396

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

A key point of the thermoelasticity problems is related to definition of the corresponding temperature fields [451]. In the mentioned monographs a heat transfer behaviour and related problems are analysed very carefully. Many particular problems are solved analytically with explicitly given formulas to define the temperature fields. We describe now some of the recent achievements in the field of solution of stationary heat transfer equations and application of the obtained results to define the corresponding temperature stresses. Good results have been achieved during the investigation of plates and shells heating using a point heating source. In the work of Kulik and Prichodskoj [395], the temperature field, during one source heating, has been obtained analytically. If we have a system of heat energy sources, a solution to the transfer equations has not been obtained yet in the analytical form. In the some of other works, mentioned in the Chapter 3, the temperature fields and the stress strain states of plates and shells are characterised by a sudden increase of temperature and deflection, and also by a sudden jump of stresses in the place of the heat source. All of the mentioned components decrease rapidly in the neighbourhood of the heat source. Due to the mathematical diﬃculties of finding an exact analytical solution to the three dimensional problems many researchers assume that a temperature changes linearly along a thickness. Investigations of non-uniform heating of plates and shells have shown that deflection and stresses are maximal in the zones of high temperature. Moving away form the zone of maximal heating the stresses decrease, and the non-uniform heating leads to the wave-shape changes of shells’ surface and stresses. Sometimes, the authors use experimental temperature distribution data during the investigation of the thermo-elasticity problems. For instance, Murray [490] has used a linear temperature distribution along the thickness and the temperature is approximated using the experimental data. The works devoted to the problem of thermal elastic stationary stresses occurring in multilayer shells belong to a diﬀerent class. During solution to these problems both diﬃculties in the field of elasticity (complex equations), and in the field of the heat transfer (each layer possesses its own heat characteristics and temperature distribution is non-uniform and sudden) occurs. For these reasons, in all works an assumption about a linear (or constant) change of temperature and stresses in each layer is taken. The layers are deformable without a contact loss and without slipping. A thermal contact between the layers is assumed to be ideal. In the work of Borovinskij and Vasilenko [127] the temperature in the interval corresponding to each layer is assumed to be constant and a solution to the obtained diﬀerential equations is sought in the form of doubled trigonometric series. In the work of Kuz’min [403] the forces yielded by a temperature variation are changed by the equivalent mechanical forces and a purely elastic problem is solved. In the work of Shubrt [614] large deflections of a two layered circle plate attacked by the temperature and pressure are analysed. The fundamental equations have been introduced using a geometrical nonlinearity in relation to the relative deflection function and to the stresses.

4.1 Introduction

397

In the works [248, 511, 560] a behaviour of multi layered shells in the nonstationary temperature fields is analysed. The investigations carried out have shown that in the zone of boundary eﬀects a rapid change of deflection and moments is observed. Increasing the orthotropy parameters the deflection and the strains increase, and the strains on the layers’ boundaries are changed. The investigation of force and temperature load’s joint influence is of great importance. In the work of Butenko, Chalilov [140], using a variational principle on the minimum of full energy functional, the equations governing the behaviour of cylindrical shells with an arbitrary force and temperature loads have been obtained. As an example, a cylindrical panel with uniformly distributed normal and temperature loads has been analysed. The obtained results have shown that the temperature action decreases the displacement values, but causes the increase of the strains occurring in the investigated shell. The works devoted to the shells statical stability constitute a separate group. It appears that thin walled structures with shells elements can often lose their loading abilities due to either global or local statical stability loss. We consider some of the works focused on this topic of investigation. In the works of Grigorenko, Bechramova [247] thermostability analysis is carried out for a rollingly supported shallow spherical shell with external pressure. The governing diﬀerentional equations have been solved using the Bubnov-Galerkin method of higher orders. The investigations have shown that the temperature jump decreases the critical values of a normal external pressure. With temperature increase of the average surface the upper critical values increase, while the lower ones decrease. The number of equilibrium forms increases and for certain temperature values a behaviour of doubled clicks occurs. Among a few works related to the problems of nonlinear thermoelasticity of shallow shells we have to mention those by Kantor (see [302]). The analysis of the problems of statical thermoelasticity has shown that the majority of them (except for those focused on the investigation of strain state of shells heated by a heating source) the heat transfer equations are not solved. The temperature distribution along the surface and along the thickness has been a priori assumed. In the works, where the heat transfer equations are solved, only a convectional heat interaction with the neighbourhood medium due to the Newton’s rule or the heat source is considered. The solutions to the statical stability problems of shells, rectangular in plane, and obtained as a result of dynamical consideration with damping and with an action of uniformly distributed heat flow with a nonlinear temperature variation along the thickness is very rarely mentioned in the references. The second large direction of the thermoelasticity research is coupled with the problems of shells’ behaviour in the nonstationary temperature field from the point of view of both quasistatical and dynamical approaches. The results of nonstationary temperature fields and the corresponding stresses investigation are given in the monographs of Lebedev [420], Maizel [453], Meland and Parkus [471], Nowacki [515], Kovalenko [354], Podstrigatch and Kolano [550].

398

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

The investigation of dynamical thermoelasticity with a quick heating has been carried out in 1950 by Danilovskaya [160]. In his work the strain deformation state of a halfspace with its surface heating has been observed. The practical importance of dynamical behaviour investigation has been recently increased due to the application of thin walled elements in spacecrafts, and according to the atomic energy needs [208]. We review some of the recent works devoted to the investigation of shells behaviour in the nonstationary thermal fields from the point of view of quasistatical and dynamical thermoelasticity theories. For the quasistatical thermoelastic problems a solution to the nonstationary heat transfer equation is of most importance. It is, however, assumed that a change of the temperature field is small enough for the deformations on each time step, to be treated as the stationary ones. However, inertial terms can be neglected in the elasticity equations. A fundamental diﬃculty in solving the quasistatical problems lies in defining the corresponding unstationary temperature fields. This topic has attracted the attention of many researchers. Among others, we mention [249, 332, 485], where the heat transfer equation is solved. In the work [332] and analytical formula of nonstationar temperature field has been obtained. The calculations of the thermoelastic strains have exhibited the following features. The thermal stresses can occur suﬃciently higher than those caused by the internal pressure. The theoretical investigations are in good agreement with an experiment. The occurrence of the variable heat transfer coeﬃcient in time [249] leads to increase of the thermal stresses. Using a point heating source [245], which moves along a rib, the stresses on the edges go to infinity. The temperature field has been calculated in the work [250] for a plate, heated from the sides by a symmetric temperature field, and using two stages of the heating process: inertial (when the heat excitation does not achieve a half) and regular (when the whole thickness is heated). For each heating stage, the temperature fields have been obtained. The obtained results have shown that maximal stretching stresses occur in the plate’s center. In the reference [153] the analysis of the plates’ deflections with ball-type and clamped supports is carried out. The results of the work [153], on the example of a circle plate, damped on the edge, and heated in the circle area, shows that for a quasistatical approach the plate’s deflection is changed in time due to a linear rule. It also increases unboundently, when the heating time tends to infinity. We consider now the thermoelastic problems in the frame of dynamics. The design element vibrations caused by temperature have been firstly analysed by Boley and Barber in 1956 for beams [117] and in 1957 for plates [116]. They have defined a relation between the maximal dynamical and the maximal quasistatical deflection and pointed out the importance of the inertial eﬀects. A dynamical behaviour of shells in the nonstationar field is rather poorly investigated. Many mathematical diﬃculties occur. First of all, a coupled solution of the heat transfer and the thermoelastic equations should be found. A majority of the works of dynamical thermoelasticity are oriented on the investigation of plates and shells heated by an unstationary source.

4.1 Introduction

399

The main attention is paid to the definition of thermal fields. In the works [129, 331, 551] the analytical formulas defining the thermal field have been obtained in the case of: (i) a band type and a point type heat source [129]; (ii) movement of the heat source with constant speed [331]; (iii) heating of the source and external medium [551]. The investigations have shown that during dynamical heating the temperature increases faster than in the quasistatical case. The thermoplastic vibrations of freely supported plates with a heating stream action have been analysed in [354]. The solution to diﬀerential equations have been sought in the form of doubled trigonometric series, and an analytical expression for a temperature distribution is obtained. The fundamental relations of elasticity theory have been obtained taking the shear eﬀect into account. It has been shown that the dynamical deflection increases twice as much as the quasistatical one, and the bending vibrations occur in the vicinity of the quasistatical deflection axis. The investigations of vibrations of plates with diﬀerent thickness have illustrated a rule here that thinner plates exhibit a larger dynamical eﬀect. It has been assumed that the heating flow has been suddenly applied to the edge surface of a plate. When a successive heat flows intensity has increased, then the vibration amplitudes and maximal deflections have decreased. Besides, thermoelastic vibrations of plates have been investigated for a coupling between deformation and temperature fields. The coupling damps vibrations. For metalic plates the influence of coupling coeﬃcient is small, but in the case of polymers it cannot be neglected. The analysis of the quasistatical and dynamical thermoelastic problems has displayed many drawbacks. Particularly, many issues related to dynamical behaviour of shells in the temperature field (essentially under a heat stream action) are still open and need serious research. The problems of dynamical stability of flexible shallow shells, rectangular in plane and nonlinear, have not been successfully solved. A diﬀerent class of works is focused on investigations of stress strain state of plates (in linear and nonlinear cases) with the inclusion of thermal properties of materials (a linear heating expansion coeﬃcient and a surface film conductance) and elastic properties (Young modulus) versus a temperature variations. We are going to review some of the works of this class. In the references [95, 286, 687] only the elastic material characteristics versus the temperature have been analysed. Similar problems appear during the investigations of glassplastic shells. The theoretical investigations have been in a good agreement with an experiment [95]. The computations have shown that for high forcing loads, large velocities and small heating durations, the walls broke down due to shear. For moderate loads’ level and for large temperatures, buckling eﬀect (even for thick shells) is observed. During the investigation of shells stability, in the case of non-uniform heating [687] as the stability criterion a transition from the initial to the flexurable equilibrium form has been taken. It has turned out that the heating strongly influences the critical axial load, that is to say, it decreases the load carrying ability of a cylindrical shell. In the reference [333], contrary to the previous approaches, the investigations

400

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

have been carried out using only thermal characteristics. The investigations have shown that increasing both the heating time and the magnitude of the surface conductance, the stress decreases. In the considered references, the dependence of either thermal or mechanical material properties versus the temperature have been considered, but their influence on the deflections and stresses has not been included. This drawback has been omitted in the works [134, 245], where the investigations have been conducted for a simultaneous inclusion of thermal dependence (the coeﬃcient of linear thermal expansion αt ), and mechanical properties (Young modulus E). The investigations have been carried out assuming a linear temperature variation along a thickness. The following conclusions have been made: (i) deflection: a lack of αt and E dependence versus the temperature T reduces the deflection by 10 − 15%; (ii) static shells stability: E(T ) does not influence the critical temperature, whereas αt (T ) has an important influence in the case of a uniform heating; for a non-uniform heating the influence of αt (T ) decreases, whereas that of E(T ) increases; (iii) eigen vibrations: the influence of αt (T ) and E(T ) slightly decreases the free vibration frequency (up to 5%); (iv) simulteneous action of the transversal load and the temperature: for the some temperatures the temperature dependence influence of αt and E diminishes the deflection. The analysis of the considered works, when thermal and mechanical material characteristics depend on the temperature, indicates a necessity of deeper research in this field. In the majority of works, the temperature change versus thickness has been taken as linear, although in reference [245] a requirement of a nonlinear one is pointed out. A solution to the problem of dynamical stability of shells with a heat impact including nonlinear thermal and mechanical material characteristics, and with a nonlinear temperature variation along the thickness is rarely met in the literature. In this chapter some of the unclear problems from the field of thermoelastic shells behaviour have been considered. They include the stability of flexurable, conical, rectangular in plane shells with a heat impact, the influence of a heat flow with diﬀerent forcing loads’ action, and influence of thermal and mechanical material characteristics versus the temperature on the shells behaviour. The following notation is applied: F w u, v a, b, h K x , Ky γ E ν

- stress function, - normal displacement of a mean surface, - displacement in the x, y directions, respectively, - shell’s size in plane and its thickness, - curvatures of a shell in the x and y directions, correspondingly, - material’s gravity, - Young modulus, - Poisson’s coeﬃcient,

4.1 Introduction

401

g - gravity acceleration, ξ - medium damping coeﬃcient, α, λg , αt - temperature transfer coeﬃcient, thermal transfer coeﬃcient, and linear heat expansion coeﬃcient, c - specific heat capacity, q - heat flow density in a normal direction to the averaged surface, q s - transversal load intensity, P x , Py - longitudinal load intensity along x and y axes, respectively, t - the time, T - temperature, T 0 - initial temperature, θ = T − T 0 - temperature’s gradient, ei j - deformation tensor components for an arbitrary point, εi j - deformation tensor components for the averaged surface, eti j - components of temperature deformation tensor for an arbitrary point, σi j - components of stress tensor, T 1 , T 2 , S - forces in the averaged surface, N1 , N2 - shear forces, M1 , M2 , M12 - bending and torsional moments, ω - frequency of forced vibrations, ϕ - phase shift, q0s , P0x , P0y , q ∗ - critical values of the quantities mentioned in the above, q s0 , P x0 , Py0 - amplitude values of the quantities mentioned above. L(∗, †) =

∂2 ∗ ∂2 † ∂ 2 ∗ ∂2 † ∂ 2 ∗ ∂2 † , + 2 2 −2 2 2 ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x ∇2k ∗ = K x

∂2 ∗ ∂2 ∗ + K y ∂y2 ∂x2

The following non-dimensional quantities are used: x=

x y z w F K x a2 , y= , z= , w= , F= , , K = x a b h h h Eh3

Ky b2 a q s a2 b2 tα ξh2 , λ = , qs = , , τ = , ξ = h b α Eh4 h2 λg T abαt0 αt q abαt0

T= , α = , q = , λg = , t 2 2 αt0 λg0 h h λg0

Ky =

402

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

M=

Mab σab a2 b2 , σ = , P = P , P = P , x x y y Eh4 Eh4 Eh3 Eh3 ωh2 . ω= α

4.2 Algorithm 4.2.1 Diﬀerential Equations, Boundary and Initial Thermoelastic Conditions We consider a flexurable shallow shell, rectangular in plane. Its mean surface is related to the orthogonal system of coordinates x, y. The coordinate normal to the average surface is denoted by z, which is defined as positive to the curvature centre direction. The being sought diﬀerential equations are obtained using the following hypotheses: 1. The material is isotropic and satisfies the Hook’s principle. 2. The Kirchhoﬀ-Love hypothesis is valid. 3. A relation between the deformations and displacements is of the second power (it is widely used in the geometrically nonlinear theory of shells). 4. Thermal (the coeﬃcient of linear thermal expansion αt ) and mechanical (the Young’s modulus E) material characteristics are the functions of temperature. In a general case, the following excitations are applied to the shell: – transversal load of q s intensity, – longitudinal loads, P x and Py , – temperature field (a heat stream with q density or a surrounding medium with the T av temperature). The heat stream density is defined as the heat amount in the time unit and related to the surface unit (it is expressed in the unit [J/m2 s]). According to the Kirchhoﬀ-Love hypothesis, full shells deformations have the form – flexurable deformations e xx = ε xx + zκ xx , e xy = ε xy + zκ xy , (x ↔ y)

(4.1)

where:

∂2 w ∂2 w (x , (x ↔ y) , , ↔ y) , κ = −2 xy ∂x∂y ∂x2 – deformations of the middle surface 2 ∂u 1 ∂w + ε xx = − K x w, (x ↔ y, υ ↔ u) , ∂x 2 ∂x κ xx = −

ε xy =

∂u ∂υ ∂w ∂w + + + . ∂y ∂x ∂x ∂y

(4.2)

(4.3)

4.2 Algorithm

403

The thermoelastic stresses, according to the Hook’s principle and the DuhamelNeuman hypothesis on the elastic and the thermal stresses splitting, have the form σ xx =

% E $

e xx + νeyy − etxx + νetyy , (x ↔ y) , 2 1−ν σ xy =

(4.4)

Ee xy . (2 (1 + ν))

The deformations yielded only by a temperature field are as follows: etxx = αt θ (x, y, z, t) , (x ↔ y) ,

(4.5)

and e xx , eyy are the deformations occurring in a shell due to the external loads.

Figure 4.1. The loaded shell element.

Let us consider the shell’s element being in equilibrium with bending (M1 , M2 ) and the rotational (M12 ) moments action, the longitudinal (T 1 , T 2 ) and the transversal (N1 , N2 ) forces (Fig. 4.1). The forces, applied to the shell (taking into account thermal and mechanical characteristics (αt and E) versus temperature) are given by the following formula: h

2 T1 =

σ xx dz = − h2

Ft

St

ε xx + νεyy − κ xx + νκyy − Nt , 2 2 1−ν 1−ν

(4.6)

404

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load h

2 S1 = S2 = S =

Ft St ε xy − κ xy ; 2(1 + ν) 2(1 + ν)

σ xy dz = − h2

h

2 M12 =

σ xy zdz = − h2

St ε xy + Dt (1 − ν) κ xy ; (1 2 + ν)

h

2 M1 =

σ xx zdz = Dt κ xx + νκyy +

− h2

St

+ νε ε − Mt , xx yy 1 − ν2

(1 ↔ 2; x ↔ y) ; where:

h

2 Dt = − h2

h

Ez2 dz; Mt = 1 − ν2

2 − h2

h

αt θEz dz; S t = 1−ν

h

− h 2

Ezdz; − h2

h

2 Nt = −

2

αt θE dz; Ft = 1−ν

2 Edz.

(4.7)

− h2

The shell’s element equilibrium equations have the form (the second order terms are omitted) [684] ∂T 1 ∂S ∂T 2 ∂S + − K x N1 = 0; + − Ky N2 = 0; ∂x ∂y ∂y ∂x ∂M2 ∂M12 ∂M1 ∂M12 + − N1 = 0; + − N2 = 0; ∂x ∂y ∂y ∂x ∂w ∂ ∂N1 ∂N2 ∂w + + K x T 1 + Ky T 2 + T1 +S + ∂x ∂y ∂x ∂x ∂y ∂ ∂w ∂w γh ∂2 w ∂w T2 +S − + q s = 0. + ξ ∂y ∂y ∂x g ∂t2 ∂t

(4.8)

Using the fourth and fifth equations of (4.8) and substituting them to the first three equations of (4.8) one obtains ∂M1 ∂M12 ∂T 1 ∂S + − kx + = 0; (4.9) ∂x ∂y ∂x ∂y ∂M2 ∂M12 ∂T 2 ∂S + − Ky + = 0; (4.10) ∂x ∂x ∂y ∂x

4.2 Algorithm

405

∂2 M12 ∂2 M2 ∂T 1 ∂w ∂2 M1 + + + 2 + K x T 1 + Ky T 2 + 2 2 ∂x∂y ∂x ∂x ∂x ∂y ∂S ∂w ∂2 w ∂S ∂w ∂2 w + T 1 2 + 2S + + ∂x ∂y ∂x∂y ∂y ∂x ∂x ∂T 2 ∂w ∂2 w γh ∂2 w ∂w + T2 2 − + q s = 0. + ξ ∂y ∂y g ∂t2 ∂t ∂y T1

(4.11)

Substituting the expressions for forces and moments (4.6) to equations (4.9), (4.10) and (4.11), is obtained the following diﬀerential thermoelastic equations for displacements ∂A3x ∂ 1 ∂2 w ∂ [A1x A7x − A2x A5x ] + A1x A8 − (1 − ν) A2x = ; (4.12) ∂x ∂y 2 ∂x∂y ∂x % ∂A3y ∂ $ ∂ 1 ∂2 w A1y A8 − (1 − ν) A2y − ; (4.13) A1y A7y − A2y A5y = ∂x 2 ∂x∂y ∂y ∂y ∂Dt ∂Dt ∂2 Dt ∂2 Dt A4x − 2 A4y − 2 2 A5x − 2 2 A5y − ∂x ∂y ∂x ∂y ! 2

St ∂2 Dt ∂2 w (1 − ν) L (w, w) + ∇2 w + K x A5x + − 2 (1 − ν) 2 ∂x∂y ∂x∂y 1 − ν ∂3 u ∂3 υ ∂3 υ ∂w ∂w ∂3 u A4x + A4y + − + Ky A5y − 3 − ν 2 − ν ∂y ∂x ∂x ∂y ∂x∂y2 ∂y3 ∂x ∂S t ∂2 S t ∂S t ∂2 S t ∂2 S t A8 1 A A + + A + A + + 6x 6y 7x 7y ∂y ∂x∂y 1 − ν2 1 − ν2 ∂x ∂x2 ∂y2 ∂2 υ 1 ∂2 u ∂w ∂2 u 1 Ft + (1 + ν) + (1 − ν) 2 + K x A7x + Ky A7y + ∂x ∂x 2 ∂y∂x 2 1 − ν2 ∂y 2 1 − ν ∂2 υ ∂2 w ∂2 w ∂2 w ∂w ∂ υ 1 + ν ∂2 u + A8 + + 2 A7x + 2 A7y + + ∂y ∂y2 2 ∂y∂x 2 ∂x2 ∂y∂x ∂x ∂y ∂Ft ∂w ∂Ft ∂w 1 ∂w 1 ∂w 1 A A A A + (4.14) + + 7x 8 7y 8 ; 2 ∂y ∂y ∂y 2 ∂x 1 − ν2 ∂x ∂x

∂Nt ∂w ∂Nt ∂w 2 2 + − qs = ∇ Mt + Nt K x + Ky + ∇ wNt + ∂x ∂x ∂y ∂y ∂w γh ∂2 w , + ξ g ∂t2 ∂x −Dt ∇4 w − 2

where:

$

% A1x = (Ft − K x S t ) ; A2x = S t − 1 − ν2 K x Dt ; 3

∂ w ∂3 w 2 ; + A3x = (Nt − K x Mt ) 1 − ν ; A4x = ∂x3 ∂x∂y2

406

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

∂υ ∂2 w ∂2 w ∂u +ν ; = + ν 2 ; A7x = ∂x ∂y ∂x2 ∂y

A5x

2 ∂2 u ∂w ∂u ∂2 υ + (1 − ν) 2 − A5x ; A6x = 2 2 + (1 + ν) ∂x∂y ∂x ∂x ∂y ∂u ∂υ ∂w ∂w (1 − ν) ; (x ↔ y) , (υ ↔ u) . A8 = + + ∂y ∂x ∂x ∂y For a uniformly distributed temperature field the equations (4.12), (4.13) and (4.14) are simplified and take the form: 2 ∂ w 1 − ν ∂ 2 u 1 + ν ∂2 υ − A2y A4x = 0, + + (4.15) A1x 2 ∂y2 2 ∂x∂y ∂x2 1 − ν ∂ 2 u 1 + ν ∂2 u ∂2 υ + − A2y A4y = 0. + (4.16) A1y 2 ∂x2 2 ∂x∂y ∂y2

2 ∂3 u Dt ∇4 w − S t (1 − ν) L (w, w) + ∇2 w + K x A5x + Ky A5y − 3 − ∂x ∂3 u Ft $ ∂3 υ ∂w ∂w ∂3 υ A4x + A4y + + 3 + ν 2 −ν K x A7x + Ky A7y + 2 ∂x ∂y ∂x ∂y ∂x∂y ∂y 1 − ν2 ∂2 υ 1 ∂2 u ∂w ∂3 u 1 (1 + ν) + (1 − ν) + + ∂x ∂x2 2 ∂x∂y 2 ∂y2 ∂2 u 1 ∂2 υ ∂2 w ∂w ∂2 υ 1 (1 + ν) + (1 − ν) + + A7x + ∂y ∂y2 2 ∂x∂y 2 ∂x2 ∂x2 ∂2 w ∂3 w γg ∂2 w ∂w 2 A8 − Nt (K x + Ky ) + ∇ wNt − q s = . (4.17) A7y + +ξ ∂x∂y g ∂t2 ∂t ∂y2 The given systems of diﬀerential equations with variable coeﬃcients for a nonuniformely ((4.12), (4.13), (4.14)) and uniformely ((4.15), (4.16), (4.17)) distributed temperature field can be used for computation of flexurable shallow shells with boundary conditions for the displacements u, υ, w. We obtain a system of diﬀerential equations in the hybrid form in relation to the deflection functions and forces and satisfying the following relations: T1 =

∂2 F ∂2 F ∂2 F (y, (x, . − P = − P t) ; T t) ; S = − x 2 y ∂x∂y ∂y2 ∂x2

(4.18)

The deformations of the average surface are obtained from the equation (4.16) and they read 1 [T 1 − νT 2 + S κ xx + (1 − ν) Nt ] , ε xx = Ft S t K xy 2 (1 + ν) , (x ↔ y) . (4.19) ε xy = S+ Ft 2

4.2 Algorithm

407

The deformations ε xx , εyy and ε xy should should satisfy the following compatibility condition: 2 2 ∂w ∂ 2 w ∂2 w ∂2 w ∂2 w ∂2 ε xx ∂2 εyy ∂2 ε xy = + − − 2 2 − K x 2 − Ky 2 . (4.20) 2 2 ∂x∂y ∂x∂y ∂y ∂x ∂x ∂y ∂y ∂x Substituting the expression for the deformation of the mean surface to (4.20) and (4.6) and expressions for the moments (4.6) to (4.11), and owing to (4.16), we get a system of diﬀerential equations with variable coeﬃcients in relation to the forcing functions and deflection for the case of nonuniformely distributed temperature field along the surface, of the form ∂2 w ∂2 w + P + y ∂x2 ∂y2 ∂ ∂Dt γh ∂2 w ∂w ∂ ∂Dt 2 − ∇ Mt + B7x + B7y + +ξ g ∂t2 ∂x ∂x ∂x ∂y ∂y ∂Dt ∂B9y 1 ∂Dt ∂B9x ∂S t +2 − B10x + 2 B1x B2x + B1y B2y + 2 ∂x ∂y ∂y ∂y ∂x (1 − ν2 )Ft ∂ ∂S t ∂ ∂S t ∂S t B10y + 2 B2 + B2 − 2 ∂y ∂x ∂x ∂y ∂y $

1 B + B B + B + B 2 B S B B − 1x 5x 1y 5y t 3x 3y 1x 1y (1 − ν2 )Ft2 ∂Ft ∂

∂Ft ∂

B1x + B1y + B1x + B1y + B2 B6 − 2 ∂y ∂y ∂x ∂x ∂B2 ∂Ft ∂B2 ∂Ft − − B2 B4 − −2S t 3 ∂x ∂y ∂y ∂x 4 4 4 4 St ∂ F ∂ F ∂2 ∂2

∂ F ∂ F (S ) 2 2 2+ 2 + 3 − B − S t B7y + t 7x 3 2 2 Ft ∂x ∂y ∂x ∂y ∂x ∂y ∂x∂y ∂x ∂y 2 2 ∂ Px ∂ Px ∇2 Nt + = qs ; − (4.21) ∂x2 ∂y2 Dt ∇4 w − ∇2k F − L (w, F) + K x P x + Ky Py + P x

∂2 w 1 2

∂2 P x ∂2 Py ∂ 2 2 St + ∇ F + 1−ν ∇ Nt − − 2 −L (S t , w)+2ν Ft ∂x∂y ∂x∂y ∂y2 ∂x 1 1 ∂Ft ∂B1x ∂Ft ∂B1y ∇2k w + L (w, w) − 2 B1x B3y + B1y B3x − 2 + − 2 ∂y ∂x ∂x ∂y Ft ∂Ft ∂B2x ∂Ft ∂B2 + = 0, (4.22) 2 (1 + ν) B2 B4 − 3 ∂y ∂x ∂x ∂y where:

408

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

∂2 F ∂2 F ∂2 w B1x = − P x − ν 2 + νP4 − S t 2 + (1 − ν) Nt ; ∂y2 ∂x ∂x 2 2 2 ∂ F ∂ Ft St ∂ w 2 ∂2 F t − ; B3x = ; + B2 = ∂x∂y 2 ∂y∂x Ft ∂x2 ∂x2 2 ∂Ft ∂S t ∂ Ft 2 ∂Ft ∂Ft ∂Ft ∂S t ; B5x = +ν ; − B4 = Ft ∂y ∂x ∂x ∂x ∂y ∂y ∂y2 2 ∂Ft ∂S t ∂Ft ∂S t ∂ w ∂2 w + ; B7x = B6 = ; + ν ∂y ∂x ∂x ∂y ∂x2 ∂y2 2 ∂ St ∂2 S t ∂2 ω (1 ; B B8x = + ν = B + − ν) ; 9x 7x ∂x2 ∂y2 ∂y2 ∂B1y ∂B1x +ν ; (x ↔ y) . B10x = ∂x ∂y

For the uniformely distributed temperature field the system of equations (4.21), (4.22) can be suﬃciently simplified using the method proposed in reference [245], which in fact is valid for the plates. It is achieved when the coordinates’ system is chosen in a way to realize the condition S t ≡ 0. For a uniform temperature distribution along the surface, the changes of S t are visible only along the thickness and they are equal in all points of the surface. The coordinate surface is shifted in relation to the averaged surface of the amount of h h * 2 2 Ezdz Edz. (4.23) δ= − h2

− h2

In this case, the centre of the cross section gravity overlaps with the coordinates origin and the function S t is equal to zero. Taking into account (4.13) in the expressions for the functions Dt , Mt , Nt and Ft (4.7) the integration intervals are changed in the following way: h2 Dt = −h1

Ez2 dz; Mt = 1−ν

h2 −h1

αt θEz dz; Nt = 1−ν

h2 −h1

αt θE dz; Ft = 1−ν

h2 Edz;

(4.24)

−h1

where:

h h h1 = − + δ, h2 = − δ. 2 2 The equations (4.21) and (4.22) for the case of a uniformely distributed temperature field along the surface and for S t = 0 have the following form Dt ∇4 w − ∇2k F − L (w, F) + K x P x + Ky Py + P x

∂2 w + ∂x2

4.2 Algorithm

∂w ∂2 w γh ∂2 w + +ξ 2 2 g ∂t ∂t ∂y 2 2 1 ∂ P x ∂ Py ∇4 F − + ∇2k w + − Ft ∂y2 ∂x2 Py

409

= qs ;

(4.25)

1 L (w, w) = 0. 2

(4.26)

The obtained systems of diﬀerential equations with variable coeﬃcients for nonuniformely (4.21), (4.22) and uniformely (4.25), (4.26) distributed temperature field are used during the computations of flexurable shallow shells with boundary conditions for the force functions and deflections. Additionally we need to aﬃx the following heat transfer equation ∂2 T ∂2 T ∂2 T 1 ∂T = 2 + 2 + 2. α ∂t ∂z ∂x ∂y

(4.27)

In the real shells’ construction we can find diﬀerent types of supports, which yield diﬀerent mathematical models of boundary conditions. It should be pointed out that the type of support often determines the choice of the initial diﬀerential equations in relation to the displacements or in the hybrid form. We consider some of the boundary conditions. For the diﬀerential equations in relation to the deflection and forces functions we have the following boundary conditions. 1. Free support w = M1 = T 1 = S = 0 for x = 0, a w = M2 = T 2 = S = 0 for y = 0, b. 2. Slip support

∂w = 0; T 1 = εyy = 0 for x = 0, a ∂x ∂w = 0; T 2 = ε xx = 0 for y = 0, b. w = 0; ∂y

(4.28)

w = 0;

(4.29)

3. Hybrid type for x = 0, a

- free support (4.28),

for x = 0, b

- slip support (4.29).

(4.30)

4. Free support on the flexurable nonstretching (noncompressed) ribbs in their tangent surfaces w = M1 = T 1 = εyy = 0 for x = 0, a, w = M2 = T 2 = ε xx = 0 for y = 0, b.

(4.31)

The components occurring in the boundary conditions for the case of uniformely distributed temperature field have the form: 2 ∂ w ∂2 w + ν 2 − Mt ; M1 = −Dt ∂x2 ∂y

410

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

ε xx =

1 [T 1 − T 2 ν + (1 − ν) Nt ] ; (1 ↔ 2; x ↔ y) . Ft

(4.32)

For the system of diﬀerential equations, with respect displacements (u, υ, w) we have the following boundary conditions. 1. Ball-type u = υ = w = M1 = 0 for x = 0, a, u = υ = w = M2 = 0 for y = 0, b. 2. Clamped support

∂w = 0 for x = 0, a, ∂x ∂w = 0 for y = 0, b. u=υ=w= ∂y

(4.33)

u=υ=w=

(4.34)

3. Hybrid boundary condition for x = 0, a for x = 0, b

- rolling support (4.33), - clamped support (4.34).

(4.35)

In order to solve a dynamical problem we need to introduce the initial conditions for t = 0 ∂w = f2 (x, y) , (4.36) w = f1 (x, y) ; ∂t where: fi (x, y), (i = 1, 2) are the initial functions of the shell’s deflection and velocities of the deflection variations. The initial and boundary conditions for the heat transfer equation (4.27) have the following form Initial conditions: for t = 0, T = T 0 . Boundary conditions: I type; temperature distribution along the surface reads T (xk , t) = f (xk , t), where xk denotes the surface point and f (xk , t) is a given function. II type; density of the heat stream through the surface. III type; temperature of the surrounding medium T av and a convection rule ∂T (xk , t) λg = q (xk , t) , ∂n where “n” denotes the normal to the surface. IV type; temperature of the surrounding medium T av and heat transfer rule between the body’s surface and the medium −λg

∂T (xk , t) = αg [T (xk , t) − T av ] . ∂n

4.2 Algorithm

411

4.2.2 Finite Diﬀerence Equations In order to solve the obtained equations we use the method of finite diﬀerences with the approximation O(h2 ) along the space coordinates. This approach yields a reduction of the partial diﬀerential equations to the ordinary diﬀerential equations. Using the finite diﬀerence method the area has been covered by a rectangular mesh with a step a x = a/n, by = b/n along the x and y axes, correspondingly (n - denotes the number of the surface parts). The initial equations in relation to the forces and deflection functions have the following form for the i, j point (node)

Φ1i j (F) = Φ2i j (w) ,

(4.37)

d2 wi j dwi j = Φ3i j (w, F) , +ξ dt dt2

(4.38)

where: Φkij (k = 1, 2, 3) are the finite-diﬀerence operators for the i, j point, i, j = 1, 2, ..., N; N 2 - denotes the number of internal nodes (the equations number). The Φkij , for the systems (4.25), (4.26) have the following form

1 $ Fi−2, j − 8 Fi−1, j + Fi+1, j + Fi, j+1 + Fi, j−1 + 2 Fi+1, j+1 + Ft % Fi+1, j−1 + Fi−1, j+1 + Fi−1, j−1 + 20Fi j + Fi+2, j + Fi, j+2 + Fi, j−2 ; $

Φ2i j (w) = 2wi, j K x + Ky + 2wi, j + wi−1, j + wi+1, j + wi, j+1 + wi, j−1 −

wi, j−1 + wi, j+1 K x + wi−1, j + wi+1, j − Ky wi+1, j + wi−1, j + wi+1, j+1 − 2 wi, j+1 + wi−1, j−1 − wi−1, j+1 + P x(i, j+1) + P x(i, j−1) + Py(i−1, j) +

" Py(i+1, j) − 2 P x + Py ; ij

$ Φ3i j (w, F) = −Dt wi−2, j − 8 wi−1, j + wi+1, j + wi, j+1 + wi, j−1 +

2 wi+1, j+1 + wi+1, j−1 + wi−1, j+1 + wi−1, j−1 + 20wi, j + wi+2, j + wi, j+2 + %

wi, j−2 + K x Fi, j+1 − 2Fi, j + Fi, j−1 + Ky Fi−1, j − 2Fi, j + Fi, j−1 +

wi−1, j − 2wi, j + wi+1. j Fi, j+1 − 2Fi, j + Fi, j−1 +

Fi−1, j − 2Fi, j + Fi+1, j wi, j+1 − 2wi, j + wi, j−1 −

2 wi+1, j+1 − wi+1, j−1 + wi−1, j−1 − wi−1, j+1 ·

Fi+1, j+1 − Fi+1, j−1 + Fi−1, j−1 − Fi−1, j+1 − K x P x(i j) − Ky Py(i j) −

P x(i j) wi, j+1 − 2wi, j + wi, j−1 − Py(i j) wi−1, j − 2wi, j + wi+1, j + q s(i j) . Φ1i j (F) =

412

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

The obtained system of ordinary diﬀerential equations with respect to time has been solved using the Runge Kutta method. The coeﬃcient Dt and Ft are the functions of the shell temperature. The latter one changes in time. On each time step (for its fixed value) the coeﬃcients Dt and Ft are constant. The solutions to the governing diﬀerential equations in relation to the force and deflection functions (4.37) and (4.38) are defined using the following algorithm. 1. The values of deflection in each i, j point from the previous step (in the first step the initial deflection values are taken) and substituted to the right hand side of the linear algebraic equations (4.37). They are solved in relation to the Fi, j using the Gauss method. 2. The obtained values of the force functions Fi, j and the deflection in each i, j point (w is taken from the previous step) are substituted to the right hand side of the system (4.38) and the integration is carried out using the Runge Kutta method. As a result we get wi j values on each step of calculations. 3. The next step in time is made, and the described procedure is repeated. The initial equations for displacements in relation to u, υ, w for the i, j point (nodes) are as follows: (4.39) ψ1i j (u) = ψ2i j (υ, w) , ψ3i j (υ) = ψ2i j (u, w) , 2 d wi j dwi j = ψ5i j (u, υ, w) . +ξ dt dt2

(4.40) (4.41)

The ψkij (k = 1, 2, 3, 4, 5) denote the finite diﬀerence operators of the i, j point. The solutions to the diﬀerential equations in displacements are defined in the following way. 1. The deflections wi j and displacements υi j in each point i, j from the previous time step (in the first step the initial w, υ values are taken) are substituted to the right hand side of (4.39), which is solved in relation to ui j by applying the Gauss method. 2. The obtained values of displacements and deflection of each point (taken from the previous step) are substituted to the right hand side of equation (4.14), which is solved in relation to the displacement υi j . 3. The obtained values of displacements ui j and υi j , and also the deflection from the previous step are substituted to the right hand side of (4.41) and the integration using the Runge Kutta method is carried out. As a result, the deflections wi j are obtained. 4. The next step is made and the described algorithm is repeated. 4.2.3 Reliability of the Results We consider a plate stability subjected to heat impact action. In this case the heat transfer equation (4.27) has the form:

4.2 Algorithm

∂2 T 1 ∂T = 2. α ∂t ∂z

413

(4.42)

The initial conditions are defined by (4.27), whereas the boundary conditions have the form: λg ∂T h = q for z = , h ∂z 2 h ∂T = 0 for z = − . (4.43) ∂z 2 The equation (4.42) for the given initial (4.27) and boundary (4.43) conditions can be solved analytically, and the solution has the following non-dimensional form 12z2 + 12z − 1 q

− τ+ T = T0 + λg 24 ⎫ ! nπ " ⎪

⎪ 2 (−1)n 2 2 ⎬ (2z + 1) exp −n π τ ⎪ cos . (4.44) ⎪ 2 2 ⎭ 2 π n=1 n The initial equations (4.25) and (4.26) (for the temperature change due to time and thickness, and for the thermal mechanical characteristics independent of the temperature) accounting transversal load, have the following non-dimensional form 4 4 ∂4 w 1 −2 ∂ w 2∂ w − ∇2k F − L (w, F) + λ + 2 + λ & ' 12 1 − ν2 ∂x4 ∂x2 ∂y2 ∂y4 2 ∂ w ∂w = qs ; κ + ξ (4.45) ∂τ ∂τ2 λ−2

∂4 F 1 ∂4 F ∂4 F + 2 2 2 + λ2 4 + ∇2k w + L (w, w) = 0, 4 2 ∂x ∂x ∂y ∂y

where: κ=

(4.46)

a2 b2 γα gEh6

is the physical geometrical parameter, and α=

λ9 . cγ

We consider the stability of free supported shells (the boundary condition (4.28)). For a given boundary condition the temperature term is presented by the following analytical expression ⎤ ⎡

⎥ q ⎢⎢⎢⎢ 96 1 2 2 ⎥ exp −n π τ ⎥⎥⎥⎦ , n = 1, 3, 5, ... . Mt = ⎢1 − 4 24λg ⎣ π n=1 n4

414

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

The above series is convergent. The computations stopped when |an / an−1 | ≤ 10−18 has been achieved. A comparison of analytical and numerical results indicate the high accuracy of the latter one. In order to make an optimal choice of n, the computational experiment has been carried out. The problems have been solved for the fixed K x and Ky parameters when the surface is divided into n2 and 4n2 parts (the Runge principle for a step estimation has been applied). The computational experiment has shown that for (K x + Ky ) ≤ 72 the distribution for 12 parts is suﬃcient. In the Fig. 4.2 a variation of deflection in the centre for a shell with the curvuture K x = Ky = 24 and with damping (ξ = 3) is presented (boundary condition (4.31)). The curves 1, 2, 3 correspond to the transversal load q s = 150, 220, 250. The solid and dashed curves correspond to the solution obtained using the Bubnov-Galerkin method for 9-th and 4-th order approximations.

Figure 4.2. Time histories of the shell centre deflection.

The dashed curves with one and two dots correspond to the solutions obtained using the method of finite diﬀerences and the surface partition into 6 × 6 and 12 × 12 parts. The solutions in the pre-critical state, obtained using the method of finite diﬀerences with 6 × 6 and 12 × 12 surface parts have been in a good agreement with the 4-th and 9-th Bubnov Galerkin approximations. In the case of a post-critical state,

4.2 Algorithm

415

the partition into 6 parts is not suﬃcient, but the 12 part partition results overlap with the 9-th order approximation of the Bubnov-Galerkin method. An error of the maximal deflection deformation does not exceed 5%. For the integration in time, a standard program using the Runge-Kutta method with the automatic step control is used. The upper error is equal to 0.01. The time step for a given error in the pre-critical shell state (small vibration amplitudes) is changed in the interval 0.01–0.025. In the post-critical state, the step achieved 0.00125. In order to find the critical load value, we need to calculate 6–8 variants (in some cases up to 12 variants) of the vibration process. In order to obtain the graphs for 24 ≤ (K x + Ky ) ≤ 72 the considered interval is divided into four parts and after finding the critical loads the graphs are drawn in five points. For the problems of the heat flow density influence on the critical load values, the latter ones are obtained for the heat stream density of 0; 0.25; 0.5; 0.75; 1 in relation to its critical values. Using the found critical values of the forcing load, the stable zone are estimated in the three dimensional coordinate system). Analogically, the unstable zones are also derived. In order to estimate reliability of the obtained results we have taken K x = 0, Ky = 48 (the 4-th boundary condition (4.31)) using only a constant transversal load without damping (ξ = 0) and without the temperature. The obtained results are compared with the solutions to the initial diﬀerential equations using the higher approximations Bubnov-Galerkin method. The results of comparison of deflections and the critical loads are given in Table 4.1. Table 4.1. Comparison of numerical eﬃciency of finite diﬀerence and the Bubnov-Galerkin methods Parameters Deflection Critical load

Finite diﬀerence method 2.98 233

Bubnov-Galerkin method 3.10 230

Diﬀerence [%] -3.87 1.30

Now we analyse the dynamical criterions applied to the stability investigation of shells subjected heating impact action. As an example we consider a cylindrical panel with the parameters K x = 0, Ky = 48 without damping (ξ = 0) and for κ = 0.393. The latter value corresponds to the shell with the sizes a = b = 1m, and h = 0.0005m, made from steel 15XM with the following mechanical and heat characteristics: ν = 0.3, E = 2.1 · 103 MPa, αt = 1.2 · 10−5 grad−1 , c = 460J/kg · K, λg = 45.4W/(m · K). The results of investigations are given in Fig. 4.3. In order to determine the critical load value (in this case the heat stream density), in the reference [684] the w(τ) dependence is proposed (Fig. 4.3a), whereas in the reference [612] the graph (Fig. 4.3b) is used. Both of them give practically the same critical load values, i.e. they describe a stability loss. The criterion of Shian, Soong

416

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.3. Deflection (a) and load (b) time histories, as well as the critical stresses versus load (c) dependence.

and Roth gives higher values of the critical load in comparison with the Volmir’s criterion. During the critical load calculation the increase of q ∗ is observed. It can be explained in the following way. An inverse time corresponding to the load’s increase is

4.2 Algorithm

417

Figure 4.4. The dependencies of σ0x and q ∗ versus K x + Ky .

observed when the shell (according to the Volmir’s criterion) is already in the postcritical state and loses its stability. Besides, it has been observed that the stability loss instant is characterized by a sudden change of stresses, and also the membrane stresses change their sign (Fig. 4.3c). When the bifurcation occurs, one of the equilibrium forms is changed to another one. The shell’s layers, where the compression stresses occurred, now are stretched. The dependence σm (q ) can be more clearly and accurately used for the q ∗ definition than the well known criterion given in [612] and [684]. It is exhibited especially in two cases. First, in the case of (K x + Ky ) ≤ 36, when the q (τ) dependence is slightly broadening, and a sudden deflection increase during the transition from pre-critical state is not observed. Second, in the case of a sign variation of the force load, when the criterion of Shian, Soong and Roth is not satisfied. Even in this case the dependence σm (q ) is clearly expressed. However, for (K x + Ky ) ≤ 24 the critical load obtained from σm (q ) is not in a good agreement with the Shian, Soong and Roth criterion. For the given values (K x + Ky ) the shell behaves as a plate, a stability loss is not observed, and time duration required to reach the first deflection maximum in the centre decreases when the load is increased. Due to Shian, Soong and Roth criterion

418

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

for (K x + Ky ) ≤ 24, the critical load is equal to zero. However, when we define the q ∗ using the graph σm (q ), then for (K x + Ky ) ≤ 24 we find a small value of the heat stream. When the membrane stresses change their sign (only for a plate) we have q ∗ = 0. Therefore, the σm (q ) dependence can be used for (K x + Ky ) > 24 to obtain the critical load similarly to the known criterions ([612], [684]), whereas for the sign changeable load this is the most reliable criterion. Using the discussed criteria, the critical loads values are found for both cylindrical and spherical shells with thermal impacts. Diﬀerent factors’ influence on the critical values of the heat stream intensity are further investigated. The dependencies q ∗ (K x + Ky ) and σ0x (K x + Ky ) express the influence of the geometrical parameters K x and Ky on the critical values of the heat stream intensity and the corresponding full stress in the shell’s centre. The following notation is applied: 1 - corresponds to the spherical shells; 2 - corresponds to the cylindrical shells; q ∗ corresponds to the solid lines; σ0x - corresponds to the dashed lines. The critical values of q have been found in the interval from 24 to 72 of the (K x + Ky ) non-dimensional units. It has been found that increasing the heat stream intensity, the corresponding full stress rapidly increases. For example, the q and σ0x values for the spherical shell (K x + Ky ) = 72 increased 4.29 and 6.5 times, correspondingly, in comparison to the shell with (K x + Ky ) = 36. It should be noted that the pre-critical strain state is characterized by relatively high absolute stress values only with a heat impact action. For instance, the stresses occurring in the cylindrical panel with the parameters K x = 0, Ky = 72 with the heat impact action are 3.77 times higher than with the separated transversal load action. We consider the surface points movement on the example of a cylindrical shell with K x = 0, Ky = 48, q = 131 (Fig. 4.5 a, b) and a spherical shell with K x = Ky = 24, q = 132 (Fig. 4.5 c, d). Because the investigated problems are symmetric, the following maps have been constructed. The dependencies of a relatively equal deflection w(x, y)/w(0.5, 0.5) have been drawn for the shell’s quarter part (0 ≤ x ≤ 0.5, 0 ≤ y ≤ 0.5) for the time values, when the deflection centre achieves its maximum or minimum. In the case of a pre-critical state of cylindrical shells, the vibrational process is characterized by a maximum deflection point movement to the symmetry axes in the direction of the zero curvuture. Then, the latter one vibrates in the neighbourhood of the shell’s centre (Fig. 4.5 a, b). In the case of spherical shells, the maximal deflection point moves along the diagonal to the shell’s centre, and then the vibrations between the shell’s centre and each of the shell’s quadrant centres occur (Fig. 4.5 c, d). When the load approaches its critical value, the vibrations are damped, and in the post-critical state the maximal deflection point approaches its centre and does not change later. It should be noted that during vibrations in the pre-critical state it is impossible to find any periodical behaviour. It is particularly evident with the increase of (K x +Ky ). Some of the points move in the anti-phase directions.

4.2 Algorithm

419

Figure 4.5. The relative deflection of the cylindrical (a, b) and spherical (c, d) shells surface for λ = 1.

The vibrations are characterized by relatively small amplitudes and deflections. The axis around which the vibrations occur, is shifted from its initial zero state in the direction of the positive deflection values. The deflections do not achieve negative values. Increasing the values of (K x + Ky ) the vibrations’ amplitude decreases, the deflections increase and the vibration axis is shifted from the initial position. For instance, for the shell K x = Ky = 36 in comparison to the shell with K x = Ky = 18 the amplitude is decreased 1.22 times, and the deflection is increased 1.26 times and achieves 2.86 of the thickness value. In the post-critical state, for the loads close to the critical values, the vibrational quasi periodic process is observed (Fig. 4.3 a). Deflections of the post-critical state are a few times higher than in the pre-critical one. For example, for the cylindrical shell with K x = 0, Ky = 60 the deflection in the post-critical state is higher of 4.02 times and achieves the value of 12.14 thickness.

420

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

4.2.4 Stationary State Method We investigate the influence of a damping ξ on the dynamical critical load values of the heating stream in the way analogical to that of a transversal load action. In order to provide an eﬀective analysis of the damping influence, we define the ξ∗ quantity. It describes a minimal value of the heat stream acting statically which monotonously approaches the q ∗ value (the dependence q ∗ (ξ) is reported in Fig. 4.6c).

Figure 4.6. The dependencies q ∗ (K x + Ky ) (a), σ0x (ξ) (b) and q ∗ (ξ) (c). 1, 3 - curves corresponding to cylindrical shells; 2, 4 - curves corresponding to spherical shells.

The solid curves correspond to the values of the heat stream intensity and the stresses during the heat impact; the dashed curves correspond to the statical action of the heat flow. It has been shown that ξ∗ value depends on the shell’s geometry (K x , Ky parameters) and the heat stream intensity. Comparing the obtained drawing q ∗ (ξ) with the analogical one established for a transversal critical load (Fig. 88 in [369]) the following conclusion can be drawn. In the case of a heat stream action the medium damping exerts essentially smaller influence on the critical load than in the case of a transversal load. For instance, the critical value of a static heat stream intensity is of 1.04 q ∗ for (K x + Ky ) = 36. Increasing the (K x + Ky ) parameters the influence of the medium damping decreases. For example, for ξ = ξ∗ the values of q ∗ and σ0x for a spherical shell (K x + Ky ) = 36 have increased in comparison to the shell with (K x + Ky ) = 18 of 3.72 and 3.98 times, respectively. (As it has been shown earlier, in the case of a

4.2 Algorithm

421

heat impact the corresponding values have increased 4.29 and 6.5 times). For ξ = ξ∗ the curves of the critical values of heat stream for cylindrical and spherical shells (see 4 and 3 drawings exhibited in Fig. 4.6a) intersect for (K x + Ky ) = 42, whereas in the case of a heat impact (see 1 and 2 drawings of Fig. 4.6a) they intersect for (K x + Ky ) = 52. Thus, increasing the ξ parameter cylindrical shells become more stable against the heat impact. The heat stream critical values become higher in comparison to the spherical shells for smaller values of (K x + Ky ). The interval of (K x + Ky ), where the stresses occurring in the cylindrical shells are higher than those occurring in the spherical shells, is increased (the drawing σ0x (K x + Ky ) shown in Fig. 4.6b). Recall the definition of following non-dimensional parameter κ is introduced. κ=

a2 b2 γα . gEh6

(4.47)

The κ parameter depends on the physical-mechanical parameters γ, E, α and the geometrical ones a, b, h. Increasing this parameter we can get the critical values of the heat stream intensity either for the shells with other physical mechanical parameters, or the q ∗ values of thinner shells. On the basis of the obtained results, the q ∗ (κ) dependence has been drawn for κ ∈ (0.393, 5.36). The upper interval κ value threshold has occurred as the shell’s thickness limit (a/b = 400). The drawing q ∗ (κ(K x + Ky )) (Fig. 4.7) indicates that increasing κ the critical value of the heat stream intensity decreases and the cylindrical shells (the dashed curves) become more stable against the heat impact action then the spherical ones (the solid curves) in the whole interval of the (K x + Ky ) change. For example, for the shells with the parameters (K x + Ky ) = 36 the q ∗ values have decreased on amount of 23.18% and by 13.28% for cylindrical and spherical shells, respectively. For (K x + Ky ) > 52 a more rapid decrease of the critical heat stream intensity is observed for cylindrical shells with a simultaneous approach of the corresponding q ∗ for the cylindrical and spherical shells. For instance, for shells with (K x + Ky ) = 72 the q ∗ value is decreased on amount of 19.52% for spherical, and on amount of 29.92% for cylindrical shells. It is visible from the time histories (w(τ) in Fig. 4.8, that curves 1 correspond to κ = κ1 , whereas curves 2 to κ = κ2 ). Increasing of κ causes a rapid increase of the vibrations’ period and the maximal pre-critical deflection. The influence of the post-critical deflection is decreased. For the spherical shell with K x = Ky = 36 the pre-critical deflection is increased of amount of 1.55 times, the period of vibrations of amount of 1.66 times, and the post-critical deflection is decreased of amount of 1.15 times. The increase of κ is accompanied by the increase of the so called initial vibration time. The latter one is defined as the initial interval of time, when the shell’s state is in the neighbourhood of the undeformable one (w 0). Then, the shell’s deflection rapidly starts to oscillate. In the case of the spherical shell with K x = Ky = 36 the initial time of vibrations is increased from τ = 0.06 for κ = κ1 , up to τ = 0.24 for κ = κ2 .

422

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.7. The critical values of the heat stream versus κ and K x + Ky .

The λ parameter describes the extensibility of a shell. Its influence on the stability loss has been investigated (for λ ∈ (1, 1.5625) for a fixed κ = κ1 values). The found critical values of the heat stream on compared with the data obtained for the case of a transverse dynamical load action. As a result, the drawings q ∗ [λ, (K x + Ky )] and q0c [λ, (K x + Ky )] are obtained (Fig. 4.9, 4.10) for spherical shells (solid curves), for beam type cylindrical shells (dashed curves) and for arcal cylindrical shells (dashed dotted curves). Increasing λ from 1 to λ1 the increase of the critical values of the heat stream intensity and the dynamical transversal load for the arc cylindrical and spherical shells is observed. For the beam type cylindrical shells the q ∗ and σ0x decrease is observed. For instance, for the shells with (K x + Ky ) = 48 the q ∗ and σ0x values for λ = λ1 achieved 1.43 and 1.53 for the spherical shells; 2.5 and 2.84 - for the arcal cylindrical shells; 0.47 and 0.53 - for the beam type cylindrical shells. The

4.2 Algorithm

423

Figure 4.8. Shell vibrations for diﬀerent values of q .

given values are relative and they are related to q ∗ and q0s for λ = 1. Increasing the parameters (K x + Ky ) a progressive increase of q ∗ and σ0x values in relation to the corresponding values of λ = 1 for the spherical shells, and the decrease of q ∗ and σ0x for the beam type cylindrical shells are observed. For the arcal cylindrical shells a progressive increase of q ∗ and σ0x with the decrease of (K x + Ky ) occurs. From the considered examples, it can be also concluded that increasing λ from 1 to λ1 a qualitative similar behaviour of the critical heat stream intensity and the transversal dynamical load is observed. For λ = 1 for the shells with (K x + Ky ) ≤ 24 no stability loss is observed, and the loaded shells behave similarly to plates. Increasing λ for the cylindrical shells a stability loss is observed for small (K x + Ky ) values. For λ = λ1 only the loaded shells with (K x + Ky ) ≤ 7 behave as plates. Comparing the obtained results of the critical loads for cylindrical and spherical shells one concludes that with the increase of λ parameter the arcal cylindrical shells become more stable in comparison with the spherical shells and essentially more

424

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.9. The dependence q ∗ [λ, (K x +Ky )] for the cylindrical shells of the beam type (dashed curves), and for the spherical shells of the arc type (dashed curves with dots).

stable than the beam type cylindrical shells with both heat impact and dynamical transversal load actions. In order to analyse time histories of the shell’s points corresponding to the extensibility parameter λ variation the curves corresponding to the equal relative deflections for the plates with parameters (K x + Ky ) = 48 in the pre-critical state for λ = λ1 are drawn. It corresponds to the time instants when a centre deflection has achieved the maximum and the minimum (Fig. 4.10) (the curves of equal relative deflection for λ = 1 have been made earlier, see Fig. 4.5). The results are presented in Fig. 4.11 (a, b – spherical shells; c, d – arc type cylindrical shells; e, f – beam type cylindrical shells).

4.2 Algorithm

425

Figure 4.10. The dependence q0s [λ, (K x + Ky )] for the spherical shells (solid curves), for the cylindrical beam type shells (dashed curves) and for the spherical arcal type shells (dashed and dotted curves).

As it has been mentioned earlier, for the square spherical shells a point of maximal deflection is shifted from the shell’s corners to its centre. Further, the vibrations between the shell’s centre and the centres of each of the shell’s quadrants are observed. In the case of rectangular spherical shells (Fig. 4.11 a, b) the maximal deflection point is moved from the corners (perpendicularly to the extensibility direction) to the symmetry axis and then moves along it to the centre of each of the shell’s halves. Further, its vibration along the symmetry axis is observed. In the case of cylindrical shells, for rectangular arcal shells (Fig. 4.11 c, d) the vibration of the maximum deflection point is similar to that of rectangular spherical shells. The extensibility coeﬃcient does not influence the maximal deflection point of the beam type cylindrical shells (Fig. 4.11 e, f). It moves from the shell’s corners

426

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.11. The curves of equal relative deflection of the shell points for λ = λ1 .

to the symmetry axis along the zero curvuture direction, and then it vibrates along the symmetry axis in the neighbourhood of the shell’s centre. As it has been seen from the examples given in Fig. 4.11, increasing λ from 1 to λ1 the maximal deflection increases and approaches the shell’s deflection arrow. Also the increase of the central point vibration is observed. We compare the vibrational process of the shell’s surface points in the case of a heat impact with that of a transversal load action. For this purpose, we have drawn (analogically to the previous cases) the curves of equal relative deflections (Fig. 4.12). Comparing the drawings a qualitative analogy to the vibrational processes of the surface point is obvious. The vibrational process of the central point for λ = λ1 for two values of the parameters (K x + Ky ) equal to 48 (Fig. 4.13) and to 72 (Fig. 4.14) for the spherical

4.2 Algorithm

427

Figure 4.12. The curves of equal relative shell surface deflections under an action of transversal load: (a, b) K x = Ky = 24, q s = 120; (c, d) K x = 48, Ky = 0, q s = 270; (e, f) K x = 0, Ky = 48, q s = 40.

shells (Fig. 4.13b, 4.14b) and for the arcal cylindrical shells (Fig. 13 c, 14 c) are analysed. It can be concluded that a transition from pre- to post-critical states occurs without a sudden deflection jump. In order to detect the critical load values in most cases only the Shian, Soong and Roth criterion and the earlier described criterion of a sudden membrane stresses change during the transition from “pre” to “post” critical states are used. The vibrational process is a quasi periodic one. It is evidently expressed for the arcal cylindrical shells during decrease of (K x + Ky ) (Fig. 4.13c). The shell loses it stability after the initial vibrational process. Increasing the (K x + Ky ) parameters the amplitudes and the pre-critical deflections increase and the periods of vibrations increase, too. Aperiodicity and a strong vibrational process of the spherical and the arc cylindrical shells is caused by relatively large loads.

428

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.13. The shell centre vibrations for diﬀerent stream heat values q and the parameter K x + Ky = 48.

We now consider the shell’s behaviour in the heat flow with the boundary condition defined by (4.29) and (4.30). For the given boundary condition the temperature term Nt is represented by the following analytical expression Nt =

q

τ. λg

(4.48)

We consider the shell central point vibrations on the example of a spherical shell with the parameters (K x = Ky ) = 24 (Fig. 4.15). (The digits correspond to

4.2 Algorithm

429

Figure 4.14. The shell centre vibrations for diﬀerent stream heat values q and the parameter K x + Ky = 48.

the boundary conditions’ number). A deflection vibrates to the time axis and is unbounded when time approaches infinity (see (4.48)). The analogical conclusion has been obtained for a plate (the linear approach) in. A deflection increase occurs for the negative values. Therefore, a sudden configuration change cannot occur and the boundary condition 2 cannot cause stability loss. The same observation is true for the hybrid boundary conditions 3.

430

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.15. Vibrations of the spherical shell centre for diﬀerent q values.

In order to analyse the surface points vibrations of the shell with the boundary conditions 2nd and 3rd, the curves of relatively equal deflections (Fig. 4.16) are drawn. The upper part of the drawing corresponds to the shell with the 2nd boundary conditions, whereas the lower one corresponds to the 3rd boundary conditions. The drawings are made for the time instants τ = 0.2 and τ = 0.6. As the graphs show, the shells’ surface shape practically does not change in time for 2nd boundary conditions. The absolute values of deflection increase, but the relative deflection values practically do not change in time, and the maximal deflection is located in the shell’s centre. As far as the shell with the boundary conditions 3 are concerned, their surface forms are complicated and change in time. From the sides, where the temperature moments occur, a zone of positive deflections with the maximum deflection located on the symmetry axis close to the shell’s edges, is observed. The thermal loads action occurring on two other sides of the shell causes occurrence of a negative surface with the maximum located on the second symmetry axis in the centres of each of the shell’s halves. A zone of positive deflections slowly vanishes with time. We investigate the shell’s behaviour when a heating flow is applied to the external surface. In this case, the boundary conditions for the heat transfer equation (4.42) have the following form: h λG ∂T = q for z = − , h ∂z 2

4.2 Algorithm

431

Figure 4.16. The relative equivalent shell deflections for diﬀerent values of the parameter q .

h ∂T = 0 for z = . (4.49) ∂z 2 Applying the Laplace transformation the heat transfer equation (4.42) with the 1st order initial and boundary conditions (4.35) is solved. As a result we obtain the following algebraic equation for the thermal field 12z2 + 12z − 1 q

− T = T0 + τ+ λg 24 ⎫ ! nπ " ⎪

⎪ 2 (−1)n 2 2 ⎬ (2z − 1) exp −n π τ ⎪ cos . (4.50) ⎪ ⎭ 2 π2 n2 n=1

The thermal terms Mt , Nt for the boundary condition 1 and 2 have the form: ⎡ ⎤

⎥ q ⎢⎢⎢⎢ 96 1 ⎥⎥⎥ , n = 1, 3, 5, ..., 2 2 ⎥ exp −n π τ Mt = 1 − (4.51) ⎢ ⎦ 24λg ⎣ π4 n=1 n4

432

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.17. Shell centre vibrations for diﬀerent heat stream density q and the parameter K x + Ky = 48.

Nt =

q

τ. λg

(4.52)

One may observe that the thermal force Nt is the same for the application of the heat stream to either external or internal surfaces, and for both cases a stability loss does not occur. We consider the vibrations of surface points (considering the shell’s central point), when the heat flow is applied to the external surface with the boundary condition 1 (Fig. 4.17). The deflection vibrates in its negative direction, a sudden change of the shell’s form cannot occur and a stability loss is not observed. Therefore, during further investigations only the shells with rolling supports (1st boundary condition) and with a thermal flow action onto the internal shell’s surface will be considered. Only in this case, a stability loss due to the heating impact can occur.

4.3 Stability of Thin Shallow Shells 4.3.1 Influence of Heat Stream Intensity Earlier we have investigated the influence of the heat stream intensity on the dynamical critical transversal load values for cylindrical and spherical shells in order to find q0 with a successive increase of q from zero to its critical value for the fixed values of (K x + Ky ).

4.3 Stability of Thin Shallow Shells

433

Figure 4.18. The critical transversal loads values q0s (a) and the critical stress σ0x (b) versus q and K x + Ky .

In result we get σ0x [q , (K x + Ky )] (Fig. 4.18 a, b) for cylindrical (the dashed curves), and for spherical shells with the parameters 24 ≤ (K x + Ky ) ≤ 72. The dynamical transversal loads lying lower than the given critical values qc are considered as pre-critical. They are not dangerous for the constructions made from

434

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

shallow shells. Contrary, in the above lying zones correspond to the post-critical values which are dangerous for the constructions. Analyzing the drawing q0c [q , (K x + Ky )] one can conclude that the cylindrical shells for (K x + Ky ) > 52 can be more stable against the dynamical transversal load action than the spherical ones. The spherical shells are more stable against to the qc actions than the cylindrical ones. Increasing the heat stream intensity magnitude we have observed the time instant, when for the same (K x + Ky ) the spherical shells loose their stability for qc = 0 only because of the heat impact action. The cylindrical shells (for the same q ) are still able to carry the dynamical transversal load. For instance, for the shells with the parameters (K x + Ky ) = 72 for q = 0 the critical transversal load for the spherical shells is 1.09 times higher than for the cylindrical shells. For q = 155 an equality of the corresponding q0s is observed; for q = 297 (the critical value of the heat stream intensity for the spherical shells with the heat impact) the q s = 0 for the spherical shells, whereas for the cylindrical ones it is equal to 72 non-dimensional units. For q s = 0 the critical values of the heat stream intensity of the cylindrical shells is 1.34 times higher than for the spherical shells. Analysing the stresses occurring in the shells σ0x [q , (K x + Ky )] (the full stresses in the shell’s centre on its internal surface) one can conclude that the critical stresses with the increase of the heat stream intensity suddenly increase for the spherical shells for (K x + Ky ) < 65. For the cylindrical shells for arbitrary (K x + Ky ) they change their sign (from extension for q = 0 to compression for q = q ∗ (qc = 0)) for σ0x for the case of only a heat impact action. Compare the stress critical values modulus for shells with only the heat impact (q s = 0) and the transversal load (q = 0) one can conclude that the stress in the case of the heat impact for (K x + Ky ) < 42 for spherical and for (K x + Ky ) < 30 for cylindrical shells are slightly lower than the corresponding stresses in the case of the transversal load. Increasing (K x + Ky ) the σ0x is higher in the case of the heat impact than in the case of the transversal load. The diﬀerence rapidly increases with the increase of (K x + Ky ). For instance, if for the shells with (K x + Ky ) = 36 the critical full stresses yielded the heat impact are lower than the corresponding stresses. Yielded by the transversal load of amount of 1.23 times for the spherical shells and of amount of 1.6 times for the cylindrical shells. For the shells with the parameters (K x + Ky ) = 72 those stresses are higher of amount of 5.54 times for the spherical shells, and of amount of 3.77 times for the cylindrical shells. We compare the vibrational processes in pre-critical state w(τ) on the example of the shells with the parameters K x = Ky = 36 (Fig. 4.2a), K x = Ky = 18 (Fig. 4.19b). The dashed curves correspond to the central point vibrations yielded by the heat impact, whereas the solid curves correspond to the transversal load. With the increase of the heat stream intensity the amplitude of the vibrational process and of the maximal possible pre-critical deflection decreases. Increasing the (K x + Ky ) parameters the amplitude diﬀerence increases, whereas the deflection amplitude decreases.

4.3 Stability of Thin Shallow Shells

435

Figure 4.19. Vibrations w(τ) of the shells with parameters K x = Ky = 36 (a) and K x = Ky = 18 (b).

For the shell with K x = Ky = 18 the amplitude and deflection decrease of amount of 1.72 and of amount of 1.35 times, respectively. Simultaneously, the vibrational process axis moves away from the zero value of the deflections, and the negative deflection values vanish. We consider the surface points’ movement presented on the drawing of diﬀerent relative deflection curves on the example of the shells with the parameters (K x + Ky ) = 48 for the case of the heat impact (earlier graph in the Fig. 4.5) and for the case of the transversal load (Fig. 4.20) (the notations correspond to those presented in Fig. 4.5). Comparing both figures one can conclude that for only the transversal load the maximal deflection point achieves the shell’s centre. With the increase of the heat stream intensity the maximal deflection is shifted for the spherical shells to each of the shell’s quadrants. For the cylindrical shells it moves along the symmetry axis in the middle points directions of the half plate. More complicated surface form

436

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.20. The equal relative deflection curves for the case of transversal load action.

character with the occurrence of the negative deflection zones for the case of only the transversal load (Fig. 4.20) is observed. In the case of the heat impact the above mentioned shape vanishes (Fig. 4.5). 4.3.2 Shells with Transversal Load and Heat Flow The stability loss investigations of thin shallow shells being under both heat impact and signchangeable transversal load actions have been carried out in order to detect the unstable zone, bounded by upper and lower critical frequencies ωu and ωl . If the excitation frequency lies inside the unstable zone, then the shell with the signchangeable load action loses its stability, even though its amplitude is smaller than the critical load value qc = const (see the previous section). The signchangeable load is accumed as the harmonic one. qc = qc0 sin (ωt + φ) .

(4.53)

A character of the unstable zone is found during a gradual increase of the heat stream intensity from zero to its critical value for diﬀerent spherical shells with the parameters (K x + Ky ) equal to 36, 48 and 72 of the dimensionless units.

4.3 Stability of Thin Shallow Shells

437

Figure 4.21. Dependence of the shell vibration frequency on the harmonic excitation amplitude q s0 and the heat stream density q .

Figure 4.22. The dependences σ x (ω) and w(ω) for q = 0, q s0 = 95 (a) and q = 0.75q ∗ , q s0 = 40 (b).

We consider the fundamental rules using the example of the shell K x = Ky = 24. A change of the unstable zone accompanying the increase of the heat stream ω(q , qc0 ) (Fig. 4.21) is observed. (The A area denotes the dependence of the constantly transversal load critical value versus the heat stream intensity).

438

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

The obtained results lead to following conclusions. Increasing the heat stream intensity, the unstable zone becomes wider and it is shifted to the side of small loads. For instance, if for q = 0 a beginning of such zone is detected for qc0 = 49, then already for q = 3/4q ∗ a contact with the plane q s0 = 0 occurs. In this plane the dependence ω(q ) is drawn. It has been observed tracing a change of upper and lower critical frequencies, that the increase of q accompanies a sudden change of ωl to ωl = 0. It means that when the heat stream intensity achieves its critical value, the unstable zone is characterized by zero lower critical frequencies and relatively large upper critical frequencies, especially for q s0 ≥ 65.

Figure 4.23. The relatively equal shell deflections for q s0 = 65 and K x = Ky = 24.

We consider a stress strain state of the shell in the transition process of the excitation frequency through the unstable zone (σ x (ω), Fig. 4.22) for q = 0, q s0 = 95 (Fig. 4.22 a) and for q = 3/4q ∗ , q s0 = 40, (Fig. 4.22 b) (σ x - solid curves; ω dashed curves). The amplitude values of the transversal loads are (approximately) taken equal to q0s for q s = const. Analyzing the obtained results one can conclude that during the transition through the unstable zone a sudden deflection and stresses increase occurs. It is clearly visible on the low boundary. It should be noted that the frequencies corresponding to the maximal values of deflection and stresses are not

4.3 Stability of Thin Shallow Shells

439

Figure 4.24. The shell vibrations for q = 0 (a) and q = q ∗ (b).

equal. The frequency corresponding to the maximal stress is shifted to the ωl side, whereas that corresponding to the maximal deflection is shifted in to the side of ωu . Increasing the heat stream intensity the stresses change their direction: their minimal values occur but deflection increases (for both stable and unstable zones). For

440

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

instance, the stresses magnitude for ω = ωl is shifted of amount of 39 dimensionless units in the negative values direction, whereas the maximal magnitude σ x of amount of 45 dimensionless units in the unstable zone. A deflection for ω = ωl is increased of 1.25 times, and in the unstable zone of 1.15 times. For q = 0 a transition through the upper boundary of the unstable zone is characterized by a slight change of the stresses and deflection in comparison to the low boundary. With the increase of q

both transitions through upper and lower boundaries become similar.

Figure 4.25. Zones of unstability for K x + Ky = 36.

Tracing the motion of the surface points, equal relative deflection (Fig. 4.23) for the same amplitude transversal load values are drawn on the upper unstable zone for q = 0 and for q = q ∗ for the times instants, when the deflections achieve their maximal or minimal values. The vibrations of the surface points are characterized by large amplitudes and by the occurrence of the negative deflection in the shell’s centre. Increasing the heat flow intensity the amplitudes decrease, the negative deflections vanish, and the maximal deflection point begins to vibrate between the shell’s centre and the centres of its each quadrants. For example, in the considered case the amplitude (for q = q ∗ ) is decreased 5.77 times in comparison with q = 0, and the maximal deflection is decreased of amount of 2.09 times. The minimal deflection is shifted from the negative values to the positive ones. The described character of the surface points’ vibrations is clearly outlined in the drawing ω(τ) (Fig. 4.24 a, q = 0; Fig. 4.24 b, q = q ∗ ). It is made for the same case as the draw of equal relative deflections (Fig. 4.23). As an example, the vibration of

4.3 Stability of Thin Shallow Shells

441

Figure 4.26. Zones of unstability for K x + Ky = 72.

Figure 4.27. Dependence of the excitation frequency ω on the heat stream density q .

the shell’s central point inside and outside the unstable zone is analysed. Stability loss does not occur on the first wave of the vibrational process. Due to the lack of the heat stream the stability loss occurs on the third wave of the vibrational process. Increasing the heat stream together with the decrease of the vibration amplitude, the period of vibration decreases, and for q = q ∗ the stability loss occurs already on the fifth wave of the vibrational process. Increasing q also the increase of the diﬀerence

442

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.28. The dependence w(ω) for the shell with parameters K x = Ky = 18 (q = 0, q s0 = 55 (a) and q = 0.75q ∗ , q s0 = 20 (b)).

between pre and post critical deflection occurs, and the vibrational axis is shifted in the positive values direction. We consider now the influence of (K x + Ky ) parameters on the shape of the unstable zones and also on the upper and lower frequencies. In the Figs. 4.25, 4.26 the unstable zones for the boundary values of the variations of (K x + Ky ) equal to 36 and 72, are presented. Increasing the parameters K x , Ky diﬀerence of the ωu values and of the width zone for q = 0 and q = q ∗ is decreased. For example, for the shell with the parameters K x = Ky = 18 (for q = q ∗ and for the constant transversal load) ωu has increased 2 times, and the zone’s width increases of amount of 3.54 times. For the shell with parameters K x = Ky = 36 the corresponding increase amounts is of 1.25 and 1.5 times. The increase of K x , Ky is accompanied by the increase of the unstable zone width on the whole change interval of the heat flow intensity from 0 to its critical value. We compare the behaviour of lower and upper frequencies with the increase of q for diﬀerent parameters (K x + Ky ). For this purpose the dependencies ω(q ) for the shells K x = Ky = 18 (curve 1), K x = Ky = 24 (curve 2) and K x = Ky = 36 (curve 3) for the amplitude values of the signchangeable transversal load equal to the statical critical values, are carried out (Fig. 4.27).

4.3 Stability of Thin Shallow Shells

443

Figure 4.29. The dependence w(ω) and σ x (ω) for the shell with parameters K x = Ky = 36 (q = 0, q s0 = 275 (a) and q = 0.75q ∗ , q s0 = 100 (b)).

Increasing (K x + Ky ), yields upper frequencies sudden increase for q = 3/4q ∗ which is then more smooth and already for (K x +Ky ) = 72 the frequency ωu becomes constant. We analyse the stress-strain state during a transition of the excitation frequency through the unstable zone when the parameters (K x + Ky ) are increased. For this aim, the dependencies σ x (ω) and w(ω) have been drawn for the boundary values of the considered interval of the (K x + Ky ) changes for the shell K x = Ky = 18 (Fig. 4.28a, q = 0, q s0 = 55; Fig. 4.28 b, q = 3/4q ∗ , q s0 = 20), and for the shell (K x + Ky ) = 36 (Fig. 4.29 a, q = 0, q s0 = 275; Fig. 4.29 b, q = 3/4q ∗ , q s0 = 100). As the figures show, the increase of (K x + Ky ) is accompanied by the increase of the stresses and deflections. The influence of the heat flow intensity is larger. We consider the influence of a phase shift on the shape of the unstable zones. The drawings ω(q s0 , q ) (Fig. 4.30) and ω(q ) (the dashed curve in Fig. 4.27) for φ = π/2 for spherical shell K x = Ky = 24 are carried out. It can be concluded that for φ = π/2 the unstable zone is larger than for φ = 0 on the whole interval of the heat flow intensity change (specially for q > 3/4q0s ). The lower and upper frequency values, used for the construction of the mentioned dependence, are given in Table 4.2 for diﬀerent values of K x , Ky , for q /q ∗ = 0, 1/4, 1/2, 3/4, 1 with (Φ = π/2) and without (Φ = 0) phase shift.

444

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load Table 4.2. Lower (ωl ) and upper (ωu ) frequencies. φ 1

K x + Ky 2

q /q ∗ ) 3 0 0.25

36

0.5 0.75 1

0 0.25

0.5 0

48

0.75

1

0 48 0.25

q s0 4 75 55 30 75 55 30 75 55 20 75 55 20 75 55 22 5 120 95 65 120 95 65 50 120 95 65 40 120 95 65 40 20 120 95 65 40 20 5 300 275 240 190 300 275 240 190 100

ωl 5 6 10 18 0 5 13 0 0 12 0 0 5 0 0 0 0 10 13 16 1 8 13 18 0 0 6 13 0 0 0 3 11 0 0 0 0 0 0 8 8 13 20 5 8 12 18 23

ωu 6 25 23 18 25 23 20 25 25 16 28 23 17 48 46 25 15 32 30 22 32 30 24 21 27 26 25 21 27 27 27 23 20 53 51 50 36 27 20 48 48 43 40 48 46 43 40 33

4.3 Stability of Thin Shallow Shells 1

2

3 0.5

0.75 48

1 π/2 0

0.25

0.5 48

0.75

1

4 300 275 240 190 100 50 300 275 240 190 100 50 300 275 240 190 100 50 5 120 95 65 50 120 95 65 50 120 95 65 40 120 95 65 40 20 120 95 65 40 20 5

5 0 1 3 5 18 26 0 0 0 0 8 15 0 0 0 0 0 0 0 0 8 15 19 0 0 13 15 0 0 5 12 0 0 0 5 13 0 0 0 0 0 0

6 53 50 43 33 28 25 47 46 45 43 30 20 63 60 50 48 33 28 23 30 27 23 22 31 27 23 21 30 27 25 23 33 31 26 21 16 103 81 58 42 28 20

445

446

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.30. The dependence ω(q s0 , q ) for the spherical shell.

4.3.3 Influence of Thermal and Mechanical Characteristics We consider the stability of thin shallow shells, rectangular in plane, rollingly supported on their edges (see boundary condition (4.28)), with a heat impact and including the temperature dependence of E and αt . We investigate the latter influence of a dynamical load on the critical values, exhibited by the heat flow intensity, and the corresponding deflections and stresses for a series of cylindrical and spherical shells. The Young’s modulus and the linear expansion coeﬃcient are the temperature functions of the form: (4.54) E = E 0 KE , α = α0t Kα ,

(4.55)

where E 0 and α0t correspond to zero temperature, and KE , Kα are the non-dimensional temperature functions. They are obtained by E and αt n-th order polynomial approximation for each material. The shell temperature is defined via the formula (4.44). The series of the expression (4.44) is converged. The computations are prolonged to reach the condition |an / an−1 | ≤ 10−8 . In the initial time (τ = 0) 397 series terms are taken. However, already for τ = 0.01, only 19 terms are needed and the terms number is decreased with increase of time τ. The fundamental equations (4.25) and (4.26) accounting (4.44), have the following form 2 4 4 ∂4 w ∂ w ∂w −2 ∂ w 2∂ w 2 = q s , (4.56) − ∇k F − L (w, F) + κ +2 2 2+λ +ξ Dt λ ∂τ ∂x4 ∂x ∂y ∂y4 ∂τ2 1 ∂4 F ∂4 F ∂4 F 1 (4.57) λ−2 4 + 2 2 2 + λ2 4 + ∇2k w + L (w, w) = 0, Ft 2 ∂x ∂x ∂y ∂y

4.3 Stability of Thin Shallow Shells

where 1 Dt = & ' 1 − ν2

h2

h2 KE (T ) z dz; Ft = 2

−h1

1 1 + δ; h2 = − δ; δ = 2 2

KE (T ) dz; −h1

1

h1 =

447

2 1 −2

⎤−1 ⎡ 1 ⎥⎥⎥ ⎢⎢⎢ 2 ⎥⎥⎥ ⎢⎢⎢ ⎥ ⎢⎢⎢ (T ) KE zdz ⎢⎢ KE (T )dz⎥⎥⎥⎥ . ⎥⎥⎥ ⎢⎢⎢ ⎦ ⎣1

(4.58)

−2

The numerical integration of (4.58) is carried out using the Simpson’s method. In order to get an optimal shell’s thickness partition 10, 20 and 40 intervals are used on the basis of the Runge’s principle. It appears that during the partition into 20 and more parts the thermal terms obtained using the Simpson’s method and the exact integration method (where E, αt are constant) practically overlap. Besides, the thermal terms, obtained during the partition into 20 and 40 intervals overlap as well, whereas during the partition into 10 and 20 intervals an error achieved 5-6%. Taking into account the earlier results we have used 20 intervals. We analyse the stability of the squared shells with the following physical mechanical characteristics (steel 15XM): ν = 0.3, E 0 = 2.1 · 105 MPa, α0t = 1.15 · 10−5 grad−1 , c = 460J/kg · K, λ0g = 476m/(n · K). The dimensionless functions Kα and KE are approximated using the experimental data via the following relations

(4.59) KE = 1 − a1 T 2 − a2 T , πT Kα = 1 − b1 T + sin , for T ≤ T 1 ; Kα = [1 + b4 T ] , for T ≤ T 1 , T1

(4.60)

where a1 , a2 , b1 , b2 , b3 , b4 are the non-dimensional approximation coeﬃcients. They have the following dimensional form a1 =

a2 b2 α2t0 abαt0 a1 = 0.2976 · 10−6 grad−2 , a2 = a2 = 0.4643 · 10−3 grad−2 , 4 h h2

b1 = 4.4269 · 10−4 grad−1 ; b2 = 0.03478; b3 = 1.2435; b4 = 1.3986grad−1 . T 1 denotes a non-dimensional value of the temperature (823◦ K), above which the αt can be approximated by a straight line. We have investigated a dynamical stability of thin shallow shells in order to determine the critical values of the heat flow intensity q ∗ with a simultaneous temperature dependence of E and αt . The q ∗ values have been compared with the data obtained for E, αt = constant (see Sect. 4.2.3). The comparison results in percents in relation to q ∗ , w, σ x for E, αt const are given on the drawings for a critical value

448

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.31. The critical heat stream q ∗ values in percents versus K x + Ky .

of the heat stream intensity (Fig. 4.31), the deflection (Fig. 4.32) and full stresses in the shell’s centre on its internal surface (Fig. 4.33). The solid curves of E, αt simultaneously depend on the temperature; the dashed ones - on the E(T ); αt = const, the dashed lines with dots correspond to αt (T ), E = const; digit 1 corresponds to the spherical shells and digit 2 corresponds to the cylindrical shells. Analysing the results for q ∗ presented in Fig. 4.31 one can conclude that when αt and E do not depend on the temperature, the values of the critical heat flow intensity

4.3 Stability of Thin Shallow Shells

449

Figure 4.32. The shell deflection in percents versus K x + Ky .

become greater. Increasing the parameters (K x + Ky ) a diﬀerence of q ∗ is increased. This behaviour can be explained in the following way. When the (K x + Ky ) are increased, then also q ∗ increases, and the shell’s temperature increases, and therefore the temperature strongly influences αt and E. This influence depends not only on (K x + Ky ), but also on the shells type. For the spherical shells the q ∗ variations are larger than for the cylindrical shells. For the cylindrical panels and for (K x + Ky ) > 50, a stabilization of q ∗ is observed.

450

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.33. The shell stresses in percents versus K x + Ky .

The maximal increase of q ∗ for (K x + Ky ) = 72 achieved 17.5% for cylindrical and 22.5% for spherical shells. However, their distribution is diﬀerent. More important is αt (T ). As it is seen from Fig. 4.31, the critical values of the heat stream intensity for E(T ) and αt (T ), E = const and αt (T ) are close to each other. The maximal diﬀerence does not exceed 3%. Therefore, the investigations can be carried out without any limitation either along the thickness, or along the surface. In the latter case (when E = const) the integral characteristics Dt , Ft are constant, and S t = 0 in (4.6) and the equation (4.25) and (4.26) are reduced to those with constant coeﬃcients. We compare the results corresponding to q ∗ . As it is shown in Fig. 4.32, the deflections change when E and αt is constant, depend on the shell’s type and on the parameters K x , Ky . When E and αt do not simultaneously depend on the temperature the deflection increase for the spherical shells for (K x + Ky ) > 60, and for the

4.3 Stability of Thin Shallow Shells

451

cylindrical panels for (K x + Ky ) > 37. When only αt (T ) is taken into account, the deflection increase is observed. The highest increase of deflections (up to 19%) is observed, when αt has been treated as temperature independent. Now, we analyse the stresses corresponding to the critical values of the heat stream intensity (Fig. 4.33). We consider only the cases of the stresses increase. When both E and αt are treated as temperature independent, the stresses step up for the spherical shells for (K x + Ky ) > 44 and for the cylindrical panels on the whole interval of the (K x + Ky ) changes. When E is treated as temperature independent the stresses are even greater. When only αt is treated as temperature independent, the stresses increase their amount only for the spherical shells for (K x + Ky ) > 56 and the stresses diﬀerence decreases. The largest height increase (of 47%) is observed, when E is treated as the temperature independent. The numerical data of the found critical heat stream intensity, deflections and stresses with and without temperature dependence of E and αt , and also the diﬀerences in percents in relation to q ∗ , w and σ x are given in Table 4.3.

Figure 4.34. Equal relative shell deflections for cylindrical (a, b) and spherical (c, d) shells.

452

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load Table 4.3. Numerical data of spherical and cylindrical shells. K x + Ky Critical E, αt values - const

36

48

60

72

36

48

60

72

q ∗ w −σ0x q ∗ w −σ0x q ∗ w −σ0x q ∗ w −σ0x

69 2.262 18.698 133 2.696 54.206 216 2.776 87.843 297 2.857 121.48

q ∗ w −σ0x q ∗ w −σ0x q ∗ w −σ0x q ∗ w −σ0x

64 2.225 16.07 131 2.593 45.595 263 3.012 105.025 400 3.121 164.1

E(T ) E, αt % - const αt (T ) Spherical shells 62 -10.6 66 2.42 +7 2.34 22.157 18.5 16.35 109 -18 131 2.857 6 2.82 49.015 -9.6 49.746 230 -20.9 219 2.776 0 2.854 67.2 -23.5 78.883 230 -22.6 314 2.676 -6.3 2.88 83.18 -31.53 104.28 Cylindrical shells 61 -4.7 65 2.234 0.4 2.66 16.006 -0.4 11.39 108 -17.5 129 2.52 -2.8 2.76 45.49 -1 41.778 217 -17.5 263 2.78 -7.7 3.169 89.271 -15 86.121 330 -17.5 419 2.728 -12.6 3.27 126.165 -23.1 121.032

%

E(T ) αt (T )

%

-4.9 3.5 -12.6 -1.5 4.6 -8.2 2 2.8 -10.2 5.7 0.81 -14.2

63 2.24 19.855 109 2.57 -56.423 167 2.684 90.039 223 2.835 122.27

-8.7 -0.97 6.2 -18 -4.67 4.1 -22.3 -3.3 2.5 -24.9 -0.77 0.65

1.5 62 -3.1 19.5 2.476 11.3 -29.5 15.292 -4.8 -1.5 109 -16.8 6.4 2.594 0.04 50.6 50.87 10.7 0. 214 -18.6 5.2 2.819 -6.4 -18 113.952 8.5 4.8 322 -19.5 4.8 2.73 -12.5 -26.4 172.41 5.1

Now we analyse the influence of E(T ) and αt (T ) on the full stresses, and independently for the membrane and bending stresses. The influence of the mentioned characteristics is diﬀerent and changes with the increase of (K x + Ky ), but a qualitative change of the membrane, bending and full stresses is similar. The maximal increase of all components of the stress state is observed at the end of the interval (K x + Ky ) = 72 for the spherical shells. In the case of bending normal stresses their amount achieves 31%, in the case of bending tangential stresses their amount achieves 23%, in the case of the normal membrane stresses their amount achieves 52%, and in the case of the membrane tangential stresses their amount achieves 21%. We investigate, how the temperature dependences of E(T ) and αt (T ) influence the vibrational process of the shell surface points. We consider the curves of equal relative deflections (Fig. 4.34) and the w(τ) dependence (Fig. 4.35) obtained for shells with (K x + Ky ) = 48 (Fig. 4.34 a, b - cylindrical shells; Fig. 4.35 a - cylindrical

4.3 Stability of Thin Shallow Shells

453

Figure 4.35. Vibrations of cylindrical (a) and spherical (b) shells.

shells, b - spherical shells). We compare the obtained characteristics with those obtained when E and αt are treated as temperature independent (Fig. 4.3, 4.5). It appears that a treatment of E and αt as temperature independent does not lead to a suﬃcient influence of the shell surface form.

454

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.36. Stresses and deflection variations versus the q parameter.

However, when both E(T ) and αt (T ) are considered as temperature independent, the relative maximum and amplitudes of vibrations are decreased. For example, for the results presented in Fig. 4.34 and Fig. 4.5 a diﬀerence in the values of relative maximum and amplitudes (in relation to E, αt - const) achieved amount of 14.8% and amount of 19.8% for the spherical shells, and 70% and 64.5% for the cylindrical shells. Comparing the vibrational process on the example of a central point (w(τ) dependence) (Fig. 4.3 and Fig. 4.34) it is clear that when E and αt are temperature dependent, then the stability loss occurs on the second wave of the vibrational process. For E, αt = const for the post-critical values of the heat flow intensity a shell loses its stability beginning from the first wave of the vibrational process. A stability loss of shells with E(T ) and αt (T ) beginning from the second wave of the vibrational process can be explained by means of a continuous heating and temperature increase in time. As a result, we have got a maximal deviation of E and αt from its initial state only on the second wave of the vibrational process. Consider the influence of temperature dependence of E and αt on the stability of flexible conical shells in the condition of a constant load and a heat impact. The investigations have been focused on the critical values of the transversal load with an increase of the heat flow intensity from zero to its critical value. The results obtained for the same q are compared with those given in Sect. 4.3.2 of this chapter for E, αt - const. The analysis has been carried out on the example of the spherical shell with (K x + Ky ) = 24.

4.3 Stability of Thin Shallow Shells

455

In the Fig. 4.36 a change of q0s versus the increase of q is presented (the solid curves - E and αt are const; the dashed curves - E and αt depend on the temperature). When E and αt are treated as temperature independent, it leads to the increase of the critical transversal load values for the same q and the diﬀerence of q0s is increased, when the heat stream intensity increases. It is mainly caused by the shell temperature increase (it means that the influence of E(T ) and αt (T ) increases), and the critical value of the heat flow intensity is lower, when E and αt depend on the temperature. Therefore, if for q = 109 non-dimensional units (q = q ∗ for E(T ) and αt (T )) a stability (with the influence of E and αt on the temperature dependence) is characterized only by the heat impact (q0s = 0). For E and αt - const, the shell can still carry the transversal load (q0s = 26). The obtained results show that for q ≤ 0.25q ∗ (q ∗ for E and αt - const) the E and αt temperature dependence practically does not influence the critical values of the transversal load. With suﬃcient practical accuracy (up to 5%), the q0s for E and αt - const can be found in the interval 0 ≤ q ≤ 60 (q 0.45q ∗ ). Now we analyse the deflections and stresses. When E and αt are temperature independent for arbitrary q it leads to the deflection decrease and to the stresses increase for q < 60. In the interval 20 < q < 37 the stresses have diﬀerent signs. Maximal increase of stresses up to 150% and the deflection decrease up to 33% are observed. For the critical value of q the absolute increase of stresses is small and does not play an essential role in the strain state during stability investigation. The numerically obtained critical values of the transversal load, deflection and stresses (with and without temperature dependence of E and αt ), and also the shift expressed in percents in the mentioned quantities in relation to q , w, σ0x for E and αt = const are presented in Table 4.4. Table 4.4. Critical values of transversal loads

E, αt const E(T ), αt (T ) % Critical values

0.25 78 2.3 -9.333 55 78 2.93 4.64 50 0 33.20 149.7 -9.9 q0c w σ0x q0c

q /q ∗ 0.5 0.75 2.28 -25.54 36 2.50 -39.90 2.71 -25.78 12 2.69 -45.69 18.9 5.1 -66.7 7.60 -14.51 w σ0x q0c w σ0x

We consider the temperature influence of E and αt on the stability of flexurable shallow shells being under the influence of a sign changeable transversal load and a heat impact. The investigations have been carried out in order to find an unstable zone on the example of the spherical shell K x = Ky = 24 with the amplitude of the transversal load equal to 65 non-dimensional units. The results have been compared for the same q with those given in Sect. 4.3.3 for E and αt = const. In Fig. 4.37 the unstable zones with the increase of q from 0 to q ∗ (q ∗ for E and αt = const) (the solid curves denote constant values of E and αt , whereas the dashed correspond to temperature dependent E(T ) and αt (T )). When E and αt depend on

456

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.37. The unstable zones in the coordinates w(q ).

the temperature, then the unstable zone increases. However, up to q = 3/4q ∗ , that influence is rather not significant. The maximal diﬀerence in width has achieved 22% for q = 0.6q ∗ , and 11% for q = 0.75q ∗ . The absolute values of diﬀerences of upper and lower frequencies do not exceed 3 non-dimensional units. With suﬃcient accuracy (up to 5%) the frequencies of the parametric vibrations for q ≤ 0.3q ∗ can be found. For q ≥ 3/4q ∗ for E and αt = const a sudden jump of high frequencies (1.85 times) is observed. For q = q ∗ and taking into account the temperature dependence of E and αt the ωg achieved its limit in infinity. It can be explained in the following manner. The q = q ∗ with E and αt = const belongs to the post-critical one (of 18%) in comparison to the critical value. Therefore, the upper frequency unboundedly increases, because for this case the stability is characterized only by the heat impact (the thermal load is post-critical, and the shell loses its stability independently of the forcing load values). The numerical calculation results are given in Table 4.5. In the earlier discussion and illustrated problems the investigations have been carried out when only the linear thermal expansion coeﬃcient and the Young’s modulus depend on the temperature. We have to mention that the temperature dependence on other characteristics, such as the heat and the temperature transfer coeﬃcient have not been considered, and a solution to the heat transfer equation is obtained for λg = const and α = const.

4.3 Stability of Thin Shallow Shells

457

Table 4.5. Results of numerical computation of ω and q /q ∗ . q s0

ω

E, αt const E(T ) αt (T ) change %

ωu ωl ωu ωl ωu ωl

0 16 22 16 22 0 0

0.25 13 24 13 25 0 4.2

q /q ∗ 0.5 6 25 4 27 33 8

0.75 0 27 0 30 0 11

1 0 50 0 0

Now we consider thin conical shells stability taking into account the temperature dependence of all material characteristics as well as the Young’s modulus. The λg and α coeﬃcients for a given material are approximated by the following expressions:

λg = λng 1 − AT 2 − BT ;

α = α0 1 − CT 2 − DT ; where: 2 abαt0 abαt0 −7 −2 ¯ ¯ A=A = 0.7 · 10−3 grad−1 ; = 0.8 · 10 grad ; B = B h2 h2 abαt0 abαt0 −5 −2 ¯ ¯ C=C = 0.11 · 10 grad ; D = D = 0.6 · 10−3 grad−1 . h2 h2 When λg and α depend on the temperature, the heat transfer equations have variable coeﬃcient and it is impossible to find an analytical expression for the temperature field. Therefore, the numerical method of finite diﬀerences with the approximation O(h2 ) along the shell’s thickness is used. It allowed to reduce the partial diﬀerential equations to the system of ordinary diﬀerential equations which has been solved using the Runge-Kutta method. The shell thickness has been divided into twenty intervals. As a result of the numerical solution to the heat transfer equation the temperature field T (z, τ) is obtained. Knowing the temperatures, the integral characteristics Ft , Dt are found and the temperature terms on each time step necessary for solution of the equations (4.55) and (4.57) are defined. The calculations have shown that the error occurred when λg and α do not depend on the temperature (for the same q ) achieved 8%, and for accounting temperature variation it achieved amount of temperature 6%. To conclude, an account of λg and α on the temperature increases a diﬀerence in the critical values of the load, deflection and stresses. For the spherical shell with the parameters K x = Ky = 24 an account of the temperature dependence of λg and α caused a decrease the heat stream intensity critical value of amount of 5.5% and the stresses of amount of 4.8%, and yields increase of deflection of amount of 2.56% in comparison to the case, when E and αt do not depend on the temperature.

458

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

4.4 Stability of Thin Conical Shells 4.4.1 Boundary Conditions and Surrounding Medium We investigate a stability of flexurable conical shells subjected to a longitudinal compressing load (constant in time) action on the shell’s frontal surface with the following boundary conditions: – The boundary condition 1 - (4.28), – The boundary condition 2 - (4.29), – The boundary condition 3 - (4.30). We consider the case when E and αt do not depend on the temperature. The fundamental equations (4.29) and (4.26) have the following non-dimensional form: 4 4 1 ∂4 w −2 ∂ w 2∂ w − ∇2k F − L (w, F) + +2 2 2 +λ & ' λ 12 1 − ν2 ∂x4 ∂x ∂y ∂y4 2 ∂ w ∂2 w ∂2 w ∂w = − K x P x + Ky Py + P x 2 + Py 2 , κ (4.61) +ξ ∂τ ∂τ2 ∂x ∂y λ−2

4 2 2 ∂4 F 1 ∂4 F 2∂ F 2 2 ∂ Px −2 ∂ P x (w, L w) = λ + 2 + λ + ∇ w + + λ . k 2 ∂x4 ∂x2 ∂y2 ∂y4 ∂y2 ∂x2

(4.62)

As dynamical stability loss criteria, the Volmir’s criterion and the criterion of the membrane stresses sudden are used. The investigations are carried out for a series of cylindrical shells with the parameters (K x + Ky ) [36; 72] with the longitudinal compressing load action along the direction of zero curvature and a lack of the thermal field. The longitudinal compressing load in the nonzero curvuture is assumed to be equal to zero. The results of the dependence P0x (K x + Ky ) are presented in Fig. 4.38a. The digits correspond to the boundary problem value, the solid curves correspond to damping coeﬃcient ξ = 0. It has been found that the boundary conditions have essential influence on the changes of the longitudinal compression load critical value, and this influence essentially the variations of the longitudinal compression load changes with a change of (K x + Ky ). Increasing the parameters (K x + Ky ) the P0x values increase. The higher influence corresponds for the 1 boundary conditions, whereas the lowest to the 3 boundary conditions. Comparing the obtained values P0x for diﬀerent boundary conditions, it has been observed that strongly stable shells correspond to the 2 boundary conditions for (K x + Ky ) < 66, whereas for (K x + Ky ) > 66 the most strongly stable shells correspond to the 1 boundary conditions. In the case of the hybrid boundary conditions 3 the shells are more stable than those with the 1 boundary conditions only if (K x + Ky ) < 58. For instance, for (K x + Ky ) = 36 the P0x values for the shells with the boundary conditions 2 and 3 are equal or greater than the P0x for shells with 1 boundary conditions of amount of 1.33 times.

4.4 Stability of Thin Conical Shells

459

Figure 4.38. The longitudinal compressing load P0x (a) and the stresses σ0y (b) versus the parameter K x + Ky , and the dependence P0x (ε) (c).

Now we consider the motion of the surface points. For this purpose, the curves with equal relative deformations (Fig. 4.39) have been determined in the time instants, when the deflection in the centre reaches its first maximum (the upper curves) and its first minimum (the lower curves) for diﬀerent boundary conditions (1, Fig. 4.39; 3, Fig. 4.38 b; 2, Fig. 4.39 c). It has been discovered that the most complicated vibrations have occurred for shells with hybrid boundary conditions 3. For the time instants, when the deflection reaches its minimum, zones with negative deflections on the surface occur. The greatest relative deflection is observed for shells with the boundary conditions 2.

460

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.39. The curves with relatively equal deflections for boundary conditions 1 (a); 2 (b); 3 (c) for K x = 0 and Ky = 48.

The deflection distribution along the coordinate axes is more clearly visible in Fig. 4.40 and Fig. 4.41. (Fig. 4.40 a, 4.41 a correspond to the shells with the boundary conditions 1; Fig. 4.41b - 2; Fig. 4.41c - 3; the digit 1 corresponds to b(a) = 1/2; the digit 2 corresponds to b(a) = 1/4; the solid curves correspond to the maximal deflection in the centre; and the dashed curves corresponds to the minimal one). The figures show that the deflection distribution along the shell (direction a) with the boundary conditions 1 and 2 are characterized by one halfwave for the arbitrary time moments. For the shells with 3rd boundary conditions, in the time instants when the deflection reaches its minimum, the deflection distribution is characterized by five halfwaves. In the case of a transversal deflection distribution (along side b) the deflection is characterized by three halfwaves for the 2 and 3 boundary conditions. We consider the vibrational process on the example of the shell’s central point with the parameters K x = 0, Ky = 48 (Fig. 4.42) for diﬀerent boundary conditions (Fig. 4.42a - 2; Fig. 4.42b - 3; Fig. 4.42c - 1). The vibrational process is characterized by relatively small deflections and amplitudes in comparison to the transversal load. The greatest deflections have been observed for shells with the boundary conditions 1, and the smallest for with the boundary conditions 2. The amplitude has achieved

4.4 Stability of Thin Conical Shells

461

Figure 4.40. Shell deflections along a direction.

its maximal value for the shells with boundary conditions 2. The vibrational process for shells with the boundary conditions 1 is characterized by large vibration periods and larger beginning of the vibration process in comparison to boundary conditions 2.

462

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.41. Shell deflections along b direction.

We investigate the influence of damping on the critical values of the longitudinal compressing load. For this purpose the dependence P0x (K x + Ky ) for ξ = ξ∗ (Fig. 4.38a - the dashed curves) is applied. ξ∗ denotes minimal value of ξ for which P0x monotonously approaches the statical value.

4.4 Stability of Thin Conical Shells

463

Figure 4.42. Vibrations of shell centre for diﬀerent boundary conditions: 2 (a), 3 (b) and 1 (c).

In what follows damping essentially influences P0x (especially the shells with the 2nd boundary conditions). P0x increases with the increase of (K x + Ky ). The statical load values P0x is achieved only for relatively large ξ values (Fig. 4.38c). For the shell with K x = 0, Ky = 72 the statical load is higher than the dynamical one for

464

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

shells with the boundary conditions 2 of amount of 1.35 times (ξ∗ = 35) and for 3 of amount of 1.22 times (ξ∗ = 50). For instance, when only the transversal load is applied (2 boundary condition) the statical value q0s is higher than the dynamical one of amount of 1.23 times (ξ∗ = 16) for the same shell. We investigate a strain state of shells for two boundary conditions with and without damping expressed by the dependence σ0y (K x + Ky ), Fig. 4.38b. The stresses appearing in the shells with the 1st boundary conditions suddenly increase when (K x + Ky ) is increased. In the case of hybrid boundary conditions 3 the stresses decrease up to (K x + Ky ) ≤ 48, but next their sudden increase occurs. However, they always remain smaller than the stresses in shells with the boundary conditions 2. For instance, σ0y for shells with 2 boundary conditions (ξ = 0) and for the parameters (K x + Ky ) = 72, the stresses for increased of amount of 1.97 times in comparison to σ0y in shells for the parameters (K x + Ky ) = 36. The stresses σ0y in the shells with the boundary conditions 3 first decreased of amount of 1.35 times in the interval (K x + Ky ) ≤ 48, and then they increased 3.3 times. The damping of a surrounding medium decreases the stresses, especially for the shells with 3rd boundary conditions. 4.4.2 Constant Compressing Load and Heat Flow The influence of the heat flow intensity on the dynamical critical longitudinal compressing load is considered. The calculations are carried out for the cylindrical shells with the parameters 36 ≤ (K x + Ky ) ≤ 72 in order to find P0x with the increase of q from zero to its critical value for the corresponding (K x + Ky ). The cylindrical shells are subjected to the longitudinal compressing load along zero curvature direction with the boundary conditions (4.53) (see dependence P0x [q , (K x + Ky )] presented in Fig.4.43). The dynamical longitudinal compressing loads lying below the given zone of the critical P0x values are pre-critical ones. When the critical q values are achieved, a stability is characterized only by the heat stream action (P0x = 0). It has been found that the heat stream intensity has essential influence on the critical value of the longitudinal load intensity, and it increases when q is increased. Comparing the obtained data with those given in Sect. 4.4.2 of this chapter (Fig. 4.18), i.e. considering the case when only transversal dynamical load is applied, one can conclude that a critical longitudinal load is more sensitive to the heat flow intensity action than q s . It is clearly seen in Table 4.6 on the example of two cylindrical shells with the parameters K x = 0, Ky = 48 and K x = 0, Ky = 72. Increase of q implies its stronger influence on the critical compressing load, especially for higher values of the parameters K x and Ky . Note, that for that case with only transversal load action, the heat stream intensity has smaller influence on q0s . The heat stream intensity has also larger influence (during the compressing longitudinal load action) on the maximal shell deflections. Increasing q a sudden deflection increase is observed. Influence of q is even more evident with the increase

4.4 Stability of Thin Conical Shells

465

Figure 4.43. The dependence P0x [q , (K x + Ky )]. Table 4.6. Kx = 0

Ky = 0

Kx = 0

Ky = 0

q

q ∗

P0x (q ) P0x (q =0)

q0s (q ) q0s (q =0)

P0x (q ) P0x (q =0)

q0s (q ) q0s (q =0)

0

1

1

1

1

0.25

0.889

0.790

0.643

0.701

0.50

0.389

0.530

0.286

0.585

0.75

0.167

0.168

0.095

0.285

1

0

0

0

0

of K x and Ky . When only the transversal load is applied, the deflections decrease. For instance, for the shell with K x = 0 and Ky = 36, the deflections increased (owing to increase of q ) from zero to its critical value of amount of 1.5 times for the compressing load, whereas for the case of a transversal load action, they it has been decreased of amount of 1.4 times. For the shell with parameters K x = 0 and Ky = 72, the deflections are changed for the considered loading cases of amount of 5.12 and 1.2 times, respectively. It again testifies the observation that the shells with the longitudinal compressing load

466

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

are more sensitive to the heat stream than those subjected to the transversal load action. We now analyse the surface points vibrations and the vibrational processes owing to with the heat stream intensity increase from zero to its critical value. The curves of equal deflections are compared (see Fig. 4.39c, Fig. 4.5a, Fig. 4.42c and Fig. 4.3a). The investigations show that increasing the heat intensity, the amplitudes and deflections of the vibrational process increase, and the relative maximum and the time of vibration beginning decrease. The relative maximum is shifted from the quadrants’ centres to the symmetry axis, which is perpendicular to zero curvature direction. For the considered shell, the amplitude and deflections increased of amount of 1.32 and 2.67 times, correspondingly (for q = q ∗ in relation to q = 0). 4.4.3 Harmonic Longitudinal Load and Heat Flow The investigation of flexurable conical shells, subjected to both heat stream and harmonic longitudinal load is carried out in order to find an unstable zone bounded by low and upper critical frequencies ωl , ωu . The applied load has the following harmonic form (4.63) P = P0 sin ωt, where P0 denotes the load amplitude.

Figure 4.44. The dependence ω[q , (P x0 + Py0 )] for the cylindrical shell.

A shape of an unstable zone is found owing to a gradual increase of the heat stream intensity from zero to its critical value. In order to compare the results with those given in Sect. 4.4.2, the same shells with (K x + Ky ) = 48 have been analysed.

4.4 Stability of Thin Conical Shells

467

Figure 4.45. The dependence ω[q , (P x0 + Py0 )] for the spherical shell.

The fundamental rules of the unstable zone variations for a cylindrical shell under action of harmonic longitudinal load action in zero curvature direction P x = P x0 sin ωt, Py = 0 is studied (see the dependence ω[q , (P x0 + Py=0 )] shown in Fig. 4.44). Recall that the A area in the figures governs the dependence of the critical value of the longitudinal load against the heat stream intensity. It is observed that increasing the heat stream intensity the unstable zone becomes wider and is slightly shifted to the direction of small amplitude values of the load. However, contrary to the case corresponding to transversal load action, the zone’s spike does not touch the plane (P x0 + Py0 ) = 0. Comparing the frequencies for q ≤ 3/4q ∗ for the harmonic longitudinal and the transversal loads (Fig. 4.44 and Fig. 4.21) one may conclude that their change character is similar, when the heat flow intensity increases up to 3/4q ∗ value. However, in the case of the heat flow intensity larger than 3/4 of its critical value, higher shells sensitivity is observed for the longitudinal load and the heat stream. For q > 3/4q ∗ an essential increase of the upper frequencies is observed. For instance, for the transversal load ωl is increased for q = q ∗ of amount of 2.27 times in comparison to q = 0. The upper frequency is increased of amount of 30 times for the longitudinal load, and became 11.76 times larger than ωu in comparison to the transversal load action. The longitudinal load action is expressed by an essentially larger increase of the upper frequency in comparison to the transversal load action. The qualitative and quantitative changes of the unstable zone essentially changes with the increase of q , when the spherical shell stability subjected to harmonic longitudinal load P x0 = Py0 = P0 action is considered (Fig. 4.45). The zone width

468

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

and the upper critical resonance frequencies are larger, even with a lack of the heat stream action. For instance, for (P x0 + Py0 ) = 12 and q = 0 both zone width and upper frequencies are higher for the spherical shell 3 of amount of 3.14 and 2.5 times in comparison to the cylindrical one. It has been found that the spherical shells subjected to the longitudinal harmonic load P x0 = Py0 = P0 action are very sensitive to the heat stream variation, and already for q > 1/2q ∗ a sudden increase of the upper critical frequencies and the zone’s width are observed.

Figure 4.46. The dependencies ω(q ) for the cylindrical (dashed curves) and spherical (solid curves) for various 1-5 loads.

For instance, for (P x0 + Py0 ) = 10 a value of the upper critical frequency for q > 3/4q ∗ is larger than for q = 0 of amount of 2.5 times, and of amount of 5 times for the cylindrical shell. For q = q ∗ the frequency ωu becomes 16 times larger than for q = 0, and 1.33 times larger than for the cylindrical shell.

4.4 Stability of Thin Conical Shells

469

The investigation yields the conclusion that the load acting on the spherical shell possesses more negative influence, as well as the heat load intensity is more strong than the load acting on the cylindrical shell with the same (K x + Ky ) and (P x0 + Py0 ) parameters. In order to detect the influence of a way of the load application (in order to define the most unprofitable case), additional investigations of cylindrical and spherical shells stability are carried out using the forcing amplitude (P x0 + Py0 ) = 10. The obtained results are presented in Fig. 4.46. The following loading cases are considered. For the cylindrical shells K x = 0, Ky = 48 (the dashed curves): 1. P x0 = P0 = 10, Py = 0, 2. P x = 0, Py0 = P0 = 10, 3. P x0 = Py0 = P0 = 5. For the spherical shells K x = Ky = 24 (the solid curves): 4. P x0 = Py0 = P0 = 5, 5. P x0 = P0 = 10, Py0 = 0. Cylindrical and spherical shells are separately considered. For cylindrical shells the most unprofitable loading corresponds to the harmonic longitudinal load action only along the nonzero curvuture direction. On the contrary, the longitudinal load action along the zero curvuture the unstable zone area.

Figure 4.47. Shell vibrations (loading type 1).

470

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.48. Shell vibrations (loading type 2).

A way of the load action practically does not influence the lower critical frequencies. However, the upper frequencies (and correspondingly, the zones width) strongly depend on the load action way. The increase of the heat stream intensity exerts larger influence during the longitudinal load action along the nonzero curvuture direction. However, for q = q ∗ a diﬀerence in the upper frequency values is not high, the upper frequencies reach large values for all types of loads. The highly expressed increase of ωu for q > 3/4q ∗ corresponds for the longitudinal load action along the direction of zero curvature action. We illustrate the above observations on the numerical examples. For q = 0 the ωu value is higher of amount of 1.48 and 3.7 times for the 2 loading case, than the ωn values corresponding to 3 and 1 loading cases correspondingly. For q > 3/4q ∗ those relations achieved amount of 1.9 and 8.25, whereas for

q = q ∗ the achieve amounts of 1.02 and 1.12. The upper frequencies for q = q ∗ are increased in comparison to q = 0 and

q > 3/4q ∗ for the 2-nd loading type of 9.05 and 3.38 times; for 3-rd loading type of 13.2 and 6.35 times and for the 1-st loading type of 30 and 25 times. For spherical shells the loading type influence is diﬀerent owing to increase of the stream intensity. For the q < 98 ( 3/4q ∗ ) the values of upper and lower critical frequencies are higher (the longitudinal load acts only in one direction). Besides, in the considered interval of q for the 5-th loading type the frequency ωl = 0 only at the interval end. In the case for the 4-th loading type, ωl = 0 for q = 1/2q ∗ . For q > 98, non-dimensional units of ωu values for the 4-th loading case (longitudinal load is applied in two directions) become larger than ωu corresponding to the 5-th

4.4 Stability of Thin Conical Shells

471

loading case. We support the above conclusions using the numerical examples. For q = 0 we have ωu of the 5-th loading case of amount of 1.24 times larger than ωu corresponding to the 4-th loading case, (for q = q ∗ it achieves amount of 1.19 times. The upper frequencies for q = q ∗ are increased correspondingly to q = 0 and q = 3/4q ∗ of amount of 10.89 and 5.82 times for the 5-th loading case, whereas for the 4-th loading case of 16 and 6.67 times, correspondingly.

Figure 4.49. Shell vibrations (loading type 3).

The numerical values of the critical frequencies for the 1-st and the 4-th loading cases and for diﬀerent amplitudes of the longitudinal load are given in Table 4.7. The frequencies, for diﬀerent loading cases P x0 = Py0 = 10 are given in Table 4.8. We consider the vibrational process of the surface points in time for diﬀerent loading cases (1 - Fig. 4.47, 2 - Fig. 4.48, 3 - Fig. 4.49, 4 - Fig. 4.50, 5 - Fig. 4.51) for time instants, when deflection in the centre reaches its maximum and minimum for q = 0 (the upper curves) and q = 3/4q ∗ (the lower curves). We analyse the surface points vibrations for cylindrical shells. As it has been seen in the figures, the surface points vibrations are complicated and depend on the way of loading. However, it is rather typical that during vibrations negative deflection zones occur in the neighbourhood of the shell’s centre, and positive deflection zones appears close to the shell’s edges. For the shells loaded using the 2-nd and 3-rd ways, pre-critical vibrations of deflections are characterized by large amplitudes and positive deflection values.

472

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.50. Shell vibrations (loading type 4).

Figure 4.51. Shell vibrations (loading type 5).

4.4 Stability of Thin Conical Shells

473

Table 4.7. Critical frequency values for diﬀerent loading types. Loading types

q in relation to q ∗ q /q ∗ 0

0.25

1

0.5

0.75

1

0

0.25 4

0.5

0.75

P x0 + Py0 15 15 10 5 15 10 5 15 10 5 15 10 5 15 10 5 1 20 15 10 6 20 15 10 6 20 15 10 6 20 15 10 6

ωl 2 2 6 11 0 3 8 0 0 7 0 0 0 0 0 0 0 0 0 6 14 0 0 4 10 0 0 0 0 0 0 0 0

ωu 24 24 20 11 24 20 11 25 23 17 27 24 20 1600 600 20 11 74 62 50 38 79 67 53 40 87 74 57 45 216 156 120 62

Maximal negative deflections are of 5-6 times smaller than the absolute values of maximal positive deflections. The shells loaded by the first way have essentially smaller values of deflection (7-8 times) in comparison to the other loading ways in both positive and negative zones. Therefore, also the vibration amplitudes are small. Increasing the heat flow intensity the vibration shift in the positive deflection direction is observed, and a decrease of maximal possible pre-critical deflections and amplitudes is noticed. For the shells loaded in the first way a reversed picture is observed. Namely deflection and amplitudes increase. The zones of negative deflections decrease with

474

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load Table 4.8. Critical frequency values for P x0 = Py0 = 10. Loading types 1 2 3 4 5

ω ωl ωu ωl ωu ωl ωu ωl ωu ωl ωu

0 6 20 5 74 6 56 6 50 8 62

0.25 3 20 3 78 4 53 4 53 7 64

q /q ∗ 0.5 0 23 0 108 0 59 0 57 5 74

0.75 0 24 0 198 0 104 0 120 0 116

1 0 600 0 670 0 660 0 800 0 675

the increase of q . For the shells loaded using the 3-rd way, the vibrations are fully shifted to the positive deflections zone.

Figure 4.52. The dependencies ω(w) and ω(σy ) for the cylindrical shell for K x = 0, Ky = 48, P x0 = 10 (deflections are denoted by dashed curves).

4.4 Stability of Thin Conical Shells

475

Table 4.9. Stress-strain states of shells. q

0

0.75q ∗

Characteristics ω σx ω σx ω σx ω σx

Loading types 1 2 3 4 5 Frequency 2.27 4.43 4.68 4.487 3.05 outside -11.0312 45.5382 52.5382 66.1082 32.2952 the zone 9.039 9.88 6.35 6.53 7.15 in the 86.534 92.5671 77.8436 96.6472 91.1287 zone 3.57 2.58 3.34 2.63 4.41 outside -7.3693 -4.861 -8.8167 -23.5604 39.4474 the zone in the 24.2094 205.317 48.7387 8.7424 138.8328 zone

For instance, the maximal possible deflection and amplitude for the shells loaded in the 2-nd and 3-rd ways are decreased, whereas for the shells loaded in the 1-st way they increased of amounts of 1.78 and 1.78 times, 1.88 and 3.2 times, and 1.51 and 1.44 times, respectively. We analyse now the surface vibrations for the spherical shells. As it has been seen in the figures, for q = 0, when the longitudinal load acts simultaneously in both directions (the 4-th loading way) the vibrations are quasi-periodic. Positive deflections on the whole surface are substituted by the negative ones. The deflection maximum occurs in the shell’s centre. When the longitudinal load acts only in one direction (the 5-th loading way) the vibrations become complicated in both space and time. The vibrational process for the spherical shells is characterized by relatively large pre-critical deflections and amplitudes of vibrations. Increasing the heat flow intensity, the vibrational process fully moves in the direction of the positive deflection. The maximum deflection point begins to vibrate between the shell’s centre and the centres of each of the shell’s quadrant. When the shell’s centre achieved its minimum, the relative maximum achieved 4.5 and 13 units for the 4-th and the 5-th loading ways, respectively. It means that the maximum deflection zones for the spherical shells (for q = 3/4q ∗ ) are clearly displayed and a sudden change of the surface form is observed. We analyse the stress strain state in the process of the excitation frequencies transition through the unstable zones. We analyse the dependencies ω(w) and ω(σ x ) for q and q = 3/4q ∗ (for deflection - the dashed curves) for a spherical shell with the parameters K x = 0, Ky = 48, P x0 = 10, Py = 0 (Fig. 4.52). A transition through the unstable zone is characterized by a sudden increase of deflections and stresses. For q = 0 the stresses change their sign. The frequencies, for which the deflections and stresses achieve their maximum, do not overlap. The point of maximal deflection is shifted in the direction of the low boundary, and the point of maximal stresses is shifted in the direction of upper boundary. Increasing the heat flow intensity, the deflections outside of zone instability are increased. Practically, the heat stream has no influence on the stresses. In the unstable zone with the increase of q , the deflections and stresses are decreased.

476

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.53. Vibration of the cylindrical shell centre for the 1st (a), 2nd (b), 3rd (c) loading types.

We compare the stress strain states for diﬀerent loading ways of cylindrical and spherical shells for the heat stream intensity q = 0 and q = 34 q ∗ on the upper boundary of the unstable zone. The results are given in Table 4.9. The results shown in Table 4.9 support the conclusions obtained during the analysis of the vibrational process in relation to deflections in the pre-critical state. Besides, it is seen, for all loading ways (except for the 1-st) and without the heat flow the transition through the upper zone’s boundary is smoother for both deflec-

4.4 Stability of Thin Conical Shells

477

Figure 4.54. The spherical shell centre vibrations for the 4th (a) and 5th (b) loading types.

tions and stresses. For the 1-st loading way for q = 0 and for all loading ways for q = 3/4q ∗ , a sudden increase of deflections and stresses is observed. Comparing the shells stress-strain state for diﬀerent loading ways, the following conclusions are obtained. The 1-st loading way corresponds to smaller stresses and deflections in comparison to the other loading ways for the cylindrical shells. The

478

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

spherical shells are the most suitable in the case of the 4-th loading way. It should be noted, that the highest stresses occur in the spherical shells subjected to the longitudinal force in one direction, and for the cylindrical shells along the nonzero curvature direction (together with the heat flow). We consider the vibrational process on the example of the shell’s central point for diﬀerent loading ways. For this purpose, the dependencies ω(τ) for the cylindrical (Fig. 4.53 a - 1-st loading way; Fig. 4.53 b - 2-nd; Fig. 4.53 c - 3-rd) and spherical (Fig. 4.54 a - 4-th; Fig. 4.54 b - 5-th) shells are displayed. The dependencies ω(τ) are obtained during the excitation frequency transition through the upper boundary of the unstable zone (in Fig. 4.53, 4.54 the solid curves correspond to the vibrational process for q = 0, whereas the dashed ones corresponds to q = 3/4q ∗ ). As it has been seen in Fig. 4.53 (for q = 0) the pre-critical state of shells with the 2-nd and the 3-rd loading ways are characterized by large deflections and amplitudes, and also by a slight deflection diﬀerence during the transition through the unstable zone. For q = 0 a stability loss occurs after a few vibrations with relatively small deflections. For a shell loaded by the 1-st (2-nd and 3-rd) way, a stability loss is observed on the third (fourth and second) wave of the vibrational process. For the shells loaded by the longitudinal harmonic load along the nonzero curvuture direction (the 2-nd and 3-rd loading way) from the halfwave in the shell’s negative deflection values. For the shells loaded by the longitudinal load along the zero curvuture direction the vibrational process begins from the halfwave located in the positive deflection area. When the heat flow intensity achieves its critical value, the vibrational process is qualitatively the same for the shells loaded in each of the considered loading ways. Amplitudes and deflections are relatively small. A stability loss is observed on the first wave of the vibrational process. Now we analyse the vibrational process of the spherical shells. As it is shown in Fig. 4.54, for q = 0 the vibrational process of the spherical shells is characterized by relatively high deflections and amplitudes and a slight diﬀerence in deflections during the transition through the unstable zone. A stability loss of the shells loaded using the 4-th way is observed already on the first wave of the vibrational process, and for the 5-th loading way it is observed on the third wave of the vibrational process. When the heat flow intensity reaches its critical value, the vibrational process of the spherical shells is characterized by relatively small amplitudes and deflections, and a sudden increase of deflection during the transition through the unstable zone occurs. A stability loss of the shells loaded using the first (fifth) way, corresponds to the first (second) wave of the vibrational process.

4.5 Stability of Flexurable Conical Shells with Convection

479

4.5 Stability of Flexurable Conical Shells with Convection 4.5.1 Problem Formulation We consider dynamical stability of shells with a convectional type heat transfer, when the Young’s modulus and the linear heat expansion coeﬃcient do not depend on the temperature. For this case, the equations (4.23) and (4.26) have the following form 4 4 ∂4 w 1 −2 ∂ w 2∂ w − ∇2k F − L (w, F) + λ + 2 + λ & ' 12 1 − ν2 ∂x4 ∂x2 ∂y2 ∂y4 2 ∂w ∂ w = qs , +ξ (4.64) κ ∂τ ∂τ2 λ−2

4 ∂4 F ∂4 F 1 2∂ F + 2 + λ + ∇2k w + L (w, w) = 0. 2 ∂x4 ∂x2 ∂y2 ∂y4

(4.65)

The unstationary convection process occurs due to the Newton’s law on the shell’s internal surface. The external and other shell’s surfaces are isolated. The boundary conditions for the heat transfer equation (4.42) for a heat exchange have the form 1 ∂T + Bi (T − T s ) = 0 for z = , ∂z 2 1 ∂T = 0 for z = − , ∂z 2

(4.66)

where Bi = (αg h)/λg ; T is the surrounding medium temperature; αg is the heat expansion coeﬃcient on the surface z = 0.5. The initial conditions for the heat transfer equations (4.42) have the form: T = T0

for τ = 0.

(4.67)

The formulated heat transfer problem (4.42) for a given boundary (4.66) and initial (4.4) conditions can be solved analytically [354]. The non-dimensional solution has the following form

1 exp −µ2n τ , Cn cos µ z + (4.68) T = T av − (T av − T 0 ) 2 n=1 where: Cn = 5

2Bi , 6 Bi (1 + Bi) + µ2n cos µn

and µn - are the roots of the equation µtgµ = Bi . The shells stability will be considered for the following boundary conditions: boundary condition 1 - (4.28); boundary condition 2 - (4.29);

480

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

boundary condition 3 - (4.30); boundary condition 4 - (4.31). The thermal moments and forces have the form Cn 5

6 µn sin µn + 2 cos µn − 2 exp −µ2 τ , Mt = −θ1 2µn n=1 ⎡ ⎤ Cn

⎥ ⎢⎢⎢ 2 ⎥ ⎢ sin µn exp −µ τ ⎥⎥⎥⎦ . Nt = θ1 ⎢⎣1 − µ n=1 n

(4.69)

(4.70)

The series, occurring in the terms of the thermal moments (4.69) and forces (4.70) are convergent. The calculations are stopped, when |an / an−1 | ≤ 10−8 . The numerical analysis proves that only six series terms are needed. When the temperature increase θ1 is positive, then the shells will work properly in the dynamical regime. Practically, this case can be realized, when the heating sources are used. When the temperature increment is negative, the shell will work in the condition of dynamical cooling. Both of the mentioned regimes are met during the shell’s constructions and need a detailed analysis. 4.5.2 Boundary and Thermal Fields Conditions We consider a vibrational process for each of the boundary conditions given in Sect. 4.5.1 for shells working in the cooling and heating conditions. The vibrational process (dependence w(τ)) will be considered on the example of the spherical shell with parameters (K x + Ky ) = 24 (Fig. 4.55). The digits correspond to the boundary condition number; the dashed curves correspond to the cooling process; the solid curves correspond to the heating process. Only for the shells with 1 boundary conditions and during heating, the stationary vibrational process is observed for the positive values. The axis of vibrations is always parallel to the initial deflection state. With the increase of the temperature, the shell loses its stability (Fig. 4.56 a). In order to detect the critical value of the temperature increment, all criteria described in Sect. 4.3.1 can be applied. For the other boundary conditions, in the heating regime, the shell subjected to the thermal forces deflects in the negative direction, and a stability loss behaviour cannot occur. The vibrational process is observed around the axis sloped to the initial deflection location, and the deflections increase infinitely with time. Similar results are obtained for a heat impact, and are given earlier in Sect. 2. Comparing the vibrational process for the boundary conditions 2 and 3, it has been observed, that the vibrational process of the shell deflection with the 3 boundary conditions is characterized by complexity and aperiodicity. An angle of the slope of the vibration axis is suﬃciently smaller than in the case of the boundary condition 2 for the same temperature values. The action of thermal moments and thermal forces for heating (cooling) is opposite. In the case of heating, the thermal moment is going to deflect the shell inside, in the curvuture direction (contrary to the thermal force action).

4.5 Stability of Flexurable Conical Shells with Convection

481

Figure 4.55. Vibrations of spherical shells.

Similar consideration can be carried out for the cooling process. We analyse the shells stress state for the given in Fig. 4.55 cases. In dependence σ x (τ) presented in Fig. 4.56, a vibrational process of full stresses in the shell’s centre is outlined (on its internal surface). Comparing the vibrations of stresses (Fig. 4.56) and deflection (Fig. 4.55) a qualitative similarity is observed. The deflection maximum (in most cases) corresponds to the stresses maximum. For the boundary conditions 2, 3 and 1 and for both heating and cooling regimes, the stresses unboundedly increase. The largest stresses are observed in the cooling process for shells with the boundary conditions 4 subjected to both temperature moments and forces action. The stresses occurring in the shells (except for the boundary condition 1) have the same sign as the corresponding deflection. We compare the shells stress state, when the parameters (K x + Ky ) are increased. For this purpose, the dependencies ω(τ) and σ x (τ) (Fig. 4.57) are drawn for the shell two temperature regimes and two boundary conditions 1 and 3. For two different (K x + Ky ) values, the pre-critical deflection for shells with the 1 boundary

482

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.56. Time histories of full stresses in the shell centre on its internal side.

condition decrease, and the maximal possible critical deflections increase. For shells with other boundary conditions, for heating and cooling processes, a sloping angle of the vibrational process axis is decreased. For the shells with the boundary conditions 1 the critical stresses increase. For the shells with other boundary conditions, the stresses decrease. In order to analyse the surface points vibrations, the curves of relatively equal deflections of the shell’s quadrant for two time instants are observed in the heating regime (Fig. 4.58, the boundary condition 2 - a, b; 3 - c) and in cooling regime (Fig. 4.59, the boundary conditions 2 - a, b; 3 - c, d; 4 - e, f). The surface configuration of the shells with the boundary conditions 2 in the heating regime and with the boundary conditions 2, 3, 4 in the cooling regime, does not practically change in time. Although the absolute values of deflections are almost unchangeable. The maximal deflection always remains in the shell’s centre. For the shell with the boundary condition 3 in the heating regime, the surface form becomes complex and changes with time. The negative and positive deflection

4.5 Stability of Flexurable Conical Shells with Convection

483

Figure 4.57. The dependencies w(τ) and σ x (τ) for the spherical shell for two diﬀerent temperature fields and two diﬀerent boundary conditions.

values are observed. They are caused by thermal moments causing positive deflections, and by thermal forces causing negative deflections. In the initial time moment the thermal moment zone occupies more than a half of the surface area. The relative positive maximum is greater of amount of 2-2.5 times in comparison to the relative negative one. A zone of positive deflection decreases. The absolute deflection values increase, and the relative positive maximum approaches zero, whereas the negative one approaches 1. 4.5.3 Critical Temperature Versus Heat Transfer Coeﬃcient We analyse a dynamical stability of flexurable conical shells, rectangular in plane, rollingly supported (boundary condition 1) with a convection on the internal surface. The heat transfer coeﬃcient, for each of the considered cases, is defined experimentally and depends on many factors, such as surface shape, its roughness, medium surrounding viscosity, velocity of the stream, the shell’s material and other factors.

484

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.58. The curves of relatively equal displacements of the shell quadrant during heating (boundary condition 2 (a, b) and 3 (c, d)).

Therefore, we need to investigate the critical temperatures versus the heat transfer coeﬃcient in the following non-dimensional form: Bi =

αg h λg

(Bi Biot number).

The investigations allow to define the critical temperature increment for cylindrical, as well as spherical, shells. We have obtained the dependencies of critical temperature increment θ10 [B1 , (K x + Ky )] (Fig. 4.60) and the corresponding full stresses σ0x [B1 , (K x + Ky )] (Fig. 4.61) on the internal surface for diﬀerent shells (24 ≤ (K x + Ky ) ≤ 72); 0.2 ≤ Bi ≤ 0.8 and 1st boundary condition. The dashed curves correspond to cylindrical shells, the solid curves to spherical shells, and the dashed curves with dots correspond to equal values of θz0 and σ0x for cylindrical and spherical shells. The temperature increments, lying below the obtained solution zones, are precritical ones and they are not dangerous for the conical shells constructions. Increasing the (K x + Ky ) parameters, the values of the critical temperature increments suddenly increase. Similarly to the heat impact action (Fig. 4.4), the cylindrical shells are more stable than the spherical ones for (K x + Ky ) > 52.

4.5 Stability of Flexurable Conical Shells with Convection

485

Figure 4.59. The curves of relatively equal displacements of the shell quadrant during heating (boundary condition 2 (a, b), 3 (c, d) and 4 (e, f)).

Increasing the parameter Bi the values of the critical temperature increments decrease. Simultaneously, an approach to θ1 for spherical and cylindrical shells is observed. A slight shift of the critical values of the temperature increment for the considered shells types occurs.

486

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.60. The dependence of θ1 [Bi , (K x + Ky )].

Increasing Bi from 0.2 o 0.8, the θ1 is decreased of amount of 3.19 and 3.22 times for the spherical and cylindrical shells, respectively, for (K x + Ky ) = 36 for (K x + Ky ) = 72 the decrease achieves 3.32 and 3.37 times, respectively. For Bi = 0.8, a diﬀerence in θ1 at the ends of the considered (K x + Ky ) interval achieved an amount of 4.65 and 6.09 times for the considered shells types. We analyse a change of full stresses on the example of σ0x in the shells centre on their interval surfaces (Fig. 4.61) with an increase of Bi parameter. A change of the

4.5 Stability of Flexurable Conical Shells with Convection

487

Figure 4.61. The dependence of σ x [Bi , (K x + Ky )].

stresses accompanying the increase of (K x + Ky ) is similar to that of the temperature increment. For Bi = 0.2 the stress, beginning from (K x + Ky ) = 55, occurring in the cylindrical shells become higher than those of spherical shells. Increase of the Bi parameter is of less importance on the critical stresses in comparison to the critical temperature. With the increase of the dimensionless heat expansion parameter, the critical stresses values are decreased. For (K x + Ky ) < 48, beginning from a certain Bi value, the stresses change its sign. A slight stresses approach for the corresponding shells types is observed only for the relatively large values of (K x + Ky ) > 58. The point of equal stresses is shifted in the direction of the upper boundary of the (K x + Ky ) interval.

488

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load Table 4.10. Minimal values of the spherical and cylindrical shell characteristics. Characteristics θ10 0.2 w0 σ0x θ10 0.4 w0 σ0x θ10 0.8 w0 σ0x

Spherical shells

Bi

36 367 2.78 12.84 198 2.86 7.766 115 3.45 -1.187

48 722 2.95 51.82 386 3.02 48.13 217 3.09 40.88

60 1242 2.95 86.9 660 3.15 84.28 362 3.4 81.4

Cylindrical shells (K x + Ky ) 72 36 48 60 72 175 361 712 1448 2300 2.949 2.99 2.98 3.00 3.09 123.69 10.32 41.18 98.2 162.565 942 105 380 794 1222 3.2 2.97 2.93 3.16 3.21 122.57 5.56 38.89 91.95 158.45 535 112 215 438 682 3.5 2.98 3.11 3.3 3.4 119.04 4.93 32.58 86.96 149.97

Figure 4.62. The curves of relative deflections of the spherical (K x = Ky = 24) and cylindrical (K x = 0, Ky = 48) shells under a convection (Bi = 0.2).

For the shells with parameters (K x + Ky ) < 48, the stresses for Bi = 0.8 change their sign. The equality of σ0x for the corresponding shells types are observed for (K x + Ky ) = 58 of non-dimensional units. To conclude, the influence of the (K x + Ky ) parameters on the critical heat loads and the corresponding stresses is stronger for the case of the convection heat transfer than for the heat impact for arbitrary values of the non-dimensional heat transfer parameter.

4.5 Stability of Flexurable Conical Shells with Convection

489

Figure 4.63. The curves of relative deflections of the spherical (K x = Ky = 24) and cylindrical (K x = 0, Ky = 48) shells under a convection (Bi = 0.4).

Figure 4.64. The curves of relative deflections of the spherical (K x = Ky = 24) and cylindrical (K x = 0, Ky = 48) shells under a convection (Bi = 0.8).

490

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

Figure 4.65. The cylindrical (a) and spherical (b) shell centre vibrations for Bi = 0.2.

The minimal values of the critical temperature increments θ10 , deflection w and the stresses σ0x for diﬀerent shells with the increase of Bi from 0.2 to 0.8 are given in Table 4.10. We consider the vibrational process of the surface points on the example of shells with (K x + Ky ) = 48. For this aim, the dependencies w(x, y)/w(0.5, 0.5) are

4.5 Stability of Flexurable Conical Shells with Convection

491

Figure 4.66. The cylindrical (a) and spherical (b) shell centre vibrations for Bi = 0.8.

derived. We compare the vibrational process with convection for Bi = 0.2 (Fig. 4.62) and with a heat impact (Fig. 4.5). The convection heat transfer is characterized by large (in comparison to the heat impact) amplitudes of vibrations and of the maximal possible pre-critical deflection.

492

4 Dynamical Behaviour and Stability of Rectangular Shells with Thermal Load

During the convection, a maximal deflection vibration between the shell’s centre and the centres of each of its quadrants is independent of each shell’s type. Now we analyse the vibrational processes of the surface points increasing Bi from 0.2 to 0.8 on the example of shells with (K x + Ky ) = 48. We compare the following results: for Bi = 0.2 (Fig. 4.62); for Bi = 0.4 (Fig. 4.63) and for Bi = 0.8 (Fig. 4.64). With the increase of the dimensionless heat transfer parameter Bi the vibrational process amplitude is increased and the values of minimal deflection are decreased, whereas those of a maximal one are increased. The relative maximum for the time instant when a deflection in the centre achieves its minimum, is increased. The point of maximal deflection is shifted from the centre of each quadrant of the spherical shells to their edges and 8 points of maximal deflections are observed. In the case of cylindrical shells, they are shifted to zero curvature direction and 4 points of maximal deflections are observed. For cylindrical shells for Bi = 0.8 for the time instant when a deflection in the centre achieves its minimum, also an area with a negative deflection is observed in the region of the shell’s centre. We consider the vibrational process on the example of the central points of shells with (K x + Ky ) = 48 on the basis of w(τ) dependence for Bi = 0.2 (Fig. 4.65) and Bi = 0.8 (Fig. 4.66 a - spherical shells; b - cylindrical shells). The figures show that increase Bi parameter practically does not influence the vibrations. However, the amplitudes and maximal deflections increase, and the axis of the vibrational process is shifted in the direction of its initial value. The time of maximal (possible) first pre-critical deflection is decreased.

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells with Thermal Load

In section 5.1, research devoted to plates and shells with rectilinear, as well as straight and curvature lines contours are reviewed. In section 5.2 theory of flexurable sectorial shells computations is introduced. It includes derivation of fundamental relations and diﬀerential equations. Then a thermal field and the “set-up” method is introduced and numerical results reliability is discussed. Stability of sectorial shells with finite deflections id studied in section 5.3, where numerous problems are carefully analysed. In section 5.4 a novel approach to study chaotic vibrations of shallow sector-type spherical shells without thermal eﬀects is proposed. Scales of vibration character of such shells being transversally and harmonically excited vs. control parameters are constructed. Scenarioto chaos are illustrated and discussed. Control of chaotic state applying synchronous action of harmonic loading torque is proposed.

5.1 Introduction Historical Review We analyse some works devoted to plates and shells computations with diﬀerent characteristics. Plates and shells with rectilinear edges The considered plates and shells consist of simple convex polygons with sides number n ≥ 3. Warburton [699] has presented the first collection of solutions to rectangular plates’ problems. Leissa [425, 427] has collected and reviewed comprehensive literature dealing with free vibration of plates, up to 1977. The stability problems of triangle plates are considered in references [4, 645, 716]. In most cases, the numerical methods are used. For instance, in the work [4] the finite diﬀerence method with triple and sixple meshes of shells is applied. The finite diﬀerence method belongs nowadays to the most universal and eﬀective methods. Its application to plates and shells with complicated shapes can be considered in various coordinate systems, i.e. rectangular or polar [321, 348]. In the works [645, 716] the finite element method is applied. The critical load values are obtained for uniformly compressed triangle plates.

494

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Experimental investigations of triangle plates are carried out using the methods of photoelasticity [141]. As an investigation object a triangle plate clamped on its contour and subjected to the uniformly distributed force action is used. The most stretching stresses have appeared on the angle bisectrix and along the plate’s angles. Trapezoidal plates are investigated by many authors [152, 462, 497]. Their stability is analysed using the Bubnov-Galerkin method [236, 557]. In the latter reference diﬀerential equations of the nonlinear theory of thin plates are used, and theoretical results are compared with the experimental ones. In reference [630] during the analysis of a trapezoidal plate with a triangle contour and finite deflection, the Bubnov-Vlasov method is used. As a function approximating a deflection, an elastic beam deflection is taken. The vibration behaviour of stiﬀened rectangular plates simply supported along the edges perpendicular to the stiﬀeners are studied by Wah [684] and Long [442, 443]. The free vibrations and dynamic response of simply supported rectangular plates has been analysed by Kirk [325], Aksu and Ali [10], and Ochs and Snowdon [517]. Smith et al. [626], Olson and Hazell [522] and Laura and Guti´errez [411] obtained natural frequencies and mode shapes of stiﬀened square and rectangular plates with damped or elastically restrained edges. Clarkson and Cicci [155], Yurkovich et al. [715], Olson and Lindberg [523], and Donaldson [183] studied a dynamic response of the stiﬀened panel structures. Bhandari et al. [103] studied stiﬀened skew plates. Bapu Rao et al. [84] experimentally studied a skew cantilever plate with stiﬀener. An analysis of cantilever plates was reported in [85, 154]. Irie et al. [280] studied trapezoidal cantilever plates with non-uniform stiﬀeners. A trapezoidal plate was transformed into a square region of unit length by an ingenious transformation of variables. The transverse deflection of the transformed square plate was expressed in a series of the products of the deflection functions of beams parallel to the edges of the plate and satisfying the boundary conditions. A strain and kinetic energies of the system were evaluated analytically, and the frequency equation was derived by the conditions for a stationary value of the Lagrangian function. The method is applied to square parallelogram or trapezoidal cantilever plates with several stiﬀeners of the same material as the plates. Nair and Durvasula [493] have analysed the vibration of a skew plate by the Ritz method, and Chopra and Durvasula [151, 152] have analysed the vibration of a trapezoidal plate by the Bubnov-Galerkin method. Conway [156], and Walkinshaw and Kennedy [696] have studied vibrations of polygonal plates by point matching, and Shahady et al. [605] have analysed the same problem by a complex variable theory. Irie et al. [283] have investigated regular polygonal plates, and have obtained natural frequencies and the mode shapes including higher order modes. In the latter work, the authors have proposed a new series-type method for estimation of the eigenvalues of non-homogeneous plates with diﬀerent boundary conditions. For this purpose, a plate is assumed to be clamped along an internal segment, and with the reaction forces and bending moments (acting on the segment) regarded as unknown harmonic forces and moments, its stationary response is expressed by the use of the Green functions. The unknown forces and moments distributed along the

5.1 Introduction

495

segment are expanded into the Fourier series with unknown coeﬃcients. The homogeneous linear equations with unknown coeﬃcients are given in a matrix form. The obtained eigenvalues and eigenvectors yield the natural frequencies and the mode shapes of the plate. Irregularly shaped plates can be build using a simply supported plate via clamping appropriate segments. The introduced theory using been applied to a cross shaped and I-shaped plate with two geometrically symmetrical axes, and to a L-shaped plate with a diagonal symmetrical axis. The natural frequencies and the mode shapes of the plates have been calculated numerically. Waller [698] has obtained Chladni’s figures of free polygonal plates experimentally. With the use of the point-matching method, Conway [156] has studied the fundamental vibration of simply supported polygonal plates, and Walkinshaw et al. [696] has obtained the frequencies of axially-symmetric vibrations of simply supported and clamped plates. Laura et al. [412, 416] and Yu [714] have analysed the same problem by a complex variable method. Applying the Ritz method, Young [706] has studied the free vibrations of a clamped square plate, and Ota et al. [525] investigated a triangular plate. Irie et al. [283] have proposed an analytical method to study the free vibrations of polygonal plates clamped at the edges. Conway [156], and Walkinshaw and Kennedy [696] have investigated analytically free vibrations of regular polygonal plates by the point-matching method, whereas Laura and co-workers [255, 605], and Yu [714] by the conformal mapping variational technique (see also Roberts [575]). Laura and Luisoni [414] have studied free vibrations of regular polygonal membranes applying the Bubnov-Galerkin method. Conway and Farnham [157] have analysed an equilateral triangular membrane, and Williams et al. [702] have studied the vibrations of a triangular plate both theoretically and experimentally. Durvasula [189], and Bauer and Reiss [90] have studied skew membranes, and the obtained results have been converted into those of an equilateral triangular membrane. Bauer and Reiss [91] have obtained the first 21 cutoﬀ frequencies and modes of a regular hexagonal wave-quide by a combination of the finite diﬀerences and numerical techniques. Free vibrations of regular polygonal plates with simply supported edges are studied using the membrane dynamical analogy by Irie et al. [279]. A regular polygonal membrane is formed on the rectangular plate by fixing several segments. The natural frequencies and mode shapes have been calculated numerically. In the work [638], the finite element method with an application of a triangle element is used to analyse nonlinear deflections of the rhomboidal plates. During investigation of a stress-strain state of non-homogeneous shells, a theoretical-experimental method is proposed in reference [601]. Structural dependence of the being sought quantities on the geometrical and physical parameters and the external loads is established. An investigation of large deflection of parallelogram plates with uniformly distributed load using the theoretical-experimental method is carried out in [105]. In reference [298], an analytical method to solve the nonlinear problems of shells deflection is outlined.

496

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

A stability of an uniformly compressed plate is investigated in reference [400] using the complex variable method. The critical load for free and clamped plates with polygon shapes is calculated. Plates and shells with straight and curvuture lines contour. A stability of elliptic, parabolic and half-circle plates, compressed on the whole contour, is analysed in reference [468]. In order to find a solution, the method of equal deflection curve coupled with the Bubnov-Galerkin method, has been used. A relative diﬀerence of the free support circle, for which the exact solution is known, has exceeded 2%. Waller [698], and Maruyama and Ichinomiya [464] have studied experimentally the free vibration of isotropic sectorial plates. Ben-Amoz [99], Westmann [700] and Rubin [579] have analysed the same problems theoretically, and also Ramakrishnan and Kunukkasseric [565], Ramaiah and Vijayakumar [564], Bhattacharya and Bhowmic [104] and Wilson and Garg [705] have studied ring shaped isotropic sectorial plates with various circular edge conditions theoretically. Rubin [578, 580] has investigated both vibration and stability of polar-orthotropic sector plates using series. Irie et al. [281] have analysed the free vibration of a ring-shaped polar-orthotropic sectorial plates by the Ritz method using a spline function as an admissible function for the plates deflection. Roberts [88], Laura and Romanelli [412] have studied free vibrations of epicycloidal membranes. Laura et al. [149, 413, 416] have determined the cutoﬀ frequencies of the wave-quides with epicycloidal cross-section. Suﬃcient engineering data, natural frequencies (the eigenvalues of vibration) are presented for epicycloidal plates by Irie et al. [284]. In reference [474] the Ritz method has been used to analyse small deflection of a half-circled plate with clamped edges. In reference [216], a solution in the polar coordinate system has been found for the plates bounded by two arcs of the concentrical circles. The plate has been subjected to normal transversal force action. In the works [344, 526], the results of stability investigation of post-critical state of plates and shells with complicated contour shapes (cutted circle, sixangle, triangle, parallelogram subjected to a transversal load and contour forces have been given). Using the hypothesis of straight verticals, in reference [32] a system of three equilibrium equations for displacements for arbitrary formed shells has been obtained. It is solved by the finite diﬀerence method. The calculations are carried out for the shells with an elliptic paraboloid shape. Cylindrical shells with a complicated form (convex and concave surface parts) are analysed in references [243, 244]. Diﬃculties due to the mathematical description of the shell’s surface are omitted applying spline approximations to the experimental data. In the reference [607], a calculation algorithm devoted conical shell analysis with a complicated contour by means of both linear orthogonal mesh and finite element methods is developed.

5.1 Introduction

497

The stress-strain state problems of isotropic plates of a circle sectorial and a circle rectangular forms are solved in the references [304, 658]. In the works [229, 230], a deflection of sectorial plates with arbitrary boundary conditions on its arc part, and with clamped straight linear boundaries is investigated. For the deflection function, the orthonormal system of special polynoms is applied. The solution is found tracing the system energy. In a frame of Kirchhoﬀ-Love hypotheses, the deflection of a thin circle sectorial plate is found. Using Euler method a diﬀerential equilibrium equation of a plate, is reduced to the nonhomogeneous one with constant coeﬃcients. A solution to the homogeneous equation is found using the method of variables separation. A particular solution has been found in the form of a trigonometric series. In the reference [530], the partial diﬀerential equations are reduced to the ordinary ones. In the work [105], in order to find a solution, the Bubnov-Galerkin procedure is used, whereas in reference [524], the finite diﬀerence method is applied. In references [254, 563], the sectorial plates are analysed, and the finite element is defined as a ring sector, whereas in reference [137], a solution of the sectorial orthotropic plate is given in the form of a trigonometric series. It follows from a brief review of the works devoted to the investigations of plates and shells with complicated plane that the sectorial plane of plates and shells is investigated rather slightly. The existing solutions are applied mainly to the sectorial plates in the frame of the linear theory. Therefore, a consideration of dynamical and statical problems of sectorial shells in a frame of non-linear geometry is required. The following notation is applied: w F r, θ U, V a h R γ ρ E ν g ω0 ε W0 α, λq , λT

- normal displacement of the mean surface; - force; - polar coordinate system; - displacements in the r, θ directions; - shell’s radius in plane; - shell thickness; - curvature radius of the mean surface; - specific gravity of the material; - material density; - Young modulus; - Poisson’s coeﬃcient; - earth acceleration; - frequency of external forcing; - damping coeﬃcient; - heat amount of the shell volume unit and the time unit; - thermal transfer coeﬃcient, heat transfer coeﬃcient and linear thermal extension coeﬃcient;

498

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

αq C q

pr , pθ , q t τ T0, T ei j M, N

- heat giving back coeﬃcient; - ideal specific heat capacity; - heat flow density in the direction of a normal to mean surface; - intensity of the external loads along the coordinates r, θ, z; - time; - non-dimensional time; - initial temperature, temperature; - components of deformation tensor for arbitrary shell’s point; - nodes number (including a contour) corresponding to the axes θ and r; εi j - deformation tensor components for the mean surface; σi j - stress tensor components; Nr , Nθ , Nrθ - forces on the mean surface; Mr , Mθ , Mrθ - bending and torsional torques; Qr , Qθ - transversal forces; θk - central angle of a sectorial shell; √ a2 b - sloping parameter: b = η Rh ; The following quantities are introduced: αq λq α2 B= h, α = , κ = 2 4 , ω0 = λq cρ ω0 h

0 Eq , γR2

RλT h b2 h2 η = 12 1 − ν2 , κ0 = , λ= 2 . λq a √ 2 ηq R r F √ w r = b , w = η , F = η 3, q = , a h 4E h Eh τ=

α z RλT εh2

RλT , , , κT = RκT , z = T = T q = q , ε = t, h h λq α h2 κ εT =

Mi j R R √ Ni j R √ R εT , M i j = η , N i j = η 2 , εi j = η εi j , h h Eh3 Eh W 0 = κ0 W0 .

5.2 Flexurable Conical Sectorial Shells Computations In this section a mathematical background of the uncoupled thermoelastic problem is given, and the methods and algorithms of sectorial shells with finite deflections analysis are proposed. The relations for deformation and displacement components,

5.2 Flexurable Conical Sectorial Shells Computations

499

and the forces and moment versus deformations are derived using the geometrical and statical Kirchhoﬀ-Love hypotheses. A relation between deformations and displacements is of second power form, and this approach is widely applied in the geometrical nonlinear theory of shells. The thermal field is defined through a solution to the three dimensional heat transfer equation for arbitrary boundary conditions independent of the elasticity equation. On the basis of the considered hypotheses and assumptions the hybrid form equations system for displacement is achieved. In the last case, all quantities are expressed by the deflection function w and the force function F. 5.2.1 Fundamental Relations, Diﬀerential Equations, Boundary and Initial Conditions Using the Vlasov’s hypotheses, the fundamental relations and equations in the polar coordinate system are introduced. For a conical shell, a geometry of its surface can be approximately treated as overlapping with a geometry of its projection. Therefore, a position of an arbitrary point on the surface in the polar coordinate system is defined by two quantities: r, θ, measured from the initial vector radius r0 . The coordinate z is directed along the normal to the mean curvature surface. Consider the a b c d shell’s element presented in Fig. 5.1.

Figure 5.1. Shell computation scheme.

500

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.2. Stresses and moments of the shell element.

The extension and shear deformations relations are similar to those of Descartes coordinate system [684], except for the relative extension along the θ axis and the angle deformation. Both of them depend on the displacements U and V.

5.2 Flexurable Conical Sectorial Shells Computations

501

Therefore, the geometrical Cauchy relations in the polar coordinate system have the form: 2 ∂U w 1 ∂w ∂V lU w 1 ∂w − + ; εθ = + − + ; εr = ∂r R 2 ∂r R 2 r ∂θ r ∂θ r ∂U ∂w ∂w ∂V V − + ; (r, θ) → − (x, y) . (5.1) + r ∂θ ∂r r r ∂θ ∂r According to the Kirchhoﬀ-Love hypotheses, full deformation of an arbitrary point along thickness er , eθ , , erθ are composed of an average surface deformation and a deflection deformation: er = εr + zκr , γrθ =

eθ = εθ + zκθ , erθ = γrθ + 2zκrθ , where:

(5.2)

1 ∂2 w ∂2 w 1 ∂w − 2 2 , κr = − 2 , κθ = − ∂r r ∂r r ∂θ ∂ 1 ∂w ∂w 1 ∂w , υr = − , υθ = − . κrθ = − ∂r r ∂θ ∂r r ∂θ

In the above, υr denotes rotation angle of a normal in the plane zOr; υθ is rotation angle of a normal along the axis r. We consider an equilibrium of the shell’s element cut by two meridial planes and two planes perpendicular to them. The meridial σr , σθ and tangential σrθ = σθr , σrz , σθz stresses appear on the element’s sides. We introduce statically equivalent forces and torques: h

h

2 Nr =

h

2 σr dz, Nθ =

h −2

2 σθ dz, Nrθ = Nθr =

h −2

h −2

h

h

2 Qr =

2 σrz dz, Qθ =

h −2 h

h −2 h

2 σr zdz, Mθ =

h −2

σθz dz;

h

2 Mr =

σrθ dz;

2 σθ zdz, Mrθ = Mθr

h −2

σrθ zdz.

(5.3)

h −2

In the above, the Nθr , Nrθ and Qr are the normal, shear and transversal forces acting in the meridial direction; Nθ , Nθr = Nrθ , Qθ are the normal, shear and transversal

502

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

forces on the radial direction; Mr and Mrθ are the bending moment and the torque in the meridial direction; Mθ , Mθr = Mrθ are the bending moment and the torque in the radial direction. The internal forces and moments are related to the length unit and to the corresponding coordinate line (a circle or a radius) of the mean surface. Positive direction of internal forces and moments, as well as of the external load, are presented in Fig. 5.2. The introduced internal forces, moments and external forces allow (instead of the shell’s space equilibrium element) for a consideration of the corresponding element of its mean surface. The deformations (5.2) consist of elastic ones caused by the stresses σrθ , σθ , σr and a purely heat deformation. Using well-known relations obtained from the Hooke’s rule for the two dimensional stress state, and taking into account only a relative heat stretching, we get: er =

σr − νσθ + αT (T − T 0 ) , E

σθ − νσr + αT (T − T 0 ) , E 2 (1 + ν) σrθ . (5.4) erθ = E Multiplying (5.4) by dz, and then by zdz, we proceed with the integration within the interval from z = − h2 to z = − h2 . Using the relations (5.2) and (5.3), we get: eθ =

εr =

Nr − νNθ Nθ − νNr 2 (1 + ν) + εT , εθ = + εT , γrθ = Nrθ ; Eh Eh Eh

(5.5)

12 (Mr − νMθ ) 12 (Mθ − νMr ) 12 (1 + ν) +κT , κθ = +κT , κrθ = + Mrθ ; (5.6) Eh3 Eh3 Eh3 where εT and κT are generalized purely heat deformations: κr =

h

h

εT =

1 h

2 αT (T − T 0 ) dz; κT = h −2

12 h3

2

αT (T − T 0 ) zdz.

(5.7)

h −2

Owing to (5.5) and (5.6), the relations between the moments, forces, and deformations are found: Nr =

Eh Eh [εr + νεθ − (1 + ν) εT ] , Nθ = [νεr + εθ − (1 + ν) εT ] , 2 1−ν 1 − ν2 1 − ν Eh γrθ ; 2 1 − ν2 Mr = D [κr + νκθ − (1 + ν) κT ] , Nrθ = Nθr =

(5.8)

5.2 Flexurable Conical Sectorial Shells Computations

Mrθ = Mθr = (1 − ν) Dκrθ .

503

(5.9)

We derive the equation governing the shell’s element motion. According to the D’Alembert principle, we add the inertial forces to both given forces and dynamical reactions of the neighbourhood elements. The equations system governing a motion of the shell’s deformed mean surface element consists of projections equation of all forces on the coordinate axes and the equations for moments in relations to those axes. The inertial terms occurring as a result of rotations are not included in the considerations. Neglecting the second order terms, the equations governing a motion of the shell’s element [684] in the polar coordinate system follow: γ ∂2 U ∂Nr r ∂Nθr + − Nθ + Pr − h 2 = 0, ∂r ∂θ g ∂t γ ∂2 V ∂Nrθ r ∂Nθ + − Nrθ + Pθ − h 2 = 0, ∂r ∂θ g ∂t ∂Qθ 1 1 ∂Qr r + − rMr + κr − rNθ + κθ − ∂r ∂θ R R 2 γ ∂ w 2rNrθ κrθ + r q − h 2 = 0, g ∂t ∂Mrθ r ∂Mθ + + Mrθ − Qθ r = 0, ∂r ∂θ

(5.10)

(5.11)

(5.12) (5.13)

∂Mrθ r ∂Mrθ + − Mθ − Qr r = 0. (5.14) ∂r ∂θ If we consider a dynamical process without elastic waves propagation, then in equations (5.10), (5.11) the inertial terms are deleted. Those equations will be satisfied by identity, when we introduce the stress function in the averaged surface [684] owing to relations: 1 ∂2 F ∂F ∂2 F + 2 2 , Nθ = 2 , Nr = ∂r r ∂r r ∂θ ∂ 1 ∂F , (r, θ) → Nrθ = Nθr = − − (x, y) , (5.15) ∂r r ∂θ where is a transition parameter from the polar coordinate system to the Descartes one. We need equal deformations condition of the form [684]: 1 ∂2 γrθ r 1 ∂2 εr 1 ∂2 εθ r 1 ∂εr = . + − r ∂r2 r ∂r r2 ∂θ2 r2 ∂r∂θ

(5.16)

Expressing in (5.16) the deformations of the mean surface by the forces given in (5.5), and taking into account (5.15) we get the deformation continuity equation: 1 2 2 1 1 ∇ ∇ F + ∇2 εT = − N (w, w) − ∇2 w, Eh 2 R

(5.17)

504

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

where: ∇2 = ∇2 ∇2 =

1 ∂2 ∂2 ∂ + + , ∂r2 r ∂r r2 ∂θ2

2 ∂4 ∂4 2 ∂3 ∂2 ∂ + + − + − ∂r4 r ∂r3 r2 ∂r2 r3 ∂r r2 ∂θ2 ∂r2

1 ∂4 4 ∂2 2 ∂3 + 4 2 + 4 4 , 2 3 r ∂θ ∂r r ∂θ r ∂θ 2 2 2 ∂ 1 ∂w 1 ∂w ∂ w ∂w + −2 . N (w, w) = 2 2 ∂r r ∂θ ∂r r ∂r r2 ∂θ2 In order to obtain the motion of the shell’s element in a hybrid form, we use the formulas (5.12)–(5.14). Reducing the forces Qr , Qθ , and expressing the moment by the formulas (5.9), and the forces by the formulas (5.15), we get: % $ 1 −D ∇2 ∇2 w + (1 + ν) ∇2 κT + N (w, F) + N (w, F) + ∇2 F + q = R 2 γ ∂w ∂w h , +ε g ∂t2 ∂t

(5.18)

where: 1 ∂2 F 1 ∂2 w ∂2 w ∂F ∂2 F ∂w + + N (w, F) = 2 + 2 − ∂r r ∂r r2 ∂θ2 ∂r r ∂r r2 ∂θ2 ∂ 1 ∂w ∂ 1 ∂F . 2 ∂r r ∂θ ∂r r ∂θ The equations (5.17)–(5.18) govern a motion of the shell’s elements. We are going to get the equations system for displacements. We express the forces of (5.10), (5.11), (5.12) by the deformations using the formula (5.8), and the deformations are expressed through the U, V, w displacements owing to (5.1). The relations between stresses and displacements read ⎧ 2 ⎪ Eh ⎪ ⎨ ∂U w 1 ∂w − + − (1 + ν) εT + Nθ = ⎪ ⎩ ∂r R 2 ∂r 1 − ν2 ⎪ ⎡ ⎢⎢ ∂V U + ν ⎢⎢⎢⎣ r∂θ r ⎧ ⎪ Eh ⎪ ⎨ ∂U U + Nr = ⎪ ⎩ r∂θ r 1 − ν2 ⎪ ⎡ ⎢⎢ ∂U − ν ⎢⎢⎢⎣ ∂r

2 ⎤⎫ ⎪ w 1 ∂w ⎥⎥⎥⎥⎪ ⎬ − + ⎥⎪ , ⎭ R 2 r∂θ ⎦⎪ 2 w 1 ∂w − + − (1 + ν) εT + R 2 r∂θ 2 ⎤⎫ ⎪ w 1 ∂w ⎥⎥⎥⎥⎪ ⎬ + ⎥⎪ , ⎭ r 2 ∂r ⎦⎪

5.2 Flexurable Conical Sectorial Shells Computations

Nrθ =

505

∂U ∂V V ∂w ∂w Eh + − + . 2 (1 + ν) r∂θ ∂r r r∂θ ∂r

(5.19)

From (5.10) we get: ν − 3 ∂V 1 + ν ∂w ∂2 U 1 − ν ∂2 U ∂U U 1 + ν ∂V − + + − + r+ + 2 2r ∂θ2 ∂r r 2 ∂r∂θ 2r ∂θ R ∂r ∂r 2 1 − ν ∂w 1 + ν ∂w ∂2 w ∂2 w ∂w 1 − ν ∂2 w ∂w + + − + 2 ∂r 2r ∂θ ∂r∂θ ∂r2 ∂r 2r ∂θ2 ∂r 2 ∂ 1 − ν2 γ 1 − ν 2 ∂2 U 1 + ν ∂w pr − − (1 + ν) εT + = 0. (5.20) 2 r∂θ ∂r Eh g E ∂t2 Proceeding in a similar way, we obtain from (5.11): 1 + ν ∂2 U 3 − ν ∂U ∂2 V 1 − ν ∂2 V 1 − ν ∂V U + + r 2 + − + + 2 ∂r∂θ 2r ∂θ 2 2 ∂r r r∂θ2 ∂r 1 + ν ∂2 w ∂w 1 − ν ∂w ∂2 w 1 ∂w ∂w + + + 2 ∂θ∂r ∂r 2 ∂θ ∂r2 r ∂θ ∂r ∂ 1 − ν2 γ 1 − ν2 ∂2 V 1 ∂w ∂2 w ν ∂w − (1 + ν) εT + pθ − − = 0. 2 r ∂θ ∂θ R ∂θ ∂θ Eh g E ∂t2

(5.21)

Reducing Qr and qθ from equation (5.22) using (5.13) and (5.14), one gets: 1 ∂2 Mr r 1 ∂2 Mθ 2 ∂2 Mrθ r 1 ∂Mθ − + + + r ∂r2 r ∂r r2 ∂θ2 r2 ∂r∂θ 1 ∂2 w 1 1 ∂w 1 ∂2 w Nr + + + + Nθ + R ∂r2 R r ∂r r2 ∂θ2 γ ∂2 w ∂ 1 ∂w + q − h 2 = 0. 2Nrθ ∂r r ∂θ g ∂t

(5.22)

Substituting in equation (5.22) the moments by displacements owing to (5.9), and substituting the forces by the displacements through (5.19), the motion equation for displacements (projected on the normal to the averaged surface) is obtained −

h2 2 2 ∇ ∇ w + (1 + ν) ∇2 κT + 12

w ∂U ∂V U + + − (1 + ν) − (1 + ν) εT + ν ∂r r∂θ r R 1 1 ∂w 1 ∂2 w ∂U ∂V U w + + 2 2 +ν + ν − (1 + ν) − (1 + ν) εT + R r ∂r r ∂θ ∂r r∂θ r R ∂ 1 ∂w ∂U ∂V U ∂w ∂w 1 (1 − ν) + − + + ∂r r ∂θ r∂θ ∂r r ∂θ ∂r r

1 ∂2 w + R ∂r2

506

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

q

1 − ν 2 γ 1 − ν 2 ∂2 w − = 0. Eh g E ∂t2

(5.23)

In order to get the equation systems (5.17)–(5.18) and (5.20)–(5.23), we need to add an unstationary three dimensional heat transfer equation in the polar coordinates system of the form: ∂2 T w0 1 ∂T = 2 + ∇2 T + . (5.24) α ∂t λq ∂z In the heat transfer equation (5.24) a mechanical coupling term does not appear, and therefore the temperature and deformation fields are not coupled. Integrating the equations (5.17)–(5.18), (5.20), (5.21), (5.23), and the heat transfer equation (5.24) use the boundary and initial conditions should be attached. Various mathematical models of the boundary conditions are applied owing to occurrence of diﬀerent support in real shell’s constructions. Sometimes a support type influences the choice of the initial diﬀerential equations. We consider some of the boundary conditions. For the equations related to w and F, the following boundary conditions can be used. 1. Rolling support on the arcal elements w = Mr = Nrθ = Nr = 0 for r = a, 0 < θ < θk .

(5.25)

2. Rolling support on the radial elements w = Mθ = Nθ = εr = 0 for θ = 0, θk , 0 < r < a.

(5.26)

3. Slip clamping along arcs w=

∂w = Nrθ = Nr = 0 for r = a, 0 < θ < θk . ∂r

(5.27)

4. Slip clamping along radiuses w=

∂w = Nθ = εr = 0 for θ = 0, θk 0 < r < a. ∂θ

(5.28)

The following boundary conditions are applied with respect displacements: 1. Stiﬀ clamping of the arcal elements w=u=ν=

∂w = 0 for r = a, 0 < θ < θk . ∂r

(5.29)

2. Stiﬀ clamping of the radial elements w=u=ν=

∂w = 0 for θ = 0, θk 0 < r < a. ∂θ

(5.30)

5.2 Flexurable Conical Sectorial Shells Computations

507

3. Rolling-unmovable clamping of the arcal elements u = ν = w = Mr = 0 for r = a, 0 < θ < θk .

(5.31)

4. Rolling unmovable clamping of the radial elements u = ν = w = Mr = 0 for θ = 0, θk 0 < r < ba.

(5.32)

In the above 2 1 ∂2 w ∂w Eh3 ∂w 1 + − (1 + ν) κT , +ν Mr = − & ' ∂r r r2l ∂θ2 12 1 − ν2 ∂r2 1 ∂2 w 1 ∂w Eh3 ∂2 w + Mθ = − & + ν 2 − (1 + ν) κT . ' 12 1 − ν2 r ∂r r2l ∂θ2 ∂r

(5.33)

(5.34)

In the case, when Qk = 2π a top of the spherical segment becomes the shell’s internal point. Because in the equation the terms with multipliers r−n (n = 1, 2, 3, 4) occur, then the integration of the equation from the point r = 0 using the numerical methods is impossible. Therefore, we need to formulate the additional conditions on the top. The solving function’s behaviour, in a close neighbourhood of the top, can be analysed in a diﬀerent way. For instance, in reference [344] using a limiting case, the equations valid in the top neighbourhood are obtained. However, they cannot be used in many cases, including our. In the reference [697] another approach is proposed. The transformation of the fundamental equation leads to exactly solved linearized equations, which can be used for r = 0. However, an analytical solution to the linearized equation cannot always be found and for our equations a similar approach can not be applied. Therefore, sometimes in the neighbourhood of the shell’s top, its geometry can be modified. For instance, in the reference [640], a neighbourhood of the top is replaced by a circle plate of constant, thickness and then the exact solutions are found. However, this approach can be used only in the case of a symmetric solution. In the monograph [696] this problem in a close top neighbourhood of the conical spherical shells is analysed. The solution to the linear equations are found in a series form. For an axially symmetric solution, only the first terms of the series are taken: w0 = A +

B 2 r + O r3 , 2

(5.35)

D 2 r + O r3 . (5.36) 2 where w0 is deflection and F0 is the stress function of the shell top. Increasing the terms number does not improve the solution accuracy. Observe that in a small neighbourhood of the shell’s top, one of the fundamental assumptions of the thin shell’s theory about smallness of h/R is not satisfied. Therefore, the obtained equation only approximately describes a real picture of the stress strain state. F0 = C +

508

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

In most cases of the numerical method applications it is assumed that the shell has a central hole with small dimensions, which only very slightly influences the obtained solution. In the reference [104] it is assumed that the shell is clamped on the circle with a small radius r0 . In order to solve the axially symmetric problems, the being sought functions in the point r = 0 are interpolated by the Lagrange 2-nd order formula of the form: f 0 = 3 f 1 − 3 f 2 + f3 ,

(5.37)

where: fi = f (ri ), ri = i∆ (i = 0, 1, 2, 3) and ∆ denotes a distance between the interpolation nodes. For an outside contour point, the following symmetry condition is satisfied: (5.38) f−1 = f1 . The numerical comparison of the results obtained using formulas (5.35)–(5.38) is carried out in the next section. The solutions found using (5.37), (5.38) only slightly diﬀer from the results obtained from (5.36). The relative errors, for the deflection function and the forces function, achieve 1% and 4%, respectively. The conditions (5.37), (5.38) are more simple. They do not need transformations of the equations, and the applied algorithms are simple, which is not true for the case of (5.35) and (5.36). In order to solve the dynamical problem we need to introduce initial conditions in time t = 0 of the form: ∂w = ϕ2 (r, θ) . (5.39) w = ϕ1 (r, θ ) ; ∂t In addition, the boundary and initial conditions for the heat transfer equation (5.39) should be formulated. 1. Temperature distribution on the surface body is assumed (1-st order boundary conditions) T (r, θ, z, t) = ψ(r, θ, z, t), (5.40) where: (r, θ, z) is the point of the body surface; ψ(r, θ, z, t) is the given function. 2. The heat flow intensity q (r, θ, z, t) (2-nd order boundary conditions) transfered through the body surface has the form ∂T (r, θ, z, t) = q (r, θ, z, t) , (5.41) ∂n where n denotes the external normal to the surface body in the point (r, θ, z). 3. Temperature distribution of the surrounding medium and a rule of convection between the surface body and the surrounding medium is assumed (3-rd order boundary conditions) $ % ∂T (r, θ, z, t) = αq T (r, θ, z, t) − T cp . (5.42) −λq ∂n The initial conditions for the heat transfer equations (5.39) have the form: −λq

t = 0,

T = T0.

(5.43)

5.2 Flexurable Conical Sectorial Shells Computations

509

All initial equations and boundary conditions will be given further in the nondimensional form. Imagine we need to find a solution to the nonlinear boundary problem governing a statical equilibrium of the sectorial shells ∇2 ∇2 w − N (w, F) − ∇2 F − 4q = 0,

(5.44)

∇2 ∇2 F + ∇2 w + N (w, w) = 0,

(5.45)

with the boundary conditions (5.25)–(5.28). A solution to the stationary problem is found via analysis of an additional nonstationary problem. For this aim, we introduce the internal and damping terms into (5.44), and we get: ∂w ∂2 w = −∇2 ∇2 w + N (w, F) + ∇2 F + 4q. +ε ∂r ∂τ2

(5.46)

Since the load q(r, θ) and the functions w, F, on the border do not depend on time, we can expect that a solution to the unstationary problem will be changed slowly in time and for τ → ∞ it will achieve a solution to the stationary problem. In order to realize a vibration damping, in equation (5.46) ε parameter is introduced. By a suitable choice of the damping parameter a stationary state has been achieved. The algorithm of the solution to the unstationary problem is given in the next section. The mentioned set-up method, does not require a special solution to the problems of statics. A solution can be easily obtained using the algorithm of computations for a dynamical problem. 5.2.2 Thermal Field and Set-Up Method Earlier, the partial diﬀerential equations (5.17), (5.18) are obtained with respect to the functions w = w(r, θ, t) and F = F(r, θ). The exact solution to these equations does not exist. However, there are methods reducing the approximate integration of equations for diﬀerent boundary conditions. Here the finite diﬀerence method is applied. The initial diﬀerential equations are substituted by algebraic equations, and then they are solved using numerical methods. The finite diﬀerence method has a wide spectrum of approximating formulas. It slightly depends on the boundary value conditions, the shell’s geometry and on the initial stress state. It is also simple in realization and suitable for the programming purpose. However, high order of the algebraic equations system belongs to one of its drawbacks. Let a projection of the mean surface of the sectorial conical shell into the coordinate plane occupying the area Ω, bounded by the contour Γ Ω = Ω + Γ = {(r, θ) , 0 ≤ r ≤ a, 0 ≤ θ ≤ θk } . We introduce a polar mesh in Ω of the form

510

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

ω1 = {ri = ihr , 0 ≤ i ≤ (N − 1) , rN−1 = a} , ω2 = θ j = jhθ , 0 ≤ j ≤ (M − 1) , θN−1 = θk , where: ω = ω1 × ω2 , L = (M + 2)(N + 2) denotes a general nodes number, including outer contour nodes. For all derivatives, the finite diﬀerence approximation O(h2 ) is applied. We introduce the following notations of the mesh operators: Λr (·) = (·) r˙, Λrθ (·) = (·)θ˙ /ri2 − (·)r˙θ˙ /ri , Λrr (·) = (·)rr , Λθθ (·) = (·)θθ , Λzz (·) = (·)zz , Λrrθ (·) = (·)rrθ , Λz (·) = (·)z˙ , Λ (·) = Λθθ (·) /ri2 + Λr (·) /ri + Λrr (·) , where: (·)rr , (·)θθ , (·)r˙θ˙ , (·)zz , (·)r˙ are known in the literature notations of diﬀerence derivatives [376]. The initial equations (5.17), (5.18) and the three dimensional heat transfer equation (5.24) can be presented in the operator form: (wττ + εwτ )i j = −Λ (Λw) + Λrr w (ΛF − Λrr F) + Λrr FΛrr (Λw − Λrr w) − Λrθ wΛrθ F + ΛF + 4 (q)i, j − (1 + ν) ΛκT , √ Λ (ΛF) = −Λrr w (Λw − Λrr w) − (Λrθ w)2 − Λw − ηΛεT , (T τ )i, j,k = Λzz T + λΛT + w0 .

(5.47) (5.48) (5.49)

The boundary conditions (5.24)–(5.43) have the form: 1. Rolling clamping of the arcal element ν wN, j = 0, Λrr w + Λr w − (1 + ν) (κT )N, j = 0, a F N, j = 0, Λr F = 0, j = 1, M − 2.

(5.50)

2. Rolling clamping of the radial element wi, j = 0, Λθθ w + ri (1 + ν) (κT )i, j = 0, Fi, j = 0, Λθθ F = 0 for j = 0, j = M − 1,

i = 0, N − 1 .

(5.51)

5.2 Flexurable Conical Sectorial Shells Computations

511

3. Slip clamping of the arcal element wN, j = 0, Λr w = 0, F N, j = 0, Λr F = 0, j = 1, M − 2.

(5.52)

4. Slip clamping of the radial element wi, j = 0, Λθ w = 0, Fi, j = 0, Λθθ F = 0, j = 0, j = M − 1,

i = 0, N − 1 .

(5.53)

The boundary conditions with respect to displacements read: 1. Stiﬀ clamping of the arcal element wN, j = uN, j = vN, j = 0, Λr w = 0, j = 1, M − 2.

(5.54)

2. Stiﬀ clamping of the radial element wi, j = ui, j = vi, j = 0, Λθ w = 0, j = 0, j = M − 1,

i = 0, N − 1 .

(5.55)

3. Rolling unmovable clamping of the arcal element ui, j = υi, j = wi, j = 0, ν Λrr w + Λr w − (1 + ν) (κT )N, j = 0, j = 1, M − 2. a 4. Rolling unmovable clamping of the radial element

(5.56)

ui, j = υi, j = wi, j = 0, Λθθ w + r (1 + ν) (κT )i, j = 0, j = 0, j = M − 1, i = 0, N − 1.

(5.57)

Since boundary conditions (5.50)–(5.53) for a hybrid problem and the conditions (5.54)–(5.57) for the problems in displacements can be combined, the hybrid boundary conditions appear in the angle points. We consider the conditions in the angle points (0, 0), (w − 1, 0), (N − 1, M − 1) (see Fig. 5.3). Suppose that the A point is the angle point. Then, the C, E points lie outside the contour, and the D, A, B are the contour ones. On the contour, for an arbitrary combination of conditions, we have wA = wB = wD = 0 (∗). For a rolling and slip clamping in the point A additional two conditions should ; ; be satisfied, for example w = 0 and = 0. The operators in explicit form in the r

θθ

nodal points are as follows: wC = wB , wD − 2wA + wE = 0. Using condition (∗) for the angle point A, all of the nodes behind the contour have zero values:

512

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.3. External and internal shell nodes.

wC = 0,

wE = 0.

Similar conditions hold for the F function. In the case for θκ = 2π we get the so called problem with a cut along a radius. In order to get a circle problem, the sewing conditions should be formulated: wi, j = wi,M+ j , Fi, j = Fi,M+ j , j = 0; −1 i = 0, N − 1.

(5.58)

The boundary conditions (5.41), (5.42) for the three-dimensional heat transfer equation read: 1-st order T i, j,K = ψi, j,K K = 0, K = P − 1, 0 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1.

(5.59)

2-st order < T = q i, j,K K = 0, K = P − 1, 0 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1.

(5.60)

n

3-st order

<

$ % T = −B T i, j,k − T cp ,

n

0 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1, 0 ≤ K ≤ P − 1.

(5.61)

Therefore, the nonlinear boundary value problem (5.17), (5.18), (5.24) is reduced to the Cauchy problem (5.47)–(5.49) with the initial conditions (5.39), (5.43). In order to compute the thermal terms in equations (5.47), (5.48) we need to determine the thermal field. It is defined by solving the diﬀerential equations of the 1-st order (5.49) with the boundary conditions (5.43). In the matrix form we have:

5.2 Flexurable Conical Sectorial Shells Computations

dT = QT, T = T 0 , τ = 0, dτ

513

(5.62)

where: T = (T )i, j,k is the vector of K = M × N × P dimension. The nonzero coeﬃcients have the form: q5 = q10 = −

1 λ , q8 = q6 , q9 = q7 , q6 = 2 2 , h2z ri hθ

2 2λ 2λ λ λ + 2 − 2 2 , q7 = 2 + , q15 = q5 . h2z hr hr 2hr ri rr hθ

(5.63)

In order to integrate equations (5.62) a multistep Adam’s method of the third order is used: 1 (23 fk − 16 fk−1 − 5 fk−2 ) ∆x, yk+1 = yk + 12 where fk denotes the right hand side of our equation on the k step, and ∆x denotes the integration step. A use of one step Runge-Kutta method of the fourth order of the form ∆x ∆x f (xk , yk ) f (xk , yk ) + 2 f xk + , yk + + yk+1 = yk + 6 2 2

⎛ f (xk ,yk ) ⎞ ⎜⎜⎜ ⎟⎟⎟ f xk + ∆x ∆x 2 , yk + 2 ⎜ ⎟⎟⎟ + 2 f ⎜⎜⎝ xk + , yk + ⎠ 2 2

f (xk ,yk ) ⎞⎤ ⎟⎟⎟⎥⎥⎥ f xk + ∆x ∆x 2 , yk + 2 ⎟⎟⎟⎥⎥⎥ , , yk + f xk + ∆x, yk + f xk + ⎠⎦ 2 2 has increased the computation time of amount on 40% in comparison to the Adam’s method (four digits after a dot). A diﬀerence in time between those methods can be explained in the following way: we need to calculate four times the right hand side of 5.62 using the Runge-Kutta method, whereas the same operation can be applied only two times using the Adam’s method. Therefore, the Adam’s method for numerical integration is further applied. In order to solve the equation (5.47) in the vector form:

dw d2 w +ε = p w, F, T , dτ dτ

(5.64)

the Adam’s third order method is used. In order to compute the right hand side of the equation (5.64) p(w, F, T ) we need the F and T vectors. The first one is a solution to the nonhomogeneous algebraic equations system (5.48), whereas the second one is a solution to the equation (5.62). The equation (5.48) has the following matrix form: AF = B,

(5.65)

514

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

where: A = (ai, j ) is a square matrix of the dimension L × L and possessing a band form. The matrix A coeﬃcients have the following form: ai, j−2 = ai, j−1 = − ai, j =

1 r4r h4θ

,

4 1 4 1 4 + 4 2 − 4 4 , 2 2 2 r hr hθ r hθ r hθ

6 2 8 8 6 1 + 2 2 + 2 2 2 − 4 2 + 4 4 , 4 hr r hr r hr hθ r hθ r hθ ai,j + 1 = ai, j−1 , ai, j+2 = ai, j−2 , ai−1, j−1 =

ai−1, j = −

+

1 , r3 hr h2θ

4 2 4 2 + − − − − , h4r rh3r r2 h2r r3 2hr r2 h2r h2θ r3 hr h2θ ai−1, j+1 = ai−1, j−1 , ai−1, j−1 =

ai+1, j = −

2 r2 h2θ h2r

2 r2 h2θ h2r

+

1 , r3 hr h2θ

4 2 4 2 − 3− 2 2+ 3 − 2 2 2 + 3 2 , 4 hr rhr r hr r 2hr r hr hθ r hr hθ ai+1, j+1 = ai+1, j−1 , ai−2, j =

1 1 − , h4r r h3r

ai+2, j =

1 1 + , 4 hr r h3r

2 ≤ i ≤ N − 3 , 2 ≤ j ≤ M − 3, where: F = (Fi, j ) is the unknown vector of dimension L, and B(bi, j ) is the known vector of dimension L. The number of equations corresponds to the unknowns number L. The boundary conditions for the function F are added to the system (5.65) for the contour and out of contour points. The system of linear algebraic equation (5.65) is solved with respect to the forcing function Fi, j using a modified Gauss method. Modification of this method has the following form. In our case, we need to solve the system (5.65) on each time step keeping the A matrix unchanged. Only a column of free terms is changed. It includes the deflection values computed on the previous step. Using the Gauss method, we get the following system of equations:

5.2 Flexurable Conical Sectorial Shells Computations

DF = C,

515

(5.66)

where: D = (di, j ) is a triple matrix of L × L dimension, and it is computed once. A transformation of the B vector to the C vector C = (Ci, j )L and the operations required to solve the system (5.66) are calculated on each step in time. This modification of the Gauss reduction decreased the time of solving algebraic equations of two times in comparison to the standard method. According to the scheme, the values of the deflection w and the temperature T in each node (i, j) taken from the previous step (on the first step the initial given values of w and T are taken) is substituted to the right hand side of equations (5.66), which is solved in relation to the forces function F. The obtained values Fi j , wi, j , T i, j,k (w and T are taken from a previous step) are substituted to the right hand side of the system (5.64), and then the integration using the Adam’s method is carried out twice. In result, a deflection value wi, j on the considered step in time is obtained. The presented algorithm has a wide range of applications and allows to find a solution for the shells with an arbitrary plane (for = 1 - spherical, circular and sectorial shells; for = 0 - spherical and cylindrical shells, rectangular in plane) of the uncoupled problem of thermoelasticity for statics, quasistatics and dynamics. Also the problems of statical and dynamical stability, and axially symmetric and axially unsymmetric problems with diﬀerent boundary conditions subjected to an arbitrary transversal load can be solved. 5.2.3 Results The proposed algorithm allows to solve a wide class of problems, where an exact solution does not exist. Therefore, a question of authenticity requires discussion. The authenticity of the numerical results is verified by a comparison with an exact solution to the model problem, and with the numerical results obtained by other authors. The model problem is prepared in the following way: the being sought functions are chosen in order to satisfy the boundary and initial conditions of the considered problem. Then, as an unknown element, the second parameter has appeared (for example, a load) which is defined via the chosen functions. The formulated in the described manner solution is compared with that obtained using the proposed program. The found parameter (the load) serves as the known value, and the values of known function are obtained. As a model problem, axially symmetric vibrations of a shallow spherical circular shell without a thermal field and with ε = 0 are considered. If we find a pair of functions w(r, t), F(r, t) which satisfy both the equation (5.17) and zero initial conditions (5.39), then this pair is said to be an exact solution to the equations (5.17), (5.18). We take 2 r2 (5.67) w (r, t) = A (t) 1 − 2 . b

516

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Substituting (5.67) into (5.17) we get the following Euler type equation: 16A2 8A 64A2 16A 48A2 F IV − 4F III + 4F II = − 2 + 2 r4 + − 6 + 4 r6 + − 8 r8 , b b b b b which posseses the following solution: r4 r6 A (t) 5b2 r2 − + 2− + F (r, t) = 2 36 3 4b 18b4 r2 r4 r6 r8 13 2 + A (t) − . − + − 144 4b2 4b4 9b6 48b8

Table 5.1. A comparison of exact and numerically obtained w and F values. wexact · 10−1 0.156249 0.155385 0.152806 0.148556 0.142706 0.135358 0.126640 0.116711 0.105759 0.093998 0.081674 0.069060 0.056457 0.044198 0.032641 0.022176 0.013218 0.006215 0.001641 0

wnumerical · 10−1 0.156208 0.155352 0.152786 0.148549 0.142710 0.135369 0.126655 0.116725 0.105769 0.094003 0.081677 0.069066 0.056474 0.044233 0.032699 0.022256 0.013313 0.006309 0.001707 0

Fnumerical · 10−2 0.431822 0.428959 0.420443 0.406484 0.387433 0.363768 0.336092 0.305115 0.271643 0.236559 0.200808 0.165366 0.131223 0.099353 0.070680 0.046051 0.026196 0.011689 0.002911 0

Fexact · 10−2 0.412573 0.409999 0.402276 0.389404 0.371628 0.34936 0.323151 0.293669 0.261686 0.228060 0.193707 0.159586 0.126667 0.095905 0.068210 0.044127 0.025299 0.011224 0.002769 0

The unknown constants are found from the boundary conditions: F (r, τ) = 0,

∂F (r, τ) = 0, for r = b. ∂r

The unknown parameter of the normal pressure is found from (5.18): 1 ∂2 w ∂w ∂4 w 2 ∂3 w 1 ∂2 w q= + 2 + +ε − + 2 4 ∂τ ∂τ r ∂r3 r2 ∂r2 ∂r

(5.68)

5.2 Flexurable Conical Sectorial Shells Computations

517

Figure 5.4. Time history of w0 (τ) under pulse (a) and sinusoidal (b) excitations.

1 ∂F ∂2 w 1 ∂w ∂2 F 1 ∂w − 2 +1 − +1 . r ∂r ∂r2 r3 ∂r ∂r r ∂r

(5.69)

To conclude, the pair of the functions (5.67), (5.68) is the exact solution of the stress strain state of the spherical shell subjected to external load (pressure) action. Substituting (5.68), (5.69) into (5.70) one gets ⎧ 2 ⎪ r2 10 2 4 4 1⎪ 64 4A2 4 6 2 ⎨

+ − 2r + 4r + 6r + q= ⎪ A 1− 2 + 4 − 2 ⎩ 4⎪ 3 b b b 3b b 3b

6 10 20 5 1 A 4 r2 − 6 r4 + 8 r6 − 10 r8 − 2 b 3b 3b b b

−

518

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.5. The load q versus normal displacement w0 .

Figure 5.6. Distribution of circumferential and radial stresses along a radius.

1 2 2 1 4 2 4 2 4 4 4 6 , A − + 2r − 4r + A 2 − 4r + 6r − 8r 3 b b b b 3b b

(5.70)

and the load is now the known quantity. We are going to find numerically w(r, t) and F(r, t) using our program. As a result we get both exact and numerical solutions. A shape of the A(t) function defines the shell’s dynamics. It has been chosen in order to satisfy zero value initial conditions. We take A(t) = t2 , N = 20, b = 2, and the step t = 0.005. The obtained numerical results have shown a high accuracy. Table 5.1 includes the exact values of w(r, t), F(r, t), and the values obtained for t = 0.125.

5.2 Flexurable Conical Sectorial Shells Computations

519

Figure 5.7. Deflection along the shell radius.

Figure 5.8. Distribution of bending circumferential Mθ and radial Mr moments along the shell radius.

A relative error for w(r, t) and F(r, t) estimation has achieved 0.0005% and 4.5%, respectively. This great diﬀerence in results is yielded by many operations necessary to solve the algebraic equations to find F(r, t). The second way to verify the results refers to the comparison with the known results. We consider a motion of a shallow spherical circular shell subjected to different forms of pulse excitation (b = 4, N = 20, ν = 0.3). The boundary condition for r = a follows: w(r, τ) = 0,

∂w (r, τ) = 0, F(r, τ) = 0, ∂r

∂F (r, τ) = 0, ∂r

i.e. it is a movable clamping. In the polar (origin) the conditions (5.37), (5.38) are satisfied. The initial conditions for τ = 0 read:

520

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

w(r, τ) = 0,

∂w (r, τ) = 0. ∂τ

In Fig. 5.4a a behaviour of the shell’s vertex w0 for the rectangular impulse with S = qτ = 0.05 (q = z, τ = 0.025) and for the triangle impulse with S = 12 qτ = 0.05 (q = z, τ = 0.025) are reported. The solid line corresponds to results given in reference [618]. The obtained results indicate that the shell’s motion does not depend on the shape of the impulse for a given boundary condition. The drawings show that the conditions (5.36), (5.37), (5.38) in the vertex are very well satisfied. Therefore, (5.38) and (5.39) can be used for spherical circular shells computations. In Fig. 5.4b the shell’s vertex behaviour can be traced (the same parameters) in the case of a sinusoidal load q = q0 sin kτ (q0 = 0.4, k = 2). The solid curve corresponds to the known solution. Again, high accuracy of the results is reported. Now, we verify the authenticity of the obtained results due to statical problems. We compare the solution found using the set-up method with that given in monograph [696]. In the latter case, the problem is solved by a reduction of the nonlinear boundary value problem to the system of algebraic nonlinear equations. In Fig. 5.5 the dependencies between the load and a normal displacement w0 of the shell’s vertex for the following parameters: b = 5, N = 10, ε = 2, ν = 0.3 are presented. The initial conditions are equal to zero, and the shell’s contour is movably clamped. The solid curve corresponds to the solution given in reference [696]. A good agreement up to q = 0.27 is obtained. In Fig. 5.6 a distribution of the radial Nr and the circle Nθ forces versus a radius are presented, whereas in Fig. 5.7 the shell deflection for a scritical load along a radius i presented. In Fig. 5.8 similar dependences to that of Fig. 5.6 for the bending moments Mr , Mθ are given. In all figures the dashed curves are obtained using our set up method, whereas the other curves correspond to the solution given in [696]. The considered examples prove the authenticity of the results obtained using the proposed program.

5.3 Stability of Sectorial Shells with Finite Deflections We begin with a definition of the critical load so called Shian, Soong and Roth criterion [612]. The time, required to achieve a suﬃcient deflection with the load increase, increases slowly up to the critical load value. For the load higher than the critical value, the maximal deflection is achieved in a very short time. Other authors take an instant of the plastic deformation occurrence as a dynamical criterion of the stability loss. In the reference [229], as a stability loss criterion, the time instant t0 is taken, when the deformations of an elastic body become unstable. It means that in time t0 we can perturbate the body only when only the displacement are changed without a change of velocity w t (t0 ) and acceleration w

t (t0 ).

5.3 Stability of Sectorial Shells with Finite Deflections

521

In reference [305], for shells rectangular in plane, the following dynamical criterion is proposed. First, the system of diﬀerential equations is reduced to the ordinary homogeneous equation in relation to an function characterizing small deviation from the initial state. A sign of the β coeﬃcient (the second power of the free vibration frequency) determines stability (β > 0) or instability (β ≤ 0) on each step.

Figure 5.9. Sectorial shell scheme.

During investigations of shells parametric vibrations, in reference [120] another criterion is proposed. The input system of diﬀerential equations is reduced to the Mathieu-Hill’s equation, and then stability is estimated. There are no adequate criteria to define a stability loss of sectorial shells which will be proposed now. Following the scheme in Fig. 5.9, let OC = R be a sphere radius, whereas AC = a1 is the diameter of horizontal shell projection. Let the arc ABC is clamped along. The shell vertex B is always clamped, because it belongs to the arc ABC. The Kantor theorem [191], indicates that the point B can not move on amount of two sags BE = H1 . Let us analyse now the displacement of the point D being the center of arc BC. In accordance with the monograph [522], the non-dimensional curvature parameter for shallow shells is defined by the relation b1 a21 /(Rh), where h is the shell thickness. In the case of small elevation shells one gets: a21 /(Rh) ≈ 8H2 /h and H2 /H1 ≈ a22 /a21 , which yields the estimation H2 ≈ H1 /h. Assuming b1 as the shell sloping parameter (see [522]), for the spherical shells the following estimation holds: H1 = b21 /2, i.e. H2 = b21 /8. The obove considerations allow one to construct the following Table 5.2. Therefore, we have introduced the geometrical H2 parameter (Fig. 5.9). It is seen in Fig. 5.9, that for the clamped sector along the ABC contour, the top B is always clamped. Therefore, according to the Kantor [305] criterion, it can not move on 2H after the stability loss. We need to analyse a displacement of the D point, which is the middle point of the arc BC.

522

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells Table 5.2. Shell parameters obtained from the scheme given in Fig. 5.9. θk π/3 π/2 π b1 11 10 9 wmax 9.6 12.7 9.3 H2 15.1 12.5 10.1 w(H2 ) w > H2 /2 w ≥ H2

1.5π 2π 8 5 19.6 19.18 8 3.125 w ≥ 2H2

5.3.1 Influence of the Sector’s Angle The dynamical problems will be solved using the governing equations (5.17), (5.18) for ε = 0. For all sectors we consider the slip clamping of the edge (5.27), (5.28) for the following fixed parameters: M = N = 10, r s /h = 200, ν = 0.3. The stress strain state of the sectorial shells in critical and pre-critical states are investigated. An applied mesh consisted of radiuses and angles is shown in Fig. 5.10. Observe that r1 = 0 represents the vertex, whereas diagonal of sector θk is situated between θ5 and θ6 . PROBLEM 1. A shell with the sectorial angle θ k = 2π.

Figure 5.10. A shell sector partition by radiuses r1 ...r10 and angles θ1 ...θ10 .

In this case we deal with a circled shell, and the sewing conditions on should be satisfied. In order to determine a stability loss, we use the Volmir’s [684] and Shian et al. [612] criteria. Both of them are in good agreement and give the same values of the critical load. The investigations have shown that for the given boundary conditions the circular spherical segment loses its stability for b = 5. Both of the used criteria are illustrated in Fig. 5.11 and 5.12, respectively and they give the same values qcr = 0.24, wmax = 19.2.

5.3 Stability of Sectorial Shells with Finite Deflections

523

Figure 5.11. Analysis of stability using Volmir criterion.

Figure 5.12. Analysis of stability using Shian, Soong and Roth criterion.

For a stress state analysis of a circular segment we, compare the shell’s behaviour for the pre-critical load q = 0.2, and for the time instants τ = 2.35; 5.1; 9.5 (the curves 1, 2, 3), as well as for critical load qcr = 0.24 for the time instants τ = 2.35; 5.1; 10.5 (the curves 1, 2, 3). For the pre-critical load, a slow increase of the deflection and forces (Fig. 5.13), a slight change of the circular forces from the negative values in the vertex neighbourhood to the positive ones on the contour (Fig. 5.15) and only negative radial stresses action (Fig. 5.14) are noticed. For the critical load, in the stability loss instant (curve 3), a sudden increase of the deflection and a decrease of the forces (Fig. 5.16) are observed, and also a sign change of the circular and radial forces (curve 3) in the area of the shell’s top (Fig. 5.17, 5.18) is exhibited. Therefore, the sector θk = 2π loses its stability for b = 5, qcr = 0.24 with the occurrence of buckling in the shell’s vertex. Besides, the stability loss is characterized by a sudden variations of the forces, and the membrane forces change their sign. The shell’s fibres with compressing stresses now become the stretching ones.

524

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.13. Deflections and stress functions distribution along a radius (pre-critical loading).

Figure 5.14. Radial stresses distribution along a radius (pre-critical loading).

Figure 5.15. Circumferential stresses distribution along a radius (pre-critical loading).

5.3 Stability of Sectorial Shells with Finite Deflections

525

Figure 5.16. Deflection and stress function distribution along a radius (critical loading).

Figure 5.17. Circumferential stresses distribution along a radius (critical loading).

PROBLEM 2.1. A shell with the sectorial angle θ k = 32 π. In order to determine a stability loss moment of a given shell we use the Shian et al. criterion [612], which is illustrated in Figs. 5.19, 5.20. According to it qcr = 0.34, wmax = 19.6, b = 8. Now we analyse the shell’s stress state. In Fig. 5.21 for the pre-critical load q = 0.3, the curves of relatively equal forces (left shading) and deflection (right shading) for which the maximal deflection is obtained are given. It is seen that the shell is dented along the angle bisectrix. A change

526

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.18. Radial stresses distribution along a radius (critical loading).

Figure 5.19. The dependence wmax (τ) for the shell θk = 1.5π for diﬀerent q (Shian, Soong and Roth criterion).

of the circular and radial forces along the angle is smooth, and the largest forces are located on the bisectrix (Fig. 5.22). In the neighbourhood of the vertex, the stretching stresses occur (Fig. 5.23, 5.24). The other shell’s part exhibits radial compression stresses. For the critical load qcr = 0.34 the curves of equal relative deflections (left shading) and the forces (right shading) are diﬀerent (Fig. 5.26). The shell loses its stability with the occurrence of a buckling concave on the bisectrix. For the critical load, the Nr character is changed along the angle (Fig. 5.25), which indicates a stability loss. Almost all radiuses on the bisectrix have positive stresses. This conclusion is supported by Fig. 5.26. Almost all stresses on the bisectrix of θ5 have a positive character. The bending stresses Mr (Fig. 5.28) and Mθ (Fig. 5.30) achieve maximal positive values on the bisectrix of the angle θ5 .

5.3 Stability of Sectorial Shells with Finite Deflections

527

Figure 5.20. The dependence q(τ) for the shell θk = 1.5π (Shian, Soong and Roth criterion).

Figure 5.21. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load q = 0.3).

528

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.22. Variation of circumferential and radial stresses along angle (pre-critical load q = 0.3).

Figure 5.23. Radial stresses distribution along a radius (pre-critical load q = 0.3).

Figure 5.24. Radial stresses distribution along a radius (critical load q = 0.34).

5.3 Stability of Sectorial Shells with Finite Deflections

529

Figure 5.25. Circumferential stresses distribution along a radius (pre-critical load q = 0.3).

Figure 5.26. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load q = 0.34).

530

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.27. Variation of circumferential and radial stresses along angle (pre-critical load q = 0.34.

Figure 5.28. Bending radial moment distribution along a radius (critical load qcr = 0.34).

5.3 Stability of Sectorial Shells with Finite Deflections

531

Figure 5.29. Sectorial shell deflection along a radius (pre-critical load q = 0.3).

Figure 5.30. Bending circumferential moment distribution along a radius (critical load qcr = 0.34).

532

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.31. Sectorial shell deflection along a radius (critical load qcr = 0.3).

It has been observed that for the critical load the maximal deflection is shifted to the shell’s vertex (Fig. 5.30). To conclude, the sector shell loses its stability for b = 8, qcr = 0.34 with the appearance of the buckling concave on the angle bisectrix close to the shell’s vertex. A stability loss is characterized by a sign change of Nr along the bisectrix θ5 .

Figure 5.32. Vibrations of diﬀerent shell points after the impulse load removal.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.33. Stress function F(τ) for diﬀerent shell points.

Figure 5.34. Circumferential stresses Nθ of diﬀerent shell points.

533

534

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

PROBLEM 2.2. Vibrations of the sectorial shell θ k = 32 π after the load removal. Let shell be subjected to the impulse with intensity q = 0.5 and its duration τ = 7.8. The shading area in all figures denotes the impulse action. We investigate the vibrations after the removal of the impulse. We trace the points 1-6 given in Fig. 5.32. With time increase, the points on the central radius and near the vertex have the largest amplitude. The point close to the arc segment possesses the smallest amplitude. A similar picture can be observed in Fig. 5.33, where the stresses exhibit a complicated vibrational process. It is seen from Fig. 5.33 that the stress function in the vertex points changes its sign in comparison to other points. The radial absolute stress value are largest near the vertex. In Fig. 5.34, the vibrations of circumferential stresses Nθ are presented. Observe that the stresses sign in the points of the arcal segment, and in other points are diﬀerent. To conclude, when the shell is compressed at the vertex along the radius, then in the other points it is extended along the radius. At the same time, it is compressed along the angle of the arcal segment, whereas in the other points it is extended. PROBLEM 3.1. A shell with θ k = π sector. In order to define a stability loss instant the Shian et al. [612] criterion will be used (Fig. 5.35, 5.36). According to it, qcr = 0.3, wmax = 9.3, b = 9. The stress state will be analysed for the pre-critical load q = 0.2, and the critical one qcr = 0.3 at the time instant of achieving a maximal deflection. For the pre-critical load (τ = 4.7) the curves of equal relative deflections, and the function of stresses show the most deformable places. It occurs on the θ5 bisectrix along the radius r7 . The radial stresses have a negative sign (Fig. 5.36). The largest Nr are obtained on the r7 radius. The circular stresses are interleaved with the zones of stretching and compressing (between the radiuses r2 -r8 ) stresses (Fig. 5.38). In the stability loss time instant, for the critical load, two buckling concaves lying symmetrically along the θ5 bisectrix have appeared. This situation is presented in Fig. 5.40, where the curves of equal relative deflections and the stress function are given (right and left shading, respectively). The maximal deflection are shifted to the shell’s vertex on r6 (Fig. 5.41). The circular stresses (Fig. 5.42) have a zone of sign interleave, but they are smaller in comparison to the central part. The stability loss time instant, is observed on the radial stresses behaviour. The stresses which occurred on θ4 , θ5 have the wave-shape character (Fig. 5.43). A decreasing of the radial stresses on θ4 between the radiuses r4 -r7 , and also the occurrence of maximal deflection stresses in that place (Fig. 5.44, 5.45) characterize the stability loss with the occurrence of two symmetric concaves. To conclude, the sectorial shell with the sector’s angle θk = π loses its stability for b = 9, q = 0.3 with the occurrence of two buckling concaves.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.35. The dependence wmax (τ) for diﬀerent q and b values.

Figure 5.36. The dependence q(τ).

535

536

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.37. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load and τ = 4.7).

Figure 5.38. Circumferential stresses distribution along a radius (pre-critical load q = 0.2).

Figure 5.39. Radial stresses distribution along a radius (pre-critical load q = 0.2).

5.3 Stability of Sectorial Shells with Finite Deflections

537

Figure 5.40. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load).

Figure 5.41. Sectorial shell deflection along a radius (pre-critical load q = 0.3).

Figure 5.42. Circumferential stresses distribution along a radius (pre-critical load q = 0.3).

538

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.43. Radial stresses distribution along a radius (critical load qcr = 0.3).

Figure 5.44. Bending circumferential moment distribution along a radius (critical load qcr = 0.3).

Figure 5.45. Bending radial moment distribution along a radius (critical load qcr = 0.3).

5.3 Stability of Sectorial Shells with Finite Deflections

539

PROBLEM 3.2. Vibrations of a sectorial shell with θ k = π after the load removal. Assume that the shell is subjected to the transversal impulse with q = 0.5 intensity and τ = 1.9 duration. We investigate the vibrations of the points 1-6, presented in Fig. 5.46, after the load’s removal. The vibrations of the deflection and stresses are qualitatively similar (Figs. 5.46, 5.47).

Figure 5.46. Vibrations of diﬀerent shell points.

Most often, the vibrations occur on the vertex but they have small amplitude. A large amplitude characterizes the points lying close to the angle bisectrix. In Fig. 5.48 the vibrations of radial stresses are presented. An interesting behaviour is observed in the case of points 3, 4. Although they lie on one radius, after the load’s removal their radial stress vibrations are shifted to the phase of π. In the Fig. 5.49 the vibrations of circular stresses are shown. After the load’s removal, in all points the radial compress stresses remain for a certain time. In the case of circular stresses after the load’s removal, the stretching stresses occur on the arcal segment. They vibrate in the antiphase manner in comparison to other points. To conclude, the vibrations of sectorial shell points with θk = π are more complicated in comparison to the vibrations of the shell θk = 32 π.

540

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.47. Stress function time histories in diﬀerent shell points.

Figure 5.48. Time histories of radial stresses for the shell θk = π.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.49. Time histories of circumferential stresses for the shell θk = π.

541

542

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

PROBLEM 4.1. A shell with the sector’s angle θ k = π2 . Again the Shian et al. criterion will be used to estimate the stability loss moment (Fig. 5.50, 5.51). According to it, we get qcr = 0.3, wmax = 12.73, b = 10.

Figure 5.50. Dependence wmax (τ) for diﬀerent q value.

We investigate the stress strain state of the shell for the pre-critical load q = 0.2 and the critical one qcr = 0.3 in the instant for achieving a maximal deflection. The occurrence of buckling convexity on the angle bisectrix characterized the pre-critical load action (Fig. 5.52). The deflection and the stress functions are similar. A distribution of the circular stresses along θ1 -θ5 are shown in Fig. 5.54. Similar to the case θk = π, on the shell’s vertex and in the arcal segment, the areas of positive stresses appear. Their magnitude on the arcal segment is larger than that on the vertex. It is caused by decrease of the angle θk . The area of negative values is narrowed and is shifted to the arcal segment. In Fig. 5.55, a change of the radial stresses along θ1 -θ5 is presented. The whole shell, except for a small area on θ2 , is in the radially compressed state.

5.3 Stability of Sectorial Shells with Finite Deflections

543

Figure 5.51. Dependence q(τ) for the shell θk = π/2.

Figure 5.52. Displacement w and stress function F distributions along an angle (pre-critical load q = 0.2).

The characteristic load is characterized by two zones of maximal values of the stress function (Fig. 5.56, the left shading), but the shell loses its stability by the occurrence of one dent on the angle’s bisectrix (the right shading). It has been found that increasing the load q = 0.3 the maximal deflection is shifted to the vertex (Figs. 5.57, 5.53). The stability loss (buckling) instant is characterized by the wave-form behaviour of the curves Nθ (Fig. 5.58) and Nr (Fig. 5.59). In the place on θ5 a change of the radial stresses occurs. The stability change is displayed also on the pictures of the change of deflection stresses Mθ (Fig. 5.60) and Mr (Fig. 5.61). The maximum of the deflection stresses is achieved on θ5 , where the dent occurs. To conclude, the sectorial shell θk = π2 loses its stability for qcr = 0.3, b = 10 with the occurrence of one dent located on the angle’s bisectrix.

544

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.53. Deflection w distribution along a radius (pre-critical state q = 0.2). N

q = 0.2 5

4

5 3

3

2

0

3

−3

0 (r1 )

1/3 (r4 )

4

2/3 (r7 )

r/a

1 (r10 )

Figure 5.54. Circumferential stresses distribution along a radius (pre-critical state q = 0.2).

Figure 5.55. Radial stresses distribution along a radius (pre-critical load q = 0.3).

5.3 Stability of Sectorial Shells with Finite Deflections

545

Figure 5.56. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load q = 0.2).

Figure 5.57. Sectorial shell deflection along a radius (pre-critical load q = 0.3).

546

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.58. Circumferential stresses distribution along a radius (pre-critical load q = 0.3).

Figure 5.59. Radial stresses distribution along a radius (pre-critical load q = 0.3).

Figure 5.60. Bending radial moment distribution along a radius (critical load qcr = 0.3).

5.3 Stability of Sectorial Shells with Finite Deflections

547

Figure 5.61. Bending circumferential moment distribution along a radius (critical load qcr = 0.3).

PROBLEM 4.2. Vibration of a sectorial shell θ k =

π 2

after the load removal.

Let the shell be subjected to the transversal impulse with intensity q = 0.5 and duration τ = 2.7. We trace the vibrations of the points 1-6 presented in Fig. 5.64. Vibrations of the function w, F are given in Figs. 5.62, 5.63. They have a complicated character. In spite of the fundamental vibrations, also small vibrations occur. The computations have displayed that the vertex points’ vibrations have a slight influence. Large amplitude vibration appears in the point 4. The vibrations of Nr (Fig. 5.64) and Nθ (Fig. 5.65) stresses have complicated forms. After the load’s removal in all points the radial compressing stresses initially occur. Then the zones of stretching and compression are interleaved. After the removal of the load, both points of circular segment and at the vertex exhibit positive circumferential stresses, and vibrations have complicated forms.

548

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.62. Vibrations of diﬀerent shell points after the impulse load removal.

Figure 5.63. Stress function F(τ) for diﬀerent shell points.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.64. Radial stresses Nr of diﬀerent shell points.

Figure 5.65. Circumferential stresses Nθ of diﬀerent shell points.

549

550

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

PROBLEM 5.1. A shell with the sector’s angle θ k = π3 . Similarly to the previously considered cases we use the Shian et al. criterion to estimate the stability loss (Figs. 5.66, 5.67) (qcr = 0.25, wmax = 9.6, b = 11).

Figure 5.66. Dependence wmax (τ) for diﬀerent q values.

Figure 5.67. Dependence q(τ) for the shell θk = π/3.

We investigate the stress state for the pre-critical load q = 0.2, and the critical one qcr in the time instant of the maximal deflection occurrence. For the pre-critical load, a change of the w, F against the angle are qualitatively similar (Fig. 5.68). The maximal deflection appears on r8 (Fig. 5.69) in the vicinity of the arcal segment. For the circular stresses the sign interleave occurs. Maximal stresses are observed on the vertex (similarly to the cases θk = 3π 2 , π), Fig. 5.70.

5.3 Stability of Sectorial Shells with Finite Deflections

551

Figure 5.68. Displacement w and stress function F distributions along an angle (pre-critical load q = 0.2).

Figure 5.69. Deflection w distribution along a radius (pre-critical state q = 0.2).

The radial stresses are only compressive (Fig. 5.71), and the smallest of them appears in the vicinity of the radial segment on θ2 . In the case of the critical load, the maximal deflections on r7 are shifted (Fig. 5.72). The pictures of relatively equal deflections and stresses are similar qualitatively (Fig. 5.73). A stability loss occurs with a dent occurrence on the angle’s bisectrix. The stability loss moment is characterized by a wave-form change of the circular stresses (Fig. 5.74), and a sign change of the radial stresses on θ5 (Fig. 5.75). A behaviour of the bending stresses along θ1 -θ5 is presented in Fig. 5.76 for Mθ , and in Fig. 5.77 for Mr . Only on θ5 the maximal values of Mr and Mθ are achieved. Therefore, the sectorial shell θk = π3 loses its stability for b = 11, qcr = 0.25 with the occurrence of one dent on the angle’s bisectrix.

552

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.70. Circumferential stresses distribution along a radius (pre-critical state q = 0.2).

Figure 5.71. Radial stresses distribution along a radius (pre-critical load q = 0.2).

Figure 5.72. Deflection w distribution along a radius (pre-critical state q = 0.25).

5.3 Stability of Sectorial Shells with Finite Deflections

553

Figure 5.73. The curves of equal relative both stress function (a) and deflection (b) values in the maximal deflection time instant (pre-critical load q = 0.2).

Figure 5.74. Distribution of circumferential stresses along a radius (pre-critical load q = 0.25).

Figure 5.75. Distribution of radial stresses along a radius (pre-critical load q = 0.25).

554

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.76. Circumferential bending moments distribution along a radius in the stability loss time instant.

Figure 5.77. Distribution of radial bending moments along a radius in the stability loss time instant.

5.3 Stability of Sectorial Shells with Finite Deflections

PROBLEM 5.2. Vibrations of a sectorial shell θ k = load.

π 3

555

after removing of the

Let a shell be subjected to the transversal load of the intensity q = 0.5 and duration τ = 2.3. We trace vibrations of the points 1-6 (Fig. 5.78) after removing the load. The deflection functions are small in the top neighbourhood during the vibrations.

Figure 5.78. Vibrations of marked shell points for the shell θk = π/3.

The point 6 lying on the angle’s bisectrix possesses the largest amplitude and the lowest vibration frequency. All the points undergo the same vibrations of stresses F (Fig. 5.79). The vibrations of the radial Nr and the circular Nθ stresses are shown in Figs. 5.80, 5.81, respectively. After removing the load, the compression area along the radius is changed by a stretching area. The largest amplitude is obtained for point 4. For the circular stresses a zone of stretching stresses occurs in the shell’s centre, whereas at the vertex neighbourhood and on arcal segment, a zone of stretching occurs. Further, the anti-phase vibrations occur. It has been observed that vibrations amplitudes of Nθ function in the vertex neighbourhood points are higher than Nr amplitudes. However, for the whole vibrational process the Nθ values will be smaller than Nr values.

556

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.79. Time histories of the stress function for the chosen shell points.

Figure 5.80. Time histories of the radial stresses Nr for the chosen shell points.

5.3 Stability of Sectorial Shells with Finite Deflections

557

Figure 5.81. Time histories of the circumferential stresses Nθ for the chosen shell points.

Figure 5.82. Critical and pre-critical values

Analysing the results, obtained in problems 1-5, the following conclusions are derived. A. Dynamical stability of sectorial shells 1. The Shian et al. criterion has been successfully used for the stability loss time instant determination for the sectorial shells with θk ∈ [ π3 , 2π]. a) the sector’s angle influence on the critical sloping parameter b (when the shell cases its stability) and on the critical load; b) the shell θk = 32 π, θk ∈ [ π3 , π] loses its stability with the occurrence of one dent on the angle’s biserctix (the shell θk = π loses its stability with the

558

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.83. Maximal deflection and shell centre deflection versus θk .

Figure 5.84. Maximal deflection distribution along a radius for a pre-critical load.

occurrence of two dents symmetrically located in relation to the angle’s biserctix). c) the stability loss instant is characterized by a sudden change of the membrane stresses (circular stresses have wave-forms, and the radial ones change their sign), where a dent occurs. 2. For the angles θk ∈ ( π3 , π2 ), the time of achieving a maximal deflection for the critical load increases, and for θk ∈ ( π2 , π) it decreases, Fig. 5.82 (the curve 1 corresponds to the pre-critical, where the curve 2 corresponds to the critical loads). For the angles θk ∈ (π, 32 π), at first the time increases, but beginning from θk = 32 π, it decreases again. All figures have been obtained for critical b parameters. 3. A maximal deflection of sector shells is achieved out of the centre, Fig. 5.83 (the curves 1, 2 for the pre-critical load correspond to the maximal deflection in the centre; curves 3, 4 are obtained also for the critical load). Increasing the angle the maximal deflection shifts to the shell’s centre (see also Fig. 5.84 for the pre-critical load, and Fig. 5.85 for the critical one). For the pre-critical loads,

5.3 Stability of Sectorial Shells with Finite Deflections

559

Figure 5.85. Maximal shell deflection dependence on a radius for a critical load.

Figure 5.86. Maximal circumferential shell stresses along a radius (a pre-critical load).

the shell is dented in the vicinity of the arcal segment between r6 -r8 , whereas for the critical ones - in the vertex vicinity between r5 -r7 . 4. For θk ∈ (π, 32 π) in the shell’s vertex neighbourhood stretching circular stresses are observed, which are two times larger than those occurring on the arcal segment. On the contray, for angles θk ∈ ( π2 , π) stretching circular stresses on the vertex are small, and the stresses on the arcal segment are higher more than

560

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

twice (Fig. 5.86). Decreasing the angle, the area of positive Nθ increases on the vertex. A zone of stretching stresses is shifted to the arcal segment, causing a sudden increase of positive stresses on that segment. Therefore, the shells with small sectors angle are more stable against the external load. In order to obtain unstability, the parameter b should be increased. B. Vibration of sectorial shells after load removing. 1. Decreasing the angle values leads to the complexity of vibrations of diﬀerent shell’s points. Together with the fundamental ones, also small vibrations of different phase occur. 2. Decreasing θk , the vertex vicinity points have very small deflection values, and their vibration frequencies are small. 3. For θk ∈ [ π2 , 32 π] the points of the shell’s centre have a maximal vibration amplitude (points 3, 4). For θk ∈ [ π3 , π2 ], the maximal amplitude occurs for point 6 and it lies on the biserctix of the arcal segment angle. 5.3.2 Set-Up Method and Determination of Critical Loads For computation purpose, we use the equations (5.45) and (5.46). In order to solve the diﬀerential equations, the algorithm presented in Sect. 5.2.2 is applied, and the thermal field is not taken into account. PROBLEM 1. Sectorial shell with the sector’s angle 2π. Let a shell be subjected to a uniform external load (pressure) with intensity of q. The shell’s edge has slip clamping (5.52), and the radial segments served using the formulae (5.58). The initial conditions (5.40) are equal to zero. The following fixed parameters have been taken: ν = 0.3, a/h = 200, ∆τ = 0.01, N = 10, ε = 2. The obtained results are compared with those obtained in reference [696], showing very high accuracy. PROBLEM 2. Sectorial shell θ k = 32 π. We consider a shell with the following initial data: uniform external pressure q; slipping clamping of arcal and radial segments (5.52), (5.53); zero initial conditions (5.40); ν = 0.3, ε = 2, M = N = 10, a/h = 200, ∆τ = 0.01. In Fig. 5.87, a dependence between the load parameter and the maximal normal shell’s displacement for diﬀerent parameter b = 5, 6, 7, 8 is presented (curves 1, 2, 3, 4, respectively). Beginning from certain b values, on the curves the limiting points occur. The calculations have proved that b = 8 in the critical value. The first limiting point on the curves 4 defines the upper critical load q+ = 0.5, when a “jump” occurs. In Fig. 5.88, the set up curves of the unstationary solutions for b = 5, q = 0.3, 0.8, 1.4, 2 (curves 1-4, respectively) are presented. The computations have shown, that a good

5.3 Stability of Sectorial Shells with Finite Deflections

561

Figure 5.87. Dependence q(wmax ) for diﬀerent slopings b.

Figure 5.88. Isoclines of deflection and wmax (τ) for b = 5.

choice of the damping parameter ε quarantees a smooth transition on the stationary solution with the accuracy of 0.005. In Fig. 5.89, the curves of relatively equal deflections for the shell with b = 6 (to the left) and b = 7 (to the right), and for b = 8 in Fig. 5.90, are displayed. We compare the curves behaviour for the same load values. For q = 0.3, the pictures qualitatively coincide. A maximal deflection is achieved on the intersection of θ5

562

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.89. Curves of equal relative shell deflections for b = 6, 7 for diﬀerent loads q.

bisectrix and the shell’s central radius. For q = 0.5, the pictures are diﬀerent: for b = 7 two zones of maximal deflections occur, symmetrically situated in relation to θ5 ; for b = 8 also to zones of maximal deflections are observed. For the load q = 0.6, the maximal deflection for the shell with b = 6 is not clearly expressed, whereas for b = 7 and b = 8 the zones of maximal deflections increase and a stability loss occurs.

5.3 Stability of Sectorial Shells with Finite Deflections

563

Figure 5.90. Curves of equal relative shell deflections for b = 8 for diﬀerent loads q.

The largest deflection zone occurs for b = 8, and this parameter is considered as the critical one for θk = 32 π. The shell’s stress state with the critical parameter b = 8 is presented in Figs. 5.91–5.94 for the pre-critical load q = 0.3, whereas in Figs. 5.95–5.98 - for the critical one q+ = 0.5. The computations shown that maximal deflections for each θ1 -θ4 are achieved on r6 for either critical or pre-critical loads (Figs. 5.92, 5.96). The circular stresses are distributed in the following form. The stretching stresses are located on the vertex vicinity and on the arcal segment (Figs. 5.93, 5.97). A zone of compressing stresses Nθ is located between the r2 -r8 radiuses. A zone of positive radial stresses is located only in the top vicinity on the radiuses r1 -r4 (Figs. 5.94, 5.97). Between radiuses r4 -r10 , a zone of compressing radial stresses occurs.

564

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.91. Deflection w and stress function F distributions along angle for the critical load.

Figure 5.92. Shell deflection along a radius for the pre-critical load.

Therefore, for the shell with the central angle θk = 32 π, a stability loss occurs for b = 8 and q+ = 0.5. The shell loses its stability with the occurrence of two dents situated symmetrically in relation to the angle’s biserctix.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.93. Circumferential stresses along a radius for pre-critical load q = 0.3.

Figure 5.94. Radial stresses along a radius for pre-critical load q = 0.3.

565

566

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.95. Deflection w and stress function F along an angle for critical load qcr = 0.5.

Figure 5.96. Deflection w along a radius for critical load q = 0.5.

Figure 5.97. Circumferential stress along a radius for critical load.

5.3 Stability of Sectorial Shells with Finite Deflections

567

Figure 5.98. Radial stress along a radius for critical load.

PROBLEM 3. A shell with a central sector’s angle π. We analyse shell with the same parameters.

Figure 5.99. Shell sloping parameter versus θk (a) and load q versus maximal displacement for b = bcr (b).

568

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.100. Curves of equal relative shell displacements for b = 5 and b = 9 for diﬀerent q values.

Figure 5.101. Curves of equal radial (a) and circumferential (b) stresses for a post-critical load q = 0.5.

In Fig. 5.99a is reported that for two shell θk = π the corresponding bcr = 9. The function q(wmax ) for the critical sloping parameter is shown in Fig. 5.99b. The dashed curves represent sets of critical points. For instance, the upper limit of the critical load is equal to q+ = 0.4 (curve 3 in Fig. 5.99b).

5.3 Stability of Sectorial Shells with Finite Deflections

569

Figure 5.102. Deflection distribution along a radius for pre-critical load.

Figure 5.103. Deflection distribution along a radius for critical load.

In Fig. 5.100 the curves of equal relative deflections for b = 5 (to the left) and b = 9 (to the right) for diﬀerent values of the transversal load q are presented. For b = 5 a maximal deflection is achieved on the intersection of the biserctix with the radius. For b = 9 the deflections picture is changed. For q = 0.3 the maximal deflection is still achieved on the biserctix, but beginning with q+ = 0.4 a zone of maximal deflection is divided into two symmetrically located in relation to θ5 . Further increase of the load leads to increase of wmax . A stress state of the shell is presented in Figs. 5.102–5.109 for the pre-critical load q+ = 0.4, and in Figs. 5.110–5.113 for the post-critical load q = 0.5. The maximal deflections in θ2 -θ5 zone are achieved on r7 (Figs. 5.102-5.104). The computations show that for q = 0.3 and q = 0.4 the curves of circular and radial stresses are qualitatively similar. A distribution of the circular stresses are characterized by the occurrence of two zones with a positive sign in the vertex vicinity and on the circular segment. A zone of compressing stresses is located between r2 -r8 (Figs. 5.104-5.107). For the radial stresses, a zone with a positive sign is not observed.

570

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.104. Deflection w and stress function F along a radius for the critical load.

Figure 5.105. Circumferential stresses along a radius for the pre-critical load q = 0.3.

In a zone from r1 to r10 , the all radial stresses are compression. The largest Nr are achieved on θ2 , θ3 on the radiuses r6 , r7 (Figs. 5.105, 5.106). Another picture is observed for the post-critical load q = 0.5. From r3 to r9 the deflection peaks occur on θ3 , which correspond to the places of two maximal

5.3 Stability of Sectorial Shells with Finite Deflections

571

Figure 5.106. Circumferential stresses along a radius for the critical load qcr = 0.4.

deflection zones symmetrically located in relation to θ5 . The radial stresses behave in a diﬀerent manner (Fig. 5.111). A distribution of Nr along θ2 , θ3 has a waveform character. In the planes, where the dents appear, the radial stresses absolute values rapidly decrease, which

572

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.107. Radial stresses along a radius for the pre-critical load q = 0.3.

Figure 5.108. Radial stresses along a radius for the critical load qcr = 0.4.

corresponds to a stability loss. For the post-critical load q = 0.5, on θ1 , θ4 , maximal compressing stresses are achieved. A behaviour of the circular stresses in relation to θi is presented in Fig. 5.113. The maximal positive stresses are achieved in the vertex vicinity, and on the arcal

5.3 Stability of Sectorial Shells with Finite Deflections

573

Figure 5.109. Radial and circumferential stresses along a radius for the critical load qcr = 0.4.

segment. The vibrations appear on θ2 and θ3 . In the places of dents, the absolute values of Nθ rapidly decrease (except for θ1 , θ4 ). A distribution of the stresses Nr and Nθ along each radius ri is presented in Fig. 5.112. Beginning from r3 , the curves ‘start to vibrate’ (in particular the curves 5, 6, 7). The smallest values are observed between θ3 and θ4 (the place of the maximal deflections).

574

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.110. Displacement w and stress function F distributions along an angle for the postcritical load q = 0.5.

Figure 5.111. Radial stresses distribution along a radius for the post-critical load.

The curves of relative equal stresses are presented in Fig. 5.114 for q = 0.5. The curves behaviour indicates the occurrence of two symmetrical dents.

5.3 Stability of Sectorial Shells with Finite Deflections

575

Figure 5.112. Radial and circumferential stresses distribution along an angle for the postcritical load q = 0.5.

Therefore, a stability loss for the shell with θk = π occurs for b = 9 and q+ = 0.4. The shell loses its stability with the occurrence of two dents symmetrically located in relation to the angle’s biserctix. The membrane stresses on θ2 , θ3 have a waveform character.

576

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.113. Circumferential stresses along a radius for the post-critical load q = 0.5.

5.3 Stability of Sectorial Shells with Finite Deflections

577

PROBLEM 4. A shell with θ k = π2 . We consider a shell with the same parameters boundary and initial conditions (only ∆τ = 0.001). The computations have shown (Fig. 5.99 - curve 2) that b = 10 corresponds to the stability loss. The first limiting point on the curve corresponds to the limiting critical load q+ = 0.35.

Figure 5.114. Deflection w and stress function F distributions along an angle for the precritical load q = 0.3.

Figure 5.115. Deflection w along a radius for the critical load qcr = 0.35.

578

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.116. Deflection w and stress function F distributions along an angle for the postcritical load q = 0.6.

For the loads q = 0.3; 0.35; 0.6 the maximal deflections are achieved on the intersection of r7 with θ4 (see Figs. 5.114–5.116). Therefore, a stability loss form corresponds to one dent on the angle’s bisectrix. For each of the loads qi = 0.3; 0.35, the curves’ Nθ and Nr character is similar in relation to θi and ri (Figs. 5.117–5.120). Contrary to the sectorial shells θk = 32 π; π, for θk = π2 a zone of the stretching circular stresses is increased up to r4 , whereas a zone of the stretched circular stresses is narrowed. Maximal circular stresses appear on the arcal segment. The greatest negative radial stresses are achieved on θ3 , θ4 .

5.3 Stability of Sectorial Shells with Finite Deflections

579

Figure 5.117. Radial stresses along a radius for the pre-critical load q = 0.3.

Figure 5.118. Radial stresses distribution along a radius (critical load qcr = 0.35).

For the post-critical load q = 0.6, the behaviour is changed. The curves of stresses are wave-forming (Fig. 5.122). The occurrence of positive radial stresses zone between r5 and r10 (Fig. 5.123) for θ5 corresponds to the shell’s extension along the bisectrix. Between r5 -r7 , along the central radius, the radial stresses achieve a minimum (close to zero) and it is subjected only to the radial stresses action (Fig. 5.124). A distribution of relative equal radial stresses for q = 0.3 and q = 0.6 are presented in Fig. 5.125. For q = 0.6, on the biserctix, a zone of stretching radial stresses occurs, yielding its bending. To conclude, for the shell θk = π2 a stability loss occurs for b = 10 and q+ = 0.35, and it corresponds to the occurrence of the dent on the angle’s biserctix.

580

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.119. Circumferential stresses distribution along a radius for the pre-critical load q = 0.3.

5.3 Stability of Sectorial Shells with Finite Deflections

581

Figure 5.120. Circumferential stresses distribution along a radius for the critical load qcr = 0.35.

Figure 5.121. Radial and circumferential stresses distributions along an angle for the precritical load q = 0.3.

582

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.122. Radial and circumferential stresses distributions along an angle for the postcritical load q = 0.6.

5.3 Stability of Sectorial Shells with Finite Deflections

583

Figure 5.123. Radial stresses along a radius for the post-critical load q = 0.6.

Figure 5.124. Circumferential stresses distribution along a radius for the post-critical load q = 0.6.

584

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.125. Curves of equal relative deflections for pre-critical (q = 0.3) and post-critical (q = 0.6) loads.

Figure 5.126. Curves of equal relative radial stresses for pre-critical (q = 0.3) and postcritical (q = 0.6) loads.

PROBLEM 5. A shell θ k = π3 . The same (as in the previous case) initial data are used (∆τ = 0.0005). Fig. 5.99 (curve 1) show that the shell loses its stability for b = 11 (q+ = 0.3). For pre-critical, critical and post-critical loads q = 0.1; 0.3; 0.4 maximal deflections on θ1 -θ4 are achieved on r8 (Figs. 5.127, 5.130, 5.134, 5.137). The pictures of

5.3 Stability of Sectorial Shells with Finite Deflections

585

Figure 5.127. Deflection w distribution along a radius for the pre-critical load q = 0.1.

Figure 5.128. Deflection w and stresses function F distributions along an angle for the critical load qcr = 0.3.

circular and radial stresses for q = 0.1; 0.3 are qualitatively the same (Figs. 5.128, 5.129, 5.131–5.133). In comparison to θk = π2 , the radial stresses in the vertex vicinity become smaller. All radial stresses have a negative sign. For the post-critical load q = 0.4 symmetrically located dents are not observed. It happened because the radial stresses on θ2 , θ3 achieve large negative values (Fig. 5.135), which does not allow the shell to buckle. In the vicinity of the circular segment, on θ4 a zone of the smallest compressing stresses is observed. Then, a buckling occurs on θ5 , and the dent is shifted more closely to the arcal segment (Fig. 5.139) in comparison to θk = π2 . Therefore, for the post-critical load the maximal deflections on θ1 -θ4 are shifted to r7 (Fig. 5.130).

586

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.129. Deflection w and stresses function F distributions along an angle for the postcritical load qcr = 0.4.

Figure 5.130. Deflection along a radius for the post-critical load q = 0.4.

A distribution of Nr and Nθ along the angle is shown in Fig. 5.138 (q = 0.3) and in Fig. 5.139 (q = 0.4). Beginning from r6 and for q = 0.4, the curves ‘start to vibrate’, which characterises a stability loss.

5.3 Stability of Sectorial Shells with Finite Deflections

587

Figure 5.131. Radial stresses along a radius for the pre-critical load q = 0.1.

Figure 5.132. Circumferential stresses distribution for the pre-critical load q = 0.1.

Figure 5.133. Radial stresses distribution for the critical load qcr = 0.3.

To conclude, for the sectorial shell θk = π3 looses stability for b = 11 and q+ = 0.3. Observe that the stability loss is the same as in previously discussed cases for θk = π/3 and θk = π/2 (compare the Figs. 5.125 and 5.137).

588

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.134. Circumferential stresses distribution along a radius for the critical load qcr = 0.3.

5.3 Stability of Sectorial Shells with Finite Deflections

589

Figure 5.135. Radial stresses distribution along a radius for the post-critical load q = 0.4.

Figure 5.136. Circumferential stresses distribution for the post-critical load q = 0.4.

590

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.137. Curves of equal relative deflections for the critical (qcr = 0.3) and post-critical (q = 0.4) loads.

Figure 5.138. Radial and circumferential stresses along a radius for the critical load qcr = 0.3.

5.3 Stability of Sectorial Shells with Finite Deflections

591

Figure 5.139. Radial and circumferential stresses along a radius for the post-critical load q = 0.4.

592

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Results of the analysis The obtained numerical results concerning stability estimation of sectorial shell with diﬀerent θk and diﬀerent sloping parameter b serve of derivation of the following conclusions

Figure 5.140. Critical loads for diﬀerent sectorial shells.

Figure 5.141. Maximal shell and shell centre deflections for diﬀerent sectorial shells.

1. A sectorial angle influences the sloping parameter, for which a sectorial shell loses its stability. The angle value decrease accompanies the increase of b parameter (Fig. 5.99).

5.3 Stability of Sectorial Shells with Finite Deflections

593

Figure 5.142. Maximal deflection distribution along a radius for a precritical load.

Figure 5.143. Maximal deflection distribution along a radius for a critical load.

2. The critical load depends on the sectorial angle and the sloping parameter b. In Fig. 5.140, it is shown how qcr (static and dynamic) changes with the decrease of the angle for the critical b values. For the same b parameter we have a more stable shell for lower angle values θk . 3. Owing to increase of the sectorial angle (for θk ∈< π/3, 1.5π >), a maximal deflection for the pre-critical (Fig. 5.142), and the critical (Fig. 5.143) loads is shifted from the shell centre into the vertex neighbourhood. For θk = 2π this deflection is achieved ion the shell vertex. 4. For the shells with θk = [π, 32 π], a stability loss is achieved through the occurrence of two dents symmetrically located in relation to the angle biserctix (Fig. 5.144, the curves 3 and 4). The shells θk = [ π3 , π] lose their stability by the occurrence of one dent lying on the θ5 biserctix (Fig. 5.144a; curves 1, 2 for θk = π3 and π2 , correspondingly). A distribution of the maximal stress Fmax along an angle for diﬀerent shells is reported in Fig. 5.144b.

594

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.144. Maximal deflection (a) and stress function (b) distributions along angle for the critical load.

Figure 5.145. Maximal circumferential shell stresses along a radius (a critical load).

5. A stability loss of the sectorial shells with θk ∈ [ π3 , 32 π] is characterized by a sudden change of stresses. The membrane stresses (θk = π, Figs. 5.111, 5.113; θk = π2 , Figs. 5.123, 5.124; θk = π3 , Figs. 5.135, 5.136) rapidly decrease or change their sign. 6. Decreasing the angle of the sectorial shell yields a change of the circular stresses distribution in the vertex vicinity and on the arcal segment (Fig. 5.145, the curves 1-4 correspond to the angles θk = π3 , π2 , π, 32 π). For θk ∈ [π, 32 π] the largest stretching stresses are achieved on the arcal segment. Small angles correspond to the increase of positive Nθ in the shell vertex, and a decrease and shift of the negative stresses to the arcal segment is to be noticed. In words, decreasing the angle of the sectorial shell is accompanied by an occurrence of the stretching circular stresses action in the vertex vicinity, because a distance between the radial segments decreases and the area of stresses distribution on the angle is shortened.

5.3 Stability of Sectorial Shells with Finite Deflections

595

Figure 5.146. Maximal radial shell stresses along a radius (a critical load).

7. Positive radial stresses (Fig. 5.146, curve 4) appear only for shells with the sectorial angle θk ∈ [π, 32 π] in the vertex vicinity. It is caused by the occurrence of the internal angle, which due to slipping clamping prevents the compressed radial stresses to be appeared. For the shells θk ∈ [ π3 , π] this eﬀect is not observed, because all corresponding angles are external. 8. Owing to a change of a shell geometry, i.e. decrease of the angle θk and an increase of b parameter (for h/a = 1/200), of the integration steps can be separately attached to each of the sectors. For θk = 2π we have ∆τ = 0.01, whereas for θk = π3 we have taken ∆τ = 0.0005. 5.3.3 Heat Impact Dynamical thermoelastic problems are complex and consist of many diﬀerent issues. In this section our attention is focused on the investigation of thin walled construction elements subjected to an impulse heat. This problem attracts the attention of many researchers. From the point of view of linear theory, this class of problems is considered in references [353, 623]. The problems of dynamical stability loss of thin shells subjected to a heat impact action, when temperature distribution along a shell thickness is nonlinear and is defined by the heat transfer equation, are analysed in references [191, 376]. Mainly uncoupled thermal and elastic relations either for plates or for rectangular shells are considered. In this section, a dynamics of sectorial shells with a heat impact action is analysed. A rule of a temperature variation along a shell thickness diﬀers from a linear one, and is defined by a solution of the three dimensional heat transfer equation.

596

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.147. Distribution of dependencies w(r/a), wrr (r/a) (a), and F(r/a), Frr (r/a) (b) for a diﬀerent radius r partition.

We investigate a behaviour of a circular spherical segment. One of its surface is attacked by a heat flow q , and the second one is ideally heat isolated. The initial shell temperature is equal to T 0 . Since the heat flow is uniformly distributed along the shell surface, then the problem is axially symmetric, and for given heat transfer conditions, three dimensional unstationary equation (5.24) takes the following form: ∂2 T 1 ∂T = 2. α ∂τ ∂z

(5.71)

5.3 Stability of Sectorial Shells with Finite Deflections

597

Figure 5.148. Shell temperature distribution along thickness for diﬀerent time instants.

Figure 5.149. Time histories of a shell vertex deflections for diﬀerent q .

We consider the case, when the heat flow is applied to the internal shell’s surface. The following boundary conditions for the heat transfer equation are applied:

598

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.150. Distribution of deflection w and stress function F along a radius in diﬀerent time instants.

Figure 5.151. Time histories of a temperature moment κT for diﬀerent stream densities q .

λq ∂T = q

h ∂z

h for z = − , 2

∂T h =0 for z = , (5.72) ∂z 2 and the boundary conditions (5.25) for a rolling support of the arcal segment have the form: ∂2 w ν ∂w − (1 + ν) κT = 0, w = 0, + ∂r2 b ∂r ∂F F = 0, = 0. (5.73) ∂r The initial conditions (5.40) equal to zero are taken. For the considered case, a numerically found solution of the three dimensional heat transfer equation using the finite diﬀerence method coincides in full with the

5.3 Stability of Sectorial Shells with Finite Deflections

599

Figure 5.152. Time histories of the vertex deflection for diﬀerent q and b = 4.

Figure 5.153. Time histories of temperature moment κT for diﬀerent q and b = 4.

analytical solution presented in reference [353], obtained using the Laplace transformation. It has the following non-dimensional form: 12z2 + z − 1 q

− τ+ T = T0 + λq 2n ! nπ " #

2 (−1)n 2 2 (2z cos π τ . + 1) exp −n 2 π2 n2

(5.74)

600

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.154. Shell temperature distribution along thickness for b = 4 and τ = 0.3.

Figure 5.155. Shell temperature distribution along its thickness in diﬀerent time instants.

We consider a shell made from steel XM15 with the following parameters: ν = 0.3, E = 210 MPa, αT = 1.2 · 10−5 , C = 460 J/kgK, λq = 45.4 W/mK, a = 20h, b = 2, N = 10, q = 0, ∆τ = 0.001. In the initial equations (5.17), (5.18), for a given unstationary temperature field (5.73), the temperatural terms ∇2 κT and ∇2 εT do not appear.

5.3 Stability of Sectorial Shells with Finite Deflections

601

Figure 5.156. Shell vertex vibrations for diﬀerent heat stream densities q .

Figure 5.157. Distributions of deflection w and stress function F along a radius in diﬀerent time instants.

An influence of the heat action on the shell is displayed through an occurrence of the temperature moment κT in the boundary condition (5.73). In order to determine an optimal area partition, a numerical experiment is carried out. The problem for fixed b = 2 and q = 0.8 for r partition into 20, 15, 10 parts

602

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.158. Temperature moment time histories for diﬀerent heat stream densities q .

is solved (Fig. 5.147). Owing to numerical experiment, a partition into 15 parts is introduced. In Fig. 5.148, the rules of temperature distribution along the shell thickness for q = 0.8 and for diﬀerent time instants τ = 0.2, 0.4, 0.6 (curves 1, 2, 3) are presented. The figures imply, that the temperature distribution along the shell thickness is nonlinear. The low surface is heated with time increase, and the temperature changes continuously in z. The temperature of the surface z = −1/2 becomes lower than that of the external surface z = 1/2. We consider a behaviour of the shell vertex w0 in time for diﬀerent heat stream intensities q = 0.4; 0.6; 0.8 (curves 1, 2, 3). After an application of the heat impact, the shell slightly vibrates in a short time duration (see Fig. 5.147), and then a deflection in the direction inversed to the q action occurs. Increasing the heat flow intensity yields the deflection increase in the central point. The shell deflection form, and the function F distributions along the radius are represented by the curves 1, 2, 3 for time instants τ = 0.2; 0.35; 0.55 for q = 0.6 (Fig. 5.148). It is seen that for an arbitrary time instants a maximal deflection is achieved on the shell’s vertex. A heat transfer into the shell is realized via the boundary conditions (5.73), due to κT action. In the Fig. 5.149 for q = 0.8; 0.6 the curves 1, 2 represent the temperature moment κT . The κT increases and achieves a maximal deflection on the vertex. Due to κT , the shell is deflected to the direction of positive deflections. A deflection change on the vertex w0 for q = 0.5; 1.0 (the curves 1, 2) and for b = 4 is presented in Fig. 5.152. It can be concluded (see Fig. 5.154) that for a shell with the sloping b = 4, for τ = 0.3 and for diﬀerent q = 0.5; 1.0 (the curves 1, 2), a temperature distribution along its thickness is of larger importance for larger q . The temperature moment κT (Fig. 5.153) behaves in a similar way to that of b = 2 (the same notation, as in Fig. 5.152, is used). We analyse a stress-strain state of a shell being under influence of the heat stream applied to its external surface. The edge and below surfaces are heat isolated.

5.3 Stability of Sectorial Shells with Finite Deflections

603

In this case the boundary conditions for the heat transfer equation (5.71) have the form: ∂T h = 0, z = , ∂z 2 λq ∂T h = −q , z = − . (5.75) h ∂z 2 Applying the Laplace transformation to (5.71) and (5.75), the following nondimensional solution is obtained: 12z2 − 12z − 1 q

− τ+ T = T0 + λq 24 ∞ ! nπ " #

8 (−1)n 2 2 (2z cos π τ . − 1) exp −n π n−1 n2 2

(5.76)

In Fig. 5.155, a rule of temperature distribution along thickness for q = 0.8 and for diﬀerent time instants τ = 0.2; 0.4; 0.6 (the curves 1, 2, 3) is displayed. The largest temperature is achieved on the surface for z = −1/2, where the heat impact is applied. In this case, the deflected shell vibrates in the direction of negative values (Fig. 5.156). The absolutely maximal deflection is achieved on the shell vertex (Fig. 5.157). The temperature moment κT achieves its certain constant value (Fig. 5.158), where the curve 1 corresponds to q = 0.4. It causes the shell’s deflection in the direction of negative deflections. Analysing the obtained results the following conclusions are formulated. For the rolling support and a uniform distribution of the temperature field along the shell surface, the shell defects in the direction opposed to the heat flow action. For b = 2.4 and q = 0.4; 0.6; 0.8; 1.0, the shell’s is pre-critical state. 5.3.4 Local Surface Load With Infinite Duration In the majority of works mainly the stress strain problems of closed spherical shells with local loads and axially symmetric behaviour are considered. The analysis of the sectorial shells for a non-axially symmetric cases and subjected to local loads actions are treated rather marginally. As an example, we consider a problem of the sectorial shell behaviour subjected to the local action surface load for the following parameters: θk = π, b = 9, q = 0, ε = 0, M = N = 10, ∆τ = 0.001; a temperature field does not appear; the initial conditions (5.40) are equal to zero; the boundary conditions are defined by (5.52) and (5.53). The local load acts on the intersection of θ7 -θ9 and r3 -r5 . We analyse a behaviour of the sectorial shell for diﬀerent time instants: τ = 1.6; 4; 10.1. For the initial time instants τ = 1.6; τ = 4 the curves of deflection (Fig. 5.159, 5.160, 5.161) and F behave in a similar manner. In the Figs. 5.163, 5.164 for each θ j ( j = 1, 10), a change of deflection along the radius ri (i = 1, 10) is displayed. Maximal deflection values are lying on r4 between θ5 -θ9 , where the local load is applied.

604

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.159. Deflection distribution along an angle for τ = 1.6.

For the time instant τ = 1.6 for a normal displacement, a zone of negative deflections occurs (Fig. 5.159), which begins from the r3 radius. The stress function is presented in Fig. 5.165. In the latter case, a zone with negative values has not occurred. We investigate a stress state of the sectorial shells for time instants τ = 1.6 and τ = 4. In Figs. 5.163, 5.164 the dependencies of the radial stresses variations versus a radius are presented. In the area of the local load action, a zone with negative Nθ is presented, and the shell is compressed. In the vicinity of the compressing zone, zones with positive Nθ also appear. A maximal one is achieved in the shell vertex vicinity. The positive stresses are going to stretch the shell in vertex vicinity and in the arcal segment directions. In Figs. 5.165, 5.166, the radial stresses Nr variations versus the radius is presented. A zone of large negative Nr values appears in the place of the load action.

5.3 Stability of Sectorial Shells with Finite Deflections

605

Figure 5.160. Deflection distribution along a radius for τ = 1.6.

We investigate the shell behaviour for the time instant τ = 10.1. It is characterized by the occurrence of negative deflections (Figs. 5.167, 5.168). The curves have a waveform character along θ2 -θ5 . A similar picture is obtained for the stress function F (Fig. 5.169). Fig. 5.170 displays the fact that a zone of stretched Nθ occurs for negative deflections. In words, the shell negative deflections corresponds to stretching stresses, whereas in the place of load action the stresses Nθ remain negative. In the Fig. 5.171 the radial stresses’ are shown, where the positive Nr appear when the shell buckles. The curves of equal relative deflections for τ = 1.6; 4; 10.1 are presented in Fig. 5.172. The area of the local load action is characterized by the sectorial shell deflection to the curvature centre. This area changes in time.

606

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.161. Deflection distribution along a radius for τ = 4.0.

Figure 5.162. Stress function F distribution along angle for τ = 1.6.

In Fig. 5.173 the curves illustrating the vibrational process in diﬀerent points of sectorial shell are presented. The curves 1, 2, 3 correspond to vibrations in the

5.3 Stability of Sectorial Shells with Finite Deflections

607

Figure 5.163. Circumferential stresses distribution along a radius for τ = 1.6.

centre of the load area in the symmetrically located point in relation to the angle θk biserctix, and in the shell centre. The figures show that the sectorial shell vibrates with bucklings (jumps).

608

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.164. Circumferential stresses distribution along a radius for τ = 4.0.

The negative deflection values appear in the vicinity of the arcal segment. A border between the negative and positive deflections is approximately located in the vicinity of the central radius.

5.3 Stability of Sectorial Shells with Finite Deflections

609

Figure 5.165. Radial stresses distribution along a radius for τ = 1.6.

Figure 5.166. Radial stresses distribution along a radius for τ = 4.0.

Comparing the time of achieving the maximal deflection for both local and uniformly distributed loads for the same initial data, one comes to the conclusion that wmax is achieved two times quicker in the case of the local load action.

610

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.167. Distribution of deflection along a radius for τ = 10.1.

Figure 5.168. Distribution of deflection along an angle for τ = 10.1.

5.3 Stability of Sectorial Shells with Finite Deflections

Figure 5.169. Stress function distribution along an angle for τ = 10.1.

Figure 5.170. Circumferential stresses distribution along a radius for τ = 10.1.

611

612

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.171. Radial stresses distribution along a radius for τ = 10.1.

Figure 5.172. Vibrational process of shell points subjected to the surface local load action.

5.3 Stability of Sectorial Shells with Finite Deflections

613

Figure 5.173. Curves of equal relative shell points deflection for three time instants τ = 1.6; 4.0; 10.1.

614

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

5.4 Chaotic Dynamics of Sectorial Shells Mathematical models of sector-type shells can be directly applied in computation of reinforced membranes which are very often sensitive elements of pressure sensors in various measuring devices. They can also be used in machine construction, etc. (see references [40, 43, 49, 54, 59, 66, 71, 72, 370, 380, 381, 391]). However, this work is devoted to the analysis of chaotic dynamics of sector-type shells and its control. 5.4.1 Statement of the problem and computational algorithm We consider a non-axially symmetric shallow spherical shell as a two-dimensional object in R2 in polar coordinates bounded by the contour Γ and defined as follows: Ω = Ω+Γ = { (r, θ, z)| r ∈ [0, rn ], θ ∈ [0, θk ], z ∈ [−h/2, h/2]}. The governing equations are given in the following form [675]: w

+ εw = −∇2 ∇2 w + N (w, F) + ∇2 F + 4q ,

(5.77)

∇2 ∇2 F = −∇2 w − N (w, w) , where: ∇2 (·) = ∇2 ∇2 (·) =

∂2 (·) 1 ∂ (·) 1 ∂2 (·) + 2 + , r ∂r ∂r2 r ∂θ2

∂4 (·) 2 ∂3 (·) 1 ∂ (·) 1 ∂ (·) + + − 2 2 + 3 r ∂r3 ∂r4 r ∂r r ∂r

2 ∂3 (·) 2 ∂4 (·) 4 ∂2 (·) 1 ∂4 (·) − 3 2 + 4 + 4 , 2 2 2 r ∂θ ∂r r ∂θ ∂r r ∂θ2 r ∂θ4 1 ∂2 F ∂2 F 1 ∂w 1 ∂2 w ∂2 w 1 ∂F + 2 2 + 2 + 2 2 − N(w, F) = 2 ∂r r ∂r r ∂θ ∂r r ∂r r ∂θ ∂ 1 ∂w ∂ 1 ∂F 2· , ∂r r ∂θ ∂r r ∂θ 2 ∂2 w 1 ∂w 1 ∂2 w ∂ 1 ∂w N(w, w) = 2 · 2 + 2 2 −2· . ∂r r ∂θ ∂r r ∂r r ∂θ

) Eg Next, the non-dimensional parameters are introduced: t = ω0 t, ω0 = , γR2 ) √

√ w η q3 R 2 g R F r ε = η h , r = b c , q = q3 = 4 E h , η = 12(1 − γE h ε, F = η Eh3 , w = √ ν2 ), b = 4 η √cRh where t denotes time; ε is the damping coeﬃcient of surrounding medium, F is the stress function, w denotes the displacement function, R and c are the main curvature radius of shell resistance contour and the radius of resistance contour in circled direction, respectively; h – shell thickness, b – sloping parameter, ν – Poisson’s coeﬃcient, r – distance between a rotation axis and a point on the shell middle surface, q – external load parameter, and ω0 – frequency of linear vibrations.

5.4 Chaotic Dynamics of Sectorial Shells

615

In equation (5.77) bars are omitted. Diﬀerentiation with respect to time t is denoted by a dash. The following boundary and initial conditions are associated with system (5.77). 1. Ball-type clamping of an arc slice w = 0,

∂F ∂2 w ν ∂w = 0, F = 0, =0. + r ∂r ∂r ∂r2

(5.78)

2. Ball-type clamping of radial slices w = 0,

∂2 w ∂2 F = 0, F = 0, =0. ∂θ2 ∂θ2

(5.79)

3. Sliding clamping of an arc slice w = 0,

∂2 F ∂w = 0, F = 0, =0. ∂θ ∂θ2

(5.80)

4. Sliding clamping of radial slices w = 0,

∂2 F ∂w = 0, F = 0, =0. ∂θ ∂θ2

(5.81)

The initial conditions are as follows: w = f1 (r, θ) = 0, w = f2 (r, θ) = 0 ,

(5.82)

for time instant t = 0. In order to reduce our continual system (5.77)-(5.82) to that of lumped parameters, a finite diﬀerence method of approximation O(∆2 ) with respect to spatial variables r and θ (Fig. 5.174) is used. The diﬀerence form of equations (5.77)-(5.82) is: −Λ (Λw) + Λrr w (ΛF + Λrr F) + Λrr F (Λw + Λrr w) − 2 · Λrθ wΛrθ F + ΛF + 4qi = (wtt + εwt )i, j , Λ (ΛF) = −Λrr w (Λw + Λrr w) + (Λrθ w)2 − Λw , where: 1 1 1 (·)r , Λrr (·) = (·)rr , Λrθ (·) = − 2 (·)θ + (·)rθ , ri ri2 ri (5.83) % $ % 1 $ 1 (·)i+1, j − (·)i−1, j , Λrr (·) = 2 (·)i+1, j − 2 (·)i, j + (·)i−1, j , Λr (·) = 2 ∆r 2 · ∆r · r i

Λ (·) = Λrr (·) + Λr (·) , Λr (·) =

∆r = b/n, ∆θ = θk /m . The boundary conditions are as follows:

616

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.174. Finite diﬀerence approximation of the investigated shell.

1. Ball-type clamping of an arc slice ν wn, j = 0, Λrr w − Λr w = 0, Fn, j = 0, Λr w = 0, j = 1, m − 1; b

(5.84)

2. Ball-type clamping of radial slices wi, j = 0, Λθθ w = 0, Fi, j = 0, Λθθ F = 0, j = 0, m, i = 0, n;

(5.85)

3. Sliding clamping of an arc slice wn, j = 0, Λr w = 0, Fn, j = 0, Λr F = 0, j = 1, m − 1;

(5.86)

4. Sliding clamping of radial slices wi, j = 0, Λθθ w = 0, Fi, j = 0, Λθθ F = 0, j = 0, m, i = 0, n.

(5.87)

Two supplementary conditions are required for system (5.83)-(5.87), i.e. in the shell top and the so called compatibility conditions. In majority of cases solved by numerical methods, it is assumed that a shell has a central hole of small diameter, which has a minor influence on the obtained results at a suﬃcient distance from the shell vertex. In our approach, while solving non-axially symmetric problems θ = 2 · π, the sought functions in point r = 0 are defined by an interpolation of second order Lagrange formula. As a result one gets f0, j = 3 · f1, j − 3 · f2, j + f3, j ,

(5.88)

5.4 Chaotic Dynamics of Sectorial Shells

617

where: fi, j = f (ri ), ri = i · h (i = 0, 1, 2, 3), 0 ≤ j ≤ m − 1 and h denotes the distance between interpolating nodes. For an out-contour point the following symmetry condition holds f−1, j = f1, j for 0 ≤ j ≤ m − 1 .

(5.89)

Compatibility conditions for non-axially symmetric problems of θ = 2 · π read: wi, j = wi,m+ j , Fi, j = Fi,m+ j for j = 0, −1, 0 ≤ i ≤ n − 1 .

(5.90)

Then, the Cauchy problem (5.83)-(5.90) is solved applying the fourth order Runge-Kutta method. Computational step is yielded by the Runge’s rule. Although in the developed algorithm the applied load can be taken in an arbitrary manner, further harmonic excitation of the form q = q0 sin(ω p t) is used, where ω p denotes the excitation frequency. 5.4.2 Static problems and reliability of results The developed numerical algorithm provides a solution to various static and dynamic problems. In order to solve static problems, the so called “set-up” method is applied [43, 203, 377]. Relations q0 (wmax ) for a shell characterized by the angle sector of θk = 32 · π (Fig. 5.175), that of θk = π (Fig. 5.176) and that of θk = π2 (Fig. 5.177) for various sloping parameters: for θk = 32 · π and θk = π we take b = 5, 6, 7, 8, 9, 10, whereas for θk = π2 sloping b = 7, 8, 9, 10, 11, 12 (for smaller sloping values the sector shell behaves like a plate) are reported in Figures 5.175-5.177. As it is shown in the graphs, beginning from a certain parameter b, the limiting points occur on the curves. Owing to the computations carried out for the shell characterized by θk = 32 π, π, 12 π, the sloping values of b = 9, 9, 11 refer to critical ones, i.e. they are associated with a “shell jump occurrence”. Table 5.3 gives curves of equal deflections (isoclines) for all considered angles θ = 12 π, π, 32 π. Let us compare the curves for critical and post-critical loads q0 for diﬀerent θ and b. For θ = π2 and θ = π, and for an arbitrary sloping parameter the analyzed pictures coincide. A maximum deflection is achieved in the point of intersection of the angle θ bisectrix and central shell radius. For θ = 32 π and for b = 7 the occurrence of a “jump” is not observed and an increasing load does not change qualitatively the analyzed pictures. For the latter sector angle and for sloping b = 9; 10 the “jump” phenomenon is shown in Figure 5.175, and the curves of equal deflections for critical and post-critical loads are clearly diﬀerent. In the case of a critical load, the maximum shell deflection occurs in the middle of bisectrix. In the case of post-critical load, two zones of maximum deflections appear. They are symmetric with respect to the angle bisectrix. Since the developed algorithm allows a wide class of problems without exact solution to be solved, checking of its reliability is highly required. Reliability of

618

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.175. Dependence q0 (wmax ) for θk = 32 π.

the obtained results relies on their comparison with numerical solutions obtained in [377]. Figure 5.178 shows the relation q(wmax ) for sector shell with movable resistance contour clamped along radial and arc directions for θk = 32 π (sloping parameter b = 5, 6, 7, 8; numbers of partition with respect to a radius and angle are n = m = 10, respectively and ν = 0.3). Curves denoted by points correspond to results given in reference 5.178, whereas solid curves correspond to our results. Good coincidence of the results is clearly indicated. 5.4.3 Convergence of a finite diﬀerence method along spatial coordinates for non-stationary problems In order to trace the behavior of spatial systems from a common point of view a concept of phase space is applied. Partial diﬀerential equations governing dynamics of the analyzed objects (5.77)-(5.82) are substituted by equations governing dynamics of lumped systems applying the finite diﬀerence method (5.83)-(5.90). Below, we outline one of the most dangerous points during a transition from PDEs to ODEs. In other words, instead of infinite dimensional systems we are going to consider finite dimensional ones. It is assumed that beginning from a certain approximation a further increase of the number of equations does not introduce anything new to the obtained results and the system behaves similarly. A crucial role plays here the dimension of an analyzed attractor. However, if the dimension of the studied attractors is bounded, truncation procedures applied to equations (5.83)-(5.90) may play

5.4 Chaotic Dynamics of Sectorial Shells

619

Figure 5.176. Dependence q0 (wmax ) for θk = π.

an important role. For instance, in the case of an unsuitable choice of radius and arc partition into m and n parts, while applying the finite diﬀerence scheme, the truncated system obtained may exhibit attractors that have properties qualitatively diﬀerent than the real system attractors. In order to trace the behavior of shells harmonically excited by q = q0 sin(ω p t), a package of routines has been developed allowing for construction of scales of vibration characters depending on the control parameters {q0 , ω p } , where ω p is fixed. We have ω p = ω0 which stands for the frequency of linear vibrations. In order to construct a scale with respect to amplitude values of more than 200 points, i.e. 2 · 102 problems of dynamics must be considered, and power spectra should be studied for each choice of the control parameters {q0 , ω0 }. The algorithm allows us to distinguish the zones of harmonic vibrations, the Feigenbaum type zones, chaotic zones as well as the zones of modified Ruelle-Takens-Newhouse scenarios. The latter one is detected and illustrated further in this paper. Below, we discuss the mentioned modified Ruelle-Takens-Newhouse scenario. After harmonic vibrations with external forcing frequency and moving along the parameter q0 , a new independent frequency appears, and a transition to chaos is realized owing to the series of combinations of two frequencies. This scenario will be analyzed in more detail. Consider vibrations of a sector-type shell movably clamped along radial and arc slice that depend on a number of partitions of integration interval with respect to

620

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.177. Dependence q0 (wmax ) for θk = 12 π.

coordinates r ∈ [0; rn ] and θ ∈ [0; θk ] using the finite diﬀerence method with approximation O(h2 ). Intervals [0; rn ] and [0; θk ] have been divided into 5; 10; 15; 20; 25 parts, i.e. the number of degrees of freedom is increased. The load is uniformly distributed along the shell surface and has the following form q = q0 sin ω p t. Both relations wmax (q0 ) and scales of vibration character {q0 , ω0 } are constructed owing to monitoring of frequency spectra and Lyapunov exponents. Power spectra S (ω p ) are depicted for bisectrix of the sector-type shell. Since vibrations in all points of intervals [0; rn ] and [0; θk ] have been synchronized, a further analysis is carried out for one point only. Relations wmax (q0 ) are constructed for two sector-type shells: θk = π, b = 9 (Fig. 5.179) and θk = 12 π, b = 10 (Fig. 5.180). Scales of vibration character versus a number of partition n = m are included under the graphs. Notation applied now and used further is given in Figure 5.179. All drawings refer to free vibration frequency of the system. Already for n = m = 15 the character of vibrations does not change with the variation of shell radius and angle partition. Only a small shift of the bifurcation zone is observed on vibration type scales. In Figure 5.179, relations wmax (q0 ) practically coincide. On the vibration scales (Figure 5.179) some points taken for further study are depicted, and the associated graphs S (ω p ) versus the number of partition n = m are given in Table 5.4. Let us consider a point on the vibration scale, which for n = m = 20 is in the bifurcation zone for the corresponding load parameters: ω0 = 0.8,

5.4 Chaotic Dynamics of Sectorial Shells

621

Figure 5.178. Dependence q0 (wmax ) for θk = 32 π.

Figure 5.179. Dependence wmax (q0 ) and scales of vibration character for b = 9 and θk = π.

θk = π, b = 9, q0 = 0.25 (Fig. 5.181) and q0 = 0.76 (Fig. 5.182). In Figure 5.181 signals w (rn /2; θk /2; t), 320 ≤ t ≤ 360 are reported, whereas Table 5.4 gives power

622

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells Table 5.3. Shell deflection isoclines for various θ, b, and q0 .

spectra S (ω0 ) for a chosen point. For n = m = 5 chaotic vibrations are observed: for n = m = 10 harmonic vibrations appear; for n = m = 5 first period doubling bifurcation occurs. Relations w (rn /2; θk /2; t) shown in Figure 5.181 coincide well. Owing to the reported results one may conclude that beginning from n = m ≥ 15 the real process is approximated adequately and hence in further consideration we take n = m = 15, i.e. on each computational step 450 first order ordinary diﬀerential and 225 linear algebraic equations are solved. Table 5.175 gives analogous results for a sector-type shell with the same parameters and amplitude of excitation q0 = 0.76. The corresponding time history versus partition number n = m is shown in Figure 5.182.

5.4 Chaotic Dynamics of Sectorial Shells

623

Figure 5.180. Dependence wmax (q0 ) and scales of vibration character for b = 9 and θk = 12 π.

Figure 5.181. Time histories for diﬀerent partition numbers n = m and q0 = 0.25.

In the above, the convergence of the proposed approach depending on n (number of partitions of r and θk ) has been studied, but a problem related to N (partition number of the set {q0 , ω0 }) remains open. The mentioned problem becomes extremely important, since chaotic dynamics depends significantly on the amplitude q0 and frequency ω p of excitation. A correct choice of N yields reliable construction of scales of vibration character of the investigated system. Initially, the following problem has been analyzed. How does the character of vibrations and the function wmax (q0 ) change with respect to the number N of interval (0; q0 ) partition. Figure 5.183 shows relations for scale 1 – N = 100; scale 2 – N = 200; scale 3 – N = 400.

624

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

Figure 5.182. Time histories for diﬀerent partition numbers n = m and q0 = 0.76.

Table 5.4. Power spectra corresponding to time histories given in Fig. 5.181 and Fig. 5.182

Figure 5.183. Dependence wmax (q0 ) and vibration scales for diﬀerent partition numbers N.

5.4 Chaotic Dynamics of Sectorial Shells

625

Comparing first and second scales, one may observe that an increasing number of partitions of the amplitude of exciting force causes small bifurcation zones. At increasing number of partitions up to N = 400 (3rd scale), the character of vibrations does not change, hence further investigations are carried out for N = 200. 5.4.4 Investigation of chaotic vibrations of spherical sector-type shells Boundary conditions Sector-type shell characterized by both the angle θ = 32 π and the space Ω is divided into n = m = 15 points and the sloping parameter b = 8. Next, we investigate how shell’s vibrations change depending on the boundary conditions: movable free contour (8, 9) and movable clamping (10, 11). Relations wmax (q0 ) are illustrated in Figure 5.184. Scales of vibration character are shown in the graphs. Consider the graph wmax (q0 ), constructed for sector-type shell harmonically loaded and with ball-type resistance contour (8, 9). For q0 ≤ 0.1 the relation wmax (q0 ) (see Figure 5.184a) is linear, whereas for q0 = 0.1 a first order discontinuity associated with the Andronov-Hopf bifurcation occurs. Graph wmax (q0 ) (see Figure 5.184b) for movably clamped shell (10, 11) increases uniformly, i.e. shell vibrations are harmonic ones. Analysis of the results given in Figures 5.184a and 5.184b, yields the following conclusion: boundary conditions essentially modify the character of vibrations.

Figure 5.184. Dependencies wmax (q0 ) for movably free (a) and clamped (b) shell contour.

A movably clamped contour is associated with complex vibrations, i.e. bifurcations and chaos interlace and also jump phenomena are observed. In the case of the clamped contour, vibrations are harmonic, and therefore a modification of boundary conditions may cancel the zones of bifurcation and chaos (compare Figures 5.184a and 5.184b). The influence of sector angle Let us investigate the influence of opening sector angle of the sector-type shell on vibrations. For this purpose one type of boundary conditions will be analyzed, i.e. movable ball-type supported contour (8, 9). Let us fix the sloping parameter b = 15, and for n = m = 15 we increase the angle θ = π/4, π/2, π, 32 π (Figure 5.185a-d, respectively). In the graph wmax (q0 ) describing the system behavior for θ = π/4, large zones of independent frequencies and their

626

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

linear combinations may be observed. Already for a small amplitude of excitation the investigated system exhibits vibrations of two frequencies. In the graph wmax (q0 ), there are a few first order discontinuities, where vibration type changes. With an increase of the angle (θ = π/2), the zone of harmonic vibrations increases, and chaos vanishes. Let us study the graph wmax (q0 ) for angle θ = π. Zones of chaotic vibrations reappear, the number of first order discontinuities increases, but also the zone of harmonic vibrations increases owing to a decrease of the bifurcation zone. Due to an increase of the sector angle θ = 32 π, again the zone of harmonic vibrations increases, and then in the chaotic zone a series of first order discontinuities appears and vibration character changes.

Figure 5.185. Dependencies wmax (q0 ) for various θk = π/4 (a); π/2 (b); π (c) and 32 π (d).

An increase of the sector angle causes an increase of the first-order discontinuities, and a diﬀerence between pre-critical and post-critical states also increases. Vibrations of sector-type shells versus sloping parameter Further on, we consider vibrations of slice of the shell movably clamped on a radial and arc sector with θk = π versus sloping b. Both relations of wmax (q0 ) and scales of vibration character were monitored (Fig. 5.186a-d). The shell associated with b = 10 works in the regime of harmonic vibrations. An increase of sloping up to b = 12 produces a zone of independent frequencies and bifurcations. For b = 15; 20 a zone of chaotic vibrations on the scales of vibration types and first order discontinuities are observed in the relation wmax (q0 ), and a zone of bifurcations and harmonic vibrations decreases.

5.4 Chaotic Dynamics of Sectorial Shells

627

Figure 5.186. Dependencies wmax (q0 ) for diﬀerent values of b: 10 (a), 12 (b), 15 (c), 20 (d) and vibration scales.

It is rather expected that an increase of the sloping parameter yields the system to a less stable state, which is manifested by a large number of stiﬀ stability losses and an increase of the chaotic zone. 5.4.5 Transitions from harmonic to chaotic vibrations Four models of transition from harmonic to chaotic vibrations are known: the Feigenbaum scenario [204], the Ruelle-Takens-Newhouse scenarios [582], the Pomeau-Manneville [556] and the Landau scenarios [409]. However, none of the mentioned scenarios has been detected during analysis of our sector-type shell subjected to uniformly distributed sign changing load for either an arbitrary angle or slope parameter. As it has been already illustrated, the magnitude of sloping and sector-angle has an important influence on the system evolution. For movably clamped sector-type shell with sloping parameter b = 12; 15; 20, the novel scenario of transition from harmonic to chaotic vibrations has been discovered, which was named as the modified Ruelle-Takens-Newhouse scenario [371]. The latter one is characterized in the following manner. First harmonic vibrations with the excitation frequency appear, and then increasing the control parameter q0 yields a new independent frequency, and transition to chaos is realized by a series of linear combinations of two frequencies. Investigating the sector-type shell subject to the action of sign changeable distributed load with a movable resistance contour, and characterized by the sector angle θ = π and the sloping parameter b = 17, the Feigenbaum scenario is manifested (see Tables 5.5 and 5.6). Andronov-Hopf bifurcation cascade (five period

628

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

doubling bifurcations) is observed after a periodic window occurred, beginning from q0 = 0.11785 (see five points in the Poincar´e map). At increasing q0 , a cascade of period doubling bifurcations is observed. In the Poincare map, five groups of points are created, and every new bifurcation causes that one more point occurs in each group. Table 5.5. Power spectra, time histories and phase portraits for diﬀerent q0 .

Note that for all initial inputs the modified Ruelle-Takens-Newhouse scenario has been found. A more detailed analysis follows (see Table 5.7):

5.4 Chaotic Dynamics of Sectorial Shells

629

Table 5.5. Cont.

Table 5.6. Bifurcation sequence.

1. q0 = 0.1. The system exhibits one-frequency harmonic vibrations with exciting frequency equal to a free system frequency. The phase portrait consists of a periodic orbit and the Poincar´e map is represented by a point. 2. At an increasing amplitude of excitation q0 = 0.105, the first independent frequency occurs, and the ratio of exciting to independent frequency is equal to ω p /ω1 = 0.9231.... An increase of the part of phase portrait exhibits seven times rotating cycle, and the Poincar´e section is composed of seven arbitrarily located points. 3. At q0 = 0.108 a new linearly independent frequency equal to ω p − ω1 is born. 4. An increase of the load q0 = 0.113 yields first Hopf bifurcation. The Poincar´e map is composed of three groups of points and four points arbitrarily located, and in the power spectrum the local maximum associated with frequency ω p = ω0 /2 is observed.

630

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

5. For the amplitude of excitation equal to q0 = 0.118, the second frequency ω2 occurs, where ω p /ω1 = ω1 /ω2 = 0.9231.... 6. While increasing the control parameter further up to q0 = 0.119, the third frequency ω3 appears, and the ratio of second to third frequencies is again equal to that of two earlier cases, i.e. ω p /ω1 = ω1 /ω2 = ω2 /ω3 = 0.9231.... Table 5.7. Bifurcation sequence.

5.4.6 Control of chaotic vibrations of flexible spherical sector-type shells It is well known that a chaotic attractor is composed of a countable set of saddle-type cycles with diﬀerent periods, and that during time evolution a phase point stays in

5.4 Chaotic Dynamics of Sectorial Shells

631

vicinity of each of them. When a saddle-type cycle begins to stabilize, the trajectory remains in its neighborhood and the system starts to move in a periodic manner. In this perspective a control is understood as a stabilizationstationary!state method of orbits imbedded into a chaotic attractor. On the other hand, the system interaction is associated with problems of synchronization control. Applying target-oriented excitations on chaotic subsets associated with synchronized motions of identical systems, some of them may be transformed into stable ones, keeping the rest as unstable and vice versa. As a result, a controlled transition from non-synchronized chaotic vibrations into the regime of full synchronization of chaos is achieved. In this work, the control is realized with a help of a target-oriented excitation of the sector-type shell uniformly loaded by both harmonic excitation q = q0 sin(ω p t) and resistance time dependent torque. This type of excitation is realized during the synchronization of frequencies. Analysis of the system behavior is carried out on the basis of numerical experiments presented in a graphical form, i.e. maximum shell deflection versus the amplitude of excitation and scales of vibration type using colors. The working regime is identified using both power spectra and Lyapunov exponents. Consider a sector-type spherical shell with ball-type movable resistance contour, the sloping parameter b = 15 and shell angle θ = 12 π; π; 32 π; 38 18 π. Figure 5.187 shows relations wmax (q0 ) and scales of vibration character of the shell subjected to uniformly distributed load q = q0 sin(ω0 t), where ω0 is the frequency of free vibrations (Figure 5.187, scales 1, 3, 5, 7).

Figure 5.187. Dependencies wmax (q0 ) and vibration scales (see text for more details).

It illustrates also the graphs presenting dynamics under the action of distributed load and time-dependent moment with a free system vibration frequency excitation (scales 2, 4, 6, 8). Dashed curves denote graphs associated with the action of dis-

632

5 Dynamical Behaviour and Stability of Flexurable Sectorial Shells

tributed time-changeable load (the excitation frequency is equal to the free vibration frequency), whereas solid curves correspond to the mentioned cases but with an additional action of the resistance torque M = M0 sin(ω0 t), where ω0 is the frequency of free vibrations. All the loads are given in Table 5.179, and the corresponding number of curves in Figure 5.187. Table 5.8. Shell parameters related to results given in Figure 5.187.

Analyzing graphs 1, 3, 5, 7, first order discontinuities are visible in the zone of chaotic vibrations, which is also indicated by the vibration scale. Additional excitation of sign-changeable moment (Figure 5.187, scales 2, 4, 6, 8) causes smoothing of drawings, vibrations are harmonic with the excitation frequency, and the amplitude of shell vibrations decreases. Note that graphs 2, 4, 6, 8 indicate the most important control of chaos results through the use of harmonic torque in the case, when the minimum deflection in the relation wmax (q0 ) is achieved in the vicinity of q0 = q0max /2. Recall that by the control of chaos we mean a transformation of the chaotic behavior of our analyzed system into another harmonic or into chaotic motions but with other properties using synchronization of frequencies.

6 Coupled Problems of Thin Shallow Shells in a Temperature Field within Kirchhoﬀ-Love Kinematic Model

Coupled problems of thin shallow shells in a temperature field within the KirchhoﬀLove kinematic model are studied in Chapter 6. After a brief introduction, fundamental assumptions and relations are introduced, and then the diﬀerential equations are derived (section 6.3). In section 6.4 boundary and initial conditions are given. The finite diﬀerence model of a solution to three dimensional heat conductivity equation is formulated in section 6.5. In section 6.6 numerical algorithm is derived. Section 6.7 includes numerous examples of investigation of stability loss of shallow rectangular shells with coupled deformation and temperature fields. One more method to solve a coupled thermoelastic is presented in section 6.8.

6.1 Introduction The recent achievements of techniques and technology (supersonic aircraft industry, spacecraft technique, nuclear power industry, and so on) moved the dynamical problems of thermoelasticity to a series of very important and forefront situated problems in the field of theory of elasticity. Nowadays it is known that a proper analysis of mechanical constructions subjected to an impulse type thermal load action cannot be carried out within the quasistatical theory. The first investigation of dynamical behaviour of an elastic body subjected to a high speed heating belongs to Danilevskaya [163], who considered in 1950 a stress-strain state of a halfspace while heating its surfaces. The first investigations of construction vibrations caused by a sudden heat impact have been carried out by Barber and Boli in 1956 for beams and in 1957 for plates [116]. In the reference [113] it has been shown that a possibility of a vibration occurrence due to non-constant thermal load increases with an increase of axial (beams) and longitudinal (plates) loads. The vibrations of spherical [366] and shallow [445] shells subjected to the supply of heating have also been analysed. It should be emphasised that both theoretical and experimental investigations of thin-walled constructions subjected to impulse type local heating have been reported in reference [29]. On the other hand, a heat conduction theory developed in a frame of assumption that the thermal and deformation fields are independent. The theory of coupled thermoelasticity links these two branches of investigation. According to Nowacki [512], the solutions obtained in a frame of linear theory of coupled thermoelasticity quantitatively do not diﬀer essentially from the solutions obtained

634

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

within theory of elasticity and theory of a heat conduction. However, the quantitative diﬀerences are remarkable. For instance, in a case of elastic waves and within the used coupled theory of thermoelasticity they exhibit a dispersion and they are damped, contrary to the behaviour governed by the theory of elasticity. Nowadays, the fundamental results are obtained mainly in a frame of linear theory.

6.2 Fundamental Assumptions and Relations A shell with the thickness h = const and the volume Ω having the boundary Γ is considered. Its middle surface is related to the orthogonal co-ordinates x, y, z. The positive direction of z axis goes in its curvature centre. The displacements in directions x, y are denoted by u(x, y, t) and υ(x, y, t), respectively. We assume that a displacement of an arbitrary point of a shell in z direction does not depend on z and for all points of a given normal element it is equal to normal displacements w(x, y) of the corresponding point of the middle surface. We assume all displacements essentially smaller then a characteristic shell dimension of the mean surface. The deformations in mean surface ε11 , ε22 , ε12 are neglected (they are small in comparison to 1). However, it does not mean that relations between displacements and deformations are linear. Below the theoretical investigations using Kirchhoﬀ-Love kinematic model is presented. In a frame of this model the following relations are valid for an arbitrary shell surface e11 = ε11 + zæ11 , e12 = ε12 + zæ12 , h h , − ≤z≤ 2 2

↔

(1, 2)

(6.1)

where εi j (i, j = 1, 2) are tangential deformations of the middle surface, χi j are the bending deformations. The following relations between deformations and displacements are taken 2 ↔ ↔ ∂u 1 ∂w − k1 w + , (x, y), (1, 2) ε11 = ∂x 2 ∂x ∂u ∂υ ∂w ∂w + + , ε12 = ∂y ∂x ∂x ∂y ↔ ∂2 w ∂2 w . æ11 = − 2 , (x, y) , æ12 = −2 ∂x∂y ∂x

(6.2)

An experiment shows that a body deformation and a heat quantity in a body are always dependent. This experimental observation leads to conclusion that a theory taking into account a coupling of deformation and thermal fields gives hope to describe the real deformation processes which occur while dynamical stability loss

6.2 Fundamental Assumptions and Relations

635

is exhibited. We assume that in an undeformable state a shell has the temperature T 0 . Due to actions of surface and mass forces, internal heat sources and a external heat change, shell starts to deformate and its temperature starts to change. By θ(x, y, z, t) = T (x, y, z, t) - T 0 a temperature increase is denoted in the point (x, y, z) and in time t. By T we mean the absolute temperature values in considered point in a given time instant. We assume that |θ/T 0 | 1. It means that a temperature change θ is so small, that it does not influence elastic and thermophysical shell material properties, which are assumed to be constant. Therefore, a linear heat transfer equation can be further considered. The following notations for isotermic quantities are used: E - Young’s modulus, ν - the Poisson’s coeﬃcient, αT - thermal expansion coeﬃcient, λq - thermal conduction coeﬃcient, ρ - density, cε - unit volume heat capacity for a constant deformation tensor. According to the Duhamel-Neuman hypothesis the Hook’s low is governed by the relations ↔

E E αT θ, (e11 + νe22 ) − 1−ν 1 − ν2 E e12 . = 2(1 − ν)

σ11 = σ12

(1, 2) (6.3)

Integrating the stresses governed by equation (6.3) in relation to z, the following forces are found h/2 T1 = −h/2

E EαT σ11 dz = (ε11 + νε22 ) − 1−ν 1 − ν2

h/2 σ12 dz = −h/2

4 1 h/2

θdz = −h/2 ↔

Eh (ε11 + νε22 − αT (1 + ν)NT ) , 1 − ν2 S =

h/2

(x, y),

Eh ε12 , 2(1 − ν)

(6.4)

where NT = h −h/2 θdz denotes the thermal force. In order to get moments the following formulas are used: h/2 M1 = −h/2

EαT σ11 dz = D(æ11 + νæ22 ) − 1−ν

D (æ11 + νæ22 − αT (1 + ν)MT ) , M12 =

D(1 − ν) æ12 , 2

h/2 dz = −h/2 ↔

(x, y), (6.5)

636

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

4 Eh3 12 h/2 where D = 2(1−ν θzdz is the thermal 2 ) is the cylindrical stiﬀness, MT = h3 −h/2 moment. Two first equations of (6.4) are solved in relation to tangential deformations of middle surface, and one obtains ε11

1 αT (T 1 − νT 2 ) + = Eh h ε12 =

h/2 θdz,

↔

(x, y),

−h/2

2(1 + ν) S. Eh

(6.6)

6.3 Diﬀerential Equations Using equations of an entropy balance and taking into account a dependence between heat flow vector components and the thermodynamical forces expressed by Fourier law, one gets the general heat conduction equation 2 ∂ θ ∂2 θ ∂2 θ EαT T 0 ∂ ∂θ (e11 + e22 + e33 ) + W0 . =− + + (6.7) cε − λq ∂t 1 − 2ν ∂t ∂x2 ∂y2 ∂z2 Note that the introduced term ∂(·) ∂t links a temperature increase with the speed of volume changes (e - first invariant of the deformation tensor, W0 (x, y, z, t) - heat amount emitted by a volume unit in a time unit). In order to get equation governing vibration of a shell in a hybrid form the following diﬀerential equations governing the shell element hdxdy motion are applied [684] (inertial forces are used only in the z direction and rotary inertia of the element of around the x, y axes) ∂T 1 ∂S + + p1 = 0, ∂x ∂y ∂T 2 ∂S + + p2 = 0, ∂x ∂y ∂w ∂ ∂Q1 ∂Q2 ∂w + + k1 T 1 + k2 T 2 + T1 +S + ∂x ∂y ∂x ∂x ∂y ∂2 w ∂ ∂w ∂w (S + T 2 ) + q = ρh 2 , ∂y ∂x ∂y ∂t 3 ∂M1 ∂M12 h ∂3 w + − Q1 = ρ , ∂x ∂y 12 ∂x∂t2 ∂M12 ∂M2 h3 ∂3 w + − Q2 = ρ , ∂x ∂y 12 ∂y∂t2

(6.8)

6.3 Diﬀerential Equations

637

where p1 , p2 , q are the external load intensities along the axes x, y, z; ρh ∂∂tw2 dxdy is h3 ∂2 w h3 ∂2 w the inertial force component along z axis; ρ 12 , ρ 12 are rotary moments of ∂x∂t2 ∂y∂t2 inertia. The transversal forces are obtained from the last two equations of (6.8) and they read h 3 ∂2 w ∂M1 ∂M12 + −ρ , Q1 = ∂x ∂y 12 ∂x∂t2 2

Q2 =

h 3 ∂2 w ∂M12 ∂M2 + −ρ . ∂x ∂y 12 ∂y∂t2

(6.9)

Let us introduce the stress function T 11 =

∂2 F − p1 (y), ∂y2

T2 =

∂2 F − p2 (x), ∂x2

S =−

∂2 F , ∂x∂y

which identically satisfies two first equations of forces equilibrium. The following equation is used additionally [684]: ∂2 ε11 ∂2 ε22 ∂2 ε12 1 = − L(w, w) − ∇2k w. + − 2 2 ∂x∂y 2 ∂y ∂x

(6.10)

Substituting the values of transversal loads into third equilibrium equation of (6.8) we get h /2 3 3 ∂ w ∂w EαT ∂ )− θzdz− Q1 = −D( 3 + ν 1 − ν ∂x ∂x ∂x∂y2 −h/2 h3 ∂ 3 w D(1 − ν) ∂3 w − ρ , 2 12 ∂x∂t2 ∂x∂y2

↔

(1, 2),

↔

(x, y),

(6.11)

and finally the following equilibrium equation is obtained

EαT 2 ∇ −D∇4 w − 1−ν

h /2 θzdz − ρ

h3 2 ∂2 w ∇ + k1 T 1 + k2 T 2 + 12 ∂t2

−h/2

∂w ∂w ∂2 w ∂w ∂ ∂w ∂ (T 1 +S ) + (S + T 2 ) + q − ρhT 1 2 = 0. ∂x ∂x ∂y ∂y ∂x ∂y ∂t Taking into account (6.6) in (6.10) one obtains 1 ∂2 T 1 ∂2 T 2 ∂2 T 2 ∂2 T 1 + − ν − ν Eh ∂y2 ∂y2 ∂x2 ∂x2

(6.12)

638

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

αT 2 ∇ h

h /2 θdz −

1 2(1 + ν) ∂2 S = − L(w, w) − ∇2k w. Eh ∂y∂x 2

(6.13)

−h/2

In case of lack of p1 and p2 the following coupled system of equations governing a motion of thin shallow shell in thermal field is obtained αT 2 1 4 ∇ F+ ∇ Eh h

h /2

1 θdz = − L(w, w) − ∇2k w, 2

−h/2

D∇4 F +

EαT 2 ∇ 1−ν

h /2 θzdz + ρ

h3 2 ∂2 w ∂2 w ∇ = q − ρh 2 + L(w, F) + ∇2k F, 2 12 ∂t ∂t

−h/2

Cε

∂θ ∂2 θ ∂2 θ EαT T 0 ∂ ∂θ − λq ( 2 + 2 + 2 ) = − (e11 + e22 + e33 ) + W0 , ∂t 1 − 2ν ∂t ∂x ∂y ∂z

(6.14)

where ∇4 = ∇2 ∇2 is the doubled two dimensional Laplace operator, and L((·), (·)) =

∂2 (·) ∂2 (·) ∂2 (·) ∂2 (·) ∂2 (·) ∂2 (·) + −2 2 2 ∂x∂y ∂x∂y ∂x ∂y ∂y2 ∂x2

represents known nonlinear operator. In the third equation of (6.14) e33 is obtained taking into account a plane stress state (σ33 = 0) ν ν 1+ν (e11 + e22 ) + αT θ. e33 = − (σ11 + σ22 ) + αT θ = − E 1−ν 1−ν

(6.15)

In this case the first invariant of the deformation tensor has the form 1 − 2ν 1+ν (e11 + e22 ) + αT θ = 1−ν 1−ν 1+ν 1 − 2ν 1 − ν (T 1 + T 2 ) + 2αT NT + z(æ11 + æ22 ) + αT θ. 1 − ν Eh 1−ν e = e11 + e22 + e33 =

(6.16)

The system of diﬀerential equations (6.14) of a hybrid type describes a coupled problem of the thin shallow thermoelastic shells with an account of rotary inertia using Kirchhoﬀ-Love model without a given law of temperature changes along the shell’s thickness. It means that we have two dimensional equations of the shell theory and three dimensional thermal conductivity parabolic type equation. When the rotary inertia are neglected the system (6.14) is reduced to the following one 2 ∂ θ ∂2 θ ∂2 θ ∂θ + 2 + 2 = −W0 + Cε + λq 2 ∂z ∂x ∂y ∂z

6.4 Boundary and Initial Conditions

639

⎫ ⎧ ⎡ ⎤ ⎪ ⎪ h/2 ⎢⎢⎢ ⎥⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2αT EαT T 0 ∂ ⎨ 1 − 2ν ⎢⎢⎢ 1 − ν 2 1+ν ⎬ ⎥ 2 ⎥ ⎥ (∇ α F − (p + p )) + θdz − z∇ w θ + , ⎢⎢⎢ ⎥⎥⎥ ⎪ 1 2 T ⎪ ⎪ ⎪ ⎪ ⎪ 1 − 2ν ∂t ⎪ 1 − ν Eh h 1 − ν ⎣ ⎦ ⎪ ⎭ ⎩ −h/2

h/2

αT 2 1 4 ∇ F+ ∇ Eh h

−h/2

1 θdz + L(w, w) + ∇2k w = 0. 2

(6.17)

Finally, below we introduce the geometrically linear system of equations of coupled problem of the shell’s theory in a thermal field 2 ∂ θ ∂2 θ ∂2 θ EαT T 0 ∂ 1 − 2ν 1 − ν 2 (∇ F− − W0 − + + λq 1 − 2ν ∂t 1 − ν Eh ∂x2 ∂y2 ∂z2 ⎫ ⎤ ⎪ h/2 ⎥⎥⎥ ⎪ ⎪ ⎪ ∂θ 2αT 1 + ν ⎥ ⎬ 2 ⎥ αT θ⎪ −(p1 + p2 )) + θdz − z∇ w⎥⎥⎥⎥ + = Cε , ⎪ ⎪ h ∂t ⎦ 1−ν ⎪ ⎭ −h/2

EαT 2 ∇ D∇ F + 1−ν

h/2

4

θzdz − ∇2k F = −ρh

−h/2

∂2 w + q. ∂t2

(6.18)

It should be emphasised that all above given systems of equations are obtained for an arbitrary distribution of thermal field along a shell thickness.

6.4 Boundary and Initial Conditions In order to solve coupled thermoelastic problem we need to attach boundary and initial conditions to the system of diﬀerential equations (6.14). As the initial conditions we take the distribution (in the initial time t = 0) of deflections and their velocities due to a given law of temperature increase of the form ∂w = ϕ2 (x, y), w|t=0 = ϕ1 (x, y), ∂t t=0 θ|t=0 = ϕ3 (x, y, z).

(6.19)

We remind a reader the mostly met types of the boundary condition for the edge x = 0: a) sliding support

∂w = 0, ε22 = 0, T 1 = 0. ∂x b) flexible in a tangential plane and non-stretched (non-damped) rib w = 0,

w = 0,

M1 = M0 (y),

T 1 = 0,

ε22 = 0.

(6.20)

(6.21)

640

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

c) free support w = 0,

M1 = M0 (y),

T 1 = 0,

S = 0.

(6.22)

Other boundary conditions can be found elsewhere [369, 686]. Remark. For a considered problem there exist a huge manifold of the possible boundary conditions, which are not reported here. For a heat conductivity equation the following four types of the boundary conditions can be formulated (I - IV types). I type. In this case a temperature distribution along a shell surface Ω1 is given θ(x, y, z, t) = f1 (x, y, z, t),

t ∈ [0, t1 ],

(x, y, z) ∈ Ω1 ,

(6.23)

as a function of the co-ordinates. II type. A temperature distribution of a heat flow density on the shell’s surface Ω1 is given ∂θ = f2 (x, y, z, t), t ∈ [0, t1 ], ∂n (6.24) (x, y, z) ∈ Ω1 , as a function of the co-ordinates. III type. On surface Ω1 a condition of a heat exchange with a surrounding medium is given ∂ + B θ(x, y, z, t) = f3 (x, y, z, t), t ∈ [0, t1 ], ∂n (x, y, z) ∈ Ω1 .

(6.25)

IV type. This case corresponds either to a heat exchange of a body surface with a surrounding medium (conventional heat exchange with a liquid) or to a heat exchange of contacting bodies with the same temperature of surfaces: −λ1

θ1S = θ2S . ∂θ1 ∂θ2 s = −λ2 s. ∂n ∂n

(6.26) (6.27)

Equation (6.26) describes a thermal field continuity condition, whereas equation (6.27) is related to energy conservation on contact surfaces of the bodies.

6.5 Solution to 3D heat conductivity equation In a case of uncoupled problem of thermoelasticity the system (6.14) splits into equations governing a motion and into heat conductivity equation, of the form

6.5 Solution to 3D heat conductivity equation

1 ∂θ ∂2 θ ∂ 2 θ ∂ 2 θ W0 , + 2+ 2 + = 2 λq α ∂t ∂x ∂y ∂z

641

(6.28)

λ

where: α = cεq is the thermal conductivity coeﬃcient. In the equation (6.28) there is no term describing a shell geometry (in fact, the heat conductivity equation is given for a plate), which is allowed (see reference [122]). The following dimensionless quantities are introduced x = a x¯,

y = b¯y,

z = b¯z,

λa =

h , a

h2 h2 ¯ h , t = τ, T = T. b 2 abαT A solution to three-dimensional unstationary heat conductivity equation possesses an important meaning for applications especially without restrictions on temperature distribution along a thickness. Taking into account the dimensionless quantities the following heat conductivity equation is obtained (bars are omitted) λb =

∂2 θ ∂2 θ ∂ 2 θ ∂θ = λ2a 2 + λ2b 2 + 2 + W0 ∂τ ∂x ∂y ∂z

(6.29)

In order to solve the obtained equation the finite diﬀerence method is applied in relation to space co-ordinates with the approximation 0(h2 ) λ2 dθi, j,k λ2a = 2 (θi−1, j,k − 2θi, j,k + θi+1, j,k ) + 2b (θi, j−1,k − 2θi, j,k + θi, j+1,k )+ dτ hx hy 1 (θi, j,k−1 − 2θi, j,k + θi, j,k+1 ) + W0i, j,k . (6.30) h2z This approach reduces the partial equation (6.29) to a set of ordinary diﬀerential equations because of time, which next is solved using the forth order Runge-Kutta method. In order to testify the reliability of the obtained results some simulations have been carried out by dividing some space into (6 x 6 x 6), (8 x 8 x 8) and (12 x 12 x 12) parts. The obtained results have also been compared with known analytical solutions obtained by Kovalenko (see reference [355], page 468), for the case of heat impact on a space boundary and convectional type of a heat exchange (boundary conditions (6.24) and (6.25)). The obtained results related to heat impact on a plate surface are reported in Table 6.1. The diﬀerence between numerical solution with partition into (12 x 12 x 12) and (8 x 8 x 8) parts in comparison to the analytical solution does not exceed 0.2% and 0.5%, respectively. Therefore, a partition of the space into (8 x 8 x 8) parts leads to accurate enough solution while solving the three dimensional unstationary heat conductivity equation.

642

6 Coupled Problems of Thin Shallow Shells in a Temperature Field Table 6.1. Comparison of various numerical scheme results. Time z = 0.5 z = 0.25 z=0 z = −0.25 z = −0.5 z = 0.5 z = 0.25 z=0 z = −0.25 z = −0.5 z = 0.5 z = 0.25 z=0 z = −0.25 z = −0.5 z = 0.5 z = 0.25 z=0 z = −0.25 z = −0.5

Exact solution [355]

(12 x 12 x 12)

(8 x 8 x 8)

(4 x 4 x 4)

τ = 0.2 67.19 39.19 21.06 11.27 8.18 67.04 39.07 20.98 11.20 8.16 66.86 38.91 20.89 11.17 8.14 65.87 38.07 20.38 10.96 8.08

τ = 0.4 97.01 68.07 47.66 35.56 31.55 96.92 67.98 47.58 35.49 31.49 96.80 67.87 47.48 35.40 31.41 96.17 67.27 46.96 34.96 32.00

τ = 0.6 124.06 94.99 74.26 61.84 57.71 123.97 94.90 74.17 61.76 57.62 123.87 94.80 74.07 61.66 57.53 123.33 94.26 73.56 61.16 57.03

6.6 Algorithm Now an algorithm for a sought solution to the unstationary problem of thermoelasticity of whipping shallow shells will be proposed. The equations (6.27) governing a shell dynamical behaviour within a kinematic Kirchhoﬀ-Love hypothesis without and rotary inertia are of the following form because of the introduced dimensionless quantities (again the bars are omitted) 4 4 ∂4 w 1 −2 ∂ w 2∂ w λ + + 2 + λ 12(1 − ν2 ) ∂x4 ∂x2 ∂y2 ∂y4 ⎛ ⎞ 0,5 0,5 ⎜⎜⎜ ⎟⎟⎟ 2 2 ∂ 1 ⎜⎜⎜ −1 ∂ ⎟ θzdz + λ 2 θzdz⎟⎟⎟⎟⎟ − L(w, w)− ⎜λ 1 − ν ⎜⎜⎝ ∂x2 ∂y ⎠ −0,5

∇2k F = q −

a2 b2 α2 ρ ∂2 w , Eh6 ∂t2

4 2 ∂4 F F 2∂ F −1 ∂ + 2 + λ + λ ∂x4 ∂x2 ∂y2 ∂y4 ∂x2

−2 ∂

λ

−0,5

4

0,5 θdz+ −0,5

6.6 Algorithm

λ

∂2 ∂y2

643

0,5 θdz = −L(w, w) − ∇2k w = 0.

(6.31)

−0,5

The following additional dimensionless quantities are introduced w = hw, ¯

¯ F = Eh3 F, λ=

a , b

q=

k1 =

h¯ k1 , a2

Eh4 q, ¯ a2 b2

ε=

k2 =

h¯ k2 , b2

α ε, ¯ h2

hλq

a2 b2 α2 ρ q¯ , æ = abαT Eh6 The heat conductivity equation (6.28) is attached to equation (6.31). The two mentioned equations govern a motion of an elastic shallow shell in a thermal field. Application of finite diﬀerences method with an accuracy of 0(h2 ) regarding to space co-ordinates to the system (6.31) result in getting a system of algebraic equations for the stress function F and a system of ordinary second order diﬀerential equations for a deflection w: −2 λ 1 (wi−2, j − 4wi−1, j + 6wi, j − 4wi+1, j + wi+2, j )+ 2 12(1 − ν ) h4x q =

2 (wi−1, j−1 − 2wi−1, j + wi−1, j+1 − 2(wi, j−1 − 2wi, j + wi, j+1 )+ h2x h2y wi+1, j−1 − 2wi+1, j + wi+1, j+1 ) +

λ2 (wi, j−2 − 4wi, j−1 + 6wi, j − h4y

−1 λ 1 4wi, j+1 + wi, j+2 ) + (MT i−1, j − 2MT i, j + MT i+1, j )+ 12(1 − ν) h2x ⎤ ⎥⎥⎥ λ 1 (M − 2M + M ) ⎦ − 2 2 (wi−1, j − 2wi, j + wi+1, j ). T i, j−1 T i, j T i, j+1 ⎥ h4y h x hy %

(6.32)

$ (Fi, j−1 − 2Fi, j + Fi, j+1 ) + (wi, j−1 − 2wi, j + wi, j+1 )(Fi−1, j − 2Fi, j + Fi+1, j )− 1 (wi+1, j+1 + wi−1, j−1 −wi+1, j−1 −wi−1, j+1 )(Fi+1, j+1 + Fi−1, j−1 −Fi+1, j−1 −Fi−1, j+1 ) − 8 d2 wi j K1 K1 (F − 2F + F ) − (F − 2F + F ) = q − æ , i, j−1 i, j i, j+1 i−1, j i, j i+1, j i, j h2y h2x dτ2 λ−2 2 (Fi−2, j − 4Fi−1, j + 6Fi, j − 4Fi+1, j + Fi+2, j ) + 2 2 (Fi−1, j−1 − 4 hx h x hy 2Fi−1, j + Fi−1, j+1 − 2(Fi, j−1 − 2Fi, j + Fi, j+1 ) + Fi+1, j−1 − 2Fi+1, j +

644

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

Fi+1, j+1 ) +

λ2 (Fi, j−2 − 4Fi, j−1 + 6Fi, j − 4Fi, j+1 + Fi, j+2 )+ h4y

λ−1 λ (NT i−1, j − 2NT i, j + NT i+1, j ) + 4 (NT i, j−1 − 2NT i, j + NT i, j+1 ) = h2x hy −

1 (wi−1, j − 2wi, j + wi+1, j ). h2x h2y

(wi, j−1 − 2wi, j + wi, j+1 ) +

(6.33)

1 (wi+1, j+1 + wi−1, j−1 − wi+1, j−1 − wi−1, j+1 )2 − 16h2x h2y

K2 K1 (wi, j−1 − 2wi, j + wi, j+1 ) − 2 (wi−1, j − 2wi, j + wi+1, j ), 2 hy hx 4 0.5 4 0.5 where: NT = −0.5 θdz, MT = 12 −0.5 θzdz are the dimensionless forces and moments, respectively. ˙ is reduced The obtained system of equations (6.31), (6.33) by a change dw dτ = w to the system of ordinary diﬀerential equations of the first order because of time, and to a system of algebraic equation for F. Similar to a case of heat conductivity equation the integration is carried out using the Runge-Kutta method of the fourth order. The boundary conditions are formulated similarly in the central diﬀerences. For instance, the condition (6.23) has the form 1 λ−1 MT 0, j , (w−1, j + w1, j ) = − 2 12(1 − ν) hx

F−1, j = F1, j .

(6.34)

It serves to define the deflections and stress function in the nodes located outside the boundary. The number of equations in the system (6.32) and (6.33) is the same as the number of equations for the boundary conditions, and it corresponds to the number of the unknown quantities. The additional equations related to a shell boundaries are not needed, because for the given above boundary conditions (6.20)–(6.23) the values of stress and deflection functions are known. In order to solve the system of equations (6.32), (6.33) governing the behaviour of a an elastic shallow shell in a thermal field the following algorithm is applied. In the first step a thermal field is defined as a result of a solution to heat conductivity equation (6.30) with the corresponding initial and boundary conditions. Integration along the thickness using the Simpson’s formulae leads to the estimation of thermal forces NT i, j and moments MT i, j . In the second step the initial deflection values in each node are substituted to the system of algebraic equations (6.33), which are solved using the Gauss method. In result a stress function field Fi, j is obtained. In the third step the obtained stress function Fi, j values in the shell nodes are substituted to equation (6.32), which is integrated by using the Runge-Kutta method. As a result the deflection field wi, j is obtained.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

645

The field of thermal forces NT i, j and moments MT i, j found in the first step serves as parameter and it is introduced into equations (6.32) and (6.33). Then the computations are carried out using the mentioned scheme, but as the initial function values serve those found in the previous step in time. It should be emphasized that because of Runge-Kutta algorithm construction a step in time should be two times smaller than in a case of shell motion equations when they are solved separately.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields An influence of a coupling between deformation and temperature fields on the stress-strain state and on the critical values of the dynamical loads will be investigated using a shallow squared (λ = 1) shell with the geometrical parameters k1 = k2 = 24 with the boundary conditions (6.23) and zero values initial conditions (6.19). The investigated shell is made from the alloy AMC with the following mechanical and thermophysical parameters [322]: E = 6.9 · 1010 Pa; γ = 0.3; ρ = 2800kg/m3 ; αT = 2.4 · 10−3 1/k; λq = 186´lBm/mk; cε = 0.84/m3 k . A dynamical behaviour of a shell with a sudden (t = 0) application of the infinite in time load impulse uniquely distributed in time of the surface z = −0.5 is investigated.

8

W(0.5;0.5) 5

6

8

4

9

4 7 2 1 0

0.2

2 0.4

6

3

0.6

t 0.8

Figure 6.1. Deflection versus time for the shells point x = y = 0.5 for coupled (dashed curves) and uncoupled (solid curves) of the thermal and deformation fields.

646

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

In Fig. 6.1 the dependencies w(t) for x = y = 0.5 obtained for both coupled (dashed curves 6-9) and uncoupled deformation and temperature fields (solid curves 1-5) are reported. It occurred that a critical dynamical load qcr = 93 in a case of coupling between deformation and temperature fields is higher in comparison to the uncoupled problem. The results obtained using dynamical criterions of Volmir, Kantor, and Shian, Soong and Roth [44] are similar in a case of considered coupled problems. In Fig. 6.1 the loads q = 90, 92, 93, 94 correspond to the curves 6 - 9. The bifurcation point of the diﬀerent curves occurred for τ = 0.1. A decrease of a dynamical critical load while uncoupled problem is considered is caused by a fact that the whole work of external forces is applied for a shell deformation. In the real shells a part of the mentioned work is spread out to the construction heating which decreases its displacements. On each of further reported figures of this section the solutions related to the uncoupled problem are denoted by solid curves. In a case of uncoupled thermoelastic problem with the loads q = 85, 88, 89 the shell has been stable (curves 1, 2, 3). The load qcr = 90 (curve 4) correspond to a critical load in a case of uncoupled problem. According to the Volmir criterion a large deflection increase is observed (from 3.2 to 6.6) when a load is increased only by value of 1. According to the Kantor criterion, a centre deflection is higher than a rise arrow of the shell. Also the Shian, Soong and Rhot criterions lead to the similar results. A time needed for the shell maximal deflection increases with a load increase (τ = 0.25 for q = 88 - curve 1; τ = 0.26 for q = 88 - curve 2; τ = 0.27 for q = 89 - curve 3; τ = 0.28 for q = 90 - curve 4). With a further load increase the time needed to achieve the maximal load starts to decrease (for q = 95 we have τ = 0.48). The critical dynamical load due to all mentioned criterions reads qcr = 90. Now a stress-strain state of a shell will be investigated when a maximal deflection is achieved (Fig. 6.2). When a coupling eﬀect between the deformation and temperature fields has been taken into consideration, the stress σ xx obtained in the shell centre for z = 0.5 was 11% lower, whereas the deflections in the shell centre were practically the same in both cases. Now the behaviour of σ xx in the stability loss moment will be analysed. For the time beginning we take the moment (Fig. 6.3) when the change of the membrane stresses occurs. The σ xx (0.5; 0.5; 0.5) values are 17% smaller in a case of coupled thermoelasticity problem. It should be noted that deflections in the shell centre for this time moment are practically the same (wc (0.5; 0.5) = 6.13; wunc = 6.12). In Figures 6.4, 6.5, 6.6 the defections of shell centre (curves 1, 2) and of shell quadrants (curves 3, 4) against time for the loads q = 80, 90, 100 are presented (solid curves correspond to uncoupled thermoelastic problem). For the load q = 80, which is a before critical one for both coupled and uncoupled thermoelastic problems, a decrease of 7% and of 10% of maximal shell

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

647

sxx 60 40

z=0.5

20 y 0

z=0 0.25

0.375

0.375

0.25

0

x=y

-20 z=-0.5

Figure 6.2. Strain state of a shell reported when a maximal deflection is achieved the left part is related to a change of σ xx along shell axis, whereas the right part is related to a change of σ xx along shell diagonal. Solid curve corresponds to the uncoupled problem (Wmax = 6.59, qcr = 90 for τ = 0.78), dashed curve corresponds to the coupled problem (Wmax = 6.60, qcr = 93 for τ = 0.66).

sx z=0.5

60 40

20 y 0

z=0 0.25

0.25

0

x=y

-20 z=-0.5

Figure 6.3. Stresses σ xx when the stability loss occurs.

deflection in a coupled thermoelastic problem is observed for the first and second wave, respectively (Fig. 6.4). Also a decrease of vibration period is clearly exhibited. For the load q = 90, which occurs to be critical for both coupled and uncoupled problems, the deflections

648

6 Coupled Problems of Thin Shallow Shells in a Temperature Field 2.0

W

1.5

2 1

1.0 4 3 0.5

0

0.2

0.4

t

0.6

Figure 6.4. Deflections of a shell centre (curves 1, 2) and deflection of a quadrant shell centre (curves 3, 4) versus time for q = 80. 8

W

1 6

4

3 2

2

4 0

0.2

0.4

0.6

t

Figure 6.5. Deflections of a shell centre (curves 1, 2) and deflection of a quadrant shell centre (curves 3, 4) versus time for q = 90.

are qualitatively diﬀerent (Figure 6.5). For a post critical load q = 100 (Fig. 6.6) the deflections corresponding to the coupled problem are smaller of 4 - 6% in comparison to uncoupled problem. A period increase is observed. A diﬀerences of σ xx for diﬀerent loads q = 100, 90, 80 are presented in Figures 6.7–6.9, respectively.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields 8

649

W

6

2

4

1 2

4 3

0

0.2

0.4

0.6

t

Figure 6.6. Deflections of a shell centre (curves 1, 2) and deflection of a quadrant shell centre (curves 3, 4) versus time for q = 100.

sxx

40

30

20

10

0

0.2

0.4

0.6

t

Figure 6.7. The changes of σ xx corresponding to deflections presented in Figure 6.4.

650

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

sxx

40

30

20

10

0

0.2

0.4

0.6

t

-10

Figure 6.8. The changes of σ xx corresponding to deflections presented in Figure 6.5.

The numerical simulations lead to conclusion that a coupling thermoelastic effect has a bigger influence on a shell stress-strain state than on deflections. It is caused by a large diﬀerence in the second derivative of deflection and stress function. For instance, for q = 80 in the time moment τ = 0.22 (Fig. 6.10) a diﬀerence of deflections (curve 1 - uncoupled thermoelastic problem, curve 2 - coupled thermoelastic problem) reached 7%, whereas a diﬀerence related to the second derivatives of deflections due to space co-ordinates (curve 3 - uncoupled thermoelastic problem, curve 4 - coupled thermoelastic problem) reaches 20%. For the stress function F a diﬀerence amount reached 4% in the centre, whereas for second derivatives - 6% (Fig. 6.12). The curves 1, 2 correspond to the F function, whereas the curves 3, 4 correspond to the second derivatives. The solid curves correspond to the uncoupled problem, whereas the dashed curves correspond to the coupled problem. The similar description has been also used for next figures. For the discussed case, a diﬀerence in stress σ xx occurring in the point (0.5; 0.5; -0.5) reached 11% (Fig. 6.14). Therefore, a coupling eﬀect between a deformation and the temperature when the before-critical load q = 80 is applied led to decrease of deflection by 7% and to the decrease of stress function (by 6%) and the stress σ xx (for instance, in the point (0.5; 0.5; -0.5) it reached 11%).

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

651

sxx 6

4

2

0

0.2

0.4

t

-2

-4

-6 Figure 6.9. The changes of σ xx corresponding to deflections presented in Figure 6.6.

The stress, deflection and its second derivative functions presented in Figures 6.11, 6.13 possess more evident diﬀerence. It is caused by a vibration period decrease while applying before critical load (Fig. 6.4). Let us compare a stress-strain state of a shell when the load q = 90 is applied in time instant τ = 0.26. In this time instant the shell deflections (coupled problem) reach their maximal value (Fig. 6.5), whereas the deflections corresponded to uncoupled problem prolong to increase. A diﬀerence in deflections (Fig. 6.15, curves 1, 2) reached 15% and 25% in relation to their second derivatives (Fig. 6.15, curves 3, 4). For the stress function the values corresponding to a coupled problem are greater (Fig. 6.16, curve 2) than for uncoupled problem (curve 1) by 7%, and for second

652

6 Coupled Problems of Thin Shallow Shells in a Temperature Field Wxx

W

t = 0.22

3 4 12 30 1 2 4 10

y

0

0.125

0.25

0

0.375

0.375

0.5

0.25

0.125

0

x=y

Figure 6.10. Deflections w and its second derivative w xx for q = 80 and τ = 0.22 (solid curves - uncoupled problem, dashed curves - coupled problem). -Wxx

W

1 3

2

2 0.08 1 0.04

4 y

0

0.125

0.25

0.375

0 0.375

0.25

x=y

-1 -0.04 -2 -0.08

Figure 6.11. Deflections w and its second derivative w xx for q = 80 and τ = 0.67 (solid curves - uncoupled problem, dashed curves - coupled problem).

derivatives (Fig. 6.16, curves 3, 4) by 20%. Stresses occurring in a shell within a coupled problem, are smaller for τ = 0.26 in comparison to uncoupled problem by 14% for σ xx (0.5; 0.5; -0.5) and by 30% for σ x (0.5; 0.5; 0.5) - Fig. 6.19. The stressstrain state of a shell for larger time (Fig. 6.18–6.20) exhibits a qualitative diﬀerence for τ = 0.5. Analysis of stress-strain state of a shell subjected to post-critical load q = 100 leads to the conclusion that deflection values and their derivatives in respect to space co-ordinates (in a case of temperature and deformation coupling) are lower by 6%

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields F

-Fxx

653

q=80

3 8 0.8 4 1 2 4 0.4

2 0.2 y 0

0.125

0.25

0

0.375

0.375

0.25

0.125

x=y 0

t = 0.22

-2 -0.2

Figure 6.12. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.10 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem). -Fxx

F

q=80

3 8 0.8 4 1

6 0.6

2 4 0.4

2 0.2 y 0

0 0.125

0.25

0.375

0.375 -2 -0.2

0.25

0.125

x=y

t = 0.67

Figure 6.13. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.11 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem).

and 4% for τ = 0.26 and by 3% and 5% for τ = 0.44 (Fig. 6.21, 6.22 - curves 1, 2 (deflection), and curves 3, 4 - second derivative of deflection).

654

6 Coupled Problems of Thin Shallow Shells in a Temperature Field sxx

q=80

20

16

12

8 z=0.5 4 y 0

0.125

0.25

0.375

0

0.375

0.25

0.125 x=y

-4 -8

z=0 z=-0.5

-12

-16

t=0.22

Figure 6.14. Stress σ xx versus position x = y (notations as in Figure 6.12).

The stress function values and its second derivatives (Fig. 6.23) in a case of coupled thermoelastic problem are higher (in modul) than corresponding values obtained using uncoupled problem by 13% and 30% in the shell centre for τ = 0.26 (curves 1, 2 - stress functions, curves 3, 4 - its second derivative). In the next considered time instant τ = 0.44 (Fig. 6.24) the values of the stress function obtained using coupled and uncoupled thermoelastic theories do not overlap because of a sign (curves 1, 2). A diﬀerence in second derivatives of F reaches 30% in the shell centre (Fig. 6.24, curves 3, 4) and 5% in the point with co-ordinates (0.19, 0.19). Note, that before a stability loss (Fig. 6.25, τ = 0.26) a maximal modulus value of stress σ xx in the shell centre for z = −0.5 is greater by 3.5% in a case of uncoupled problem. For the time instant τ = 0.46 (Fig. 6.26), when a shell is in a post-critical state and its deflections are maximal, the stresses obtained using a coupled theory are lower by

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

655

-Wxx W q=90

3 40 4

4 1

30 3

2 20 2

10 1

t=0.26

y 0

0.125

0.25

0.375

0.5

0.375

0.25

0.125

0

x=y

Figure 6.15. Deflections w and its second derivative w xx for q = 90 and τ = 0.26 (solid curves - uncoupled problem, dashed curves - coupled problem).

Fxx F 2

q=90

12 0.6

1 8 0.4 4 0.2 y 0

0.125

0.25 0.375

0

0.375

0.25

0.125

0

x=y

-4 3

-8

t=0.26

4

Figure 6.16. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.10 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem) for q = 90, τ = 0.26.

6% in the shell centre for z = 0.5 those obtained by neglecting a coupling between temperature and deformation. Therefore, an account of coupling between temperature and deflections leads to qualitative and quantitative diﬀerences in a stress-strain state in geometrically

656

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

-Wxx W

q=90

1 60 3 3

40 2

20 1 y 0

0.125

2

0

4

0.375

0.375 2

0.25

0.125

0

x=y

t=0.50

-20 -1

Figure 6.17. Deflections w and its second derivative w xx for q = 90 and τ = 0.50 (solid curves - uncoupled problem, dashed curves - coupled problem). Fxx F

q=90

1 0.6 8 0.4 4 0.2 4

y 0

0.125

0.25

0 0.375

3

0

x=y

-4 -0.2 3

t=0.50

Figure 6.18. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.10 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem) for q = 90, τ = 0.50.

non-linear problems of shell theory. In linear problems the coupling eﬀect plays less important role. A key diﬀerence between coupled and uncoupled thermoelastic theories is exhibited by an occurrence of a thermal field caused by deformations. In Fig. 6.27 and Fig. 6.28 a temperature distribution of surfaces z = −0.5 (above), z = 0 and z = 0.5 (below) for time instants τ = 0.22 and τ = 0.38 for the applied load q = 80 is shown.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields s

q=90

40

xx

657

z=0.5 20

y 0

0.25

0.375

0.375

0

0

0.125

x=y

z=0 -20

z=-0.5

t=0.26

Figure 6.19. Stress σ xx versus position x = y (notations as in Figure 6.12) for q = 90, τ = 0.26. sxx

40

q=90

z=0.5 20

y 0

z=-0.5 0.125

x=y

0

0

z=0.5

z=0 -20

-40

t=0.50

z=-0.5

Figure 6.20. Stress σ xx versus position x = y (notations as in Figure 6.12) for q = 90, τ = 0.50.

In the left upper side of figures a low of temperature distribution along thickness in the shell centre is shown. It should be emphasised that temperature changes versus shell thickness are strictly non-linear. The shell temperature strongly changes with time, maximal values of shell temperature on the surfaces z = −0.5 and z = 0.5 are shifted and also a sign change of θ occurs (Fig. 6.27, 6.28). On the surface z = 0.5 for τ = 0.22 the maximal temperature occurs in the middle of a shell side, and for τ = 0.38 - in the middle of the axis linking the shell centre with a side middle. A shell temperature distribution when the post-critical load q = 100 is applied is shown in Fig. 6.29 (τ = 0.22) and Fig. 6.30 (τ = 0.46). Contrary to the before critical load q = 80 (Fig. 6.27) in this case a maximal temperature on the surface z = 0.5 occurs in the shell centre in time τ = 0.22 (Fig. 6.29). For the post-critical load q = 100 the temperature in the shell centre changes sign along thickness, which does not occur for the pre-critical load (Figs. 6.27, 6.29).

658

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

-Wxx W 80 8

q=100

3 4 1 40 4

2

20 2 y 0

0.125

0.25

0.375

x=y

0

0.375

0.25

0 t=0.26

-20 -2

Figure 6.21. Deflections w and its second derivative w xx for q = 100 and τ = 0.26 (solid curves - uncoupled problem, dashed curves - coupled problem).

-Wxx W 80 8 2

3

q=100

1

4 40 4

20 2 y 0

x=y

0 0.125

0.25

0.375

0.375 -20 -2

0.25

0.125

0

t=0.44

Figure 6.22. Deflections w and its second derivative w xx for q = 100 and τ = 0.44 (solid curves - uncoupled problem, dashed curves - coupled problem).

2

-Fxx F

q=100

4 0.4

1

2 0.2 y 0

0 0.125

0.25

0.375

0.375

0.25

0.125

0

x=y

-2 -0.2 3

4 -4 -0.4 t=0.26

Figure 6.23. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.10 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem) for q = 100, τ = 0.26. -Fxx F

8 0.2 2 4 0.1

y 0

0.125

0.25

0.375

4 3

0 -4

0.375

0.25

0.125

0

x=y

1 -0.1

-8 -0.2

t=0.44

Figure 6.24. Stress function (curves 1, 2) and its second derivative (curves 3, 4) corresponding to Figure 6.10 (solid curves correspond to uncoupled problem, whereas dashed curves correspond to coupled problem) for q = 100, τ = 0.44.

660

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

sxx 50

q=100

z=0.5

40 30 20 10 y 0

0.125

0.25

0.375

0

0.375

0.25

0.125

0

x=y

z=0

-10

-20 -30

z=-0.5

-40 t=0.26

Figure 6.25. Stress σ xx versus position x = y (notations as in Figure 6.12) for q = 100, τ = 0.26.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

661

sxx q=100 z=0.5

40 30

20 10 y 0

z=0

0.125

0.25

0.375

0

0.375

0.25

x=y 0

-10 -20

-30 z=-0.5

-50

t=0.44

Figure 6.26. Stress σ xx versus position x = y (notations as in Figure 6.12) for q = 100, τ = 0.44.

662

6 Coupled Problems of Thin Shallow Shells in a Temperature Field 0.5

z

0.25 0

0.2 0.4

q(0.5;0.5;z)

0.6

y

-0.25

0.5

-0.5

q(x ;y;0.5) 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

x

0.5

y

q(x ;y;0) 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

x

0.5

y

q(x ;y;-0.5) 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

0.5

x

Figure 6.27. Temperature distribution on the surfaces z = −0.5, z = 0.5 for q = 80 and τ = 0.22.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields z

663

0.5 0.25

-0.05 0

q(0.5;0.5;z)

0.05

y

-0.25

0.5

-0.5

q(x ;y;-0.5) 0.2 0.1 0

x

-0.1 -0.2

y 0.5 q(x ;y;0) 0.2 0.1

x

0

0.5

-0.1 -0.2

0.5 q(x ;y;0.5) 0.2 0.1 0 -0.1

0.5

x

Figure 6.28. Temperature distribution on the surfaces z = −0.5, z = 0.5 for q = 80 and τ = 0.38.

664

6 Coupled Problems of Thin Shallow Shells in a Temperature Field z 0.5

-1.0

-0.5

0

1.0

DT(0.5;0.5;z)

1.5

y 0.5

-0.5

DT(x ;y;0.5) 2.0 1.0 0 1.0

x 0.25

0.5

y 0.5

DT(x ;y;0) 2.0 1.0 0 1.0

x 0.25

0.5

y 0.5

DT(x ;y;0.5) 2.0 1.0 0 1.0

x 0.25

0.5

Figure 6.29. Temperature distribution on the surfaces z = −0.5, z = 0.5 for q = 100 and τ = 0.22.

6.7 Stability Rectangular Shells with Coupled Deformation and temperature fields

665

0.5 z 0.15 -0.5

q(0.5;0.5;z)

0.25 0.5

0

y 0.5

-0.5

q(x ;y;-0.5) 1.0

0.25

0.5 0

x 0.5

-0.5

y 0.5

q(x ;y;0) 1.0

0.25

0.5 0

x

-0.5

y 0.5

q(x ;y;-0.5) 1.0 0.5 0

x

-0.5

Figure 6.30. Temperature distribution on the surfaces z = −0.5, z = 0.5 for q = 100 and τ = 0.46.

666

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

6.8 Additional Method Let us consider one of the methods of reducing of 3D thermoelastic problem to the 2D one. In a case of linear thermoelastic problems an application of an operator method for thin-walled structures has been considered by Podstrigatch [555], within a heat transfer theory. The first invariant of deformation tensor has the form e = e11 + e22 + e33 .

(6.35)

The component e33 is found from a condition of in-plane stress state (σ33 = 0): ν e33 = − (σ11 + σ22 ) + αT θ. E

(6.36)

We are going to obtain the equations of coupled thermoelastic problem without taking into account the physical non-linearities. Substituting the values of σ11 and σ22 into (6.36) for physically linear body we get e33 = −

ν 1+ν (e11 + e22 ) + αT θ. 1−ν 1−ν

(6.37)

Substituting (6.36) to (6.35) the following first invariant of deformation tensor is obtained 1+ν 1 − 2ν (e11 + e22 ) + αT θ = e= 1−ν 1−ν 1+ν 1 − 2ν 1 − ν (T 11 + T 22 ) + 2αT NT + z(æ11 + æ22 ) + αT θ. (6.38) = 1 − ν Eh 1−ν Substituting (6.38) into the heat transfer equation one gets KT

∂ 2θ ∂ 2θ ∂ 2θ EαT T 0 ∂ ∂θ − KT ( 2 + (e11 + e22 + e33 ) + W0 , + 2)= 2 ∂t 1 − 2v ∂ t ∂x ∂y ∂z

(6.39)

and taking into account the following notations α1 = γ=

KT , C

γ0 =

αT ET 0 , C(1 − 2ν)

γ0 (1 − 2ν) , 1+ν (1 − ν)(1 + γ0 αT 1−ν )

M = æ11 + æ22 ,

N=

β=

D2 = ∇2 −

α1 , 1 + αT γ0 1+ν 1−ν 1∂ , β ∂t

1−ν (T 11 + T 22 ) + 2αT NT , Eh

D21 =

NT =

1 h

1∂ , β ∂t h/2 θdz.

(6.40)

−h/2

The following operator form of the heat transfer equation is obtained ∂ 2θ + D2 θ − γD21 (N + zM) = 0. ∂ z2

(6.41)

6.8 Additional Method

667

Using the symbolic method, the solution to equation (6.41) has the form ⎧ ⎫ ⎧ ⎫ ⎪ D21 ⎪ D21 ⎪ D21 sin x3 D ⎪ ⎨ ∂θ ⎬ ⎨ ⎬ −γ M + cos zD θ| − γ N + γ (N + zM). (6.42) θ= ⎪ ⎪ ⎪ ⎪ z=0 z=0 ⎩ D ⎩ ∂ z D2 ⎭ D2 ⎭ D2 After integration because of z one obtains 1 NT = h

h/2 θdz = −h/2

⎧ ⎫ 2 sin h2 D ⎪ D21 ⎪ ⎨ ⎬ 1 D1 θ| − γ N + N ⎪ ⎪ z=0 ⎩ ⎭ 2 h γ D2 D 2D

12 MT = 3 h

(6.43)

h/2 θzdz = −h/2

⎧ ⎧ ⎫ ⎫ h 2 D21 ⎪ D21 ⎪ h ⎪ 12 sin 2 D ⎪ ⎨ ∂θ ⎨ ∂θ ⎬ 12 ⎬ D1 D −γ M cos −γ M M. (6.44) − −γ ⎪ ⎪ ⎪ ⎪ 2 ⎩ ∂z z=0 h2 D2 h2 D ⎩ ∂z z=0 D2 ⎭ h2 D2 d2 ⎭ D2 Eliminating θ|z=0 and ∂θ ∂z z=0 from (6.44) and (6.43) and substituting to (42) the temperature is expressed by the integral characteristics: θ (x, y, z, t) =

h 2D sin h2 D

cos zDNT +

sin zD h3 D2 MT − h 24 sin 2 D − h2 D cos h2 D

⎫ ⎧ h3 3 h ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 8 D ctg 2 D − 3⎪ M. ⎪ ⎪ ⎪ h h ⎭ ⎩ 1 − ctg D 2 2 Satisfying the boundary conditions on the surfaces z = ± h2 : ∂θ ∂z z=−0.5 and 0, the following equations are obtained D2 γ 12 D

⎫ ⎧h ⎪ ⎪ ⎪ ⎪ γ D21 ⎬ ⎨ 2 D cos zD − 1 N − ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ sin h D 3 D2 2

(6.45)

∂θ ∂z z=+0.5

=

ctg h2 D h3 D3 h h MT + γ D21 N− − D2 NT + 2 24 1 − h2 Dctg h2 D 2 γ D21 3 D2

⎫ ⎧ h3 3 h ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 8 D ctg 2 D − 3 M = 0, ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 1 − h ctg h D 2

ctg h2 D h3 D3 h h 2 D NT + MT − γ D21 N− 2 24 1 − h2 Dctg h2 D 2 ⎫ ⎧ h3 3 h ⎪ ⎪ ⎪ γ D21 ⎪ ⎬ ⎨ 8 D ctg 2 D − 3 M = 0. ⎪ ⎪ ⎪ ⎪ 2 ⎭ 3 D ⎩ 1 − h ctg h D 2

(6.46)

2

(6.47)

2

They lead to the equation governing the thermal stresses NT D2 NT − γ D21 N = 0

(6.48)

668

6 Coupled Problems of Thin Shallow Shells in a Temperature Field

and to the equation governing the thermal moments MT h2 D2 MT − 12MT − γ h2 D21 N = 0.

(6.49)

The equation (6.49) has been obtained taking into account two first terms of the series of ctg h2 D, which corresponds to a cubic distribution of temperature along the thickness (6.44). According to the earlier applied dimensionless parameters, equations (6.48) and (6.49) read 2 α2T ET 0 ∂ −1 ∂ 2 F ∂ 2 NT ∂ 2F −1 ∂ NT λ − +λ +λ 2 = λ1 λ2 λ C ∂t ∂ x2 ∂ y2 ∂ x2 ∂y ⎞ ⎛ 1 + ν ⎜⎜⎜⎜ 2γ0 αT ⎟⎟⎟⎟ ∂ NT 1 + γ0 αT , (6.50) ⎜1 + ⎟ 1+ν ⎠ ∂ t 1−ν ⎝ 1 + γ0 αT 1−ν α2T ET 0 ∂ −1 ∂ 2 w ∂ 2 MT ∂ 2 MT ∂ 2w λ λ1 λ2 λ−1 − 12M = + λ + + λ T C(1 − ν) ∂ t ∂ x2 ∂ y2 ∂ x2 ∂ y2 1 + ν ∂ MT ) . (6.51) 1 − ν ∂t In addition, the motion equation and the continuity equation should be attached to equations (6.50) and (6.51). (1 + γ0 αT

1 ∇2 ∇2 w − ∇2k F − L(w, F)+ 12(1 − ν2 ) 2 a2 b2 ρ α21 ∂ 2 w 1 ∂ 2 MT −1 ∂ MT + + λ ) = q, (λ Eh3 ∂ t2 12(1 − v2 ) ∂ x2 ∂ y2

∇2 ∇2 F = −λ−1

∂ 2 NT 1 ∂ 2 NT −λ − ∇2k w − L(w, w). 2 2 ∂x ∂ y2

(6.52)

As a result, we got the system (6.49)–(6.50) consisting of four partial diﬀerential equations which govern a shell motion taking into account a coupling between temperature and deformation fields. Contrary to the proposed earlier integral-diﬀerential equations obtained in section 3, the proposed system of equations is already a differential one and possesses one dimension. The system is an approximating one because in the series of ctg h2 D two first terms are taken which practically realise a cubic type rule of temperature distribution along the shell thickness. However, the obtained system of equations is of a hybrid (parabolic and hyperbolic) type. The boundary and initial conditions should be attached to the system (6.50)– (6.52). In order to solve the mentioned system of equations as well as in order to integrate the integro-diﬀerential system due to the space co-ordinates the method of finite diﬀerences with approximation 0(h2 ) is used. It leads to the system of ordinary diﬀerential equations in relation to wi j , NT i j , MT i j and to the system of algebraic equations in relation to Fi j . Again the system of algebraic equations is solved using the Gauss method on each of the time step, whereas the system of diﬀerential

6.8 Additional Method

669

equations is solved using the Runge-Kutta method with automatic choice of the integration step. In addition, contrary to the earlier method, where the values of thermal stresses occurring in the 3D heat transfer equation have been taken from a previous step in time, this approach does not possess the mentioned restriction. In order to estimate an influence of the given assumption the stress-strain state of a shell will be considered (the shell is made from alloy AMC and the boundary conditions (6.22) are applied). We assume that on the surfaces x = 0, y = 0, x = 1, y = 1 the 1-st order boundary condition for a temperature is given (θ = 0, it means that NT = MT = 0). The shell in the initial time t = 0 is subjected to an infinite in time uniformly distributed on a surface impulse with the intensivity q = 81 (k1 = k2 = 24). The calculation results are given in Table 6.2. Table 6.2. Stress-strains state for diﬀerent time instants. Time Method Function w(0.5; 0.5) w(0.25; 0.25) F(0.5; 0.5) F(0.25; 0.25) w xy (0.5; 0.5) w xy (0.25; 0.25) F xy (0.5; 0.5) F xy (0.25; 0.25) NT (0.5; 0.5) MT (0.25; 0.25)

t = 0.1 1 2 0.99 0.68 0.38 0.15 -2.34 -8.94 -4.63 -1.83 0.124 -0.101

0.99 0.68 0.38 0.15 -2.34 -8.94 -4.63 -1.83 0.126 -0.101

t = 0.2 1 2 1.78 1.07 0.63 0.23 -12.86 -10.52 -8.46 -2.38 0.228 0.298

1.78 1.07 0.63 0.23 -12.86 -10.52 -8.46 -2.38 0.300 0.310

t = 0.3 1 2 0.85 0.68 0.33 0.14 0.90 -9.78 -3.44 -1.93 0.092 0.353

0.85 0.68 0.33 0.14 0.94 -9.77 -3.44 -1.93 0.093 0.344

t = 0.4 1 2 0.02 0.03 0.00 0.004 -0.75 -0.36 -0.03 -0.12 0.001 0.080

0.02 0.03 0.00 0.004 -0.74 -0.37 -0.02 -0.12 0.001 0.084

The digit 1 corresponds to solution of the system (6.50)–(6.52), the digit 2 corresponds to solution of the integral-diﬀerential equations system. The results prove a very high accuracy of the presented approaches to solve the coupled thermoelastic problems of whipping shells subjected to transversal impulse type loads (in a similar way other types of transversal loads, and boundary and initial conditions can be considered).

7 Novel Solution Method for a System of Linear Algebraic Equations

In this chapter a novel optimal and exact method of solving large systems of linear algebraic equations is proposed. In the approach under consideration the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. The proposed method makes it possible to benefit from the essential advantages of both the direct method (universality, finitness of a computational process, exactness) and the iterational one (minimal amount of operational storage). Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in R3 is considered, where boundary value problems of the 1st, 2nd or 3rd order, or their combinations are taken into account. The governing diﬀerential equations are reduced to algebraic ones with the help of the finite element and the boundary element methods for diﬀerent meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a non-homogeneous shallow physically and geometrically non-linear shell subject to transversal uniformly distributed load. The partial diﬀerential equations are reduced to a system of non-linear algebraic equations with the error of O(h2x1 + h2x2 ). The linearization process is realized through either Newton method or diﬀerentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

7.1 Introduction It is obvious that a vast number of problems in physics, mechanics and technology is modelled through linear and non-linear partial diﬀerential equations (PDEs, equations of mathematical physics). In the next step, PDEs are usually reduced to linear or non non-linear set of algebraic equations applying either finite element or finite diﬀerence methods. In what follows only the method enabling solution of a large number of algebraic equations is considered. There are two classical approaches to solve the mentioned problem, i.e. direct and iterational methods. Perhaps the most popular are the Gauss reduction for a general system, and the relaxation or matrix relaxation methods usually applied for triple diagonal or block-diagonal matrices. The mentioned methods belong to the direct ones. Their eﬃciency depends on the system equation order and on the matrix structure.

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7 Novel Solution Method for a System of Linear Algebraic Equations

From the point of view of iterational methods, a system of equations is treated as the first order operator equations of the form Ax = b, where: A = (aik ) is a squared matrix of dimension n × n, x − (x1 , x2 , ..., xn ) stands for the sought vector, b = (b1 , b2 , ..., b4 ) is a given vector (right hand side). In the case of the mentioned iterational methods, the system of algebraic equations can be treated as the first order operator equations with the operator acting in an n- dimensional space Hn (A : Hn → Hn ), x, b ∈ Hn . It is worth noticing that the application of the general theory makes it possible to prove convergence of iterations for Seidel’s and upper relaxation methods with the minimal constraints used by operator A. Usually, two types of approach are applied: (i) for known boundaries γ1 > 0, and γ1 ≥ γ2 for a spectrum of operator A lying in a certain energy space H p ; (ii) for the case when boundaries γ1 and γ2 are unknown. However, more eﬀective in application seems to be the triangle variational method. For all numerical methods, the reduction of the infinite dimensional problem to that of the finite dimension plays a key role. It is expressed by the fact that a computational algorithm should yield a solution of the initial problem with a given accuracy ε > 0 through a finite number Q(k) of actions. Moreover, the algorithm should be practically realized, i.e. should yield a solution to the problem within the required computer time. Furthermore, the number of actions (and hence the time of solution) Q(ε) should be minimal for the considered problem. Algorithms with the mentioned properties are called economical. It is obvious that the choice of a numerical method of solving a system of linear algebraic equations depends on many circumstances, i.e. on the matrix A properties, on the computational type applied, etc. A computational type is one of the possible formulations of the problem, like finding a solution to one special problem of A x = b, or finding solutions to a few variants of problem A x = b with the same matrix A and diﬀerent right hand sides B. It may happen that a non-optimal choice to the problem with one matrix A x = b can be suitable for multi-variant computations. Note that for multi-variant computation one may decrease the average number of operations for one variant if some quantities are conserved and not computed once more for each variant. It depends on a computer and its operating storage. The choice of a computational algorithm should depend on the computational type, on the volume of the operating storage, and on the considered system order. In this work, the presented method of solution of large algebraic equation systems oﬀers reduction of the infinite dimensional problem of mathematical physics to the finite dimensional one through the finite diﬀerence approximation. The used approach yields a band matrix possessing only a few diagonals with non-zero elements. Direct methods of solving those systems matched with the transformation of input matrix A, do not enable arrival at similar results, i.e. the possibility of solution of the system of high order algebraic equations oﬀered by them is rather doubtful.

7.2 Elimination method for equations

673

7.2 Elimination method for equations Consider a direct method [597], further referred to as the elimination method for equations, making it possible to find the system solution as a point of intersection of hyperplanes and requiring a minimal operating storage volume. The latter property is rarely considered by the standard direct methods [210]. The idea of the proposed method will be illustrated and clarified when applied to a system of linear algebraic equations (SLAE). Consider first SLAE I of the general form n

aik xk = bi ,

(i = 1, 2, ..., n) .

(7.1)

k=1

The process of the proposed elimination consists of the following few steps. Step 1. Assume j = n. Solve the j-th equation of system (7.1) j times with respect to xn , by keeping variables x1 , x2 , ...xn−i fixed. As a result, the set M j consisting of points {X k }, k = k = 1, 2, ..., j with the coordinates X 1 (c, c, ..., c, xn1 ), X 2 (p, c, ..., c, xn2 ), X 3 (c, p, ..., c, xn3 ), .

.

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X (c, c, ..., c, p, j

xnj ),

is obtained, where c, p are arbitrary numbers (it is suitable to take c = 1, p = −1). Step 2. Compute i = j − 1 and construct set M, of points that are intersections of straight lines going through points {X j , X k }, k = 1, 2, ..., i, and the hyperplane defined by the i-th system of algebraic equations of the form ⎧ n ⎪ ⎪ ⎪ ⎪ ail xl = bi , ⎪ ⎨ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ⎩ x = x j + (xk − x j )λ , l l l l k

(7.2) l = 1, 2, ..., n ,

is solved. System (7.2) yields

n

λk = bi − l=1

ail xlj

* n l=1

ail (xlk − xlj )

and the coordinates of the k-th point of set Mi are computed in the following way x˜lk = xlj + (xlk − xlj )λk ,

l = 1, 2, ..., n.

674

7 Novel Solution Method for a System of Linear Algebraic Equations Table 7.1. Coordinates of points in set Mi (for i = n − 1).

x1 x2 ··· xn−3 xn−2 xn−1 xn

X1 c c ··· c c 1 xn−1 xn1

X2 x12 c ··· c c 2 xn−1 xn2

X3 c x23 ··· c c 3 xn−3 xn3

··· ··· ··· ··· ··· ··· ··· ···

X i−1 c c ··· i−1 xn−3 c i−1 xn−1 xni−1

Xi c c ··· c i xn−2 i xn−1 xni

Set Mi of points {X k }, k = 1, 2, ..., i, belongs to the hyperplane defined through the i-th equation of system (7.1). The coordinates of the points of the mentioned set satisfy the last n − i equations of system (7.1). In Table 7.1, coordinates of set Mi for i = n − 1 are reported. Step 3. Assume j = i and go to step 2. Repeating computations n − 1 times, one gets a point in an n-dimensional space, which is a solution to system (7.1). The above considerations allow the conclusion that the above algorithm is suitable for solving SLAE through computations of the matrix rows. Observe also that for a chosen parametrization of set Mn and a given method k , xik , applied for the construction of the straight lines, only the coordinates xk−1 k k xi+1 , ..., xn should be determined; other coordinates preserve values equal to c. It is worth noticing that in the proposed novel approach becomes four times smaller, which enables a considerable decrease in the volume of the computer operating storage used to keep coordinates of points X k computational time. Consider SLAE II with a band matrix. l

a j j+i ,x j+i = b j , j = 1, 2, ..., n, 1 ≤ j + i s ≤ n, i s ∈ I ,

(7.3)

s=1

where: l – number of non-zero diagonals, I – set consisting of l elements, i.e. ordered numbers of non-zero band diagonals with an account of sign with respect to the main diagonal. The following parameters are further introduced: l1 , m1 – numbers of non-zero diagonals and width of the band part lying under the main diagonal, respectively; l2 , m2 – numbers of non-zero diagonals and width of the band part lying over the main diagonal; m = m1 + m2 + 1 – band width. Observe that m1 = −I(1), m2 = I(l) . The algorithm for solving SLAE (7.3) runs as follows. Step 1. Assume j = n and solve the j-th equation of (7.3) m1 + 1 times with respect to Xn introducing variables xn−m1 , xn−m1 +1 , ..., xn−1 . As a result, set M j of points X k in the (m1 + 1)-th dimensional space of variables xn−m1 , xn−m1 +1 , ..., xn is obtained

7.2 Elimination method for equations

675

X j−m1 (c, c, ..., c, c, xnj−m1 ), X j−m1 +1 (p, c, ..., c, c, xnj−m1 +1 ), ................................................. X j−1 (c, c, ..., p, c, xnj−1 ), X j (c, c, ..., c, p, xnj ). Note that it is advisable to take c = 1, p = −1. Step 2. Compute i = j − 1. If i > m1 , then set M j is modified by introducing the variable x j−m1 −1 = c, and the points

X j−m1 −1 = X j−m1 , X j−m1 (p, c, c, ..., c, c, xnj−m1 ) .

˜ j of points A new set M {X k },

k = j − m1 − 1, j − m1 , ..., j

is obtained in the space of variables xn−m1 −1 , xn−m1 , ..., xn of the form X j−m1 −1 (c, c, c, ..., c, c, xnj−m1 −1 ), X j−m1 (p, c, c, ..., c, c, xnj−m1 ), .............................................. .............................................. X j−1 (c, c, c, ..., p, c, xnj−1 ), X j (c, c, c, ..., c, p, xnj ). Construct a set of points attained through intersection of the hyperplane defined by the ( j − m) -th equation of system (7.3) with the straight lines going through points {X j , X k }, k = j − m1 − 1, j − m1 , ..., j − 1. For this purpose i, or m1 + 1 if i > m1 , systems of equations ⎧ l ⎪ ⎪ ⎪ ⎪ ⎪ aii+is xi+is = bi , ⎪ ⎨ (7.4) ⎪ s=1 ⎪ ⎪ ⎪ ⎪ j j ⎪ ⎩ xi+is = x + (xk − x )λk , s = 1, 2, ..., l, 1 ≤ i + i s ≤ n i+i s

i+i s

i+i s

are solved. System (7.4) yields * n j n λk = bi − aii+is xi+is l=1

l=1

j k aii+is (xi+i − xi+i ). s s

Then coordinates of the k-th intersection point are computed

(7.5)

676

7 Novel Solution Method for a System of Linear Algebraic Equations

x˜rk = xrj + (xrk − xrj )λk , r = k, i, i + 1, ..., i + m2 , r ≤ n.

(7.6)

Step 3. Assume j = i and return to step 2. In Table 7.2, coordinates of the points of set M j for m1 < j n − m2 , are reported. Notice that for the repeated index combination the following notation is introduced: k = j − m1 , m = j + m2 . Remark. The already mentioned advantages with respect to the chosen parametrization of the input points and straight line construction methods hold also for this case. Table 7.2. Coordinates of points M j (for m1 < j n − m2 ).

xk−1 xk ··· x j−2 x j−1 xj x j+1 ··· xm−1 xm

X k−1 c c ··· c c k−1 xj xk−1 j+1 ··· k−1 xm−1 xmk−1

Xk p c ··· c c xkj xkj+1 ··· k xm−1 xmk

X k+1 c xkk+1 ··· c c k+1 xj xk+1 j+1 ··· k+1 xm−1 xmk+1

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···

X j−1 c c ··· j−1 x j−2 c x jj−1 j−1 x j+1 ··· j−1 xm−1 xmj−1

Xj c c ··· c j x j−1 x jj j x j+1 ··· j xm−1 xmj

Step 4. By repeating the computations through steps (2,3) n − 1 times, a point in the space with coordinates x1 , x2 , ..., xm2 is obtained. The space with obtained coordinates overlaps with the values of the unknown x1 , x2 , ..., xm2 occurring in the solution of system (7.3). Step 5. The order of the initial system is decreased first by neglecting m2 equations and by putting the found values of the m2 unknowns into the remaining equations. Notice that during this operation the parameters of the truncated band matrix do not change. Step 6. Parameters of the next group are chosen from the m2 unknowns when the computation through steps (1-3) of the truncated system is carried out. The described procedure of the successive decrease in the system order is repeated until all values of n unknowns are found. Generally speaking, one may require to get n and m2 as integers. Notice that by adding a zero diagonal to the compressed band matrix, parameter m2 may be introduced arbitrarily. Using the presented method (see [597]) it is possible to apply other variants of SLAE solution to the band matrix as well . The proposed novel algorithm is stable with respect to the errors introduced by roundings. In this sense, as practical computations show, this method is close to

7.2 Elimination method for equations

677

that of the Gauss elimination technique for unknowns with a partial choice of the main element. For instance, solving the system of equations considered in reference [597] (page 61), for the matrix conditioned by ω = 4.7 · 105 , the proposed method gives exact decimal digits, i.e. of order one less than the Gauss method. This result is obtained owing to a particular symmetry of the fundamental computational formulas (7.5), (7.6) and a homogeneity in computations of all unknowns. Owing to the algorithm one may conclude that in order to store coordinates of the points of set M j one may isolate independently on the order of the sought system, i.e. the set consisting of (m1 + 2)(m2 + 3) elements. From this point of view, the considered direct method is similar to iterational methods owing to the required volume of operational storage. Furthermore, the proposed method is characterized by a real computational process cycling and a weak coupling of the algorithm due to constant and full refreshments of transitional results. Indeed, giving the corresponding coordinates of the points of manifold M for a certain j < n, one may begin the process of solution of the system of equations beginning from the equation with the number i = j − 1. The mentioned property accounts for an essential increase of the solution process on a computer owing to the elimination of computation repetitions for truncated systems. Namely, while solving of an input system of order n we remember the coordinates of the points of the sets: M j , j = n + 1 − lm2 ,

l = 1, 2, ..., n/m2 − 2,

j ≥ m2 + 1Q .

In order to store the values of coordinates, it is advisable to use external computer memory. Next, after a successive truncated system of order n − km2 , k = 1, 2, ... has been formulated, the corresponding set M j , of points j = 2m2 + 1 + (k − 1)m2 , is created. The discussed algorithm for solving SLAE with a band matrix is also suitable for solving boundary value problems using the method of finite diﬀerences in a rectangular parallelepiped, since the matrix of SLAE possesses a band with a regular structure. The latter property allows the use of one and only one set I for all equations. For the space with a solution to a boundary value problem slightly diﬀering from the canonical one, an artificial way of introducing fictitious equations for the mesh nodes to SLAE is as follows: xi = 0, which supplements the given space to a canonical one. Let us dwell now for a while on the computation of elements of set I ordered by the numbers of non-zero diagonals of a band of the SLAE matrix with respect to its main diagonal. In what follows, a three dimensional case is considered. Let n1 , n2 , n3 be the numbers of mesh nodes in directions ox1 , ox2 , ox3 , respectively. For approximation of partial derivatives of second order in the node {i, j, k}, the 7-th point pattern is used: {i, j, k; i ± 1, j, k; i, j ± 1, k; i, j, k ± 1}, where 1 ≤ i ≤ n1 , 1 ≤ j ≤ n2 , 1 ≤ k ≤ n3 . Observe that the number of non-zero band

678

7 Novel Solution Method for a System of Linear Algebraic Equations

diagonals is equal to that of nodes in the pattern l = 7. The position of non-zero diagonals with respect to the main diagonal of the band matrix is defined using ordered numbers of the unknowns corresponding to the nodes appearing in the pattern. Numbering of unknowns is defined by the number of mash nodes during SLAE formulation. Although various variants are possible, only those are interesting which formulate a band matrix of a regular structure, i.e. one set I gives the position of non-zero coeﬃcients in all equations. The variant associated with the successive picking of nodes in directions ox1 , ox2 , ox3 , and corresponding to the formula for the computation of unknown number n in the node {i, j, k} reads si, j,k = i + ( j − 1)n1 + (k − 1)n1 n2 .

(7.7)

Formula (7.7) yields i1 = si, j,k−1 − si, j,k = −n1 n2 , i2 = si, j−1,k − si, j,k = −n1 , i3 = si−1, j,k − si, j,k = −1 , i3 = si−1, j,k − si, j,k = −1 ,

(7.8)

i4 = si, j,k − si, j,k = 0 , i5 = si+1, j,k − si, j,k = 1 , 6

= si, j+1,k − si, j,k = n1 ,

i7 = si, j,k+1 − si, j,k = n1 n2 . For the last band matrix parameters one gets m1 = n1 n2 , m2 = m1 , m = 2n1 (n2 + 1), n = n1 n2 n3 .

(7.9)

The formula used for the successive picking of nodes along directions ox3 , ox1 , ox2 , i.e. the formula applied to compute an unknown number in node {k, i, j} is: sk,i, j = j + (k − 1)n3 + ( j − 1)n1 n3 ,

(7.10)

where elements i s of set I read i1 = −n1 n3 , i2 = −n3 , i3 = −1, i4 = 0, i5 = 1, i6 = n3 , i7 = n2 n3 ,

(7.11)

and the band parameter matrix follows m1 = n1 n3 , m2 = m1 , m = 2n1 n3 + 1 .

(7.12)

Finally, the last variant of the successive node picking in directions ox2 , ox3 , ox1 corresponds to the formula for computation of the unknown number in node { j, k, i}, i.e.

7.2 Elimination method for equations

s j,k,i = j + (k − 1)n2 + (i − 1)n2 n3 ,

679

(7.13)

where elements i s of set I read i1 = −n2 n3 , i2 = −n2 , i3 = −1, i4 = 0, i5 = 1, i6 = n2 , i7 = n2 n3 ,

(7.14)

and the band element matrix has the form m1 = n2 n3 , m2 = m1 , m = 2n2 n3 + 1 .

(7.15)

It follows from the algorithm description that both the parameter m-band SLAE matrix width and the system order n determine the dimension of the operating storage and the machine time required to solve the equations on a computer. Therefore, if values n1 , n2 , n3 are diﬀerent, then the variant of mesh nodes preserving the minimal matrix band width should be chosen. It is clear that the last picking procedure should be associated with the maximal number of nodes in direction n x . In practice, when the SLAE matrix in the form of a compressed band is built (only non-zero elements are considered), only one picking variant of the shell nodes (independently of the values of n1 , n2 , n3 ) is taken. Therefore, minimization of the band width is introduced, which is reduced to the exchange of elements in rows, exchange of rows and exchange of column elements of free terms. In the considered example the correspondence between elements of the band row is given in Table 7.3 for three variants of picking of mesh nodes. Relations (7.7), (7.10), (7.13) are used for the exchange of rows, and of column elements of free terms, which define simultaneously the ordering numbers of the system equations for the various variants of picking of the shell nodes. Table 7.3. Exchange of mesh nodes. Pattern Row node element i, j, k − 1 1 i, j − 1, k 2 i − 1, j, k 3 i, j, k 4 i + 1, j, k 5 i, j + 1, k 6 i, j, k + 1 7

Variant of the nodes exchange i, j, k k, i, j j, k, i 1 3 2 2 1 3 3 2 1 4 4 4 5 6 7 6 7 5 7 5 6

If one considers the boundary value problem for a system of diﬀerential equations with respect to a few sought functions, then with the use of finite diﬀerence method the structure of the SLAE matrix becomes dependent on the choice of the numbering of unknowns.

680

7 Novel Solution Method for a System of Linear Algebraic Equations

In the case of successive numeration of unknowns into groups corresponding to sought functions, the matrix has a block structure. The band width overcreases the system order, which makes the considered algorithm non-eﬀective. Note that numeration of similar unknowns with a step equal to the number of sought functions corresponds to successive numeration of all (related to a mesh node) unknowns, and the SLAE matrix possesses a regular band-type structure. The band width is essentially smaller than the system order, and hence the proposed method can be applied. In practice, both numeration methods mentioned above are applied. The first one is used in the computation of non-zero elements of the equation matrix. The second one is applied during transformation of the matrix to a form suitable for the application of the direct solution method. It is worth noticing that using the described approach the general properties of the matrix transformation algorithm are applied independently of the number of the sought functions. The procedure consists of the following steps: Step 1. Elements of set I, i.e. ordering numbers of non-zero diagonals of the matrix with respect to the main diagonals are computed. For this purpose (analogously to (7.7), (7.8)) ordering numbers of the unknowns corresponding to the nodes of the diﬀerence pattern scheme are computed first with respect to each group of the same unknowns in accordance with the first numbering order. Step 2. Elements of set Ik are computed, i.e. the relative numbers of these unknowns for the cases when the unknowns lie on the main diagonal associated with the first, the second, ..., the k-th group of the unknowns named the same. Step 3. A non-order set I matching all sets Ik is constructed. Further, an increasing ordering of the elements of set I is carried out and the number of elements of set I with parameters l1 , l2 , m1 , m2 , m is computed. Step 4. The reconstruction of the initial block-band compressed matrix is initiated through changes of row elements, rows and free column elements with respect to the second numbering way of unknowns; first elements of all blocks, second elements of all blocks, and so on; first rows of blocks corresponding to the equations of the boundary value problem, second rows of these blocks, etc. Elements of the column of free terms are transformed in the way similar to that applied to the rows. Step 5. Then a band type matrix in the compressed form is built in the following way: ordering numbers of the row elements from the block of the input matrix are defined through the condition of equality of the corresponding elements of sets Ik , I. Notice that in its essential part the resulting band is filled with zero elements. Consider an example of the SLAE matrix transformation occurring during the solving of the boundary value problem for the system of two diﬀerential equations in partial derivatives of fourth order with variable coeﬃcients in two-dimensional space using the method of finite diﬀerences. Denote by n1 , n2 the numbers of mesh nodes in directions ox1 , ox2 , respectively. The derivatives in mesh node {i, j} are approximated through diﬀerence relations on the pattern consisting of 13 nodes (see Figure 7.1).

7.2 Elimination method for equations

681

Figure 7.1. The pattern of 13 nodes

The following coordinates of the local nodes {i s , j s }, s = 1, 2, ..., 13, are introduced i1 = i − 2, j1 = j; i2 = i − 1, j2 = j − 1; i3 = i − 1, j3 = j; i4 = i − 1, j4 = j + 1; i5 = i, j5 = j − 2; i6 = i, j6 = j − 1; i7 = i, j7 = j; i8 = i, j8 = j + 1; i9 = i, j9 = j + 2; i10 = i + 1, j10 = j − 1; i11 = i + 1, j11 = j; i12 = i + 1, j12 = j + 1; i13 = i + 2, j13 = j. The input matrix in the compressed form (non-zero diagonals) is built owing to the first numbering way for unknowns, and it describes a two-dimensional packet of dimension kn · kl , where k = 2 – number of sought functions; n = n1 n2 – number of mesh nodes for which equations are constructed; l = 13 – pattern dimension. The discussed matrix is a block diagonal one consisting of k2 blocks and possessing 39 non-zero diagonals. Local index s of the pattern node defines a position in the row of a coeﬃcient standing by the corresponding unknown in the equation for node {i, j}. The computation is carried out with respect to the beginning of the corresponding block. Then, elements of the following sets are computed: i1,k ∈ I1 , i2,k ∈ I2 , k = 1, 2, ..., 26 . For this purpose, the relative members of all unknowns corresponding to pattern nodes and appearing in equations for node {i, j} are computed: i1,k = lk − l13 , i2,k = lk − l14 ,

682

7 Novel Solution Method for a System of Linear Algebraic Equations

where: even k : lk = 2((i s − 1)n2 + j s ) ; odd k : lk−1 = 2((i s − 1)n2 + j s ) − 1, s = 1, 2, ..., 13. The set I = I1 ∪ I2 is constructed and its elements are ordered in an increasing manner. The corresponding results for n2 = 5 are reported in the first four columns of Table 7.4. Table 7.4. Comparison of obtained results (see text). N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

I1 -20 -19 -12 -11 -10 -9 -8 -7 -4 -3 -2 -1 0 1 2 3 4 5 8 9 10 11 12 13 20 21

I2 -21 -20 -13 -12 -1 -10 -9 -8 -5 -4 -3 -2 -1 0 1 2 3 4 7 8 9 10 11 12 19 20

I -21 -20 -19 -13 -12 -11 -10 -9 -8 -7 -5 -4 -3 -2 -1 0 1 2 3 4 5 7 8 9 10 11 12 13 19 20 21

k=1 2 3 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 23 24 25 26 27 28 30 31

k=2 1 2 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 23 24 25 26 27 29 30

As seen from this Table, the number of non-zero diagonals of the band matrix is equal to 31. This value does not depend on n2 , but is defined only through a type of

7.2 Elimination method for equations

683

the pattern chosen for approximation of derivatives and by a number of the functions sought in the boundary value problem. Comparing elements of sets I1 , I2 and I, one obtains a rule of distribution of the row elements in the modified matrix during construction of the band matrix in the compact form. The correspondence between the ordering numbers of non-zero coeﬃcients for the even and odd rows of the modified matrix, and rows of the band matrix are reported in the last two columns of Table 7.4. The free positions in the row of the matrix are filled out by zeros. Notice that for arbitrary n1 , n2 this correspondence is the same. As known, when solving numerically boundary value problems for PDEs using finite element methods, there appear SLAE associated with matrices including a relatively small number of non-zero elements most of which remain on the main diagonal, i.e. in this case the equations dealt with are associated with the band matrix of irregular structure (the so called cutting-oﬀ matrix). However, there exists a modification of elimination the method for equations due to which it is possible to solve SLAE of a similar form. The cutting-oﬀ matrix will be introduced through two sets: the set of non-zero matrix elements and the corresponding set of indices, i.e. relative numbers of non-zero elements with respect to the elements lying on the main diagonal, which are associated with the indices. In what follows, contrary to the band matrix of a regular structure requiring only one row of indices for non-zero elements (set I), in our case this type of row is formulated for each of the equations (set Ii ). Here lies the fundamental diﬀerence in the application of the elimination method for the equations elimination of SLAE using the cutting-oﬀ matrix. Let us comment on parameters m1 , m2 . Being the same for all equations, they are computed initially as m1 = max(−Ii (1)), m2 max(Ii (li )), where l1 denotes the number of non-zero coeﬃcients in the i-th equation. Therefore, the row of non-zero coeﬃcients and the row of indices for each equation are supplemented, if necessary, by zeros in the case of elements, and by Ii (1) = −m1 , Ii (li ) = m2 in the case of indices, where li is the value of li satisfying the above requirement. As a result, SLAE with a band matrix of non-regular structure is obtained. Note that the formulation of the cutting-oﬀ matrix with the help of two packets is optimal with respect to the memory storage volume since for the index conservation it is suﬃcient that the machine word be 2 bytes long. The corresponding computation shows, for instance, that for the conservation of a band-type matrix of the 27-th order and a regular structure obtained through approximation of the Laplace operator in a 3D space by finite diﬀerence equations, 5 832 bytes are required in a general case, 1 512 bytes are required in the case of its

684

7 Novel Solution Method for a System of Linear Algebraic Equations

compressed form (only non-zero elements are considered), and finally 1 350 bytes are needed when the matrix is represented by two packets. It is worth noticing that the method of elimination extends essentially possibility of computer computation needed to solve large order SLAE through reaching a match between the advantages of both direct (universality, finiteness of the computational process, exactness) and iterational (minimal requirement of the storage volume) methods. Let us finally emphasize particularities of the SLAE matrix in each separate case, which can also be treated as an advantage of the method.

7.3 Numerical solution of a three-dimensional equation of elliptic type Many stationary processes of diﬀerent physical properties lead to PDEs of elliptic type, to mention only the stationary problem of current distribution in a medium, problems of electrostatics and magnetostatics of the theory of plates and shells, problems of the theory of elasticity or the theory of filtration, etc. Exact solution to boundary value problems for elliptic equations can be found in rare cases only. Therefore, numerical methods are usually applied to solve diﬀerential equations of elliptic type (linear or nonlinear). In the latter case, equations are linearized through diﬀerentiation along the parameter [606] or using the “set-up” method [201]. Hence, the problem is finally reduced to the solution of the system of linear algebraic equations (SLAE). Moreover a majority of methods for non-linear problem solving are reduced to a series of linear systems. Application of the method of finite diﬀerences to elliptic equations results in SLAE with a band-type matrix. In the case of PDEs, the matrix (SLAE) possesses the cutting-oﬀ band, with only a few non-zero elements (see section 7.2 of this work). Let us briefly model the problem of stationary heat distribution in a certain volume G with surface Γ of three-dimensional space x = (x1 , x2 , x3 ). The heat transfer equation is governed by the Fourier principle. A vector of heat stream density W is proportional to the temperature gradient V = V(x), such that W = −K gradV,

(7.16)

where K = K(x) is the heat transfer coeﬃcient. The density of the heat stream is equal to the amount of the heat stream a unit of time passing through a unit area of an isotermic surface. To derive the equation governing the heat balance for a certain volume U ∈ G and surface S , let the distributed heat sources exist inside volume U with density ϕ(x), where ϕ(x)dU is the heat amount occuring in volume dU. Let Wn describe the vector W projection onto external normal n to surface S . The heat balance equation is governed by the known rule, i.e. the total heat stream passing through surface S

7.3 Numerical solution of a three-dimensional equation of elliptic type

685

Wn dS S

should be equal to the heat amount of ϕ(x)dU U

appearing in volume U, i.e.

Wn dS =

Using the Gauss formula

S

to the form

S

Wn dS =

ϕ(x)dU.

(7.17)

U

div, balance equation (7.17) is transformed

U

(divW − ϕ(x))dU = 0 .

(7.18)

U

As volume U is arbitrary, therefore if ϕ(x) and divW are continuous functions of point x = (x1 , x2 , x3 ), then (7.18) yields W = ϕ(x) .

(7.19)

Substituting here expression (7.16) for the vector of heat stream W, the following equation for the stationary temperature V = V(x) is obtained LV = div(KgradV) = −ϕ(x) ,

(7.20)

or in the equivalent form 3

(K(x)V(x), xα ), xα = −ϕ(x) ,

(7.21)

α=1

where K is the function of point x = (x1 , x2 , x3 ). Heat transfer equation (7.21) is obtained under an asumption of isotropy of the heat transfer process. If a heat coeﬃcient depends on direction and is a tensor (an isotropic medium), then, instead of (7.21) the following equation is taken 3

(Kαβ V(x), xβ ), xα = −ϕ(x) .

(7.22)

α1 β=1

If Kαβ ≡ O and α β, then equation (7.22) reads 3

(Kα V(x), xα ), xα = −ϕ(x) .

(7.23)

α=1

Equation (7.21) holds for all internal points of space G. Additional conditions of the following form are attached to boundary Γ.

686

7 Novel Solution Method for a System of Linear Algebraic Equations

1. Temperature is given: V(x) = g(x) for x ∈ Γ; 2. Heat stream is given: K ∂V ∂n = g(x) for x ∈ Γ; 3. Heat transfer Newton’s rule is applied: K

dV = æV + g(x), dn

x ∈ Γ,

where æ = æ(x) > 0 .

Consider the boundary value problem for the equation of the stationary heat transfer in a rectangular parallelepiped of a three-dimensional space G¯ = {0 ≤ xα ≤ lα , α = 1, 2, 3} . In this case, the method of finite diﬀerences is used; the construction of the diﬀerence equation on the mesh with a variable step is realized through the method of functional approximation [592]. Consider boundary value conditions 1, 2 and 3 in various states, and if necessary, symmetry q of solution is accounted: 3

¯ (Kα (X)V(x), xα ), xα = −ϕ(x), x ∈ G;

(7.24)

α=1

⎧ Kα (x)V(x), xα = æ−α (x)(V(x) − V0 (x)), xα = 0, ⎪ ⎪ ⎪ ⎪ ⎨ −K ⎪ α (x)V(x), xα = æ+α (x)(V(x) − V0 (x)), xα = lα ⎪ ⎪ ⎪ ⎩ ,

(7.25)

where: Kα (x) – heat transfer coeﬃcient in direction xα ; x±α (x) – heat exchange coeﬃcient with surrounding medium on walls xα = 0, lα ; Vα (x) – temperature of surrounding medium. It is known that a solution to boundary value problem (7.24), (7.25) satisfies the functional minimum: ⎞ ⎛⎜ ⎟⎟ ⎜⎜⎜ 3 2 ⎜⎜⎝ kα V,xα − 2Vϕ⎟⎟⎟⎟⎠dx+ I(u) = G lβ lγ 3

α=1 0

α=1

V 2 − 2g−α V xα =0 + (V 2 − 2g+α V) x

α =lα

dxβ dxγ , α β γ ,

(7.26)

0

where: g±α (x) =æ ±α (x)V0 (x) Indeed, variating functional (7.26) with respect to V, V, xα , integrating by parts the terms with δV, xα and comparing variation of the functional to zero, relations (7.17) and (7.25) are obtained as the necessary conditions to minimize functional (7.26). Further, instead of V and K, v and k will be used. Let us introduce the following fundamental mesh ω ¯ =

3 = α=1

ω ¯ α,

7.3 Numerical solution of a three-dimensional equation of elliptic type

687

ω ¯ α = {xαi , i = 0, 1, 2, ..., Nα − 1, Nα } – mesh on interval [0, lα ]. The set of interval mesh nodes is denoted by 3 = ω= ωα , α=1

where: ωα = {xαi , i = 1, 2, ..., Nα − 1}. Let us introduce the following notations hαi = xαi − xαi−1 , i = 1, 2, ...Nα , ⎧ 0, 5α1 , i = 0, ⎪ ⎪ ⎪ ⎪ ˜hαi = ⎨ 0, 5(hαi−1 + hαi , i = 1, 2, ..., Nα − 1, ⎪ ⎪ ⎪ ⎪ ⎩ 0, 5h αNα ,i=Nα ; vi, j,k = v(x1i, x2 j, x3k ), k1i±0,5, j,k = 0, 5(k1i, j,k + k1i±1, j,k ), k2i, j,±0,5,k = 0, 5(k2i, j,k + k2i, j,±1,k ), k3i, j,k±0,5 = 0, 5(k3i, j,k + k3i, j,k±1 ). All integrals occurring in functional (7.26) are substituted by quadratures. If the under integral function includes a partial derivative, then the centre triangle formula is applied with respect to the corresponding variable (in other cases the trapezoid rule is used). Derivatives are approximated through diﬀerence approximations, for instance by v, x1 (x1i − 0, 5h1i x2 j, x3k ) = (vi, j,k − vi−1, j,k )/h1i . In what follows the function with n variables vi, j,k is obtained In (v) =

N3 N1 N2

k1i−0,5, j,k (vi, j,k − vi−1, j,k )2 h˜ 2 j h˜ 3k /h1i +

i=1 j=0 k=0 N3 N2 N1

k2i, j−0,5,k (vi, j,k − vi, j−1,k )2 h˜ 1i h˜ 3k /h2 j +

i=0 j=1 k=0 N3 N2 N1

k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )2 h˜ 1i h2i /h3k +

i=0 j=0 k=1 N3 N2 N1

ϕi, j,k vi, j,k h˜ 1i h˜ 2 j h˜ 3k +

i=0 j=0 k=0 Nβ Nγ 3 α=1 j=0 k=0

(æ−α v(xα0 , xβ j , xγk )2 − 2g−α v(xα0 , xβ j , xγk )+

(7.27)

688

7 Novel Solution Method for a System of Linear Algebraic Equations

æ+α v(xαNα , xβ j , xγk )2 − 2g+α v(xαNα , xβ j , xγk )h˜ β j h˜ γk , α β γ, n=

3 =

(Nα + 1).

α=1

As known the minimum of this function is achieved at the point, where its partial derivatives are equal to zero with respect to vi, j,k . Let us construct a system of diﬀerence equations of order n to define values vi, j,k realizing the minimum of the functional (7.26) and being a solution to boundary value problem (7.24), (7.25). The following four cases are considered. ¯ 1. Internal mesh nodes of space G. Terms (7.27) including vi, j,k are reported below S 1 = (k1i−0,5, j,k (vi, j,k − vi−1, j,k )2 /h1i + k1i+0,5, j,k (vi+1, j,k − vi, j,k )2 /h1i+1 )h˜ 2 j h˜ 3k + (k2i, j−0,5,k (vi, j,k − vi, j−1,k )2 /h2 j + (k2i, j+0,5,k (vi, j+1,k − vi, j,k )2 /h2 j+1 )h˜ 1i h˜ 3k + (k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )2 /h3k ) k3i, j,k+0,5 (vi, j,k+1 − vi, j,k )2 /h3k+1 )h˜ 1i, h˜ 2 j − 2ϕi, j,k vi, j,k h˜ 1i h˜ 2 j h˜ 3k . By diﬀerentiating S 1 with respect to v( i, j, k) and comparing its derivative to zero, the following diﬀerence equation is obtained at node {i, j, k} of the mesh owing to the division by 2h1i h2 j h3k (k1i+0,5, j,k (vi+1, j,k − vi, j,k )/h1i+1 − k1i−0,5 j,k (vi, j,k − vi−1, j,k )/h1i )/h˜ 1i + (k2i, j+0,5,k (vi, j+1,k − vi, j,k )/h2 j+1 − k2i, j−0,5,k (vi, j,k − vi, j−1,k )/h2 j )/h˜ 2 j + (k3i, j,k+0.5 (vi, j,k+1 − vi, j,k )/h3k+1 − k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )/h3k )/h˜ 3k = −ϕi, j,k , 1 ≤ i ≤ N1 − 1, 1 ≤ j ≤ N2 − 1, 1 ≤ k ≤ N3 .

(7.28)

7.3 Numerical solution of a three-dimensional equation of elliptic type

689

2. Node of the parallelepiped wall: i = 0, 1 ≤ j ≤ N2 − 1, 1 ≤ k ≤ N3 − 1 . Terms (7.27) (including variable vi, j,k for i = 0) follow S 2 = k1i+0,5, j,k (vi+1, j,k − vi, j,k )2 h˜ 2 j h˜ 3k /h1i+1 + (k2i, j−0,5,k (vi, j,k − vi, j−1,k )2 /h2 j + k2i, j+0,5,k (vi, j+1,k − vi, j,k )2 /h2 j+1 )h˜ 1i h˜ 3k + (k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )2 /h3k + k3i, j,k+0,5 (vi, j,k+1 − vi, j,k )2 /h3k+1 )h˜ 1i h˜ 2 j − 2ϕi, j,k vi, j,k h˜ 1i h˜ 2 j h˜ 3k + (æ−1 v2i, j,k − 2g−1 vi, j,k )h˜ 2 j h˜ 3k . By diﬀerentiating S 2 with respect to vi, j,k , and comparing the corresponding derivative to zero, the following diﬀerence equation (after division by 2h˜ 1i h˜ 2 j h˜ 3k ) is obtained at the parallelepiped node x1 = 0: (k1i+0,5, j,k (vi+1, j,k − vi, j,k )/h1i+1 − æ−1 vi, j,k )/h˜ 1i + (k2i, j,+0,5,k (vi, j+1,k − vi, j,k )/h2 j+1 − k2i, j−0,5,k (vi, j,k − vi, j−1,k )/h2 j )/h˜ 2 j + (k3i, j,k+0,5 (vi, j,k+1 − vi, j,k )/h3k+1 − k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )/h3k )/h˜ 3k = −ϕi, j,k − g−1 /h˜ 1i .

(7.29)

Diﬀerence equations at other parallelepiped wall nodes are constructed analogously. 3. Mesh nodes belong to the parallelepiped i = j = 0, 1 ≤ k ≤ N3 − 1. By proceeding in a way similar to that applied in the previous case, the following diﬀerence equation is obtained: (k1i+0,5, j,k (vi+1, j,k − vi, j,k )/h1i+1 − æ−1 vi, j,k )h˜ 1i + (k2i, j+0,5k (vi, j+1,k − vi, j,k )/h2 j+1 − æ−2 vi, j,k )/h˜ 2 j + (k3i, j,k+0,5 (vi, j,k+1 − vi, j,k )/h3k+1 − k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )/h3k )/h˜ 3k = −ϕi, j,k − g−1 /h˜ 1i − g−2 /h˜ 2 j .

(7.30)

690

7 Novel Solution Method for a System of Linear Algebraic Equations

4. A node of the mash coincides with the parallelepiped vertex i = 0, N1 ; j = 0, N2 ; k = 0, N3 . Proceeding again in the previous case, one arrives at the following corresponding equations: for i = j = k = 0 (k1i+0.5, j,k (vi+1, j,k − vi, j,k )/h1i+1 − æ−1 vi, j,k )/h˜ 1i + (k2i, j+0.5,k (vi, j+1,k − vi, j,k )/h2 j+1 − æ−2 vi, j,k )/h˜ 2 j + (k3i, j,k+0.5 (vi, j,k+1 − vi, j,k )/h3k+1 − æ−3 vi, j,k )/h˜ 3k = −ϕi, j,k − g−1 /h˜ 1i − g−2 /h˜ 2 j − g−3 /h˜ 3k ;

(7.31)

for i = N1 , j = N2 , k = N3 (−æ−1 vi, j,k − k1i−0,5, j,k (vi, j,k − vi−1, j,k )/h1i )/h˜ 1i + (−æ−2 vi, j,k − k2i, j−0.5,k (vi, j,k − vi, j−1,k )/h2 j )/h˜ 2 j + (−æ−3 vi, j,k − k3i, j,k−0,5 (vi, j,k − vi, j,k−1 )/h3k )/h˜ 3k = −ϕi, j,k − g+1 /h˜ 1i − g+2 /h˜ 2 j − g+3 /h˜ 3k .

(7.32)

Thus, all possible cases occurring while formulating of the diﬀerence equations on the mesh with variable step for the 3rd boundary value problem have been discussed. Let us discuss briefly the peculiarities involved in the formulation of diﬀerence equations in the case when the following Dirichlet conditions are given on some parallelepiped walls, i.e. v(x)| xα =0,Iα = vu (x), where vu (x) is the given function. One of the possible ways to include a condition of such a type is to formulate the following corresponding equation associated with nodes vi, j,k = vu . The latter one allows for construction of both an algorithm and computer programs universal with respect to the boundary condition type. Indeed, the drawback of the approach is associated with conservation of the order of the diﬀerence equation system. Furthermore, if it is known a priori that the solution to problem (7.24), (7.25) possesses one, two or three plane symmetry xα = lα /2, then these conditions are recommended to be included while diﬀerence are being formulated, equations, since they essentially decrease the order of the investigated system. If the solution is symmetric with respect to, say, the plane x1 = 1 /2, then we take

7.3 Numerical solution of a three-dimensional equation of elliptic type

691

h˜ 1N1 = h1N1 , k1N1 +0,5, j,k = k1N1 −0,5, j,k . In what follows, in the diﬀerence equations formulated for the nodes of the plane xα = α /2 accounting the symmetry property v(1 /2 + h1N1 , x2 , x3 ) = v(1 /2 − h1N1 , x2 , x3 ), the coeﬃcient standing by the unknown vi−1, j,k is doubled. The wall node is treated as an internal one in this case. The solution to boundary value problem (7.24), (7.25) consists of two steps: formulation of the system of diﬀerence equations with a band-type compressed matrix having a regular structure (in the operating storage only non-linear matrix diagonals are conserved); solution to the obtained system using the elimination method for equations [210]. Let us discuss the computations of the first stage. In order to identify a particular problem the following items are required: node numbers Nα with respect to each direction xα , α = 1, 2, 3; packets of steps of the mesh h1i , h2 j , h3k ; formulas for kα (x), g±α (x), ϕ(x) computations, and vu (x) (if necessary); values of symmetry indicators IS , JS , KS ; packet of 6 elements for to formulate the type of the boundary condition on each of the 6 parallelepiped walls; values of χ±α (x). Having this information, the operator coeﬃcients are computed k1i, j,k , k1i−0,5, j,k , and the matrix of diﬀerence equations is formulated due to formulas (7.28)-(7.32). Analogous operations are carried out with k2 (x), k3 (x). In what follows, the matrix of coeﬃcients of the diﬀerence equations is obtained, which corresponds to the third boundary value problem. Then free terms of the equation system are computed, and (if necessary) the equations involving the Dirichlet conditions are formulated. Recall that when solving numerically the boundary value problems, the reality estimation of computer yielded results is in the focus of attention. Therefore the correct derivation of appropriate relations together with suitable formulation of the variational problem should be emphasized, and then an application of the exact method of solution to the system of diﬀerence equations with double precision should be carried out. During analysis, a series of computations to verify the mentioned reliability have been carried out. 1. Model problem The possibilities of both the developed algorithm and the system subroutines are checked through solution to problem (7.24), (7.25), with k1 = 1 + 2/3c( x˜1 + x˜2 + x˜3 ), k2 = 1 + 2/3c(3/4 − x˜1 − x˜2 − x˜3 ), k3 = 1 + 2/3c(1/2 − x˜1 − x˜2 + x˜3 );

where:

692

7 Novel Solution Method for a System of Linear Algebraic Equations

x˜α = (xα − 1/2)2 , α = 1, 2, 3 . The exact solution reads vu (x) = (1/2 − x˜1 − x˜2 )x32 /2, and hence ϕ(x) = −x32 (2 + 4/3c(3/8 + x˜1 − x˜2 )) + (1/2 − x˜1 − x˜2 ) (1 + 2/3c(1/2 − x˜1 − x˜2 + x˜3 + 2x3 (x3 − 1/2))), g−2 (x1 , 0, x3 ) = −3/4 − x˜1 + 2/3c(1/2 − x˜1 − x2 )x32 /2, g−3 (x1 , 0, x3 ) = 0, g+3 (x1 , x2 , 1) = (1/2 − x˜1 − x˜2 ) (3/2 + 3/2c(3/4 − x˜1 − x˜2 )) assuming that æ±α (x) = 1. Let us introduce the Dirichtet condition on the wall x1 = 0. The symmetry of solution is taken into account with respect to the plane x1 = 1/2, x2 = 1/2. The solution is defined in the space G¯ = {0, ≤ xα ≤ 1, α = 1, 2, 3}. In Table 7.5, the results of the solution to this problem for c = 2048 are reported for the mesh with a constant step. An analysis of the given results yields the conclusion that during the change of the mesh step, the approximated solution converges to the exact one. Variations of the operator coeﬃcients in the considered case is high, i.e. 1 ≤ kα (x) ≤ 1024, α = 1, 2, 3.

Table 7.5. Numerical solution of the model problems at point: x1 = .5, x2 = .5, x3 = x3k .

k 1 2 3 4 5

x3k 0 0.25 0.5 0.75 1.0

3×3×5 0 0.0112 0.504 0.120 0.223

Mesh 5×5×5 0 0.0120 0.0523 0.122 0.0224

9×9×9 0 0.0147 0.0599 0.136 0.243

Exact solution 0 0.0156 0.0625 0.141 0.250

Notice that for c = 0 the numerical solution with accuracy greater than 5 meaningful digits overlaps with the exact solution. 2. Problem without a source The convergence of the solution to the problem without a source (owing to the decreasing mesh step) is testified through the results included in Table 7.3. The following parameters are taken: k1 = k2 = k3 = 207, ϕ(x) = 0

7.3 Numerical solution of a three-dimensional equation of elliptic type

693

Table 7.6. Numerical solution of the model problems at point: x1k = x2k = x3k .

k 1 2 3 4

x3k 0 0.25 0.5 0.75

3×3×3 0 0.378

Mesh 5×5×5 0 0.0038 0.0375 0.114

9×9×5 0 0.0060 0.0444 0.127

Exact solution 0 0.0068 0.0469 0.132

and the following boundary conditions are applied: v(0, x2 , x3 ) = 0, v, x2 (x1 , 0, x3 ) = 0 k3 v, x3 (x1 , x2 , 0) = 10(v − 100), k3 v, x3 (x1 , x2 , l3 ) = 10v . The following symmetry solution with respect to the planes is taken into account: x1 = l1 /2, x2 = l2 /2; l1 = l2 = 1m., l3 = 0.02 m . Owing to the analysis of the results reported in Table 7.7, it is clear that without sources the numerical solution is obtained with an eﬃcient accuracy on the mesh 5 × 5. Table 7.7. Numerical solution to the problem without source.

k x1k 1 0 2 0.125 3 0.25 4 0.375 5 0.5

Mesh Recipes 5×5×5 9×9×5 0 0 9.17 9.2 x1k = x2k 15.3 15.3 x3 = 3 18.8 18.8 19.9 20.0

3. Problem with a point source The convergence of the solution to the problem with a point source obtained during the decrease of the mesh step is considered on the basis of the following problem ⎧ ⎪ ⎪ ⎨ 0, k1 = k2 = k3 = 1, ϕ(x) ⎪ ⎪ ⎩ 4 · 106 , for x1 = x2 = 1 /2, x3 = 3 ; with the attached boundary conditions: v(0, x2 , x3 ) = 0, v, x2 (x1 , 0, x3 ) = 10v, v, x3 (x1 , x2 , 0) = 0, v, x3 (x1 , x2 , l3 ) = 0; 1 = 2 = 1, 3 = 0.02.

694

7 Novel Solution Method for a System of Linear Algebraic Equations

The following symmetry condition with respect to the planes x1 = 1 /2, x2 = 2 /2 is applied. In Tables 7.8, 7.9 the results of the solution to this problem on four meshes: 3 × 3 × 5, 5 × 5 × 5×, 9 × 9 × 5, 17 × 17 × 5, is given. For the three partitions, the mesh step (in the vicinity of the node with the source) is taken to be constant and equal to the step of partition 17 × 17 × 5. In all four cases, the constant volume power is achieved. The changeable step allows application of the mentioned actions in a relatively simple manner by introduction of additional nodes with respect to x1, x2 in the first three cases, i.e. the solution is obtained for the shells: 4 × 4 × 5, 6 × 6 × 5 and 10 × 10 × 5. Owing to the given results, stable convergence is achieved during the mesh step variation. Maximal diﬀerences are observed at the node with the source, but it is less than 1% for two neighbouring partitions. Table 7.8. Numerical results to the problem with point source.

k 1 2 3 4 5

Mesh Recipes k3k 3 × 3 × 5 5 × 5 × 5 9 × 9 × 5 17 × 17 × 5 0 264.5 300.9 321.0 324.3 0.25 269.5 305.7 325.9 329.2 x1 = 0.5 0.5 284.4 320.6 340.9 344.2 x2 = 0.5 0.75 310.9 347.2 367.5 370.8 1 351.7 388.0 408.3 411.6

Table 7.9. Numerical results to the problem with point source.

k 1 2 3 4

Mesh Recipes k1k 3 × 3 × 5 5 × 5 × 5 9 × 9 × 5 17 × 17 × 5 0 0 0 0 0 0.25 12.05 12.25 12.31 x2k = x1k 0.5 36.6 38.9 39.6 39.8 x3k = 3 0.75 90.0 92.0 92.5

4. Comparison of solutions obtained through various methods Finally, we present the results of computation through the finite diﬀerence method for two problems, for which both analytical and boundary value solutions are known. In Table 7.10, the results of the solution to the 1st boundary value problem for the Laplace equation for the cube with ribs of unit length are reported.

7.3 Numerical solution of a three-dimensional equation of elliptic type

695

The boundary condition is as follows: v(−0.5, x2 x3 ) = 2, v(0.5, x2 , x3 ) = 1, v(x1 , −0.5, x3 ) = 0, v(x1 , 0.5, x3 ) = 0, v(x1 , x2 , −0.5) = 0, v(x1 , x2 , 0.5) = 0.

(7.33)

Table 7.10. Solution to the 1st boundary value problem for Laplace equation.

x1k -0.375 -0.250 -0.125 0 0.125 0.250 0.375

Boundary element Finite diﬀerence method method N = 12 N = 24 5 × 3 × 3 9 × 5 × 5 17 × 9 × 9 1.637 1.472 1.413 1.426 1.044 0.979 0.926 0.953 0.963 0.678 0.661 0.653 0.657 0.5 0.5 0.5 0.5 0.5 0.478 0.472 0.471 0.472 0.597 0.566 0.545 0.555 0.558 0.770 0.770 0.740 0.746

Analytical solution 1.430 0.967 0.659 0.5 0.472 0.560 0.748

In Table 7.11, the results of the solution to the Laplace equation with a hybrid boundary condition for the rectangular parallelepiped are reported. The boundary conditions are: v(−0, 5, x2 , x3 ) = 10, (v, x1 +5v)| x1 =0,5 = 0, (v, x2 +5v)| x2 =±1 = 0, (v, x3 +5v)| x3 =±1 = 0. Notice that when numerical solutions are constructed the symmetry with respect to two planes, x2 = 0, x3 = 0, is taken into account. Table 7.11. Solution of the Laplace equation with hybrid boundary conditions (x3k = x2k ).

x1k -0.25 0 0 0.25 0.25 0.25 0.25

x2k 0 0 0.5 0 0.25 0 0.75

Boundary element method N = 24 N = 48 7.387 7.282 4.827 4.84 3.745 3.843 2.816 2.843 2.612 2.658 2.0 2.073 1.050 1.144

Finite elemenet method 5×3×3 9×5×5 17×9×9 7.155 7.234 7.257 4.724 4.804 4.829 3.821 3.835 3.841 2.778 2.825 2.839 2.645 2.655 2.105 2.094 2.091 1.202 1.186

Analytical solution 7.259 4.837 3.843 2.844 2.658 2.089 1.180

696

7 Novel Solution Method for a System of Linear Algebraic Equations

The reported results show that the solution obtained through the finite diﬀerence method during the mesh step decrease converges to the analytical solution, and its error is small for the appropriate partition. The results of the numerical experiments yield the conclusion that our proposed method and the associated subroutines allow eﬃcient numerical solution of three dimensional boundary problems for the stationary heat transfer equation without any essential limitations. The same conclusion holds when the problems of plates and shells are considered.

7.4 Computation of geometrically non-linear non-homogenous shallow shells with mixed boundary condition along their sides One of the important computation problems connected with shallow non-homogeneous and geometrically non-linear shells concerns conditions of the boundary value problems variations along a supporting side. In order to derive appropriate relations, a system consisting of two identical shells stiﬀened by ribs along two sides linked by a hinge is considered (see Figure 7.2). The shell material is assumed to be isotropic but non-homogeneous, i.e. shear modulus G and Poisson coeﬃcient µ are functions of point coordinates x = (x1 , x2 , x3 ), or, according to the theory of small elastic-plastic deformations, they depend on the stress-strain material state at the considered point.

Figure 7.2. Investigated shell.

Let us introduce the following notation: x1 , x2 , x3 – rectangular system of coordinates (Fig. 7.2); α, b, h – shell dimension and thickness, respectively; k1 , k2 – curvatures; u, v, w – components of displacements of mean surface point; q – transversal load intensity. Following the Kirchhoﬀ-Love hypothesis the components of shell and rib deformation, read eii = εii + χii x3 (i = 1, 2),

7.4 Computation of geometrically non-linear non-homogenous shallow shells

e12 = ε12 + χ12 x3

697

(7.34)

where: ε11 = u,x1 − k1 w + w2,x1 /2, ε22 = v,x2 − k2 w + w2,x2 /2,

ε12 = U,x2 + v,x1 + w,x1 w,x2

(7.35)

ε11 , ε22 , ε12 – length and stream deformations of the mean surface, χ11 = −w x1 x1 , χ22 = −w x2 x2 , χ12 = −w x1 x2 ,

(7.36)

where: χ1 1, χ2 2, χ1 2 – variations of curvature and torsion. In accordance with the Hook principle for a flat strain state and (7.34), the coupling between stressed σik and deformation eik is presented in the following form σ11 =

6 2G 5 ε11 + χ11 x3 + µ(ε22 + χ22 x3 ) 1−v

(1 ↔ 2),

σ12 = G(ε12 + 2χ12 x3 ).

(7.37)

Integration of (7.37) with respect to x3 yields, the following expressions for forces and movements: T 11 = c00 ε11 + c10 ε22 + c01 χ11 + c11 χ22 , T 22 = c00 ε11 + c00 ε22 + c11 χ11 + c01 χ22 ,

(7.38)

T 12 = 0, 5(c00 − c10 )ε12 + (c01 − c11 )χ12 , M11 = c01 ε11 + c11 ε22 + c02 χ11 + c12 χ22 , M22 = c11 ε11 + c01 ε22 + c12 χ11 + c02 χ22 ,

(7.39)

M12 = 0, 5(c01 − c11 )ε12 + (c02 − c12 )χ12 ; where: h/2 cik (x1 , x2 ) = −h/2

2G(x)µi (x)x3k 3 dx , 1 − µ(x)

(i = 0, 1; k = 0, 1, 2).

The deformation energy of the considered system reads ⎡ 0 b α b 0 ⎢⎢⎢ ⎢⎢⎢ Adx1 dx2 + Adx1 dx2 + (H1− + P−1 )| x2 =0 dx1 + ∃ = 0.5 ⎢⎢ ⎣ −α 0

α 0

where:

0

−α

0

(H1+ + P+1 )| x2 =0 dx1 +

b 0

⎤ ⎥⎥⎥ ⎥ (H2− + H2+ + P)| x1 =0 dx2 ⎥⎥⎥⎥ , ⎦

(7.40)

698

7 Novel Solution Method for a System of Linear Algebraic Equations

A = T 11 ε11 + T 22 ε22 + T 12 ε12 + M11 χ11 + M22 χ22 + 2M12 χ12 ; H1± = D±1 ε211 + A±1 w2,x1 x1 + B±1 v2,x1 x1 + C1± w2,x1 x2 ; H2± = D±2 ε222 + A±2 w2,x2 x2 + B±2 u2,x2 x2 + C2± w2,x1 x2 ; P±1 = α±1 w2,x2 + β±1 w2 , P = α[w,x1 ]2 + βw2 ; where: D±1 , A±1 , B±1 , C1± – rib stiﬀness on elongation (compressing), bending in a vertical plane, bending in a horizontal plane, and torsion, respectively; α±1 , α – stiﬀness of hinge rib-shell, rib-rib interaction, respectively, β±1 , β – elasticity of support under the rib; [w,x1 ] – jump of derivative w,x1 on the line x1 = 0. Let us introduce the following functional 0 b

α b qwdx1 dx2 −

I(u, v, w) = ∃ − −α 0

qwdx1 dx2 . 0

(7.41)

0

Variating (7.41) with respect to u,v,w, and carrying out transformations with the help of integration by parts, one gets 0 b

α b ˜ 1 dx2 + Adx

δI = −α 0

0

˜ 1 dx2 + Adx 0

0

(R1 | x2 =0,b + (S 1− + β−1 wδw + α−1 w,x2 δw,x2 )| x2 =0 )dx1 +

−α

α

(R1 | x2 =0,b +(S 1+ + β+1 wδw + α+1 w,x2 δw,x2 )| x2 =0 )dx1 +

0

b

(R2 | x1 =−α,0 + R2 | x1 =0,α +(S 2− + S 2+ + α[w,x1 ]δ[w,x1 ] + βwδw)| x1 =0 )dx2 +

0

Q−1 || x1 =−α,0 + (Q+2 + Q−2 )|| x1 =0 + Q+1 || x1 =0,α − x2 =0

x2 =0,b

x2 =b

2M12 δw|| x1 =−α,0 − 2M12 δw|| x1 =0,α = 0, x2 =0,b

x2 =0,b

where: A˜ = −(T 11,x1 + T 12,x2 )δu − (T 12,x1 + T 22,x2 )δv− (M11,x1 x1 + M22,x2 x2 + 2M12,x1 x2 + k1 T 11 + k2 T 22 + (T 11 w,x1 + T 12 w,x2 ),x1 + (T 12 w,x1 + T 22 w,x2 ),x2 − q)δw ; R1 = T 12 δu + T 22 δv + (M22,x2 + 2M12,x1 + T 22 w,x2 )δw − M22 δw,x2 , S 1± = −(D±1 ε11 ),x1 δu + (B±1 v,x1 x1 ),x1 x1 δv + (−k1 D±1 ε11 ) − (D±1 ε11 w,x1 )+

(7.42)

7.4 Computation of geometrically non-linear non-homogenous shallow shells

699

(A±1 w,x1 x1 )δw − (C1± w,x1 x2 )δw x2 , Q±1 = D±1 ε11 δ − (B±1 v,x1 x1 ),x1 δv + (B±1 v,x1 x1 )δv,x1 + (D±1 ε11 w,x1 − (A±1 w,x1 x1 ),x1 )δw + A±1 w,x1 x1 δw,x1 + C1± w,x1 x2 δw,x2 , (1 ←→ 2, u ←→ v, x1 ←→ x2 ). Let us consider some of the relations that follow from the previous considerations. Diﬀerential equations governing the equilibrium state of a non-homogeneous shell with respect to displacements are obtained by comparison to zero of the coeﬃcients standing by variations of u, v, w in doubled integral (7.42), and by taking into account relations (7.35), (7.38), (7.39). However, they are not explicitly given owing to their complexity. Boundary conditions for the side x2 = 0 follow from the examinations of the under integral expressions. If the displacements on the contour are not given, then the boundary conditions are defined by comparison to zero of the coeﬃcients standing by variations of u, v, w, w,x2 . Conditions for the coupling of solutions on the line x1 = 0 are derived from the following expression: R2 | x1 =−0 − R2 | x1 =0 + (S 2− + S 2+ + α[w,x1 ]δ[w,x1 ] + βwδw)| x1 =0 = (−[T 11 ] + (B2 u,x2 x2 ),x2 x2 )δu + (−[T 12 ] − D2 ε22 )δv+ (−[T 13 ] − D2 k2 ε22 − (D2 ε22 w,x2 ),x2 + (A2 w,x2 x2 ),x2 x2 + βw)δw+ − (−M11

+ − (C2− w−,x1 x2 ),x2 − α[w,x1 ]δw−,x1 + (M11 − (C2+ w+,x1 x2 ),x2 + α[w,x1 ]δw+,x1 = 0,

where [T 1i ] = T 1i | x1 =0 − T 1i | x1 =−0 ,

(i = 1, 2, 3),

T 13 = M11,x1 + 2M12,x2 + T 11 w,x1 ; B2 = B−2 + B+2 , D2 = D−2 + D+2 , A2 = A−2 + A+2 . This means that the following conditions should be satisfied on the line of the solution coupling: [T 11 ] = (B2 u,x2 x2 ),x2 x2 , [T 12 ] = −D2 ε22 ; = −D2 k2 ε22 − (D2 ε22 w,x2 ),x2 + (A2 w,x2 x2 ),x2 x2 + βw [M11 ] = (C2+ w,x1 x2 ),x2 + (C2− w−,x1 x2 ),x2 ; + − − M11 + (C2+ w,x1 x2 ),x2 + (C2− w,x1 x2 ),x2 , α[w,x1 ] = −M11

(7.43)

as well as the conditions on the continuity of displacements [u] = [v] = [w] = 0. Taking k2 = ε22 = u = v = T 11 w,x1 = 0 in (7.43), the conditions for the solution coupling for a plate are obtained [592].

700

7 Novel Solution Method for a System of Linear Algebraic Equations

Let us determine agreement conditions in the shell angle x1 = 0, x2 = 0. For this purpose, the forms related to an angle are considered in (7.42), i.e. (−Q+1 − Q+2 − 2M12 δw)|| x1 =0 = (−D+1 ε11 + (B+2 u,x2 x2 ),x2 )δu+ x2 =0

((B+1 v,x1 x1 ),x1 − D+2 ε22 )δv + (−2M12 − D+1 ε11 w,x1 + (A+1 w,x1 x1 ),x1 − D+2 ε22 w,x2 + (A+2 w,x2 x2 ),x2 δw − B+1 v,x1 x1 δv,x1 − B+2 u,x2 x2 δu,x2 − (A+1 w,x1 x1 C2+ w,x1 x2 )δw,x1 − (A+2 w x2 x2 + C1+ w,x1 x2 )δw,x2 = 0. One may derive now, for instance, the following compatibility condition in the angle: −D+1 ε11 + (B+2 u,x2 x2 ),x2 = 0, −D+2 ε22 + (B+1 v,x1 x1 ),x1 = 0; − + (A+1 w,x1 x1 ),x1 + (A+2 w,x2 x2 ),x2 − D+1 ε11 w,x1 − D+2 ε22 w,x2 = 0. −2M12

B+1 v,x1 x1 = 0, B+2 u,x2 x2 = 0 ; A+1 w,x1 x1 + C2+ w,x1 x2 = 0; A+2 w,x2 x2 + C1+ w,x1 x2 = 0 . Notice that if the ribs are excluded, then only one compatibility condition M12 = 0 remains. Let us derive compatibility conditions for the point x1 = 0, where the change of boundary conditions occurs. They are obtained from the terms of (7.42) with respect to the point x1 = 0, x2 = 0, and the coeﬃcients with index 2 are equalled to zero (homogeneous shell): (Q−1 − Q+1 )|| x=0 = (D−1 ε−11 − D+1 ε+11 )δu + ((B+1 v+,x1 x1 ),x1 − (B−1 v−,x1 x1 ),x1 )δv+ x2 =0

(−D+1 ε+11 w+,x1

+ D−1 ε−11 w−,x1 + (A+1 w+,x1 x1 ),x1 − (A−1 w,x1 x1 ),x1 )δw + B−1 v−,x1 x1 δv−,x1 −

B+1 v+,x1 x1 δv+,x1 +(C1− w−,x1 x2 −C1+ w,x1 x2 )δw,x2 +A−1 w−,x1 x1 δw−,x1 −A+1 w+,x1 x1 δw+,x1 = 0 . (7.44) As an example let us consider the compatibility condition for a hinged support of the shell side x2 = 0 with diﬀerent stiﬀness: D−1 ε11 − D+1 ε+11 = 0, (B+1 v+,x1 x1 ),x1 − (B−1 v−,x1 x1 ),x1 = 0, B+1 v+,x1 x1 = 0, C1− w−,x1 x2 − C1+ w+,x1 x2 = 0 .

(7.45)

Owing to relations (7.44), (7.45), the necessity of application of the compatibility condition appears only for the shells supported by ribs. Let us introduce the following force function F,x2 x2 = T 11 ; F,z1 x2 = −T 12 ; F,x1 x1 = T 22 , and let us derive the equations in a hybrid form with respect to force F and deflection w. For this purpose the deformation continuity equation and the third equation of (7.42) are applied to yield

7.4 Computation of geometrically non-linear non-homogenous shallow shells

701

ε11,x2 x2 + ε22,x1 x1 − ε12,x1 x2 + ∇2k w + 0, 5L(w, w) = 0; M11,x1 x1 + M22,x2 x2 + 2M12,x1 x2 + ∇2k F + L(w, F) − q = 0.

(7.46)

By solving (7.38) with respect to εik one gets ε11 = AF,x2 x2 + A ∗ F,x1 x1 + Bw,x1 x1 = B ∗ w,x2 x2 , ε22 = A ∗ F,x2 x2 + AF,x1 x1 + B ∗ w,x1 x1 + Bw,x2 x2 , ε12 = −4A2 F,x1 x2 + 4B2 w,x1 x2 , where: A = A1 + A2 , A∗ = A1 − A2 , B = B1 + B2 , B∗ = B1 − B2 , A1 = 0.5/(c00 + c10 ), A2 = 0.5/(c00 − c10 ) , B1 = (c01 + c11 )A1 , B2 = (c01 − c11 )A2 .

(7.47)

Taking inot account (7.47) in (7.39), one obtains M11 = BF,x2 x2 + B ∗ F,x1 x1 − Cw,x1 x1 − C ∗ w,x2 x2 ; M22 = B ∗ F,x2 x2 + BF,x1 x1 − C ∗ w,x1 x1 − Cw,x2 x2 ; M12 = −2B2 F,x1 x2 − (C − C∗)w,x1 x2 ,

(7.48)

where: C = c02 − B∗1 − B∗2 , C ∗ = c12 − B∗1 − B∗2 ; B∗1 = (c01 + c11 )B1 , B∗2 = (c01 − c11 )B2 . Equations (7.46) with an account of (7.47), (7.48) read A∗ F,x1 x1 + AF,x2 x2 + Bw,x1 x1 + B ∗ w,x2 x2 ),x2 x2 + (AF,x1 x1 + A ∗ F,x2 x2 + B ∗ w,x1 x1 + Bw,x2 x2 ),x1 x1 − (−4A2 F,x1 x2 + 4B2 w,x1 x2 ),x1 x2 + ∇2k w + 0.5L(w, w) = 0 ;

(7.49)

(BF,x1 x1 + B ∗ F,x2 x2 − C ∗ w,x1 x1 − Cw,x2 x2 ),x2 x2 + (B ∗ F,x1 x1 + BF,x2 x2 − Cw, x1 x1 −C ∗ ω,x2 x2 ),x1 x1 − 2(−2B2 F,x1 x2 − (C − C∗)w,x1 x2 ),x1 x2 + ∇2k F + L(w, F) − q = 0 .

(7.50)

In what follows, the algorithm for solving the stability problem of physically non-linear flexible shallows shell with mixed boundary conditions and without ribs is discussed. Step 1. In the space occupied by the shell, a rectangular mash with equal step {x1i , x2 j , x3k } is introduced. Assuming in the beginning that the shell is unloaded, modulus Gi jk the shell nodes is introduced. Step 2. At the nodes of mesh surface x3 = 0 coeﬃcients Cα β (7.40) are computed. Then, substituting the partial derivatives by diﬀerence expressions with error Q(h2x1 +

702

7 Novel Solution Method for a System of Linear Algebraic Equations

h2x2 ), instead of system equations (7.49) the system of non-linear algebraic equations with respect to wi j, Fi j , i.e. deflection and stress function formulated at nodes of the mean surface, is obtained. Boundary conditions are considered while out-contour nodes are being eliminated from the equation system. If the mesh node overlaps with the point where the boundary conditions change, for instance in the case of hingeclamping, then the clamping is understood as the value of derivative with respect to a normal solution to system of algebraic equations for a given load q(x1 x2 ) defined with the help of the Newton or diﬀerentiation along the parameter methods [606], and then the elimination method is applied. Step 3. New values of Gi jk and µi jk are computed at the mesh nodes in accordance with the theory of small elastic-plastic deformation [43, 369]. Modulus of elongation and modulus of volume deformation K are found using formulas of the theory of small elastic-plastic deformation E=

1 3K − 2G 9KG , µ= . 3K + G 2 3K + G

Notice that in this theory it is assumed that K does not depend on the deformable state at a point, since in the case of a body which is non-homogeneous before deformation K = K(x1 , x2 , x3 ), and for a body homogeneous before deformation K = K0 = const. The shear modulus is defined through the formula G=

1 σi (ei ) , 3 ei (x)

and G is called the cutting modulus. In order to compute the shell, dependence σi (ei ) should be explicitly given (intensity of stress σi versus strain (ei )). Some of the diagrams σi (ei ) are reported in reference [369]. Then the procedure goes back to step 2. The computations are repeated until the obtained solution overlaps with the previous one keeping the assumed accuracy. Step 4. The load q = q + hq is increased and the computations start at step 2. As a result, dependence q(w) yielding the critical values is obtained. On a basis of the introduced algorithm the program in Fortran has been developed . As an example, a square shell made of aluminium and with parameters k1 = k2 = 18; λ = a/h = 50 ; q = const [369] has been considered. Let us apply the Mises flow condition (σi = σ s ), and the dependence σi (ei ) taken in the form σi = σS (1 − exp(−ei /eS )), where: e s = σ s /3σ0 is the artificial intensity of the flow deformation. The problem is considered symmetric with respect to x1 , x2 for four types of the boundary conditions, namely: 1. hinged support on the contour;

7.4 Computation of geometrically non-linear non-homogenous shallow shells

703

2. hinged support in an angle, and clamping in the middle of the side over the interval 1/4a; 3. clamping in an angle, and hinged support in the middle of the side over the interval 1/4a; 4. clamping along a contour. The stress function on the side x2 = 0 is F,x1 x1 = 0, F,x1 x2 = 0, whereas for the deflection w = 0, w x2 = 0 or M22 = 0.

Figure 7.3. Load versus shell centre deflection (see text for more details).

Consider a few computational results. In Figure 7.3, dependencies of load versus centre deflection represented by curves 1-4 corresponding to the considered boundary conditions are reported. Analysing the behaviour of the curves one may conclude that the value of the upper critical load depends essentially on the position of the clamped part. Figure 7.4 shows influence of the boundary conditions on the distribution of plasticity zones on the upper shell surface x3 = −h/2 for the loads corresponding to deflection w = 0, 4 in the shell centre. The largely developed plasticity zones are achieved for hinged support. Owing to clamping of the shell point in the middle of its side, the plastic zones are sharply decreased and their position is subject to change. The minimal areas of the plasticity zones are achieved in the case of clamping along the whole length. Figure 7.5 displays graphs of moments M on the side x1 = 0 for the following cases: 2 – solid curve; 3 – dashed curve; 4 – dashed-dotted curve. Deflection in the shell centre achieved in all cases is w = 0.4. In Figures 7.3 and 7.4, digits 1-4 should be considered simultaneously. For the given computational model, contrary to [665], a jump of the moments is observed, which is close to real observations. The results of section [592], make the elimination method for equations eﬀective also for solving of the complicated PDEs like non-linear equations governing

704

7 Novel Solution Method for a System of Linear Algebraic Equations

Figure 7.4. Influence of boundary conditions on plasticity zone distribution (see text for more details).

Figure 7.5. Moment M11 vs x2 on shell side x1 = 0 (see text).

elastic-plastic problems of shells with finite deflections. Here on each loading step one has to solve many times a large system of linear algebraic equations, which, without any doubt, exerts significant impact on the chosen computational algorithm.

8 Mathematical Approaches to Coupled Termomechanical Problems

In section 8.1 the equations of thermo-elastic plate are analysed. The suﬃcient conditions of existence, uniqueness and continuity dependence on initial data of the Cauchy problem solutions for diﬀerential-operational equation of mixed type (a part of the equation of hyperbolic type, and a part of parabolic type) are given. If the operational coeﬃcients are suitably chosen, the investigated equation can be used to obtain a diﬀerential equation describing vibrations of a plate - the modified Germain-Lagrange equation of hyperbolic type. Moreover, in order to define the temperature field, one can use a three-dimensional equation of thermal conductivity (a parabolic equation). In Section 8.2 a coupled thermo-mechanical problem of non-homogeneous shells with variable thickness and variable Young modulus (the so-called Timoshenko type model) is considered. The problem is reduced to uniformly correct problem in the form of a first order diﬀerence equation. In addition, a similar approach can easily be applied to the Kirchhoﬀ-Love model. The boundary conditions for a non-homogeneous first order operator-diﬀerential equation, with a unique solution, are derived in Section 8.3. This is motivated by a coupled thermomechanical problem. Two important theorems are formulated.

8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem 8.1.1 Introduction Let H be a Hilbert space, let L, M be a self-adjoint operators negatively defined in H with the domains D(M) ⊂ D(L), and let C be an operator bounded in H. It is assumed that the operator L commutes with C and with the resolvent M (the definition of commuting is given in [574] chapter 8, §1 p.322). Let us consider the following system on the interval [0,T] W

(t) + L2 W(t) + αLCθ(t) = q1 (t), θ (t) − Mθ(t) − βC ∗ LW (t) = q2 (t),

(8.1)

(α, β > 0) with unknown functions W and θ. We take H = L2 (Ω) as the space of functions defined in the bounded domain with the

706

8 Mathematical Approaches to Coupled Termomechanical Problems

h h piecewise smooth boundary ∂Ω in the space Ω = Ω1 × [− , ] ⊂ R3 (Ω1 ⊂ R2 , h > 2 2 0) and having the summing up square norm. As M we take the operator which is λ2 ∂2 λ2 ∂2 ∂2 realization of the formal diﬀerential expression 0 2 + 0 2 + 2 (λ0 0), and with ∂x ∂y ∂z the attached boundary condition θ |∂Ω = 0. As L operator we take the operator being a closure of a tensor product of the operators L1 and E, where L1 is generated in ∂2 ∂2 L2 (Ω1 ) by the expression γ2 = γ( 2 + 2 ) (γ > 0) with the boundary condition ∂x ∂y h h W|∂Ω1 = 0. E is the identity operator in L2 (− , ), i.e. L is the closure of the operator 2 2 ∂2 ∂2 2 generated in L2 (Ω) by the expression γ = γ( 2 + 2 ) and defined on smooth ∂x ∂y h h functions from L2 (Ω) which are equal to zero on ∂Ω1 × [− , ]. The operator C 2 24 h 4h is defined by the formula Cθ = −2h zθ(x, y, z)dz. Then C ∗ g = z −2h g(x, y, z)dz. If 2 2 L, M, C are chosen in this way, and the numbers α, β, γ are properly chosen, one can get from (8.1) the known plate equation governing its vibrations ([44], chapter I, §1, p.25), with the following parameters: F = 0, k x = ky = 0, P x = Py = 0; it is called the thermal conductivity equation. Furthermore, in order to simplify the considerations we assume in (8.1) that α = β = 1. Observe that practically the introduced considerations do not change other values of α and β. In order to reduce the system (8.1) into one equation of the first order we introduce the operator ⎞ ⎛ ⎜⎜⎜ 0 E 0 ⎟⎟⎟ ⎟ ⎜⎜⎜ 2 A0 = ⎜⎜ −L 0 −LC ⎟⎟⎟⎟ , ⎠ ⎝ 0 C∗ L M > = H ⊕ H ⊕ H. Here and further E means the identity operator acting in the space H in the respective space. We emphasize that the commutativity of the operators C and L yields the commutativity of C ∗ and L∗ = L [574] (chapter 8, §1, p. 323). > The operator Lemma 1. The operator A0 allowes of closing of A in the space H. > A−1 exists, is bounded and defined in the whole space H. Proof. The operational matrix ⎞ ⎛ ⎜⎜⎜ CM −1C ∗ −L−2 −L−1CM −1 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ E 0 0 ⎟⎟⎠ ⎜⎜⎝ −1 ∗ −1 0 M −LM C > A direct check shows that its product (of any defines the bounded operator in H. order) with the operational matrix defining the operator A0 gives the unit operational matrix. All claims of the lemma result from this. Moreover, the operator A−1 is defined by the last operational matrix. Further on, the operator and its closure will be tagged with one symbol. If we assume ν = w , and u will denote the column

8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem 707

⎛ ⎞ ⎜⎜⎜ w ⎟⎟⎟ ⎜⎜⎜⎜ ν ⎟⎟⎟⎟ , ⎜⎝ ⎟⎠ θ > can be written in the following then the homogeneous system (1) in the space H way: (8.2) u (t) = Au(t). −2 > > Furthermore, we denote by L the operator in H, defined by the equation ⎞ ⎛ −2 ⎜⎜⎜ L 0 0 ⎟⎟⎟ ⎟ ⎜ −2 > L = ⎜⎜⎜⎜ 0 L−2 0 ⎟⎟⎟⎟ . ⎠ ⎝ 0 0 L−2 It results from the commuting properties of the operators that also > L−2 and A−1 commute. 8.1.2 Basic assumption In order to formulate the basic assumption we need to define the following spaces and the attached operators. The smoothness of a solution to the homogeneous equation (8.2) depends on that if the initial condition u(0) belongs to the domains of the operators > L and A. Observe that in applications the operator > L is simpler than the operator A and therefore the spaces are constructed with regard to the powers of > L. However, we can not completely omit the operator A. Observe also that one can use the powers of A only (without > L). We define the scalar product by the formula (x, y)F ∗ = (A∗ x, A∗ y) on the manifold F ∗ = D(A∗ ). As the operator A∗ possesses the inverse one defined everywhere, then F ∗ is a complete space and x F ∗ ≥ x H>. This is why F ∗ can be considered as > [100] (chapter 1, §1, p.46.). We shall the space with a positive norm in relation to H (−1) > > denote by H the space with a negative norm, constructed using the pair F ∗ , H. ∗ ∗ > The operator A maps continuously and bijectively F into H. This is why the > acts from H > to H >(−1) continuously and bijectively, and it conjugated operator A serves as the extension of A. >(+1) = D(A) by the equation We shall define the scalar product on the set H (+1) > >(+1) appears as the full space and the operator A maps H (x, y)H>(+1) = (Ax, Ay). H > continuously and bijectively (one-to-one). to H L−1 is self-adjoint and Because the operator > L−1 commutes with A, A∗ , and > −1 >(+1) and H >(−1) > then > bounded in H, L is self-adjoint and bounded in the spaces H −1 −1 > ). The Hilbert space (the > L operator is extended due to continuity in the space H >α = H >α0 , H >α(−1) (−∞ < α < ∞) [100] >α(+1) , H scales will be denoted by the symbols H > >(+1) , H (chapter 3, §6, p.253), and they are generated by the operator > L acting in H (−1) > , correspondingly. As it is known [100] (chapter 3, §6, p. 253) for α ≥ 0 and H >α(β) (β = −1, 0, 1) continuously and bijectively, and >(β) into H the operator > L maps H α+1 (β) >(β) (α ≥ 0). We > >−α to H L allows for continuous and bijective extension from H −α−1 shall denote this extension as > L in order not to introduce redundant denotations.

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8 Mathematical Approaches to Coupled Termomechanical Problems

8.1.3 Main results >α , and H >α into H >α(−1) (α ≥ 0) continuously and > map H >α(+1) into H The operators A, A bijectively, respectively. The same theorem is true for α < 0, and for the operators > >> > respectively. These Lα A> L−α , which serve as the extension of A and A, L−α and > Lα A > extensions are also denoted by A and A. >α(β) (β = +1, 0; −∞ < α < ∞) there exists THEOREM 8.1 For any element u0 ∈ H >(β−1) , conone function u(t) continuous in the closed interval [0, T] in the space H α−2 (β) > , diﬀerentiable on tinuous in the half open from the left side interval (0, T] in H α−4 >(β−1) and satisfying both the equation (0, T] in H α−4 > u (t) = Au(t),

(8.3)

and the condition u(0) = u0 . (The continuity is understood in the strong sense in a suitable space). Observe that from the formulated theorem one can conclude that, in particular, >(+1) the function u(t) is a weak solution of the equation (8.2) (in the sense for u0 ∈ H 4 given in reference [100], §3, p. 76). The proof of Theorem 1 is based on the quoted below estimates of the resolvent of the operator A in a way similar to that presented in [367] (chapter 1, §3, page 78), and the Laplace inverse transformation. Let λ = σ + iτ. We define the operators 1 = 1 (λ), 2 = 2 (λ) in H in the following way: 1 = (λ2 E + L2 ) − λL2C(M − λE)−1C ∗ , 2 = (M − λE) − λL2 (L2 + λ2 E)−1C ∗C. Lemma 2. If σ = Reλ > 0 then the operators 1 , 2 have bounded inverses: −1 1 ≤ Proof. Let S =

1 , σ|λ|

−1 2 ≤

1 . σ

1 1 . Then for any f ∈ D(L2 ) we have: λ (S f, f ) = (σ + iτ)( f, f ) +

1 (σ − iτ)(L2 f, f )− | λ |2

−((M − σE + iτE) | M − λE |−1 LC ∗ f,

| M − λE |−1 LC ∗ f ).

It results from this and from the negative definition of M that Re(S f, f ) ≥ σ f 2 , and therefore S f ≥ σ f . Analogously we get S ∗ g ≥ σ g . This is why the operator S −1 exists, is bounded, defined on the whole H, and S −1 ≤ σ1 . The equation −λL2 (L2 + λ2 E)−1 = −L2 (λL2 + | λ |2 λE) | L2 + λ2 E |−2 yields Re(−λL2 (L2 + λ2 E)−1C ∗Cg, g) ≤ 0 (g ∈ H). From the above and taking into account the negative definition of the operator M we get: Re(2 f, f ) ≤ −σ( f, f )( f ∈

8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem 709

D(2 )). This is why | Re(2 f, f ) |≥ σ f 2 , which yields 2 f ≥ σ f . Analogously, we get ∗2 f ≥ σ f . Two last inequalities prove the lemma on the operator 2 . Lemma 3. If Reλ ≥ 0 then the operator A possesses the resolvent R(λ) which is defined by the matrix R(λ) = ⎛ 2 −1 −1 ∗ −1 ⎞ −−1 − −1 ⎟⎟⎟ ⎜⎜⎜ −λ −1 1 +L 1 C(M − λE) C 1 1 LC(M − λE) 2 −1 −1 −1 ⎜⎜⎜ L 1 −λ1 −λ 1 LC(M − λE)−1 ⎟⎟⎟⎠ . ⎝ 3 2 2 −1 −1 ∗ 2 2 −1 −1 ∗ −1 −L (L + λ E)

2 C

λL(L + λ E)

2 C

2

Proof. By direct check and taking into account the commutativity of L and M −1 one can verify that the expression R(λ)(A0 − λE) gives a unit operator matrix. Be> Indeed, let such sides, the space of values of the operator A0 − λE is dense in H. ∗ > the element x = (x1 , x2 , x3 ) ∈ H occurs, that (A0 − λE) x = 0. This equation is equivalent to the following three: −λx1 − L2 x2 = 0,

−C ∗ Lx2 + (M − λE)x3 = 0.

x1 − λx2 + CLx3 = 0,

Finding x1 from the first equation, x3 from the last one and by substituting in the second one, we get 1 (λ)x2 = 0. It results from Lemma 2 that x2 = 0, and therefore x1 = x3 = 0. This is why x = 0. Lemma 4. If σ = Reλ > 0 are large enough the following estimations hold R(λ) ≤ k

|λ| , σ

(8.4)

1 . R(λ)> L−2 ≤ k |λ|

(8.5)

Here and further k denotes constant (independent of λ), but diﬀerent, generally speaking, for diﬀerent inequalities. Proof. First we are going to prove the inequality L2 −1 1 ≤

|λ| . σ

(8.6)

As in the proof of Lemma 2, if g ∈ H, the following holds (S L−2 g, g) = (σ + iτ)(L−2 g, g) +

1 (σ − iτ)(g, g)− | λ |2

−((M − σE − iτE) | M − λE |−1 C ∗ g, | M − λE |−1 C ∗ g). σ | λ |2 . Thus (as in Lemma 2) we see that L2 S −1 ≤ 2 σ |λ| and this is why (8.6) is satisfied. Let us denote the elements of operator matrix defining R(λ) by ai j . Then: Then Re(S L−2 g, g) ≥

a11 ≤

k , σ

a12 ≤

1 , |λ|σ

a13 ≤

k , |λ|σ

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8 Mathematical Approaches to Coupled Termomechanical Problems

a21 ≤

|λ| , σ

a11 L−2 ≤

k , |λ|

a21 L−2 ≤

1 , |λ|σ

1 , σ k , ≤ |λ|σ

a22 ≤

a12 L−2

a22 L−2 ≤

a23 ≤

k , |λ|

k , σ

a13 L−2 ≤

k , | λ |2 σ

a23 L−2 ≤

k . |λ|σ

The estimations of elements of the first two rows of the matrix R(λ) immediately result from (8.6) of Lemma 2, the inequalities L(M − λE)−1 ≤ k, and (M − λE)−1 ≤

1 , |λ|

(8.7)

and also from the following two expressions 1 1 a11 = − E − L2 −1 1 , λ λ

1 1 −1 ∗ a22 L−2 = − L−2 + −1 + −1 1 C(M − λE) C , λ λ 1

which can be verified using the elementary transformations. In order to evaluate the elements of the third row of the matrix R(λ) we shall add some supporting inequalities. Because for any f ∈ D(L2 ) we have Re(( λ1 L2 + λE) f, f ) ≥ σ( f, f ), then ( λ1 L2 + λE)−1 ≤ σ1 . This yields (L2 + λ2 E)−1 ≤

1 . |λ|σ

(8.8)

From this inequality (using the elementary transformations) we get L2 (L2 + λ2 E)−1 ≤ We shall prove that

|λ| . σ

(8.9)

L−1 2 ≤ m,

(8.10) −1

−1

where m does not depend on λ. For any f ∈ D(2 L ) we have (2 L f, f ) = − (σ + iτ)(L−1 f, f ) − (σ − iτ) | λ |2 (Lg, g) − (σ + iτ)(L3 g, g), (ML−1 f, f ) 3 where g = | L2 + λ2 E |−2 C ∗C f . It results from the negative definition of L that Re(2 L−1 f, f ) ≥ Re(ML−1 f, f ) ≥ k f 2 . The last inequality results from the positive definition of the limited operator LM −1 , inversed to ML−1 . Therefore, the inequality (8.10) has been satisfied. The equation −1 2 2 −1 ∗ −1 = (M − λE)−1 L−1 + (L−1 −1 2 L 2 )(L + λ E) C Cλ(M − λE)

and the inequalities (8.7), (8.8), (8.10) yield −1 ≤ −1 2 L

k . |λ|

(8.11)

8.1 Existence and Uniqueness of Solution of One Coupled Plate Thermomechanics Problem 711

Now the inequalities (8.8), (8.9), (8.10), (8.11) and Lemma 2 allow to obtain the estimation of the elements of the third row of the matrix R(λ): a31 ≤ k

|λ| , σ

a32 ≤

k , σ

a33 ≤

1 , σ

k k k , a32 L−2 ≤ , a33 L−2 ≤ . |λ|σ |λ|σ |λ| The proof of Lemma results from the obtained ai j estimates. Let’s notice that some evaluations can be even more improved, for instance ( (( k (a12 L−2 (( ≤ . However, they will be not used further. | λ |2 Now we prove directly the theorem. Following [367] (chapter 1, §3, page 78) we are looking for the solution of the equation (8.2) in the form a31 L−2 ≤

u(t) = −

1 2πi

σ+i∞

eλt R(λ)u(0)dλ,

σ−i∞

where the integration takes place along the straight line, parallel to imaginary axis and passing through the point (σ, 0), where σ > 0 is suﬃciently large. >(+1) , i.e. u0 = R(0)> > and we consider L−4 z0 (z0 ∈ H), We assume first that u0 ∈ H 4 the function σ+i∞ 1 eλt R(λ)R(0)> L−4 z0 dλ = u(t) = − 2πi σ−i∞

1 2πi

=−

σ+i∞

σ−i∞

1 1 L−4 z0 − R(0)> L−4 z0 )dλ. eλt ( R(λ)> λ λ

(8.12)

Here, while passing from the first integral to the second one, the resolvent identity has been used. The existence of the integral, the continuity of u(t) on [0, T] in > and the identity u(0) = R(0)> >+2 (and correspondingly, in H) L−4 z0 result the space H directly from the Lemma 4. Applying the integration by parts to the first integral in (8.12) and taking into consideration Lemma 4 we get 1 1 u(t) = 2πi t

σ+i∞

eλt R2 (λ)R(0)> L−4 z0 dλ.

σ−i∞

>(+1) and It results from the above that u(t) is continuous on (0, T ] in the space H > it possesses the continuous derivative on (0, T ] in H 1 1 u (t) = − 2πi t2

σ+i∞

σ−i∞

1 1 e R (λ)R(0)L z0 dλ + 2πi t λt 2

>−4

σ+i∞

eλt λR2 (λ)R(0)> L−4 z0 dλ,

σ−i∞

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8 Mathematical Approaches to Coupled Termomechanical Problems

and satisfies the equation (8.2). The uniqueness of the solution of the Cauchy problem for the equation (8.2) follows from the estimation (8.4) and the results presented in reference [367] (chapter 1, §3, page 81). >α(β) (β = +1, 0; −∞ < α < ∞). Then x0 = Aβ−1 > >(+1) Lα−4 u0 ∈ H Now, let u0 ∈ H 4 (here A0 denotes E). Because the operators R(λ) and > Lα commute and from the previous considerations we get the function 1 >1−β >4−α A L u(t) = − 2πi

σ+i∞

eλt R(λ)x0 dλ

σ−i∞

with all properties mentioned in the formulated theorem. The uniqueness of such Lα−4 u(t) is the solution of the function results from the fact that the function Aβ−1 > equation (8.2). Theorem 1 has been proved. Remark. It results from the above proof that in order to solve the equations (8.2) and (8.3), the estimation u(t) H>α(β−1) ≤ k u(0) H>(β) (β = +1, 0; −∞ < α < α+2 ∞) holds, where k does not depend on t. It defines the continuous dependence of solution on the initial data in the corresponding spaces. Let us pass to the consideration of the non-homogeneous equations > + f (t). u (t) = Au(t)

(8.13)

By U(t) we denote the operator 1 U(t)x = − 2πi

σ+i∞

eλt R(λ)xdλ,

0 ≤ t ≤ T,

σ−i∞

defined for all x, for which the last integral exists. The Lemma 4 and the equation 1 U(t)R(0)> L−2 z = − 2πi

σ+i∞

σ−i∞

1 L−2 zdλ eλt (R(λ) − R(0))> λ

>α(β) → H >(β−1) (β = +1, 0; −∞ < α < imply that for every fixed t, the operator U(t) : H α−2 ∞) is bounded and the operator function U(t) is strongly continuous on [0, T ]. For >α(β) , the function u(t) = U(t)u0 has the properties given in Theorem 1. any u0 ∈ H THEOREM 8.2 Let for any t ∈ [0, T ], f (t) belongs to the domain of the operator > Lα A1+β (β = +1, 0; −∞ < α < ∞) and let the function > Lα A1+β f (t) be continuous in > H. Then the function t u(t) = U(t − s) f (s)ds (8.14) 0

>(β) , is diﬀerentiable on [0, T ] in the space is continuous on [0, T ] in the space H α−2 >(β−1) and satisfies the equation (8.13). H α−2

8.2 On the Solution of a Coupled Thermo-mechanical Problem

713

Proof. Let g(t) = > Lα A1+β f (t). Following the steps given in the reference [662] one can conclude that the function (8.14) and its derivative have the form t u(t) =

L−α g(s)ds, U(t − s)A−1−β >

0

u (t) = A−1−β > L−α g(t) +

t

L−α g(s)ds = Ut (t − s)A−1−β >

0

t = f (t) +

> > AU(t − s)A−1−β > L−α g(s)ds = f (t) + Au(t).

0

The existence of integrals in these equations results from the continuity of g and the properties of U(t). In what follows u(t) is the solution of the equation (8.13). The theorem has been proved. Remark. The continuity condition of the function > Lα A1+β can be weakened if the function f is smooth enough. Finally, we acknowledge the recent developments of a similar problem considered in our work and presented in references [410, 662] (see also the extended bibliography therein).

8.2 On the Solution of a Coupled Thermo-mechanical Problem for Non-homogeneous Timoshenko-type Shells 8.2.1 Introduction and Statement of the Problem As it is well known (see, for example [42]), the thermo-mechanical equations governing the dynamics of the Timoshenko shell model have the form ∂u 1 ∂ ∂v ∂2 u Eh − k − k w + µ w − ρh 2 − x y ∂x ∂y ∂t 1 − µ2 ∂x h/2 ∂u ∂v αT ∂ 1 ∂ Eh + + E θdz = p1 , 2(1 + µ) ∂y ∂y ∂x 1 − µ ∂x

−h/2

∂v 1 ∂ ∂u ∂2 v Eh − ky w + µ − kx w ρh 2 − ∂y ∂x ∂t 1 − µ2 ∂y

−

h/2 ∂v ∂u 1 ∂ αT ∂ Eh + + E θdz = p2 , 2(1 + µ) ∂x ∂x ∂y 1 − µ ∂y −h/2

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8 Mathematical Approaches to Coupled Termomechanical Problems

∂ 2 ∂ 2 ∂2 w 1 ∂w ∂w k Eh ψ x + + k Eh ψy + − ρh 2 + 2(1 + µ) ∂x ∂x ∂y ∂y ∂t Eh ∂u ∂v − k x w + (ky + µk x ) − ky w + (k x + µky ) ∂x ∂y 1 − µ2 EαT (k x + ky ) 1−µ

h/2 θdz = q1 ,

−h/2

∂ψy ∂ψ x 1−µ ∂ ∂ψ x ∂ψy ∂ ρh3 ∂2 ψ x D + µ − D + + − 12 ∂t2 ∂x ∂x ∂y 2 ∂y ∂y ∂x h/2 k2 Eh αT ∂ ∂w ψx + + E zθdz = 0, 2(1 + µ) ∂x 1 − µ ∂x −h/2

∂ψy ∂ψy ∂ψ x ∂h3 ∂2 ψy ∂ψ x 1−µ ∂ ∂ D +µ − D + + − 12 ∂t2 ∂y ∂y ∂x 2 ∂x ∂x ∂y h/2 k2 Eh αT ∂ ∂w ψy + + E zθdz = 0, 2(1 + µ) ∂y 1 − µ ∂y −h/2

∂θ − λq ∆θ+ ∂t ∂ψ x ∂ψy EαT T 0 ∂ ∂u ∂v + − (k x + ky )w + z + = q2 , (8.15) 1 − µ ∂t ∂x ∂y ∂x ∂y where u, v, w, ψ x , ψy , θ are known functions. The attached boundary conditions will be defined later. The following standard notation is used: T 0 - initial temperature; E = E(x, y) > 0 - Young modulus; µ - Poisson’s coeﬃcient (0 ≤ µ ≤ 0.50; ρ = ρ(x, y) > 0 - material density; λq - heat transfer coeﬃcient; n˜ ε - specific heat capacity corresponding to a constant deformation tensor; αT - linear thermal expansion coeﬃcient; h/2 z 1 1 = f2 dz; 2 h h k n˜ ε (1 + ε)

−h/2

f (z, h) - function describing a distribution of tangent stresses along the thickness of a shell; h = h(x, y) - variable shell thickness; θ = θ(x, y, z, t) - shell temperature increase; u = u(x, y, t), v = v(x, y, t), w = w(x, y, t) - components of the displacement vector of the point (x, y) in the mean surface and a deflection at the time instant t; ψ x (x, y, t), ψy (x, y, t) - rotation angles of the normal to the mean surface in the planes xz and yz, respectively; k x , ky - initial curvatures corresponding to the coordinates x, y; p1 , p2 , q1 - external load intensities along the axes x, y, z; q2 - specific heat capacity power of the sources situated within the shell.

8.2 On the Solution of a Coupled Thermo-mechanical Problem

715

8.2.2 Method The system (8.15) is considered for t ≥ 0 in a three-dimensional space Ω = Ω1 × (−h/2, h/2), where Ω1 ⊂ 2 is the bounded space with a piece-wise boundary (all of the functions appearing in (8.15) are also assumed to be suﬃciently diﬀerentiable). In order to reduce (8.15) to a diﬀerence-operator equation we introduce the following Hilbert spaces H xy = L2 (Ω1 ), H xyz = L2 (Ω), spanned by measurable functions having integral square norm and defined in the spaces Ω1 and Ω, respectively. We also use the notation H1 = H xy ⊕ H xy , H2 = H xy , H3 = H xy ⊗ H xy , and H4 = H xyz . In addition, we introduce the following diﬀerential and matrix-diﬀerential expressions (T denotes transposition): ⎛ 1 ∂ µ ∂ ∂ ∂ ∂ ∂ 1 ∂ ∂ ⎞ ⎟⎟ Eh ∂x + 12 ∂y Eh ∂y 1 ⎜⎜⎜ 1−µ 1−µ ∂x Eh ∂y + 2 ∂y Eh ∂x ⎟ ⎜⎝ µ ∂x K˜ 1 = − ⎠= ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ ⎟ 1 + µ 1−µ ∂y Eh ∂x + 2 ∂x Eh ∂y 1−µ ∂y Eh ∂y + 2 ∂x Eh ∂x 1 − 2(1 − µ) K˜ 2 = − ⎛ ⎜⎜ K˜ 3 = − ⎜⎜⎜⎝ 1+µ − 2

∂ ∂x ∂ ∂y

∂ ∂x ∂ ∂y

∂ 0 Eh ∂x 0 0

⎛ ∂ ⎜⎜⎜ ∂x 1 ⎜⎝ ∂ − 0 2(1 + µ) − ∂y

∂ ∂y

∂ ∂y ∂ ∂x

⎞ ⎛ ∂ ∂ ⎞ ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ ∂x − ∂y ⎟⎠ Eh ⎜⎝ ∂ ⎠, ∂ ⎟ ∂y

∂x

∂ ∂ ∂ ∂ Eh 2 k2 2 Eh + Eh + + k + 2µk k k , x y x y 2(1 + µ) ∂x ∂x ∂y ∂y 1 − µ2

1−µ ∂ ∂ ∂ ∂ ∂x D ∂x + 2 ∂y D ∂y ∂ ∂ ∂ ∂ µ ∂y D ∂x + 1−µ 2 ∂x D ∂y

⎞ ⎛ k2 Eh ∂ ∂ ∂ ∂ ⎟ µ ∂x D ∂y + 1−µ ⎟⎟ + ⎜⎜⎜⎜⎜ 2(1+µ) 2 ∂y D ∂x ⎟ 1−µ ∂ ⎠ ⎝ 0 ∂ ∂ ∂ ⎟ ∂y D ∂y + 2 ∂x D ∂x

0 k2 Eh 2(1+µ)

⎛ k2 Eh ⎛ ∂ ⎛ ∂ ∂ ⎞ ∂ ⎞ ⎟⎟⎟ ⎜⎜⎜ ∂x ⎟⎟⎟ ⎜⎜⎜ 2(1+µ) − ∂y 1 − µ ⎜⎜⎜ ∂x ∂y ⎜⎝ ∂ ∂ ⎟⎠ D ⎜⎝ ∂ − ⎠ + ⎜⎜⎝ ∂ ⎟ − ∂y ∂x 0 2 0 ∂y ∂x λq ∂2 ∂2 ∂2 , + + K˜ 4 = − T 0 ∂x2 ∂y2 ∂z2 T ∂ 1 ∂ P˜ 12 = Eh(k Eh(k + µk ), + µk ) , x y y x ∂y 1 − µ2 ∂x 1 ∂ ∂ , Eh(k , P˜ 21 = − Eh(k + µk ) + µk ) x y y x ∂x ∂y 1 − µ2 ∂ 2 1 ∂ 2 ˜ P23 = − k Eh, k Eh , 2(1 + µ) ∂x ∂y T 1 ∂ ∂ k2 Eh , k2 Eh P˜ 32 = , 2(1 + µ) ∂x ∂y EαT ∂ ∂ ˜ , . Q=− 1 − µ ∂x ∂y

∂ 0 D ∂x 0 0

∂ ∂y

⎞ ⎟⎟⎟ ⎟⎟⎠ = 0

k2 Eh 2(1+µ)

⎞ ⎟⎟⎟ ⎟⎟⎠ ,

716

8 Mathematical Approaches to Coupled Termomechanical Problems

Analysis of the expressions K˜ 1 , K˜ 2 , K˜ 3 , K˜ 4 leads to the conclusion that they are formally self-adjoint. The minimal operators generated by those operators in the spaces H1 , H2 , H3 , H4 are positive well defined. We are going to attach to the system (8.15) the boundary conditions in such a way that the operators K1 , K2 , K3 , K4 generated by the expressions K˜ 1 , K˜ 2 , K˜ 3 , K˜ 4 and the mentioned boundary conditions in the spaces H1 , H2 , H3 , H4 will be self-adjointed and positive defined (observe that, for instance, Dirichlet type conditions satisfy our requirement). The maximal operators generated by the expressions P˜ 21 , P˜ 12 , P˜ 23 , P˜ 32 , Q˜ serve as the operators P21 , P12 , P23 , P32 , Q. The operators P21 , P23 , Q map from H xy ⊕ H xy into H xy , whereas the operators P12 , P32 map from H xy into H xy ⊕ H xy . Note, that expressions P˜ 21 and P˜ 12 , as well as P˜ 23 and P˜ 32 are formally self-adjoint to each other. The expression formally self-adjoint to Q has the form T ∂ ∂ αT E, E . 1 − µ ∂x ∂y By Gi we denote the product operators on the spaces Hi (i = 1, 2, 3, 4) with the functions ρh, ρh, ρh3 /12, n˜ ε (1+ε)/T 0 , respectively. It is obvious, that Gi are bounded and possess everywhere bounded inverse operators. In addition, we introduce the following bounded operators from H xy into H xyz : the operator B1 assigns to each function f ∈ H xy the same function f , but considered as an element from H xyz , B2 the product operator of the functions from Hzy with z. It is not diﬃcult to check that self-adjoint operators B∗1 , B∗2 to the operators B1 and B2 : H xyz → H xy map according to the formulas: h/2 ∗ g(x, y, z)dz, B1 g(x, y, z) = −h/2

B∗2 g(x, y, z)

h/2 =

zg(x, y, z)dz,

g ∈ H xyz .

−h/2

Let us denote S2 =

EαT (kz + ky )B1 , 1−µ

S 1 = B1 Q1 ,

S 3 = B2 Q1 .

S 1 is regarded as the operator mapping from H1 into H4 , and S 3 is regarded as the operator mapping from H3 into H4 . Since B1 , B2 are bounded operators, then it follows from the results given in the monograph ([574], Chapter 8, §1) that S 1∗ = Q∗1 B∗1 , S 3∗ = Q∗1 B∗2 . Let us denote by U, Ψ the column vectors U = (u, v)T , Ψ = (ψ x , ψy )T and using the operators, the homogeneous system (8.15) can be presented in the following form introduced earlier: G1

d2 U(t) + K1 U(t) + P12 W(t) + S 1∗ θ(t) = 0, dt2

8.2 On the Solution of a Coupled Thermo-mechanical Problem

G2

717

d2 w(t) + K2 w(t) + P21 U + P23 Ψ + S 2∗ θ(t) = 0, dt2

d2 Ψ (t) + K3 Ψ (t) + P32 w + S 3∗ θ(t) = 0, dt2 d d G4 θ(t) + K4 θ(t) − [S 1 U(t) + S 2 w(t) + S 3 Ψ (t)] = 0. (8.16) dt dt Observe that for the system (8.16) the spaces defining the operators are linked by the expressions G3

D(P12 ) ⊃ D(K21/2 ),

D(P21 ) ⊃ D(K11/2 ),

D(P23 ) ⊃ D(K31/2 ),

D(P32 ) ⊃ D(K21/2 ),

D(S i ) ⊃ D(Ki1/2 ) (i = 1, 2, 3),

and that the system (8.16) represents a particular case of the more general system d 2 ξi = −Ni ξi + Pi j ξ j − S i∗ θ0 2 dt j=1 n

Fi

F0

(i = 1, 2, . . . , n),

n n d dθ0 = −N0 θ0 + P jξ j + S jξ j dt dt j=1 j=1

(8.17)

with known functions ξi = ξi (t), θ0 = θ0 (t). here Fi , Ni are the positive defined selfadjoint operators on the Hilbert spaces Hi (i = 0, 1, 2, . . . , n), the Fi are bounded, the Pi j are linear and closed operators acting from H j into Hi (i, j = 1, 2, . . . , n), S i , Pi are linear and closed operators from Hi into H0 (i = 1, 2, . . . , n). The defining spaces of those operators satisfy the conditions D(Pi j ) ⊃ D(N 1/2 j ), D(Pi ) ⊃ D(Ni1/2 )

D(S i ) ⊃ D(Ni1/2 ), (i, j = 1, 2, . . . , n).

Let H = H1 ⊕H2 ⊕· · ·⊕Hn , and by N, F we denote the operators generated in H

by matrices with the diagonals Ni , Fi (i = 1, 2, . . . , n), respectively (other elements are equal to zero). By P˜ we denote the operator generated by the matrix (Pi j )ni, j=1 , S˜ = (S 1 , . . . , S n ), P˜ 0 = (P1 , . . . , Pn ). Finally, denoting by ξ˜ the column of unknown functions (ξ1 , . . . , ξn )T we reduce (8.17) to the system of two equations F F0

d2 ξ˜ = −N ξ˜ + P˜ ξ˜ − S˜ ∗ θ, dt2

dθ0 d ˜ = −N0 θ0 + P˜ 0 ξ˜ + Sξ . dt dt

(8.18)

˜ θ = F 1/2 θ0 We introduce the following change of variables in (8.18): ξ = F 1/2 ξ, 0 −1/2 −1/2 −1/2 ˜ −1/2 −1/2 PF and we denote K = F NF ,P = F , M = F N0 F −1/2 , P0 = F −1/2 P˜ 0 F −1/2 , S = F −1/2 S˜ F −1/2 . The system (8.18) then takes the form

718

8 Mathematical Approaches to Coupled Termomechanical Problems

d2 ξ = −Kξ + Pξ − S ∗ θ, dt2 dθ d = −Mθ + P0 ξ + (S ξ), 0 ≤ t < ∞. (8.19) dt dt The system (8.19) is analysed in the space H ⊕ H0 . In this system K, M are the positive defined self-adjoint operators in the spaces H , H0 , respectively. P is the linear closed operator on H , and S , P0 are linear closed operators form H

into H0 . In addition, the following relations are specified: D(P) ⊃ D(K 1/2 ), D(S ) ⊃ D(K 1/2 ), D(P0 ) ⊃ D(K 1/2 ). Observe, that the system (8.19) has been considered already in reference [128] but from a diﬀerent point of view. We propose now a diﬀerent approach to solve (8.19) than that given in [128]. In order to reduce (8.19) to a first order equation let us introduce in the space H = H ⊕ H ⊕ H0 the matrix operator ⎞ ⎛ ⎜⎜⎜ 0 E 0 ⎟⎟⎟ ⎟ ⎜⎜⎜ A0 = ⎜⎜ −K + P 0 −S ∗ ⎟⎟⎟⎟ ⎠ ⎝ P0 S −M (here and below E denotes the identity operator from a corresponding space). By η we denote the column vector η = (ξ, ξ , θ)T . Then the system (8.19) is reduced to the following equation (8.20) η = A0 η. Let K˜ be the operator on H, defined by the matrix ⎛ ⎞ ⎜⎜⎜ K 0 0 ⎟⎟⎟ ⎜ ⎟ K˜ = ⎜⎜⎜⎜ 0 E 0 ⎟⎟⎟⎟ . ⎝ ⎠ 0 0 E The change ζ = K˜ −1/2 η transforms (8.20) into equation ζ = A1 ζ,

(8.21)

where A1 = K˜ 1/2 A−1/2 . Denote by A2 , A3 the operators, defined by the matrices 0 ⎞ ⎞ ⎛ ⎛ ⎜⎜⎜ 0 ⎜⎜⎜ K 1/2 0 ⎟⎟⎟ 0 0 0 ⎟⎟⎟ ⎟ ⎟ ⎜ ⎜ 0 −S ∗ ⎟⎟⎟⎟ , A3 = ⎜⎜⎜⎜ PK −1/2 0 0 ⎟⎟⎟⎟ . A2 = ⎜⎜⎜⎜ −K 1/2 ⎠ ⎠ ⎝ ⎝ −1/2 0 S −M 0 0 P0 K It is easy to check that A1 = A2 + A3 . The conditions applied to P, P0 , lead to the conclusion that A3 is a bounded operator. On the other hand, the properties of the operators S , M, K allow us to show that A2 is a dissipative operator ([367], Chapter 1, §4), (i.e., for all x ∈ D(A2 ) Re(A2 f, f ) ≤ 0). The following lemma leads to the conclusion, that the closure A¯ 2 of the operator A2 is a maximal dissipative operator. Lemma. The operator A¯ 2 possesses everywhere defined and bounded inverse.

8.3 On the Solvable Operators Generated by Uniformly Correct Problems

719

Proof. Observe first that the operator S K −1/2 is bounded. The equation (K −1/2 × S ) = S ∗∗ K −1/2 ([574], Chapter 8, §4) yields the bound of (K −1/2 S ∗ )∗ . Therefore, the operator K −1/2 S ∗ allows for a continuous extension into the whole space. In what follows the operator-matrix ⎞ ⎛ ⎜⎜⎜ −K −1/2 S ∗ M −1 S K −1/2 −K −1/2 K −1/2 S M −1 ⎟⎟⎟ ⎟⎟⎟ ⎜⎜⎜ K −1/2 0 0 ⎟⎟⎠ ⎜⎜⎝ 0 M −1 M −1 S K −1/2 ∗ ∗

defines a bounded operator on H, and hence it allows for a continuous extension into the whole space H. It can be checked directly that its product (of an arbitrary order) with the operator-matrix defining the operator A2 , yields the unit operator matrix. Hence the lemma is proved. Taking into account the maximal dissipative extension of the operator A¯ 2 , the bound of A3 and the results given in reference ([367], Chapter 1, §4), one obtains the following theorem. Theorem. The Cauchy problem for equation (8.21), where the operator A1 is replaced by its closure, is uniformly correct.

8.3 On the Solvable Operators Generated by Uniformly Correct Problems 8.3.1 Introduction Coupled problems of plate thermomechanics (Timoshenko and Kirchhoﬀ types models) [43] can be reduced in a Hilbert space to the following first order diﬀerentialoperator equation (8.22) y = Ay = f (t) with the uniformly correct Cauchy problem for the associated homogeneous equation [370]. In this section the boundary conditions for equation (8.22) such that it has only one solution are formulated. 8.3.2 Method Let H be a Hilbert space with the scalar product (·, ·) and norm · ; L2 (H; 0, b) is a space spanned by the measurable functions having integral square norm on [0, b] ⊂ and defined in the space H; A is a closed linear operator in H, being the carring operator of a C0 class semi-group. The latter condition is equivalent to the uniform correctness of the Cauchy problem for the equation y Ay = 0 ([367], page 64). On the set D(L ) composed of strongly diﬀerentiable functions y(t) with values in D(A) and satisfying the property l[y] = y − Ay ∈ L2 (H 0, b), the operator L is

720

8 Mathematical Approaches to Coupled Termomechanical Problems

defined: L y = l[y]. A closure A of the operator L will be called a maximal operator generated in L2 (H; 0, b) by the expression l. Analogously, a minimal operator L0 is defined as the closure of L defined as a contraction into the functions y(t) ∈ D(L ) set satisfying the condition y(0) = y(b) = 0. Applying the proposition given in ([232], page 163), an operator M is said to be solvable, if L0 ⊂ M ⊂ L and there exists M −1 , which is bounded and defined on the whole space L2 (H; 0, b). Let U(t) be the semi-group generated by A, and let the following norm on H be introduced: b 2 x− = (U(s)x, U(s)x)ds ≤ α(b) x2 , x ∈ H. 0

Denoting by H− the compliment of the norm H, observe that the inequality b U(t)x2− =

b U(s)U(t)2 ds ≤ c

0

U(s)x2 ds = c x2− 0

˜ implies that the semi-group U(t) expands continuously up to the semi-group U(t) in H− . Note also that if a sequence {xn } in H converges to x0 ∈ H− in H− , then the sequence {U(t)xn } is the fundamental one in L2 (H; 0, b) and is convergent to the ˜ ˜ limit U(t)x 0 . Hence, the function U(t)x0 belongs to L2 (H; 0, b). Owing to the results obtained here and the results given in ([367], page 166) regarding the solution of the non-homogeneous equation (8.22), the following description of the domain of existence of the L operator is given. Lemma. The operator L space D(L) is composed of the functions defined via the relation t ˜ U(t − s) f (s)ds, (8.23) y(t) = U(t)x 0+ 0

where: x0 ∈ H− , f ∈ L2 (H; 0, b), and Ly = f . In addition, one more norm in H is introduced via the relation (x∗− )2

b =

b (U ∗ (s)x, U ∗ (s)x)ds =

0

(U(s)U ∗ (s)x, x)ds ≤ β(b) x2 ,

(x ∈ H),

0

and the compliment in H with respect to this norm is denoted by H−∗ . Observe that the space H−∗ can be treated as the negative one with respect to H0 = H. A corresponding positive space is denoted by H+∗ ([100], page 46). The operator Ib , being the extension of the operator b U(s)U ∗ (s)ds, 0

8.3 On the Solvable Operators Generated by Uniformly Correct Problems

721

onto H−∗ maps H−∗ into H+∗ continuously and mutually uniquely. Proceeding in a similar way, one may prove that U ∗ (t) allows for the extension of ∗ ˜ U (t) onto H−∗ and the operator x → U˜ ∗ (t)x maps H−∗ continuously into L2 (H; 0, b). Therefore, the corresponding conjugated operator b f →

U(s) f (s)ds

(8.24)

0

represents a continuous mapping from L2 (H; 0, b) into H+∗ . Furthermore, defining as f the following function f (s) = U˜ ∗ (s)Ib−1 x, where x ∈ H+∗ , one may conclude that (8.24) maps L2 (H; 0, b) onto the whole space H+∗ . Now using a simple change of variables it is not diﬃcult to establish that the operator b U(b − s) f (s)ds f → 0

˜ maps L2 (H; 0, b) continuously into H+∗ . Assuming f (s) = U(s)x (x ∈ H− ) and ap˜ − s)U(s) ˜ ˜ ˜ plying the equality U(b = U(b), the relation U(b)x ∈ H+∗ is obtained, and ˜ hence the operator U(b) maps H− continuously into H+∗ . The following limiting maps γ1 : D(L) →→ H− , γ2 : D(L) → H+∗ are introduced for the functions y ∈ D(L) by the formulas γ1 y = y(0) and γ2 y = y(b), respectively. Owing to the relation (8.23) and the operators considered in the above, the following properties of γ1 , γ2 are yielded: (i) for any arbitrary elements h1 ∈ H− , h2 ∈ H+∗ , there exists a function y ∈ D(L), that γ1 y = h1 , γ2 y = h2 ; (ii) γ1 , γ2 are continuous on D(L) with the norm of L; (iii) the contraction γ1 onto the kernel KerL is the unique self map onto H− . To conclude, the quadrable (H− , H+∗ , γ1 , γ2 ) defines a space of limiting values of the operator L in the sense of the work [130] (see also the attached bibliography therein). Note also that the results obtained in reference [130] yield a description of the existence space of any solvable contraction of the operator L. THEOREM 8.3 An operator M is solvable if and only if there exists a bounded operator N: H+∗ → H− , that D(M) consists of only the elements y ∈ D(L), which satisfy the following condition

˜ y(0) = N y(b) − U(b)y(0) . Now let M be a contraction of L onto a set of functions y ∈ D(L), satisfying the condition S 1 y(0) = S 2 y(b), where: S 1 : H− → B0 , S 2 : H+∗ → B0 are bounded linear operators, and B0 is an arbitrary Banach space. Let the bounded operator C is introduced, C: H− → B0 via ˜ the relation C = S 1 − S 2 U(b). THEOREM 8.4 The operator M is solvable if and only if the operator C is invertible and the operator C −1 S 2 is everywhere defined.

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Zenisek A., Finite element methods for linear coupled thermoelasticity. Equadiﬀ. 5th Czech. Conf. Diﬀer. Equations and Appl., Bratislava-Leipzig 1981-1982, 387-390. Zhuravlev A.G., On the conditions of simple loading in non-homogeneous medium. Vestnik MGU, Series of Mathematics and Mechanics 1962, (6) 39-42. Ziegler G., Introduction to Theory of Construction Stability. Mir, Moscow, 1971. Zienkiewicz O.C., Finite Element Method (3-rd edition). McGraw Hill Book Company Ltd., London, 1989. Zimcik D.G., Tennyson R.S., Stability of circular cylindrical shells under transient axial impulsive loading. Proceed. AIAA/ASME/ASCE/AHS 20-th Structures, Structural Dynamics and Materials Conf., St. Louis. Apr. 4-6, 1979, 275-281. Zino I.E., Tropp E.A., Asymptotic Method in the Heat Transfer and Thermoelastic Problems. LGU Press, Saint Petersburg, 1978, in Russian. Zorski H., On a certain property of thermoelastic media. Bull. Acad. Polon. Sci., Ser. Techn. 1958, 6(6) 331-339. ˇ Zigalko Ju.P., Statics and Dynamics of Thin Plates and Shells with Local Loads. Soviet School of Young Scientists. Kazan’, 1983, 64-67, in Russian. ˇ Zuravilov A.A., Stability of orthotropic triangle plate. Izv. VUZ, Stroitielstvo i Architektura, No 11, 1985, 47-51, in Russian.

Index

adiabatic process, 147 algebraic equation, VII, 21, 191, 229, 233–235, 314, 328, 412, 431, 509, 513–515, 519, 622, 643, 644, 668, 671–673, 684, 702, 704 algorithm, V, VI, 1, 2, 5, 39, 44, 48–50, 53, 54, 133, 136, 149, 190, 191, 209, 232, 267, 272, 314, 317, 319, 320, 323, 325, 327, 329, 333, 342, 385, 395, 412, 496, 498, 508, 509, 515, 560, 614, 617, 619, 633, 642, 644, 645, 672, 674, 676, 677, 679, 680, 690, 691, 701, 702, 704 aluminium alloy, 61, 117 amplitude, 79, 292, 327, 335, 336, 338, 339, 345, 353, 356, 382–387, 399, 401, 415, 419, 427, 434–438, 440–442, 454, 455, 460, 466, 467, 469, 471, 473, 475, 478, 491, 492, 534, 539, 547, 555, 560, 619, 622, 623, 625, 626, 629–632 analytical expression, 185, 399, 413, 428, 457 Andronov-Hopf bifurcation, 625, 627 – cascade, 627 angle, 154, 155, 236, 237, 270, 292, 327, 328, 480, 482, 494, 498, 500, 501, 511, 522, 525, 526, 528, 530, 532, 534, 539, 542, 543, 550, 551, 555, 557–560, 564, 566, 567, 574, 575, 577–579, 581, 582, 585, 586, 592–595, 604, 606, 607, 610, 611, 617, 618, 620, 625–627, 631, 700, 703, 714 anti-phase direction, 418 approximation, V, 1, 6, 9, 21, 26–29, 45, 46, 48, 50, 130, 133–135, 144, 149, 151, 186–189, 191, 193, 194, 197, 198, 209–217, 230, 233–235, 270, 320, 322, 325, 326, 328–330, 353, 386, 389, 411, 414, 415, 446, 447, 457, 496, 510, 615,

616, 618, 620, 641, 668, 672, 677, 683, 686, 687 arc, 291, 336, 422, 424, 427, 497, 521, 534, 615, 616, 618, 619, 626 arc type cylindrical shell, 424 arcal – element, 506, 507, 510, 511 – segment, 534, 539, 542, 550, 555, 559, 560, 563, 573, 578, 585, 594, 598, 604, 608 assumption, VI, 21, 54, 109, 132, 138, 144, 147, 151, 152, 154, 160, 273, 274, 276, 283, 284, 298, 304, 396, 499, 507, 633, 634, 669, 707 asymptotic method, 268 axially symmetrical heating, 7 ball type condition, 238 band matrix, 672, 674, 676–678, 682, 683 beam type cylindrical shell, 422, 424 beam type cylindrical shell, 422–425 bending moment, 153, 494, 502, 520, 554 bifurcation zone, 620, 625, 626 biharmonic equation, VI, 267, 315, 319 bijectively, 707, 708 bilinear – form, 279 – transformation, 299 bisectrix, 494, 525, 526, 532, 534, 539, 542, 543, 551, 555, 562, 578, 579, 617, 620 body, 3, 7, 11, 12, 82, 89, 129, 130, 136, 144, 147, 154–156, 183–185, 267, 272, 276, 278, 287, 308, 336, 410, 508, 520, 633, 634, 640, 666, 702 boundary conditions, VI, VII, 1, 4, 8, 11, 13–15, 18, 19, 28, 30, 31, 33, 39, 50, 53, 55, 80, 83, 95, 103, 114, 117, 118, 133, 146, 147, 149, 150, 160, 167–173,

768

Index

175–177, 184–187, 210, 211, 213–215, 217, 218, 228, 230–236, 238, 244–249, 251–254, 256, 262, 264, 265, 269, 271, 275, 277–279, 291, 294, 296, 297, 305, 307, 311, 314, 315, 320, 324–326, 360, 361, 374, 395, 406, 409, 410, 413, 429–431, 458–461, 463, 464, 479–483, 494, 497, 499, 506, 508–512, 514–516, 522, 597, 598, 602, 603, 615, 625, 640, 641, 644, 645, 667, 669, 693, 695, 699–705, 714, 716, 719 bounded inverse, 708, 716, 718 Bubnov-Galerkin method, 7, 235, 270, 325, 326, 337, 397, 414, 415, 494–496 Bubnov-Vlasov method, 494 buckling, 152, 270–272, 274, 292, 335, 336, 361, 366, 373, 382, 386, 399, 523, 526, 532, 534, 542, 543, 585, 607 Cauchy – inequality, 302 – problem, VII, 270, 328, 329, 512, 617, 705, 712, 719 central – angle, 498, 564 – force, 13 chaotic – dynamic, VI, 614, 623 – state, 493 characteristic – heat time, 305, 306 – mechanical time, 79, 305, 306 characteristics, 2, 6, 8, 26, 39, 41, 45, 47, 57, 131, 132, 268, 273, 274, 283, 284, 297, 304–309, 323–325, 334, 336, 337, 359, 361, 366, 368, 369, 371, 373, 395, 396, 399, 400, 402, 403, 413, 415, 447, 450, 452, 453, 456, 457, 488, 493, 667 Chebyshev’s acceleration, 1, 41, 45, 46, 186, 192 Chladni’s figure, 495 circumferential stresses, 524, 525, 529, 533, 534, 536, 537, 541, 544, 546, 547, 549, 552, 553, 557, 565, 570, 571, 573, 575, 576, 580–583, 587–591, 607, 608, 611 civil, V, 151 clamping edge, 238 closed – cylindrical shell, VI, 270, 342

– linear operator, 719 closure, 706, 718–720 coeﬃcient, VII, 3, 4, 6, 9, 18, 26, 29, 33, 40, 44, 50, 53, 79, 114, 117, 130, 134, 138, 151, 154, 155, 157, 163, 166, 168, 170, 172, 175, 177–179, 183–185, 201, 203, 204, 206, 208, 211, 231, 232, 234, 236, 271–274, 276–278, 283, 285, 296, 297, 305–307, 309, 316, 317, 319, 320, 325, 329–331, 337, 360, 366, 369–375, 398–402, 406, 407, 409, 412, 425, 446, 447, 450, 456–458, 479, 483, 484, 495, 497, 498, 513, 514, 521, 614, 635, 641, 678, 680, 681, 683–686, 691, 692, 696, 699–701, 705, 714 combined – dynamical load, 338 – load, 338, 385–388 commutativity, 706, 709 compactness, 39, 135, 297 compatibility, 21, 57, 58, 149, 152, 167, 171–174, 176, 177, 209–211, 213, 215, 230, 234, 407, 616, 617, 700 compliment, 720 composite shell, 163 compressing, 270, 395, 458, 459, 462, 464, 465, 523, 534, 547, 563, 569, 572, 585, 604, 698 conical shell, VI, 284, 286, 296, 297, 304, 337, 373, 395, 454, 457, 458, 466, 483, 484, 496, 499, 509 conjugated operator, 707, 721 conservative elastic system, 2 constant coeﬃcient, 283, 320, 450, 497 continuity deformation, 290, 307 continuous – function, 22, 181, 685 – mapping, 721 – media, 5, 47 – medium, 5 – thermal load, VI – trilinear form, 299 continuously, 153, 602, 707, 708, 720, 721 continuously and mutually uniquely, 721 control, VI, 327, 337, 415, 493, 614, 619, 627, 630–632

Index convection, VI, 19, 55, 185, 278, 395, 397, 410, 479, 483, 488, 489, 491, 492, 508, 641 convectional heat transfer, 19 convergence, 1, 2, 4–6, 41, 43–46, 101, 134–136, 149, 186, 189–191, 193, 206, 320, 323, 326, 328, 331, 332, 342, 618, 623, 672, 692–694 – estimation, 206 convergent approximation, 21 coordinate, V, 6, 9, 18, 25, 46, 47, 55, 56, 114, 137, 149, 151–153, 155, 158, 163, 164, 184, 236, 268, 283, 285, 288, 291, 292, 296, 297, 322, 323, 326, 328, 335, 340, 342, 345–348, 352, 355, 362, 391–394, 402, 408, 411, 415, 456, 460, 493, 496–503, 506, 509, 614, 618, 620, 673–677, 681, 696, 714 copper, 327 corner, 1, 11, 13, 14, 18, 21, 28, 32, 51, 54, 58, 70, 149, 171–173, 239, 242, 246, 264, 425 – point, 149, 171–173, 239, 242, 246, 264 coupled problem, VI, 3, 5, 130, 146, 515, 633, 638–640, 646–648, 650–656, 658, 659, 719 coupling, VI, 1–3, 6–9, 39, 117, 121, 124, 126–129, 136, 138, 144, 146–149, 152, 167, 180, 182, 183, 304, 395, 399, 506, 634, 645, 646, 650, 652, 655, 656, 668, 677, 697, 699 coupling term, 395, 506 criterion of the membrane stresses, 458 critical – impulse, 271, 386 – time, 270, 375 – value, 338, 358, 386, 397, 401, 415, 418–422, 432–434, 436–438, 442, 446, 447, 450, 451, 454–458, 462, 464–467, 478, 480, 485, 520, 557, 560, 645, 702 cubicoid, 1, 5, 8, 13, 18, 19, 27, 42, 46, 55, 83, 86, 135 curvatures, 152, 161, 283, 285, 295, 400, 696, 714 cutted circle, 496 cylindrical – panel, 397, 415, 418, 449, 451

769

– shell, VI, 7, 267, 270, 271, 274–276, 305, 306, 309, 322, 325, 327, 334, 335, 338–340, 342, 345, 358, 360, 362, 364, 366, 368, 377, 382, 385, 386, 388, 397, 399, 418–425, 427, 434, 435, 448, 449, 452–454, 458, 464, 466–469, 474, 476–478, 484–488, 492, 496, 515 cylindrical – – shell, 768 cylindrical – shell, 767

damping, V, 6, 11, 304, 329, 330, 332, 333, 397, 401, 414, 415, 420, 458, 462–464, 497, 509, 561, 614 damping coeﬃcient, 329, 401, 458, 497, 614 deflection, VI, 48, 49, 151–153, 159, 167, 232, 234, 236, 237, 239–243, 250, 253–255, 257–260, 262–265, 270, 294, 322, 324–326, 328, 329, 332, 333, 335–342, 344–346, 352, 353, 359, 361, 362, 364, 368, 372–385, 387, 389–393, 396–400, 406, 407, 409–412, 414–419, 421, 424–427, 429–432, 434–436, 438–440, 442, 443, 446, 448–452, 454, 455, 457, 459–462, 464–466, 471, 473–478, 480–483, 488–499, 501, 507, 508, 514, 515, 519, 520, 523–527, 529, 531, 532, 534, 536, 537, 539, 542–545, 550–553, 555, 558–564, 566, 569–571, 573, 577, 578, 584–586, 590, 592–594, 597–599, 601–606, 608–610, 613, 617, 622, 631, 632, 639, 643–656, 658, 700, 702–704, 714 deformation – fields, 2, 8, 9, 121, 136, 146, 148, 267, 395, 506, 633, 645, 668 – tensor, 401, 498, 635, 636, 638, 666, 714 density, 9, 19, 184, 185, 234, 277, 401, 402, 410, 415, 432, 437, 441, 497, 498, 635, 640, 684, 714 derivative, 4, 10, 17, 22, 25, 26, 87, 114, 117, 135, 152, 153, 159, 160, 186, 228–230, 234, 279, 282, 283, 288, 298, 313, 321, 328, 336, 372, 373, 510, 650–656, 658, 659, 677, 680, 683, 687–689, 698, 701, 702, 711, 713

770

Index

diﬀerence – approximation, 1, 9, 21, 26, 29, 130, 133, 510, 616, 672, 687 – schemes, VI, 4, 5, 8, 47, 149, 194, 202, 315, 320, 321, 331 Diﬀerential Equation, 636 diﬀerential equation, V–VII, 1–4, 6–9, 11, 13, 14, 17, 21, 40, 46, 48, 130, 132–134, 149, 159, 166, 167, 186, 283, 297, 300, 320, 325, 326, 328, 329, 337, 395, 396, 399, 402, 406, 407, 409–412, 415, 457, 493, 494, 497, 506, 509, 512, 521, 560, 618, 633, 636, 638, 639, 641, 643, 644, 668, 669, 671, 679, 680, 684, 699, 705 dimensionless unite power, 113 dimensions, 2, 323, 359, 508 Dirichlet model, 41 Dirichlet’s homogeneous boundary conditions, 4 discontinuous coeﬃcient, 4 displacement, 4, 5, 9, 11, 13, 15, 53, 54, 59–63, 65–82, 87–89, 92, 94, 98, 104, 109, 113–115, 117, 121, 127, 134, 136–139, 142, 144–146, 148, 152, 159, 160, 167, 277, 278, 283, 285, 287, 291–293, 334–336, 353, 382, 385, 397, 400, 402, 405, 406, 409, 410, 412, 484, 485, 496–500, 504–506, 511, 518, 520, 521, 543, 551, 560, 567, 568, 574, 604, 614, 634, 646, 696, 699, 714 dissipation energy, 11, 12 dissipative operator, 718 dissipative operator, 718 distribution of temperature, 4, 138, 139, 145, 668 domain, 705, 707, 712, 720 Duhamel-Neuman – hypothesis, 284, 403, 635 – law, 130 Duhamel-Neuman’s dependence, 13 durability, V, 2 dynamic – behaviour, 7, 18, 53, 55, 134, 138 – problems, 1, 3, 4, 8, 48, 53, 57, 58, 79, 617 edge, 7, 8, 11, 13, 14, 18–20, 25, 27, 29, 32, 39, 51, 53–55, 57, 58, 60, 68–70, 79, 80, 87, 127, 146, 148, 153, 159, 167, 168,

170, 171, 174, 177, 180, 194, 218, 227, 238, 328, 360, 366, 398, 399, 430, 446, 471, 492–496, 522, 560, 602, 639, 713 eigenfunction, 315–317 elasticity – potential, 12 – problem, VI, 1, 4–8, 30, 40, 53, 121, 133, 278, 279, 304, 332, 372, 396, 646 electrical conductivity, 327 electronic device, V, 2 electrostatic, 684 electrotechnic, V, 151 elliptic – equation, V, VI, 1, 49, 149, 684 – paraboloid shape, 496 energy – estimation, 29, 33 – inequalities, 4 engineering, V, 151, 496 engineers, V, 151 entropy flux, 10, 13, 15, 148 – component, 13 epicycloidal plate, 496 equilibrium – equation, 160, 290, 293, 295, 404, 496, 497, 637 – position, 271, 353 estimations, 29, 30, 36, 297, 332, 709, 710 Euler’s system, 14 excitation, V, 7, 59, 87, 88, 109, 114, 152, 258, 261–263, 267, 269–271, 273, 275, 277, 279, 283, 285, 307, 313, 314, 320, 325, 327, 336, 398, 402, 436–438, 441, 443, 475, 478, 517, 519, 617, 622, 623, 626, 627, 629–632 existence, VII, 4, 163, 267, 275, 279, 297, 300, 304, 314, 705, 711, 713, 720, 721 experimental – data, 274, 327, 396, 447, 496 – investigation, 270–272, 274, 307, 494, 633 explicit method, 1, 45, 46, 189, 192 exponent, 274, 304, 356, 357, 620, 631 external – pressure, 269, 270, 275, 342, 343, 360, 362, 366, 386–390, 397, 560 – process, 135 Feigenbaum

Index – scenario, 627 – type zone, 619 finite diﬀerence – approximation, 510, 616, 672 – equation, VI, 313, 395, 683 – method, VI, 2, 4–6, 48, 57, 149–151, 186, 232–235, 312, 320, 326, 342, 411, 493, 496, 497, 509, 598, 615, 618, 620, 641, 671, 679, 694–696 first order boundary condition, 184, 185 five halfwaves, 460 flat, 3, 5, 296, 388, 697 flexurable – conical shell, VI, 395, 458, 466, 483 – sectorial shell, VI, 493 – sectorial shell, VI flow density, 401, 415, 498, 640 fluid, 5, 185 flying object, V, 2 force, 2, 9, 12, 13, 18, 38, 50, 53, 57, 59, 61, 62, 66–69, 71, 72, 76–80, 87, 89, 90, 92, 94, 114, 130–132, 135, 138, 147, 153, 157–160, 168, 180, 183, 232, 234, 254, 257, 267, 269–271, 285, 287, 288, 295, 322, 325, 327, 330, 335, 336, 344, 345, 350–352, 368, 371, 373–376, 378, 382, 385, 389–394, 396, 397, 401, 403, 405, 406, 409, 411, 412, 417, 432, 478, 480, 481, 483, 494, 496–499, 501–505, 508, 515, 520, 523, 525, 526, 614, 625, 635–637, 644–646, 697, 700 formula, V–VII, 1–4, 8, 9, 22, 31, 38, 39, 43–45, 47, 48, 50, 51, 53, 54, 76, 80, 99, 109, 114, 124, 131, 135, 144, 146, 149, 152, 153, 155–157, 160, 167, 188–193, 233, 267, 269, 273, 275–279, 283, 287, 292, 297, 298, 308, 314–316, 320, 322–324, 329, 332, 336–338, 361, 364, 373, 395, 396, 398, 399, 403, 446, 479, 504, 507–509, 512, 515, 560, 603, 616, 633, 635, 640, 644, 672, 677, 678, 683, 685, 687, 690, 691, 702, 705–708, 712, 716, 719, 721 Fourier’s series, 3, 4 fourth order boundary condition, 185 free – edge, 19, 53, 55, 57, 68, 69, 79, 80 – support, 167, 409, 413, 496, 640

771

frequency, 79, 305, 306, 337, 339, 342, 345, 373, 379, 400, 401, 436–439, 441, 443, 456, 467, 468, 470, 473–475, 478, 494, 497, 521, 555, 614, 617, 619, 620, 623, 627, 629–632 functional, 8, 10, 13, 14, 21, 131, 168, 170–172, 174–176, 178, 181, 275, 297, 328, 337, 338, 397, 686–688, 698 fundamental function, 236, 238–242, 244–249, 251–253, 259, 261 Gateaux derivative, 279, 282, 283 Gauss – formula, 160, 685 – reduction, 515, 671 generalized derivative, 10, 26 generalized Fourier’s law, 6 generalized mathematical, 4 geometrical – nonlinear shells theory, 269 – nonlinearity, 396 Germain-Lagrange equation, VII, 705 Green’s – function, 3, 297 – operator, 300 Gronwille’s lemma, 302 half-circle plate, 496 Hamilton-Ostrogradski’s principle, 12 harmonic and analytical functions, 3 harmonic loading torque, 493 heat – capacity, 307, 401, 498, 635, 714 – conductivity equation, VI, 6, 53, 82, 86, 98, 109, 117, 633, 640, 641, 643, 644 – flow, VI, 272, 274, 278, 296, 324, 325, 377, 379, 395, 397, 399–401, 415, 420, 428, 432, 440, 442, 443, 446–448, 454, 455, 464, 467, 473, 475, 476, 478, 498, 508, 596, 597, 602, 603, 636, 640 – flux, 15, 19, 82, 109 – impact, 7, 8, 19, 55–57, 83, 87, 102, 103, 109, 114, 117, 121, 122, 124, 132, 136–138, 267, 268, 273, 274, 338, 366, 368, 373, 382, 389, 400, 412, 418, 420, 421, 424, 426, 434–436, 446, 454–456, 480, 484, 488, 491, 595, 602, 603, 633, 641 – loads, VI, 2, 395, 488

772

Index

– source, V, VI, 3, 5, 8, 9, 13, 19, 53, 55, 82, 86, 87, 109, 113–115, 126, 127, 150, 152, 217, 218, 227, 257, 277, 396, 397, 399, 635, 684 – stream density, 185, 402, 415, 432, 437, 441, 684 – transfer, VI, VII, 6, 19, 48–50, 53, 55, 58, 84, 149–152, 183–186, 194, 209, 253, 267, 268, 272–274, 277, 296, 304, 306, 307, 309, 313, 320, 323, 324, 366–368, 395–398, 409, 410, 412, 430, 431, 456, 457, 479, 483, 484, 488, 491, 492, 497, 499, 506, 508, 510, 512, 595–598, 602, 603, 635, 666, 669, 671, 684–686, 696, 714 – wave, 127 Heaviside’s function, 117, 147 Hilbert and Sobolev’s space, 5 Hilbert’s space, 10, 135, 297 homogeneous, VII, 3, 4, 6, 11, 40, 130, 135, 149, 153, 162–164, 167, 168, 171, 183, 202, 279, 294, 305, 310, 329, 337, 388, 494, 495, 497, 513, 521, 671, 696, 699, 700, 702, 705, 707, 712, 716, 719, 720, 770 Hook’s law, 154, 163, 284 hybrid boundary condition, 410, 429, 458, 459, 464, 511, 695 hybrid-type variational equation, VI hyperbolic equation, 1, 4, 30 hyperplane, VII, 671, 673–675 I-shaped plate, 495 impact, 1, 2, 4, 6–8, 19, 31, 55–57, 83, 87, 89, 102, 103, 109, 114, 117, 119, 121, 122, 124–126, 132, 136–139, 141, 142, 144, 147, 267, 268, 270, 273, 274, 338, 366, 368, 372, 373, 382, 389, 390, 400, 412, 415, 418, 420, 421, 424, 426, 432, 434–436, 446, 454–456, 480, 484, 488, 491, 595, 602, 603, 633, 641, 704 imperfection, 152, 292, 293, 295, 296, 382–386, 388 implicit – method, 1, 41, 44–46, 186 – scheme, 47 impulse, 6, 269, 271–274, 304, 325–327, 336–338, 340, 342, 343, 353–358, 379,

381, 382, 385, 386, 388, 390, 520, 532, 534, 539, 547, 548, 595, 633, 645, 669 inequality, 36, 38, 39, 48, 152, 188, 192, 206, 207, 276, 302, 323, 331, 332, 709, 710, 720 inertial coeﬃcient, 374, 375 initial – boundary problem, 3, 4 – condition, VI, 8, 18, 32, 53, 57, 79, 95, 109, 114, 133, 135, 137, 138, 144, 275, 278, 297, 311, 314, 324, 325, 329, 336, 337, 410, 413, 479, 506, 508, 512, 515, 518–520, 560, 577, 598, 603, 615, 633, 639, 645, 668, 669, 707 instability, 53, 54, 271, 274, 275, 334, 336–338, 340, 475, 521 integral – identity, 1, 21, 22 – square norm, 719 – square norm, 715 integro-diﬀerential system, 668 interaction, 5, 6, 11, 267, 270, 327, 366, 382, 392, 397, 631, 698 isothermal energy, 12 isothermic theory, 273 isotropic medium, 3, 685 iterative method, V, 1, 41, 45, 46, 59, 134 jump, 271, 340, 344, 346, 396, 397, 427, 456, 560, 607, 617, 625, 698, 703 kinematic boundary conditions, 291 kinetic energy, 12, 287, 288, 290 Kirchhoﬀ-Love – hypothesis, 151, 163, 273, 304, 402, 642, 696 – kinematic model, VI, 633, 634 Kirchhoﬀ-Love’s method, 60 Krylov-Bogolubov method, 7 L-shaped plate, 495 Lam´e coeﬃcient, 163 Landau scenario, 627 Laplace operator, 183, 683 Laplace operator, 371, 638 Laplace’s transformation, 3 lemma, 299, 300, 302, 303, 706, 708–712, 718–720, 771

Index linear – algebraic equation, VII, 412, 514, 622, 671–673, 684, 702, 704 – theory, 132, 151–153, 270, 273, 275, 334, 402, 494, 497, 499, 595, 633, 634 – thermal expansion, 276, 369, 400, 402, 456, 714 load-carrying ability, 269, 338 loads with infinite duration, 358 logarithmic scale, 356, 357 long circular cylinder, 6 longitudinal, VI, 270, 271, 285, 305, 322, 326, 330, 342, 344, 345, 359, 360, 391, 395, 401–403, 458, 459, 462, 464–471, 475, 478, 633 – compression load, 458 – load, VI, 395, 401, 402, 464, 466, 467, 469–471, 475, 478 Lusternik algorithm, 190 Lusternik’s algorithm, 44 magnetostatic, 684 matching, VII, 269, 494, 495, 671, 680 material’s non-homogeneity, 1, 11 mathematical – model, V, 87, 129, 150, 167, 272, 304, 333, 409, 506, 614 – physics, 3, 4, 671, 672 – treatment, V, VII Mathieu-Hill – problem, 269 matrix structure, 317, 671 mean surface, 400, 402, 407, 497, 498, 502, 503, 509, 634, 696, 697, 702, 714 measurable function, 715, 719 measurements, 2 mechanical – excitation, 109, 114 – impact, 1, 2, 7, 117, 119, 121, 124, 126, 137, 147 mechanical and temperature excitation, V, 152 mechanics, 5, 39, 297, 323, 671, 705, 719 medium’ temperature, 15 membrane stresses, 417, 418, 427, 452, 458, 558, 575, 594, 646, 769 mesh, VII, 1, 9, 10, 14, 21, 23, 26, 29, 30, 33, 35, 39, 46, 50, 57, 59, 186, 187, 194, 196, 206, 207, 209, 216, 228,

773

229, 232–234, 312–315, 322, 334, 342, 411, 493, 496, 509, 510, 522, 671, 677, 679–681, 686–694, 696, 701, 702 minimal deflection, 440, 492 modified Germain-Lagrange equation, VII, 705 modified relaxation method, 267, 330 modified Ruelle-Takens-Newhouse scenario, 619, 627, 628 moment, 102, 104, 153, 157, 161–164, 169, 257, 271, 272, 282, 285, 287, 295, 306, 322, 334–337, 343–345, 348, 349, 352, 368, 370, 371, 373, 374, 376, 377, 390–392, 394, 397, 401, 403, 405, 407, 430, 460, 480, 481, 483, 494, 499, 500, 502–505, 519, 520, 525, 530, 531, 538, 542, 546, 547, 551, 554, 598, 599, 601–603, 631, 632, 635–637, 644–646, 650, 668, 703, 704 motion, 2, 12, 47, 304, 306, 309, 314, 320, 324, 330, 334, 335, 337, 373, 375, 392, 440, 459, 503–505, 519, 520, 631, 632, 636, 638, 640, 643, 645, 668 multi-dimensional Poisson’s equation, 196 multi-dimensional summation, 30 negative deflection, 435, 436, 440, 459, 471, 473, 478, 483, 492, 603–605, 608 Newton method, VII, 329, 330, 671 Newton’s law, 5, 479 nodal point, 511 node, 22, 23, 25, 27, 29, 33, 36, 39, 46, 51, 53, 54, 206, 207, 211, 229–231, 233, 234, 312, 313, 317, 322, 323, 326, 342, 411, 412, 498, 508, 510–512, 515, 617, 644, 677–681, 687–691, 694, 701, 702 non-compressed, 167 non-dimensional – approximation, 447 – unit, 418, 434, 455, 456, 470, 488 non-dimensional time, 498 non-extended, 167 non-homogeneity, 1, 11, 773 non-isothermal, 87 non-linear dynamics, 8 non-stationary – problems, 47, 55, 618 – temperature field, 82 non-uniform stiﬀener, 494

774

Index

nonmonotonic dependencies, 274 norm, 4, 9, 19, 26, 39, 57, 114, 117–119, 121, 124, 126–129, 138, 141, 143, 144, 147, 153, 159, 163, 268, 273, 283, 284, 288, 291, 299, 321, 373, 397, 400–402, 410, 452, 496–499, 501, 505, 508, 516, 518, 520, 560, 604, 634, 684, 702, 706, 707, 714, 715, 719–721, 772 numerical method, 5, 133, 134, 150, 457, 493, 507–509, 616, 672, 684 one-dimensional – equations, 4 – relation, 30 operator, VII, 26, 50, 136, 158, 183, 192, 194, 196, 199–204, 210, 230, 275, 279–283, 296, 297, 299–302, 310, 315–317, 319–321, 330–332, 371, 411, 412, 510, 511, 638, 666, 672, 683, 691, 692, 705–710, 712, 715–721, 768, 771, 772 – convolution, 279 ordinary diﬀerential equation, 300, 326, 411, 412, 457, 641, 644, 668 orthotropic – material, 169, 284, 296 – shell, 285, 288 out-contour node, 230, 231, 234, 702 parabolic – equation, VII, 4, 82, 705 parallelepiped, VII, 18, 671, 677, 686, 689–691, 695 parallelogram, 494–496 partial diﬀerential equation, VII, 21, 411, 457, 497, 509, 618, 668, 671 partition, 414, 415, 447, 522, 596, 601, 602, 618–620, 622–625, 641, 694, 696 period, 7, 45, 150, 269, 272, 292, 305, 356, 373, 379, 418, 419, 421, 427, 441, 461, 475, 480, 622, 627–631, 647, 648, 651 phase – portrait, 628, 629 – shift, 401, 443 photoelasticity, 494 physical phenomena, 4, 40, 130 physically non-linear problem, 138 Pisman-Pakford method, 190 Poincar´e map, 628, 629

point heating source, 396, 398 Poisson equation, 41, 187, 194 polar-orthotropic sector plate, 496 polygonal plate, 494, 495 polymer, 399 Pomeau-Manneville, 627 positive defined operator, 299 post-critical – state, 336, 414, 415, 417–419, 427, 496, 626 – zone, 384 power spectra, 619, 620, 622, 624, 628, 631 pre-critical state, 414, 417, 418, 424, 434, 476, 478, 522, 544, 551, 552, 603 proof, 3, 29, 30, 136, 207, 209, 297, 300, 304, 706, 708, 709, 711–713, 719 quasistatic, 1–3, 5–8, 39, 48, 54, 95, 103, 104, 109, 131, 151, 267, 268, 271, 278, 304, 327, 336, 356, 370, 372, 373, 397–399, 515 radial – element, 506, 507, 510, 511 – stress, 518, 523, 524, 526, 528, 530, 534, 536, 538–540, 542–544, 546, 549, 551–553, 556, 563, 565, 567, 569–572, 574, 578, 579, 583–585, 587, 589, 595, 604, 605, 609, 612 reciprocal – coupling, 6, 124, 126–129, 148 – dependence, 25 reciprocity, 3 rectangular, VI, 6, 42, 149, 152, 163, 164, 174, 177, 184, 188, 189, 192, 235, 269, 270, 274, 322, 325, 326, 332, 336, 343, 354, 356, 357, 360, 395, 397, 399, 400, 402, 411, 425, 446, 483, 493–495, 497, 515, 520, 521, 595, 633, 645, 677, 686, 695, 696, 701 relative – deflection, 333, 338, 339, 342, 345, 346, 352, 361, 368, 372–378, 381, 383, 387, 390, 396, 419, 424, 426, 430, 435, 436, 440, 452, 459, 488, 489, 526, 534, 569, 584, 590, 605 – error, 60, 508, 519 – shell deflection, 326, 341, 384, 392, 393, 451, 562, 563

Index – thickness, 59, 61–65, 79 relaxation, 1, 41, 43, 45, 46, 50, 59, 87, 98, 149, 186–188, 192–194, 210, 233, 235, 267, 328–330, 332, 333, 671, 672, 773 reliability, VI, 54, 55, 234, 267, 395, 415, 493, 617, 641, 691 resistance contour, 614, 618, 625, 627, 631 resolvent, 705, 708, 709, 711 rib, 1, 11, 167, 171, 177, 180, 211, 213, 398, 639, 694, 696, 698, 700, 701 rigid fixing, 19, 89 ring, 6, 46, 72–82, 87, 89, 91, 92, 94, 95, 102–104, 109, 110, 386, 496, 497 rolling support, 305, 326, 338, 343, 410, 432, 506, 598, 603 rotation angle, 501, 714 rotational shell, 7, 8, 272 Runge’s law, 46 Runge-Kutta’s method, 1, 2, 47, 48, 53, 57, 79, 82, 98, 105, 109, 114, 133, 134, 136 scalar product, 707, 719 second order boundary condition, 184, 185, 211 sector-type shell, 614, 619, 620, 622, 625–627, 630, 631 sectorial shell, VI, 493, 497, 498, 509, 515, 521, 522, 531, 532, 534, 537, 539, 543, 545, 547, 551, 555, 557, 560, 578, 587, 592, 594, 595, 603–607, 771 Seidel method, 188, 189, 193 Seidel’s method, 1, 41–43, 45, 134, 136, 194 self-adjoint, 705, 707, 716–718 semi-group, 719, 720 set-up method, 493, 509, 520, 617, 684 set-up technique, VI shallow – anisotropic shells, VI, 149 – shell, VI, 149, 152, 177, 183, 209, 232, 253, 395, 397, 399, 402, 406, 409, 434, 436, 446, 447, 455, 521, 633, 638, 642–644, 696 shear – force, 401 – modulus, 285, 696, 702 simple point, 1 Simpson’s method, 314, 447 singular point, 1, 13, 14, 58, 340 singularities, 9, 14, 268, 269, 272, 273

775

sixangle, 496 skew cantilever plate, 494 sliding clamping, 167, 171, 615, 616 slip support, 409 sloping parameter, 498, 521, 557, 567, 568, 592, 593, 614, 617, 618, 625–627, 631 smoothness, 4, 5, 707 Sobolev space, 298 space problem, 3 spatial coordinate, V, 618 spherical shell, 5, 337, 395, 397, 418, 420–425, 427, 428, 430, 432–436, 443, 446, 448–455, 457, 467–469, 475–478, 480, 481, 483, 484, 487, 488, 492, 493, 507, 517, 521, 603, 614, 631 stability – condition, 1, 5, 48, 320–323, 326, 332 – loss, VI, 150, 183, 267, 270–274, 276, 334, 338, 340, 342, 354, 356–358, 385, 386, 388, 395, 397, 415, 417, 422, 423, 429, 432, 436, 441, 454, 458, 478, 480, 520–523, 525, 526, 532, 534, 542, 543, 550, 551, 554, 557, 558, 562, 564, 572, 575, 577–579, 586, 587, 593–595, 627, 633, 634, 646, 647, 654 – problem, 232, 272, 273, 275, 328, 334, 336, 338, 397, 493, 701 – zone, 338, 382, 388, 389 stabilization, 273, 449, 631 static hybrid problem, 5 statically equivalent force, 501 stationary, VI, VII, 2, 5, 6, 8, 47–50, 53–55, 57, 58, 82, 87, 103, 109, 117, 126, 129, 131, 149–152, 183, 184, 186, 202, 209, 232, 252, 267, 268, 273, 274, 283, 296, 297, 305, 323, 328, 329, 331, 366–368, 370, 375, 395–398, 479, 480, 494, 506, 509, 560, 561, 596, 600, 618, 631, 641, 642, 671, 684–686, 696 – state method, VI, 395 strain, V, VI, 5–7, 9, 11, 12, 54, 55, 57, 59–61, 70, 72, 78–80, 87, 89, 98–100, 102–104, 109, 110, 121, 130, 131, 134, 136, 137, 149–153, 183, 232, 235, 236, 239, 242, 244, 246, 252, 259, 264, 336, 342, 344, 395–399, 418, 438, 443, 455, 464, 475–477, 494, 495, 497, 507,

776

Index

517, 522, 542, 602, 603, 633, 645–647, 650–652, 655, 669, 672, 696, 697, 702 – tensor, 9, 130 stream action, 395, 399, 420, 464, 468 stress – distribution, 116–119, 121, 138, 141, 143 – function, 153, 158, 159, 161, 343, 347, 352, 400, 503, 507, 524, 525, 527, 529, 533, 534, 536, 537, 540, 542, 543, 545, 548, 551, 553, 556, 564, 566, 570, 574, 577, 578, 594, 598, 601, 604–606, 611, 614, 637, 643, 644, 650, 651, 653–656, 659, 702, 703 stress-strain state, V, VI, 5–7, 54, 55, 57, 59–61, 70, 72, 78–80, 87, 89, 98–100, 102–104, 109, 110, 121, 130, 136, 137, 149–152, 183, 232, 236, 239, 242, 244, 246, 252, 264, 342, 344, 395, 443, 475, 477, 495, 497, 602, 633, 645, 646, 650–652, 655, 669, 696 stresses propagation, 268 stretching, 386, 388, 398, 409, 494, 502, 523, 526, 534, 539, 547, 555, 559, 560, 563, 578, 579, 594, 605 strong elliptic operator, 299 summation, 29, 30, 33, 36, 317, 773 surface, 6, 7, 9, 10, 13, 14, 18, 19, 41–43, 53, 57, 59, 61, 66–72, 76–80, 83, 87, 89, 90, 92, 94, 113, 114, 124, 126, 130, 131, 134, 135, 138, 141, 147, 151–153, 155, 157–163, 169, 170, 172, 183–185, 189, 218, 227, 242, 257, 259, 262, 268, 270, 278, 282, 283, 285, 288, 291, 296, 297, 306, 323, 325, 327, 336, 338, 359, 360, 366–368, 384, 385, 388, 396–402, 406–411, 414, 418, 419, 426, 427, 430, 432, 434, 435, 440, 448, 450, 452, 453, 458, 459, 466, 471, 475, 479, 481–484, 486, 490, 492, 496–499, 501–503, 505, 508, 509, 596, 597, 602, 603, 612, 614, 620, 633–636, 640, 641, 645, 656, 657, 662–665, 667, 669, 684, 696, 697, 701–703, 714 – impulse, 669 symmetrical, 5, 7, 67, 89, 94, 109, 113, 114, 121, 136, 305, 495, 534, 558, 562, 564, 569, 571, 574, 575, 585, 593, 607

symmetry condition, 279, 282, 283, 311, 508, 617, 694 synchronous, 493 Taylor’s series, 26, 194, 196, 211, 212, 228, 276 technological instrumentation, 2 temperature – conditions, V, 94, 152 – distribution, 19, 20, 31, 48, 50, 53, 55, 56, 82–84, 86, 87, 94, 104, 109, 113–115, 117, 126, 138, 144, 151, 184, 202, 273, 274, 278, 296, 323, 324, 366–368, 370, 396, 397, 399, 408, 410, 508, 595, 597, 600, 602, 603, 640, 641, 656, 657, 662–665, 668 – field, V, VI, 6, 8, 11, 53, 55, 56, 82, 87, 104, 117, 138, 144, 146, 149, 150, 152, 153, 164, 183, 217–220, 222–224, 226, 227, 232, 252–255, 257, 259, 262–264, 268, 273, 274, 283, 296, 297, 304, 314, 323, 325, 331, 366, 367, 395–399, 402, 403, 406–409, 457, 483, 600, 603, 633, 645, 646, 705 temperature’s gradient, 401 theorems, VII, 3, 4, 7, 207, 275, 304, 705 theory of plates, V, 8, 273, 684 thermal – energy, 12 – flux, 4 – impact, 7, 31, 89, 117, 121, 142, 144, 418 – load, VI, 53, 267, 275, 276, 305, 306, 366, 375–378, 430, 456, 633 – stress, 2, 4, 53, 150, 287, 307, 366, 398, 403, 667, 669 thermal – – load, 768 thermal-stress state, 257 thermoelastic vibration, 2, 267, 399 thermoplasticity, 2, 7, 39 thermosensitive excitation, V thermosensitive shell, VI, 267, 304 thermostability, 273, 397 thick two-layered sphere, 5 thickness, VII, 53–56, 58–65, 67, 79, 87, 138, 151, 152, 157, 272, 273, 285, 288, 293, 296, 305, 306, 314, 323, 324, 327, 337, 367, 368, 382, 384, 385, 396–400, 408, 413, 419, 421, 447, 450, 457, 497,

Index 501, 507, 521, 595, 597, 600, 602, 603, 614, 634, 638, 639, 641, 644, 657, 668, 696, 705, 714 third order boundary condition, 185, 211 three-dimensional plate, 9, 11, 18, 30, 35, 40, 53, 109, 114, 130, 136 three-dimensional problems, V, 2, 49, 79 time – history, 333, 342, 373, 374, 517, 622 – of forcing action, 305 – of heat action, 305 Timoshenko type model, VII, 705 top relaxation, 187, 192, 194 torsional moment, 401 trajectory, 12, 48, 631 transversal load, 150, 152, 164, 236, 238, 254, 264, 329, 330, 395, 400–402, 413–415, 418, 420, 422, 424, 426, 427, 432–438, 440, 442, 454, 455, 460, 464–467, 496, 515, 555, 569, 637, 669, 696 transverse deflection, 494 trapezoidal cantilever plate, 494 triangle – contour, 494 – plate, 493, 494 – variational method, 672 triangular, 41, 189, 190, 192, 193, 495 two-dimensional – case, 30 – dynamic, 5 – problem, 5, 109, 273 uniformly – correct, VII, 705, 719 – distributed load, VII, 55, 57, 236, 257, 495, 609, 631, 671 uniqueness, VII, 4, 304, 335, 705, 712 unstability, 269, 335, 440, 441, 560 unsymetric deformations, 274 upper relaxation, 1, 41, 43, 45, 46, 50, 59, 87, 98, 149, 193, 194, 672 variable coeﬃcient, 4, 201, 204, 273, 297, 320, 337, 406, 407, 409, 457, 680 variables transformation, 196 variational – equations, VI, 11, 149, 162, 166, 267, 275 – method, 8, 11, 133, 275, 287, 297, 672

777

– principle, 2, 6, 151, 275, 397 vector projection, 288 vertex, 520–523, 526, 532, 534, 539, 542, 543, 547, 550, 555, 559, 560, 563, 569, 572, 585, 593–595, 597, 599, 601–604, 616, 690 vibration, VI, VII, 2, 3, 5, 48–50, 79, 267, 269, 271–273, 297, 305, 306, 328, 336, 337, 339, 340, 353–355, 386, 395, 398–401, 415, 418, 419, 421, 423, 425–430, 432, 434, 435, 437, 439–442, 452–454, 456, 459–461, 463, 466, 469–478, 480–482, 490–496, 509, 515, 521, 532, 534, 539, 547, 548, 555, 560, 573, 601, 606, 612, 614, 619–627, 629–633, 636, 647, 651, 705, 706, 776 vibrational process, 395, 418, 426, 427, 434, 435, 441, 452, 454, 460, 461, 466, 471, 475, 476, 478, 480–482, 490–492, 534, 555, 606, 612 Volmir’s criterion, 416, 417, 458 volume, 9, 53, 59, 61, 62, 66, 68, 69, 71, 72, 76, 79, 87, 89, 90, 114, 115, 130, 132, 135, 136, 153, 158, 192, 273, 278, 497, 634–636, 672–674, 677, 683–685, 694, 702 wall, 8, 9, 15, 18–20, 27, 31, 40, 54, 55, 58, 59, 68, 69, 71, 72, 79, 80, 83, 84, 87, 89, 109, 113, 114, 117, 124, 126, 136–138, 141, 146, 147, 183, 267, 270, 272–274, 276, 290, 296, 305, 366, 382, 388, 397–399, 495, 496, 595, 633, 666, 686, 689–692 wave – equation, 3, 4 – process, 6, 272 wave-shaped form, 345 weak compactness, 39 work, V, 2–8, 12, 14, 30, 36, 39, 45–48, 55, 62, 114, 130, 131, 133–136, 144, 151, 159, 168, 183, 194, 268, 270–275, 278, 279, 283, 287, 288, 296, 297, 299, 315, 320, 321, 328, 334–337, 340, 341, 353, 368, 396–400, 480, 493–497, 603, 614, 626, 631, 646, 672, 684, 713, 721 Young modulus, VII, 151, 163, 266, 399, 400, 497, 705, 714