Nonlinear Continuum Mechanics and Large Inelastic Deformations

Nonlinear Continuum Mechanics and Large Inelastic Deformations

SOLID MECHANICS AND ITS APPLICATIONS Volume 174

Series Editors:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

Yuriy I. Dimitrienko

Nonlinear Continuum Mechanics and Large Inelastic Deformations

123

Prof. Yuriy I. Dimitrienko Bauman Moscow State Technical University 2nd Baumanskaya St. 5 105005 Moscow Russia [email protected]

ISSN 0925-0042 ISBN 978-94-007-0033-8 e-ISBN 978-94-007-0034-5 DOI 10.1007/978-94-007-0034-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010938719 c Springer Science+Business Media B.V. 2011  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Nonlinear continuum mechanics is the kernel of the general course ‘Continuum Mechanics’, which includes kinematics of continua, balance laws, general nonlinear theory of constitutive equations, relations at singular surfaces. Moreover, in the course of nonlinear continuum mechanics one also considers the theory of solids at finite (arbitrary) deformations. This arbitrariness of deformations makes the equations describing the behavior of continua extremely complex – nonlinear (so that sometimes the term ‘strongly nonlinear’ is used), as the relationships contained in them cannot always be expressed in an explicit analytical way. If we drop the condition of the arbitrariness of continuum deformations and consider only infinitesimal deformations – usually the deformations till 1%, then the situation changes: the equations of continuum mechanics can be linearized. Hence for solving the applied problems one can exploit the wide range of analytical and numerical methods. However, many practical tasks demand the analysis not of the infinitesimal, but just the arbitrary (large) deformations of bodies, for example, such tasks include the rubber structure elements design (shock absorbers, gaskets, tires) for which the ultimate deformations can reach 100% and even higher. The various tasks of metal working under high pressure also belong to that class of problems, where large plastic deformations play a significant role, as well as the dynamical problems of barrier breakdown with a striker (aperture formation in the metal barrier while the breakdown is an example of large plastic deformations). Within this class of problems one can also find many problems of ground and rock mechanics, where there usually appears the need to consider large deformations, and modelling the processes in biological systems such as the functioning of human muscular tissue, and many others. The theory of infinitesimal deformations of solids appeared in the XVII century in the works by Robert Hooke, who formulated one of the main assumptions of the theory: stresses are proportional to strains of bodies. Translating the assertion into mathematical language, this means that relations between stresses and displacements gradients of bodies are linear. Nowadays the theory of infinitesimal deformations is very deeply and thoroughly elaborated. On the different parts of this theory such as elasticity theory, plasticity theory, stability theory and many others there are many monographs and textbooks.

v

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Preface

But the well-known Hooke’s law does not hold for finite (or large) deformations: the basic relations between stresses and displacement gradients become ‘strongly non-linear’, and they cannot always be expressed analytically. The basis of finite deformations theory was laid in the XIX century by the eminent scientists A.L. Cauchy, J.L. Lagrange, L. Euler, G. Piola, A.J.C. Saint-Venant, G.R. Kirchhoff, and then developed by A.E.H. Love, G. Jaumann [28], M.A. Biot, F.D. Murnaghan [41] and other researchers. The works by M. Mooney and R.S. Rivlin written in the 1940s of the XX century contributed much to the formation of finite deformations theory as an independent part of continuum mechanics. The fundamental step was made in 1950–1960s of the XX century by the American mechanics school, first of all by B.D. Coleman [9], W. Noll and C. Truesdell [43, 54–56], who considered the nonlinear mechanics from the point of view of the formal mathematics. According to D. Hilbert, they introduced the axiomatics of nonlinear mechanics which structured the system of accumulated knowledge and made it possible to formulate the main directions of the further investigations in this theory. Together with R.S. Rivlin and A.J.M. Spencer [10, 13, 51] they elaborated the special mathematical apparatus for formulation of relationships, generalizing Hooke’s law for finite deformations, namely the theory of nonlinear tensor functions. And also the tensor analysis widely used in continuum mechanics was considerably adapted to the problems of nonlinear mechanics. Equations of continuum mechanics got the invariant (i.e. independent of the choice of a reference system) form. The further development of this direction was made by A.C. Eringen, A.E. Green, W. Zerna, J.E. Adkins and others [1–7, 11, 12, 14–27, 29, 30, 32–35, 38–40, 42, 47–50, 52, 53, 57–60]. The role of Russian mechanics school in the development of contemporary nonlinear continuum mechanics principles is also quite substantial. In 1968 the first edition of the fundamental two-part textbook ‘Continuum Mechanics’ by L.I. Sedov was published, which is still one of the most popular books on continuum mechanics in Russia. Outstanding results in the theory of finite elastic deformations were obtained by A.I. Lurie [36, 37], who wrote the principal monograph on the nonlinear theory of elasticity and systemized in it the problem classes of the theory of finite elastic deformations allowing for analytical solutions. Also the considerable step was made by K.F. Chernykh [8], who developed the theory of finite deformations for anisotropic media and elaborated the methods for solving the problems of nonlinear theory of shells and nonlinear theory of cracks. One can also mention the works by mechanics scientists: B.E. Pobedrya, V.I. Kondaurov, V.G. Karnauhov, A.A. Pozdeev, P.V. Trusov, Yu.I. Nyashin and many others who made considerable contributions to the theory of viscoelastic, elastoplastic and viscoplastic finite deformations. This book is based on the lectures which the author has been giving for many years in Moscow Bauman State Technical University. The book has several fundamental traits: 1. It follows the mathematical style of course exposition, which assumes the usage of axioms, definitions, theorems and proofs. 2. It applies the tensor apparatus, mostly in the indexless form, as the latter combined with the special skills is very convenient in usage, and does not shade

Preface

3.

4.

5.

6. 7.

vii

the physical essence of the laws, and permits proceeding to any appropriate coordinate system. It uses the divergence form of dynamic equations of deformation compatibility, that made it at last possible to write the complete system of balance laws of nonlinear mechanics in a single generalized form. The theory of constitutive equations being the key part of nonlinear mechanics is for the first time exposed with the usage of all energetic couples of tensors, which were established by R. Hill [26] and K.F. Chernykh [8] and ordered by the author [12], and also with quasi-energetic couples of tensors found by the author [12]. To derive constitutive equations of nonlinear continuum mechanics, the author applied the theory of nonlinear tensor functions and tensor operators, elaborated by A.J.M. Spencer, R.S. Rivlin, J.L. Ericksen, V.V. Lokhin, Yu.I. Sirotin, B.E. Pobedrya, the author of this book and others. The bases of theories of large elastic, viscoelastic and plastic deformations are explored from the uniform position. The book uses a ‘reader-friendly‘ style of material exposition, which can be characterized by the presence of quite detailed necessary mathematical calculations and proofs.

The axiomatic approach used in this book differs a bit from the analogous ones suggested by C. Truesdell [56] and other authors. The system of continuum mechanics axioms in the book is composed so as to minimize their total number, and give each axiom a clear physical interpretation. That is why the axioms by C. Truesdell connected with the logic relations between bodies are not included in the general list, the axioms on the bodies’ mass are united into one axiom, the mass conservation law, and analogously the axioms on the existence of forces and inertial reference systems are united into one axiom, the momentum balance law. Though, the last axiom is split into the two parts: first the Sect. 3.2 considers the case of inertial reference systems, and then Sect. 4.10 deals with non-inertial ones. Unlike the axiomatics by C. Truesdell [56], in this book the axiom system includes so called principles of constitutive equations construction which play a fundamental role in the formation of a continuum mechanics equations system. The axiomatic approach to the exploration of continuum mechanics possesses at least one merit – it permits the separation of all the values into two categories: primary and secondary. These are introduced axiomatically and consequently within the continuum mechanics there is no need to substantiate their appearance. The secondary category includes combinations of the first category’s values. The axiomatic approach allows us also to distinguish from continuum mechanics statements between the definitions and corollaries of them (theorems); this is extremely useful for the initial acquaintance with the course. To get acquainted with the specific apparatus of tensor analysis the reader is recommended to use the author’s book ‘Tensor Analysis and Nonlinear Tensor Functions’ [12], which uses the same main notations and definitions. All the references to the tensor analysis formulas in the text are addressed to the latter book.

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Preface

This book covers the fundamental classical parts of nonlinear continuum mechanics: kinematics, balance laws, constitutive equations, relations at singular surfaces, the basics of theories of large elastic deformations, large viscoelastic deformations and large plastic deformations. Because of limits on space, important parts such as the theory of shells at large deformations, and the theory of media with phase transformations were not included in the book. I would like to thank Professor B.E. Pobedrya (Moscow Lomonosov State University), Professor N.N. Smirnov (Moscow Lomonosov State University) and Professor V.S. Zarubin (Bauman Moscow State Technical University) for fruitful discussions and valuable advice on different problems in the book. I am very grateful to Professor G.M.L. Gladwell of the University of Waterloo, Canada, who edited the book and improved the English text. I also thank my wife, Dr. Irina D. Dimitrienko (Bauman Moscow State Technical University), who translated the book into English and prepared the camera-ready typescript. I hope that the book proves to be useful for graduates and post-graduates of mathematical and natural-scientific departments of universities and for investigators, academic scientists and engineers working in solid mechanics, mechanical engineering, applied mathematics and physics. I hope that the book is of interest also for material science specialists developing advanced materials. Russia

Yuriy Dimitrienko

Contents

1

Introduction: Fundamental Axioms of Continuum Mechanics . . . . . . . . . . .

1

2 Kinematics of Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Material and Spatial Descriptions of Continuum Motion.. . . . . . . . . . . . . 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.2 Material and Spatial Descriptions.. . . . . . . . . .. . . . . . . . . . . . . . . . . ı 2.1.3 Local Bases in K and K . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.4 Tensors and Tensor Fields in Continuum Mechanics . . . . . . . ı 2.1.5 Covariant Derivatives in K and K . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.6 The Deformation Gradient . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.7 Curvilinear Spatial Coordinates.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2 Deformation Tensors and Measures . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.1 Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.2 Deformation Measures . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.3 Displacement Vector.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector .. . . . . . . . . 2.2.5 Physical Meaning of Components of the Deformation Tensor . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.6 Transformation of an Oriented Surface Element .. . . . . . . . . . . 2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor .. . . . . . . 2.3 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.1 Theorem on Polar Decomposition . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.2 Eigenvalues and Eigenbases . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.3 Representation of the Deformation Tensors in Eigenbases . 2.3.4 Geometrical Meaning of Eigenvalues . . . . . .. . . . . . . . . . . . . . . . . 2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum . . . . . . . 2.4 Rate Characteristics of Continuum Motion.. . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.2 Total Derivative of a Tensor with Respect to Time . . . . . . . . .

5 5 5 9 9 11 13 14 16 24 24 25 25 26 28 30 33 36 36 40 42 44 45 49 49 50 ix

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2.4.3 2.4.4 2.4.5

Differential of a Tensor .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Properties of Derivatives with Respect to Time .. . . . . . . . . . . . The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.6 Eigenvalues of the Deformation Rate Tensor . . . . . . . . . . . . . . . 2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor .. . . . . . . . . . . . . . . . 2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point . . . . . . . 2.4.9 Kinematic Meaning of the Vorticity Vector . . . . . . . . . . . . . . . . . 2.4.10 Tensor of Angular Rate of Rotation (Spin) .. . . . . . . . . . . . . . . . . 2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.12 Trajectory of a Material Point, Streamline and Vortex Line .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.13 Stream Tubes and Vortex Tubes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Co-rotational Derivatives .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.1 Definition of Co-rotational Derivatives .. . . .. . . . . . . . . . . . . . . . . 2.5.2 The Oldroyd Derivative (hi D ri ) . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.3 The Cotter–Rivlin Derivative (hi D ri ) . . . . .. . . . . . . . . . . . . . . . . 2.5.4 Mixed Co-rotational Derivatives .. . . . . . . . . . .. ................ ı 2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.6 The Derivative in the Eigenbasis (hi D pi ) of the Left Stretch Tensor . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.7 The Jaumann Derivative (hi D qi ). . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.9 Spin Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.10 Universal Form of the Co-rotational Derivatives.. . . . . . . . . . . 2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient . . . . . . .

53 54

3 Balance Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 The Mass Conservation Law .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.1 Integral and Differential Forms . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.2 The Continuity Equation in Lagrangian Variables . . . . . . . . . . 3.1.3 Differentiation of Integral over a Moving Volume .. . . . . . . . . 3.1.4 The Continuity Equation in Eulerian Variables . . . . . . . . . . . . . 3.1.5 Determination of the Total Derivatives with respect to Time . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.6 The Gauss–Ostrogradskii Formulae . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 The Momentum Balance Law and the Stress Tensor .. . . . . . . . . . . . . . . . . 3.2.1 The Momentum Balance Law. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2.2 External and Internal Forces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

89 89 89 90 91 92

2.5

56 58 59 60 62 63 65 73 75 77 77 79 80 81 81 82 83 83 84 85 85

93 94 95 95 97

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3.2.3 3.2.4 3.2.5 3.2.6

3.3

3.4

3.5

3.6

Cauchy’s Theorems on Properties of the Stress Vector .. . . . 98 Generalized Cauchy’s Theorem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .101 The Cauchy and Piola–Kirchhoff Stress Tensors . . . . . . . . . . .102 Physical Meaning of Components of the Cauchy Stress Tensor .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .103 3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .107 The Angular Momentum Balance Law .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.2 Tensor of Moment Stresses . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .110 3.3.3 Differential Form of the Angular Momentum Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .111 3.3.4 Nonpolar and Polar Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .112 3.3.5 The Angular Momentum Balance Equation in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .113 The First Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .114 3.4.1 The Integral Form of the Energy Balance Law.. . . . . . . . . . . . .114 3.4.2 The Heat Flux Vector .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .116 3.4.3 The Energy Balance Equation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .117 3.4.4 Kinetic Energy and Heat Influx Equation . .. . . . . . . . . . . . . . . . .118 3.4.5 The Energy Balance Equation in Lagrangian Description .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .119 3.4.6 The Energy Balance Law for Polar Continua . . . . . . . . . . . . . . .121 The Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.2 Differential Form of the Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126 3.5.3 The Second Thermodynamic Law in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .127 3.5.4 Heat Machines and Their Efficiency .. . . . . . .. . . . . . . . . . . . . . . . .128 3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles . . .132 3.5.6 Truesdell’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .136 Deformation Compatibility Equations .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.1 Compatibility Conditions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.2 Integrability Condition for Differential Form . . . . . . . . . . . . . . .142 3.6.3 The First Form of Deformation Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .142 3.6.4 The Second Form of Compatibility Conditions .. . . . . . . . . . . .143 3.6.5 The Third Form of Compatibility Conditions .. . . . . . . . . . . . . .145 3.6.6 Properties of Components of the Riemann–Christoffel Tensor . . . . . . . . .. . . . . . . . . . . . . . . . .146 3.6.7 Interchange of the Second Covariant Derivatives .. . . . . . . . . .148 3.6.8 The Static Compatibility Equation.. . . . . . . . .. . . . . . . . . . . . . . . . .148

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3.7

3.8 3.9

Dynamic Compatibility Equations .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.1 Dynamic Compatibility Equations in Lagrangian Description.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.2 Dynamic Compatibility Equations in Spatial Description ..151 Compatibility Equations for Deformation Rates . . . . . .. . . . . . . . . . . . . . . . .152 The Complete System of Continuum Mechanics Laws .. . . . . . . . . . . . . . .155 3.9.1 The Complete System in Eulerian Description.. . . . . . . . . . . . .155 3.9.2 The Complete System in Lagrangian Description . . . . . . . . . .156 3.9.3 Integral Form of the System of Continuum Mechanics Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157

4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .161 4.1 Basic Principles for Derivation of Constitutive Equations.. . . . . . . . . . . .161 4.2 Energetic and Quasienergetic Couples of Tensors . . . .. . . . . . . . . . . . . . . . .162 4.2.1 Energetic Couples of Tensors . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .162 I

4.2.2

The First Energetic Couple .T; ƒ/ . . . . . . . . . .. . . . . . . . . . . . . . . . .164

4.2.3

The Fifth Energetic Couple .T; C/ . . . . . . . . .. . . . . . . . . . . . . . . . .165

4.2.4

The Fourth Energetic Couple . T ; .U  E// . . . . . . . . . . . . . . . . .166

4.2.5

The Second Energetic Couple .T; .E  U1 // . . . . . . . . . . . . .167

4.2.6 4.2.7

The Third Energetic Couple . T ; B/ . . . . . . . .. . . . . . . . . . . . . . . . .167 General Representations for Energetic Tensors of Stresses and Deformations .. . . . .. . . . . . . . . . . . . . . . .168 Energetic Deformation Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors . . . . . . . . . . .175 Quasienergetic Couples of Stress and Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .176

4.2.8 4.2.9 4.2.10

V

IV

II

III

I

4.2.11

The First Quasienergetic Couple .S; A/ . . . .. . . . . . . . . . . . . . . . .177

4.2.12 4.2.13

The Second Quasienergetic Couple .S; .E  V1 // . . . . . . . .178 The Third Quasienergetic Couple .Y; TS /.. . . . . . . . . . . . . . . . .178

4.2.14

The Fourth Quasienergetic Couple . S ; .V  E// . . . . . . . . . . .179

4.2.15 4.2.16 4.2.17 4.2.18

The Fifth Quasienergetic Couple .S; J/ . . . . .. . . . . . . . . . . . . . . . .180 General Representation of Quasienergetic Tensors . . . . . . . . .180 Quasienergetic Deformation Measures . . . . .. . . . . . . . . . . . . . . . .182 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors .. . . . . . . . . . . .183 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 The Generalized Form of Representation of the Stress Power . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .186

4.2.19

4.2.20

II

IV

V

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4.2.21

4.3

4.4

4.5

4.6

4.7

4.8

Representation of Stress Power in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .187 4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient . . . . . . . . .188 The Principal Thermodynamic Identity . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.1 Different Forms of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.2 The Clausius–Duhem Inequality .. . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.3 The Helmholtz Free Energy .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.4 The Gibbs Free Energy .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .199 4.3.5 Enthalpy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .201 4.3.6 Universal Form of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .202 4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .203 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.1 Active and Reactive Variables . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.2 The Principle of Thermodynamically Consistent Determinism .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .206 4.4.3 The Principle of Equipresence . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 4.4.4 The Principle of Local Action . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 Definition of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.1 Classification of Types of Continua . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.2 General Form of Constitutive Equations for Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .210 The Principle of Material Symmetry . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.1 Different Reference Configurations.. . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.2 H -indifferent and H -invariant Tensors . . . .. . . . . . . . . . . . . . . . .215 4.6.3 Symmetry Groups of Continua . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .219 4.6.4 The Statement of the Principle of Material Symmetry.. . . . .220 Definition of Fluids and Solids. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.1 Fluids and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.2 Isomeric Symmetry Groups .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .222 4.7.3 Definition of Anisotropic Solids . . . . . . . . . . . .. . . . . . . . . . . . . . . . .226 4.7.4 H -indifference and H -invariance of Tensors Describing the Motion of a Solid . . . . . . . . . . .. . . . . . . . . . . . . . . . .228 4.7.5 H -invariance of Rate Characteristics of a Solid . . . . . . . . . . . .231 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua . . . . . . . .. . . . . . . . . . . . . . . . .236 4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids . . . . . . . . .236 4.8.2 Scalar Indifferent Functions of Tensor Argument.. . . . . . . . . .237 4.8.3 Producing Tensors of Groups . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .239

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4.8.4 4.8.5

Scalar Invariants of a Second-Order Tensor .. . . . . . . . . . . . . . . .240 Representation of a Scalar Indifferent Function in Terms of Invariants .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .243 4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua . . . . . . . . . . . . .244 4.8.7 Quasilinear and Linear Models An of Elastic Continua . . . .249 4.8.8 Constitutive Equations for Models Bn of Elastic Continua . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253 4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua . . . . .254 4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua . . . . . . . . . . . . . .262 4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases .. . . . . . . . . . . . . . . .265 4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua ‘in Rates’ . . . .. . . . . . . . . . . . . . . . .271 4.8.13 Application of the Principle of Material Symmetry to Fluids . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .276 4.8.14 Functional Energetic Couples of Tensors.. .. . . . . . . . . . . . . . . . .283 4.9 Incompressible Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .287 4.9.1 Definition of Incompressible Continua . . . . .. . . . . . . . . . . . . . . . .287 4.9.2 The Principal Thermodynamic Identity for Incompressible Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .288 4.9.3 Constitutive Equations for Ideal Incompressible Continua .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .289 4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids . . . . . . .. . . . . . . . . . . . . . . . .291 4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases .. . . . . . . . . . . . . . . . .292 4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .295 4.9.7 Linear Models of Ideal Incompressible Elastic Continua.. .296 4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis.. .298 4.10 The Principle of Material Indifference.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.1 Rigid Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.2 R-indifferent and R-invariant Tensors .. . . . .. . . . . . . . . . . . . . . . .301 4.10.3 Density and Deformation Gradient in Rigid Motion.. . . . . . .302 4.10.4 Deformation Tensors in Rigid Motion .. . . . .. . . . . . . . . . . . . . . . .303 4.10.5 Stress Tensors in Rigid Motion . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .304 4.10.6 The Velocity in Rigid Motion .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .306 4.10.7 The Deformation Rate Tensor and the Vorticity Tensor in Rigid Motion.. .. . . . . . . . . . . . . . . . .306

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4.10.8 4.10.9 4.10.10 4.10.11

Co-rotational Derivatives in Rigid Motion .. . . . . . . . . . . . . . . . .307 The Statement of the Principle of Material Indifference.. . .312 Material Indifference of the Continuity Equation .. . . . . . . . . .313 Material Indifference for the Momentum Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .313 4.10.12 Material Indifference of the Thermodynamic Laws . . . . . . . .316 4.10.13 Material Indifference of the Compatibility Equations . . . . . .318 4.10.14 Material Indifference of Models An and Bn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .319 4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .320 4.10.16 Material Indifference for Incompressible Continua .. . . . . . . .322 4.10.17 Material Indifference for Models of Solids ‘in Rates’ . . . . . .322 4.11 Relationships in a Moving System . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.1 A Moving Reference System . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.2 The Euler Formula .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .326 4.11.3 The Coriolis Formula .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .327 4.11.4 The Nabla-Operator in a Moving System . .. . . . . . . . . . . . . . . . .329 4.11.5 The Velocity Gradient in a Moving System . . . . . . . . . . . . . . . . .330 4.11.6 The Continuity Equation in a Moving System . . . . . . . . . . . . . .330 4.11.7 The Momentum Balance Equation in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .331 4.11.8 The Thermodynamic Laws in a Moving System .. . . . . . . . . . .331 4.11.9 The Equation of Deformation Compatibility in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .332 4.11.10 The Kinematic Equation in a Moving System . . . . . . . . . . . . . .335 4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System . . . . . . . . . .335 4.11.12 Constitutive Equations in a Moving System . . . . . . . . . . . . . . . .335 4.11.13 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .338 4.12 The Onsager Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .339 4.12.1 The Onsager Principle and the Fourier Law.. . . . . . . . . . . . . . . .339 4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .341 4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law .. . . . . . . . . . .. . . . . . . . . . . . . . . . .343 4.12.4 The Fourier Law for Fluids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .344 4.12.5 The Fourier Law for Solids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .345 5 Relations at Singular Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1 Relations at a Singular Surface in the Material Description .. . . . . . . . . .347 5.1.1 Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1.2 The First Classification of Singular Surfaces.. . . . . . . . . . . . . . .347 5.1.3 Axiom on the Class of Functions across a Singular Surface.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .350

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5.2

5.3

5.4

5.1.4

The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface.. . . . . . . . . . . . . .352

5.1.5 5.1.6

Relations at a Coherent Singular Surface in K . . . . . . . . . . . . . .355 Relation Between Velocities of a Singular

ı

ı

Surface in K and K . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .357 Relations at a Singular Surface in the Spatial Description.. . . . . . . . . . . .358 5.2.1 Relations at a Coherent Singular Surface in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .358 5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface . . . . .360 Explicit Form of Relations at a Singular Surface . . . . .. . . . . . . . . . . . . . . . .362 5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration .. . .362 5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration . . . . . .363 5.3.3 Mass Rate of Propagation of a Singular Surface .. . . . . . . . . . .363 5.3.4 Relations at a Singular Surface Without Transition of Material Points . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .365 The Main Types of Singular Surfaces . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.1 Jump of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.2 Jumps of Radius-Vector and Displacement Vector.. . . . . . . . .367 5.4.3 Semicoherent and Completely Incoherent Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .369 5.4.4 Nondissipative and Homothermal Singular Surfaces . . . . . . .369 5.4.5 Surfaces with Ideal Contact . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .370 5.4.6 On Boundary Conditions .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .372 5.4.7 5.4.8

ı

Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .372 Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .374

6 Elastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1 Closed Systems in the Spatial Description . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1.1 RU VF -system of Thermoelasticity .. . . . . .. . . . . . . . . . . . . . . . .377 6.1.2 RVF -, RU V -, and U V -Systems of Dynamic Equations of Thermoelasticity . . . . . . . . . . . . . . . . .381 6.1.3 T RU VF -system of Dynamic Equations of Thermoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .384 6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description . . . . .385 6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .388 6.2 Closed Systems in the Material Description.. . . . . . . . . .. . . . . . . . . . . . . . . . .390 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description . . . . . . . . . . . .390

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U V - and U -systems of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .394 6.2.3 T U VF -system of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .395 6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description . . . . . .396 Statements of Problems for Elastic Continua at Large Deformations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .399 6.3.1 Boundary Conditions in the Spatial Description .. . . . . . . . . . .399 6.3.2 Boundary Conditions in the Material Description . . . . . . . . . .402 6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .405 6.3.4 Statements of Thermoelasticity Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408 6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations .. . . . . . . . . . . . . . . .410 6.3.6 Conditions on External Forces in Quasistatic Problems .. . .412 6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description . . . . . . . . .. . . . . . . . . . . . . . . . .413 6.3.8 Variational Statement of Quasistatic Problem in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .417 6.3.9 Variational Statement for Incompressible Continua in the Material Description .. . . . . .. . . . . . . . . . . . . . . . .417 The Problem on an Elastic Beam in Tension .. . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.1 Semi-Inverse Method .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.2 Deformation of a Beam in Tension . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.3 Stresses in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .422 6.4.4 The Boundary Conditions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.5 Resolving Relation 1  k1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.6 Comparative Analysis of Different Models An . . . . . . . . . . . . .425 Tension of an Incompressible Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .430 6.5.1 Deformation of an Incompressible Elastic Beam . . . . . . . . . . .430 6.5.2 Stresses in an Incompressible Beam for Models Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .431 6.5.3 Resolving Relation 1 .k1 / . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.4 Comparative Analysis of Models Bn . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.5 Stresses in an Incompressible Beam for Models An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .436 Simple Shear .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.1 Deformations in Simple Shear . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.2 Stresses in the Problem on Shear . . . . . . . . . . .. . . . . . . . . . . . . . . . .439 6.6.3 Boundary Conditions in the Problem on Shear . . . . . . . . . . . . .441 6.2.2

6.3

6.4

6.5

6.6

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6.6.4

6.7

6.8

Comparative Analysis of Different Models An for the Problem on Shear . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .442 6.6.5 Shear of an Incompressible Elastic Continuum . . . . . . . . . . . . .443 The Lam´e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .446 6.7.1 The Motion Law for a Pipe in the Lam´e Problem . . . . . . . . . .446 6.7.2 The Deformation Gradient and Deformation Tensors in the Lam´e Problem . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .448 6.7.3 Stresses in the Lam´e Problem for Models An . . . . . . . . . . . . . . .449 6.7.4 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .450 6.7.5 Boundary Conditions of the Weak Type .. . .. . . . . . . . . . . . . . . . .451 6.7.6 Boundary Conditions of the Rigid Type .. . .. . . . . . . . . . . . . . . . .453 The Lam´e Problem for an Incompressible Continuum . . . . . . . . . . . . . . . .454 6.8.1 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.3 Equation for Hydrostatic Pressure p. . . . . . . .. . . . . . . . . . . . . . . . .456 6.8.4 Analysis of the Problem Solution .. . . . . . . . . .. . . . . . . . . . . . . . . . .456

7 Continua of the Differential Type . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .461 7.1 Models An and Bn of Continua of the Differential Type . . . . . . . . . . . . . .461 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type .. . . . . . .. . . . . . . . . . . . . . . . .461 7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type .. . .. . . . . . . . . . . . . . . . .463 7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type .. . . . . . .465 7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .467 7.1.5 Models Bn of Solids of the Differential Type .. . . . . . . . . . . . . .470 7.1.6 Models Bn of Incompressible Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .471 7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .472 7.2 Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . . . .473 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .473 7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type .. . . . . .. . . . . . . . . . . . . . . . .474 7.2.3 Simultaneous Invariants for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .476 7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type .. . . . . . . . . .. . . . . . . . . . . . . . . . .478

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Viscous Coefficients in Model AV of a Fluid of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .479 7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type . . . . . . . . . . . . . . .480 7.2.7 Constitutive Equations for Incompressible Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .481 7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . .481 Models Cn and Dn of Continua of the Differential Type .. . . . . . . . . . . . .482 7.3.1 Models Cn of Continua of the Differential Type .. . . . . . . . . . .482 7.3.2 Models Cnh of Solids with Co-rotational Derivatives . . . . . . .484 7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids . . . . . . . .. . . . . . . . . . . . . . . . .485 7.3.4 Viscosity Tensor in Models Cnh . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .488 7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids . . . . . . . . . . . . . . . .489 7.3.6 Models Dnh of Isotropic Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .491 The Problem on a Beam in Tension.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.1 Rate Characteristics of a Beam . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.2 Stresses in the Beam .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.3 Resolving Relation .k1 ; kP1 / . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.4 Comparative Analysis of Creep Curves for Different Models Bn . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type .. . . . . . . . . . .496 7.2.5

7.3

7.4

8 Viscoelastic Continua at Large Deformations . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1 Viscoelastic Continua of the Integral Type . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.1 Definition of Viscoelastic Continua.. . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.2 Tensor Functional Space . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .498 8.1.3 Continuous and Differentiable Functionals . . . . . . . . . . . . . . . . .499 8.1.4 Axiom of Fading Memory . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .503 8.1.5 Models An of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .505 8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua . . . . . . . .506 8.1.7 General Representation of Functional of Free Energy in Models An . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .507 8.1.8 Model An of Stable Viscoelastic Continua .. . . . . . . . . . . . . . . . .510 8.1.9 Model An of a Viscoelastic Continuum with Difference Cores . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .511 8.1.10 Model An of a Thermoviscoelastic Continuum .. . . . . . . . . . . .513 8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .514 8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .. . .516 8.2.1 Principal Models An of Viscoelastic Continua .. . . . . . . . . . . . .516

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Principal Model An of an Isotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .518 8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum . .. . . . . . . . . . . . . . . . .519 8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .520 8.2.5 Quadratic Models An of Thermoviscoelastic Continua . . . .522 8.2.6 Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .522 8.2.7 Representation of Linear Models An in the Boltzmann Form .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .525 8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .528 8.2.9 Linear Models An for Isotropic Viscoelastic Continua.. . . .530 8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .531 8.2.11 Linear Models An of Orthotropic Viscoelastic Continua .. .532 8.2.12 The Tensor of Relaxation Functions .. . . . . . .. . . . . . . . . . . . . . . . .533 8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .535 8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations.. . . . . . . . . . .539 8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua . . . . . . . . . . .. . . . . . . . . . . . . . . . .542 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.1 Models An of Incompressible Viscoelastic Continua .. . . . . .549 8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .551 8.3.4 Models Bn of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .552 8.3.5 Models An and Bn of Viscoelastic Fluids . .. . . . . . . . . . . . . . . . .554 8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua .. . . . . . . . . . .556 Statements of Problems in Viscoelasticity Theory at Large Deformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.1 Statements of Dynamic Problems in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.2 Statements of Dynamic Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .562 8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description .. . . . . . .564 8.4.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .566 8.2.2

8.3

8.4

Contents

8.5

8.6

xxi

The Problem on Uniaxial Deforming of a Viscoelastic Beam . . . . . . . . .568 8.5.1 Deformation of a Viscoelastic Beam in Uniaxial Tension .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .568 8.5.2 Viscous Stresses in Uniaxial Tension .. . . . . .. . . . . . . . . . . . . . . . .569 8.5.3 Stresses in a Viscoelastic Beam in Tension . . . . . . . . . . . . . . . . .569 8.5.4 Resolving Relation 1 .k1 / for a Viscoelastic Beam . . . . . . . .570 8.5.5 Method of Calculating the Constants B . / and  . / . . . . . . . .571 8.5.6 Method for Evaluating the Constants m, N l1 , l2 and ˇ, m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .574 8.5.7 Computations of Relaxation Curves .. . . . . . .. . . . . . . . . . . . . . . . .575 8.5.8 Cyclic Deforming of a Beam. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .577 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .579 8.6.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming .. . . . . . .. . . . . . . . . . . . . . . . .579 8.6.2 Fast and Slow Times in Multicycle Deforming . . . . . . . . . . . . .580 8.6.3 Differentiation and Integration of Quasiperiodic Functions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .580 8.6.4 Heat Conduction Equation for a Thin Viscoelastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .581 8.6.5 Dissipation Function for a Viscoelastic Beam . . . . . . . . . . . . . .582 8.6.6 Asymptotic Expansion in Terms of a Small Parameter . . . . .583 8.6.7 Averaged Heat Conduction Equation .. . . . . .. . . . . . . . . . . . . . . . .584 8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .585 8.6.9 Regimes of Dissipative Heating Without Heat Removal . . .585 8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal . . . . . . . . . . .. . . . . . . . . . . . . . . . .586 8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies .. . . . . . . . . . . . .588

9 Plastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1 Models An of Plastic Continua at Large Deformations .. . . . . . . . . . . . . . .591 9.1.1 Main Assumptions of the Models. . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua .. . . . . . . . . . . . . .594 9.1.3 Corollary of the Onsager Principle for Models  An of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .597 9.1.4 Models An of Plastic Yield. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .598 9.1.5 Associated Model of Plasticity An . . . . . . . . . .. . . . . . . . . . . . . . . . .599 9.1.6 Corollary of the Principle of Material  Symmetry for the Associated Model An of Plasticity . . . . . .603  9.1.7 Associated Models of Plasticity An for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .605

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9.1.8 9.1.9

9.2

9.3

The Huber–Mises Model for Isotropic Plastic Continua . . .607  Associated Models of Plasticity An for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .610 9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum .. . .. . . . . . . . . . . . . . . . .612  9.1.11 Associated Models of Plasticity An for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .614 9.1.12 The Orthotropic Unipotential Huber–Mises Model for Plastic Continua.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .616 9.1.13 The Principle of Material Indifference for Models An of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .618 Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.1 Representation of Stress Power for Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua .. . . . . . . . . . . . . .627 9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .629  9.2.4 Associated Models Bn of Plastic Continua . . . . . . . . . . . . . . . . .631 9.2.5 Corollary of the Principle of Material  Symmetry for Associated Model Bn of Plasticity . . . . . . . . . .632  9.2.6 Associated Models of Plasticity Bn with Proper Strengthening . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.7 Associated Models of Plasticity Bn for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.8 Associated Models of Plasticity Bn for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .637  9.2.9 Associated Models of Plasticity Bn for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .638 9.2.10 The Principle of Material Indifference for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .639 Models Cn and Dn of Plastic Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .640 9.3.1 General Representation of Constitutive Equations for Models Cn of Plastic Continua .. . . . . . . . . . . . . .640 9.3.2 Constitutive Equations for Models Cn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .643 9.3.3 General Representation of Constitutive  Equations for Models Dn of Plastic Continua . . . . . . . . . . . . . .646 9.3.4 Constitutive Equations for Models Dn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .650 9.3.5 The Principles of Material Symmetry and Material Indifference for Models Cn and Dn .. . . . . . . . . . . . . .651

Contents

9.4

9.5

9.6

9.7

9.8

9.9

xxiii

Constitutive Equations of Plasticity Theory ‘in Rates’ . . . . . . . . . . . . . . . .652 9.4.1 Representation of Models An of Plastic Continua ‘in Rates’ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .652 Statements of Problems in Plasticity Theory .. . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.1 Statements of Dynamic Problems for Models An of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.2 Statements of Quasistatic Problems  for Models An of Plasticity . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .657 The Problem on All-Round Tension–Compression of a Plastic Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .659 9.6.1 Deformation in All-Round Tension–Compression .. . . . . . . . .659 9.6.2 Stresses in All-Round Tension–Compression .. . . . . . . . . . . . . .660 9.6.3 The Case of a Plastically Incompressible Continuum . . . . . .661 9.6.4 The Case of a Plastically Compressible Continuum . . . . . . . .662 9.6.5 Cyclic Loading of a Plastically Compressible Continuum .664 The Problem on Tension of a Plastic Beam . . . . . . . . . . .. . . . . . . . . . . . . . . . .666 9.7.1 Deformation of a Beam in Uniaxial Tension .. . . . . . . . . . . . . . .666 9.7.2 Stresses in a Plastic Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .667 9.7.3 Plastic Deformations of a Beam . . . . . . . . . . . .. . . . . . . . . . . . . . . . .668 9.7.4 Change of the Density . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .670 9.7.5 Resolving Equation for the Problem .. . . . . . .. . . . . . . . . . . . . . . . .671 9.7.6 Numerical Method for the Resolving Equation . . . . . . . . . . . . .672 9.7.7 Method for Determination of Constants H0 , n0 , and s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .675 9.7.8 Comparison with Experimental Data for Alloys . . . . . . . . . . . .676 9.7.9 Comparison with Experimental Data for Grounds .. . . . . . . . .677 Plane Waves in Plastic Continua .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.1 Formulation of the Problem .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.2 The Motion Law and Deformation of a Plate . . . . . . . . . . . . . . .680 9.8.3 Stresses in the Plate. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .681 9.8.4 The System of Dynamic Equations for the Plane Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .682 9.8.5 The Statement of Problem on Plane Waves in Plastic Continua.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .684 9.8.6 Solving the Problem by the Characteristic Method . . . . . . . . .685 9.8.7 Comparative Analysis of the Solution for Different Models An . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .689 9.8.8 Plane Waves in Models AIV and AV . . . . . . . .. . . . . . . . . . . . . . . . .691   9.8.9 Shock Waves in Models AI and AII . . . . . . . .. . . . . . . . . . . . . . . . .693 9.8.10 Shock Adiabatic Curves for Models AI and AII . . . . . . . . . . . .695 9.8.11 Shock Adiabatic Curves at a Given Rate of Impact .. . . . . . . .697 Models of Viscoplastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 9.9.1 The Concept of a Viscoplastic Continuum .. . . . . . . . . . . . . . . . .699

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9.9.2 9.9.3 9.9.4 9.9.5 9.9.6

Model An of Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 Model of Isotropic Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .701 General Model An of Viscoplastic Continua .. . . . . . . . . . . . . . .702 Model An of Isotropic Viscoplastic Continua .. . . . . . . . . . . . . .703 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type . . . . . . . .703

References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .707 Basic Notation . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .708 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .713

Chapter 1

Introduction: Fundamental Axioms of Continuum Mechanics

Continuum mechanics, including nonlinear continuum mechanics, studies the behavior of material bodies or continua. We can mathematically define a body as follows: it is a set B consisting of elements M called material points. The concept of a material point in continuum mechanics is primary, i.e. axiomatic, as is the concept of a geometrical point in elementary geometry. The mathematical description of a body B in continuum mechanics starts from the following definition. Definition 1.1. A material body B, for which there is a one-to-one correspondence e between each material point M 2 B and its image in some metric space X , i.e. W e W B ! W.B/ e W  X ; or e a D W.M/;

M 2 B;

a 2 X;

(1.1)

is called a continuum. The set of all continua B is called the universe U. The definition of one-to-one correspondence one can find, for example, in [31]. A metric space X is characterized by the presence of a distance function l.M; N /, with the help of which one can measure the distance between any two points M and N of a body B [31]. The one-to-one correspondence between material points and points of a metric space X allows us to consider not the material body itself but only its image. Below, we draw no distinction between a material point and its image. Definition 1.1 should be complemented with three fundamental axioms. e Axiom 1 (on continuity). The image W.B/ of a body B is a continuous set (a continuum) in space X . The concept of a continuum was considered in [31]. Axiom 1 introduces the main model of continuum mechanics, namely a continuous set, that is an idealization of real bodies consisting of discrete atoms and molecules. Physically, there is a limiting distance lmin such that for l 6 lmin the neighborhood of a material point M 2 B is empty. However, in a continuum (and in its image), due to the properties of a continuous set [31], any infinitesimal "-neighborhood U" .A/ of a point A 2 W.B/  X contains an infinite number Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 1, c Springer Science+Business Media B.V. 2011 

1

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1 Introduction: Fundamental Axioms of Continuum Mechanics

of other points of this medium. In this sense, a continuum is only a model of a real body; and computed results in continuum mechanics referred to real bodies for distances between points l 6 lmin may prove to be incorrect. Nevertheless, Axiom 1 is of great importance: it allows us to apply the methods of tensor analysis in metric spaces. Axiom 2 (on an Euclidean space). As a metric space X , in which continua are considered, we can choose a three-dimensional Euclidean metric space E3a , i.e X D E3a . An Euclidean point space E3a (called also an affine Euclidean space) is a set of points M; N ; : : : ; where there exists a mapping, uniquely assigning each ordered pair of points M; N to an element (vector) y of Euclidean vector space E3 adjoined ! to E3a (it is also denoted by MN D y). In Euclidean space E3 , there are operations of addition of elements (vectors) x C y, multiplication by a real number y,  2 R and scalar product of vectors [31], that can be given by a metric matrix gij relative to some basis ei in E3 : x  y D gij x i y j , where x i and y j are coordinates of the vectors with respect to the same basis: x D x i ei , y D y i ei . Moreover, in E3 we can introduce the vector product of vectors a D ai ei and b D b i ei as follows: cDabD

1 p g ijk ai b j ek D p  ijk ai bj ek : g

(1.2)

Here ijk and  ijk are the Levi–Civita symbols [12] (they are zero, if at least two of the indices i; j; k are coincident; and they are equal to 1, if the indices i; j; k form an even permutation, and are equal to (1) if the indices form an odd permutation), and g D det .gij / is the determinant of the metric matrix. Thus, Axiom 2 allows us to describe materials points of a continuum with the help of instruments of Euclidean point spaces: in the space E3a we can introduce a rectangular Cartesian coordinate system O eN i , being common for all continua and consisting of some point O (the origin of the coordinate system) and an orthonormal basis eN i . In this system O eN i , every material point M is uniquely assigned to ! its radius-vector x D OM. The distance l.M; N / between points M and N is ! measured by the length of vector MN ! ! ! l.M; N / D jMN j D .MN  MN /1=2 D jyj D .y  y/1=2 : The length l.M; N / specifies a metric in the space E3a ; such a space is called a metric space. In a metric space we define the concept of a domain V , and in an Euclidean point space, the concepts of a plane and a straight line. In addition, in a metric space there are concepts of convergence of a point sequence, continuity and differentiability of functions etc.; Axiom 2 allows us to apply these concepts of mathematical analysis to continua. Due to isomorphism (one-to-one correspondence) of Euclidean point spaces of the same dimension as a space E3a , we can always consider the space of elementary

1 Introduction: Fundamental Axioms of Continuum Mechanics

3

e .B/ in E3a Fig. 1.1 A real body B and its image W geometry E3a , where points are usual geometric points, and vectors are directed straight-line segments in the space. The space E3a allows us to show different objects of continuum mechanics geometrically. Example 1.1. Figure 1.1 shows schematically a real material body B and its image e W.B/ in the space E3a . t u Consider the pair .M; t/, where M 2 B, and t 2 RC0 is some nonnegative real number. This pair is an element of the Cartesian product of the sets B  RC0 . Axiom 3 (on the existence of absolute time). For every body B there exists a mapping W W B  RC0 ! VN  E3a ; in the form of a function A D W.M; t/;

M 2 B;

t 2 RC0 ;

A 2 VN  E3a :

(1.3)

The parameter t is called the absolute time (or simply time). Notice that both Axioms 1 and 3 establish relations between points M and A. To avoid the ambiguity one should assume that the following consistency condition is satisfied: the mapping (1.1) coincides with (1.3) at some time t D t1 e W.M/ D W.M; t1 / 8M 2 B:

(1.4)

Axiom 3 allows us to describe the motion of a body B, which is defined as ! changes of the radius-vectors x D OM of material points M of the body in a coordinate system O eN i being common for all M 2 B with time t. For the same material point M at different times t1 and t2 we have, in general, distinct radius-vectors x1 and x2 , respectively, in the system O eN i . Example 1.2. Figure 1.2 shows the motion of a real body B and its images W.B; t1 / and W.B; t2 / in the space E3a at times t1 and t2 , respectively. t u The absolutism of time means that the time t is independent of the radius-vector x of a point M in the coordinate system O eN i , i.e., in physical terms, the time varies

4

1 Introduction: Fundamental Axioms of Continuum Mechanics

Fig. 1.2 The motion of a body B

in the same course for all material points M. In a physical sense, this axiom remains valid only for the motion of bodies with speeds which are considerably smaller than the speed of light; otherwise, relativistic effects become essential, and Axiom 3 ceases to describe actual processes adequately. Relativistic effects are not considered in this work.

Chapter 2

Kinematics of Continua

2.1 Material and Spatial Descriptions of Continuum Motion 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law Let us consider a continuum B. Due to Axiom 2, at time t D 0 there is a oneto-one correspondence between every material point M 2 B and its radius-vector ! ı x D OM in a Cartesian coordinate system O eN i . Denote Cartesian coordinates of ı ı ı the radius-vector by x i (x D x i eN i ) and introduce curvilinear coordinates X i of the same material point M in the form of some differentiable one-to-one functions ı

ı

x i D x i .X k /: ı

(2.1)

ı

Since x D x i eN i , the relationship (2.1) takes the form ı

ı

x D x.X k /:

(2.2)

Let us fix curvilinear coordinates of the point M, and then material points of the continuum B are considered to be numbered by these coordinates X i . For any motion of the continuum B, coordinates X i of material points are considered to remain unchanged; they are said to be ‘frozen’ into the medium and move together with the continuum. Coordinates X i introduced in this way for a material point M are called Lagrangian (or material). Due to Axiom 3, at every time t there is a one-to-one correspondence between ! every point M 2 B with Lagrangian coordinates X i and its radius-vector x D OM with Cartesian coordinates x i , where x and x i depend on t. This means that there is a connection between Lagrangian X i , and the Cartesian x i coordinates of point M and time, i.e. there exist functions in the form (1.3) x i D x i .X k ; t/

8X k 2 VX :

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 2, c Springer Science+Business Media B.V. 2011 

(2.3)

5

6

2 Kinematics of Continua

These functions determine a motion of the material point M in the Cartesian coordinate system O eN i of space E3a . The relationships (2.3) are said to be the law of the motion of the continuum B. Coordinates x i in (2.3) are called Eulerian (or spatial) coordinates of the material point M. Since x D x i eN i and the coordinate system O eN i is the same for all times t, the equivalent form of the motion law follows from (2.2): x D x.X k ; t/:

(2.4)

Since the consistency conditions (1.4) must be satisfied, from (2.2) and (2.4) we get the relationships ı

x.X k ; 0/ D x.X k /;

ı

x i .X k ; 0/ D x i .X k /:

(2.5)

Here the initial time t D 0 is considered as the time t1 in (1.4), because just at time t D 0 we introduced Lagrangian coordinates X i of point M. Unless otherwise stipulated, functions (2.3) are assumed to be regular in the domain VX  R3 for all t, thus there exist the inverse functions X k D X k .x i ; t/ 8x i 2 Vx  R3 : ı

The closed domain V D W.B; 0/ in a fixed coordinate system O eN i , which is occupied by continuum B at the initial time t D 0, is called the reference configuı

ration K, and the domain V D W.B; t/ occupied by the same continuum B at the time t > 0 is called the actual configuration K. Figure 2.1 shows a geometric picture of the motion of a continuum from the ı

reference configuration K to the actual one K at time t in space E3a .

Fig. 2.1 The motion of a continuum: positions of continuum B and material point M in reference and actual configurations

2.1 Material and Spatial Descriptions of Continuum Motion

7

It should be noticed that if the continuum motion law (2.3) (or (2.4)) is known, then one of the main problems of continuum mechanics (to determine coordinates of all material points of the continuum at any time) will be resolved. However, in actual problems of continuum mechanics this law, as a rule, is unknown and must be found by solving some mathematical problems, whose statements are to be formulated. One of our objectives is to derive these statements. Example 2.1. Let us consider a continuum B, which at time t D 0 in the reference ı

ı

ı

configuration K is a rectangular parallelepiped (a beam) with edge lengths h1 , h2 ı

and h3 , and in an actual configuration K at t > 0 the continuum is also a rectangular parallelepiped but with different edge lengths: h1 , h2 and h3 . We assume that corresponding sides of both the parallelepipeds lie on parallel planes, and for one of the sides, which for example is situated on the plane .x 2 ; x 3 /, points of diagonals’ ı

intersection in K and in K are coincident (Fig. 2.2). Then the motion law (2.3) for this continuum takes the form x ˛ D k˛ .t/ X ˛ ; ı

˛ D 1; 2; 3;

(2.6)

ı

i.e. coordinates x i ; x i D X i of any material point M in K and K are proportional, ı

and k˛ .t/ D h˛ .t/=h˛ is the proportion function. The motion law (2.6) is called the beam extension law. t u ı

Example 2.2. In K, let a continuum B be a rectangular parallelepiped oriented as shown in Fig. 2.3; its motion law (2.3) has the form

Fig. 2.2 Extension of a beam

Fig. 2.3 Simple shear of a beam

8

2 Kinematics of Continua

8 1 1 2 ˆ ˆ <x D X C a.t/X ; x2 D X 2; ˆ ˆ :x 3 D X 3 ;

(2.7)

ı

where x i D X i , a.t/ is a given function. In K this continuum B has become a parallelepiped, all cross-sections of which are planes orthogonal to the Ox 3 axis and are the same parallelograms. This motion law is called simple shear; the tangent of the shear angle ˛ is equal to a. t u ı

Example 2.3. Consider a continuum B, which in K is a rectangular parallelepiped (a ı

beam) shown in Fig. 2.4; under the transformation from K to K this parallelepiped changes its dimensions without a change in its angles (as in Example 2.1) and rotates by an angle '.t/ in the plane Ox 1 x 2 around the point O (Fig. 2.4). The motion law for the continuum is called the rotation of a beam with extension. In this case Eq. (2.1) have the form ı

ı

x i D F0ij x j ;

xj D X j ;

(2.8)

where the matrix F0ij is the product of two matrices, the rotation matrix O0 and the stretch matrix U0 : F0ij D O0i k U0kj ;

U0ij

0 1 k1 0 0 D @ 0 k2 0 A ; 0 0 k3

O0i j

F0ij

0 cos '  sin ' D @ sin ' cos ' 0 0

1 0 0A ; 1

0 1 k1 cos ' k2 sin ' 0 D @ k1 sin ' k2 cos ' 0 A ; 0 0 k3

and k˛ .t/ D h˛ .t/= h0˛ are the proportion functions characterizing the ratio of ı

lengths of the beam edges in K and K (as in Example 2.1).

Fig. 2.4 Rotation of a beam with extension

t u

2.1 Material and Spatial Descriptions of Continuum Motion

9

2.1.2 Material and Spatial Descriptions In continuum mechanics, physical processes occurring in bodies are characterized by a certain set of varying scalar fields  D .M; t/, vector fields a D a.M; t/, and tensor fields of the nth order n .M; t/. We will consider tensors and tensor fields in detail in Sect. 2.1.4 (see also [12]). Since in the Cartesian coordinate system O eN i a material point M corresponds to both Lagrangian coordinates X i and Eulerian coordinates x i , varying scalar and vector fields can be written as follows: .X i ; t/ D .X i .x j ; t/; t/ D e .x j ; t/ D e .x; t/; a.X i ; t/ D e a.x j ; t/ D e a.x; t/;

(2.9)

With the help of the motion law (2.3) (or (2.4)), we can pass from functions of Lagrangian coordinates to functions of Eulerian coordinates in formulae (2.9). In continuum mechanics the tilde e is usually omitted (we will do this below). For a fixed time t in (2.9), we obtain stationary scalar and vector fields. If in (2.9) a material point M is fixed, and time t changes within the interval 0 6 t 6 t 0 , then we get an ordinary scalar function  D .M; t/ and vector function a D a.M; t/ depending on time. According to relationships (2.9), there are two ways to describe different physical processes in continua. In the material (Lagrangian) description of a continuum, all tensor fields describing physical processes are considered as functions of X i and t. In the spatial (Eulerian) description, all tensor fields describing physical processes are functions of x i and t. Both the descriptions are equivalent. It should be noted that for solids we more often use the material description, where it is convenient to fix coordinates X i of a material point M and to observe its motion at different times t. For gaseous and fluid continua, Eulerian description is more convenient; when an observer fixes a geometric point with coordinates x i and monitor the material points M passing through this point x i at different times t.

ı

2.1.3 Local Bases in K and K Using the motion law (2.4) and relationship (2.1), at every material point M with coordinates X i in the actual and reference configurations we can introduce its local basis vectors: rk D

@x @x i eN i D Qik eN i ; D k @X @X k

ı

rk D

ı

ı

ı @x @x i eN i D Qik eN i ; D k k @X @X

(2.10)

10

2 Kinematics of Continua

Fig. 2.5 Local basis vectors in reference and actual configurations

where ı

ı

Q ik D @x i =@X k ;

Qik D @x i =@X k ;

P ik D @X i =@x k ;

P ik D @X i =@x k

ı

ı

(2.11)

are Jacobian matrices and inverse Jacobian matrices. ı Here and below all values referred to the configuration K will be denoted by ı superscript ı . As follows from the definition (2.11), local bases vectors rk and rk are directed tangentially to corresponding coordinate lines X k (Fig. 2.5). ı

ı

In K and K introduce metric matrices gkl , g kl and inverse metric matrices g kl ,

ı kl

g

as follows: j

gkl D rk  rl D Qik Q l ıij D ı

ı

ı

ı

g kl D rk  rl D

@x i @x j ıij ; @X k @X l

ı

@x i @x j k ıij ; g kl glm D ım ; @X k @X l

ı

ı

k g kl g lm D ım ;

(2.12)

and also vectors of reciprocal local bases ri D g i m rm ;

ı

ı

ı

ri D g i m rm ;

(2.13)

which satisfy the reciprocity relations ri  rj D ıi j ;

ı

ı

ri  rj D ıi j ;

(2.14a)

q ı ı rn  rm D g nmk rk ;

(2.14b)

and also the following relations: rn  r m D

p

k

g nmk r ;

ı

ı

2.1 Material and Spatial Descriptions of Continuum Motion

11

With the help of the mixed multiplication, i.e. sequentially applying scalar and vector products to three different local bases vectors, we can determine the volumes ı

jV j and jV j constructed by these vectors: q q ˇ ˇ ı ı ı ˇ ˇ jV j D r1  .r2  r3 / D g D det .g ij / D ˇ@x k =@X i ˇ; p jV j D r1  .r2  r3 / D g D j@x k =@X i j: ı

ı

ı

ı

(2.15)

ı

It should be noted that although local bases ri and ri have been introduced in ı

different configurations K and K, they correspond to the same coordinates X i (if one consider the same point M); therefore each of the bases can be carried as a ı

rigid whole into the same point in K or in K. Due to this, we can resolve any vector ı ı field a.M/ for each of the bases ri , ri , ri and ri : ı ı

ı ı

a D ai ri D ai ri D ai ri D ai ri :

(2.16) ı

If curvilinear coordinates X i are orthogonal, then the vectors ri are orthogonal as ı ı ı ı well: (ri  rj D ıij ), and matrices g ij and g ij are diagonal; hence we can introduce q ı ı Lam´e’s coefficients H ˛ D g ˛˛ (˛ D 1; 2; 3) and the physical orthonormal basis ı b ı ı ı ı r˛ D r˛ =H ˛ D r˛ H ˛ :

(2.17)

Components of a vector a with respect to this basis are called physical: ı b ı b a D ai ri :

(2.18) ı

The actual basis ri is in general not orthogonal even if the basis ri is orthogonal; therefore we cannot introduce the corresponding physical basis in K. One can introduce a physical basis in K not with the help of ri , but with the help of another special basis (see Sect. 2.1.7).

2.1.4 Tensors and Tensor Fields in Continuum Mechanics ı

ı

For different local bases ri , ri , ri , ri or eN i at every point M, and with the help of formulae given in the work [12] we can introduce different dyadic (tensor) bases: ı ı ı ı ı ri ˝rj , ri ˝ rj , ri ˝rj , ri ˝ rj , ri ˝rj etc., which are equivalence classes of vector sets consisting of 2  3 D 6 vectors (for example, r1 ˝ rj D Œr1 rj r2 0r3 0, where Œ  is the notation of an equivalence class), and ˝ is the sign of tensor product. A field

12

2 Kinematics of Continua

of second-order tensor T.M/ can be represented as a linear combination of dyadic basis elements: ı

ı

ı ı ı ı T D T ij ri ˝ rj D T ij ri ˝ rj D TN ij eN i ˝ eN j D T ij ri ˝ rj :

(2.19)

During the passage from one basis to another, tensor components T ij are transformed by the tensor law: ı

ı

ı

TN ij D P ik P jl T kl D P ik P jl T kl : ı

(2.20)

ı

Metric matrices g i m , gi m , g i m and g i m are components of the unit (metric) tensor E with respect to different bases: ı

ı

ı

ı

ı

ı

E D gi m ri ˝ rm D g i m ri ˝ rm D gi m ri ˝ rm D g i m ri ˝ rm :

(2.21)

For a second-order tensor T, in continuum mechanics one often uses the transpose tensor TT D T ij rj ˝ ri and the inverse tensor T1 , where T1  T D E. The inverse tensor exists only for a nonsingular tensor (when det T ¤ 0). The determinant of a tensor is defined by the determinant of its mixed components matrix: det T D det T ij . Besides second-order tensors, in continuum mechanics one sometimes uses tensors of higher orders [12]. To introduce the tensors, we define polyadic bases by induction: ri1 ˝ : : : ˝ rin ; the bases are equivalence classes of vector sets consisting of n  3 D 3n vectors. A field of nth order tensor n .M/ can be represented by a linear combination of polyadic basis elements: n

ı

ı

ı

 D i1 :::in ri1 ˝ : : : ˝ rin D i1 :::in ri1 ˝ : : : ˝ rin ; ı

where i1 :::in and i1 :::in are components of the nth order tensor with respect to the corresponding polyadic basis. For fourth-order tensors, analogs of the tensor E are the first, second and third unit tensors defined as follows: I D ei ˝ ei ˝ ek ˝ ek D E ˝ E; II D ei ˝ ek ˝ ei ˝ ek ; III D ei ˝ ek ˝ ek ˝ ei ;

(2.22)

and also the symmetric fourth-order unit tensor D

1 .II C III /: 2

 D ijkl ei ˝ ej ˝ ek ˝ el ;

ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

(2.22a)

2.1 Material and Spatial Descriptions of Continuum Motion

13

We can transpose fourth-order tensors as follows: 4

.m1 m2 m3 m4 / D i1 i2 i3 i4 rim1 ˝ rim2 ˝ rim3 ˝ rim4 ;

where .m1 m2 m3 m4 / is some substitution, for example, 4 .4321/ D i1 i2 i3 i4 ri4 ˝ ri3 ˝ ri2 ˝ ri1 :

ı

2.1.5 Covariant Derivatives in K and K ı

Introduce the following nabla-operators in configurations K and K, respectively: ı

@ ; @X k

r D rk ˝

ı

r D rk ˝

@ : @X k

(2.23)

Applying the nabla-operators to a vector field, we get the gradients of a vector in ı

K and K: r ˝ a D rk ˝ ı

ı

r ˝ a D rk ˝

ı

@a @X k

D rk ai rk ˝ ri ; ı

ı

ı ı

ı

ı

ı ı

ı ı

ı

D r k ai rk ˝ ri D r k ai rk ˝ ri D r k ai rk ˝ ri ; (2.24)

@a @X k

where we have denoted the following covariant derivatives in different tensor bases ı

in configurations K and K: ı

ı

r k ai D rk ai D

ı

@a i @X k @ai @X k

ı

ı

ı

ı

 m a ; ik m

r k ai D

 m a ; ik m

rk ai D

ı

@ai @X k @ai @X k

ı

ı

C  ikm am C  ikm am :

(2.25) ı

ı

Here ijm and  m ij are the Christoffel symbols in configurations K and K. For the Christoffel symbols the following relations (see [12]) hold: 

@gkj @gki @gij C  i j @X @X @X k



m ij

1 D g km 2

ı

0 1 ı ı ı @g ij 1 ı km @ @g kj @g ki A: D g C  2 @X i @X j @X k

m ij

;

(2.26)

ı

Contravariant derivatives in K and K are introduced as follows: ı

ı

ı

ı

ı

r k ai D g km r m ai ;

r k ai D gkm rm ai :

(2.27)

14

2 Kinematics of Continua ı

The covariant derivatives (2.25) are components of the second-order tensors r ˝ a and r ˝ a, therefore during the passage from the local basis ri to another one they are transformed by the tensor law (2.20). ı

ı

The nabla-operators r and r in K and K can be applied to a tensor field n .X i / of nth order: r ˝ n  D rk ˝

ı ı @ n ı ı ı D r k i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; k @X

r ˝ n  D rk ˝

@ n D rk i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; @X k

ı

ı

ı

(2.28) ı

ı

where rk i1 :::in and r k i1 :::in are the covariant derivatives in K and K, respectively: n ı ı @ ı i1 :::in X ı is ı i1 :::is Dm:::in i1 :::in rk  D  C  mk  : (2.29) @X k sD1 In the same way we can define operations of scalar product of the nabla-operator in ı

K (the divergence of a tensor): ı

ı

r  n  D rk 

ı ı @ n ı ı D r k ki2 :::in ri2 ˝ : : : ˝ rin ; k @X

(2.30)

ı

and vector product of the nabla-operator in K (the curl of a tensor): ı

ı

r  n  D rk 

@ n 1 ijk ı ı ı ı ı D q  r i j i2 :::in rk ˝ ri2 ˝ : : : ˝ rin : ı @X k g

(2.31)

2.1.6 The Deformation Gradient Consider how a local neighborhood of a point M is transformed during the passage ı

ı

from configuration K to K. Take an arbitrary elementary radius-vector d x connectı

ing in K two infinitesimally close points M and M0 (Fig. 2.6). In configuration K, these material points M and M0 are connected by the elementary radius-vector d x. ı The vectors d x and d x can always be resolved for local bases: ı

d x.X k / D

@x @x ı ı dX k D rk dX k ; d x.X k / D dX k D rk dX k : k k @X @X

(2.32)

2.1 Material and Spatial Descriptions of Continuum Motion

15

Fig. 2.6 Transformation of an elementary radius-vector during the passage from the reference configuration to the actual one

ı

On multiplying the first equation by rm and the second – by rm , we get rm  d x D rm  rk dX k D dX m ;

ı

ı

ı

ı

rm  d x D rm  rk dX k D dX m :

(2.33) ı

Substitution of dX m (2.33) into the first equation of (2.32) yields d x D rk ˝ rk  d x. Changing the order of the tensor and scalar products (that is permissible by the ı tensor analysis rules), we get the relation between d x and d x: ı

ı

d x D F  d x:

(2.34)

Here we have denoted the linear transformation tensor ı

F D rk ˝ rk ;

(2.35)

called the deformation gradient. As follows from (2.34), the deformation gradient ı connects elementary radius-vectors d x and d x of the same material point M in ı

configurations K and K. Definition (2.19) allows us to give a geometric representation of the deformation ı gradient: if ri are considered as the left vectors and ri – as the right vectors, then, by formulae of Sect. 2.1.4 (see [12]), the tensor F takes the form ı

ı

ı

ı

F D ri ˝ ri D Œr1 r1 r2 r2 r3 r3 : According to the geometric definition of a tensor (see Sect. 2.1.4), the tensor ı F can be represented as equivalence class of the ordered set of six vectors ri ; ri (Fig. 2.7).

16

2 Kinematics of Continua

Fig. 2.7 Geometric representation of the deformation gradient

Besides F, in continuum mechanics one often uses the transpose tensor FT , the inverse tensor F1 and the inverse to the transpose tensor F1T : ı

ı

FT D rk ˝ rk D rk ˝

@x @X k

ı

ı

D r ˝ x; ı

F1T D .rk ˝ rk /T D rk ˝ rk D rk ˝

ı

F1 D rk ˝ rk ; ı

@x @X k

ı

D r ˝ x:

(2.35a)

It follows from (2.35) that ı

ı

F  ri D rk ˝ rk  ri D rk ıik D ri :

(2.36)

i.e. the deformation gradient transforms local bases vectors of the same material ı

point M from K to K. Theorem 2.1. The transpose deformation gradient FT connects gradients of an arı

bitrary vector a in K and K: ı

r ˝ a D FT  r ˝ a;

ı

r ˝ a D F1T  r ˝ a:

(2.37)

H To derive formulae (2.37), we apply the definitions (2.24) and (2.35): r ˝ a D ri ˝

ı @a @a @a ı ı D rj ıji ˝ D rj ˝ rj  ri ˝ D F1T  r ˝ a: N (2.38) i i i @X @X @X

2.1.7 Curvilinear Spatial Coordinates Notice that the choice of Cartesian basis O eN i as a fixed (immovable) system in the spatial (Eulerian) description of the continuum motion is not a necessary condition. For some problems of continuum mechanics it is convenient to consider a moving ! system O 0 eN 0i with the origin at a moving point O 0 (x0 D OO 0 ) and a moving orthonormal basis eN 0i (Fig. 2.8), which is connected to eN i by the orthogonal tensor Q: eN 0i D Q  eN i :

(2.39)

In this case, instead of Cartesian coordinates x of a point M in the basis eN i one consider its Cartesian coordinates e x i in basis eN 0i : i

e x D x  x0 D e x i eN 0i :

(2.40)

2.1 Material and Spatial Descriptions of Continuum Motion

17

Fig. 2.8 Moving bases eN0i ande ri and curvilinear spatial coordinates e X i in moving system O 0 eN 0i

Let Q ij be components of the tensor Q with respect to the basis eN i : Q D Qij eN i ˝ eN j ;

(2.41)

then relation (2.39) takes the form eN 0i D Qji eN j ;

(2.42)

and coordinates e x i and x i are connected as follows: j e x i eN 0i D e x i Q i eN j ; x D x  x0 D .x i  x0i /Nei D e

x j ; @x i =@e x j D Qij ; x i  x0i D Qij e

@e x j =@x i D P ji :

(2.43)

Instead of Cartesian coordinates e x i , we can consider special curvilinear coordinates k 0 e X with the origin at point O : e k /; x i .X e xi D e

(2.44)

which, due to (2.27), are connected to x i by the relations e k /  x i .X e k ; t/ or X ej D X e j .x i ; t/: x i D x0i .t/ C Qij .t/e x j .X

(2.45)

The dependence on t in the relations is defined by functions x0i .t/ and Q ij .t/ (i.e. only by the motion of system O 0 eN 0i ), which are assumed to be known in continuum mechanics. e i are no longer Lagrangian (material): at different times they Coordinates X correspond to different material points. However, it is often convenient to choose

18

2 Kinematics of Continua ı

e i coincident with X i in the reference configuration K. In this case we coordinates X have the relations ı e j ; 0/: x i .X i / D x i .X j ; 0/ D x i .X

(2.46)

With the help of transformation (2.45) we can use the spatial description in coore i as well when consider the functions dinates X e i ; t/I a.X a D a.x i ; t/ D e

(2.47)

e i are called curvilinear spatial coordinates. therefore coordinates X Introduce local vectors e ri D

@x @x j D eN : ei ei j @X @X

(2.48)

x D x, and In particular, the basis eN 0i may be fixed (Fig. 2.9), then eN 0i D eN i , e ej D X e j .x i / are independent of t; the basis e curvilinear spatial coordinates X ri is independent of t as well, and from (2.46) and (2.48) it follows that the basis ı coincides with ri : ı @x j @x j ı e Nj D ri D (2.49) e eN j D ri : ei @X i @X ı

When the basis eN 0i is moving, basese ri and ri are no longer coincident. e i and defined The vectors e ri are directed tangentially to the coordinate lines X simultaneously with ri at every point M at any time t > 0. A change of vectorse ri in time is defined only by the motion of basis eN 0i , because from (2.42), (2.43), and (2.48) it follows that j eN 0i D Q i

 @X ek ek ek  @X @x j @X e e e r r rk ; D D k k @x j @e x i @x j @e xi

(2.50)

e k =@e e k  @X x i is independent of t according to (2.44). and the matrix P i

Fig. 2.9 Curvilinear spatial coordinates e X i and Lagrangian coordinates X i for the fixed basis 0 eNi D eN i

2.1 Material and Spatial Descriptions of Continuum Motion

19

The bases vectors ri and e ri are connected as follows: ri D

ek ek @x @x @X @X e D D rk : e k @X i @X i @X i @X

(2.51)

Just as in Sect. 2.1.2, we define the metric matrix e g ij and the inverse metric matrix e g ij : e g ij D e ri e rj ; e g ij e g jk D ıki ; (2.52) and the reciprocal basis vectors e g i ke ri D e rk D

ei ei @X @X j N D e eN 0k : @x j @e xk

(2.53)

According to formulae (2.51) and (2.52), we find the relation between matrices gij and e gkl : gij D ri  rj D

el el e k @X e k @X @X @X e e r D e g kl : r k l @X i @X j @X i @X j

(2.54)

The inverse matrix g ij is found from (2.54) by the rule of matrix product inversion (see Exercise 2.1.13): gij D

@X i @X j kl e g : e k @X el @X

(2.55)

From (2.51), (2.53) and (2.55) we can find the relation between vectors of reciprocal bases ri and e ri : ri D g ij rj D

e ri :

em @X i @X j kl @X @X i k e e r r : e g D m e k @X el ek @X j @X @X

(2.56)

Let there be a tensor n , then it can be resolved for the basis ri and for the basis n

ei1 :::ine  D i1 :::in ri1 ˝ : : : ˝ rin D  rin : ri1 ˝ : : : ˝e

(2.57)

On substituting (2.51) into (2.57), we derive transformation formulae for tensor ei : components during the passage from coordinates X i to X e i1 e in e i1 :::in D j1 :::jn @X : : : @X :  @X j1 @X jn

(2.58)

e of covariant differentiation in coordinates X ei : Introduce the nabla-operator r @ e De r ri e @X i

(2.59)

20

2 Kinematics of Continua

and contravariant derivatives of components e a i of a vector a D e aie ri in coordii e : nates X @e ai @e ai i eke eke ai D e m e a ; r a D Ce  ikme am : (2.60) r m ik ek ek @X @X ei The Christoffel symbols e m g ij by the relaij in coordinates X are connected to e tions which are similar to (2.26). e i and X i (in Theorem 2.2. The results of covariant differentiation in coordinates X the configuration K) are coincident: e ˝ n ; r ˝ n D r

e  n ; r  n D r

e  n : r  n D r

(2.61)

H Prove the first formula in (2.61). Due to (2.23), we have r ˝ n  D ri ˝

el @ n @X i k @ n  @X @ n e ˝ n : (2.62) e D D ıkl rk ˝ Dr r ˝ i i k l e e @X el @X @X @X @X

The remaining two formulae in (2.61) can be derived in the same way (see Exercise 2.1.8). N Going to components of a tensor n  with respect to bases ri and e ri , from (2.58) we get the relation between the covariant derivatives: ei  e j1 :::jn : ri j1 :::jn D r

(2.63)

e i into X j : Determine the tensor H transforming coordinates X ı e ij e rj D H rj D HN ij eN i ˝ eN j : ri ˝e H D rj ˝e

(2.64)

Then we get the relations (see Exercises 2.1.10 and 2.1.11): ı

ej e ri D H e ri D H i rj ;

ı

e ij /1e ri D H1T e r i D .H rj :

(2.65)

e i are often chosen orthogonal, then the bases e The coordinates X ri and e ri are ij orthogonal as well, and the matrices e g ij and e g are diagonal; and we can introduce the physical (orthonormal) basis: b e ˛; e r˛ D e r˛ = H

(2.66)

p e g ˛˛ are Lam´ qe’s coefficients, which are in general not coincident ı ı e r˛ with the coefficients H ˛ D g ˛˛ . Tensor components with respect to the basis b are called physical: b eijb e e TDT ri ˝ b rj : (2.67) e˛ D where H

Relations between physical and covariant components of a tensor are determined by the known formulae (see [12]).

2.1 Material and Spatial Descriptions of Continuum Motion

21

Exercises for 2.1 2.1.1. With the help of formulae (2.10), (2.12), (2.13), and (2.17) show that if the motion law of a continuum describes extension of a beam (2.6) (see Example 2.1), ı then the local basis vectors ri and the metric matrices have the forms ı

ı

ri D ei ; r˛ D k˛ eN ˛ ;

ri D ei ;

r˛ D .1=k˛ /Ne˛ ;

ı

˛ D 1; 2; 3;

ı

g ij D ıij ; g ij D ı ij ; 1 1 0 2 0 2 k1 0 0 k1 0 0 .gij / D @ 0 k22 0 A ; .g ij / D @ 0 k22 0 A ; 0 0 k32 0 0 k32 i.e.

g˛ˇ D k˛2 ı˛ˇ ; g ˛ˇ D k˛2 ı˛ˇ ; H˛ D

p

g˛˛ D k˛ ; b r˛ D e ˛ :

2.1.2. Show that if the motion law of a continuum describes a simple shear (see Example 2.2), then the local basis vectors and the metric matrices have the forms ı

ri D ei ;

ı

r i D ei ;

ı

ı

g ij D ıij ;

r1 D eN 1 ; r2 D aNe1 C eN 2 ;

gij D ı ij ; r3 D eN 3 ;

r1 D eN 1  aNe2 ; r2 D eN 2 ; r3 D eN 3 ; 0 0 1 1 a 0 1 C a2 a ij 2 gij D @a 1 C a 0A ; g D @ a 1 0 0 0 0 1

1 0 0A : 1

2.1.3. Show that if the motion law describes rotation of a beam with extension (see Example 2.3), then with introducing the rotation tensor O0 and the stretch tensor U0 : 3 X O0 D O0i j eN i ˝ eN j ; U0 D k˛ eN ˛ ˝ eN ˛ ; ˛D1

we can rewrite the beam motion law in the tensor form ı

x D F0  x;

F0 D O0  U0 :

Show that the local basis vectors and metric matrices for this problem have the forms ri D F0ki eN k ;

ı

ri D eN i ;

22

2 Kinematics of Continua

gij D F0ki F0lj ıkl

0 2 1 k1 cos2 ' C k22 sin2 ' .k12  k22 / cos ' sin ' 0 D @ .k12  k22 / cos ' sin ' k12 sin2 ' C k22 cos2 ' 0 A ; 0 0 k32 g D k1 k2 k3 ;

1 k22 sin2 ' C k12 cos2 ' .k12  k22 / cos ' sin ' 0 gij D @ .k12  k22 / cos ' sin ' k22 cos2 ' C k12 sin2 ' 0 A : 0 0 k32 0

2.1.4. Using the property (2.14) of reciprocal basis vectors, show that the following relations hold: @X i @X i k ı ri D ı eN k ; ri D P ik eN k D eN : @x k @x k 2.1.5. Show that F, FT , F1 and F1T in the Cartesian coordinate system take the forms FD

@x m ı

@x i

eN m ˝ eN i ;

FT D

@X k ı

eN i ˝

@x i

@x m @x m i N D e eN ˝ eN m ; m ı @X k @x i

ı

F

1

ı

@x m @X k i @x m N N D ˝ D e e eN m ˝ ei ; m @x i @x i @X k ı

F1T D

ı

@X k i @x m @x m i N N ˝ D e e eN ˝ em : m @x i @x i @X k

2.1.6. Substituting (2.54), (2.52), and (2.55) into (2.12), derive formula (2.55). 2.1.7. Prove that

ı

ri D F1T  ri :

2.1.8. Derive the third formula of (2.61). ı

2.1.9. Prove that for any scalar function '.X i / its gradients in K and K are connected by the relationship ı

r ' D F1T  r ': 2.1.10. Show that formulae (2.65) follow from (2.64). 2.1.11. Using (2.47), show that in formulae (2.64) the tensor H has the following components with respect to bases eN i and e ri : ı ı k i ek ei @x i @X @x i @X k e i @x k @X i i 1 N e ij /1 D @x @X : N ; . H / D ; H D ; . H H jD k j j e k @xı j e j @xı k @X j @x k @X @x j @X @X

2.1 Material and Spatial Descriptions of Continuum Motion

23

ı

2.1.12. Introducing the notation F ij for components of the deformation gradient F ı

ı

ı

ı

ı

ı

ı

with respect to basis ri : F D F ij ri ˝ rj D F ij ri ˝ rj ; show that formula (2.36) yields ı

ı

rj D F ij ri : 2.1.13. Show that the Levi-Civita symbols are connected by the relations ijk  ijk D 6; p

ijk  i lm D ıjl ıkm  ıkl ıjm ;

p g ijk D .1= g/  mnl gmi gnj glk :

ijk  ijl D 2ıil ; ijk T jk D 0

where T jk are components of an arbitrary symmetric tensor: T jk D T kj . 2.1.14. Using relations (2.14a), show that the local bases vectors are connected by the relations q ı ı p  ı ı r˛  rˇ D g r ; r˛  rˇ D g r ; ˛ ¤ ˇ ¤  ¤ ˛: 2.1.15. Show that the unit fourth-order tensors I , II and III defined by formulae (2.22) have the following properties: I   T D I1 .T/E; I1 .T/ D T   E; II   T D TT ; III   T D T;

TD

1 .T C TT /; 2

and I   4  D E ˝ E   4 ; III   4  D 6 ;

II   4  D .2134/ ;

   4 D

1 .2134/ C /; . 2

for arbitrary second-order tensor T and fourth-order tensor 4 . As follows from these formulae, the tensor III is the ‘true’ unit fourth-order tensor. 2.1.16. Show that components of the symmetric unit fourth-order tensor  with respect to a tetradic basis have the form D

1 .ei ˝ el ˝ ei ˝ el C ei ˝ el ˝ el ˝ ei / D ijkl ei ˝ ej ˝ ek ˝ el : 2 ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

2.1.17. Show that for any second-order tensor T and for any vector a the following formula of covariant differentiation hold: r  .T  a/ D T   .r ˝ a/T C a  r  T:

24

2 Kinematics of Continua

2.2 Deformation Tensors and Measures 2.2.1 Deformation Tensors Besides F, important characteristics of the motion of a continuum are deformation tensors, which are introduced as follows: 1 gij 2 1 AD gij 2 1  ı ij g ƒD 2 1  ı ij JD g 2 CD

ı ı ı ı ı  g ij ri ˝ rj D "ij ri ˝ rj ;  ı  g ij ri ˝ rj D "ij ri ˝ rj ; ı ı ı ı  gij ri ˝ rj D "ij ri ˝ rj ;   gij ri ˝ rj D "ij ri ˝ rj :

(2.68)

Here C is called the right Cauchy–Green deformation tensor, A – the left Almansi deformation tensor, ƒ – the right Almansi deformation tensor, and J – the left Cauchy–Green tensor. As follows from the definition of the tensors, covariant components of C and A are coincident, but they are defined with respect to different tensor bases. Components "ij are called covariant components of the deformation tensor. Contravariant components of the tensors ƒ and J are also coincident and called contravariant components "ij of the deformation tensor, but they are defined with respect to different tensor bases of the tensors ƒ and J. Notice that the deformation tensor components "ij D

 1 ı gij  g ij ; 2

"ij D

 1  ı ij g  gij ; 2

(2.69)

have been defined independently of each other, therefore the formal rearrangement of indices is not permissible for these components, i.e. "Mkl D "ij gi k g jl ¤ "kl ;

"Mkl D "ij gi k gjl ¤ "kl :

(2.70)

Thus, when there is a need to obtain contravariant components from "ij and covariant components from "ij , one should use the notation "M kl and "Mkl . We will also use the notation ı ı ı ı ı ı "kl D "ij g i k g jl ; "kl D "ij g i k g jl : (2.71) Theorem 2.3. The deformation tensors C, A, ƒ and J are connected to the deformation gradient F as follows: C D 12 .FT  F  E/;

A D 12 .E  F1T  F1 /;

ƒ D 12 .E  F1  F1T /;

J D 12 .F  FT  E/:

(2.72)

2.2 Deformation Tensors and Measures

25 ı

H Let us derive a relation between C and F. Having used the definitions of gij , g ij and F, we get   1 ıi 1 1 ı ı ı r ˝ ri  rj ˝ rj  E D .FT  F  E/: .ri  rj /ri ˝ rj  E D 2 2 2 (2.73) The remaining relations of (2.72) can be proved in the same way:   1 1 ı ı AD E  ri ˝ ri  rj ˝ rj D E  F1T  F1 ; 2 2   1 1 ı ı E  ri ˝ ri  rj ˝ rj D E  F1  F1T ; ƒD 2 2   1 1 ıi ı j ri ˝ r  r ˝ rj  E D F  FT  E : N JD (2.73a) 2 2

CD

2.2.2 Deformation Measures Besides the deformation tensors, we define deformation measures: the right Cauchy– Green measure G and the left Almansi measure g: ı

ı

G D gij ri ˝ rj D FT  F D E C 2C; ı

g D g ij ri ˝ rj D F1T  F1 D E  2A;

(2.74)

and also the left Cauchy–Green measure g1 and the right Almansi measure G1 : ı

g1 D g ij ri ˝ rj D F  FT D E C 2J; ı

ı

G1 D gij ri ˝ rj D F1  F1T D E  2ƒ:

(2.75)

2.2.3 Displacement Vector Introduce a displacement vector u of a point M from the reference configuration to the actual one as follows (Fig. 2.10): ı

u D x  x:

(2.76)

Theorem 2.4. The deformation tensors and the deformation gradient are connected to the displacement vector u by the relations ı

F D E C .r ˝ u/T ; ı

FT D E C r ˝ u;

F1 D E  .r ˝ u/T ; F1T D E  r ˝ u;

(2.77)

26

2 Kinematics of Continua

Fig. 2.10 The displacement vector of a point M from the reference configuration to the actual one

and also CD AD ƒD JD

  ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u ; 2  1 r ˝ u C r ˝ uT  r ˝ u  r ˝ uT ; 2  1 r ˝ u C .r ˝ u/T  r ˝ uT  r ˝ u ; 2  ı ı ı 1 ı r ˝ u C r ˝ uT C r ˝ uT  r ˝ u : 2

(2.78)

H The definition (2.76) of the displacement vector and the properties (2.35) of the deformation gradient yield ı

ı

ı

FT D r ˝ x D r ˝ .x C u/ ı

D ri ˝

ı

ı ı ı @x ı ı C r ˝ u D ri ˝ ri C r ˝ u D E C r ˝ u: i @X

Then the tensor C takes the form   ı ı 1 T CD .E C r ˝ u/  .E C r ˝ u /  E 2   ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u : D 2

(2.79)

(2.80)

In a similar way, we can prove the remaining relations of the theorem. N

2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector ı

The displacement vector u can be resolved for both bases ri and ri : ı ı

u D ui ri D ui ri :

(2.81)

2.2 Deformation Tensors and Measures

27

The derivative with respect to X i can be determined in both the bases as well: ı ı ı @u k k D r i u rk D ri u rk : @X i

(2.82)

Then the displacement vector gradients take the forms ı

ı

r ˝ u D ri ˝

ı ı ı ı ı ı @u ı ı k i i k D r u r ˝ r D r u ri ˝ rk ; i k @X i

@u D ri uk ri ˝ rk D r i uk ri ˝ rk : @X i Substitution of these expressions into (2.77) gives r ˝ u D ri ˝

  ı ı ı ı ı ı ı F D ıik C r i uk rk ˝ ri D F ki rk ˝ ri :

(2.83)

(2.84)

(2.85)

Here we have introduced components of the deformation gradient in the reference configuration: ı ı

ı

F ki D ıik C r i uk :

(2.86)

The transpose gradient FT has the components ı

ı

ı

ı

ı

ı

FT D F ki ri ˝ rk D F ik rk ˝ ri ; ı

ı

ı

F ik D ıik C r k ui ; ı

(2.87)

(2.88)

ı

where .F ik /T D F ki . In a similar way, one can find the expression for the inverse gradient    k F1 D ıik  ri uk rk ˝ ri D F 1 i rk ˝ ri

(2.89)

and for the inverse–transpose gradient F1T D .F 1 /ki ri ˝ rk D .F 1 /ik rk ˝ ri ;

(2.90)

where their components with respect to the actual configuration are expressed as follows: .F 1 /ik D ıik  r k ui ; (2.91) .F 1 /ki D ıik  ri uk : Thus, we have proved the following theorem.

(2.92)

28

2 Kinematics of Continua

Theorem 2.5. Components of the deformation gradients F, FT , F1 and F1T in ı

local bases of configurations K and K are connected to components of the displacement vector u by relations (2.86), (2.87), (2.91), and (2.92). On substituting formulae (2.83) and (2.84) into (2.78) for C and A and comparing them with (2.68), we get   ı ı ı ı ı ı 1 ı ı "ij D r i u j C r j u i C r i uk r j uk ; 2  1 "ij D ri uj C rj ui  ri uk rj uk ; 2

(2.93)

– The expressions for covariant components of the deformation tensor in terms of ı

components of the displacement vector with respect to K and K. In a similar way, substituting (2.83) and (2.84) into (2.78) for ƒ and J, we obtain   ı ı ı ı ı ı 1 ı i ıj r u C r j u i C r k u i r k uj ; 2   1 r i uj C r j u i  r k u i rk uj "ij D 2 "ij D

(2.94)

– The relations between contravariant components of the deformation tensor and ı

components of the displacement vector in K and K. Then with using relations (2.69), (2.93) and (2.94), we can find the connection between the metric matrices and displacement components: ı

ı ı

ı

ı

ı ı

ı

ı ı

ı

ı

ı

ı

ı

ı

ı

gij D g ij C r i uj C r j ui C r i uk rj uk D g ij Cri uj Crj ui ri uk rj uk ; (2.95) ı ı

g ij D g ij C r i uj C r j ui C r k ui r k uj D g ij Cr i uj Cr j ui r k ui rk uj : (2.96) Thus, we have proved the following theorem. Theorem 2.6. Components of the deformation tensor "ij , "ij and metric matrices gij , g ij are connected to components of the displacement vector u by relations (2.93)–(2.96).

2.2.5 Physical Meaning of Components of the Deformation Tensor Let us clarify now a physical meaning of components of the deformation tensor:   1 1 ı ı ı gij  g ij D ri  rj  ri  rj : "ij D (2.97) 2 2

2.2 Deformation Tensors and Measures

29

By the definition of the scalar product (see [12]), we have "˛ˇ ı

where ı

˛ˇ

and

 1 D jr˛ jjrˇ j cos 2

ı

ı

 jr˛ jjrˇ j cos

˛ˇ



ı

;

˛ˇ

(2.98) ı

˛ˇ

ı

are the angles between basis vectors r˛ , rˇ and r˛ , rˇ in K and

K, respectively.

ı

ı

Consider elementary radius-vectors d x and d x in configurations K and K, and ı introduce their lengths ds and d s, respectively: ds 2 D d x  d x;

ı

ı

ı

d s 2 D d x  d x:

(2.99)

ı

Since d x is arbitrary, we can choose it to be oriented along one of the basis vectors ı r˛ . Then d x will be directed along the corresponding vector r˛ as well, because ı under this transformation r˛ becomes r˛ for the same material point M with Lagrangian coordinates X k . In this case we have ˇ @xı ˇ ı ˇ ˇ jd xj D d s ˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ ; @X ˇ @x ˇ ˇ ˇ jd xj D ds˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ : @X ı

ı

(2.100)

Hence ı

ı

ds˛ =d s ˛ D jr˛ j=jr˛ j D ı˛ C 1;

(2.101)

where ı˛ is called the relative elongation. Formula (2.101) yields ı

jr˛ j D jr˛ j.1 C ı˛ /:

(2.102)

On substituting this expression into (2.98), we get "˛ˇ D

 1 ı ı jr˛ jjrˇ j .1 C ı˛ /.1 C ıˇ / cos 2

Consider the case when ˛ D ˇ, then

˛ˇ

D

ı ˛ˇ

˛ˇ

 cos



ı ˛ˇ

:

(2.103)

D 0 and

ı

"˛˛ D

  g  1 ı 2 jr˛ j .1 C ı˛ /2  1 D ˛˛ .1 C ı˛ /2  1 : 2 2

(2.104)

30

2 Kinematics of Continua ı

Let coordinates X i be coincident with Cartesian coordinates x i , then g ˛ˇ D ı˛ˇ ; and for infinitesimal values of the relative elongation, when ı˛  1, we obtain "˛˛  ı˛ ;

(2.105)

i.e. "˛˛ is coincident with the relative elongation. In general, "˛˛ is a nonlinear function of corresponding elongations. Consider ˛ ¤ ˇ and assume that X i D x i , then we get

ı

˛ˇ

D =2; and from (2.103)

1 ı ı jr˛ jjrˇ j.1 C ı˛ /.1 C ıˇ / cos ˛ˇ 2q q 1 ı ı D g ˛˛ g ˇˇ .1 C ı˛ /.1 C ıˇ / sin ˛ˇ 2 1 D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ ; 2

"˛ˇ D

(2.106)

ı

where ˛ˇ D ˛ˇ  ˛ˇ D . =2/  ˛ˇ is the change of the angle between basis vectors r˛ and rˇ . For small relative elongations, when ı˛  1, and small angles

˛ˇ  1, from (2.106) we get "˛ˇ  ˛ˇ =2;

(2.107)

i.e. "˛ˇ is a half of the misalignment angle of the basis vectors.

2.2.6 Transformation of an Oriented Surface Element In actual configuration K consider a smooth surface †, which contains two coordinate lines X ˛ and X ˇ . Then we can introduce the normal n to the surface † as follows: 1 n D p r˛  rˇ : e g

(2.108)

Here e g D det .e g ˛ˇ /, and e g ˛ˇ is the two-dimensional matrix of the surface (˛; ˇ D 1; 2): (2.109) e g ˛ˇ D r˛  rˇ (it is not to be confused with the metric matrix gij D ri  rj ). In configuration K consider a surface element d† constructed on elementary radius-vectors d x˛ , which are directed along local basis vectors, i.e. d x˛ D r˛ dX ˛ (Fig. 2.11). The value p d† D e g dX ˛ dX ˇ (2.110)

2.2 Deformation Tensors and Measures

31

Fig. 2.11 Introduction of oriented surface element n d†

is called the area of the surface element d† constructed on vectors d x˛ and d xˇ . Then formula (2.108) takes the form n d† D r˛ dX ˛  rˇ dX ˇ D d x˛  d xˇ ;

(2.111)

where n d† is called the oriented surface element. Show that the normal n defined by formula (2.108) is a unit vector. According to the property (2.14b) of the vector product of basis vectors and the results of Exercise 2.1.13, we can rewrite Eq. (2.111) in the form p n d† D r˛  rˇ dX ˛ dX ˇ D g˛ˇ r dX ˛ dX ˇ p D .1= g/ ijk g˛i gˇj rk dX ˛ dX ˇ

(2.112)

(there is no summation over ˛; ˇ). Thus, n d†  n d† D

p

1 g ˛ˇ r  p  ijk g˛i gˇj rk /.dX ˛ dX ˇ /2 g j

D .˛ˇ k  ijk g˛i gˇj /.dX ˛ dX ˇ /2 D.ı˛i ıˇ ıˇi ı˛j /g˛i gˇj .dX ˛ dX ˇ /2 2 D .g˛˛ gˇˇ  g˛ˇ /.dX ˛ dX ˇ /2 D e g.dX ˛ dX ˇ /2 D d†2 :

(2.113) Hence, n  n D 1:

ı

ı

The surface element d† in K corresponds to the surface element d † in K, which ı ı is constructed on elementary radius–vectors d x˛ and d xˇ : ı

ı

ı

ı

ı

ı

n d † D r˛ dX ˛  rˇ dX ˇ D r˛  rˇ dX ˛ dX ˇ : ı

(2.114)

ı

Here n is the unit normal to d †. ı Since r D F1T  r , we get p ı n d† D g ˛ˇ F1T  r dX ˛ dX ˇ q ı ı ı D g=g F1T  r˛  rˇ dX ˛ dX ˇ q ı ı ı D g=g F1T  n d †: Thus, we have proved the following theorem.

(2.115)

32

2 Kinematics of Continua ı

ı

ı

Theorem 2.7. The oriented surface elements n d † and n d† in K and K are connected by the relation q n d† D

ı ı

g=g n  F

1

q ı ı ı d † D g=g F1T  n d †: ı

(2.116)

With the help of the deformation measures we can derive formulae connecting ı the normals n and n to the surface element containing the same material points in K ı

and K. Multiplying Eq. (2.116) by itself and taking the formula n  n D 1 into account, we get d†2 D

ı ı g ı ı 1 1T ı 2 1 ı 2 . . n  F  F  n/ d † D n  G  n/ d † : ı ı g g

g

Thus,

q

ı

d†=d † D

ı

ı

ı

g=g .n  G1  n/1=2 :

(2.117)

(2.118)

ı

On the other hand, expressing n from (2.116) and then multiplying the obtained relation by itself, we obtain ı

ı

d †2 D

ı

g g .n  F  FT  n/ d†2 D .n  g1  n/ d†2 ; g g

Thus, we find that

q

ı

d †=d† D

ı

g=g .n  g1  n/1=2 :

(2.119)

(2.120)

On introducing the notation ı

ı

ı

ı

ı

k D .n  G1  n/1=2 D .n  F1  F1T  n/1=2 ; k D .n  g1  n/1=2 D .n  F  FT  n/1=2 ;

(2.121)

relations (2.119) take the form ı

d †=d† D

q

q ı ı g=g k D g=g .1=k/: ı

Thus, we get

(2.122)

ı

k D 1=k:

(2.123)

Substitution of (2.119) and (2.120) into (2.116) gives the desired relations ı

ı

kn D F1T  n;

ı

k n D FT  n:

(2.124)

2.2 Deformation Tensors and Measures

33

2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor Components gij of the metric matrix are connected to components of the deformation tensor "ij by relation (2.69). In continuum mechanics, one often needs to know the expression of the inverse metric matrix gij in terms of "ij (but not in terms of "ij ). To derive this relation, we should use the connection between components of a matrix and its inverse (see [12]): g ij D

1 imn jkl   gmk gnl : 2g

(2.125)

ı

For g ij , we have the similar formula ı

g ij D

1 ı

2g

ı

ı

 imn  jkl g mk g nl :

(2.126)

ı

On substituting the relations (2.69) between gmn , g mn and "mn into (2.126), we get 1 imn jkl ı ı   .g mk C 2"mk /.g nl C 2"nl / 2g 1 imn jkl ı ı ı ı   .g mk g nl C 2g mk "nl C 2g nl "mk C 4"mk "nl /: D 2g

gij D

(2.127)

Removing the parentheses, modify four summands in (2.127) in the following way. The first summand with taking formula (2.126) into account gives the maı ı trix g ij .g=g/. To transform the second and the third summands, we should use the formulae q  q  ı ı ı ı ı jkl 1= g  D g t sp g jt g ks g lp : (2.128) ı

ı

ı

ı

ı

ı

ı

ı

.1=g/ imn  jkl g mk D imn t sp g jt g ks g lp gmk D  imn t mp g jt g lp ı

ı

ı

ı

ı

ı

D .ıti ıpn  ıpi ıtn /g jt g lp D g ij gnl  g j n g i l : ı

ı

ı

ı

ı

ı

.1=g/imn  jkl gmk "nl D .gij g nl  g j n g i l /"nl :

(2.129)

(2.130)

Formula (2.128) follows from the relation p

p g ijk D .1= g/  mnl gmi gnj glk

(see [12]), and relationship (2.129) has been obtained by using formula (2.128) and the properties of the Levi-Civita symbols (see Exercise 2.1.13).

34

2 Kinematics of Continua

On substituting formula (2.130) into (2.127), we get ı

g g ij D g

!  ı ı 2 ı ı g C 2 g ij g nl  2gi l gj n "nl C ı  imn jkl "mk "nl : g ı ij

(2.131)

Finally, we should express the determinant g D det .gij / in terms of "ij . To do this, we multiply relation (2.131) by gij and take formula (2.69) into account: ı

ı ij

ı ij ı nl

3g D g g C 2.g g

ı il ıjn

 g g /"nl C

2 ı

! 

imn jpl



g

ı

"mp "nl .g ij C 2"ij /: (2.132)

Thus, we get  ı ı ı ı 3g D g 3 C 4gnl "nl C .2=g/imn jpl g ij "mp "nl

 ı ı ı ı ı ı C 2g ij "ij C 4.g ij g nl  gi l gj n /"nl "ij C .4=g/ imn jpl "ij "mp "nl : (2.133)

Modifying the third summand on the right-hand side by formula (2.130) and introducing the notation ı

I1" D g nl "nl ; I2" D

1  ı ij ı nl ı i l ı j n  ı g g g g "ij "nl ; I3" D det ."ij gi k /; (2.134) 2

from (2.133) we get the desired formula ı

g D g.1 C 2I1" C 4I2" C 8I3" /:

(2.135)

Here we have taken account of formula (2.125) for the matrix determinant and also ı the relation I3" D .1=g/det ."ij /. Thus, we have proved the following theorem. Theorem 2.8. The inverse metric matrix gij is expressed in terms of components ı "ij of the deformation tensor and g ij by formulae (2.131) and (2.135). Formulae (2.131) and (2.135) allow us to find the expression of contravariant components "ij of the deformation tensor in terms of "ij . It follows from (2.69) that  ı g  gı gı  ı ı  ı 1 ı ı  "ij D.1=2/ g ij  g ij Dg ij g ij g nl  g i l g j n "nl   imn  jkl "mk "nl : 2g g g (2.136) Substitution of formulae (2.93) or (2.94) into (2.131) and (2.136) gives the expresı sion for components "ij in terms of components of the displacement vector ui or ui .

2.2 Deformation Tensors and Measures

35

Exercises for 2.2 2.2.1. Using the results of Exercise 2.1.1, show that the deformation gradient F and its inverse F1 for the problem on a beam in tension (see Example 2.1) have the forms F D FT D

3 X

k˛ eN ˛ ˝ eN ˛ ;

F1 D F1T D

˛D1

3 X 1 eN ˛ ˝ eN ˛ : k ˛D1 ˛

For this problem, the deformation tensors are determined by the formulae CDƒD

3 1X 2 .k  1/Ne˛ ˝ eN ˛ ; 2 ˛D1 ˛

ADƒD

3 1X .1  k˛2 /Ne˛ ˝ eN ˛ ; 2 ˛D1

the deformation measures are determined as follows: G D g1 D

3 X

k˛2 eN ˛ ˝ eN ˛ ;

g D G1 D

˛D1

3 X

k˛2 eN ˛ ˝ eN ˛ ;

˛D1

and components of the deformation tensor take the forms "˛ˇ D

1 2 .k  1/ı˛ˇ ; 2 ˛

"˛ˇ D

1 .1  k˛2 /ı˛ˇ : 2

2.2.2. Using the results of Exercise 2.1.2, show that for the problem on a simple shear we have the following formulae for the deformation gradient: ı

F D F ij eN i ˝ eN j D E C aNe1 ˝ eN 2 ; F1 D E  aNe1 ˝ eN 2 ; i.e.

FT D E C aNe2 ˝ eN 1 ;

F1T D E  aNe2 ˝ eN 1 ;

0

ı

F ij

1 1a0 D @0 1 0A ; 001

det F D 1;

for the deformation tensors: C D .a=2/O3 C .a2 =2/Ne22 ;

A D .a=2/O3  .a2 =2/Ne22 ;

ƒ D .a=2/O3  .a2 =2/Ne21 ;

J D .a=2/O3 C .a2 =2/Ne21 ;

and for components of the deformation tensor: 1 0 a=2 0 ."ij / D @a=2 a2 =2 0A ; 0 0 0 0

1 0 a=2 0 ."ij / D @a=2 a2 =2 0A : 0 0 0 0

36

2 Kinematics of Continua

Here we have introduced the notation O3  eN 1 ˝ eN 2 C eN 2 ˝ eN 1 ;

eN 2˛  eN ˛ ˝ eN ˛ ;

˛ D 1; 2; 3:

2.2.3. Using the formulae from Example 2.3 (see Sect. 2.1.1), show that for the problem on rotation of a beam with extension, the deformation gradient has the form F D F0ij eN i ˝ eN j D cos '

2 X

k˛ eN ˛ ˝ eN ˛ C k3 eN 3 ˝ eN 3 C sin 'k2 .Ne2 ˝ eN 1  eN 1 ˝ eN 2 /:

˛D1

2.2.4. Using formulae (2.36), (2.85)–(2.92) and the results of Exercise 2.1.7, show that local basis vectors are connected to displacements by the relations ı

ı

ı

ri D rk .ıik C r k ui /;

ı

ri D rk .ıik  ri uk /:

2.2.5. Using formulae (2.104) and q (2.106), show that the physical components of b ı ı ı the deformation tensor "˛ˇ D "˛ˇ = g ˛˛ g ˇˇ are connected to relative elongations ı˛ and angles ˛ˇ by the relations 1 b ı "˛˛ D ..1 C ı˛ /2  1/; 2

1 b ı "˛ˇ D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ : 2

e i the expression 2.2.6. Show that in the basis e ri of curvilinear coordinate system X 1 of the tensor F in terms of r ˝ u can be rewritten in the form similar to (2.89)– (2.92): u De uke rk ;

e 1 /ki e F1 D .F ri ; rk ˝e

eie e 1 /ki D ıik  r .F uk :

2.2.7. Using formula (2.34), show that the following relationships hold: ı

ı

jd xj2 D d x  G  d x;

ı

jd xj2 D d x  g  d x:

2.3 Polar Decomposition 2.3.1 Theorem on Polar Decomposition According to (2.36), the tensor F can be considered as a tensor of the linear transı ı formation from the basis ri to the basis ri . Since the vectors ri and ri are linearly independent, the tensor F is nonsingular. Then for this tensor the following theorem is valid.

2.3 Polar Decomposition

37

Theorem 2.9 (on the polar decomposition). Any nonsingular second-order tensor F can be represented as the scalar product of two second-order tensors: FDOU

or F D V  O:

(2.137)

Here U and V are the symmetric and positive-definite tensors, O is the orthogonal tensor, and each of the decompositions (2.137) is unique. H Prove the existence of the decomposition (2.137) in the constructive way, i.e. we should construct the tensors U, V and O. To do this, consider the contractions of the tensor F with its transpose: FT  F and F  FT . Both the tensors are symmetric, because .FT  F/T D FT  .FT /T D FT  F and .F  FT /T D .FT /T  FT D F  FT ; (2.138) and positive-definite: a  .FT  F/  a D .a  FT /  .F  a/ D .F  a/  .F  a/ D b  b D jbj2 > 0

(2.139)

for any non-zero vector a, where b D F  a. Since any symmetric positive-definite tensor has three real positive eigenvalues [12], eigenvalues of tensors FT  F and ı

F  FT can be denoted as 2˛ and 2˛ . These tensors are diagonal in their eigenbases, i.e. they have the following forms: FT  F D

3 X

ı

ı

ı

2˛ p˛ ˝ p˛ ;

F  FT D

˛D1

3 X

2˛ p˛ ˝ p˛ :

(2.140)

˛D1

ı

Here p˛ are the eigenvectors of the tensor FT  F and p˛ – of the tensor F  FT , which are real-valued and orthonormal: ı

ı

p˛  pˇ D ı˛ˇ ;

p˛  pˇ D ı˛ˇ :

(2.141)

The right-hand sides of (2.140) are the squares of certain tensors U and V defined as UD

3 X

ı

ı

ı

ı

˛ p˛ ˝ p˛ ; ˛ > 0I

VD

˛D1

3 X

˛ p˛ ˝ p˛ ; ˛ > 0:

(2.142)

˛D1

Here signs at ˛ are always chosen positive. In this case, the following relations are valid: F T  F D U2 ;

F  FT D V2 :

(2.143)

38

2 Kinematics of Continua

The constructed tensors V and U are symmetric due to formula (2.142) and positivedefinite, because for any nonzero vector a we have aUaD

3 X

ı

ı

ı

˛ a  p˛ ˝ p˛  a D

˛D1

3 X

ı

ı

˛ .a  p˛ /2 > 0;

(2.144)

˛D1

ı

as ˛ > 0. In a similar way, we can prove that the tensor V is positive-definite. Both the tensors V and U are nonsingular, because, under the conditions of the theorem, the tensor F is nonsingular. And from (2.143) we get .det U/2 D det U2 D det .FT  F/ D .det F/2 ¤ 0:

(2.145)

Then there exist inverse tensors U1 and V1 , with the help of which we can construct two more new tensors ı

O D F  U1 ;

O D V1  F;

(2.146)

which are orthogonal. Indeed, ı

ı

OT  O D .F  U1 /T  .F  U1/ D U1  FT  F  U1 D U1  U2  U1 D E: (2.147) ı

According to [12], this means that the tensor O is orthogonal. In a similar way, we can show that the tensor O is orthogonal as well. ı

Thus, we have really constructed the tensors U and O, and also V and O, the product of which, due to (2.146), gives the tensor F: ı

F D O  U D V  O:

(2.148) ı

Here U and V are symmetric, positive-definite tensors, O and O are orthogonal tensors. Show that each of the decompositions (2.148) is unique. By contradiction, let there be one more resolution, for example ı

But then

ee U: FDO

(2.149)

FT  F D e U2 D U 2 ;

(2.150)

hence, e U D U, because the decomposition of the tensor FT  F for its eigenbasis is ı

ı

˛ are chosen positive by the condition. The coincidence of unique. Signs at ˛ and e ı

ı

e and O are coincident as well, because U and e U leads to the fact that O ı

ı

e DFe O U1 D F  U1 D O:

(2.151)

2.3 Polar Decomposition

39

This has proved uniqueness of the decomposition (2.148). We can verify uniqueness of the decomposition F D V  O in a similar way. ı

Finally, we must show that the orthogonal tensors O and O are coincident, i.e. formula (2.137) follows from (2.148). To do this, we construct the tensor ı

ı

ı

F  OT D O  U  OT :

(2.152)

Due to (2.148), this tensor satisfies the following relationship: ı

ı

ı

O  U  OT D V  O  OT :

(2.153)

ı

The tensor O  OT is orthogonal, because ı

ı

ı

ı

ı

ı

.O  OT /T  .O  OT / D O  OT  O  OT D O  OT D E:

(2.154)

Then the relationship (2.153) can be considered as the polar decomposition of the ı

ı

tensor O  U  OT . This tensor is symmetric, because ı

ı

ı

ı

ı

ı

.O  U  OT /T D .OT /T  .O  U/T D O  U  OT :

(2.155)

Then the formal equality ı

ı

ı

ı

O  U  OT D O  U  OT

(2.156)

is one more polar decomposition. However, as was shown above, the polar decomposition is unique; hence, the following relationships must be satisfied: ı

ı

ı

V D O  U  OT and O  OT D E: ı

(2.157) ı

Thus, the orthogonal tensors O and O are coincident: O D O. N The tensors U and V are called the right and left stretch tensors, respectively, and O is the rotation tensor accompanying the deformation. The tensor F has nine independent components, the tensor O – three independent components, and each of the tensors U and V – six independent components. Remark 1. Since the rotation tensor O is unique in the polar decomposition, from formula (2.157) we get that the stretch tensors U and V are connected by means of the tensor O: V D O  U  OT ; U D OT  V  O: (2.157a) t u

40

2 Kinematics of Continua

Theorem 2.10. The Cauchy–Green and Almansi deformation tensors can be expressed in terms of the stretch tensors U and V as follows: 1 2 1 .U  E/; A D .E  V2 /; 2 2 1 1 ƒ D .E  U2 /; J D .V2  E/: 2 2 CD

(2.158)

H To see this, let us substitute the polar decomposition (2.137) into (2.72), and then we get the relationships (2.158). N

2.3.2 Eigenvalues and Eigenbases Theorem 2.11. Eigenvalues of the tensors U and V defined by (2.142) are coincident: ı

˛ D ˛ ;

˛ D 1; 2; 3;

(2.159)

ı

and eigenvectors p˛ and p˛ are connected by the rotation tensor O: ı

p˛ D O  p˛ :

(2.159a)

H To prove the theorem, we use the definition (2.142) and the first formula of (2.157a): VD

3 X

˛ p˛ ˝ p˛ D O  U  OT D

˛D1

where

3 X

ı

ı

ı

˛ O  p˛ ˝ .O  p˛ / D

˛D1

3 X

ı

˛ p0˛ ˝ p0˛ ;

˛D1 ı

p0˛ D O  p˛ :

According to the relationship, we have obtained two different eigenbases of the tensor V and two sets of eigenvalues, that is impossible. Therefore, ı

ı

p0˛ D O  p˛ D p˛ and ˛ D ˛ ; as was to be proved. N Due to (2.141), both the eigenbases are orthogonal. Therefore, reciprocal vectors ı of the eigenbases do not differ from p˛ and p˛ : p˛ D p˛ ;

ı

ı

p˛ D p˛ :

(2.160)

2.3 Polar Decomposition

41 ı

The important problem for applications is to determine ˛ , p˛ and p˛ by the given deformation gradient F. To solve the problem, one should use the following method: 1. Construct the tensor U2 D FT  F (or V2 D F  FT ) and find its components in some basis being suitable for a considered problem; for example, in the Cartesian basis eN i : U2 D .UN 2 /i eN i ˝ eN j and V2 D .VN 2 /i eN i ˝ eN j : j

j

2. Find eigenvalues of the matrix .UN / j by solving the characteristic equation 2 i

det .U2  2˛ E/ D 0;

(2.161)

which in the basis eN i takes the form det ..U 2 /ij  2˛ ıji / D 0:

(2.161a)

ı

3. Find eigenvectors p˛ of the tensor U and eigenvectors p˛ of the tensor V from the equations ı

ı

U2  p˛ D 2˛ p˛ ; V2  p˛ D 2˛ p˛ ;

(2.162)

written, for example, in the basis eN i : ı

b j˛ D 0; ..UN 2 /ij  2˛ ıji /Q

b j˛ D 0; ..VN 2 /ij  2˛ ıji /Q

(2.162a)

ı

b j˛ are Jacobian matrices of the eigenvectors: b j˛ and Q where Q ı

ı

b j˛ eN j ; p˛ D Q

b j˛ eN j : p˛ D Q

(2.163)

ı

b j˛ and Q b j˛ , one should consider only independent To determine the matrices Q equations of the system (2.162a) and the normalization conditions (2.141): jp˛ j D 1;

ı

jp˛ j D 1;

(2.164)

which are equivalent to the quadratic equations ı

ı

b i˛ Q b j˛ ıij D 1; Q

b j˛ ıij D 1: b i˛ Q Q

(2.164a)

42

2 Kinematics of Continua

4. Write the dyadic products (2.142) and find resolutions of the tensors U and V for the eigenbases; for example, for the Cartesian basis eN i : UD

3 X

ı

ı

b i˛ Q b j˛ eN i ˝ eN j ; V D ˛ Q

˛D1

3 X

b i˛ Q b j˛ eN i ˝ eN j : ˛ Q

˛D1

Exercises 2.3.2–2.3.4 show examples of determination of the tensors U and V. Remark 2. Notice that a solution of the quadratic equations (2.164a) may be not ı

b i˛ , this ambigub i˛ and Q unique due to the choice of signs of matrix components Q ity is resolved by applying one more additional condition, namely the condition of ı coincidence of the vectors p˛ and p˛ when t ! 0C : ı

t ! 0C ) p˛ .t/ D p˛ .t/;

˛ D 1; 2; 3:

ı

b i˛ , the ambiguity of the sign choice remains. However, if there For the matrix Q ı

is a field of eigenvectors p˛ .x; t/, then this ambiguity may be retained only at one point x0 at one time, for example, t D 0; and for the remaining x and t, a sign at b i˛ is chosen from the continuity condition of the vector field pı ˛ .x; t/ (for continQ ı

uous motions). If the eigenvector field p˛ .x0 ; 0/ contains the vectors eN ˛ , then the ı remaining ambiguity is resolved by the condition p˛ .x0 ; 0/ D eN ˛ : The ambiguity of a solution of the system (2.162a), (2.164a) may also appear, if at some time t1 at a point x the eigenvalues ˛ .t1 / prove to be triple. In this case, ı

b i˛ .t1 / are determined, as a rule, by passage to b i˛ .t1 / and Q values of the matrices Q the limit: b i˛ .t/; b i˛ .t1 / D lim Q Q t !t1

ı

ı

b i˛ .t1 / D lim Q b i˛ .t/; ˛ D 1; 2; 3: Q t !t1

In the case of double eigenvalues ˛ , these formulae are applied only to their correı

b i˛ and Q b i˛ . sponding matrix components Q

t u

2.3.3 Representation of the Deformation Tensors in Eigenbases ı

ı

Theorem 2.12. In the tensor bases p˛ ˝pˇ and p˛ ˝ pˇ , the Cauchy–Green tensors C and J, the Almansi tensors A and ƒ, and the deformation measures G, g1 and G1 , g have the diagonal form: CD

3 X 1 2 ı ı .˛  1/p˛ ˝ p˛ ; 2 ˛D1

ƒD

3 X 1 ı ı .1  2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

2.3 Polar Decomposition

43

3 X 1 .1  2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

AD

JD

3 X 1 2 .  1/p˛ ˝ p˛ I (2.165a) 2 ˛ ˛D1

and GD

3 X

ı

ı

G1 D

2˛ p˛ ˝ p˛ ;

˛D1

g

1

D

3 X

3 X

ı

ı

2 ˛ p˛ ˝ p˛ ;

˛D1

2˛ p˛

˝ p˛ ;

gD

˛D1

3 X

2 ˛ p˛ ˝ p˛ :

(2.165b)

˛D1

H On substituting formulae (2.142) into (2.158), we get (2.165a). Formulae (2.165b) follow from (2.165a) and (2.74), (2.75). N Similarly to formulae (2.165), we can introduce new deformation tensors by deı ı termining their components with respect to the bases p˛ ˝ pˇ or p˛ ˝pˇ as follows: 3 X

ı

MD

ı

ı

f .˛ /p˛ ˝ pˇ ;

MD

˛D1

3 X

f .˛ /p˛ ˝ pˇ ;

(2.166)

˛D1

where f .˛ / is a function of ˛ . If f .1/ D 0, then we get the deformation tensors; and if f .1/ D 1, then we get the deformation measures. Among the tensors (2.166), the logarithmic deformation tensors and measures ı

HD

P3

ı ˛D1 lg ˛ p˛ ı ı H1 D H

ı

˝ pˇ ;

HD

C E;

P3

˛D1

lg ˛ p˛ ˝ pˇ ;

H1 D H C E;

(2.167)

are the most widely known; they are called the right and left Hencky tensors, and also the right and left Hencky measures, respectively. ı With the help of the eigenvectors p˛ and p˛ we can form the mixed dyads 3 X ˛D1

ı

p˛ ˝ p˛ D

3 X

p˛ ˝ p˛  O D

˛D1

3 X

! p˛ ˝ p

˛

 O D E  O:

(2.168)

˛D1

Here we have used the properties (2.159a) and (2.160), and the representation of the unit tensor E in an arbitrary mixed dyadic basis. Thus, the rotation tensor O accompanying the deformation can be expressed in the eigenbasis as follows: OD

3 X ˛D1

ı

ı

p˛ ˝ p˛ D pi ˝ pi :

(2.169)

44

2 Kinematics of Continua

On substituting (2.169) and (2.142) into (2.137) and taking (2.141) into account, we get the following expression of the deformation gradient in the tensor eigenbasis: FDOUD

3 X

ı

p˛ ˝ p˛ 

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ D

ˇ D1

˛D1

3 X

ı

˛ p˛ ˝ p˛ :

(2.170)

˛D1

According to (2.170), the transpose FT and inverse F1 gradients are expressed as follows: F D T

3 X

ı

˛ p˛ ˝ p˛ ;

F

1

D

˛D1

3 X

ı

1 ˛ p˛ ˝ p˛ :

(2.171)

˛D1

2.3.4 Geometrical Meaning of Eigenvalues ı

ı

Vectors of eigenbases p˛ and p˛ are connected by the transformation (2.159a). In K ı ı take elementary radius-vectors d x˛ oriented along the eigenbasis vectors p˛ , then in K they correspond to radius-vectors d x˛ : ı

ı

ı

ı

d x˛ D p˛ jd x˛ j;

d x˛ D F  d x˛ :

(2.172)

Substitution of (2.170) into (2.172) yields d x˛ D

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ  p˛ jd x˛ j D ˛ jd x˛ jp˛ ;

(2.173)

ˇ D1

i.e. the elementary radius-vectors d x˛ in K will be also oriented along the corresponding eigenbasis vectors p˛ . ı ı Denote lengths of the vectors d x˛ and d x˛ by d s ˛ and ds, respectively, and derive relations between them: ı

ı

ı

ı

ı

ds˛2 D d x˛  d x˛ D d x˛  FT  F  d x˛ D jd x˛ j2 p˛  FT  F  p˛ ı ı

ı

ı

D d s 2˛ p˛  G  p˛ D d s 2˛ 2˛ :

(2.174)

Here we have used Eqs. (2.165b) and (2.172). Formula (2.174) proves the following theorem. Theorem 2.13. Eigenvalues ˛ (principal stretches) are the elongation ratios for material fibres oriented along the principal (eigen-) directions: ı

˛ D ds˛ =d s ˛ :

(2.175)

2.3 Polar Decomposition

45

2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum ı

In K, consider a small neighborhood of the material point M contained in a conı tinuum; then every point M0 , connected to M by the elementary radius-vector d x (Fig. 2.12), will be connected to the same point M by radius-vector d x in K. These radius-vectors are related as follows: ı

d x D F  d x:

(2.176) ı

The relation can be considered as the transformation of arbitrary radius-vector d x into d x. Rewrite the relation (2.176) in Cartesian coordinates: ı dx i D FNmi d x m ;

(2.177)

where FNmi are components of the deformation gradient with respect to the Cartesian basis (see Exercise 2.1.5): ı (2.178) FNmi D .@x i =@x m /; ı

which depend only on coordinates x m of the point M, but they are independent of ı coordinates d x m of its neighboring points M0 . Therefore the transformation (2.177) ı is a linear transformation of coordinates d x m into dx i , i.e. this is an affine transformation. As follows from the general properties of affine transformations, straight lines ı

and planes contained in a small neighborhood in K will be straight lines and planes in actual configuration K. Parallel straight lines and planes are transformed into ı

parallel straight lines and planes. Therefore if a small neighborhood in K is chosen

Fig. 2.12 Transformation of a small neighborhood of the point contained in a continuum

46

2 Kinematics of Continua

to be a parallelogram, then in K the neighborhood will be a parallelogram as well (although angles between its edges, edge lengths and orientation of planes in space may change). ı

Since a second-order surface in K (and, in general, a surface specified by an algebraic expression of arbitrary nth order) is transformed into a surface of the same ı

order in K, a small spherical neighborhood in K is transformed into an ellipsoid in actual configuration K (Fig. 2.12). ı As follows from formula (2.101), the ratio of lengths ds˛ =d s ˛ of an arbitrary ı

vector (or of elementary radius-vector d x in K and K) is independent of the initial ı ı length d s ˛ of the vector (because the relative elongation ı˛ is independent of d s ˛ ). According to the polar decomposition (2.137), the transformation (2.176) from ı

K to K can always be represented as the superposition of two transformations: ı

d x D O  d x0 ;

ı

ı

d x0 D U  d x;

(2.179)

realized with the help of the stretch tensor U and the rotation tensor O, or d x D V  d x0 ;

ı

d x0 D O  d x:

(2.180) ı

The stretch tensor U, which has three eigendirections p˛ , transforms a small neighborhood of the point M with compressing or extending the neighborhood ı along these three directions p˛ . The tensor O rotates the neighborhood deformed ı ı along p˛ as a rigid whole until the direction of p˛ becomes the direction of p˛ . If ı

ı

one use the left stretch tensor V, so rotation of axes p˛ in K till their coincidence with p˛ is first realized, and then compression or tension of the neighborhood occurs along the direction p˛ . The result will be the same as for U. ı If a point M˛ is connected to M by radius-vector d x˛ oriented along the ı eigendirection p˛ (which is unknown before deformation), then in K the point M˛ will be connected to M by radius-vector d x˛ oriented along the corresponding eigendirection p˛ . ı

If a small neighborhood of point M is chosen to be a sphere in K (see Fig. 2.12), then in K the sphere becomes an ellipsoid with principal axes oriented along the eigendirections p˛ . Thus, the transformation of a small neighborhood of every point M contained in a continuum under deformation can always be represented as a superposition of tension/compression along eigendirections and rotation of the neighborhood as a rigid whole, and also displacement as a rigid whole.

2.3 Polar Decomposition

47

Exercises for 2.3 2.3.1. Using the formula (2.157a), show that the following relations between V and U hold: Vm D O  Um  OT ; Um D OT  Vm  O for all integer m (positive and negative). 2.3.2. Using the results of Exercises 2.1.1 and 2.2.1, show that for the problem on tension of a beam, eigenvalues ˛ are ˛ D k˛ ;

˛ D 1; 2; 3:

The stretch tensors U and V are coincident and have the form UDVD

3 X

k˛ e˛ ˝ e˛ ;

˛D1 ı

and eigenvectors p˛ and p˛ coincide with e˛ : ı

p˛ D p˛ D e˛ ;

˛ D 1; 2; 3:

The rotation tensor O for this problem is the unit one: O D E. 2.3.3. Using the results of Exercises 2.1.2, 2.2.2 and Remark 2, show that for the problem on simple shear (see Example 2.2 from Sect. 2.1.1), the tensors U2 and V2 are expressed as follows: U2 D FT  F D E C aO3 C a2 e2i D .UN 2 /ij eN i ˝ eN j ; V2 D E C aO3 C a2 e2i D .VN 2 /ij eN i ˝ eN j ; 0 0 1 1 a 0 1 C a2 a 2 i 2 i .UN / j D @a 1 C a2 0A ; .VN / j D @ a 1 0 0 0 0 1

1 0 0A ; 1

eigenvalues ˛ are 2˛ D 1 C b˛ jaj; ˛ D 1; 2I b1 D

a p C 1 C a2 =4; 2

b2 D

3 D 1;

a p  1 C a2 =4; 2

ı

eigenvectors p˛ and p˛ (a > 0) are 1 ı ı p˛ D p .Ne1 C b˛ eN 2 /; p3 D eN 3 ; 2 1 C b˛

48

2 Kinematics of Continua

1 1 p1 D q .b1 eN 1 C eN 2 /; p2 D  q .b2 eN 1 C eN 2 /; p3 D eN 3 ; 2 1 C b1 1 C b22 the stretch tensors U and V are U D UN ij eN i ˝ eN j D U0 eN 21 C U1 O3 C U2 eN 22 C eN 23 ; V D VN ij eN i ˝ eN j D U2 eN 21 C U1 O3 C U0 eN 22 C eN 23 ; 0

UN ij

1 U0 U 1 0 B C D @U1 U2 0A ;

VN ij

0 0 1 b1ˇ

Uˇ D

0 1 U2 U1 0 D @U1 U0 0A ; 0 0 1

p

p 1 C b1 a b2ˇ 1 C b2 a C ; 1 C b12 1 C b22

ˇ D 0; 1; 2;

and the rotation tensor O has the form O D ON ij eN i ˝ eN j D cos '.Ne21 C eN 22 / C sin '.Ne1 ˝ eN 2  eN 2 ˝ eN 1 /; 0

ON ij cos ' D

1 cos ' sin ' 0 D @ sin ' cos ' 0A ; 0 0 1

b2 b1  ; 1 C b12 1 C b22

sin ' D

b12 b22  : 2 1 C b1 1 C b22

Show that functions b1 .a/ and b2 .a/ satisfy the following relationships: b1 C b2 D a;

b1 b2 D 1;

b12 C b22 D 2 C a2 :

Show that at a D 0 for the considered problem the following equations really hold: b1 D 1;

b2 D 1;

1 D 2 D 3 D 1;

1 1 ı ı p1 D p1 D p .Ne1 C eN 2 /; p2 D p2 D p .Ne1  eN 2 /: 2 2 2.3.4. Using the results of Exercise 2.2.3, show that for the problem on rotation of a beam with tension (see Example 2.3 from Sect. 2.1.1), eigenvalues ˛ have the form ˛ D k˛ ;

˛ D 1; 2; 3;

and eigenvectors ı

p˛ D eN ˛ ;

p˛ D O0  eN ˛ ;

˛ D 1; 2; 3:

2.4 Rate Characteristics of Continuum Motion

49

Using formulae from Exercise 2.1.3 and data from Example 2.3, show that tensors U, V, O, and also C, A, ƒ and J have the form U D U0 D

3 X

k˛ eN ˛ ˝ eN ˛ ;

O D O0 D O0i j eN i ˝ eN j ;

˛D1

V D O0  U0  OT0 D V0 eN 21 C V1 O3 C V2 eN 22 C k3 eN 23 D V0 ij eN i ˝ eN j ; 0 1 V0 V 1 0 i V 0 j D @ V1 V 2 0 A ; 0 0 k3 V0 D k1 cos2 ' C k2 sin2 ';

V1 D .k1  k2 / cos ' sin ';

V2 D k1 sin2 ' C k2 cos2 '; 3 X 1 2 1 2 .k  1/Ne˛ ˝ eN ˛ ; C D .U0  E/ D 2 2 ˛ ˛D1

ƒD AD

3 X 1 1 / D .E  U2 .1  k˛2 /Ne˛ ˝ eN ˛ ; 0 2 2 ˛D1

1 1 .E  V2 / D .ı ij  gij /Nei ˝ eN j ; 2 2

JD

1 2 1 .V  E/ D .gij  ıij /Nei ˝ eN j ; 2 2

where metric matrices gij and g ij are determined by formulae from Exercise 2.1.3. We should take into consideration that the tensors C and ƒ do not feel the beam rotation – they are coincident with the corresponding tensors for the problem on pure tension of the beam. Show that if we change the sequence of transformations (i.e. we first rotate and then extend the beam), then the tensors A and J do not feel the rotation.

2.4 Rate Characteristics of Continuum Motion 2.4.1 Velocity The velocity (vector) of the motion of a material point M with Lagrangian coordinates X i is determined as the partial derivative of the radius-vector x.X i ; t/ with respect to time at fixed values of X i : v.X i ; t/ D

@x i ˇˇ .X ; t/ˇ i : X @t

(2.181)

50

2 Kinematics of Continua

Velocity components vN i with respect to the basis eN i have the form v D vN i eN i D

@x i eN i ; @t

vN i D

@x i j .X ; t/: @t

(2.182)

2.4.2 Total Derivative of a Tensor with Respect to Time Any vector field a.x; t/ (and also scalar or tensor field) varying with time, which describes some physical process in a continuum, can be expressed in both Eulerian and Lagrangian descriptions with the help of the motion law (2.3): a.x; t/ D a.x.X j ; t/; t/:

(2.183)

Determine the derivative of the function with respect to time at fixed X i (i.e. for a fixed point M): @a ˇˇ @a ˇˇ @a @x j ˇˇ (2.184) ˇ i D ˇ iC j ˇ : @t X @t x @x @t X i Definition 2.1. The partial derivative of a varying vector field a (2.183) with respect to time t at fixed coordinates X i is called the total derivative of the function (2.183) with respect to time: da @a ˇˇ aP  (2.185) D ˇ : dt @t X i According to formulae (2.182), (2.11), and (2.23), the second summand on the right-hand side of (2.184) can be rewritten as follows: @a @a @x j @a @a D vN i eN i  eN j P kj ˝ k D vrk ˝ k D vr ˝a: (2.186) D vN j P kj j k @x @t @X @X @X Then the relationship (2.184) yields da @a D C v  r ˝ a; dt @t

(2.187)

where we have introduced the notation for the partial derivative with respect to time which will be widely used below: @a i ˇˇ @a D .x ; t/ˇ i : x @t @t

(2.188)

In formula (2.187) the vector a is considered as a function a.x j ; t/. It is evident that if a is considered as a function of .X j ; t/, then from the definition (2.185) we get ˇ da i @ ˇ .x ; t/ D a.X j ; t/ˇ i : X dt @t

(2.189)

2.4 Rate Characteristics of Continuum Motion

51

The total derivative .d a=dt/ is also called the material (substantial, individual) derivative with respect to time, .@a=@t/ in (2.187) is the partial (local) derivative with respect to time, and v  r ˝ a is the convective derivative. The material derivative d a=dt characterizes a change of the vector field a in a fixed material point M, the local derivative determines a change of values of a in time at a fixed point x in space, and from (2.186) we get that the convective derivative characterizes a change of the field due to transfer of the material particle M from a point x to a point x C vdt in space. If we choose the vector v as a, then the relationship between the displacement u and the velocity v vectors has the form vD

dx du @u D D C v  r ˝ u: dt dt @t

(2.190)

Similarly to formula (2.185), we can define the total derivative of the nth-order tensor n  with respect to time: n

ˇ n P D d n .x i ; t/ D @  .X i ; t/ˇˇ :  Xi dt @t

(2.191)

Theorem 2.14. The total derivative (2.191) of a varying tensor field n .x i ; t/ can be written as a sum of local and convective derivatives: d n @ n C v  r ˝ n : D dt @t

(2.192)

H Proof of the theorem is similar to the proof of the relationship (2.187). Details are left as Exercise 2.4.6 N Let us consider now the question on components of the total derivative tensor. P are connected with Theorem 2.15. Components of the total derivative tensor n  ı the corresponding components of a tensor n  with respect to stationary bases ri , eN i and e ri and a moving basis ri as follows: ˇ d ı i1 :::in @ ı ˇ  D i1 :::in .X i ; t/ˇ i ; X dt @t

(2.193)

@ N i1 :::in i @ N i1 :::in i d N i1 :::in D  .x ; t/ C vN k k  .x ; t/;  dt @t @x

(2.194)

@ e i1 :::in e i d e i1 :::in ek  e i1 :::in .X e i ; t/; D  .X ; t/ Ce vk r  dt @t

(2.195)

n X @ d i1 :::in  D i1 :::in .X i ; t/ C .i1 :::k:::in rk vi˛ /.X i ; t/; dt @t ˛D1

(2.196)

52

2 Kinematics of Continua

where n

ı P D d  N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D d i1 :::in .X i ; t/rıi1 ˝ : : : ˝ rıin  dt dt d e i1 :::in e i d i1 :::in i D .X ; t/e ri1 ˝ : : : ˝e rin D .X ; t/ri1 ˝ : : : ˝ rin :   dt dt (2.197)

In formula (2.196), the component i1 :::k:::in as the ˛th superscript has index k in place of i˛ . H To prove the theorem, we resolve the tensor n  for different bases: n

ı

N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D i1 :::in .X i ; t/rı i1 ˝ : : : ˝ rıin D e i1 :::in .X e i ; t/e D ri1 ˝ : : : ˝e rin D i1 :::in .X i ; t/ri1 ˝ : : : ˝ rin ;

(2.198)

and choose arguments of components of the tensor n  as in formula (2.198). ı Then, substituting the resolution (2.198) for the basis ri into the definition ı (2.191), we get the expression (2.193), because d ri =dt D 0. On substituting the resolution (2.198) for the basis eN i into the relationship (2.192), we obtain n

N i1 :::in @ N i1 :::in m P D @  eN i1 ˝ : : : ˝ eN in C vN k eN k  m  eN ˝ eN i1 ˝ : : : ˝ eN in : (2.198a) @t @x

It is evident that formula (2.194) follows from (2.198a). In a similar way, substituting the resolution (2.198) for the basis e ri into the relae, and also the tion (2.192) and using the property (2.61) of nabla-operators r and r equation @e ri =@t D 0 (for the stationary basise ri see Sect. 2.1.7), we get n

e i1 :::in P D @ em  e i1 :::ine e ri1 ˝ : : : ˝e rk  r rm ˝e  rin Ce vke ri1 ˝ : : : ˝e rin I (2.199) @t

and formula (2.195) follows from (2.199) at once. Finally, substituting the resolution (2.198) for the moving basis ri into the definition of the total derivative (2.192), we obtain n i1 :::in ˇ X @ri ˇˇ ˇ P D @ i1 :::i˛ :::in ri1 ˝ : : : ˛ ˇ i : : : ˝ rin :  ˇ i ri1 ˝ : : : ˝ rin C X @t @t X ˛D1 (2.200) Due to the definition (2.10) of local bases vectors and the definition (2.181) of the velocity, we have n

@ri˛ j ˇˇ @2 x ˇˇ @v ˇˇ .X ; t/ˇ j D D ˇ ˇ D ri˛ vk rk : X @t @t@X i˛ X j @X i˛ X j

(2.201)

2.4 Rate Characteristics of Continuum Motion

53

On substituting (2.201) into (2.200) and then collecting components at the same elements of the polyadic basis, we derive the formula (2.196). N It should be noted that arguments of the resolutions (2.198) and of the derivatives of tensor components (2.193)–(2.196) have been chosen in the specific way.

2.4.3 Differential of a Tensor Definition 2.2. For a tensor field n .x i ; t/, the following object d n D

d n dt dt

(2.202)

is called the differential of a tensor field (or the differential of a tensor) n .x i ; t/. According to formula (2.192) for the total derivative of a tensor with respect to time, we get that the differential of a tensor can be written in the form  d n .x i ; t/ D

 @ n C v  r ˝ n  dt: @t

(2.203)

According to (2.190), the relation (2.203) takes the form d n D

@ n dt C d x  r ˝ n  @t

(2.204)

When a tensor field is stationary (i.e. @ n =@t D 0), the differential of the tensor field has the form b d n  D d x  r ˝ n : (2.205) For stationary tensor fields b d n  D d n , but in general these differentials are not coincident. According to Theorem 2.15, components of the tensor d n  with respect to the ı fixed basis ri are written as follows: n

ı

d  D d

j1 :::jn ı

ı

rj1 ˝ : : : ˝ rjn ;

ı

d

ı

j1 :::jn

d j1 :::jn dt: D dt

(2.206)

From (2.202) and (2.187) we get the following expression for the differential of a vector:   da @a i d a.X ; t/ D dt D C v  r ˝ a dt; (2.207) dt @t and from (2.205) we have db a D .v  r ˝ a/dt D .r ˝ a/T  d x:

(2.208)

54

2 Kinematics of Continua ı

In particular, if a D x, then, by formulae (2.208) and (2.35a), we obtain

or

b ı ı d x D .r ˝ x/T  d x D F1  d x;

(2.209)

b ı d x D F  d x:

(2.210)

On comparing formulae (2.210) with (2.34), we find that the elementary radiusı vector d x, introduced in Sect. 2.1 and connecting two infinitesimally close material b ı points M and M0 , coincides with the vector d x in the notation (2.205).

2.4.4 Properties of Derivatives with Respect to Time Let us establish now important properties of partial and total derivatives of vector fields with respect to time. Theorem 2.16. The partial derivative of the vector product of basis vectors with respect to time has the form @rˇ @r˛ @ .r˛  rˇ / D  rˇ C r˛  : @t @t @t

(2.211)

H Determine the derivative of the vector product of two local basis vectors with respect to time: @ @ @ j j .r˛  rˇ / D .Qi˛ eN i  Q ˇ eN j / D .Qi˛ Q ˇ /Nei  eN j @t @t @t j @Q ˇ @Qi˛ eN i  Qjˇ eN j C Qi˛ eN i  eN j : D @t @t With use of relation (2.10) we really get (2.211). N Theorem 2.17. For arbitrary continuously differentiable vector fields a.x; t/ D aN i .x k ; t/Nei and b.x; t/ D bN i .x k ; t/Nei , we have the formulae @a @b @ .a  b/ D bCa ; @t @t @t

(2.212)

@ @a @b .a ˝ b/ D ˝bCa˝ ; @t @t @t

(2.213)

@a @b @ .a  b/ D bCa : @t @t @t H A proof is similar to the proof of Theorem 2.16. N

(2.214)

2.4 Rate Characteristics of Continuum Motion

55

Theorem 2.18. The total derivatives of the vector and scalar products of two arbitrary vector fields a.x; t/ and b.x; t/ with respect to time have the forms d da db .a  b/ D bCa ; dt dt dt

(2.215)

da db d .a  b/ D bCa : dt dt dt

(2.216)

H To prove formula (2.215), one should use the property of the total derivative (2.187): @ d .a  b/ D .a  b/ C v  r ˝ .a  b/: dt @t Modify the first summand by formula (2.212) and the second summand – by the formula r ˝ .a  b/ D .r ˝ a/  b  .r ˝ b/  a [12], then we get d @a @b .a  b/ D b  a C v  .r ˝ a/  b  v  .r ˝ b/  a: dt @t @t Collecting the first summand with the third one and the second summand with the fourth one, and using the property (2.187) of the total derivative of a vector, we obtain d da db da db .a  b/ D b aD bCa : dt dt dt dt dt Formula (2.216) can be proved in a similar way. N p Theorem 2.19. The total derivative of g with respect to time is connected to the divergence of the velocity v by d p p p g D g ri vi D g r  v: dt

(2.217)

H Let us differentiate the second relation of (2.15) with taking formula (2.211) into account: d d p gD r1  .r2  r3 / dt dt  2    @2 x @2 x @ x D  .r2  r3 / C r1   r3 C r1  r2  : @t@X 1 @t@X 2 @t@X 3 (2.218) Since

@v @2 x D D ri v D ri vj rj ; @t@X i @X i

56

2 Kinematics of Continua

we get d p p g D r1 v g  r1 C r1  .r2 v  r3 / C r1  .r2  r3 v/: dt

(2.219)

Here we have used the relations from Exercise 2.1.14. According to the definition of the vector product (1.2), we obtain r1  r2 v  r3 D r1 

p

g ijk r2 vi ı3j rk D

p

g i 31 r2 vi D

p g r2 v2 :

(2.220)

On substituting (2.220) into (2.219), we really get formula (2.217). N

2.4.5 The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor ı

Consider elementary radius-vectors d x and d x connecting two infinitesimally close ı

points M and M0 in configurations K and K, respectively. Determine the velocity of the point M0 relative to the configuration connected to the point M. To do this, determine the velocity differential b d v:   T  2 2 ı @ x @v T ı @ x @ ıi ıi ı ı i b  d x D r ˝ v  d x: d v D d xD i dX D i ˝ r  d xD r ˝ @t @X @t @X @t @X i (2.221) Here we have used the second equation of (2.33), the definition of the gradient (2.24) and formula (2.181). In a similar way, using the first equation of (2.33): dX i D d v: ri  d x, we get one more expression for the vector b b d v D .r ˝ v/T  d x:

(2.222)

The second-order tensor .r ˝ v/T is called the velocity gradient, which connects the relative velocity b d v of an elementary radius-vector d x to the vector d x itself: b d v D L  d x;

L D .r ˝ v/T :

(2.223)

Just as any second-order tensor (see [12]), the tensor L can be represented by a sum of the symmetric tensor D and the skew-symmetric tensor W: L D D C W:

(2.224)

The symmetric deformation rate tensor D is determined as follows: DD

1 .r ˝ v C r ˝ vT /: 2

This tensor has six independent components.

(2.225)

2.4 Rate Characteristics of Continuum Motion

57

The skew-symmetric vorticity tensor W is determined as follows: WD

1 .r ˝ vT  r ˝ v/: 2

(2.226)

Since the tensor W is skew-symmetric and has three independent components, we can put the tensor W in correspondence with the vorticity vector ! connected to the tensor (see [12]) as follows: !D

1 W   "; 2

W D !  E:

(2.227)

where " is the Levi-Civita tensor, which has the third order (see [12]). This tensor is determined as follows: 1 " D p  ijk ri ˝ rj ˝ rk : g

(2.228)

On substituting (2.224)–(2.227) into (2.222), we prove the following theorem. Theorem 2.20 (Cauchy–Helmholtz). The velocity v.M0 / of an arbitrary point M0 in a neighborhood of the material point M consists of the translational motion velocity v.M/ of the point M, the velocity !  d x of rotation as a rigid whole and the deformation rate D  d x, i.e.

or

b dv D !  dx C D  dx

(2.229)

v.M0 / D v.M/ C !  d x C D  d x C o .jd xj/:

(2.229a)

Example 2.4. Determine the tensor L for the problem on tension of a beam (see Example 2.1), substituting (2.182) into (2.223): LT D eN i

3 3 X @ X P ˛ @ i N N ˝ v D e ˝ X D e k kP˛ eN ˛ ˝ eN ˛ D L: ˛ ˛ @X i @X i ˛D1 ˛D1

Since the velocity gradient L in this case proves to be a symmetric tensor, from (2.225) and (2.226) it follows that D D L;

W D 0:

Thus, in this case ! D 0.

t u

Example 2.5. Determine the tensor L for the problem on simple shear (see Example 2.2), substituting formula (2.182) into (2.223): LT D eN i ˝

@v @Nvj i @Nv1 N N e eN 2 ˝ eN 1 D aN D ˝ e D P e2 ˝ eN 1 : j @X i @X i @X 2

58

2 Kinematics of Continua

According to formulae (2.225) and (2.226), we get D D .a=2/.N P e1 ˝ eN 2 C eN 2 ˝ eN 1 /; j

j

i i W D .a=2/.N P e1 ˝ eN 2  eN 2 ˝ eN 1 / D .a=2/.ı P ei ˝ eN j : 1 ı2  ı2 ı1 /N

Using formula (2.227), we determine the vorticity vector !D

aP aP aP 1 W   " D .ı1i ı2j  ı2i ı1j /j i k eN k D .21k  12k /Nek D  eN 3 ; 2 4 4 2 t u

which is orthogonal to the shear plane.

2.4.6 Eigenvalues of the Deformation Rate Tensor Just as any symmetric tensor, the deformation rate tensor D has three orthonormal real-valued eigenvectors and three real positive eigenvalues (see [12]). Denote the eigenvectors by q˛ (these vectors, in general, are not coincident with p˛ ) and the eigenvalues – by D˛ . Then the tensor D can be resolved for its dyadic eigenbasis as follows: 3 X D˛ q˛ ˝ q˛ ; q˛  qˇ D ı˛ˇ : (2.230) DD ˛D1

Take in the actual configuration K an elementary radius-vector d x˛ , connecting points M and M0 , so that the vector is oriented along the eigenvector q˛ of the tensor D; then, similarly to (2.172), we can write d x˛ D q˛ jd x˛ j;

jd x˛ j D .d x˛  d x˛ /1=2 :

(2.231)

Apply the Cauchy–Helmholtz theorem (2.229) to the elementary radius-vector: b d v ˛ D !  d x˛ C D  d x˛ :

(2.232)

Multiplying the left and right sides of the equation by d x˛ and taking account of the property of the mixed derivative d x˛  .!  d x˛ / D 0, we get d x˛  b d v ˛ D d x ˛  D  d x˛ :

(2.233)

Substituting in place of D its expression (2.230) and in place of d x˛ their expressions (2.231), we obtain d x˛  b d v˛ D jd x˛ j2

3 X

Dˇ q˛  qˇ ˝ qˇ  q˛ D D˛ jq˛ j2 :

ˇ D1

Here we have used the property (2.230) of orthonormal vectors q˛ .

(2.234)

2.4 Rate Characteristics of Continuum Motion

59

Modify the scalar product on the left-hand side as follows: @ 1 @ @ d x˛  b d v˛ D d x˛  d x ˛ D .d x˛  d x˛ / D jd x˛ j jd x˛ j: @t 2 @t @t

(2.235)

On comparing (2.234) with (2.235), we obtain the following theorem. Theorem 2.21. Eigenvalues D˛ of the deformation rate tensor D are the rates of relative elongations of elementary material fibres oriented along the eigenvectors q˛ : D˛ D

1 @ jd x˛ j: jd x˛ j @t

(2.236)

2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor Modify the right-hand side of (2.232) as follows: b d v˛ D !  d x˛ C D  d x˛ D .!  q˛ C D˛ q˛ /jd x˛ j;

(2.237)

and the left-hand side of (2.232) with taking (2.236) into account:   @ @ @jd x˛ j @q˛ @q˛ b : q˛ C jd x˛ j Djd x˛ j D˛ q˛ C d v˛ D d x˛ D .jd x˛ jq˛ /D @t @t @t @t @t (2.238) On comparing (2.237) with (2.238), we get the following theorem. Theorem 2.22. The vorticity tensor W (or the vorticity vector !) connects the rate of changing the eigenvectors q˛ to the vectors q˛ themselves: qP ˛ D

@q˛ D !  q˛ D W  q˛ : @t

(2.239)

Using formula (2.239), we can resolve the tensor W for the eigenbasis q˛ of the deformation rate tensor as follows: WD

3 X ˛D1

qP ˛ ˝ q˛ D qP i ˝ qi :

(2.240)

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2 Kinematics of Continua

2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point If in configuration K at time t we consider an elementary radius-vector d x connecting two infinitesimally close material points M and M0 , then for infinitesimal time dt the radius-vector is transformed into radius-vector d x0 in configuration K.t C dt/ (Fig. 2.13): d x D x0 .t/  x.t/;

d x0 D x0 .t C dt/  x.t C dt/;

(2.241)

where x.t/ and x0 .t/ are radius-vectors of the points M and M0 in configuration K.t/, respectively; and x.t C dt/ and x0 .t C dt/ – in configuration K.t C dt/. Displacements of points M and M0 for infinitesimal time are defined by the velocity vectors v.M/ and v.M0 /, respectively: x.t C dt/  x.t/ D v.M/ dt;

x0 .t C dt/  x0 .t/ D v.M0 / dt:

(2.242)

Formulae (2.241), (2.242), and simple geometric relations (see Fig. 2.13) give v.M0 /dt  v.M/dt D d x0  d x:

(2.243)

On substituting (2.243) into (2.229a), we obtain the relation between elementary radius-vectors d x0 and d x: d x0 D d x C dt!  d x C dtD  d x C dt o.jd xj/:

(2.244)

The relation (2.244) can be considered as the transformation of coordinates dx i ! dx 0i in a small neighborhood of the point contained in a continuum. Since dt! and dtD are independent of d x and d x0 , so the transformation is

Fig. 2.13 Infinitesimal transformation of an elementary radius-vector

2.4 Rate Characteristics of Continuum Motion

61

linear, i.e. affine. The relation (2.244) can be represented as a superposition of two transformations up to an accuracy of o.jd xj/: d x00 D AD  d x; d x0 D Q!  d x00 ;

AD D E C dtD;

(2.245)

Q! D E C dt!  E:

(2.246)

The tensor AD is symmetric and has three eigendirections, which are coincident with the eigendirections q˛ of the deformation rate tensor D. So just as the tensor U, the tensor AD transforms a small neighborhood of a point M by extending or compressing the neighborhood along the principal directions q˛ . The material segments jd x00˛ j oriented along the eigendirections q˛ retain their orientation under the transformations (2.245), but their lengths vary as follows: d x˛ D jd x˛ jq˛ ;

d x00˛ D .1 C D˛ dtjd x˛ j/q˛ D .1 C D˛ dt/d x˛ :

The tensor Q! (2.246) is orthogonal up to an accuracy of values .dt/2 , because Q!  QT! D .E C dtW/  .E C dtWT / D E  .dt/2 W2 :

(2.247)

Here we have taken into account that the vorticity tensor W is skew-symmetric. Thus, the transformation (2.246) determined by the tensor Q! is rotation of the M-point neighborhood as a rigid whole for infinitesimal time dt. The vorticity vector ! forming the tensor Q! can be considered as instantaneous angular rate of rotation of the small neighborhood as a rigid whole, or as instantaneous angular rate of rotation of the eigentrihedron q˛ of the deformation rate tensor relative to the fixed basis eN i . This fact will be considered in detail in Sect. 2.5.7. On uniting the properties of the transformations (2.245) and (2.246), we can make the following conclusion. Theorem 2.23. The infinitesimal transformation of a small neighborhood of the point contained in a continuum is a superposition of tension-compression of the neighborhood along the eigendirections q˛ and rotation of the axes q˛ as a rigid whole about the axis with the direction vector !. ı

Thus, we have the certain analogy between the eigendirections p˛ of the tensor U ı and the directions q˛ of the tensor D: elementary material fibres oriented along p˛ and along q˛ remain mutually orthogonal and undergo only tension-compression. ı

ı

The axes p˛ remain mutually orthogonal under any finite transformations from K to K, but q˛ – only under infinitesimal transformations from K.t/ to K.t C dt/.

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2 Kinematics of Continua

2.4.9 Kinematic Meaning of the Vorticity Vector Just as any orthogonal tensor, the orthogonal tensor Q! of infinitesimal rotation from K.t/ to K.t C dt/ can be represented in the form (see [12]) Q! D E cos.d'/ C .1  cos.d'//e ˝ e  e  E sin.d'/;

(2.248)

where d' is the infinitesimal angle of rotation of the trihedron q˛ about the axis with the direction vector e. Since values of d' are infinitesimal, we have Q! D E  e  Ed':

(2.249)

On comparing (2.249) with (2.246), we get !D

d' e; dt

j!j D

d' ; dt

(2.250)

i.e. the vorticity vector ! is really oriented along the instantaneous rotation axis e, and the length j!j is equal to the instantaneous angular rate of rotation of the trihedron q˛ of the deformation rate tensor. Let us consider now the question: relative to what system the vorticity vector ! defines the rotation rate. To answer the question, we introduce another orthogonal rotation tensor OW D qi ˝ eN i ;

(2.251)

which transforms the Cartesian trihedron eN i as a rigid whole into the orthonormal trihedron qi : qi D OW  eN i :

(2.252)

The tensor OW is a function of time t, because qi D qi .t/. According to (2.240) and (2.252), the vorticity tensor W takes the form P W  OT : W D qP i ˝ qi D O W

(2.253)

With the help of (2.253) we can represented the orthogonal tensor Q! as follows: P W  OT : Q! D E C dtW D E C dt O W

(2.254)

Thus, at each time t two orthogonal tensors OW and Q! connect local neighı

borhoods of a point M in the reference K and actual configurations K.t C dt/. ı

If in K we consider an elementary radius-vector d x00 , then in K we find its

2.4 Rate Characteristics of Continuum Motion

63

ı

corresponding radius-vector d x00 obtained with the help of the rotation tensor OW , and in K.t C dt/ – radius-vector d x0 : ı

d x00 D OW  d x00 ;

d x0 D Q!  d x00 :

A fixed observer connected to the Cartesian trihedron eN i sees both the transformations: finite rotation for time t, which is described by the tensor OW , and instantaneous rotation of a local neighborhood for time dt, which is described by the infinitesimal rotation tensor Q! Thus, the vorticity vector ! is the vector of instantaneous angular rate of rotation of the trihedron q˛ relative to the trihedron eN i . Comparing (2.246) with (2.254) (or (2.253) with (2.227)), we get P W  OT D !  E: O W

(2.255)

2.4.10 Tensor of Angular Rate of Rotation (Spin) P W  OT , where OW is the orthogonal In Sect. 2.3 we have introduced the tensor O W P  QT can be set up for any orthogonal tensor tensor of rotation. Such tensor  D Q Q depending on time t. P  QT is skew-symmetric, because The tensor Q

i.e.

P  QT C Q  Q P T D .Q  QT / D .E/ D 0; Q

(2.256)

P  QT /T D Q  Q P T D Q P  QT D :  T D .Q

(2.257)

This tensor characterizes the angular rate of rotation of the orthonormal trihedron hi formed with the help of Q: hi D Q  eN i ; (2.258) relative to the Cartesian trihedron eN i . P  QT we can form the tensor (2.254): Indeed, with the help of the tensor Q P  QT ; Q! D E C dt Q

(2.259)

which, according to (2.247), is the orthogonal tensor of infinitesimal rotation; and this tensor can be represented in the form (2.248) or (2.249): Q! D E  d'e  E;

(2.260)

where d' is the infinitesimal angle of rotation of the trihedron hi about the axis with the direction vector e. Comparing (2.259) with (2.260), we get the expression P  QT D  d' e  E; Q dt

(2.261)

64

2 Kinematics of Continua

P T characterizes the instantaneous which makes clear the sentence that the tensor QQ angular rate d'=dt of rotation of the trihedron hi about the axis e. P  QT is called the tensor of angular rate of rotation or the spin. The tensor Q Expressing the tensor Q from (2.258) in terms of the bases hi and eN i (h˛ D h˛ , ˛ D 1; 2; 3, as the vectors are orthonormal): Q D hi ˝ eN i ;

(2.262)

we get another representation of the spin: P  QT D hP i ˝ hi : Q

(2.263)

Thus, we have proved the following theorem. Theorem 2.24. The spin connects the rates hP i and the vectors hi defined by formula (2.258) as follows: P  QT /  hi : hP i D .Q (2.264) T P  Q is a skew-symmetric tensor, we can introduce the correSince the spin Q sponding vorticity vector !h : P  QT D !h  E Q

(2.265)

(from formulae (2.261) and (2.265) it follows that !h D .d'=dt/e); then formula (2.264) takes the form hP i D !h  hi : (2.266) j

Resolving the vector !h for the orthonormal basis hi : !h D ! h hj ; we get one more representation of formula (2.264): j j hP i D ! h hj  hi D j i k ! h hk :

(2.267)

This formula can also be rewritten in the form hP ˛ D !hˇ h ;

˛ ¤ ˇ ¤  ¤ ˛:

(2.268)

Taking different orthogonal tensors (or orthonormal bases) as Q (or hi ), we obtain different spins: ı

(1) If we choose eigenvectors of the stretch tensor U as hi , i.e. hi D pi , then, according to (2.263), the corresponding spin U takes the form ı

ı

P U  OT D pi ˝ pi ; U D O U and formula (2.264) yields

P ı ı pi D U  pi :

ı

OU D pi ˝ eN i ;

(2.269)

(2.270)

2.4 Rate Characteristics of Continuum Motion

65

(2) If hi D pi , then the corresponding spin V and the rotation tensor OV have the forms P V  OT D pP i ˝ pi ; OV D pi ˝ eN i ; V D O (2.271) V pP i D V  pi :

(2.272)

(3) If hi D qi , then the corresponding spin W coincides with the vorticity tensor W (see (2.253)): P W  OT D qP i ˝ qi D W; W D O W

(2.273)

qP i D W  qi :

(2.274)

(4) If we take the rotation tensor O accompanying deformation as Q, then, as shown ı in (2.159a), the tensor O connects two moving bases pi and pi : ı

pi D O  pi :

(2.275)

The tensor O can be expressed in terms of OV and OU as follows: ı

ı

O D pi ˝ pi D pi ˝ eN i  eN j ˝ pj D OV  OTU :

(2.276)

The corresponding spin  has the form P  O T D .O P V  OT C OV  O P T /  OU  O T DO U U V T T T P P D OV  OV C OV  OU  OU  OV D V  OV  U  OTV :

(2.277)

Unlike the cases (1)–(3), the spin tensor  characterizes the angular rate of roı tation of the trihedron pi relative to the moving trihedron pi , but not relative to the trihedron eN i being fixed. Therefore, for the cases (1)–(3) the spins characterize the total angular rate, and for the case (4) – the relative rate.

2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients In continuum mechanics, one often needs the relations between rates of the deformation tensors (and also measures) and the velocity gradients L D .r ˝ v/T and ı T ı L D r ˝v : Let us derive these relations.

(2.278)

66

2 Kinematics of Continua

Theorem 2.25. The rates of varying the gradient FP and the inverse gradient .F1 / ı

are connected to L and L by the relations FP D L  F;

.F1 / D F1  L;

ı

FP D L; ı

.F1 / D F1  F1T  L:

(2.279)

H Differentiating the relationships (2.35a) with respect to time t and taking the definition of the velocity (2.181) into account, we get ı

FP T D r ˝ v D FT  r ˝ v;

FP D .r ˝ v/T  F:

(2.279a)

ı

According to the definitions of tensors L (2.223) and L (2.278), from (2.279a) we obtain formulae (2.279). Differentiating the identity .F  F1 / D EP D 0; we find that FP  F1 D F  .F1 / I whence we get .F1 / D F1  FP  F1 :

(2.280)

On substituting the first two formulae of (2.279) into (2.280), we obtain .F1 / D F1  .r ˝ v/T ;

.F1T / D .r ˝ v/  F1T ;

(2.281)

i.e. the third and the fourth relationships of (2.279) hold as well. N According to formulae (2.225), (2.279a), and (2.281), we find that the rate of the deformation gradient is connected to the deformation rate tensor D by the relations DD

1 P 1 .F  F C F1T  FP T /; 2

1 D D  .F  FP 1 C FP 1T  FT /: 2

(2.282)

Here and below we will use the notation FP 1  .F1 / . P A, P ƒ, P JP and deformation Theorem 2.26. The rates of the deformation tensors C, ı P gP , .G1 / and .g1 / are connected to the velocity gradients L and L measures G, by the relationships 8
P D D  LT  A  A  L; A JP D D C L  J C J  LT ;

(2.283)

2.4 Rate Characteristics of Continuum Motion

and 8
gP D LT  g  g  L;

:.G1 / D 2F1  D  F1T ;

.g1 / D L  g1 C g1  LT I

67

(2.284)

and also 8  ı  ı ı ı ˆ P D .1=2/ FT  L C LT  F ; JP D .1=2/ L  FT C FT  L ; C ˆ ˆ ˆ <   ı ı P D .1=2/ .E  2A/  L  F1 C F1T  LT  .E  2A/ ; A ˆ ˆ   ˆ ı ı ˆ :ƒ P D .1=2/ F1  L  .E  2ƒ/ C .E  2ƒ/  LT  F1T ;

(2.285)

and 8  ı  ı ı ı ˆ T T 1 1T T P ˆ P G D F  L C L  F; g D  g  L  F C F  L  g ; ˆ ˆ <  ı ı  .G1 / D  F1  L  G1 C G1  LT  F1T ; ˆ ˆ ˆ ı ı ˆ : 1  .g / D L  FT C FT  L:

(2.286)

H To prove formula (2.283), we must differentiate the relationships (2.72) with respect to t and apply formulae (2.279): P D 1 .FP T  F C FT  F/ P D 1 .FT  LT  F C FT  L  F/ C 2 2 1 D FT  .LT C L/  F D FT  D  F; 2 P D 1 .FP 1T  F1 C F1T  FP 1 / A 2 1 T 1T 1  F C F1T  F1  L/ D .L  F 2 1 D .LT  .E  2A/ C .E  2A/  L/ D D  LT  A  A  L; 2

(2.287)

(2.288)

P D  1 .FP 1  F1T C F1  FP 1T / D 1 .F1  L  F1T C F1  LT  F1T / ƒ 2 2 1 1 D F  .L C LT /  F1T D F1  D  F1T ; (2.289) 2

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2 Kinematics of Continua

1 1 JP D .FP  FT C F  FP T / D .L  F  FT C F  FT  LT / 2 2 1 D .L  .E C 2J/ C .E C 2J/  LT / D D C L  J C J  LT : 2

(2.290)

Formulae (2.284) follow from (2.283), if we have used the connections (2.74) and (2.75) between the deformation tensors and measures. ı

Formulae (2.285) follow from (2.287)–(2.290), if we have gone from L to L: ı

L D L  F1 :

(2.291)

Using the connections (2.74) and (2.75) between the deformation tensors and measures, from (2.285) we get formulae (2.286). N ı P V P and the velocity gradients L and L are more Relations between the tensors U, ı

complicated. To derive them, we should use representations of L and L in terms of the eigenbasis vectors. Theorem 2.27. The following expressions for the velocity gradients hold: ı

ı

LT D r ˝ v D

3 X @ ı ı P ˛ p˛ ˝ p˛ C U  FT  FT  V ; r ˝xD @t ˛D1

! ˛ ı P ˛ ı˛ˇ C Uˇ ˛ p˛ ˝ pˇ C V ; ˛ ˇ

3 X

L D .r ˝ v/T D

˛;ˇ D1

(2.292)

(2.293)

ı

ı

where U˛ˇ are components of the tensor U with respect to the eigenbasis p˛ : ı

ı

U D pi ˝ pi D

3 X

ı

ı

ı

U˛ˇ p˛ ˝ pˇ ;

ı

ı

ı

ı

ı

U˛ˇ D p˛  U  pˇ D p˛  pˇ : (2.294)

˛D1

H To prove formula (2.292), we should consider the first formula of (2.279a) and substitute into this formula the expression (2.171) for F in the eigenbasis: ı

L D FP T D T

3 X

ı

ı

ı

.P ˛ p˛ ˝ p˛ C ˛ .p˛ ˝ p˛ C p˛ ˝ pP ˛ //:

(2.295)

˛D1

Using formulae (2.270) and (2.272), from (2.295) we derive the relationship (2.292). To prove formula (2.293), we use formulae (2.292) and (2.291), having expressed F1 in the form (2.171); then we get

2.4 Rate Characteristics of Continuum Motion ı

L D L  F1 D

3 X

69

! ı ı ı P ˛ p˛ ˝ p˛ C ˛ .p˛ ˝ p˛ C pP ˛ ˝ p˛ / 

˛D1

D

3 X ˛;ˇ D1

˛ ı  ı P ˛ ı˛ˇ p˛ ˝ p˛ C .p  p /p˛ ˝ pˇ ˇ ˇ ˛ ˇ

3 X

ı

1 ˇ pˇ ˝ pˇ

ˇ D1

! C

3 X

pP ˛ ˝ p˛ : (2.296)

˛D1

ı

Here we have taken into account that the vectors p˛ are orthonormal. Using formulae (2.270) and (2.272), from (2.296) we derive the formula (2.293). N From formula (2.293) it follows that the deformation rate tensor D and the vorticity tensor can be represented in terms of the eigenbasis p˛ as follows: !   3 X ˇ ı 1 ˛ P ˛ DD ı˛ˇ C  Uˇ ˛ p˛ ˝ pˇ ; (2.297) ˛ 2 ˇ ˛ ˛;ˇ D1

  3 ˇ ı 1 X ˛ C Uˇ ˛ p˛ ˝ pˇ C V : WD 2 ˇ ˛

(2.298)

˛;ˇ D1

Here we have taken into account that the tensors U and V are skew-symmetric. Denote components of the tensor D with respect to the basis p˛ by D˛ˇ : DD

3 X

D˛ˇ p˛ ˝ pˇ ; D˛ˇ D p˛  D  pˇ :

(2.299)

˛;ˇ D1

Then from (2.297) and (2.299) we get that diagonal components of the deformation rate tensor D˛˛ with respect to the eigenbasis p˛ determine the relative rates of lengthening the material fibres oriented along the eigenvectors p˛ (compare with formula (2.236)): D˛˛ D P ˛ =˛ D d sP˛ =ds˛ ;

˛ D 1; 3I

(2.300)

ı

and off-diagonal components D˛ˇ are connected to U˛ˇ by the relations D˛ˇ

1 D 2

2˛  2ˇ ˛ ˇ

!

ı

Uˇ ˛ ;

˛ ¤ ˇ:

(2.301) ı

From formulae (2.299) and (2.301) we can express the components U˛ˇ in terms of components of the deformation rate tensor: ı

U˛ˇ D

ı 2˛ ˇ D˛ˇ ; ˛ ¤ ˇI U˛˛ D 0: 2 2 ˇ  ˛ ı

(2.302)

The diagonal components U˛˛ are equal to zero, because the tensor U is skewsymmetric.

70

2 Kinematics of Continua

On substituting the relationships (2.302) into (2.298), we find the expression for ı

components V ˛ˇ of the tensor V with respect to the basis p˛ in terms of the tensors W and D (and, hence, in terms of the velocity gradient L): ı

V ˛ˇ D p˛  V  p˛ D p˛  W  pˇ  where V D

3 X

2˛ C 2ˇ 2˛  2ˇ

D˛ˇ ; ˛ ¤ ˇ;

(2.303)

V ˛ˇ p˛ ˝ pˇ :

(2.304)

˛;ˇ D1

Remark. The expressions (2.302) and (2.303) are valid only if the eigenvalues are not multiple: ˛ ¤ ˇ (˛ ¤ ˇ; ˛; ˇ D 1; 2; 3). If within the interval Œt1 ; t2  all three eigenvalues are coincident: ˛ D  (˛ D 1; 2; 3), then the stretch tensors are ı ı spherical: U D pi ˝ pi D E, V D E, and the eigenbases are not uniquely ı defined: as pi and pi we can take any orthonormal triple of vectors. In particular, ı one of the bases can be taken as fixed 8t 2 Œt1 ; t2 , for example, pi can be chosen ı as coincident with pi .t1 /; and the second basis pi can depend on time t. In this ı case pi  0 8t 2 Œt1 ; t2 , and from (2.294) and (2.298) it follows that within the considered time interval: ı

U˛ˇ D 0; V D W;

U D 0;

(2.305)

V ˛ˇ D p˛  W  pˇ ; ˛; ˇ D 1; 2; 3:

(2.306)

These relationships take the place of formulae (2.302), (2.303) in this case. If within the time interval Œt1 ; t2  only two of three eigenvalues are coincident, ı ı for example, ˛ D ˇ , then their corresponding eigenvectors p˛ and pˇ are not ı

uniquely defined as well: only their orthogonality to the vector p , corresponding ı

ı

to the third eigenvalue  , is given. Then we can extend the definition of p˛ and pˇ ı

so that p˛  pˇ D 0 8t 2 Œt1 ; t2 . In this case it follows from (2.294) that the only ı

ı

ı

component U˛ˇ vanishes, but U˛ ¤ 0 and Uˇ ¤ 0. It follows from (2.298) that the component V ˛ˇ is determined by the formula V ˛ˇ D p˛  W  pˇ ; ı

ı

U˛ˇ D 0:

(2.307)

ı

The remaining components U˛ , Uˇ and V ˛ , Vˇ are determined by formulae (2.302) and (2.303). If the situation with multiple roots appears only at some time t, then the values ı

of U˛ˇ .t/ and V ˛ˇ .t/ can be determined by passing to the limit.

t u

2.4 Rate Characteristics of Continuum Motion

71

Substituting formulae (2.294) and (2.298) into Eq. (2.277), and taking expressions (2.269) and (2.271) for OU and OV into consideration, we obtain the representation of the spin  in the basis p˛ :  D W

3 2 2 3 X ı 1 X ˛ C  ˇ ı ı ı Uˇ ˛ p˛ ˝pˇ  U˛ˇ .pi ˝Nei / p˛ ˝ pˇ .Nej ˝pj /: 2 ˛ ˇ ˛;ˇ D1

˛;ˇ D1

(2.308) Introducing the notation for direction cosines ı

ı

l ˛ˇ D p˛  eN ˇ ;

l˛ˇ D p˛  eN ˇ ;

(2.309)

substituting (2.302) into (2.308) and collecting like terms, we obtain the following expression of the spin  in terms of W and D (i.e. in terms of L): e  D W C ; eD 

3 X

e  p ˝ p ; 

e  D 

3 X

2 ˛;ˇ D1 ˇ

3 X

e ˛ˇ ; 

˛;ˇ D1

;D1

e  D 

(2.310)

1 ..2˛ C 2ˇ /ı˛ ıˇ  2˛ ˇ l˛ lˇ /D˛ˇ :  2˛

P .U1 / and V, P .V1 / are Theorem 2.28. Rates of the deformation measures U, connected to the velocity gradient L by the formulae e  O C OT  .D C  e T /  F/; P D 1 .FT  .D C / U 2 1 e  O C OT  .D   e T /  F1T /; .U1 / D  .F1  .D  / 2 P D 1 ..L C /  V C V  .LT C T //; V 2 1 1  .V / D ..  LT /  V1 C V1  .T  L//: 2

(2.311)

H Let us express the tensors V and U from the polar decomposition (2.137): U D OT  F;

V D F  OT :

(2.312)

Since U and V are symmetric tensors, these expressions can be rewritten in the symmetrized form UD

1 T .F  O C OT  F/; 2

VD

1 .F  OT C O  FT /: 2

(2.313)

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2 Kinematics of Continua

Let us differentiate these relationships: P CO P T  F C OT  F/; P P D 1 .FP T  O C FT  O U 2 PTCO P  FT C O  FP T /: P D 1 .FP  OT C F  O V 2

(2.314)

On substituting formulae (2.279) and the expression (2.277) for the spin  into (2.314), we obtain P  OT  O C OT  O  O P T  F C OT  L  F/ P D 1 .FT  LT  O C FT  O U 2 D

1 T .F  .LT C /  O C OT  .L C T /  F/; 2

(2.315)

PTCO P  OT  O  FT C O  FT  LT / P D 1 .L  F  OT C F  OT  O  O V 2 D

1 ..L C /  V C V  .LT C T //: 2

Taking formulae (2.310) and (2.224) into consideration, we find that e D D C : e LT C  D D C WT C W C 

(2.316)

Substituting (2.316) into (2.315), we really get the first and the third formulae of (2.311). The remaining two formulae in (2.311) can be proved in a similar way. N In continuum mechanics the deformation tensors B and Y are applied, which have no explicit expression; they are defined by their derivatives and initial values: P P  U1 C U1  U/; P D 1 .U B.0/ D 0; B 2

(2.317)

P D 1 .V P P  V1 C V1  V/; Y Y.0/ D 0: (2.318) 2 After substitution of the expressions (2.312), formula (2.317) takes the form 1 PT P P  U1 C U1  .FP T  O C FT  O//  F C OT  F/ BP D ..O 2 D

1 T .O  . C L/  F  U1 C U1  FT  .LT C T /  O/ 2

D

1 T .O  . C L/  O C OT  .LT C T /  O/; 2

(2.319)

2.4 Rate Characteristics of Continuum Motion

73

and formula (2.318) is rewritten as follows: P T /  V1 C V1  .O P  FT C O  FP T // P D 1 ..FP  OT C F  O Y 2 D

1 .L  F  OT  V1 CF  OT  T  V1 CV1    O  FT C V1  O  FT  LT / 2

D

1 .L C V    V1 C V1    V C LT /: 2

(2.320)

P Finally, we get the following expressions for BP and Y: BP D OT  D  O;

(2.321)

P D D C 1 .V  T  V1 C V1    V/: Y 2

(2.322)

2.4.12 Trajectory of a Material Point, Streamline and Vortex Line Having fixed coordinates X i of a material point M in the motion law (2.3), we get the parametric equation of a certain curve, where time t is a parameter: 0 6 t 6 t 0:

x i D x i .X k ; t/;

(2.323) ı

The origin of the curve at t D 0 is a point with Cartesian coordinates x i .X k / of the ı

material point M in K, and the end of the curve at t D t 0 is a point with Cartesian coordinates x i .X k ; t 0 / of the point M in K.t 0 / (Fig. 2.14). The curve (2.323) is called the trajectory of the point M in the Cartesian coordinate system O eN i . In the spatial description, the trajectory (2.323) at fixed X k is a solution of the kinematic equation (2.190): dx i =dt D vN i .x j ; t/; with the initial condition t D0W

Fig. 2.14 Trajectory of material point M

0 < t 6 t0 ı

x i D xi :

(2.324)

74

2 Kinematics of Continua

Here vN i .x j ; t/ are the velocity components with respect to the Cartesian basis eN i , which are assumed to be known. Let a field of velocities v.x j ; t/ D vN i eN i be given. Fix a time t, and take a point M1 with Eulerian coordinates x1i and Lagrangian coordinates X1k . Then a streamline passing through the point M1 is the curve x i D x i .X k ; /;

1 6 6 2 ;

(2.325)

which at its every point x i has a tangent being parallel to the velocity v.x i ; t/ at the considered point and at the considered time. The equation of the streamline has the form dx i =d D vN i .x j ; t/; D 1 W

1 < 6 2 ;

x D x1i : i

(2.326)

Thus, the trajectory of a material point and the streamline are described, in general, by different equations, and so they are not coincident. However, if the motion of a continuum is steady-state within time interval t1 6 t 6 t 0 , then in Eulerian description all the partial derivatives of all values, describing the motion, with respect to time vanish, in particular @v.x i ; t/=@t D 0. So the trajectory equations (2.324) and the streamline equation (2.326) become coincident within the interval t1 6 t 6 t 0 , if they have at least one common point M1 : dx i =d D vN i .x j /; D 1 W

t 1 D 1 < 6 2 D t 0 ;

x i D x1i D x i .X1k ; t1 /:

(2.327)

In other words, in the steady-state motion a material point M moves along a streamline: at time t D t1 its coordinates are the same as coordinates of point M1 at parameter value D 1 , and at time t D t2 they are the same as coordinates of point M2 at parameter value D 2 (Fig. 2.15). Multiplying the Eq. (2.326) by the basis vectors eN i , we can rewrite the streamline equation in the vector form d x D v.x; t/d ; 1 < 6 2 ; x D x1 ; D 1 :

Fig. 2.15 The streamline

(2.328)

2.4 Rate Characteristics of Continuum Motion

75

Let us define now a vortex line passing through a point M1 : this is a curve, which at its every point x i has a tangent being parallel to the vorticity vector !.x j ; t/ at the considered point and at the fixed time t. The vortex line is described by the equation d x D !.x; t/d ; x D x1 ;

1 < 6 2 ; D 1

(2.329)

2.4.13 Stream Tubes and Vortex Tubes Consider a curve L in coordinates x i and draw a streamline through each point of the curve L. If L is not a streamline itself, then we get a surface †v , at each point of which the velocity v lies on the tangent plane to the surface. This surface is called the stream surface. Let fv .x i / D 0 (2.330) be the equation of the stream surface. Since the vector r f is normal to the surface †v [12], so it is orthogonal to the velocity v, i.e. we have the relation v  r fv D 0;

(2.331)

which is a partial differential equation for determination of the function f .x i / by the known velocity field v.x i ; t/ at fixed t. If a curve L is closed, then the set of streamlines drawn through its points is called the stream tube. Let a curve L be not a vortex line. Drawing a vortex line through each point of the curve L, we obtain the vortex surface †! , which is described by the equation f! .x i / D 0. This relation is a solution of the differential equation !  r f! D 0:

(2.332)

If L is a closed curve, then the surface †! is called the vortex tube.

Exercises for 2.4 2.4.1. Show that the tensors QU and QV are orthogonal. ı

2.4.2. Using formulae (2.284) and (2.124), show that for the coefficient k determined by formula (2.121) we have the following relationship: ı

ı

k  D k.n  D  n/:

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2 Kinematics of Continua

2.4.3. Using formulae (2.124), (2.279) and the result of Exercise 2.4.2, show that a rate of changing the normal n is determined as follows: nP D  n  n  L;

 D n  D  n:

2.4.4. Using the results of Exercise 2.4.2, show that for the coefficient k determined by formula (2.121) we have the following equation: kP D k.n  D  n/: 2.4.5. Show that the transformations (2.245) of infinitesimal tension–compression (2.245) and infinitesimal rotation (2.246) are commutative up to terms of order .dt/2 : Q!  AD  d x D A!  QD  d x; while transformations of a small neighborhood determined by the tensors O and U or O and V are not commutative in general. 2.4.6. Prove Theorem 2.14. 2.4.7. Using the representation (2.142) for tensors U and V and formulae (2.270) and (2.272), show that rates of stretch tensors are expressed as follows: P D U

3 X

ı ı P D P ˛ p˛ ˝ p˛ C U  U  U  U ; V

˛D1

3 X

P ˛ p˛ ˝ p˛ C V  V  V  V :

˛D1

2.4.8. Show that expressions for rates of the Hencky tensors (2.167) have the form ı

H D

3 P 3 P X X ı ı ˛ ı ˛ ı p˛ ˝ p˛ CU  H HU ; H D p˛ ˝p˛ CV HHV :   ˛ ˛ ˛D1 ˛D1

2.4.9. Using representations (2.294) and (2.304) for tensors U and V , and also Eq. (2.300) and the result of Exercise 2.4.8, show that rates of the Hencky tensors can be expressed in the form ı

H D

3 X

ı

ı

D˛ˇ p˛ ˝ pˇ C

˛;ˇ D1

P D DC H

3 X ˛;ˇ D1

! 2˛ ˇ ˇ ı ı lg  1 D˛ˇ p˛ ˝ pˇ ; 2ˇ  2˛ ˛

˛¤ˇ

3 X ˛;ˇ D1 ˛¤ˇ

! 2˛ ˇ ˇ lg  1 .p˛  D  pˇ /p˛ ˝ pˇ : 2ˇ  2˛ ˛

2.5 Co-rotational Derivatives

77 ı

P derived in Exercise 1.4.9 can be rewrit2.4.10. Show that expressions for H and H ten as follows: ı ı P D 4 XH  D; H D 4 XH   D; H where the following fourth-order tensors are denoted: 4

XH

ı

XH D X H

ijkl

ıi ijkl p

ı

˝ pj ˝ pk ˝ pl ;

8  ˆ < 2˛ ˇ lg ˇ ˛ˇ kl ; 2ˇ  2˛ ˛ D ˆ : ; ˛ˇ kl

4

XH D X H

˛ ¤ ˇ; ˛ D ˇ;

i ijkl p

˝ pj ˝ pk ˝ pl ;

˛ˇ kl D.1=2/.ı˛k ıˇ l C ı˛l ıˇ k /:

2.4.11. Show that relations (2.294) and (2.302) for U , Eq. (2.303) and (2.304) for V and Eq. (2.310) for  can be rewritten as follows: U D 4 U   D;

V D 4 V   D C W;

e   D C W;  D 4

where 4

ı

ı

U D Uijkl pi ˝ pj ˝ pk ˝ pl ;

4

V D Vijkl pi ˝ pj ˝ pk ˝ pl ;

4e

U˛ˇ kl

e ijkl pi ˝ pj ˝ pk ˝ pl ; D 8 8 2 2 ˆ ˆ < 2˛ ˇ ˛ˇ kl ; ˛ ¤ ˇ; < ˛ C ˇ  2 2 ˛ˇ kl ; ˛ ¤ ˇ; V ˛ˇ kl D 2ˇ  2˛ D ˇ  ˛ ˆ ˆ :0; : ˛ D ˇ; 0; ˛ D ˇ; e ˛ˇ D V ˛ˇ    .l ˛ lˇ  l˛ lˇ /; ˛ ¤ ˇ;  ¤ ;  2  2

if ˛ D ˇ (or  D ), then the first (or the second) summand vanishes. 2.4.12. Using the definitions (2.225) and (2.226) of the tensors D and W, and also the properties of unit tensors, show that D, W and L are connected by the formulae e   L;  e D .1=2/.III  II /: D D    L; W D 

2.5 Co-rotational Derivatives 2.5.1 Definition of Co-rotational Derivatives Besides the total derivative d a=dt introduced in Sect. 2.4.1 and partial derivative of vectors and tensors with respect to time @a=@t, so-called co-rotational derivatives

78

2 Kinematics of Continua

are of great importance in continuum mechanics. They determine rates of changing tensors relative to some moving basis hi , i.e. the relative rates. Let in a actual configuration K.t/ there be some moving bases hi or hi and arbitrary varying scalar .X i ; t/, vector a.X i ; t/ and second-order tensor T.X i ; t/ fields with the following components with respect to the bases: a D a i hi D ai hi ;

(2.333)

T D T ij hi ˝ hj D Tij hi ˝ hj D T ij hi ˝ hj D Ti j hi ˝ hj :

(2.334)

Since any scalar function .X i ; t/ is not connected to any basis (moving or fixed), it is evident that the co-rotational derivative of the function must be coincident with the total derivative with respect to time: h

D P:

(2.335)

For a vector a and a tensor T we can introduce co-rotational derivatives ah and T as vectors or tensors, components of which with respect to the same basis hi coincide with rates of changing vector a and tensor T components, respectively: h

ah D

dai hi ; dt

Th D

d ij T hi ˝ hj : dt

(2.336)

If we consider the basis hi , then for the basis we can determine other corotational derivatives: dai i (2.337) h; aH D dt TH D

d Tij hi ˝ hj : dt

(2.338)

Thus, the co-rotational derivative ah (or Th ) determines the rate of varying a vector a (or a tensor T) for an observer moving together with the basis hi . For the observer, the basis hi is fixed, and hence in (2.336) the basis is not differentiated with respect to time. In a similar way, the derivatives aH and TH determine the rates of changing a and T for an observer moving together with the basis hi . We can determine the co-rotational derivatives of a second-order tensor T in mixed moving dyadic bases hi ˝ hj and hi ˝ hj , respectively: Td D

d i T hi ˝ hj ; dt j

TD D

d j i T h ˝ hj : dt i

(2.339)

Since vector components ai and ai can always be expressed in the form ai D a  hi ;

ai D a  hi ;

(2.340)

2.5 Co-rotational Derivatives

79 j

and tensor components Tij , T ij , T ij and Ti , with the help of the scalar product of (2.334) by hi or hj , can be written as follows: T ij D hi Thj ;

Tij D hi Thj ;

T ij D hi Thj ;

j

Ti D hi Thj ; (2.341)

so rates of changing vector and tensor components in (2.336), (2.337), and (2.339) can be represented in the explicit form: dai d hi da i D h Ca ; dt dt dt

d hi dai da D  hi C a  ; dt dt dt

(2.342)

and also d T ij D hi dt d Tij D hi dt d i T D hi dt j d j T D hi dt i

dT j d hi h C  T  hj C hi  T  dt dt dT d hi   hj C  T  hj C hi  T  dt dt 

d hj ; dt d hj ; dt

d d hi d hj T  hj C  T  hj C hi  T  ; dt dt dt d d hi d  T  hj C  T  hj C hi  T  hj : dt dt dt 

(2.343)

Here total derivatives d a=dt and d T=dt are determined by the rules (2.187) and (2.192), respectively. Rates of changing basis vectors d hi =dt and d hi =dt are defined by the choice of basis hi or hj . Taking different bases as hi and hj , we get different co-rotational derivatives. Let us consider the most widely used bases.

2.5.2 The Oldroyd Derivative (hi D ri ) If we choose the general local vector basis ri as hi , then the derivative ah D aOl (or Th D TOl ) determines the rate of changing a (or T) relative to the Lagrangian coordinate system X i moving together with the continuum. This derivative is called the Oldroyd derivative [44]. The derivative d ri =dt is determined as follows: @v @v d ri @2 x d hi D D ri rj ˝ j D ri r ˝v D .r ˝v/T ri : (2.344) D D i i dt dt @t@X @X @X In this case, as a basis hi we consider the reciprocal local basis ri , the derivative of which d ri =dt with respect to time has the form d i d .r ˝ ri / D E D 0; dt dt

(2.345)

80

2 Kinematics of Continua

or

d ri d ri ˝ ri D ri ˝ D ri ˝ .r ˝ v/T  ri : dt dt

(2.346)

Multiplying the equation by rj from the right, we get d rj d hj D D rj  .r ˝ v/T D .r ˝ v/  rj : dt dt

(2.347)

On substituting the expressions (2.347) for the derivatives d hi =dt and d hj =dt into (2.342), we get the formula for the Oldroyd derivative in the basis ri : aOl D

dai da i d ri D  r ˝ ri  a  .r ˝ v/  ri ˝ ri D a  a  r ˝ v; (2.348) dt dt dt

TOl D

d Tij d T ij ri ˝ rj D ri ˝ rj dt dt

D ri ˝ r i 

d T  rj ˝ rj  ri ˝ ri  .r ˝ v/T  T  rj ˝ rj dt

ri ˝ ri  T  .r ˝ v/  rj ˝ rj D

d TT  r ˝ v.r ˝ v/T  T: (2.349) dt

Here we have taken into account that ri ˝ ri D E. Thus, we have proved the following theorem. Theorem 2.29. The Oldroyd derivative is related to the total derivative with respect to time as follows (for a vector a and for a tensor T, respectively): aOl D aP  a  r ˝ v;

TOl D TP  T  r ˝ v  .r ˝ v/T  T:

(2.350)

2.5.3 The Cotter–Rivlin Derivative (hi D ri ) If we choose the reciprocal local basis ri as a moving basis hi , then the derivative aH (or TH ) characterizes the rate of changing a (or T) relative to the basis ri moving together with the Lagrangian coordinate system X i . This derivative is called the Cotter–Rivlin derivative [10]. Because of formulae (2.342) and (2.347), we get the following theorem. Theorem 2.30. The Cotter–Rivlin derivative is related to the total derivative as follows (for a vector a and for a tensor T, respectively): aH  aCR D TH  TCR D

dai i r D aP C .r ˝ v/  a; dt

d Tij i r ˝ rj D TP C r ˝ v  T C T  .r ˝ v/T : dt

(2.351) (2.352)

2.5 Co-rotational Derivatives

81

2.5.4 Mixed Co-rotational Derivatives Since any vector a is defined by its components with respect to a vector basis, for example, in a moving basis hi or hj , so for the vector in the moving bases we can determine only two co-rotational derivatives: by Oldroyd and by Cotter–Rivlin. Any second-order tensor T is defined by its components with respect to a dyadic basis. Therefore, besides the Oldroyd and Cotter–Rivlin derivatives, which specify the rates of changes of a tensor T in moving dyadic bases ri ˝ rj and ri ˝ rj , by formulae (2.339) we can determine two more derivatives in moving mixed dyadic bases: d T ij d j i Td D (2.353) ri ˝ rj ; TD D T r ˝ rj : dt dt i On substituting the expressions (2.344) and (2.347) into (2.343), we get the following formulae for the rates of changing mixed components of the tensor T: d i T D ri  TP  rj  ri  .r ˝ v/T  T  rj C ri  T  .r ˝ v/T  rj ; dt j d j T D ri  TP  rj C ri  .r ˝ v/  T  rj  ri  T  .r ˝ v/  rj : dt i

(2.354)

Having substituted (2.354) into (2.353), we get the following theorem. Theorem 2.31. The mixed derivatives (2.353) are connected to the total derivative by the relations Td D TP  L  T C T  L;

TD D TP C LT  T  T  LT :

(2.355)

The derivatives (2.353) are called the left and right mixed co-rotational derivatives, where L D .r ˝ v/T is the velocity gradient (see (2.223)). It should be noticed that, unlike other co-rotational derivatives considered in this paragraph, the mixed derivatives Td and TD do not form a symmetric tensor when they are applied to a symmetric tensor T. This fact explains a scarcer application of mixed derivatives in continuum mechanics.

ı

2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor ı

If we choose the eigenbasis pi of the right stretch tensor U as a moving basis hi , ı then, since pi are orthonormal, we get that hi and hj are coincident: h˛ D h˛ , ˛ D 1; 2; 3, and jhi j D 1. At every time, the moving coordinate system defined by

82

2 Kinematics of Continua ı

the trihedron pi executes an instantaneous rotation, which is characterized by the spin U (2.269), and due to (2.270) we have d hi P ı ı ı ı D pi D U  pi D pi  TU D pi  U : dt

(2.356)

On substituting (2.356) into (2.343), we get ah  a U D

dai ı ı ı ı ı p D aP  pi ˝ pi C a  U  pi ˝ pi D aP C a  U ; dt i

Th  TU D

T ij ı d T ij ı ı ı pi ˝ pj D pi ˝ p dt dt j

ı

ı

ı

ı

ı

ı

ı

(2.357)

ı

P  pj ˝ pj  pi ˝ pi  U  T  pj ˝ pj D p i ˝ pi  T ı ı ı ı C pi ˝ pi  T  U  pj ˝ pj D TP  U  T C T  U :

(2.358)

The co-rotational derivative of a vector a (or a tensor T) determined by (2.358) is called the right derivative relative to the eigenbasis. Thus, we have proved the following theorem. Theorem 2.32. The right derivative relative to the eigenbasis is connected to the total derivative as follows (for a vector a and for a tensor T, respectively): aU D aP C a  U ;

TU D TP  U  T C T  U :

(2.359)

2.5.6 The Derivative in the Eigenbasis (hi D pi ) of the Left Stretch Tensor Take the eigenbasis pi of the left stretch tensor V as a moving basis hi and define the following co-rotational derivatives aH  a V D

dai pi ; dt

TH  TV D

d T ij pi ˝ pj ; dt

(2.360)

called the left derivatives in the eigenbasis. Theorem 2.33. The left derivatives (2.360) in the eigenbasis are connected to the total derivative with respect to time by the following relations (for a vector a and for a tensor T, respectively): aV D aP  V  a;

TV D TP  V  T C T  V :

(2.361)

2.5 Co-rotational Derivatives

83

H A proof follows from (2.337), (2.342), and (2.343), because from (2.272) we have d hi D pP i D V  pi : dt

(2.362)

ı

Since the bases pi and pi are orthonormal, all the co-rotational derivatives ı ı relative to the mixed dyadic bases pi ˝ pi , pi ˝ pi coincide with TU or TV , respectively. N

2.5.7 The Jaumann Derivative (hi D qi ) If we choose the eigenbasis of the deformation rate tensor as a moving basis hi D qi (it should be noted that the basis qi is also orthonormal and coincides with qi ), then from (2.336) we get the co-rotational Jaumann derivatives: a h  aJ D

dai qi ; dt

Th  TJ D

d T ij qi ˝ qj : dt

(2.363)

Theorem 2.34. The Jaumann derivatives (2.363) are connected to the total derivatives with respect to time by the relations (for a vector a and for a tensor T, respectively) aJ D aP C a  W;

(2.364)

TJ D TP  W  T C T  W:

(2.365)

H According to the relationship (2.274), we get d hi D qP i D W  qi ; dt therefore, due to formulae (2.337) and (2.342) we find aJ D aP  qi ˝ qi C a  W  qi ˝ qi D aP C a  W: In a similar way, we can prove the relation (2.365). N

2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis Let hi be a moving orthonormal basis. In this case we denote co-rotational derivatives by the following way: ah  aQ and Th  TQ . Due to orthonormalization of

84

2 Kinematics of Continua

the basis hi , the total derivatives of a and T with taking account of (2.333), (2.334), and (2.336) can be written as follows: aP 

da dai d hi D hi C ai D aQ C ai !h  hi ; dt dt dt dh d hj  d T ij i TP D hi ˝ hj C T ij ˝ hj C hi ˝ dt dt dt

(2.366)

D TQ C T ij !h  hi ˝ hj  hi ˝ hj  !h : Here we have used formula (2.266) for derivative hP i of the moving basis, where !h is the vorticity vector giving a rotation of the basis hi relative to the fixed basis eN i (see (2.258) and (2.265)). With taking account of (2.333) and (2.334), formulae (2.366) can be written in the form aQ D aP  !h  a;

P  !h  T C T  !h : TQ D T

(2.367)

It should be noticed that if a D !h , then ! P h D !hh ;

(2.367a)

because !h  !h D 0 due to properties of the vector product.

2.5.9 Spin Derivative Take an arbitrary orthonormal basis hN i at a point M of a continuum in K. The trihedron must have the only property that at any time t the basis hN i rotates with the instantaneous angular rate, which is equal to the rotation rate of the trihedron pi ı relative to the trihedron pi . As shown in Sect. 2.4.10, the instantaneous rotation of P  OT determined by (2.277). the trihedron is characterized by the spin tensor  D O Then we can define the co-rotational derivative in the basis, which is called the spin derivative (of a vector a and of a tensor T, respectively): dai N hi ; dt

(2.368)

d T ij N hi ˝ hN j : dt

(2.369)

ah  aS D Th  TS D

Theorem 2.35. The spin derivative is related to the total derivative with respect to time as follows (for a vector a and for a tensor T, respectively): aS D aP C a  ;

(2.370)

TS D TP    T C T  :

(2.371)

2.5 Co-rotational Derivatives

85

H A proof of Theorem 2.35 follows from (2.342), (2.343) and the relation NPhi D   hN i ;

(2.372)

which is a consequence of (2.264). The relation (2.371) follows from (2.343). N

2.5.10 Universal Form of the Co-rotational Derivatives On comparing formulae (2.350), (2.351), (2.352), (2.359), (2.361), (2.365), (2.370), and (2.371), we can notice that all the representations of the co-rotational derivatives and also the total derivative with respect to time can be written in the universal form: P  Zh  T  T  ZT ; ah D aP  Zh  T; Th D T h h D f ; Ol; CR; U; V; J; S g;

(2.373)

where tensors Zh have the following representation for different h: Zh D f 0; L; LT ; U ; V ; W;  g; h D f ; Ol; CR; U; V; J; S g:

(2.374)

Since tensors U , V and  are linearly expressed in terms of W and D (see Exercise 2.4.11), so tensors Zh can be written as linear functions of W and D: Zh D 4 ZDh   D C 4 ZW h   W:

(2.375)

Table 2.1 gives expressions for fourth-order tensors 4 ZDh and 4 ZW h , where tensors 4 e are defined in Exercise 2.4.11. U , 4 V and 4 

2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient In Sect. 2.4.11 we have derived the relationships between rates of deformation tensors and velocity gradient L. Similar connections also exist between co-rotational derivatives of the tensors and L. Let us establish them. Table 2.1 Expressions of tensors 4 ZDh , 4 ZW h , and 4 Eh for different co-rotational derivatives h  Ol CR U V J S 4 4 4e  ZDh 0 III III 4 U V 0 4 ZW h 0 III III 0 III III III 4 Eh 0 2 2 0 0 0 0

86

2 Kinematics of Continua

P J, P gP , On substituting representations (2.283), (2.284), and (2.311) for rates A, P and .V1 / into formula (2.373), we get .g1 / , V Ah D D  .Zh C LT /  A  A  .ZTh C L/; Jh D D  .Zh  L/  J  J  .ZTh  LT /; gh D .Zh C LT /  g  g  .ZTh C L/; .g1 /h D .Zh  L/  g1  g1  .ZTh  LT /;     1 1 Vh D  Zh  .L C /  V  V  ZTh  .LT C T / ; 2 2     1 1 .V1 /h D  Zh  .  LT /  V1  V1  ZTh  .T  L/ ; 2 2 h D f ; Ol; CR; U; V; J; S g: (2.376) From these relationships we can find the following expressions: .V  E/h D D  4 Eh   D  .Zh C ZTh /     1 1 T T T  Zh  . C L/  .V  E/  .V  E/  Zh  . C L / ; 2 2 (2.377) .E  V1 /h D D C 4 Eh   D C Zh C ZTh     1 1  Zh  .  LT /  .E  V1 /  .E  V1 /  ZTh  .T  L/ : 2 2

Here we have denoted the co-rotational derivative of the metric tensor by Eh D 4 Eh   D:

(2.378)

The tensor 4 Eh differs from zero-tensor only when h D fCR; Olg (see Exercise 2.5.3), its expressions are given in Table 2.1 (see Sect. 2.5.10). Since tensors Zh and  are linearly expressed in terms of W and D (see formulae (2.375), (2.310), (2.302)–(2.304)), so on the right-hand sides of Eqs. (2.376) there are also linear functions of W and D, their explicit expressions will be given in Sect. 4.2.22.

Exercises for 2.5 2.5.1. Show that the mixed co-rotational derivatives, the left and the right co-rotational derivatives relative to the eigenbasis and also the Jaumann and spin derivatives satisfy the differentiation rules of scalar products:

2.5 Co-rotational Derivatives

87

.A  B/h D Ah  B C A  Bh ; . A/h D

h

AC

Ah ;

.a  A/h D ah  A C a  Ah ; h D f; d; D; U; V; J; S g;

and the Oldroyd and Cotter–Rivlin derivatives do not satisfy this rule. 2.5.2. Show that for the co-rotational derivatives, the following rules of differentiation of scalar products of two vectors a and b and also of two tensors T and B remain valid: .a  b/h D .a  b/ D ah  b C a  bh ;

h D fU; V; J; S gI

.T   B/h D .T   B/ D Th   B C T   Bh ;

h D fd; D; U; V; J; S g:

2.5.3. Show that the following co-rotational derivatives of the unit tensor E give the zero-tensor: Eh D 0; h D f; d; D; U; V; J; S g; and the Oldroyd and Cotter–Rivlin derivatives of E are different from zero: ECR D 2D;

EOl D 2D:

2.5.4. Using formulae (2.283) and (2.352), show that the Cotter–Rivlin derivatives of the left Almansi deformation tensor A and of the left Almansi measure g have the form gCR D 0: ACR D D; 2.5.5. Using formulae (2.283) and (2.350), show that the Oldroyd derivatives of the right Cauchy–Green tensor J and of the right deformation measure g1 have the form JOl D D; .g1 /Ol D 0: 2.5.6. Using the expressions for the tensors U (2.142), C and ƒ (2.165), and also G and G1 (2.165), show that we can write the right derivative relative to the eigenbaı sis p˛ in the form UU D

3 X

ı ı P ˛ p˛ ˝ p˛ ;

˛D1

ƒU D

3 P X ˛ ı ı p˛ ˝ p˛ ; 3  ˛D1 ˛

GU D 2CU ;

CU D

3 X

ı ı ˛ P ˛ p˛ ˝ p˛ ;

˛D1

.U1 /U D

3 P X ˛ ı ı p˛ ˝ p˛ ; 2  ˛D1 ˛

.G1 /U D 2ƒU :

88

2 Kinematics of Continua

2.5.7. Using the expressions for the tensors V (2.142), for A and J (2.165), and also for g and g1 (2.165), show that we can write the left derivative relative to the eigenbasis p˛ in the form VV D

3 X

P ˛ p˛ ˝ p˛ ;

AV D

˛D1

JV D

3 X

˛ P ˛ p˛ ˝ p˛ ;

3 P X ˛ p˛ ˝ p˛ ; 3 ˛D1 ˛

gV D 2AV ;

.g1 /V D 2JV :

˛D1

2.5.8. Show that the Oldroyd and Jaumann derivatives of a second-order tensor T are connected by the relationship TOl D TJ  T  D  D  T: 2.5.9. Show that if for an arbitrary symmetric tensor T its co-rotational derivatives are equal to zero: Th D 0; h D fd; D; U; V; J; S g; then the first invariant of the tensor: I1 .T/ D T   E has its stationary value, i.e. IP1 .T/ D 0: Show that for the co-rotational Oldroyd and Cotter–Rivlin derivatives this statement is not valid. 2.5.10. Using the results of Exercise 2.4.3, show that the co-rotational derivatives of the normal vector n satisfy the following relations: nCR D  n;

 D n  D  n;

nOl D  n  n  L  L  n;

nJ D  n  n  D:

2.5.11. Show that the following co-rotational derivatives of a symmetric tensor give a symmetric tensor: if A D AT ; then .Ah /T D Ah ; h D fU; V; J; S; Ol, CRg; and also a skew-symmetric tensor, if they are applied to a skew-symmetric tensor: if B D BT ; then .Bh /T D Bh ; h D fU; V; J; S; Ol; CRg: The mixed co-rotational derivatives h D d; D have no such properties.

Chapter 3

Balance Laws

3.1 The Mass Conservation Law 3.1.1 Integral and Differential Forms Let us consider fundamental laws of continuum mechanics, and complement the set of Axioms 1–3 (see Introduction) with new axioms. Axiom 4 (The mass conservation law). For every continuum B, there exists a scalar function M.B; t/ (i.e. the transformation M W U  RC0 ! RC ), called the m a s s of the continuum, which has the following properties:  It is positive: M > 0,  It is additive: M.B1 C B2 ; t/ D M.B1 ; t/ C M.B2 ; t/; 8B1 ; B2 , and 8t > 0,  It is invariant relative to any transformation of coordinates (2.1) and relative to

any motion (2.3). Due to the third property, the mass in any actual configuration remains constant: M.B; t/ D const:

(3.1)

Remark. The mass conservation law is valid only for a continuum containing the same material points for a considered time interval Œ0; t. If a continuum B loses or acquires material points as time goes on (in this case phase transformations are said to occur in the continuum (see Sect. 5.1)), then the mass conservation law in the form (3.1) is no longer valid. The mass conservation law does not hold as well, if we exclude Axiom 3 and consider the motions with speeds close to the light speed; however, relativistic phenomena are not considered in classical continuum mechanics. t u The law (3.1) can be written in another form dM=dt D 0:

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 3, c Springer Science+Business Media B.V. 2011 

(3.2)

89

90

3 Balance Laws

Since the mass is additive, M may be expressed as follows: Z M D d m;

(3.3)

V

where d m is the mass of an elementary volume dV involving a material point M belonging to the considered continuum V . Definition 3.1. The ratio  D d m=dV

(3.4)

is called the density of substance at the point M. Since the mass M and volume dV are positive, the mass d m and density  are always positive as well:  > 0; d m > 0: (3.4a) On substituting (3.3) and (3.4) into (3.2), we can rewrite the mass conservation law (3.2) in the form Z d dV D 0: (3.5) dt V

Having applied this relationship to an elementary volume, we get ı

ı

 dV D  d V D const;

(3.6) ı

ı

ı

where  and  are the densities in configurations K and K, respectively, and d V is ı

the elementary volume in the configuration K. The relationship (3.5) is called the mass conservation law in the integral form, and (3.6) – in the differential form.

3.1.2 The Continuity Equation in Lagrangian Variables ı

ı

Consider in K an elementary volume d V constructed on elementary radius-vectors ı ı oriented along the local basis vectors d x˛ D r˛ dX ˛ (see Sect. 2.2.6). This volume corresponds to a volume dV constructed on vectors r˛ dX ˛ in the actual configuraı

tion K. The volumes d V and dV are determined by formulae (2.15): q ˇ @xı k ˇ ı ˇ ˇ 1 2 3 d V D r1  .r2  r3 /dX dX dX D g dX dX dX D ˇ i ˇdX 1 dX 2 dX 3 ; @X ˇ @x k ˇ p ˇ ˇ dV D r1  .r2  r3 /dX 1 dX 2 dX 3 D g dX 1 dX 2 dX 3 D ˇ i ˇdX 1 dX 2 dX 3 : @X (3.7) ı

ı

ı

ı

1

2

3

On substituting (3.7) into (3.6), we get the following theorem.

3.1 The Mass Conservation Law

91 ı

Theorem 3.1. A change of the density in going from K to K can be determined by one of the following equations: ı

 D 

s

g ı

D

g

ˇ @x k ˇ ˇ ˇ D ˇ ı ˇ D det F: ı j n i j@x =@X j @x j@x k =@X i j

(3.8)

Unlike (3.6), these equations are called the continuity equations in Lagrangian variables. ı The ratio of the elementary volumes in configurations K and K, which follows from (3.7), is also widely used: ı

q

dV =d V D

ı

g=g:

(3.9)

3.1.3 Differentiation of Integral over a Moving Volume Consider a vector field a.x i ; t/, which is a continuously differentiable function of x i and t in a domain V .t/, 8t > 0, where the domain V .t/ contains the same material points (such a domain is usually called a moving volume). i Let us integrate the field R a.x ; t/ over the domain V .t/ and determine the derivad tive of the integral: dt V .t / a dV: ı

To do this, we should change variables in the integral: x i ! x i , where x i 2 V , ı

ı

ı

i k i x q 2 V , and, according to (3.8), the Jacobian of the transformation is j@x =@x j D ı g=g. Then we get Z Z s ı d g d a dV : a dV D (3.10) ı dt dt g V .t /

ı

V

Since this change in variables means the transformation from the configuration K ı

ı

to K, where the domain V is time–independent, the derivative with respect to t may be introduced under the integral sign: d dt

Z

Z a dV D V .t /

ı

V

1   Z p ı d B g C ı 1 d p p da gC g dV : a q @ q aA d V D dt ı ı dt dt g g ı 0

V

(3.10a)

92

3 Balance Laws

On substituting (2.187) and (2.217) into (3.10a), we get d dt

Z q

Z a dV D

ı

V .t /

  ı @a g=g ar  v C C v  r ˝ a dV @t ı

V

Z s

D

g ı

ı

g



@a C r  .v ˝ a/ @t



ı

Z 

dV D

@a C r  .v ˝ a/ @t

 dV:

V

V

(3.11) ı

The last equality we have obtained by the reverse substitution x i ! x i . Thus, we have proved following theorem. Theorem 3.2 (The Rule of Differentiation of the Integral over a Moving Volume). For any vector field a.x i ; t/ specified in V .t/ 8t > 0 and being a continuously differentiable function of x i and t, the following relationship holds: d dt

Z 

Z a.x i ; t/ dV D

@a Cr v˝a @t

 dV:

(3.12)

V

V .t /

Taking in formula (3.12) '.x i ; t/Ne˛ as a vector field a.x i ; t/, where eN ˛ is one of the Cartesian basis vectors, and '.x i ; t/ is a scalar field, and factoring eN ˛ outside the integral sign on the left and right sides, we get the formula for differentiation of the integral of a scalar field: d dt

Z 

Z '.x i ; t/ dV D

@' C r  .'v/ @t

 dV:

(3.13)

V

V .t /

3.1.4 The Continuity Equation in Eulerian Variables Putting ' D  in formula (3.13) and using the mass conservation law (3.5), we obtain  Z  @ C r  v dV D 0 (3.14) @t V

– the mass conservation law in Eulerian variables. Unlike the law (3.5) which is formulated for a volume V of a continuum containing at different times t > 0 the same material points, the law (3.14) holds for an arbitrary geometric domain V of space E3a , which at different times t may involve different material points. Since formula (3.14) holds true for an arbitrary domain V  E3a , the integrand must vanish. So we have proved the following theorem.

3.1 The Mass Conservation Law

93

Theorem 3.3. If the function , satisfying the mass conservation law (3.5), and the velocity v are continuously differentiable in V .t/ for all considered times t > 0, then at every point x of the domain V  E3a of a continuum the following relation holds 8t > 0: @ C r  v D 0: (3.15) @t Equation (3.15) is called the continuity equation in Eulerian variables. According to the definition (2.187) of the total derivative with respect to time, Eq. (3.15) can be rewritten in the form @ d C v  r  C r  v D C r  v D 0; @t dt

(3.16)

1 d D r  v:  dt

(3.17)

or

Due to (3.17), the continuity equation takes the form 

d dt

  1 D r  v: 

(3.18)

3.1.5 Determination of the Total Derivatives with respect to Time With the help of the continuity equation (3.15) we will prove several relationships which are frequently used in continuum mechanics. Theorem 3.4. If there is a tensor field n .x; t/ of the nth order, which is continuously differentiable in a domain V .t/ 8t > 0, then the following relations hold:   @ n d n D   C r  .v ˝ n /; dt  @t @ d  .n / D . n / C r  .v ˝ n /: dt @t

(3.19) (3.20)

H According to the definition of the total derivative with respect to time (2.192), we have     n  d n  @ n  D C v  r ˝ dt  @t    @ n @ n   2 D   2 .v  r  ˝ n / C v  r ˝ n : @t  @t  (3.21)

94

3 Balance Laws

Collecting the second and the third summands and taking the following consequence of the continuity equation into account: .@=@t/ C v  r  D r  v, we finally get   d n @ n @ n   C r  .v ˝ n /: (3.22) D C .r  v/ n  C v  r ˝ n  D dt  @t @t To prove formula (3.20), it is sufficient to make the substitution n  !  n  in (3.19). N

3.1.6 The Gauss–Ostrogradskii Formulae In continuum mechanics, the Gauss–Ostrogradskii formulae are frequently used for tensor fields k .x/ of arbitrary kth order in reference and actual configurations: Z

Z

ı

ı

ı

n ˝ k d † D ı

†

Z

ı

r ˝ k d V ; ı

Z

V ı

ı

ı

†

Z

n  k d † D

ı

Z

ı

r  k d V ;

Z

r ˝ k  dV: (3.23) V

n  k  d† D

ı

†

Z n ˝ k  d† D

†

V

r  k  dV: V

According to the second formula of (3.23), Eq. (3.12) takes the form Z Z Z @a d a dV D dV C n  v ˝ a d†: dt @t V .t /

(3.24)

V

(3.25)

†

Exercises for 3.1 3.1.1. Show that the continuity equation (3.8) can be written in the differential form: p d  d g D p : dt g dt 3.1.2. Using formulae (3.10), (3.19), and (3.20), show that the following integral relations (called the rule of differentiation of the integral of a tensor field over a moving volume) hold:    Z Z Z  n d n @  d n n  dV D  dV D C r  .v ˝ / dV dt dt  @t V .t /

V .t /

Z

D V

@ n dV C @t

Z

V

n  .v ˝ n / d†; †

3.2 The Momentum Balance Law and the Stress Tensor

d dt

Z

Z  n  dV D V .t /



d n dV D dt

D

95

@ n . / C r  .v ˝ n / @t

 dV

V

V .t /

Z

Z 

@ n . / dV C @t

V

Z n  .v ˝ n / d†:

†

3.1.3. Using formulae (3.19), show that for an arbitrary vector field a.X k ; t/ the following relation holds: 

d dt

  d a D a C a.r  v/:  dt

3.1.4. Similarly to formula (3.10), prove that for a tensor field k .x/ of arbitrary kth order the following formula of change of variables is valid: Z

Z k

k

 dV D

ı

ı

 d V :

ı

V

V

3.2 The Momentum Balance Law and the Stress Tensor 3.2.1 The Momentum Balance Law Definition 3.2. The vector Z

Z v dm D

ID V

v dV;

(3.26)

V

is called the momentum of a continuum. Remark 1. As in the mass conservation law (3.5), a domain V .t/ contains the same material points for all considered time interval. t u Remark 2. It should be noted that since the velocity v D d x=dt of a material point M, due to (2.181), has been introduced in a certain coordinate system O eN i , where x.M/ is the radius-vector of the point M, the momentum vector I is connected to this coordinate system. If we have chosen another coordinate system RO 0 e0i , then, applying formula (3.26), we obtain another momentum vector I0 D V v0 dV, where v0 D d x0 =dt, and x0 .M/ is the radius-vector of the same material point M t u but in the system O 0 e0i .

96

3 Balance Laws

This remark is of great importance for understanding the following axiom. Axiom 5 (The momentum balance law). For any two continua B and B1 at any time t, there exists a vector function F D F .B; B1 ; t/ (i.e. the transformation F W U  U  RC0 ! E3a ), which is possible to be zero-valued (i.e. F D 0). This vector function is called the force of interaction of the bodies B and B1 , and has the following properties:  It is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 ; B1 D B10 [ B100,  The rate of changing the momentum I of a continuum B at any time t is equal to the vector F .B; t/ D F .B; B e ; t/ which is the summarized vector of external forces, acting onto the body B (where B e D U n B is the outside of the body B): d I=dt D F :

(3.27)

The relationship (3.27) is called the momentum balance law. Remark 3. Since the vector I has been introduced in a certain coordinate system O eN i , the momentum balance law (3.27) is written for this coordinate system O eN i . In another coordinate system O 0 e0i the vector d I0 =dt may be different from d I=dt, and the law (3.27) also changes its form. The momentum balance law for such systems will be given in Sect. 4.10.11. The coordinate systems, in which the momentum balance law has the form (3.27), are called inertial (the system O eN i is also inertial). The existence of inertial coordinate systems is declared by the first Newton law, and Eq. (3.27) expresses the second Newton law. In continuum mechanics both these laws are introduced by Axiom 5 together with Axioms 1–3 on the existence of the special coordinate system O eN i . In more detail, noninertial coordinate systems will be considered in Sect. 4.9. u t As a body B, consider the elementary volume dV of a continuum involving a point M. Then, by Axiom 5, the summarized vector of external forces denoted by d F acts on dV. Here the following two cases are possible: (1) M is an interior point of the domain V . (2) M is on the surface † of the domain V (in this case the intersection of the surface † with closure of the domain d V is denoted by d†). Definition 3.3. The vector fD

dF dF D dm  dV

(3.28)

3.2 The Momentum Balance Law and the Stress Tensor

97

is called the specific mass force and the vector s D d F=d†

(3.29)

is called the specific surface force. Here d m is the mass of an elementary volume dV of a continuum. Since the vector F is additive, we have the following relations for the whole continuum occupying the volume V in K: Z Fm D V

F D F m C F †; Z Z f d m D f dV ; F † D s d†: V

(3.30) (3.31)

†

The vector F m is called the summarized vector of external mass forces acting on the considered continuum, and F † is called the summarized vector of external surface forces. Due to (3.30), the momentum balance law (3.27) takes the integral form: Z Z Z d v dV D f dV C s d†: (3.32) dt V

V

†

3.2.2 External and Internal Forces Mass f and surface s forces are external forces for a continuum V , if they are induced by objects not belonging to the considered continuum V (i.e. by external objects). External mass forces may be of the following types: (1) the gravity force f D g† eN , where g† is the acceleration in a free fall at a surface of a planet (for the Earth g†  9:8 kg=m  s2 ), and eN is the normal vector to the planet surface; (2) external forces of inertia caused by the motion of a body relative to a moving coordinate system (see Sect. 4.11); (3) electromagnetic forces. External surface forces are forces of interaction of two continua contacting with each other, for example, at impact of two solid bodies. Besides external forces, there are internal forces considered in continuum mechanics. Modify Eq. (3.32). According to the rule (3.12) and the continuity equation (3.15), we have d dt

Z 

Z v dV D V

@ v C r  .v ˝ v/ @t

 dV

V

Z 

D V

@ @v v C  C .r  v/v C v  r ˝ v @t @t



Z dV D



dv dV : dt

V

(3.33)

98

3 Balance Laws

Fig. 3.1 Internal surface forces on the surface †0

Then Eq. (3.32) takes the form Z D

  Z dv  f dV C s d† D 0: dt

V

(3.34)

†

Hence, the acceleration .d v=dt/ is the specific mass force but it is an internal force caused by inertia effects. Therefore, it is also called the internal inertia force. Consider an example of internal surface forces. Take an arbitrary continuum volume V and divide the volume by a surface †0 into two parts V1 and V2 (Fig. 3.1). Let n be a normal vector to †0 at a point M 2 †0 , and the vector be directed outwards from V1 . Then each of volumes V1 and V2 may be considered as a separate continuum undergoing the action of the external forces F 1 and F 2 . If we consider the surface element d† 2 †0 , which contains the point M, then for the domain V1 , according to (3.29), the surface element d† undergoes the action of the surface force d F 1 (or the specific surface force s1 ), and for the domain V2 the same surface element d† undergoes the action of the surface force d F 2 (or the specific surface force s2 ). Denote these specific surface forces as follows: tn D s1 D d F 1 =d†

and tn D s2 D d F 2 =d†:

(3.35)

The vectors tn and tn are called the stress vectors. They are specific internal surface forces relative to the whole volume V of the continuum (because they are defined for interior points M of the continuum).

3.2.3 Cauchy’s Theorems on Properties of the Stress Vector The split of forces into external and internal ones is relative: the same forces may be internal or external with respect to different subdomains of a considered continuum. Let us write the momentum balance equation (3.34) for the whole volume V and for its separate parts V1 and V2 :

3.2 The Momentum Balance Law and the Stress Tensor

V1

Z

99

  Z Z dv  f1  dV C s1 d† C tn d† D 0; dt

Z

†1

†0

  Z Z dv dV C s2 d† C tn d† D 0;  f2  dt

V2

†2

(3.36)

(3.37)

†0

where fi and si are the forces acting in the volumes Vi and on their surfaces †i , i.e. f D fi in Vi

and s D si on †i :

Since all the functions si and fi are continuous, so having subtracted Eqs. (3.36) and (3.37) from (3.34), we get Z .tn C tn / d† D 0:

(3.38)

†0

Since the surface †0 is arbitrary, we get tn C tn D 0. Thus, we have proved the following theorem. Theorem 3.5 (The first Cauchy theorem – on continuity of the stress vector). For the same point M, which is an interior point of a volume V , the stress vectors defined relative to the surface elements nd†0 and .nd†0 / differ only in sign: tn D tn ;

(3.39)

i.e. the field tn .x/ is continuous in the volume V . Remark. The result (3.39) is a consequence of the assumption that there are no discontinuities of the functions, which was made at the derivation of Eq. (3.34). When the functions are not continuous (for example, for shock waves in a continuum), Eq. (3.39) does not hold. t u Let us consider one more important property of the stress vector. At any point M we may construct an elementary volume dV in the form of a tetrahedron (Fig. 3.2),

Fig. 3.2 The properties of the internal stresses

100

3 Balance Laws

whose edges are oriented along vectors dx˛ D r˛ dX ˛ . Let d†˛ be areas of three sides lying on the coordinate planes, and d†0 be the area of the inclined side of the tetrahedron. The exterior normal vector to the surface element d†0 is denoted by n. On the surfaces d†˛ , the normal vectors are defined as .r˛ =jr˛ j/, because the reciprocal basis vectors r˛ are orthogonal to the planes d†˛ . Areas of the sides d†0 and d†˛ are connected by the relation n d†0 

3 X r˛ d†˛ D 0: jr˛ j ˛D1

(3.40)

This relation follows, for example, from the equation Z

Z n d† D †

Z n  E d† D

†

r  E dV D 0;

(3.41)

V

applied to the tetrahedron, where † and V are the total area and the total volume of the tetrahedron, respectively. Here we have used the Gauss–Ostrogradskii formula (3.23). The scalar product of (3.40) by r˛ yields jr˛ jn  r˛ d†0 D d†˛ :

(3.42)

Having applied Eq. (3.34) to the tetrahedron, we get tn d†0 C

3 X ˛D1

  dv dV D 0; t˛ d†˛ C  f  dt

(3.43)

where t˛ is the stress vector on the surface element d†˛ with the normal .r˛ =jr˛ j/, and t˛ – with the normal .r˛ =jr˛ j/. Taking formulae (3.39) and (3.42) into account, we have   d v jdV j tn  n  r˛ jr jt˛ D  f  : dt d†0 ˛D1 3 X

˛

(3.44)

Since dV=d†0 is an infinitesimal value, we get the following theorem. Theorem 3.6 (The second Cauchy theorem). The stress vector tn on an arbitrary surface element with a normal n is expressed in terms of stress vectors t˛ on three coordinate areas as follows: tn D

3 X ˛D1

n  r˛ jr˛ jt˛ :

(3.45)

3.2 The Momentum Balance Law and the Stress Tensor

101

Theorems 3.5 and 3.6 have been proved by Cauchy in another form. Since p jr˛ j D .r˛  r˛ /1=2 D g ˛˛ ;

(3.46)

Equation (3.45) takes the form tn D n  T;

(3.47)

where T is the second-order tensor called the Cauchy stress tensor: TD

3 X

r˛ ˝ t˛ D ri ˝ ti ;

˛D1

t ˛  t˛

p g ˛˛ :

(3.48) (3.49)

Thus, Theorem 3.6 may be enunciated in another way. Theorem 3.6a (The Cauchy theorem). For a continuous in V [ † stress vector field s.x/ D tn .x/ satisfying Eq. (3.32), there always exists a tensor field T.x/ satisfying the relation (3.47) in V [ †.

3.2.4 Generalized Cauchy’s Theorem Theorem 3.6a may be stated for an arbitrary vector or even tensor field ˆ.x; t/ satisfying the equation, which is similar to (3.32). Theorem 3.6b. Let there be a continuously differentiable tensor field m A.x; t/, m > 0, and a continuous tensor field m C.x; t/ in a volume V , and there also be a continuous in V [ † vector field m Bn .x; t/ depending on the choice of normal vector field n.x; t/, which satisfy the equation d dt

Z

Z  A dV D e V

Z  C dV C

m

m

e V

m

e  V; Bn d† 8V

(3.50)

e †

e is the boundary of the domain V e, then the field m Bn .x; t/ may depend where † on the field n.x; t/ only linearly, i.e. there exists a tensor field mC1 B.x; t/ of the .m C 1/th order, such that in V [ † the following relation holds: m

Bn D n 

mC1

B:

(3.51)

H A proof of this theorem is similar to the proof of Theorem 3.6, where as the e we should take the tetrahedron, whose edges are oriented along the local domain V basis vectors ri (see Exercise 3.2.3). N Equation (3.50) is called the balance equation.

102

3 Balance Laws

3.2.5 The Cauchy and Piola–Kirchhoff Stress Tensors The Cauchy stress tensor T defined by formulae (3.47)–(3.49), as a second-order tensor, can be resolved for any tensor basis, for example: T D T ij ri ˝ rj D Tij ri ˝ rj :

(3.52)

The definition (3.48) allows us to give a geometric representation of the Cauchy stress tensor. Indeed, if we take vectors ri as the left vectors and consider ti as the right vectors, then we can represent the tensor T in terms of equivalency classes (see Sect. 2.1.4 and [12]): T D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 : (3.53) According to the geometric definition of a tensor (see [12]), the tensor T can be represented as the ordered set of six vectors ri ; ti with the common origin at a considered point M (Fig. 3.3), where the basis vectors ri are defined. The Cauchy stress tensor T is defined on a deformable surface element d† in the actual configuration. We can determine the stress tensor on the corresponding ı

nondeformed surface element d † in the reference configuration as well. To do this, ı

write out the relation (2.116) which connects oriented surface elements in K and K: q n d† D

ı

ı ı

g=g n  F1 d †0 ;

(3.54)

and consider the stress vector tn on the surface element d†: q tn d† D n  T d† D

ı ı

ı

ı

ı

ı

ı

g=g n  F1  T d † D n  P d † D tn d †:

(3.55)

Here we have introduced the tensor q ı P D g=g F1  T;

Fig. 3.3 Geometric representation of the Cauchy stress tensor

(3.56)

3.2 The Momentum Balance Law and the Stress Tensor

103

Fig. 3.4 Geometric representation of the Piola–Kirchhoff stress tensor P

called the first Piola–Kirchhoff stress tensor, which is defined on the nondeformed ı

surface element d †. ı The vector tn is called the Piola–Kirchhoff stress vector; this vector is connected to the tensor P by the Cauchy formula (3.47): ı tn

ı

D n  P:

(3.57)

With the help of the relations (2.35) and (3.48), the expression (3.56) can be represented in terms of equivalency classes (see Sect. 2.1.4) ı

ı

ı ı ı ı ı ı

P D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 ;

(3.58)

where the following vectors have been denoted ı ˛

t D t˛

q

ı

g˛˛ g=g:

(3.59)

With the help of this expression we can also give a geometric representation of the ı Piola–Kirchhoff stress tensor in the basis ri of the reference configuration (Fig. 3.4).

3.2.6 Physical Meaning of Components of the Cauchy Stress Tensor Formulae (3.53) and (3.58) give a geometric representation of the tensors T and P. bij of the Cauchy stress tensor Let us explain a physical meaning of components T with respect to an orthonormal basis. Let there be a local basis ri in K, then, using the orthogonalization process, we can always construct an orthogonal basis e ri , and, normalizing the vectors e ri , we obtain the orthonormal basis b ri called the physical basis. In this basis the Cauchy bij , which are coincident with T bij : stress tensor (3.52) has components T bijb TDT ri ˝b rj :

(3.60)

104

3 Balance Laws

Fig. 3.5 Physical meaning of components of the Cauchy stress tensor

Take an arbitrary material point M 2 V and consider an elementary volume dV 2 V involving this point and having the form of a cube whose sides are orthogonal to the vectorsb r˛ (Fig. 3.5). Denote the area of the cube side orthogonal tob r˛ (˛ D 1; 2; 3) by d†˛ . Since we consider the elementary volume dV, the stress tensor T.x/ is the same at every point x 2 dV and x 2 d†˛ (˛ D 1; 2; 3). Each of the sides d†˛ undergoes the action of the surface force d F ˛ connected to the stress vector tn on this side by the relation (3.35): on d†˛ W

dF˛ D tn D n  T: d†˛

(3.61)

ri : Resolving the vectors d F ˛ and n for the basisb d F ˛ D d F˛i b ri ;

n Db ni b ri ;

(3.62)

from (3.61) we get on d†˛ W

d F˛i T ji D b Db nj b T ˛i ; d†˛

˛ D 1; 2; 3;

(3.63)

because on d†˛ : b n˛ D 1, b nˇ D b n D 0, ˛ ¤ ˇ ¤  ¤ ˛. bij of Formula (3.63) allows us to explain the physical meaning of components T the Cauchy stress tensor: ˇ



˛ b˛˛ D d F˛ ; T b˛ˇ D d F˛ ; T b˛ D d F˛ ; ˛ ¤ ˇ ¤  ¤ ˛: on d†˛ W T d†˛ d†˛ d†˛ (3.64) ˛˛ b is the ratio of the corresponding normal component d F˛˛ The normal stress T of the surface force d F ˛ , acting on the surface element d†˛ , to the area of this b˛ˇ , b surface element; and the tangential stresses T T ˛ are the ratios of tangential ˇ  components d F˛ , d F˛ of the same force d F ˛ , acting on the surface element d†˛ , to the area of the same surface element.

Remark 1. Since the tensor T is the same in the whole cube dV, by Theorem 3.5, the forces d F ˛ on the opposite sides of the cube differ only in sign. But the normals n

3.2 The Momentum Balance Law and the Stress Tensor

105

on these sides differ also only in sign, therefore relations (3.61) on these sides are the same as well as the relations (3.64). t u Remark 2. Due to Remark 1, there are three different relations in formulae (3.64) on each of three sides d†˛ , i.e. we have nine different relations: for three normal b˛˛ and six tangential stresses T b˛ˇ , ˛ ¤ ˇ. The pair tangential stresses stresses T b˛ˇ and T bˇ ˛ , defined on different sides d†˛ and d†ˇ , in general, are not coinciT dent (Fig. 3.6). Below, we will know that for many problems of continuum mechanics the stress tensor proves to be symmetric T D TT (due to additional assumptions); it is the b˛ˇ D T bˇ ˛ . only case when pair tangential stresses coincide: T t u b ij of the Piola– Let us clarify now the physical meaning of components P b ı Kirchhoff stress tensor P with respect to the orthonormal basis ri of the reference configuration: b ı ı b ijb ri ˝ rj : (3.65) PDP ı

ı

Take a material point M in K and consider an elementary volume d V being a cube ı b ı and containing this point. Let sides of the cube d †˛ be orthogonal to the vectors r˛ (Fig. 3.7).

Fig. 3.6 Difference of pair tangential stresses

Fig. 3.7 Physical meaning of components of the Piola–Kirchhoff stress tensor

106

3 Balance Laws ı

In configuration K a distorted volume dV corresponds to the cube d V . According to the geometric representation of transformation of a small neighborhood of a material point (see Sect. 2.3.5), the volume dV is a parallelepiped, in general, with inclined sides d†˛ remaining plane. ı

ı

For elementary volumes d V and dV the stress tensors T.x/ and P.x/ are the same ı

ı

at every point x 2 d V and x 2 d V . The deformed (but plane) side d†˛ undergoes the action of the surface force ı

ı

d F ˛ . Carry the force in the parallel way in K onto the corresponding side d †˛ , ı

then the relations (3.55) on the side d †˛ take the form 0 1 0 1 d F ˛ @ d†˛ A d† ı ˛ D tn @ ı A D n  P: ı d†˛ d †˛ d †˛

ı

on d †˛ W

(3.66)

b ı ı Resolving the vectors d F ˛ and n for the basis ri : ı b ı d F ˛ D d F i˛ ri ;

b ı ı ı b n D ni ri ;

from (3.66) we get ı

ı

on d †˛ W

d F i˛ b ı b ij D P b ˛i ; D ni P d†˛

˛ D 1; 2; 3I

(3.67)

ı b b b ı ı ı because on †˛ : n˛ D 1, and nˇ D n D 0. As follows from (3.67), ı

ı

ı

 b ˛ D d F ˛ ; ˛ ¤ ˇ ¤  ¤ ˛: on d †˛ W P ı d †˛ (3.68) ˛˛ b Thus, according to (3.68), the normal stress P is the ratio of normal compo˛ b ˛˛ D d F ˛ ; P ı d †˛

ı

ˇ b ˛ˇ D d F ˛ ; P ı d †˛

ı

nent d F ˛˛ of the surface force d F ˛ , acting on deformed surface element d†˛ , to ı

the area of corresponding undistorted surface element d †˛ . The tangential stresses ı ı b ˛ˇ , P b ˛ are the ratios of tangential components d F ˇ˛ , d F ˛ of the same surface P force d F ˛ , acting on the deformed surface element d†˛ , to the area of undistorted ı

surface element d †˛ . Remarks 1 and 2 for the tensor P are also valid with the exception of the fact that the tensor P remains nonsymmetric even if T is symmetric.

3.2 The Momentum Balance Law and the Stress Tensor

107

3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions Substituting the notation (3.35) for internal surface forces and Cauchy’s formula (3.47) into (3.32), we get one more widely used form of the momentum balance law: Z

Z

d dt

v dV D V

Z n  T d† C

†

f dV:

(3.69)

V

Having substituted (3.35) and (3.47) into (3.34):  Z  Z dv   f dV D n  T d†; dt V

(3.70)

†

and then having transformed the surface integral into the volume one by the Gauss– Ostrogradskii formula (3.24), we obtain  Z  dv   f  r  T dV D 0: dt

(3.71)

V

Since the volume V is arbitrary, the integrand must always vanish. Thus, we have proved the following theorem. Theorem 3.7. If the functions F, v, T and f, which satisfy the momentum balance law (3.69) and depend on x i ; t, are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following momentum balance equation in Eulerian description (i.e. in K) holds: 

dv D r  T C f: dt

(3.72)

According to the property (3.20) of the total derivative, we can rewrite the momentum balance equation in the divergence form: @v C r  v ˝ v D r  T C f: @t

(3.73)

Modify the momentum balance equation (3.70) with taking (3.55) into account: Z ı

V



dv  f dt ı



ı

Z

dV 

ı

ı

n  P d † D 0: ı

†

(3.74)

108

3 Balance Laws

With the help of the Gauss–Ostrogradskii theorem (3.24), we can rewrite this equation in the form   Z   ı ı ı dv   f  r  P d V D 0: dt

(3.75)

ı

V ı

Since the volume V is arbitrary, we obtain the following theorem. Theorem 3.8. If the conditions of Theorem 3.7 are satisfied, then at every point ı

M 2 V for all considered times t > 0 the following momentum balance equation ı

in Lagrangian description (i.e. in K) holds: ı dv



ı

ı

D f C r  P:

dt

(3.76)

Since in Lagrangian description the total derivative with respect to time coincides with the partial one (see (2.189)), the divergence form of the momentum balance ı

equation in K coincides with (3.76): ı @v



@t

ı

ı

D r  P C f:

(3.77)

Exercises for 3.2 3.2.1. Using formulae (2.35a) and (3.56), show that the Piola–Kirchhoff tensor P ı ı has the components ..=/T ij / with respect to the mixed dyadic basis ri ˝ rj : ı

ı

P D .=/T ij ri ˝ rj : ı

3.2.2. Using the results of Exercises 3.2.1 and 2.1.12, show that components P ij of the tensor P with respect to the reference configuration basis can be written in the form ı

ı

ı

P D P ij ri ˝ rj ;

ı

ı

ı

F D F jm rj ˝ rm ;

ı

ı

ı

P ij D .=/T i m F jm :

3.2.3. Prove Theorem 3.6b by the same method as we used in the proof of Theorem 3.6.

3.3 The Angular Momentum Balance Law

109

3.3 The Angular Momentum Balance Law 3.3.1 The Integral Form Consider the next fundamental law of continuum mechanics. Definition 3.4. The vectors k0 D

Z V

0m 0†

Z x  v dm D

Z D

V

Z

D

x  v dV; Z

x  f dm D V

x  f dV;

(3.78)

V

x  tn d†; 0 D 0m C 0† ;

†

are called as follows: k0 is the angular momentum vector of a continuum, 0m is the vector of mass moments of a continuum, 0† is the vector of surface moments of a continuum, 0 is the vector of moments of a continuum. Axiom 6 (The angular momentum balance law). For any continuum B there are two additive vector-functions: k00 .B; t/ called the own angular momentum vector and 00 .B; t/ called the vector of own moments, and also the positive constant s > 0 called the spin constant, such that 8t > 0 the following equation holds: dk D ; dt

(3.79)

where k is the vector of the total angular momentum of the continuum, and  is the total moment vector: k D k0 C

1 00 k ; s

 D 0 C 00 :

Remark. Axiom 6 postulates the existence of two new vectors: k00 and 00 , similarly to Axiom 5 introducing the force vector F . Physical interpretation of the vectors k00 and 00 is more complicated than for the force F (for example, effects caused by the gravity force or the inertia force are intuitively clear and well known). The appearance of the vectors k00 and 00 can be caused by electromagnetic effects in continua, which have an ordered magnetic structure (for example, ferromagnetic materials). However, the vectors k00 and 00 can appear for some continua, where there are no electromagnetic effects (such continua are often called the Cosserat continua). It should be noted that although the spin constant s in the law (3.79) seems to be unnecessary (in place of .1=s /k00 we could introduce the vector k00 ), its appearance is very essential; this will be explained in Sect. 3.4.6. t u

110

3 Balance Laws

Since the vectors k00 and 00 are additive, so, according to Sect. 3.2.1, we obtain the following classification. Definition 3.5. The specific own angular momentum is the vector km , the specific own mass moment is the vector hm , and the specific own surface moment is the vector h† , which are defined at a point M of a continuum as follows: d k00 ; dm

km D

hm D

d 00 ; dm

h† D

d 00 : d†

(3.80)

Since the vectors k00 and 00 are additive, for the whole considered volume of a continuum we have Z (3.81) k00 D km dV; 00 D 00m C 00† ; V

where the following notation has been introduced: 00m

Z D

hm dV;

00†

V

Z D

h† d†;

(3.81a)

†

On substituting (3.78) and (3.81) into (3.79), the angular momentum balance law takes the integral form d dt

 Z Z  Z  x  v C km dV D .x  f C hm / dV C .x  tn C h† / d†: s

V

V

†

(3.82)

3.3.2 Tensor of Moment Stresses Equation (3.82) has the form of (3.50), if we introduce the following notation: A D x  v C km ;

C D x  f C hm

and

B D x  tn C h† :

Then to Eq. (3.82) we can apply Theorem 3.6b, which asserts the existence of a e t/, defined in V [ †, such that the following relation second-order tensor field M.x; being an analog of (3.47) holds:

With new notation

e h† C x  tn D n  M:

(3.83)

e C T  x; MDM

(3.84)

this result can be written in another form.

3.3 The Angular Momentum Balance Law

111

Theorem 3.9. For vector fields tn and h† being continuous in V [ † and satisfying Eqs. (3.32) and (3.82), there exists a second-order tensor field M.x; t/ such that in V [ † the following relation holds: h† D n  M:

(3.85)

The tensor M is called the tensor of moment stresses.

3.3.3 Differential Form of the Angular Momentum Balance Law ı

At first, consider the left-hand side of (3.82) and pass from V to V : d dt

Z ı

  ı 1 ı  x  v C km d V s

V

d D dt Z D V

Z ı

Z   ı  1 1 d km  ı ı d  x  v C km d V D  dV .x  v/ C s dt s dt ı

ı

V

V

Z   dv dv 1 d km  1 d km   vvCx dV D  x  dV ; C C dt s dt dt s dt V

(3.86) as v  v D 0. Modify the right-hand side of Eq. (3.82) by using the Gauss–Ostrogradskii theorem (3.24) and formula (3.47): Z Z Z Z x  tn d† D x  .n  T/d† D  n  .T  x/ d† D  r  .T  x/ dV : †

†

†

V

(3.87)

The nabla-operator r in (3.87) is determined by the formula @ .T  x/ D .r  T/  x C ri  T  ri @X i D x  .r  T/ C "   T:

r  .T  x/ D ri 

(3.88)

Substitution of (3.88) into (3.87) yields Z

Z x  tn d† D

†

Z x  r  T dV 

V

"   T dV: V

(3.89)

112

3 Balance Laws

Having substituted (3.89) and (3.86) into (3.82), we get Z V



 Z dv 1  d km x   f  r  T dV C dV dt  s dt V Z Z Z D hm dV C r  M dV  "   T dV: V

V

(3.90)

V

Here we have used the definition of the tensor of moment stresses M (3.84). Due to formula (3.72), Eq. (3.90) takes the final form Z  V

  d km  hm  r  M C "   T dV D 0: s dt

(3.91)

Since the volume V is arbitrary, we get the following theorem. Theorem 3.10. If the functions F, v, T, f, satisfying Eq. (3.69), and also the functions km , hm , M, satisfying Eq. (3.90), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following angular momentum balance equation holds:  d km D hm C r  M  "   T: s dt

(3.92)

Thus, the angular momentum balance equation includes only one classical mechanical characteristic, namely the stress tensor T, the remaining functions km , hm and M, as it was mentioned above, either have a nonmechanical nature or are caused by nonclassical mechanical properties.

3.3.4 Nonpolar and Polar Continua For classical continua, for which there are not any nonmechanical values, we have km D 0;

hm D 0;

M D 0:

(3.93)

Such continua are called nonpolar, and for them it follows from (3.93) and (3.92) that 3 X p "T D g .T ˛ˇ  T ˇ ˛ /r D 0; (3.94) ˛D1

˛¤ˇ¤ ¤˛

and hence T D TT ; i.e. the Cauchy stress tensor for nonpolar continua is symmetric.

(3.95)

3.3 The Angular Momentum Balance Law

113

It should be noted that the corresponding Piola–Kirchhoff tensor is not symmetric even for nonpolar continua; that follows from its definition (3.56). Thus, for nonpolar continua the angular momentum balance law reduces to the condition (3.95) that the Cauchy stress tensor T is symmetric. A continuum, for which values of km , hm and M are different from identically zero: km ¤ 0; hm ¤ 0; M ¤ 0; (3.96) is called polar. For polar continua, the Cauchy stress tensor T is not symmetric.

3.3.5 The Angular Momentum Balance Equation in the Material Description The integral form of the angular momentum balance equation (3.82) can be represented in the material description. To do this, we should modify the left-hand side of formula (3.82) according to (3.86): d dt

Z V

   Z  ı 1 1 d ı  x  v C km dV D  x  v C km d V s dt s ı

V

  ı dv 1 d km  x dV C dt s dt

Z

ı

D ı

(3.97)

V

ı

ı

and introduce Lagrangian tensor of moment stresses M in K similarly to the Piola–Kirchhoff stress tensor: ı

ı

ı

n  M d† D n  M d †;

(3.98)

ı

ı

and the specific own surface moment h† in K: ı

ı

ı

h† D n  M:

(3.99) ı

Then the surface integral in (3.82) can be written in K as follows: Z

Z .x  tn C h† / d† D †

Z n  .T  x C M/ d† D

†

ı

ı

ı

n  .P  x C M/ d †; ı

†

(3.100)

114

3 Balance Laws

and Eq. (3.82) takes the form  Z  ı d 1 ı  x  v C km d V dt s ı

V

Z

Z

ı

ı

 .x  f C hm / d V C ı

ı

ı

ı

n  .P  x C M/ d †:

(3.101)

ı

V

†

Exercises for 3.3 3.3.1. Show that from (3.101) we can get the angular momentum balance equation in the material description ı

ı ı  d km ı D hm C r  M  "   .F  P/: s dt

3.3.2. Show that for nonpolar continua the angular momentum balance equation in the material description (see Exercise 3.3.1) takes the form "   .F  P/ D 0

F  P D PT  F T

or

and coincides exactly with (3.96). 3.3.3. Using equation (3.98) and the formula of transformation of an oriented surı

face element, show that the tensors M and M are connected by the relation q ı ı M D g=g F1  M:

3.4 The First Thermodynamic Law 3.4.1 The Integral Form of the Energy Balance Law The mass, momentum and angular momentum balance laws describe the motion of a continuum. If we need thermal effects to be taken into account, then we must use thermodynamic laws. Let us consider nonpolar continua, for which the relations (3.93) are satisfied. Definition 3.6. The scalar function Z Z jvj2 vv KD dm D  dV; 2 2 V

V

is called the kinetic energy of a continuum V .

jvj2 D v  v

(3.102)

3.4 The First Thermodynamic Law

115

Definition 3.7. The power of external mass forces Wm and the power of external surface forces W† are the following scalar functions: Z

Z

Wm D

f  v dm D V

Z f  v dV;

W† D

V

tn  v d†:

(3.103)

†

Axiom 7 (The first thermodynamic law (or the energy balance law)). For any continuum B there are two scalar additive functions: U.B; t/ called the internal energy of the continuum and Q.B; t/ called the heating rate for the continuum such that 8t > 0 the following equation is satisfied: dE D W C Q; dt

(3.104)

where E is called the total energy of the continuum and consists of U and K: E D U C K;

W D W m C W† :

(3.105)

Remark. There are different statements of the thermodynamic law, and the above is called Truesdell’s statement [54, 55]. The statement is convenient, because its form corresponds to Axioms 4–6. Moreover, this statement (unlike others) is really universal, i.e. it is independent of the type of a continuum. t u Since the functions U and Q are additive, similarly to Sect. 3.2.1 we can introduce the corresponding specific functions. Definition 3.8. The specific internal energy is the function e, the specific heat flux from mass sources is the function qm and the specific heat flux from surface sources is the function q† , which are defined at every point M of a continuum as follows: eD

dU ; dm

qm D

dQ ; dm

q† D

dQ : d†

(3.106)

Since the functions Q and U are additive, for a whole continuum we have Z Qm D V

Q D Qm C Q † ; Z Z qm d m D qm dV; Q† D q† d†; V

Z

Z e dm D

U D V

(3.107)

†

e dV: V

Substitution of Eqs. (3.102), (3.103), (3.105)–(3.107) into (3.104) yields the energy balance law in the integral form

116

3 Balance Laws

d dt

Z

  Z Z jvj2  eC dV D .f  v C qm / dV C .tn  v C q† / d†: (3.108) 2

V

V

†

Using on the left-hand side of (3.108) the differentiation rules for a volume integral (see Exercise 3.1.2): d dt

Z

   Z  jvj2 de dv  eC dV D  dV; Cv 2 dt dt

V

(3.109)

V

we obtain the following form of the energy balance law: Z

    Z d 1   e C jvj2 C f  v C qm dV C .tn  v C q† / d† D 0: (3.110) dt 2

V

†

3.4.2 The Heat Flux Vector Equation (3.108) has the form (3.50), if in (3.50) we put A D e C jv2 j=2;

C D f  v C qm

and B D tn  v C q† :

Then we can apply Theorem 3.6b to this equation; the theorem claims the existence of the vector .q/ such that in V [ † the following relation holds: q† D n  q:

(3.111)

The vector q is called the heat flux vector. Prove the analog of Theorem 3.6 allowing us to determine components of the vector q. As was made in Sect. 3.2.3, at a point M we construct an elementary volume dV in the form of a tetrahedron, whose edges are oriented along vectors r˛ dX ˛ . Then for the volume all formulae (3.40)–(3.42) hold. Applying Eq. (3.110) to the tetrahedron, when there is no action of any mass or surface forces on the continuum (i.e. tn D 0, f D 0), we get q† d†0 C

   d jvj2 q˛ d†˛ C  qm  eC dV D 0: dt 2 ˛D1 3 X

(3.112)

Here q˛ is the heat flux from surface sources on the surface element d†˛ with the normal .r˛ =jr˛ j/, and q† – on the surface element d†0 .

3.4 The First Thermodynamic Law

117

Then with taking (3.42) into account, we have    d jvj2 dV q† C n  r˛ jr jq˛ D  qm  : eC dt 2 d† 0 ˛D1 3 X

˛

(3.113)

Since the value dV=d†0 is infinitesimal, we get the following theorem. Theorem 3.11. On every surface element with a normal n, the heat flux from surface sources q† is expressed in terms of heat fluxes on three coordinate surface elements: 3 X p n  r˛ g ˛˛ q˛ : (3.114) q† D  ˛D1

Comparing (3.114) with (3.111), we find components of the heat flux vector: qD

3 X

r˛

p g ˛˛ q˛ D ri q i ;

q ˛ D q˛

p g ˛˛ :

(3.114a)

˛D1

3.4.3 The Energy Balance Equation Let us consider now formula (3.110) and transform the surface integrals by the Gauss–Ostrogradskii formula (3.24) with taking (3.111) into account as follows: Z Z Z q† d† D  n  q d† D  r  q dV; Z

†

Z

tn  v d† D †

†

V

Z

n  .T  v/ d† D †

(3.115) r  .T  v/ dV:

V

Substituting the expression (3.115) into (3.108), we obtain another integral form of the energy balance law: d dt

Z V

  Z Z jvj2  eC dV D .f  v C qm / dV C n  .T  v  q/ d†: (3.116) 2 V

†

Substitution of formulae (3.115) into (3.110) yields   Z   d jvj2 =2 de  f  v  qm C r  q  r  .T  v/ dV D 0: (3.117)  C dt dt V

Since the volume V is arbitrary, we get the following theorem.

118

3 Balance Laws

Theorem 3.12. If the functions F, v, T, f, qm , q and e, satisfying Eq. (3.116), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the energy balance equation has the following differential form: d  dt

  jvj2 eC D f  v C qm  r  q C r  .T  v/: 2

(3.118)

According to formula (3.20), we can rewrite the energy balance equation in the divergence form @ C r  .v/ D r  q C r  .T  v/ C f  v C qm ; @t

DeC

jvj2 ; (3.119) 2

where  is the specific total energy of the continuum.

3.4.4 Kinetic Energy and Heat Influx Equation The scalar product of the momentum balance equation (3.72) by v yields 

dv  v D f  v C .r  T/  v: dt

(3.120)

Using the property of covariant differentiation (see Exercise 2.1.17), we can reduce this equation to the form 1 d jvj2  D f  v C r  .T  v/  T   .r ˝ v/T : 2 dt

(3.121)

Definition 3.9. The scalar function Z W.i / D 

T   .r ˝ v/T dV

(3.122)

V

is called the power of internal surface forces of a continuum V . Integrating Eq. (3.121) over a volume V and using Definitions 3.6, 3.7, 3.9, and the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2), we get the following theorem. Theorem 3.13 (The kinetic energy balance equation). A change of the kinetic energy of a continuum is equal to the summarized power of external and internal forces acting on the continuum: dK D W C W.i / : dt

(3.123)

3.4 The First Thermodynamic Law

119

Subtracting the kinetic energy equation (3.121) from the energy balance equation (3.118), we obtain the differential heat influx equation 

de D T   .r ˝ v/T C qm  r  q: dt

(3.124)

This equation can be rewritten in the divergence form @e C r  .ev/ D r  q C T   .r ˝ v/T C qm : @t

(3.125)

Subtracting Eq. (3.123) from (3.104), we get the heat influx equation in the integral form d (3.126) U C W.i / D Q: dt

3.4.5 The Energy Balance Equation in Lagrangian Description Let us write now the energy balance equation in Lagrangian description. Use the equation in the integral form (3.108) and transform the surface integrals as follows: Z

Z .tn  v C q† / d† D

†

Z q D

n  .T  v  q† / d† † ı

ı ı

g=g n  .F1  T  v  F1  q† / d † D

ı

Z

  ı ı n  P  v  q d†:

ı

†

†

(3.127) Here we have used the definition (3.56) of the Piola–Kirchhoff tensor and introduced the heat flux vector in the reference configuration: q

ı

qD

ı

g=g F1  q:

(3.128)

ı

Theorem 3.14. Heat flux vectors q and q are connected by the relation ı

ı

ı

ı

ı

q  n d † D q  n d†:

(3.129)

H Indeed, multiplying (3.128) by n d † and using formula (2.116), we obtain ı

ı

ı

q  n d† D

q

ı

ı

ı

g=g q  F1T  n d † D q  n d†: N

120

3 Balance Laws

Then Eq. (3.108) in the reference configuration takes the form    Z  ı ı ı ı jvj2 ı d ı ı  eC  f  v  qm  r  .P  v/ C r  q d V D 0: dt 2

(3.130)

ı

V

ı

Since the volume V is arbitrary, the integrand vanishes. Thus, we have proved the following theorem. ı

Theorem 3.15. Under the conditions of Theorem 3.12, at every point M 2 V for all considered times t > 0 we have the energy balance equation in Lagrangian description ı



d dt

  ı ı ı jvj2 ı ı eC D f  v C qm C r  .P  v/  r  q: 2

(3.131)

Similarly to formula (3.120), we can write the kinetic energy balance equation in the reference configuration ı dv



dt

ı

ı

 v D f  v C v  .r  P/;

(3.132)

with the help of which Eq. (3.131) takes the form ı

e

ı ı ı d ı D P   .r ˝ v/T C qm  r  q: dt

(3.133)

This equation is called the heat influx equation in material (Lagrangian) description. Since in Lagrangian description the total derivative coincides with the partial one, the divergence form of the heat influx equation does not differ from (3.133): ı

ı @e



ı

ı

ı

D r  q C P   .r ˝ v/T C qm :

@t

(3.134) ı

Integrate the heat influx equation (3.133) over the domain V and take into account that the internal energy U and the heating rate Q, according to (3.107), can also be represented in Lagrangian description: Z U D Z Qm D

Z e dV D

V

ı

Q D Q m C Q† ;

ı

Z

qm dV D V

ı

e d V ; V

ı

ı

qm d V ; ı

V

Z Q† D

Z q† d† D

†

ı

ı

q † d †; ı

†

(3.135)

3.4 The First Thermodynamic Law

121

here we have used Eq. (3.129). As a result, from (3.133) we get the heat influx equation in the integral form (3.126) again, where the power of internal surface forces is defined by the following relations in Eulerian and Lagrangian descriptions, respectively: Z W.i / D 

Z T   .r ˝ v/T dV D 

ı

ı

P   .r ˝ v/T d V :

(3.136)

ı

V

V

According to relation (3.55), we can represent the power of external forces (3.103) in Lagrangian description as follows: Z Wm D

Z f  v dV D

Z

ı

ı

f  v d V ; W† D ı

V

Z tn  v d† D

ı

 v d †: (3.137)

ı

†

V

ı tn

†

3.4.6 The Energy Balance Law for Polar Continua Let us consider now polar continua, for which the relations (3.96) hold, i.e. the values of km , hm and M are different from identically zero. The energy balance law for polar continua differs from the corresponding law (3.104), (3.105) for nonpolar continua. Similarly to (3.102), introduce the kinetic energy of the own motion of a continuum: Z Z jkm j2  Ks D dm D jkm j2 dV; jkm j2 D km  km : (3.138) 2s s V

V

Similarly to (3.103), introduce the power of mass moments Wms and the power of surface moments W†s : Z Wms D

hm  km dV; V

Z W†s

D

Z

h†  km d† D †

(3.139)

n  M  km d†:

(3.140)

†

The energy balance law for a polar continuum has the universal form (3.104), but the total energy E and the summarized power W of a continuum are defined not by (3.105) but by the formulae E D U C K C Ks ;

W D Wm C Wms C W† C W†s :

(3.141)

122

3 Balance Laws

As a result, we obtain the following form of the energy balance law for a polar continuum: d .U C K C Ks / D Wm C Wms C W† C W†s C Q: dt

(3.142)

From (3.142) with the help of (3.138)–(3.140) we get the integral form of the law: Z  Z d jvj2 jkm j2   eC C dV D .f  v C hm  km C qm / dV dt 2 2s V V Z C .t†  v C h†  km C q† / d†: (3.143) †

Similarly to Sect. 3.4.2, the heat flux vector q for polar continua is introduced by formulae (3.111) and (3.114). Thus, in Eq. (3.143) we can transform the surface integral to the volume one, then we obtain the following differential form of the energy balance law for a polar continuum (the energy equation): 

d dt

  jvj2 jkm j2 eC D f  v C hm  km C qm  r  q C r  .T  v C M  hm /: C 2 2s (3.144)

Remark. The energy equation (3.143) clarifies the meaning of introduction of the spin constant s in the angular momentum balance law (3.79): if in (3.79) or in (3.92) we eliminated the constant s by introducing the vector e km D km =s instead of km , then the energy equation (3.144) would include the constant s all the same. This constant s plays a role of the energetic equivalent between the kinetic energy of a body and the kinetic energy of the own body motion caused by the own angular momentum for polar continua. t u According to relation (3.20), the energy equation (3.144) takes the divergence form @ " C r  .v" C q  T  v  M  km / D f  v C hm  km C qm ; @t jvj2 jkm j2 : "DeC C 2 2s

(3.145)

Let us formulate now an analog of the kinetic energy balance equation (3.123) for the kinetic energy of own motion Ks . Consider the angular momentum balance equation (3.92) and multiply the equation by km : d km  km  D hm  km C km  r  M  km  3 "   T: s dt

(3.146)

According to the property of the nabla-operator from Exercise 2.1.17, we can reduce Eq. (3.146) to the form

3.4 The First Thermodynamic Law

123

 d jkm j2 D hm  km C r  .M  km /  M   .r ˝ km /T  km  3 "   T: (3.147) s dt Integrating this equation over V and using the Gauss–Ostrogradskii formula and the differentiation rule of an integral over a moving volume (3.12), from (3.147) we get dKs D Wms C W†s C W.is / C Ws : dt

(3.148)

Here we have introduced the following notation: Z W.is / D  M   .r ˝ km /T dV

(3.149)

V

– the power of internal surface moments and Z Ws D 

km  3 "   T dV

(3.150)

V

– the power of internal surface forces for the own body motion. Relation (3.148) is called the kinetic energy balance equation for the own body motion. Subtracting Eqs. (3.123) and (3.148) from the energy balance law (3.142), we get the integral heat influx equation for polar continua: dU D Q  W.i /  W.is /  Ws : dt

(3.151)

Subtracting Eqs. (3.147) and (3.121) from the energy equation (3.144), we obtain the differential heat influx equation for polar continua: 

de D T   .r ˝ v/T C M   .r ˝ km /T C km  3 "   T  r  q C qm : (3.152) dt

If we assume km D 0, M D 0 and hm D 0, then the heat influx equation in the forms (3.151) and (3.152) coincides with the corresponding equations (3.124) and (3.126) for nonpolar continua. ı

According to the definition (3.97) of Lagrangian tensor of moment stresses M, we can rewrite the heat influx equation (3.152) in the material description ı de



dt

ı

ı

ı

ı

ı

ı

D P   .r ˝ v/T C M   .r ˝ km /T C km  3 "   .F  P/  r  q C qm : (3.153)

This assertion should be proved as Exercise 3.4.1.

124

3 Balance Laws

Exercises for 3.4 3.4.1. Prove the relation (3.153).

3.5 The Second Thermodynamic Law 3.5.1 The Integral Form To state the second thermodynamic law, we introduce a new local characteristic of a continuum which is called a temperature. Axiom 8 (on the existence of an absolute temperature). For every material point M of any continuum B for all t > 0 there is a scalar positive function .X i ; t/ D .x i ; t/ > 0;

(3.154)

called an absolute temperature. The axiom on the existence of an absolute temperature is sometimes called the zero thermodynamic law. It should be noted that we have introduced a temperature as local but not integral characteristic of a continuum (unlike mass M or angular momentum). Although we R can always introduce the integral temperature ‚, for example, as ‚ D V d m, but this value does not appear in any fundamental laws of continuum mechanics. However, there are integral characteristics containing a temperature, which are of great importance for formulating the second thermodynamic law. Definition 3.10. The entropy production by external mass sources is the scalar function QN m and the entropy production by external surface sources is the scalar function QN † , which are defined for a volume V as follows: QN m D

Z V

qm dm D 

Z V

qm dV; 

QN † D

Z

q† d† D  

†

Z

nq d†: (3.155) 

†

Remark. If by analogy with (3.106) we consider the entropy production by external mass sources d QN m , supplied to elementary volume dV by mass d m, then from (3.155) and (3.106) we get d QN m D qm d m= D dQ=;

(3.155a)

i.e. d QN m is the heat influx due to external mass sources divided by temperature  of the volume dV heated. Thus, d QN m characterizes the efficiency of heating the volume dV, and d QN † characterizes the efficiency of heating the surface element d†. u t

3.5 The Second Thermodynamic Law

125

Axiom 9 (The second thermodynamic law). For any continuum B there are two scalar additive functions: H.B; t/ called the entropy of the continuum and QN  .B; t/ called the entropy production by internal sources, such that for all t > 0 the following equation holds: dH D QN C QN  ; (3.156) dt where we have denoted the summarized entropy production by external sources as follows: (3.157) QN D QN m C QN † ; and a value of QN  is always nonnegative (the Planck inequality): QN  > 0:

(3.158)

Since the functions H and QN  are additive, we can introduce the corresponding specific functions at every point M 2 V . Definition 3.11. The specific entropy  and the specific internal entropy production q  are defined for every point M of a continuum as follows: D

dH ; dm

q D 

d QN  : dm

(3.159)

As follows from the Planck inequality (3.158), and also the inequalities (3.154) and (3.4a), the function q  is always nonnegative: q  > 0:

(3.158a)

This inequality is also called the Planck inequality. Since the functions H and QN  are additive for a whole continuum, we have QN  D

Z

q dm D 

V

Z

q  dV; H D 

V

Z

Z  dm D V

 dV:

(3.160)

V

Two relations (3.156) and (3.158) are equivalent to the Clausius inequality N dH=dt > Q:

(3.161)

On substituting (3.155) and (3.160) into (3.156), we obtain the integral form of the second thermodynamic law d dt

Z

Z  dV D

V

V

.qm C q  / dV  

Z †

nq d†: 

(3.162)

126

3 Balance Laws

3.5.2 Differential Form of the Second Thermodynamic Law Transform the surface integral in (3.162) by the Gauss–Ostrogradskii formula (3.24):   Z  Z Z q r q 1 nq d† D dV: (3.163) d† D r  Cqr     †

V

V

Then Eq. (3.162) takes the form Z   q  d .qm C q  / dV D 0:   Cr  dt  

(3.164)

V

Since the volume V is arbitrary, we get the following theorem. Theorem 3.16. If the functions , q, q  , qm and , satisfying Eq. (3.162), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V the second thermodynamic law has the following differential form (the entropy balance equation): 

q d qm C q  C D r  : dt  

(3.165)

According to the property (3.20), we can rewrite Eq. (3.165) in the divergence form  @ qm C q  q D C r  v C : (3.165a) @t   According to the equation r

q 

D

1 1 r  q  2 q  r ;  

formulae (3.165) and (3.165a) can be rewritten in the form 

d D r  q C qm C w ; dt

(3.166)

or

@ C r  v D .1=/r  q C .1=/.qm C w /: @t Here we have introduced the function w D q  C

q  r ; 

(3.166a)

(3.167)

called the dissipation function. Besides the Planck inequality (3.158), there is one more important inequality in continuum mechanics.

3.5 The Second Thermodynamic Law

127

Axiom 10. At every point M of a continuum for all t > 0 the Fourier inequality holds: q  r  6 0: (3.168) Rewrite (3.167) in the form q  D w 

q  r : 

(3.169)

Equation (3.169) means that the specific internal entropy production is caused by two factors: dissipation (i.e. irreversible conversion of part of mechanical energy to heat energy) and nonuniform heating of a continuum (when there are irreversible processes of heat transfer from hotter parts of the continuum to its colder ones). The first cause corresponds to the function w , and the second one corresponds to the function 1  q  r   wq : (3.170)  A mathematical expression for the dissipation function w will be given in Chap. 4; however, it should be noted here that if w is independent of the temperature gradient r  (in Chap. 4 we will show that this assumption always holds), then, ı

considering the motion of a continuum from K to K only with uniform temperature field  D .t/ (r   0 in V ), from (3.169) and the Planck inequality we get the dissipation inequality w > 0: (3.171) Since the function w is independent of r , the inequality (3.171) holds under nonuniform heating as well.

3.5.3 The Second Thermodynamic Law in the Material Description In the material description, Eq. (3.162) takes the form Z

ı d

 ı

dt

ı

Z

dV D ı

V

ı

.qm C q  / ı dV  

V

Z ı

ı

ı

nq ı d †; 

(3.172)

†

then with the help of the usual manipulations we get the second thermodynamic law in the material description ı d



dt

ı

D r 

ı

q 

!

ı qm

C

C q : 

(3.173)

128

3 Balance Laws

This equation can be rewritten in the form ı d



dt

ı

ı

ı

ı

D r  q C qm C w ;

where

ı

q ı  r > 0 

ı

ı

(3.174)

w D q  C

(3.175)

ı

is the dissipation function in K. The Clausius inequality (3.161) in the material (Lagrangian) description retains its form, but entropy H and entropy production QN by external sources, defined by (3.160), in Lagrangian description take the forms Z H D

Z  dV D

Z V

QN D QN m C QN † ;

ı

V

QN m D

ı

ı

 d V ;

qm dV D 

Z ı

V

QN  D

V ı

qm ı dV ; 

QN † D

Z

q† d† D 

†

Z

q  dV D 

V

Z ı

Z ı

ı

q† ı d †; (3.176) 

†

ı

q  ı dV : 

V

3.5.4 Heat Machines and Their Efficiency The present-day heat machines are facilities, which convert heat into mechanical work (for example, internal-combustion engines, jet and turbo-jet engines, gasturbine engines, nuclear engines etc.), or, on the contrary, convert mechanical work into heat (for example, some heating and cooling devices), or use both the conversions in turn (for example, diesel engines). The operation of all the heat machines is based on the thermodynamic laws. Conventionally, the structure of a heat machine consists of a vessel with solid walls, which is filled with a working body of volume V (it is usually gas or fluid). Heat machines can be split into the three main types: (1) Machines, for which the domain V .t/ is varying but material points of a ı

working body are the same, i.e. V D const. (2) Machines, for which V remains unchanged with time but material points of a working body, occurring in V , are different. (3) Machines, for which the domain V is varying and material points occurring in V are different.

3.5 The Second Thermodynamic Law

129

Fig. 3.8 The scheme of three main types of heat machines

These three types of heat machines are shown in Fig. 3.8. The first type involves cooling and heating facilities, and also machines operating on the Carnot cycle (see below). The second type includes air-jet and rocket engines etc. The third type involves gas-turbine, turbo-jet, internal-combustion engines etc. Lagrangian description proves to be more convenient for computations of heat machines of the first type, and Eulerian description – for the ones of the second and third types. Further, we will consider heat machines only of the first type; machines of the second and third types can be considered in a similar way (see Exercise 3.5.2). All heat machines are characterized by their efficiency. To introduce the concepts of heat, mechanical work and the efficiency, we should return to the integral statements (3.126) and (3.161) of the thermodynamic laws. Consider an arbitrary time interval t1 6 t 6 t2 and integrate Eqs. (3.126) and (3.161) with respect to t from t1 to t2 ; as a result, we obtain U C A.i / D C;

(3.177)

H > CN :

(3.178)

Here we have introduced the notation U D U.t2 /  U.t1 /; Zt2 A.i / D

Zt2 Z

Zt2

Zt2 Z

ı

ı

ı

V

ı

Zt2 Z

f  v d V dt C t1

ı

V

(3.179)

P   r ˝ vT d V dt; t1

W .t/ dt D t1

ı

W.i / .t/ dt D  t1

A.e/ D

H D H.t2 /  H.t1 /I

(3.180) ı

ı

tn  v d † dtI

t1

ı

†

130

3 Balance Laws

Zt2 C D

CN D

Zt2 Z Q.t/ dt D

qm d V dt C ı

t1

t1

Zt2

Zt2 Z

N Q.t/ dt D

t1

Zt2 Z

ı

ı

t1

t1

V

ı

qm ı d V dt C 

ı

ı

ı

†

Zt2 Z t1

V

ı

q † d † dt; (3.181)

ı

q† ı d † dt; 

ı

†

where we used Lagrangian representation of the integral values U , H , W.i / , Q and QN (see (3.135), (3.136), and (3.176)). The value U is called the change of the internal energy of a body B, H – the change of the entropy of a body B, A.i / – the mechanical work done by a body B (or simply the work), C – the heat supplied to a body (or simply the heat), CN – the entropy influx to a body B, A.e/ – the work done on a body B (or the work done by external forces). ı

Let us use Lagrangian description in variables X i ; t, where X i 2 V .t/, t 2 ı

Œt1 ; t2 , and divide the four-dimensional integration domain V  .t1 ; t2 / of volume ı

integrals into subdomains V t ˛ such that: (1) in each of these subdomains a sign of the scalar function qm remains unchanged, i.e. either qm > 0, or qm < 0, and (2) ı

ı

ı

[V t ˛ D V  .t1 ; t2 /. In a similar way, divide the integration domain †  .t1 ; t2 / ˛

ı

ı

of surface integrals into subdomains †t ˛ such that: (1) in each †t ˛ a sign of the ı

ı

ı

ı

function q † remains unchanged, i.e. either q † > 0, or q † < 0, and (2) [†t ˛ D ˛

ı

†  .t1 ; t2 /. Then the heat C supplied to the body can be split into two nonnegative parts: C D CC  C ; Zt2 Z C˙ D t1 ˙ qm

ı ˙ qm

Zt2 Z

ı

d V dt C

ı

t1

V

D .jqm j ˙ qm /=2 > 0;

ı q˙ †

(3.182) ı

ı

q˙ † d † dt > 0;

(3.183)

ı

† ı

ı

D .jq † j ˙ q † /=2 > 0:

(3.184)

Here CC is the heat absorbed by the body (or supplied to the body), and C is the heat released by the body (or withdrawn from the body). Since the temperature  of a body is nonnegative, we can also split the entropy influx CN to the body into two nonnegative parts: CN D CN C  CN  ; ı ˙ ı ı Rt2 R q Rt2 R qı ˙ † m CN ˙ D  d V dt C  d † dt > 0: t1

ı

V

t1

ı

†

(3.185) (3.186)

3.5 The Second Thermodynamic Law

131

Substituting (3.182) into (3.177), we can rewrite the heat influx equation in the form C D CC  A.i /  U:

(3.187)

Assume that CC ¤ 0, then we can introduce the ratio of the mechanical work done by the body B to the heat supplied: kB D A.i / =CC ;

(3.188)

that is called the efficiency of the body B for the time interval Œt1 ; t2  considered. According to the axiom (3.154), in each body B for any time interval Œt1 ; t2  the conditions 0 < .X i ; t/ < C1 are satisfied; then we can introduce minimum and maximum values of the temperature of a body: ı

0 < min 6 .X i ; t/ 6 max < C1; 8X i 2 V ; 8t 2 Œt1 ; t2 :

(3.189)

From relations (3.189) we obtain ı

ı

ı

C C qm = > qm =max ı

ı

ı

C qC † = > q † =max ı

ı

ı

  and qm = 6 qm =min ; ı

 and q  † = 6 q † =min ;

(3.190)

ı

C   , qC because qm † , qm , q † ,  and  are nonnegative. Substituting (3.190) into the integrals (3.183), (3.186) and taking into account that the integrands are nonnegative, we find that

CN C > CC =max ;

CN  6 C =min :

(3.191)

From formulae (3.178), (3.182), and (3.191) we get H > CN D CN C  CN  >

CC C  : max min

(3.192)

Substitution of the expression (3.187) into (3.192) yields  H > CC

1 max



1 min

 C

1 .A.i / C U /: min

(3.193)

Dividing this relation by CC and multiplying by min (both the values are nonnegative), we obtain   A.i / U Hmin min C C 1 6 : (3.194) CC CC max CC According to the definition of the efficiency kB (3.188), we get the final Truesdell estimate for the efficiency:

132

3 Balance Laws

kB 6 1 

min U  min H  : max CC

(3.195)

Notice that since a sign of the third summand on the right-hand side of (3.195) is unknown, we cannot claim that in any process for each body kB < 1. This assertion, which is well known from the course of physics, holds for special processes of motion and heating of a body. Consider them in the next paragraph.

3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles One can say that in a body B there occurs a locally adiabatic process in the subdoı

ı

main V t ˛ , if the heat influx due to mass sources into V t ˛ is zero: ı

qm  0 8.X i ; t/ 2 V t ˛ :

(3.196)

ı

If in some subdomain †t ˛ the heat influx due to surface sources is zero: ı

ı

q †  0 8.X i ; t/ 2 †t ˛ ;

(3.197) ı

then one can say that a locally adiabatic process occurs in the subdomain †t ˛ of the body B. If in a body B within the time interval Œt1 ; t2  the following conditions are satisfied: ı ı ı (3.198) qm  0 8X i 2 V ; q †  0 8X i 2 †; then one can say that there occurs an adiabatic process in the restricted sense in the body B. Finally, if in a whole body B within the time interval Œt1 ; t2  the following conditions are satisfied: ı

qm  0 8X i 2 V ;

ı

ı

q  0 8X i 2 V ;

(3.199)

then one can say that there occurs an adiabatic process in the body B in the broad sense (or simply an adiabatic process). Since (3.196) and (3.197) follow from the condition (3.199) (but not conversely), one can say that if in a body B an adiabatic process in the broad sense occurs, then in this body there occurs also an adiabatic process in the restricted sense (but not conversely). One can say that in a body B there occurs a locally isothermal process in the ı

subdomain V t ˛ , if the temperature in this subdomain remains constant: ı

.X i ; t/  const 8.X i ; t/ 2 V t ˛ :

(3.200)

3.5 The Second Thermodynamic Law

133

If the temperature remains constant within a time interval Œt1 ; t2  in a whole body B: ı

.X i ; t/  const 8X i 2 V ; 8t 2 Œt1 ; t2 ;

(3.201)

then one can say that in the body B there occurs an isothermal process. One can say that in a body B there occurs a thermodynamic cycle within the time interval Œt1 ; t2 , if the following conditions of periodicity are satisfied simultaneously: U D U.t2 /  U.t1 / D 0; H D H.t2 /  H.t1 / D 0; CC ¤ 0;

(3.202)

ı

.X i ; t2 /  .X i ; t1 /  0 8X i 2 V : Notice that not for every continuum we may ensure that the conditions (3.202) be satisfied, and, thus, not for every body a thermodynamic cycle can be realized. However, there are whole classes of bodies, for which the conditions (3.202) are satisfiable. Example 3.1. Consider a continuum B, where the specific internal energy e and the specific entropy  uniquely depend only on the density  and temperature : e D e.; /;

 D .; /:

(3.203)

In Chap. 4 we will show that such a model of a continuum describes an ideal fluid or gas. If relations (3.203) hold, then the conditions (3.202) can easily be satisfied by giving such external heat and mechanical actions that the fields of temperature .X i ; t/ and density .X i ; t/ in the body B be the same at t D t1 and t D t2 : .X i ; t1 / D .X i ; t2 /, .X i ; t1 / D .X i ; t2 /. Indeed, in this case the following conditions are satisfied: e.X i ; t1 / D e..X i ; t1 /; .X i ; t1 // D e..X i ; t2 /; .X i ; t2 // D e.X i ; t2 /; and hence

Z U.t1 / D

ı

ı

Z

e.X ; t1 / d V D i

ı

ı

ı

e.X i ; t2 / d V D U.t2 /; ı

V

V

i.e. U D 0. The condition H D 0 can be satisfied in a similar way. A thermodynamic cycle, occurring in an ideal fluid (gas), at every material point t u X i is assigned to some closed curve on the plane .; / (Fig. 3.9). Example 3.2. Consider a continuum B, where e and  are one-valued functions of temperature  and the deformation gradient F: e D e.F; /;

 D .F; /:

(3.204)

134

3 Balance Laws

Fig. 3.9 A thermodynamic cycle at the material point X i of an ideal fluid

ı

Since density  is a function of F ( D =det F due to (3.8)), the relations (3.203) are a particular case of (3.204). If dependences (3.204) cannot be reduced to (3.203), then they correspond to the model of an ideal solid (these models will be considered in detail in Chap. 8). For an ideal solid, one can readily ensure the conditions (3.202) to be satisfied by using relations (3.204). For this, it is sufficient that the following periodicity conditions for the cycle be satisfied: ı

F.X i ; t1 / D F.X i ; t2 /; .X i ; t1 / D .X i ; t2 / 8X i 2 V : A thermodynamic cycle, occurring in an ideal solid, at every material point X i is also assigned to some closed curve in the generalized seven-dimensional space .FN ij ; t/, where FN ij are components of the tensor F with respect to the Cartesian basis eN i . t u One can say that a generalized Carnot cycle occurs in a body B, if the following conditions are satisfied simultaneously: (1) In the body B there occurs a thermodynamic cycle within Œt1 ; t2 . ı

ı

(2) The domains V  Œt1 ; t2  and †  Œt1 ; t2  of the body B consist only of the subı

ı

domains V t ˛ and †t ˛ , where there occur locally adiabatic or locally isothermal processes. (3) Locally isothermal processes may be only of the two types: either with the minimum temperature min and heat absorbtion: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D min ; qm < 0; (3.205) ı

.X ; t/ D min ; q † < 0; i

or with the maximum temperature max and heat release: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D max ; qm > 0; ı

.X i ; t/ D max ; q † > 0:

(3.206)

3.5 The Second Thermodynamic Law

135

Fig. 3.10 The typical picture of location of subdomains in the generalized Carnot cycle

ı

ı

The subdomains V t ˛ and †t ˛ , where there occur locally isothermal processes, have no common points (and even their closures have no common points too) due ı

to continuous differentiability of the temperature .X i ; t/ within V  Œt1 ; t2 . ı

ı

Figure 3.10 shows a typical location of the subdomains V t ˛ and †t ˛ in the generalized Carnot cycle. One can say that in a body B there occurs an uniform thermomechanical process within the time interval Œt1 ; t2 , if fields of its mechanical and thermodynamic values are uniform, i.e. they are independent of coordinates X i but may depend on time t: ı

.X i ; t/ D .t/ 8X i 2 V ; 8t 2 Œt1 ; t2 ;  D f ; q; F; T; P; qm ; e; ; ; q  g:

(3.207)

(Notice that the velocity v and displacement u vectors of material points in an uniform process may depend on the coordinates X i .) The generalized Carnot cycle, which occurs in a body under the conditions of a uniform thermomechanical process, is called simply the Carnot cycle. The Carnot cycle, according to the definitions given above, consists only of isothermal and adiabatic processes in a body B, which alternate with each other; here within the time subinterval T0 2 Œt1 ; t2 , when the process is adiabatic in the whole body, the rate of heating Q, according to (3.198) and (3.107), is zero; within the time subinterval T 2 Œt1 ; t2 , when the temperature in the whole body reaches its minimum value min , the rate of heating Q is negative; and within the time subinterval TC 2 Œt1 ; t2 , when the temperature in the whole body has its maximum value max , the rate of heating Q is positive:

136

3 Balance Laws

Fig. 3.11 The elementary Carnot cycle

T0 W TC W T W

Q.t/ D 0;

 D max ; Q.t/ > 0;  D min ; Q.t/ < 0:

(3.208)

Figure 3.11 shows the elementary Carnot cycle. This cycle consists of four processes: two isothermal and two adiabatic ones. Other Carnot cycles for the same body B can differ from the elementary Carnot cycle only by the number of isothermal sections.

3.5.6 Truesdell’s Theorem Theorem 3.16a (of Truesdell). (1) If for the body B there exists a thermodynamic cycle with conditions (3.202) and given values min and max in this cycle, then the efficiency reaches its maximum possible value min kBmax D 1  < 1; (3.209) max only in generalized Carnot cycles. (2) Among all bodies B, for which the thermodynamic cycle (3.202) is possible under the conditions of an uniform thermomechanical process (3.207), the efficiency reaches its maximum value (3.209) for such bodies, in the cycle of which there is no entropy production by internal sources: QN   0

8t 2 Œt1 ; t2 :

(3.210)

H (1) To prove the first assertion of Truesdell’s Theorem we consider an arbitrary thermodynamic cycle; for this cycle, relationships (3.187) and (3.188) yield A.i / D CC  C ; and hence kB D 1 

C : CC

(3.211)

(3.212)

3.5 The Second Thermodynamic Law

137

As follows from (3.211) and (3.209), in order for the efficiency to reach its maximum value kB D kBmax in the thermodynamic cycle, it is necessary and sufficient that the following condition be satisfied: C =CC D min =max :

(3.213)

Since, according to (3.202), for any thermodynamic cycle H D 0, relation (3.192) with use of (3.213) takes the form CC C  D 0I max min

0 > CN C  CN  > that is possible only if

(3.214)

CN C D CN  :

(3.215)

Due to this relation, the system of two inequalities (3.191) becomes C CC CC 6 CN C D CN  6 D : max min max

(3.216)

Since the left-hand and right-hand sides of (3.216) coincide, the inequality signs in this relation should be replaced by the equality sign. Thus, the following two equations must hold: C CC D CN C ; D CN  ; (3.217) max min or with use of the notations (3.183) and (3.186) Zt2 Z t1



ı

1 1   max



ı

ı

 qm



1 min



1 

Zt2 Z

ı

dV dt C t1

V

Zt2 Z t1

ı C qm



ı



ı

1 1   max



ı

d† dt D 0;

†

Zt2 Z

dV dt C t1

V

ı

qC †

ı

q †



1 min

ı



1 



(3.218) ı

d† dt D 0:

†

Since all integrands in these equations are non-negative, the sum of integrals is zero only if the integrands vanish; i.e. the following equations must hold:  1 1 D 0;   max   1 1  qm  D 0; min 

C qm



8X i 2 V ;

 1 1 D 0;   max   1 1 ı q†  D 0; min  ıC



q†

8t 2 Œt1 ; t2 :

(3.219)

138

3 Balance Laws

These equations are satisfied if at least one of the cofactors vanishes, i.e. if in ı

subdomains V t ˛ the following conditions are satisfied: ı

in V t ˛ W ı

qm > 0;  D max ; or qm < 0;  D min ; or qm D 0; ı

ı

ı

at †t ˛ W q † > 0;  D max ; or q † < 0;  D min ; or q † D 0:

(3.220) (3.221)

But, according to (3.205) and (3.206), this means that the thermodynamic cycle considered is a generalized Carnot cycle. Thus, we have shown that the requirement that the efficiency reach its maximum kB D kBmax in a thermodynamic cycle leads to the requirement that this cycle be a Carnot cycle, and vice versa. (2) To prove the second assertion of the theorem we use the fact established above that the efficiency reaches its maximum value at the generalized Carnot cycles; hence, under the conditions of uniform thermomechanical processes it is sufficient to prove only that the relation (3.210) holds for simple Carnot cycles. Take into account that for a thermodynamic cycle H D 0, and represent H as a sum of three terms: Zt2 0 D H D H.t2 /  H.t1 / D

HP dt D

Z

HP dt C

T0

t1

Z

HP dt C

TC

Z

HP dt; (3.222)

T

where the subsets T0 , TC and T are defined by the conditions (3.208) in an uniform thermomechanical process. Within TC , according to (3.208),  D max and Q > 0, then due to the Clausius inequality (3.161), we have within TC W

HP > QN D max Q > 0:

(3.223)

e C of the set TC , the Assume that within some finite time interval being a subset T strict inequality in (3.223) holds: HP > max Q. Then, integrating (3.223) over the whole TC , we obtain the strict inequality Z

HP dt > max

TC

Z Q dt D max CC :

(3.224)

TC

(The case, when the inequality HP > max Q is satisfied only at an isolated point, is excluded, because we assume that all the functions H.t/, Q.t/ and .t/ are continuous.) Due to the Clausius inequality (3.161) and (3.208), Z T

HP dt >

Z T

QN dt D min

Z

T

Qdt D min C :

(3.225)

3.5 The Second Thermodynamic Law

139

(Here we have usedR the notations (3.183) R t and (3.184), which for uniform processes reduce to C D  T Qdt D .1=2/ t12 .jQj  Q/dt.) Addition of two inequalities (3.224) and (3.225) yields Z

HP dt C

TC

Z

HP dt > max CC  min C D 0;

(3.226)

T

because we consider the Carnot cycle, for which the conditions (3.213) are satisfied. As follows from (3.222) and (3.226), the integral over T0 must be negative: Z

HP dt < 0:

(3.227)

T0

P However, due to the R Clausius inequality (3.161), within T0 the conditions H > P N Q D Q D 0 and T0 H dt > 0 must be satisfied; therefore, the inequality (3.227) e C : HP > max Q is not true, is impossible. Thus, the made assumption that within T and from (3.223) it follows that within the whole TC the following equality must hold: HP D max Q: (3.228) TC W In a similar way, one can prove that within T W

HP D min Q:

(3.229)

Then, integrating (3.228) over TC and (3.229) over T and substituting the results into (3.222), we obtain Z 0 D H D

HP dt C max CC  min C D

T0

Z

HP dt:

(3.230)

T0

Here we have taken into account once more that the condition (3.213) is satisfied for Carnot’s cycle. As noted above, due to the Clausius inequality (3.161), within T0 we have HP > 0; then (3.230) yields T0 W HP D 0 D Q: (3.231) Collecting the three relations (3.228), (3.229), (3.231), and taking (3.208) into account, we find that in the Carnot cycle with the maximum efficiency for the whole time interval considered the following relation holds: HP D Q D QN

8t 2 Œt1 ; t2 :

(3.232)

140

3 Balance Laws

This relationship, according to (3.156), means that there is no entropy production QN  by internal sources, i.e. the condition (3.210) is really satisfied. N Remark 3. Since the assertion (2) of Truesdell’s theorem holds for uniform thermomechanical processes, for which, by (3.207), the gradient of a temperature field is zero in a whole body B: ı

ı

r   0 8t 2 Œt1 ; t2 ;

8X i 2 V I

(3.233)

so it follows from (3.175), (3.176), and (3.210) that the body B, satisfying the assertion (2), must be nondissipative, i.e. in this body, at least, within the interval Œt1 ; t2  the dissipation function is zero: w  0 8t 2 Œt1 ; t2 ;

ı

8X i 2 V :

(3.234)

The processes, for which in a whole body the conditions (3.233) and (3.234) are satisfied simultaneously, i.e. the specific internal entropy production q  is identically zero, are often called reversible; and the processes, in which at least one of these conditions is violated, are called irreversible. t u

Exercises for 3.5 3.5.1. Using formula (3.128) and the result of Exercise 2.1.9, show that the dissipaı tion functions w and w , defined by (3.167) and (3.175), are connected by ı ı

w D .=/w : 3.5.2. For heat machines of the second and third types, by analogy with (3.179)– (3.181) introduce the concepts of H , A.i / , C and CN and prove that Truesdell’s estimate (3.195) is valid. Introduce the concepts of adiabatic and isothermal processes by analogy with the definition from Sect. 3.5.5 and also the concept of a thermodynamic cycle for Œt1 ; t2 , having replaced the last relation in (3.202) by the two ones V .x i ; t1 / D V .x i ; t2 /;

.x i ; t1 /  .x i ; t2 / D 0 8x i 2 V;

(which mean the coincidence of domains V occupied by a body at times t1 and t2 and coincidence of temperature fields in the domains). Prove Truesdell’s theorem for thermodynamic cycles occurring in heat machines of the second and third types.

3.6 Deformation Compatibility Equations

141

3.6 Deformation Compatibility Equations 3.6.1 Compatibility Conditions So we have completed the formulation of balance laws of continuum mechanics. However, the conditions of continuity of the motion of bodies, which have been used in Chap. 2, can also be rewritten in the form of some formal ‘conservation law’. This formal law is of great importance for closing the equation system in continuum mechanics; thus, it will be consider in Sects. 3.6–3.8. Formulate the requirement for continuity of a motion as follows: let it be known ı

that in configuration K we can put every material point M of a continuum in one-toı one correspondence with its radius-vector x.X k / in the Cartesian coordinate system O eN i ; then we must define the conditions, under which there exists a one-valued function (the displacement vector u.X k ; t/) connecting positions of the point M in ı

K and K. If in K some discontinuities appear (cracks, pores, etc.) (Fig. 3.12), then the motion is no longer continuous. Definition 3.12. Necessary and sufficient conditions of the existence of a onevalued vector-function u.X k ; t/ for a continuum V are called deformation compatibility conditions (equations) for the continuum V . ı

If there is no one-valued vector-function u.X k ; t/ for all X k 2 V , this means that in K we cannot introduce one-valued radius-vector x.X k ; t/. Thus, the configuration K does not belong to Euclidean space E3a . The converse statement holds as well. So we have proved the following theorem. Theorem 3.17. The deformation compatibility conditions for a continuum V are equivalent to the condition that an actual configuration of the continuum belong to Euclidean space E3a . Remark. At first sight it seems strange that an usual medium with cracks does not belong to an Euclidean space. But this continuum is not quite usual, because it is

Fig. 3.12 The example of violation of the compatibility conditions

142

3 Balance Laws

defined in actual configuration K in such a way that in the reference configuration ı

K this medium is assigned to a continuum already without discontinuities. If we defined a considered continuum with cracks in another way, namely as belonging ı

e then each material point of the continto some new reference configuration K, uum would have its individual radius-vector and such a continuum would belong to Euclidean space E3a . t u

3.6.2 Integrability Condition for Differential Form There are two types of compatibility equations: dynamic and static. In this paragraph we consider static compatibility equations. To derive the equations we should consider some differential form 3 X

A˛ dX ˛ ;

(3.235)

˛D1

where A˛ D A˛ .X i / are smooth functions of variables X j . As known from the course of mathematical analysis, this form gives a total differential dA if and only if the following integrability conditions hold @Aˇ @A˛ D ; ˇ @X ˛ @X

˛; ˇ D 1; 2; 3:

(3.236)

A˛ dX ˛ :

(3.237)

In this case we have the representation dA D

3 X ˛D1

This expression can be considered as an equation in differentials. The equation can be resolved for A if and only if the conditions (3.236) are satisfied.

3.6.3 The First Form of Deformation Compatibility Conditions ı

Let A be radius-vectors x and x of the same material point M. For them the relation (3.237) is written as follows: d x D ri dX i ;

ı

ı

d x D ri dX i :

(3.238)

3.6 Deformation Compatibility Equations

143

These relations can be considered from different points of view. On the one hand, ı

if the motion law is known and in K there is a common Cartesian coordinate system, then the following relations hold ı

ı

x D x.X j /; x D x.X j ; t/;

(3.239)

which are smooth functions of their arguments. Then Eqs. (3.238) are consequences of the relations (3.239). This approach has been used above. On the other hand, if basis vectors ri are known and radius-vector x is unknown ı (by the condition, the radius-vector x is always known, and it is a one-valued function), then, similarly to the expression (3.237), the first equation of (3.238) is an equation in differentials for radius-vector x. A solution of this equation always exists, because the integrability conditions (3.236) are satisfied. Indeed, @rˇ @r˛ @x @x D D D : ˇ ˇ ˛ ˛ ˇ @X ˛ @X @X @X @X @X

(3.240)

Thus, if there exist local basis vectors at every point X i in K, then there also exist radius-vectors of these points in K. If every material point M with coordinates X i in K may be uniquely assigned to ı its radius-vector x, this means that there exist one-valued functions u.X k ; t/ D x x ı

of displacements of the points from K to K. Conversely, if there exist one-valued displacements u.X k ; t/ of material points ı

ı

from K into K, then in K we can always introduce the radius-vector x D x C u and the local basis vectors ri D @x=@X i . Thus, we have proved the following theorem. Theorem 3.18. The deformation compatibility conditions for a continuum are satisfied if and only if in K there exist local basis vectors ri , i.e. functions having the following properties:  they are linearly independent at every point X i ;  they are one-valued and smooth 8X i 2 V ;  they have a vector potential x: ri D @x=@X i .

3.6.4 The Second Form of Compatibility Conditions ı

As A, we choose local bases vectors ri and ri . For them, the relation (3.237) becomes   ı ı @ri ı j m j m d ri D dX j : dX D .

r / dX ; d r D

r (3.241) i ij m ij m @X j

144

3 Balance Laws

If the motion law is known (the second equation in (3.239)) and the deformation compatibility conditions are satisfied, then relations (3.241) follow from the motion law (3.239). If the Christoffel symbols ijm are known, then (3.241) are equations for functions ri . The integrability conditions (3.236) for these equations have the form (the indices j and n change places) @ @ . m rm / D . m rm /: @X n ij @X j i n

(3.242)

Removing the parentheses, we obtain @ ijm

@ imn k rm C imn mj rk ; @X j  k m k m C ij kn  i n kj rm D 0:

(3.243)

@ imn m m C ijk kn  ikn kj D 0; @X j

(3.244)

k r C ijm mn rk D n m

@X  m @ ij

@ imn  @X n @X j

Since rm are arbitrary, we have m  Rnji

@ ijm @X n



m is some notation for the present. where Rnji Using the condition (3.244), we can formulate the following theorem.

Theorem 3.19. The deformation compatibility conditions are satisfied if and only if in K there exist Christoffel symbols ijm satisfying the integrability conditions (3.244). H Indeed, if there exist ijm satisfying the conditions (3.244), then the integrability conditions (3.242) remain valid. Hence the first form in (3.241) has a differential and there exist local basis vectors ri . Then, according to Theorem 3.18, the deformation compatibility conditions will be satisfied. Conversely, if the compatibility conditions are satisfied, then, according to Theorem 3.18, there exist vector functions ri . Performing all manipulations (3.241)– (3.244), we verify that the conditions (3.244) hold. N ı

ı

ı

Notice that in K there always exist vectors x and ri . Hence we can perform similar manipulations for the second form in (3.241) too; as a result, we obtain the equations     ı ı ı ı @ @ m m

ij rm D

i n rm (3.245) @X n @X j and

ı

ı

Rm nji



@ m ij @X n

ı

ı ı ı ı @ m k m in  C kij m kn  i n kj D 0: j @X

(3.246)

3.6 Deformation Compatibility Equations

145

ı

m Values Rnji and Rm nji are called components of the Riemann–Christoffel fourthı

ı

order tensors 4 R and 4 R in configurations K and K, respectively. We will verify that these are really components of tensors in paragraph 3.6.6.

3.6.5 The Third Form of Compatibility Conditions ı

Introduce the Christoffel symbols of the first kind ijk and ijk , which are connected ı

to ijm and m ij , called the Christoffel symbols of the second kind, by the relations (see [12])

ijk D gkm ijm ;

ı

ı

ı

ijk D g km m ij :

(3.247)

The Christoffel symbols of the first kind and the metric matrix are connected by the formula   @gkj 1 @gi k @gij

ijk D C  ; (3.248) 2 @X j @X i @X k which follows from (2.26). Interchanging indices i and k in (3.248) and summing the result with ijk , we get

ijk C kj i D

@gi k 1 C @X j 2



@gkj @gkj @gij @gij  C  i k k @X @X i @X @X

 D

@gi k ; @X j

(3.249)

Purely covariant components of the Riemann–Christoffel tensor are defined as follows:   @. ijl g ml / @. ikl g ml / m  Rnjik D Rnji gmk D gmk C gsl . ijl snk  inl sjk /: (3.250) @X n @X j According to the equation gmk

@gml @gmk D gml D gml . mnk C knm /; @X n @X n

(3.251)

we can rewrite Rnjik in the form Rnjik D

@ ijk @ i nk   ijl . mnk C knm /g ml @X n @X j C g ml inl . kjm C mjk / C g sl . ijl snk  inl sjk /

D

@ ijk @ i nk   gml . inl kjm  ijl knm /: @X n @X j

(3.252)

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3 Balance Laws

Substituting in place of derivatives of ijk their expressions (3.248) in terms of the metric matrix components, we get Rnjik D  2  @2 gkj @2 gij @2 gi k @2 gkn @2 gi n @ gi k 1 C    C 2 @X j @X n @X i @X n @X k @X n @X n @X j @X i @X j @X k @X j C gml . inl kjm  ijl knm /; or Rnjik

1 D 2



@2 gkj @2 gij @2 gi n @2 gkn  C  @X i @X n @X k @X n @X i @X j @X k @X j



C g ml . inl kjm  ijl knm / D 0:

(3.253) ı

In the same way we can find the expression for Rnjik in terms of components of ı

the metric matrix g ij : ı

Rnjik

1 0 ı ı ı ı @2 g ij 1 @ @2 g kj @2 g kn @2 g i n A D  C  2 @X i @X n @X i @X j @X k @X j @X k @X n   ı ı ı ı ı C g ml inl kjm  ijl knm D 0:

(3.254)

Theorem 3.19a. The deformation compatibility conditions are satisfied if and only if in K there exists a metric matrix gij satisfying the Eqs. (3.253). H If the matrix gij satisfying Eqs. (3.253) is known, then their equivalent relations (3.244) hold. As shown above, the relations (3.244) define a space to be Euclidean; and, according to Theorem 3.19, this means that the deformation compatibility conditions are satisfied too. Conversely, if the deformation compatibility conditions are satisfied, then relations (3.244) hold and in Euclidean space the coefficients ijm have a potential being gij ; thus, relations (3.248) hold too. Then their equivalent relations (3.253) remain valid as well. N

3.6.6 Properties of Components of the Riemann–Christoffel Tensor We can check that functions Rnjik are symmetric in pairs of indices n; j and i; k: Rnjik D Ri knj ;

(3.255)

3.6 Deformation Compatibility Equations

147

and also skew-symmetric in indices n; j and i; k: Rnjik D Rnjki ;

Rnjik D Rj ni k :

(3.256)

Thus, among 81 components Rnjik there are only six independent ones. They are usually chosen as follows: R1212 ; R2323 ; R3131 ; R1223 ; R1231 ; R2331 ;

(3.257)

the remaining components either are equal to zero or can be expressed in terms of ı

components (3.257). Functions Rnjik have the same properties. ı

With the help of components Rnjik and Rnjik we can set up the following fourthorder tensors: 4

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

4

ı

ı

ı

ı

ı

ı

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

(3.258)

ı

which are called the Riemann–Christoffel tensors in K and K. ı

We can verify that Rnjik and Rnjik are components of a tensor, for example, in the following way. ı ı Consider an arbitrary vector a D ak rk D ak rk and determine its covariant derivative @ak C iks as (3.259) ri ak D @X i and the second covariant derivative rj ri ak D

@ iks s @ @2 ak k k m m k .r a / C

r a 

r a D C a i i m jm ji @X j @X j @X i @X j !  m  @a @ak @as m s k C iks j C jkm C

a C ms as :  jmi is i @X @X @X m (3.260)

Interchange indices i and j and write the difference: ! k k @

@

js m is  C jkm ims  ikm js as D Rj i sk as : rj ri ak  ri rj ak D @X j @X i (3.261) It follows from (3.261) that Rjki s are components of a tensor, because as and the left-hand side of (3.261) are tensor components.

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3 Balance Laws

3.6.7 Interchange of the Second Covariant Derivatives In Euclidean space E3a , the following relations have been established: ı

Rkji s D Rjki s D 0;

(3.262)

(see formulae (3.253) and (3.254)); then from (3.261) it follows that the second covariant derivatives may be interchanged. So we have proved the following theorem. Theorem 3.20. If the deformation compatibility conditions (3.253), and also (3.254) are satisfied, then the second covariant derivatives may be interchanged: ı

ı ı

ı

ı

ı

rj ri ak D ri rj ak and r j r i ak D r i r j ak :

(3.263)

This differentiation rule remains valid for tensors of any order.

3.6.8 The Static Compatibility Equation Just as in Sect. 3.6.5, we assume that in K there is a metric matrix gij satisfying ı

Eq. (3.253), and in K the continuity conditions (i.e. conditions (3.254)) are satisfied. Then in place of gij we can consider components of the deformation tensor: "ij D

1 ı .gij  g ij / 2

(3.264)

and express the deformation compatibility conditions in terms of "ij . Subtracting (3.254) from (3.253), we obtain ı

Rnjik  Rnjik D

@2 "kj @2 "kn @2 "i n @2 "ij  C  @X i @X n @X i @X j @X k @X j @X k @X n ı

ı

ı

ı

ı

Cgml . inl kjm  ijl knm /  g ml . inl kjm  ijl knm / D 0; (3.265) where

ı

ijk D ijk C "ijk ;

"ijk D

@"jk @"i k @"ij C  : j i @X @X @X k

(3.266)

The inverse metric matrix g ml can be expressed in terms of deformation tensor ı

ı

components according to formulae (2.131). Since all the functions inl and g ml in the reference configuration are assumed to be known, the relations (3.265), with

3.7 Dynamic Compatibility Equations

149

taking (3.266) and (2.131) into account, state the system of six scalar equations (because among components Rnjik there are only six independent ones) for six scalar functions "ij . These equations are called static equations of compatibility (or deformation compatibility equations). Thus, we have proved the following theorem. Theorem 3.21. The deformation compatibility conditions for a continuum are satisfied if and only if in K the deformation tensor components "ij satisfy Eqs. (3.265). Finally, let us formulate one more important theorem. Theorem 3.22. The deformation compatibility equations (3.265) have the solution   ı ı ı ı ı ı ı 1 ı ı kl r i uj C r j ui C r i uk r j ul g "ij D : 2

(3.267)

Sometimes this result is said as follows: a solution of Eqs. (3.265) admits a potential, i.e. six functions "ij are expressed in terms of covariant derivatives of three ı

functions ui . H To prove the theorem we can replace the covariant derivatives by the partial ones: ı ı

ı

ı

ı

r i uj D uj= i  m ij um ;

(3.268)

and, substituting the expressions (3.268) and (3.267) into Eq. (3.265), verify that this equation will be identically satisfied. However, this method is very clumsy. So let us prove the theorem in another way. Let there be a solution of Eqs. (3.265) (functions "ij ). According to Theorem 3.21, this means that the deformation compatibility equations are satisfied. Then, by ı Definition 3.12, there exists a one-valued vector-function u.X k ; t/ D x  x beı ing the displacement vector of material points. By components of this vector ui , according to formulae (2.93), we can always find components e "ij , which coincide with "ij (3.267). Indeed, ife "ij ¤ "ij , then this means that there exist two distinct solutions of the deformation compatibility equations (3.265) and their corresponding, by Definition 3.12, different displacement vectors; that is impossible due to uniqueı

ness of displacements u.X k ; t/ of a continuum under the transformation from K to K. Thus,e "ij D "ij , and formula (3.267) really holds. N

3.7 Dynamic Compatibility Equations 3.7.1 Dynamic Compatibility Equations in Lagrangian Description The deformation compatibility equations can be written in one more equivalent form, namely in terms of the velocity.

150

3 Balance Laws ı

Consider Eq. (2.77) connecting F to r ˝ uT , and differentiate the equation with respect to t, taking the definition (2.181) of the velocity v into account: ı d ı @2 u @v d FT ı ıi D : r ˝ u D ri ˝ r ˝ D r ˝ v D dt @X i @t @X i dt

(3.269)

Then we get the dynamic compatibility equation in Lagrangian description: ı d FT D r ˝ v: dt

(3.270)

Theorem 3.23. The deformation compatibility conditions are satisfied if and only if in K there is a tensor field F.X i ; t/ satisfying the following conditions:  det F ¤ 0 at every point X i ;  F.X i ; 0/ D E at t D 0;  the field F has a vector potential, i.e. there exists a vector field v such that equa-

tion (3.270) holds 8t > 0 and 8X i 2 V . H Let the deformation compatibility conditions be satisfied, then, by Definition 3.12, ı

there is a displacement vector u.X i ; t/ with its gradient r ˝ u. On making the manipulations (3.269), we verify that Eq. (3.270) remains valid. Prove that the converse assertion is valid. Let there R tbe a vector-function v satisfying Eq. (3.270). Consider the function e u.X i ; t/ D 0 v.X i ; / d . This function satisfies the equation ı

ı

Zt

r ˝e uDr˝

Zt v d D

0

Zt r ˝ v d D

0

d FT d  D FT  E: d

(3.271)

0 ı

Then with the help of this function we can construct the radius-vector e x D x Ce u. As a result, the tensor F takes the form ı

ı

u D ri ˝ FT D E C r ˝ e

ı

@.x C e u/ ı D ri ˝e ri ; i @X

where e ri D @e x=@X i . But this means that the vector e u is the desired displacement vector u, and F is the deformation gradient, because they satisfy all the kinematic relations: (2.35), (2.77), etc. Thus, the displacement vector really exists u, and hence the deformation compatibility conditions are satisfied. N

3.7 Dynamic Compatibility Equations

151

3.7.2 Dynamic Compatibility Equations in Spatial Description At first, let us prove the following auxiliary statement. Theorem 3.24. If the continuity equation (3.15) holds, then the deformation gradient satisfies the following relationship: r  .F/ D 0:

(3.272)

H Represent the deformation gradient in the dyadic basis: F D F ij ri ˝ rj :

(3.273)

Let us write the following formula for the divergence of a tensor (see [12]): ı

 @ 1 @  p   g F ij rj D p r  .F/ D p i g @X g @X i

q

ı ıi



gr

:

(3.274)

q ı ı Here we have used the continuity equation  D  g=g, and also the evident relations ı

ı

F i k rk D F jk ri  rj ˝ rk D ri  F D .ri  rk / ˝ rk D ri :

(3.275)

Differentiating (3.274) by parts, we get 1 0 q ı q q  ıi ı q g @  B  ıi ı @r C ı ı s ıi ı ı s ıi r  .F/ D p @ r C g g

r  g

r D 0: D p A is is g @X i @X i g ı

(3.276) Here we have used the properties of the Christoffel symbols (see [12]). N Modify Eq. (3.270) with use of (2.37): d FT D FT  r ˝ v; dt

(3.277)

and use the continuity equation (3.15) multiplied by FT : @ T F C FT r  v D 0: @t

(3.278)

Multiply (3.277) by  and use the definition (2.187) of the total derivative with respect to time: 

d FT @FT D C v  r ˝ FT D FT  r ˝ v: dt @t

(3.279)

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3 Balance Laws

On summing Eqs. (3.278) and (3.279), we get @FT C r  .v ˝ FT /  FT  r ˝ v D 0: @t

(3.280)

Tensor multiplication of Eq. (3.272) by v (i.e. r  .F/ ˝ v D 0) and summation of the obtained expression with (3.280) give the dynamic compatibility equation in the spatial description:   @FT C r  v ˝ FT  F ˝ v D 0: @t

(3.281)

In order for the deformation compatibility conditions be satisfied in K, it is necessary and sufficient that Eq. (3.281), as well as (3.270), hold. Transforming the first and the second summands in (3.281) by formula (3.20), we can rewrite the dynamic compatibility equation in the form 

d FT D r  .F ˝ v/: dt

(3.282)

Exercises for 3.7 3.7.1. Show that the dynamic compatibility equation (3.270) can be written in the divergence form ı

d FT =dt D r  .E ˝ v/: 3.7.2. Using the relation F1T  F1 D E, show that the dynamic compatibility equation (3.270) can be rewritten as follows: d F1T =dt D .r ˝ v/  F1T :

3.8 Compatibility Equations for Deformation Rates Equations (3.270) and (3.281) express a condition of the existence of displacements u, when a field of the deformation gradient F is given. The deformation compatibility conditions can also be formulated as conditions for a field of the deformation rate tensor D. ı

Theorem 3.25. If a continuum does not contain discontinuities in K and the symmetric deformation rate tensor D (2.225) is given in K, then the continuum does not contain discontinuities in K if and only if the following compatibility conditions for deformation rates are satisfied:

3.8 Compatibility Equations for Deformation Rates

Ink D D 0:

153

(3.283)

Here the differential operator of incompatibility (see [12]) has been introduced as follows: Ink D D r  .r  DT / D .1=g/ ijk  mnl ri rm Dj n rk ˝ rl ;

(3.284)

or in the component form .Ink D/kl D .1=g/ ijk  ˛ˇ l .ri r˛ Djˇ  ri rˇ Dj˛ /; ˛ ¤ ˇ ¤ l:

(3.285)

H Show the necessity of (3.283). If a continuum in K does not contain discontinuities, then there exist one-valued functions of coordinates: a radius-vector x, a displacement vector u and a velocity v. According to the definitions of the tensor D (2.225) and of the operator Ink (3.284), we get Ink D D

1 ijk mnl .ri rm rj vn C ri rm rn vj /rk ˝ rl D 0:   2g

(3.286)

Here we have used the property (3.263) of interchange of covariant derivatives ri rj and rm rn , and also the property of contraction of the Levi-Civita symbols with components of an arbitrary symmetric tensor (see Exercise 2.1.13). Prove the sufficiency of the conditions (3.283). Let the symmetric tensor D, satisfying the conditions (3.283), be given. Then we can write the following differential form similar to (3.235): d ! D .r  D/  d x D .r  D  ri / dX i ;

(3.287)

which is a total differential if and only if the following integrability conditions of the type (3.236) are satisfied: @ @ .r  D  ri / D .r  D  rj /: j @X @X i

(3.288)

These conditions with use of the definition (2.31) of the tensor curl .r  D/ take the form     1 1 rj p  psk rp Dsi rk D ri p  psk rp Dsj rk : (3.289) g g p Due to Ricci’s theorem (see [12]), the metric matrix gij and g can be placed outside the covariant derivative sign; therefore the relation (3.289) is rewritten as follows: psk .rp rj Dsi  rp ri Dsj / D 0: (3.290)

154

3 Balance Laws

On comparing formulae (3.290) with (3.285), we obtain that condition (3.288) is equivalent to the condition r  .r  D/T D 0 (3.291) (formula (3.283)). Thus, the conditions (3.288) are always satisfied, because the condition (3.291) is assumed to be satisfied and hence the integral form (3.287) is a total differential, and ! is a one-valued function !.x/. Then the total differential can be written in the form d ! D r ˝ !T  d x: (3.292) It follows from (3.287) and (3.292) that the curl of the tensor D under the condition (3.291) can always be represented as the transpose gradient of some vector !: r  D D r ˝ !T :

(3.293)

With the help of the vector ! by formula (2.47) we can form the second-order tensor p W D !  E D g mi k ! m ri ˝ rk ; (3.294) which is skew-symmetric, because mi k ! m D mki ! m . With the help of D and W we can write one more differential form: d v D .D C W/  d x:

(3.295)

Similarly to the integrability conditions (3.291) for the form (3.287), the integrability conditions for the form (3.295) can be represented as follows: r  .D C W/T D 0:

(3.296)

But this condition is always satisfied, because r  .D C W/T D r ˝ !T  r  W D r ˝ !T  r  .!  E/ D r ˝ !T C .r  !/E  r ˝ !T C !  r ˝ E  ! ˝ r  E D r ˝ !T  r ˝ !T D 0: (3.297) Here we have used formula (3.294) and the property of the operator r  .!  E/ (see [12]), and also taken into account that r ˝ E D 0, r  E D 0 and r  ! D 0 due to r m ! m D r ˝ !T   E D r  D   E 1 1 D p  ijk ri Djl rk ˝ rl   rm ˝ rm D p ijk ri Djk D 0: g g (3.298) Thus, the form (3.295) is a total differential, and there exists a vector v, being a one-valued function of x, hence by (2.222) we can represent its differential in the form d v D .r ˝ v/T  d x: (3.299)

3.9 The Complete System of Continuum Mechanics Laws

155

On comparing (3.299) with (3.295), we get that r ˝ vT can be represented as a sum of symmetric D and skew-symmetric W tensors: r ˝ vT D D C W;

(3.300)

but this decomposition is unique, therefore by (2.225) and (2.226) we get D D .1=2/.r ˝ vT C r ˝ v/; W D .1=2/.r ˝ v  r ˝ vT /;

(3.301) (3.302)

that proves the existence of the vector v, satisfying Eq. (3.301) and hence being the velocity. N As known from Sect. 2.4.5, the tensor W, defined by (3.294) and (3.302), is called the vorticity tensor, and ! – the vorticity vector.

3.9 The Complete System of Continuum Mechanics Laws 3.9.1 The Complete System in Eulerian Description Consider a nonpolar continuum. The system of continuum mechanics laws in the spatial description (3.18), (3.72), (3.118), (3.165), and (3.282) can be rewritten in the generalized form 

d AN˛ D r  BN˛ C C˛ ; dt

˛ D 1; : : : ; 6;

(3.303)

where the following generalized vectors are denoted: 1 1= C B v C B C B 2 =2 e C jvj C B AN˛ D B C; C B  C B A @ u FT 0

0

1 v B T C B C B C T  v  q B C BN ˛ D B C; B q= C B C @ A 0 F ˝ v

1 0 C B f C B C B B f  v C qm C C˛ D B C: B.qm C q  /= C C B A @ v 0 (3.304)

Here index ˛ D 1 corresponds to the continuity equation, ˛ D 2 – to the momentum balance equation, ˛ D 3 – to the energy balance equation, ˛ D 4 – to the entropy balance equation, ˛ D 5 – to the kinematic equation, ˛ D 6 – to the dynamic compatibility equation.

0

156

3 Balance Laws

Equation (3.303) at ˛ D 5 has been obtained from (2.190) with the help of its multiplication by : du  D v: (3.305) dt This relation is called the kinematic equation. The system (3.303) is called the universal system of continuum mechanics laws in total differentials. This system can be reduced to the divergence form. To do this, we can collect the corresponding divergence forms of Eqs. (3.15), (3.73), (3.119), (3.165a), and (3.281), or can transform the left-hand side of the system (3.303) with the help of the continuity equation as follows: 

@AN˛ C v  r ˝ AN˛ C @t



 @AN˛ @ C r  v AN˛ D C r  v ˝ AN˛ ; @t @t ˛ D 2; : : : ; 6: (3.306)

Then together with the continuity equation, the following complete system of continuum mechanics laws in the spatial description in the divergence form holds: @ A˛ C r  .v ˝ A˛  B˛ / D C˛ ; @t where

0

1 1 B C v B C B C 2 Be C jvj =2C A˛ D B C; B C  B C @ A u T F

˛ D 1; : : : ; 6;

(3.307)

0

1 0 B T C B C B C BT  v  qC B˛ D B C: B q= C B C @ A 0 F ˝ v

(3.308)

In particular, the kinematic equation (3.305) has the divergence form @u C r  v ˝ u D v: @t

(3.309)

Thus, we have proved the following theorem. Theorem 3.26. The complete system of continuum mechanics laws in Eulerian description can be represented in the universal form in total differentials (3.303) and in the equivalent divergence form (3.307).

3.9.2 The Complete System in Lagrangian Description In Lagrangian description, the system of continuum mechanics laws (3.8), (3.76), (3.131), (3.173), (2.190), and (3.270) can also be written in the universal form

3.9 The Complete System of Continuum Mechanics Laws ı ı d A˛



dt

ı

ı

ı

D r  B ˛ C C˛ ;

157

˛ D 1; : : : ; 6;

(3.310)

where two more generalized vectors appear: 0

0

1 ı .=/ det F B C v B C B C ı 2 B e C jv j=2 C A˛ D B C; B C  B C @ A u T F

1

0 P

B C B C B ıC ı BP  v  q C C: B˛ D B ı B q= C B C B C 0 @ A ı E ˝ v

(3.311)

Since in a Lagrangian coordinate system the total derivative d=dt coincides with the partial derivative @=@t, the equation set (3.310) already has the divergence form Remark. It should be noted, that the first equation in the system (3.310) at ˛ D 1 has been obtained by differentiation of the continuity equation (3.8) with respect to Lagrangian variables. In this case we must complement this differential equation with the initial condition det F D 1. Then Eq. (3.310) at ˛ D 1 with this initial conı dition always has a solution, which is the relation (3.8): .=/ det F D 1. Hence, as ı

ı

function A1 in (3.311) we can always use its actual value: A1 D 1. This means that ı

ı

the generalized vectors A˛ in (3.311) and A˛ in (3.308) are coincident: A˛ D A˛ ; ˛ D 1; : : : ; 6: t u

3.9.3 Integral Form of the System of Continuum Mechanics Laws The system of continuum mechanics laws in the integral form (3.5), (3.69), (3.116), and (3.162) can be written in the universal form as well. Theorem 3.27. The system of continuum mechanics laws in the differential form (3.307) in the spatial description is equivalent to the integral form d dt

Z

Z A˛ dV D V

Z n  B˛ d† C

†

C˛ dV;

˛ D 1; : : : ; 6:

(3.312)

V

H To prove the theorem for ˛ D 1; : : : ; 4, it is sufficient to substitute the generalized vectors A˛ , B˛ and C˛ into (3.312) and to write Eq. (3.312) for each of ˛ D 1; : : : ; 4. As a result, we obtain the relations (3.5), (3.69), (3.116), and (3.162) proved above. For ˛ D 5; 6, we should integrate Eqs. (3.309) and (3.281) over V and then apply the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2) and the Gauss–Ostrogradskii theorem (3.24). N

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3 Balance Laws

In the same way we can prove the following theorem (see Exercise 3.9.2). Theorem 3.28. The system of continuum mechanics laws in the differential form in the material description (3.310) is equivalent to the integral form d dt

Z

ıı

ı

Z

A˛ d V D ı

ı

ı

ı

n  B˛ d † C ı

V

Z

†

ı

ı

C˛ d V ;

˛ D 1; : : : ; 6:

(3.313)

ı

V

In spite of the above-mentioned equivalence of differential and integral forms of the continuum mechanics laws, there is a considerable difference between them: in the integral form the angular momentum balance equation (3.82) even for nonpolar continua is not identically satisfied, while the corresponding equation in the differential form (see Sect. 3.3), reduced to symmetry of the stress tensor T, has been excluded from the system (3.303). Thus, the integral form (3.312) must be complemented with the law (3.82); in other words, the index ˛ ranges from 1 to 7 for the generalized vectors A˛ , B˛ and C˛ in (3.312): 1 0 0 1 1 1 0 0 C B B T C B C v f C B B B C C Be C jvj2 =2C BT  v  qC B fvCq C C B B B C m C C B B B C C A˛ D B C ; B˛ D B q= C ; C˛ D B.qm C q  /= C : (3.314)  C B B B C C C B B B C C u 0 v C B B B C C A @ @ F ˝ v A @ A FT 0 xv T  x xf 0

The integral form (3.313) in the material description must consist of seven equaı

ı

tions as well; and the generalized vectors A˛ and B ˛ have the forms 1 ı .=/ det F C B v C B C B 2 j=2 e C jv C B ı C B A˛ D B C;  C B C B u C B T A @ F xv 0

Exercises for 3.9 3.9.1. Prove Theorem 3.27. 3.9.2. Prove Theorem 3.28.

0

0 P

1

C B C B B ıC BP  v  q C ı C B ı C B˛ D B B q= C : C B 0 C B C Bı @ E ˝ v A P  x

(3.315)

3.9 The Complete System of Continuum Mechanics Laws

159

3.9.3. Consider an arbitrary varying domain VQ .t/  V , whose boundary points Q Q †.t/ move with velocity c D d x† =dt, x† 2 †.t/ (see Sect. 5.1.6), and show with use of formula (2.12) that in this case the complete system of continuum mechanics laws in the divergence form (3.307) yields one more integral formulation d dt

Z

Z A˛ dV C

VQ

Z n  ..v  c/ ˝ A˛  B˛ / d† D

Q †

C˛ dV:

(3.316)

VQ

In the particular case when the domain VQ is fixed, the last formula reduces to d dt

Z

Z A˛ dV C

VQ

Z n  .v ˝ A˛  B˛ / d† D

Q †

C˛ dV: VQ

(3.317)

Chapter 4

Constitutive Equations

4.1 Basic Principles for Derivation of Constitutive Equations Consider nonpolar continua. The equation system (3.307) consists of 18 scalar equations (each of the vector equations in (3.307) is equivalent to three scalar equations, and each of the tensor ones is equivalent to nine scalar equations), but involves 29 scalar unknowns: ; v; u; T; e; ; ; q; F; q  :

(4.1)

The functions qm and f are usually assumed to be known when electromagnetic effects are absent. Thus, the set (3.307) is incomplete. To close the equation system (3.307) we need additional relations. These relations are called constitutive equations, because just they specify the type of a continuum (the universal balance laws do not differ types of continua, they are the same for all bodies). If constitutive equations are given in any way, then a continuum model is said to be specified. Derivation of constitutive equations is based on some additional principles, i.e. on physical assumptions of a general character, which cannot be formulated in the form of partial differential equations. We will accept the following basic principles:      

the principle of thermodynamically consistent determinism the principle of local action the principle of equipresence the principle of material indifference the principle of material symmetry the Onsager principle

Besides them, other additional principles may be formulated for special models of a continuum.

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 4, c Springer Science+Business Media B.V. 2011 

161

162

4 Constitutive Equations

4.2 Energetic and Quasienergetic Couples of Tensors 4.2.1 Energetic Couples of Tensors Before formulating the above-mentioned principles, let us give definitions of special stress and deformation tensors, which are of great importance for the theory of constitutive equations in continuum mechanics. Consider the power of internal forces W.i / (3.122) and introduce the stress power w.i / and the elementary work d 0 A of stresses: Z w.i / D T  .r ˝ v/T ;

W.i / D 

V

w.i / dV ;

d 0 A D w.i / dt:

(4.2)

ı

According to relation (2.37) between the gradients r ˝ v and r ˝ v, and also the dynamic compatibility equation (3.270), we can find one more representation for w.i / : ı

w.i / D T   r ˝ vT D T   .r ˝ vT  F1 / D .F1  T/  

dF : dt

(4.3)

Here we have used the formula of rearrangement of three tensors in the scalar product (see [12]): A   .B  C/ D .C  A/   B D .A  B/   C:

(4.4)

The following properties of contraction of two arbitrary tensors (see [12]) will be frequently used below: A   B D AT   BT ;

(4.5)

1 .A   B C AT   BT /; 2

(4.6)

A   B D AS   BS C AK   BK :

(4.7)

ABD

Here AS , BS are symmetric tensors and AK , BK are skew-symmetric tensors: AS D .A C AT /=2;

AK D .A  AT /=2:

According to the definition (3.56) of the Piola–Kirchhoff tensor and formula (4.3), we obtain the following expression for w.i / in terms of the tensor P: ı

w.i / D .=/ P  

dF : dt

(4.8)

4.2 Energetic and Quasienergetic Couples of Tensors

163

The representation (4.8) of the stress power as the contraction of some stress tensor and the rate of some tensor describing deformation proves to be not unique. There are several different such representations. Since they are of great importance for derivation of constitutive equations in continuum mechanics, let us find the representations. To derive the representations, we use the definition (4.2) and formula (4.7) with substitutions A ! T; B ! L D r ˝ vT ; then we get w.i / D TS   D C TK   W;

(4.9)

where D is the deformation rate tensor (2.225), W is the vorticity tensor (2.226) introduced by the decomposition (2.224) of the velocity gradient L; TS and TK are symmetric and skew-symmetric parts of the tensor T: TS D

1 .T C TT /; 2

TK D

1 .T  TT /: 2

(4.10)

For nonpolar continua, the tensor T is symmetric, and Eq. (4.9) takes the form w.i / D T   D:

(4.11)

Theorem 4.1. There exists a denumerable set of couples of symmetric tensors .n/ .n/

. T ; C/, with the help of which the stress power w.i / can be expressed in the form d .n/ C C TK   W; dt

.n/

w.i / D T  

n 2 Z:

(4.12)

If the tensor T is symmetric, then the expression (4.12) reduces to the form .n/

w.i / D T  

d .n/ C; dt

n 2 Z:

(4.13) .n/

.n/

Tensors T are called the energetic tensors of stresses, and C – the energetic tensors of deformations. Here Z is the set of integers, and the index n may be positive or negative integer, or zero. To emphasize the special status of the energetic couples we introduce the following notation: particular values of index n will be denoted by Roman numerals: I, II, III, IV, V etc., or I, II, III etc. The first five couples of energetic tensors I

I

II II

III III

IV IV

V V

are the most important, namely .T; C/, .T; C/, .T; C/, . T; C/ and .T; C/; they will be called principal, because these couples are most often used in solving different problems, and these energetic tensors may be expressed explicitly in terms of the stress and deformation tensors introduced before (see Table 4.1). For the symmetric tensor T, the couples at n D I, III, IV and V were derived by Hill [26], and the couple at n D II was found by K.F. Chernykh [8]. The special

164

4 Constitutive Equations

Table 4.1 Principal energetic couples of tensors Energetic deformation Energetic stress Number n .n/ .n/ of couple tensors T tensors C I FT  TS  F ƒ D 12 .E  U2 /  TS  O C OT  TS  F/

II

1 .FT 2

III

OT  TS  O

IV

1 .F1  TS  O C 2 1 S 1T

V

F

.n/

Energetic deformation .n/

measures G  12 G1

E  U1

U1

B

G

III

OT  TS  F1T / U  E

T F

CD

1 .U2 2

U  E/

1 G 2

.n/

.n/

notation T and C and the method of ordering of these tensors, when each tensor C is expressed in terms of the corresponding power of the right stretch tensor U, were suggested by the author [12]. Prove the theorem for each of the energetic couples separately.

I

4.2.2 The First Energetic Couple .T; ƒ/ Consider the expression TS   D and substitute in place of the tensor D its represenP tation (2.282) in terms of the deformation gradient rate F: DD

1 P 1 .F  F C F1T  FP T /: 2

(4.14)

Then .i /

wS D TS  D D

1 S T   .FP  F1 C F1T  FP T /: 2

(4.15)

Differentiation of the identities F  F1 D E and FT  F1T D E gives FP  F1 D F  FP 1 ;

F1T  FP T D FP 1T  FT :

(4.16)

On substituting these expressions into (4.15), we get .i /

wS D

1 S 1 T   F  FP 1 C TS   FP 1T  FT : 2 2

(4.17)

Rewrite the tensors FP 1T and FP 1 as the scalar products of themselves by the metric tensor E: FP 1T D E  FP 1T D .F  F1 /  FP 1T;

FP 1 D FP 1  E D FP 1  .F1T  FT/: (4.18)

4.2 Energetic and Quasienergetic Couples of Tensors

165

Substitution of these expressions into (4.17) yields / w.i S D

1 S 1 T   .F  F1  FP 1T /  FT C TS   .F  FP 1  F1T /  FT : 2 2

(4.19)

Applying the rule (4.4) of rearrangement of three arbitrary tensors in the scalar product, we obtain 1 T S 1 .F  T  F/   .F1  FP 1T / C .FT  TS  F/   .FP 1  F1T / 2 2 1  1 P 1T P 1 1T  P D FT  TS  F   CF F F F D .FT  TS  F/   ƒ: 2 (4.20)

/  w.i S D

Here we have used the expression (2.72) for the right Almansi deformation tensor ƒD

1 1 .E  F1  F1T / D .E  U2 /; 2 2

and P D ƒ

 1  P 1 1T F F C F1  FP 1T : 2

(4.21)

(4.22)

Introducing the new tensor I

T D FT  TS  F;

(4.23)

from (4.9) and (4.20) we obtain I

P C TK   W: w.i / D T   ƒ

(4.24)

Thus, the representation (4.12) really exists, and the first energetic couple of tensors I

I

I

is .T; C/ D .T; ƒ/:

V

4.2.3 The Fifth Energetic Couple .T; C/ Introduce the following new tensor: V

T D F1  TS  F1T

(4.25)

166

4 Constitutive Equations

and rewrite the expression (4.15) as follows: .i /

wS D D

1 1 S P .F  T   F C TS  F1T   FP T / 2 1 1 S 1T   FT  FP C F1  TS  F1T   FP T  F/ .F  T  F 2 V

D T

 V 1 T P P F  F C FP T  F D T   C: 2

(4.26)

Here we have used the property (4.4) of the scalar product of tensors, have written the tensor TS in the form TS D TS  E D TS  F1T  FT ;

TS D E  TS D F  F1  TS ;

(4.27)

and have taken account of the expression for the derivative of the right Cauchy– Green deformation tensor C (2.287):   P D 1 FT  FP C FP T  F : C 2

(4.28)

Thus, there is one more energetic couple of tensors: V

V

V

.T; C/ D .T; C/:

IV

4.2.4 The Fourth Energetic Couple .T ; .U  E// V

Rewrite the expression (4.26), passing from T to TS by formula (4.25) and from C to U by (2.158). Then we get V

/ P w.i S D TCD

1 1 S 1T P CU P  U/: .F  T  F /   .U  U 2

(4.29)

Use the rules (4.4) of rearrangement of three tensors in the scalar product: / w.i S D

1 1 S 1T P C 1 U  F1  TS  F1T   U: P UU F T F 2 2

(4.30)

On taking the polar decomposition (2.137) into account: F D O  U;

F1 D U1  OT ;

F1T D O  U1 ;

(4.31)

4.2 Energetic and Quasienergetic Couples of Tensors

167

we finally get / w.i S D

 IV IV 1  1 S P D T   C : F  T  O C OT  TS  F1T   U 2 IV

Here we have introduced the fourth energetic stress tensor T: IV

TD

 1  1 S F  T  O C OT  TS  F1T ; 2 IV

(4.32) IV

P coupled for which is the tensor C D UE. We have taken into account that C D U, P as E D 0. II

4.2.5 The Second Energetic Couple .T; .E  U1 // Rewrite the first energetic couple (4.24), replacing the tensor ƒ by its expression (2.158): ƒ D 12 .E  U2 /. Then from (4.24) we get  1 1  1 T S / P CU P 1  U1 U w.i S D  F T F U 2 1 T S P 1  1 U1  FT  T  F   U P 1 : D  F  T  F  U1   U 2 2 According to the property (4.31) of the polar decomposition, we obtain II

II

/  w.i S D TC :

Here we have introduced the second energetic stress tensor II

TD

 1 T S F  T  O C OT  TS  F ; 2 II

(4.33)

II

coupled for which is the tensor .E  U1 / D C, as C D .U1 / .

III

4.2.6 The Third Energetic Couple . T ; B/ We can derive one more energetic couple from Eq. (4.30), if in place of F1 and F1T substitute their polar decompositions (4.31): .i /

wS D

1 1 P C 1 U  U1  OT  TS  O  U1   U: P U  OT  TS  O  U1  U   U 2 2

168

4 Constitutive Equations

Changing the order in the scalar product, we get .i /

wS D

1 T S P  U1 C 1 OT  TS  O   U1  U: P O T OU 2 2

Hence, there exists the third energetic couple III

/ P w.i S D T   B; III

where T is the third energetic stress tensor: III

T D OT  TS  O;

(4.34)

III

and B D C is the the third energetic deformation tensor determined by its derivative and its initial value at t D 0 (see (2.317)).

4.2.7 General Representations for Energetic Tensors of Stresses and Deformations .n/

Theorem 4.2. Each of the energetic deformation tensors C can be expressed in terms of the corresponding power of the right stretch tensor as follows: .n/

CD

1 .UnIII  E/; 8n 2 Z; n ¤ III; .n  III/

(4.35)

.n/

and the energetic stress tensors T satisfy the equations IV

TD

nIV X .n/ 1 UnkIV  T  Uk ; n  III

if n > IVI

(4.35a)

IIn 1 X nIICk .n/ k U  T U ; III  n

if n 6 II:

(4.35b)

kD0

II

TD

kD0

Here and below in such expressions at particular values of n we must replace Roman numerals by corresponding Arabic ones and then perform arithmetic operations. For example, the relation (4.35) at n D I has the form I

CD

1 1 .U13  E/ D .E  U2 /: 13 2

4.2 Energetic and Quasienergetic Couples of Tensors

169

H (1) A proof of formula (4.35) for n D I, II, IV, and V is evident, if we write out the formula for these values of n and compare the obtained expressions with the .n/

tensors C from Table 4.1. .n/

.n/

(2) In order to show that the tensors C (when n > V) with tensors T also constitute .n/

energetic couples of the form (4.12), we should determine the derivative of C : .n/

C D

1 1 P  UnV C : : : P  UnIV C U  U .U  : : :  U/ D .U n  III „ƒ‚… n  III nIII

P  U C UnIV  U/ P D : : : C UnV  U

nIV X 1 P  UnIVk : Uk  U n  III kD0

.n/

So, if there exists energetic stress tensor T , then it satisfies the equation .n/

.n/

T   C D

nIV 1 .n/ X k P T  U  U  UnIVk n  III kD0

D

nIV X .n/ 1 UnIVk  T  Uk n  III

! P   U:

kD0

IV

Since the tensor .U  E/ D C is assigned to the fourth energetic stress tensor IV

IV

.n/

T, the expression in parentheses should form the tensor T. Thus, for all T (when n > V) the formula (4.35a) really holds. While n D V, formula (4.35a) takes the form IV

TD

IV

V V 1 .U  T C T  U/: 2

V

Since the tensors T and T have been introduced before according to Table 4.1, we should verify the last formula. Substitution of expressions (4.32) and (4.25) into this formula yields 1 1 S 1 .F  T  O C OT  TS  F1T / D .U  F1  TS  F1T C F1  TS  F1T  U/: 2 2 This formula is valid due to the polar decomposition (2.137) as OT D U  F1 . Hence formula (4.35a) really holds when n D V.

170

4 Constitutive Equations .n/

(3) To prove formula (4.35b) we determine the derivative of tensors C when n < I: .n/

C D

IIn 1 1 X 1 k .U1 „ƒ‚…  : : :  U1 / D .U /  .U1 /  .U1 /IInk : n  III n  III kD0

IIIn

Then .n/

.n/

T  C D

IIn 1 X nIICk .n/ k U  T U III  n

!   .U1 / :

kD0

II

II

Since the tensor .E  U1 / D C is assigned to the second energetic stress tensor T, II

the expression in parentheses just forms the tensor T. Hence formula (4.35b) holds for all n < I. When n D I, formula (4.35b) takes the form II

TD

I 1 1 I .U  T C T  U1 /: 2

We can verify this formula by substituting expressions (4.33) and (4.23) into the last equation. N .n/

A general relation between T and T is more complicated than the relation (4.35) .n/

.n/

between C and U; however, the tensors T and TS are evident to be connected by the linear relation .n/

.n/

T D 4 E 1   TS ;

n 2 Z;

(4.36)

.n/

where 4 E 1 are the inverse tensors of energetic equivalence (fourth-order tensors), whose expressions for the principal energetic couples are given in Table 4.2. These representations should be derived as Exercise 4.2.11. The notation .FT ˝ F/.1432/ for the transpose fourth-order tensor has been introduced in Sect. 2.1.4. .n/

Expressions for the tensors of energetic equivalence 4 E can be found by inverting the relations (4.36): .n/

.n/

TS D 4 E   T ; Table 4.2 Inverse tensors of energetic equivalence

n 2 Z:

(4.37)

.n/

E 1 .FT ˝ F/.1432/

Number n I

4

II III IV V

˝ O C OT ˝ F/.1432/ .O ˝ O/.1432/ 1 .F1 ˝ O C OT ˝ F1T /.1432/ 2 .F1 ˝ F1T /.1432/ 1 .FT 2 T

4.2 Energetic and Quasienergetic Couples of Tensors

171

Substituting the relationship (4.35a) into formula (4.37) where n D IV, we obtain .n/

IV

the expression for tensors 4 E (when n > V) in terms of the tensor 4 E: 4

nIV 1 4 IV X nkIV E .U ˝ Uk /.1432/ ; n  III

.n/

E D

n > V:

(4.38a)

kD0

.n/

In a similar way, we can find the expression for tensors 4 E when n 6 I: 4

IIn 1 4 II X nIIk E .U ˝ Uk /.1432/ ; III  n

.n/

E D

n 6 I:

(4.38b)

kD0

.n/

ı

Components of all the tensors 4 E with respect to the eigenbases pi and pi can be expressed in terms of only eigenvalues ˛ . .n/

Theorem 4.3. The tensors of energetic equivalency 4 E , connecting the tensors TS .n/

and T with the help of the linear relations (4.37), have the form 4

3 X

.n/

E D

.n/

ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.38)

˛;ˇ D1

where I

E ˛ˇ D

II III IV V 2˛ ˇ 1 2 ; E ˛ˇ D ; E ˛ˇ D 1; E ˛ˇ D ; E ˛ˇ D ˛ ˇ ; ˛  ˇ ˛ C ˇ ˛ C ˇ .n/

E ˛ˇ

nIV X 2˛ ˇ D ˛nkIV kˇ ; .˛ C ˇ /.n  III/

n > V;

kD0

.n/

E ˛ˇ

IIn X 2 D nIICk k ˛ ˇ ; ˛ C ˇ

n 6 I:

(4.39)

kD0

.n/

H (a) Consider the case when n D I, and represent the tensors F, O, TS and T in ı the eigenbases pi and pi (see formulae (2.169)–(2.171)): FD

3 X

ı

˛ p˛ ˝ p˛ ; FT D

˛D1

3 X ˛D1

TS D

3 X ˛;ˇ D1

.p/

3 X

ı

˛ p˛ ˝ p˛ ; O D

T˛ˇ p˛ ˝ pˇ ;

ı

p˛ ˝ p˛ ;

˛D1 .n/

T D

3 X .n/ı

T

˛;ˇ D1

.p/ ı ˛ˇ p˛

ı

˝ pˇ ;

(4.40)

172

4 Constitutive Equations .n/ı

.p/

where T˛ˇ are components of the tensor TS with respect to the basis p˛ , and T .n/

.p/ ˛ˇ

ı

are components of the tensor T with respect to the basis p˛ . On substituting these representations into formula (4.36) at n D I, we get I

TD

3 X

Iı

ı

.p/ ı

T ˛ˇ p˛ ˝ pˇ D FT  TS  F

˛;ˇ D1

D

3 X

ı

ı

.p/ ˛ p˛ ˝ p˛  T! p ˝ p!  ˇ pˇ ˝ pˇ D

3 X ˛;ˇ D1

˛;ˇ;;!D1

.p/ ı

ı

˛ ˇ T˛ˇ p˛ ˝ pˇ :

Here we have taken into account that the basis p˛ is orthonormal (see (2.141)). Hence, Iı

.p/ T .p/ D T˛ˇ ˛ ˇ : ˛ˇ

Use once again the representation of the tensor TS in the basis p˛ and take into account that Iı

ı

I

ı

T .p/ D p˛  T  pˇ ; ˛ˇ then TS D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

.p/ T˛ˇ p˛ ˝ pˇ D

3 X ˛;ˇ D1

Iı 1 p˛ ˝ T .p/ ˝ pˇ ˛ˇ ˛ ˇ

I ı 1 ı p˛ ˝ p˛  T  pˇ ˝ pˇ ˛ ˇ I 1 ı ı .p˛ ˝ p˛ ˝ pˇ ˝ pˇ /.1432/   T ˛ ˇ I 1 ı ı .p˛ ˝ pˇ ˝ pˇ ˝ p˛ /   T: ˛ ˇ

I

Thus, the representation (4.37) with the tensor 4 E (4.38), (4.39) really holds at n D I. (b) Consider the case when n D II. According to the representations (4.40), we can rewrite relation (4.36) for n D II as follows:

4.2 Energetic and Quasienergetic Couples of Tensors 3 X

II

TD

IIı

ı

1 T .F  T  O C OT  T  F/ 2

ı

T .p/ p ˝ pˇ D ˛ˇ ˛

˛;ˇ D1

D

1 2

3 X

173

ı

ı

.p/ .˛ p˛ ˝ p˛  T! p ˝ p!  pˇ ˝ pˇ

˛;ˇ;;!D1 ı

3 1 X ı .p/ ı .˛ C ˇ /T˛ˇ p˛ ˝ pˇ : 2

ı

.p/ C p˛ ˝ p˛  T! p ˝ p!  ˇ pˇ ˝ pˇ / D

˛;ˇ D1

Thus, we get IIı

.p/

T ˛ˇ D

1 .p/ .˛ C ˇ /T˛ˇ : 2

The further proof is analogous to the proof for n D I. The theorem for n D III; IV and V can be proved in the same way (see Exercise 4.2.18). (c) To prove the theorem for the cases when n > V and n 6 I, one should use formulae (4.38), (4.39) at n D II and IV, and also the expression (2.142) for the tensor U and substitute them into (4.38a) and (4.38b). .n/ı

In proving the theorem, we have established that components T .n/

ı

.p/ ˛ˇ

of the tensor

.p/

T with respect to the basis p˛ and components T˛ˇ of the tensor T with respect to the basis p˛ are connected by the relations Iı

.p/ T .p/ D ˛ ˇ T˛ˇ ; ˛ˇ IIIı

T

.p/ ˛ˇ

.p/ D T˛ˇ ;

IVı

T

.p/ ˛ˇ

D

IIı

T .p/ D ˛ˇ

1 .p/ ; .˛ C ˇ /T˛ˇ 2

1 1 1  .p/ T ; C 2 ˛ ˇ ˛ˇ

Vı

T .p/ D ˛ˇ

1 T .p/ ; ˛ ˇ ˛ˇ

(4.41)

or .n/

.p/

.n/ı .p/ ˛ˇ ;

T˛ˇ D E ˛ˇ T

8n 2 Z: N

4.2.8 Energetic Deformation Measures .n/

Besides the energetic tensors C, introduce the energetic deformation measures .n/

GD

1 UnIII ; n  III

8n 2 Z;

n ¤ III;

which are the corresponding powers of the right stretch tensor U.

(4.42)

174

4 Constitutive Equations .n/

.n/

One can readily establish the relation between G and C : .n/

.n/

CDG

1 E: n  III

(4.43) .n/

.n/

Since the derivative of the metric tensor is zero, the derivative tensors G  and C  are coincident: .n/ 

.n/

C D G :

(4.44)

Introduce the third energetic deformation measure III

III

G D E C C:

(4.43a)

Due to (2.317), the measure satisfies the equation III

1 dG D dt 2



 dU dU ;  U1 C U1  dt dt

III

G.0/ D E;

(4.43b)

and also satisfies the general equation (4.44). Due to (4.44), the relation (4.12) for w.i / takes the form .n/

.n/

w.i / D T   G  C TK   W;

8n 2 Z:

(4.45)

If the tensor T is symmetric, then this formula reduces to the form .n/

w.i / D T  

d .n/ G; dt

8n 2 Z;

(4.46) .n/ .n/

i.e. besides the couples of the energetic stress and deformation tensors . T ; C/, there .n/ .n/

are couples of the energetic stress tensors and deformation measures . T ; G/ (see Table 4.1). .n/

It should be noted that besides the tensors G we can introduce a set of tensors of .n/ P  0, which also satisfy the form G C N, where N is an arbitrary constant-tensor: N .n/

the relation (4.44). However, only the tensors G, which were already used above, have some physical meaning: I II IV V 1 1 G D  G1 ; G D U1 ; G D U; G D GI 2 2

(4.47)

4.2 Energetic and Quasienergetic Couples of Tensors

175

they are the right Cauchy–Green and Almansi deformation measures, and also the .n/ .n/

.n/

right stretch tensor and its inverse. All the tensors T , C and G are symmetric. The principal energetic deformation measures for n D I; : : : ; V are given in Table 4.1. For the first time, formulae (4.35) and (4.42), and also the existence of energetic deformation measures have been derived in the systematized form by the author in [12].

4.2.9 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors In continuum mechanics, three principal invariants of a second-order tensor (see [12]) are frequently used; they are determined as follows: .n/

.n/

.n/

I1 . C/ D E   C; I2 . C / D

.n/ .n/ .n/ 1 2 .n/ .I1 . C /  I1 . C 2 //; I3 . C / D det. C /: 2

(4.48)

The concept of an invariant will be considered in detail in Sect. 4.8.4. .n/

Now, let us derive relations between principal invariants I˛ . C / (˛ D 1; 2; 3) of .n/

.n/

the energetic tensors C and principal invariants I˛ . G/ of the energetic deformation measures. .n/

.n/

Theorem 4.4. The principal invariants of the tensors C and G are connected by the relations .n/

.n/

3 ; n  III .n/ .n/ .n/ 2 3 ; I2 . C / D I2 . G/  I1 . G/ C n  III .n  III/2 I1 . C / D I1 . G/ 

.n/

.n/

.n/

.n/

I2 . G/ 1 I1 . G/  ; I3 . C / D I3 . G/   n  III .n  III/2 .n  III/3 .n/ .n/ 3 ; I1 . G/ D I1 . C/ C n  III .n/ .n/ .n/ 2 3 I2 . G/ D I2 . C / C ; I1 . C / C n  III .n  III/2 .n/

.n/

.n/

.n/

I1 . C / I2 . C / 1 I3 . G/ D I3 . C / C C C ; n  III .n  III/2 .n  III/3 n D I; II; IV; V:

(4.49)

176

4 Constitutive Equations

H To prove the theorem, one should express the first, second and third powers of the .n/

.n/

tensors C in terms of powers of the tensors G: .n/

.n/

CDG

.n/

1 E; n  III

.n/

2 .n/ 1 GC E; n  III .n  III/2 .n/ .n/ .n/ 3 .n/2 3 1 C 3 D G3  G C G E: 2 n  III .n  III/ .n  III/3 C 2 D G2 

.n/

.n/

.n/

Write out the following invariants of the tensors: I1 . G/, I1 . G 2 / and I1 . G 3 /, and then, according to the formulae (see [12])  1 3 I1 .T/  3I1 .T/I1 .T2 / C 2I1 .T3 / ; 6  1 I1 .T3 /  I13 .T/ C 3I1 .T/I2 .T/ ; I3 .T/ D 3

I3 .T/ D det .T/ D

(4.50)

.n/

we get the desired expressions for the first, second and third invariants of C . The derivation should be performed in detail as Exercise 4.2.10. N

4.2.10 Quasienergetic Couples of Stress and Deformation Tensors .n/

With the help of the left stretch tensor V, we can introduce couples of the tensors S .n/

and A as well. However, in this case, the stress power w.i / depends, in addition, on .n/ .n/

the derivative of the rotation tensor OT . Therefore, such couples . S ; A/ are called quasienergetic. .n/

.n/

Definition 4.1. The tensors S and A, defined by Table 4.3, are called the principal quasienergetic stress and deformation tensors, respectively. Theorem 4.5. The stress power w.i / (4.2) can always be expressed in terms of each of the quasienergetic tensor couples: .n/

w.i / D S  

d .n/ ı d T A C S   O C TK   W; 8n 2 Z: dt dt

(4.51)

4.2 Energetic and Quasienergetic Couples of Tensors

177

Table 4.3 Principal quasienergetic couples of tensors Quasienergetic Quasienergetic deformation Number n .n/ .n/ tensors A of couple stress tensors S AD EV

 V2 /

I II

V  TS  V 1 .V  TS C TS  V/ 2

III IV V

TS Y 1 1 S S 1 .V  T C T  V / V E 2 J D 12 .V2  E/ V1  TS  V1

1 .E 2 1

Quasienergetic deformation .n/

measures g  12 g1 1 V III

g V 1 g 2

If the tensor T is symmetric, then the expression for w.i / takes the form .n/

w.i / D S  

d .n/ ı d T A CS O ; dt dt

8n 2 Z:

(4.51a) ı

.n/ .n/

Here S , A are symmetric second-order quasienergetic tensors; and the tensor S is the same for all the couples, and it is called the rotation tensor of stresses: ı

SD

1 .V  TS  V1  V1  TS  V/  O: 2

(4.52)

Let us prove the theorem for each of the principal quasienergetic couples separately.

I

4.2.11 The First Quasienergetic Couple .S; A/ Consider the first energetic couple (4.24) and pass from the tensor U2 to V2 with the help of the relations U2 D OT  V2  O (see Exercise 2.3.1): I I 1 T S / S  P 2 w.i S D T   D D T  ƒ D  F  T  F   U 2   1 P : P T  V2  O C OT  V P 2  O C OT  V2  O D  FT  TS  F   O 2 (4.53)

According to the polar decomposition (2.137) and the rules (4.4) of rearrangement of tensors in the scalar product, we obtain / w.i S D 

1  1 S P T C V  TS  V   V P 2 V T VOO 2  P  OT : CV  TS  V1   O

(4.54)

178

4 Constitutive Equations

Taking into account that O  OT D E and, hence, P  OT D O  O P T; O

(4.55)

we obtain the following final expression of the form (4.51): / w.i S

  1 2  ı P T DS  V CSO : 2 I

I

I

(4.56) I

Here the first quasienergetic tensors S, A are defined as follows: S D V  TS  V ı

I

and A D A D 12 .E  V2 / D 12 .E  F1T  F1 /, and the tensor S is defined by formula (4.52). Thus, we have proved the existence of the first quasienergetic couple I

I

I

.S; A/ D .S; A/:

II

4.2.12 The Second Quasienergetic Couple .S; .E  V1 // P 2 to V P 1 in (4.55), we get Going from the derivative V ı   P T  1 V  TS  V   V1  V P 1 C V P 1  V1 w.i / D S   O 2 ı 1 T P  .V  TS C TS  V/   V P 1 : D SO 2

(4.57)

Here we have used the rule (4.4) again. As a result, we find the second quasienergetic couple: ı

II

P T; w.i / D S   .E  V1 / C S   O

(4.58)

II

where the second quasienergetic stress tensor S has been introduced as follows: II

SD

1 .V  TS C TS  V/; 2

(4.59)

II

and the second quasienergetic deformation tensor is A D .E  V1 /.

4.2.13 The Third Quasienergetic Couple .Y; TS / P Consider the third energetic couple (see Sect. 4.2.6) and pass from the tensor U P to V: 1 T S / 1 P P w.i C U1  U/ S D O  T  O   .U  U 2

4.2 Energetic and Quasienergetic Couples of Tensors

D

179

1 T S P T  V  O C OT  V P  OT  V1  O P  O C OT  V  O/ O  T  O   .O 2 1 P T  V  O C OT  V P P  O C OT  V  O/: C OT  TS  O   OT  V1  O.O 2

Changing the order in the scalar product of tensors in each of the summands, we get .i /

wS D

1 S P T C V1  TS   V P  OT P C V1  TS  V   O .T   O  O 2 P T C TS  V1   V P  OT /: (4.60) P C TS   O CV  TS  V1   O  O

According to the relation (4.55), we see that the first and the last terms are canceled in pair. Thus, Eq. (4.60) yields ı

/ S PT P w.i S DT YCSO :

(4.61)

where we have introduced the new tensor Y similar to the tensor B, which is defined by its derivative (see (2.318)): P  V1 C V1  V/; P D 1 .V P Y 2

Y.0/ D 0:

(4.62) III

III

Thus, we have proved the existence of the third quasienergetic couple: .A; S/ D .Y; TS /:

IV

4.2.14 The Fourth Quasienergetic Couple . S ; .V  E// Due to (4.62), formula (4.61) takes the form / w.i S D

ı 1 1 S P TI P CSO .V  T C TS  V1 /   V 2

(4.63)

that proves the existence of the fourth quasienergetic couple: IV

IV

ı

/  PT w.i S D S A CSO ;

where the fourth quasienergetic tensors are defined as follows (see Table 4.3): IV

SD

1 1 S .V  T C TS  V1 /; 2

IV

A D V  E:

(4.63a)

180

4 Constitutive Equations V

4.2.15 The Fifth Quasienergetic Couple .S; J/ Rewrite the relation (4.63) as follows: / w.i S D

1 1 S P C TS  V1   V P  V  V1 / .V  T   V1  V  V 2 ı ı P T D V1  TS  V1   1 .V  V P T: P CV P  V/ C S   O CS O 2

Hence, there exists the fifth quasienergetic couple: ı

V

/ PT P w.i S DSJCSO ;

where the fifth quasienergetic stress and deformation tensors are defined as follows: V

S D V1  TS  V1 ;

V

ADJD

1 2 1 .V  E/ D .F  FT  E/: 2 2

(4.64)

For the first time, the principal quasienergetic couples have been derived by the author in [12].

4.2.16 General Representation of Quasienergetic Tensors .n/

Theorem 4.6. Each of the quasienergetic deformation tensors A can be expressed in terms of the corresponding power of the left stretch tensor V as follows: .n/

AD

1 .VnIII  E/; .n  III/

8n 2 Z;

n ¤ III;

(4.65)

nIV X .n/ 1 VnkIV  S  Vk ; n  III

if n > IVI

(4.65a)

IIn 1 X nIICk .n/ k V  S V ; III  n

if n 6 II:

(4.65b)

.n/

and quasienergetic stress tensors S satisfy the equations IV

SD

kD0

II

SD

kD0

H The theorem for n DI, II, IV and V can be proved by immediate comparison of .n/

formula (4.65) with the tensors A from Table 4.3, and for n > V and n < I – by the same method as was used in the proof of Theorem 4.2. N

4.2 Energetic and Quasienergetic Couples of Tensors Table 4.4 Inverse tensors of quasienergetic equivalency

181

Number n of couple I II III IV V

4

.n/ 1

Q .V ˝ V/.1432/ 1 .V ˝ E C E ˝ V/.1432/ 2  1 .V1 ˝ E C E ˝ V1 /.1432/ 2 .V1 ˝ V1 /.1432/

.n/

The quasienergetic stress tensors S can be represented in the general form (see Exercise 4.2.17) .n/

.n/

S D 4 Q 1   TS ;

(4.66)

.n/

where 4 Q 1 are the inverse tensors of quasienergetic equivalency, whose expressions for n D I; : : : ; V are given in Table 4.4. .n/

Expressions for the tensors of quasienergetic equivalency 4 Q can be obtained by inverting the relations (4.66): .n/

.n/

TS D 4 Q   S :

(4.67)

To invert them, we use the representations of the tensors in the eigenbasis pi of the left stretch tensor V, as it was made in Sect. 4.2.7. .n/

Theorem 4.7. The tensors of quasienergetic equivalency 4 Q, connecting the tensors .n/

TS and S with the help of the linear relations (4.67), have the form 4

.n/

QD

3 X

.n/

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.68)

˛;ˇ D1 .n/

where E ˛ˇ are expressed by formulae (4.38), (4.39). H Consider only the case when n D I; for the remaining cases a proof is analogous and should be performed as Exercise 4.2.19. .n/

Expressing the tensors S in the eigenbasis pi : .n/

S D

3 X ˛;ˇ D1

.n/ .p/ S ˛ˇ p˛

˝ pˇ ;

(4.69)

182

4 Constitutive Equations

and using the representations (4.40) for TS and (2.142) for V, we can rewrite the relation (4.66) for n D I as follows: 3 X

I

SD

I

˛;ˇ D1

S .p/ p ˝ pˇ D V  TS  V ˛ˇ ˛

3 X

D

3 X

.p/ ˛ p˛ ˝ p˛  T! p ˝ p!  ˇ pˇ ˝ pˇ D

˛;ˇ D1

˛;ˇ;;!D1

.p/

˛ ˇ T˛ˇ p˛ ˝ pˇ ;

I

.p/ .p/ i.e. S ˛ˇ D ˛ ˇ T˛ˇ : Then

T D S

3 X

.p/ T˛ˇ p˛

˝ pˇ D

˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

3 X ˛;ˇ D1

I 1 p˛ ˝ S .p/ ˝ pˇ ˛ˇ ˛ ˇ

I 1 p˛ ˝ p˛  S  pˇ ˝ pˇ ˛ ˇ I 1 .p˛ ˝ pˇ ˝ pˇ ˝ p˛ /   S: N ˛ ˇ

Remark. It follows from the performed proof and formula (4.41) that components I

.p/

Iı

I

.p/

I

S ˛ˇ and T ˛ˇ of the tensors S and T are coincident. The same result is valid for all .n/

.n/

the tensors S and T (see Exercises 4.2.18 and 4.2.19): .n/ S .p/ ˛ˇ

.n/ı

D T

.p/ : ˛ˇ



(4.70)

4.2.17 Quasienergetic Deformation Measures .n/

Similarly to the measures G, let us introduce quasienergetic deformation .n/

measures g : .n/

g D

1 VnIII ; n  III

n 2 Z;

n ¤ III;

(4.71)

being the corresponding powers of the left stretch tensor V. The third quasienergetic III

deformation measure g is introduced as follows: III

III

g D E C A:

(4.71a)

4.2 Energetic and Quasienergetic Couples of Tensors

183

Due to (4.62), the measure satisfies the equation III

dg 1 D dt 2 .n/



 dV dV 1 1 ; V CV  dt dt

III

g .0/ D E:

(4.71b)

.n/

The tensors g and A are connected by the relation .n/

.n/

AD g 

1 E; n  III

(4.72)

(when n D III the multiplier (nIII) is replaced by 1), therefore for all the couples .n/ 

.n/

A D g :

(4.73)

According to this relation, the power (4.51) can be expressed in another form .n/

ı

.n/

P T C TK   W; w.i / D S   g  C S   O

n 2 Z:

(4.74)

For the symmetric tensor T, this expression reduces to .n/

.n/

ı

P T; w.i / D S   g  C S   O

n 2 Z:

(4.74a)

.n/

The tensors g at n D I, II, IV, V are the left Cauchy–Green and Almansi deformation measures, and also the left stretch tensor and its inverse: 1 I g D  g1 ; 2

II

g D V1 ;

IV

g D V;

V

gD

1 g: 2

(4.75)

All the tensors are symmetric.

4.2.18 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors ı

Theorem 4.8. The rotation tensor of stresses S (4.52) can be represented in terms of the quasienergetic couples of the stress and deformation tensors as follows: ı

.n/ .n/

.n/ .n/

S D . A  S  S  A /  O;

8n 2 Z;

n ¤ III:

(4.76)

184

4 Constitutive Equations

H We first consider the principal quasienergetic couples when n D I; II; IV; V. ı

The definition of the tensor S (4.52) ı

SD

1 .V  TS  V1  V1  TS  V/  O 2

(4.77)

can be rewritten in the following four equivalent forms: ı

1 ..V  TS  V/  V2  V2  .V  TS  V//  O; 2 ı 1 S D ..V  TS C TS  V/  V1  V1  .V  TS C TS  V//  O; 2 ı 1 S D .V  .V1  TS C TS  V1 /  .V1  TS C TS  V1 /  V/  O; 2 ı 1 2 S D .V  .V1  TS  V1 /  .V1  TS  V1 /  V2 /  O: (4.78) 2 SD

.n/

Comparison of these relations with the definition of the quasienergetic tensors S .n/

and A (see Table 4.3) gives the formulae ı

I I I 1 I .S  .E  2A/  .E  2A/  S/  O; 2 ı II II II 1 II S D .S  .E  A/  .E  A/  S/  O; 2 ı IV IV IV 1 IV S D ..A C E/  S  S  .E C A//  O; 2 ı V V V V 1 S D ..E C 2A/  S  S  .E C 2A//  O: 2

SD

(4.79) I

I

Since the scalar product by the unit tensor is always commutative: S  E D E  S etc.; so from (4.79) we obtain the desired relation (4.76). Let us consider now the case when n > V. Since for n D IV formula (4.76) has already been proved, we can use it and substitute relationship (4.65a) into this formula: ı

IV IV

IV IV

IV

IV

S  OT D A  S  S  A D V  S  S  V .n/ .n/ .n/ .n/ 1 V  .VnIV  S C VnV  S  VC : : : CV  S  VnV C S  VnIV / D n  III .n/ .n/ .n/ .n/ 1  .VnIV  S CVnV  S  VC : : : CV  S  VnV C S  VnIV /  V: nIII

4.2 Energetic and Quasienergetic Couples of Tensors

185

If we remove parentheses in this formula, all the terms are canceled there except the first and last ones; therefore, we get ı

S  OT D .n/

.n/ .n/ .n/ .n/ .n/ .n/ 1 .VnIII  S  S  VnIII / D A  S  S  A ; n  III

.n/

as E  S D S  E. Thus, formula (4.76) really holds for n > V. The theorem for the case when n < I can be proved in a similar way with the help of formula (4.65b). N From this theorem we can get the following significant corollary, which will be used below. ı

Theorem 4.9. The rotation tensor of stresses S is zero-tensor if and only if the ten.n/

.n/

sors S and A commutate with each other.

4.2.19 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors Relations between the density  and principal invariants of the energetic and quasienergetic deformation tensors and measures will be frequently used below. Let us establish them. ı

Theorem 4.10. For every continuum the ratio of densities = is a one-valued function of the third principal invariant of the energetic and quasienergetic measures .n/

.n/

I3 . G/ and I3 . g /, and also a function of principal invariants of the energetic and .n/

.n/

quasienergetic deformation tensors I˛ . C / and I˛ . A /: .n/

.n/

I3 . G/ D I3 . g / D

1 ı .=/IIIn ; .n  III/3

(4.80)

ı

.n/

.n/

.n/

ı

.n/

.n/

.n/

= D .1 C .n  III/I1 . C/ C .n  III/2 I2 . C / C .n  III/3 I3 . C //1=.IIIn/ ; (4.81) = D .1 C .n  III/I1 . A / C .n  III/2 I2 . A / C .n  III/3 I3 . A//1=.IIIn/ : (4.82) To prove formula (4.80), one should use the continuity equation (3.8) and the polar decomposition (2.137): ı

= D det F D det .O  U/ D det U D det V:

(4.83)

186

4 Constitutive Equations .n/

Using the definition (4.42) of the measures G and formula (4.83), we verify that .n/

formula (4.80) for I3 . G/ is valid: 

.n/

I3 . G/ D det

1 UnIII n  III

 D

1 1 ı .detU/nIII D .=/IIIn : 3 .n  III/ .n  III/3 (4.84)

Similarly, according to the definition (4.75) of quasienergetic measures and formula (4.83), we can prove the remaining relations in formula (4.80): .n/



I3 . g / D det

1 VnIII n  III

 D

1 1 ı .detV/nIII D .=/IIIn : 3 3 .n  III/ .n  III/ (4.85)

On substituting the sixth formula of (4.50) into (4.80), we obtain the relations .n/

ı

(4.81) between .=/ and I˛ . C /. Since there are connections between the invariants .n/

.n/

I˛ . g / and I˛ . A /, which are similar to (4.49) (see Exercise 4.2.8), so values of ı

.n/

.=/ and I˛ . A/ are also connected by relations (4.82) similar to (4.81). N

4.2.20 The Generalized Form of Representation of the Stress Power Consider only nonpolar continua, when the tensor T is symmetric; then TS D T

and

TK  0;

(4.86)

and, according to formulae (4.13), (4.46), (4.51a), and (4.74a), the stress power w.i / (4.2) can be expressed in the following equivalent forms: w.i / w.i /

.n/

.n/

dC D T  ; dt .n/

n 2 Z;

(4.87)

n 2 Z;

(4.88)

.n/

dG ; D T  dt .n/

w.i /

ı dA d OT D S  CS ; dt dt

w.i /

ı dg d OT D S  CS ; dt dt

.n/

.n/

n 2 Z;

(4.89)

n 2 Z:

(4.90)

.n/

The stress power w.i / is said to be written in the form An , Bn , Cn or Dn , if we use expressions (4.87), (4.88), (4.89) or (4.90), respectively.

4.2 Energetic and Quasienergetic Couples of Tensors

187 .n/

Introduce the generalized energetic stress tensors T G , the generalized energetic .n/

.n/

deformation tensors C G , the generalized energetic deformation measures G G and the generalized rotation tensor of stresses SG as follows:

.n/

TG

.n/

GG

8 .n/ ˆ ˆ T; ˆ ˆ ˆ ˆ .n/ ˆ
if G D A; if G D B; if G D C; if G D D;

.n/

CG

8 .n/ ˆ ˆ ˆ C; ˆ ˆ ˆ .n/ ˆ < G; D .n/ ˆ ˆ ˆ A; ˆ ˆ ˆ ˆ .n/ :g;

if G D A; if G D B; if G D C;

SG D

if G D D;

8 ˆ 0; ˆ ˆ ˆ ˆ <0; ı

ˆ S; ˆ ˆ ˆ ˆ :ı S;

if G D A; if G D B; if G D C; if G D D; if G D A; if G D B; if G D C;

(4.91)

if G D D;

where G is the index taking on the letter values G D A; B; C , and D. Then all the expressions (4.87)–(4.90) can be written in the single generalized form .n/

.n/

w.i / D T G

d CG d OT  C SG   ; dt dt

n 2 Z; G D A; B; C; D:

(4.92)

4.2.21 Representation of Stress Power in Terms of Co-rotational Derivatives Two symmetric tensors M and N are called coaxial, if they have the same eigenbasis / / / (but their eigenvalues .M and .N may in general be different): p.M ˛ ˛ ˛ MD

3 X ˛D1

/ .M / / .M ˝ p.M ˛ p˛ ˛ ;

ND

3 X

/ .M / / .N ˝ p.M ˛ p˛ ˛ :

˛D1

/ Since the eigenbasis vectors p.M are orthogonal with each other, the scalar product ˛ of coaxial tensors M and N is independent of order of the multipliers: MN D NM, i.e. the tensors M and N commutate with each other. The converse assertion holds as well: commuting symmetric tensors are coaxial (see Exercise 4.2.24).

188

4 Constitutive Equations .n/

.n/

.n/

.n/

Theorem 4.11. Let the tensors T and C, and also S and A be either coaxial in pairs or commuting in pairs for each n, then the stress power w.i / (4.92) can be represented in terms of the co-rotational derivatives as follows: .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

w.i / D T   C h D T   G h D S   A h D S   g h ;

(4.93)

where h D f ; U; V; J; S g. H Notice that all the co-rotational derivatives, mentioned in the theorem, have the same structure (see (2.373)): .n/ h

.n/

.n/

.n/

C D C   Zh  C  C  ZTh ;

(4.94)

where Zh D f 0; U ; V ; W;  g is a second-order tensor (see (2.374)). Then, due to the property (4.3) of contraction of three tensors, we obtain .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

T   C h D T   C   T   Zh  C  T   C  ZTh .n/

D T   C   . C  T/   Zh  .T  C/   ZTh : .n/

.n/ .n/

.n/

(4.95) .n/ .n/

Since the tensors C and T are either commuting or coaxial, so T  C D C  T , and the tensor Zh is either skew-symmetric or zero-tensor. Then we find that .n/

.n/

.n/

.n/

T   C h D T   C  D w.i / ;

(4.96)

as was to be shown. .n/

.n/

Using commutativity of the tensors A and S and Theorem 4.9, we find that ı ı P T D 0. Then, following the manipulations made above, we get S  0 and S   O .n/

.n/

ı

.n/

.n/

P T D S   A h: N w.i / D S   A  C S   O

(4.97)

4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient In Sect. 2.5.11 we have derived the relations between the co-rotational derivatives (and the total derivative with respect to time) of the Almansi and Cauchy–Green deformation tensors and measures, on the one hand, and the velocity gradient L, on the other hand. Using Eqs. (2.376) and (2.377), we can readily pass to the energetic

4.2 Energetic and Quasienergetic Couples of Tensors

189

and quasienergetic deformation tensors and measures and rewrite these relations in the generalized form .n/

.n/

.n/

C G h D 4 X Gh   D C 4 Y Gh   W; h D f; Ol; CR; d; D; U; V; J; S gI G D A; B; C; DI n D I; : : : ; V; (4.98) .n/

.n/

where 4 Y Gh and 4 X Gh are the fourth-order tensors, their general expressions for G D C; D have the forms 4 4

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

X Gh D 4 X 0Gh  .4 M.1342/  C G /.1423/  C G  4 M.2134/ ; h h e .1342/  C G /.1423/  C G  4 M e .2134/ ; Y Gh D 4 Y 0Gh  .4 M h h G D C; D;

.n/

n D I; : : : ; V:

(4.99)

.n/

The tensors 4 X 0Gh and 4 Mh are given in Table 4.5. Table 4.5 shows the following tensors: tensors 4 ZDh and 4 ZW h determined by e determined in Exercise 2.4.11, and tensor formula (2.375) and Table 2.1, tensor 4  4 Eh determined by Table 2.1 (here n D I; : : : ; V). .n/

III

The tensors 4 X Dh and 4 Y have the forms 4

.n/

X Dh D 0; n D I; II; IV; VI 4

III

YD

4

III

III

XDh D 4 XC h ;

1 1 .V ˝ V  V ˝ V1 /: 2

(4.100)

For the energetic deformation measures (G D A; B) we usually apply only the total derivatives with respect to time: 4

.n/

.n/

X Gh D 4 X ;

4

.n/

Y Gh D 0; G D A; B:

(4.101)

Table 4.5 Tensors included in formula (4.99) at different n n I II

4

.n/ X 0Gh

III III  4 Eh  .2134/ .4 ZDh C 4 ZDh / III

III IV V

e III C 4 Y   4  4 III C Eh C .2134/ C4 ZDh C 4 ZDh III

.n/

.n/

.n/

Mh 4Z Dh C III 4 e C III / ZDh  12 .4 

4

eh M 4Z W h  III 4 ZW h  III

Y 0Gh 0 0

4

4

4

4

ZDh 4 e  III / ZDh  12 .4  4

ZDh  III

ZW h 4 ZW h  III 4

ZW h  III

4

III

Y 0 0

190

4 Constitutive Equations .n/

Here the fourth-order tensors 4 X are the same for deformation measures and tensors, they differ only for different n: 4

V

X D .FT ˝ F/.1432/ ;

4

III

X D .OT ˝ O/.1432/ ;

4

I

X D .F1 ˝ F1T /.1432/ ;

1 T e .1342/  O/.1423/ C OT  .. C 4  e .2134/ /.1342/  F/.1423/ /; .F  . C 4  2 II 1 4 e .1342/ O/.1423/ C OT ..  4  e .2134/ /.1342/ F1T /.1423/ /: X D .F1 .  4  2 (4.102)

4

IV

XD

Formula (4.101) should be proved as Exercise 4.2.22. Thus, the rates of the energetic deformation tensors and measures are independent of the vorticity tensor; they are linear tensor functions only of the deformation rate tensor: .n/ C G

.n/

D 4 X   D;

G D A; B:

(4.103)

If the tensors D and W are expressed in terms of L (see Exercise 2.4.12): D D    L;

e D 1 .III  II /;  2

e   L; WD

(4.104)

.n/

then relation (4.98) connects C hG to L: .n/ C hG

.n/

D 4 B Gh   L; G D A; B; C; DI

h D f; Ol; CR; U; V; J; S g;

n D I; : : : ; V;

(4.105)

where 4

.n/

.n/

.n/

e B Gh D 4 X Gh    C 4 Y Gh   :

(4.106)

Exercises for 4.2 4.2.1. Let the symmetric Cauchy stress tensor T in an actual configuration have the following components (3.52): T D T ij ri ˝ rj D Tij ri ˝ rj . Using the relations .n/

between T and T (see Table 4.1) and Eq. (2.36), show that the energetic tensors have the same components but with respect to other tensor bases: I

ı

ı

T D Tij ri ˝ rj ; IV

TD

V

ı

II

ı

T D T ij ri ˝ rj ;

1 ij ı ı T .ri ˝b rj Cb ri ˝ rj /; 2

TD III

1 ı ı Tij .ri ˝b rj Cb ri ˝ rj /; 2

ri ˝b ri ˝b T D Tijb rj D T ijb rj ;

4.2 Energetic and Quasienergetic Couples of Tensors

191

where b r i D OT  r i ;

b ri D OT  ri : .n/

4.2.2. Show that the quasienergetic stress tensors S have the same components Tij , T ij as the Cauchy tensor T D T ij ri ˝ rj D Tij ri ˝ rj ; but with respect to other tensor bases: I

III

S D Tij rM i ˝ rM j ; IV

SD

II

S D T D Tij ri ˝ rj ;

SD

1 ij T .ri ˝ rM j C rM i ˝ rj /; 2

where

ı

1 Tij .ri ˝ rM j C rM i ˝ rj /; 2

V

S D T ij rM i ˝ rM j ; ı

rM i D O  ri ;

rM i D O  ri : .n/

4.2.3. Using the fact that the components S .p/ are determined with respect to the ˛ˇ .n/ı

ı

– with respect to the basis pi , and taking formula (2.159a) into basis pi , and T .p/ ˛ˇ account, show that all the energetic and quasienergetic stress tensors are connected with the help of the rotation tensor O accompanying deformation as follows: .n/

.n/

S D O  T  OT ;

n D I, II, III, IV, V : .n/

.n/

4.2.4. Show that all the energetic tensors C and G (n D I, II, IV, V) have the ı following resolutions for the eigenbasis p˛ : .n/

CD

3 1 X nIII ı ı .  1/p˛ ˝ p˛ ; n  III ˛D1 ˛ .n/

.n/

GD

3 1 X nIII ı ı  p˛ ˝ p˛ ; n  III ˛D1 ˛

.n/

and all the quasienergetic tensors A and g (n D I, II, IV, V) have the following resolutions for the basis p˛ : .n/

AD

3 1 X nIII .  1/p˛ ˝ p˛ ; n  III ˛D1 ˛

.n/

g D

3 1 X nIII  p˛ ˝ p˛ : n  III ˛D1 ˛

4.2.5. Using formulae (2.36), (2.74), (2.75), (2.312), (4.47) and the results of .n/

Exercise 2.1.7, show that components of the energetic deformation measures G are the metric matrices gij and g ij in different tensor bases: I 1 1 ı ı G D  G1 D  g ij ri ˝ rj ; 2 2

II 1 ı ı G D  g ij .ri ˝b rj Cb ri ˝ rj /; 2

192

4 Constitutive Equations IV

GD

1 ı ı rj Cb r i ˝ rj /; gij .ri ˝b 2

1 1 ı ı G D gij ri ˝ rj : 2 2

V

GD

.n/

4.2.6. Show that components of the quasienergetic deformation measures g (n D I; II; IV; V) are the metric matrices gij , gij in different tensor bases: 1 1 1 I II g D  g1 D  g ij rM i ˝ rM j ; g D  g ij .Mri ˝ rj C ri ˝ rM j /; 2 2 2 1 1 1 IV V g D gij .Mri ˝ rj C ri ˝ rM j /; g D g D gij rM i ˝ rM j : 2 2 2 I

V

4.2.7. Show that the energetic tensors A and A have components "ij and "ij with respect to the following bases: I

V

A D A D "ij rM i ˝ rM j ; A D J D "ij rM i ˝ rM j : .n/

.n/

4.2.8. Using the results of Exercise 2.3.1, show that the tensors G and g are connected by the relation .n/

.n/

g D O  G  OT ;

n D I, II, IV, VI .n/

.n/

and, using formulae (4.72) and (4.43), show that the tensors C and A are connected by the similar relation .n/

.n/

A D O  C  OT ; .n/

n D I, II, IV, V:

.n/

.n/

.n/

Using these relations between A and C, and also between g and G, show that the principal invariants of the energetic and quasienergetic deformation tensors and measures coincide: .n/

.n/

I˛ . A / D I˛ . C/;

.n/

.n/

I˛ . g / D I˛ . G/;

˛ D 1; 2; 3:

4.2.9. Show that the bases rM i , rM i andb rj ,b rj , introduced in Exercises 4.2.1 and 4.2.2, have the following properties: ı

ı

ı

rM i D g ij rM j ; b r i D g ijb r i b rj D g ij ; b ri b rj D gij : rj ; rM i Mrj D g ij ; rM i Mrj D g ij ; b 4.2.10. Prove Theorem 4.4. .n/

4.2.11. Prove that the energetic stress tensors T can always be expressed in the form (4.36).

4.2 Energetic and Quasienergetic Couples of Tensors

193 .n/

.n/

4.2.12. Using the result of Exercise 2.5.6, show that the tensors C and C U , and .n/

.n/

also G and G U are commutative in pairs: .n/ .n/

.n/

.n/

.n/ .n/ U

C  C U D C U  C;

GG

.n/

.n/

D G U  G;

n D I; II; IV; V: .n/

.n/

.n/

Using the result of Exercise 2.5.7, show that the tensors A and A V , and also g .n/

and g V are commutative in pairs: .n/ .n/ V

AA

.n/

.n/

D AV  A;

.n/ .n/V

g  g

.n/

.n/

D gV  g:

4.2.13. Using the results of Exercises 2.1.1, 2.2.1, and 2.3.2, show that in the .n/ .n/ .n/

.n/

problem on tension of a beam (see Example 2.1) the tensors C , G, A and g have the forms .n/

.n/

CDAD .n/

3 3 .n/ X 1 X nIII .k˛  1/Ne˛ ˝ eN ˛  CN ˛ eN ˛ ˝ eN ˛ ; n  III ˛D1 ˛D1

.n/

GD g D

III

III

CDAD

3 X

3 1 X nIII k eN ˛ ˝ eN ˛ ; n  III ˛D1 ˛ III

III

lg k˛ eN ˛ ˝ eN ˛ ; G D g D

˛D1

n D I; II; IV; VI 3 X

.1 C lg k˛ /Ne˛ ˝ eN ˛ :

˛D1

Using formulae (4.80)–(4.82), show that for this problem the principal invariants .n/

.n/

ı

I˛ . C /, I˛ . G/ and change in density J D = have the forms ! 3 3 X X .n/ III 1 1  k˛nIII  3 ; I1 .C/ D lg k˛ ; J D ı D ; I1 . C / D n  III ˛D1 k1 k2 k3  ˛D1 3 .n/ 1 X nIII 1 k˛ ; I3 . G/ D .k1 k2 k3 /nIII : I1 . G/ D n  III ˛D1 .n  III/3 .n/

Using the results of Exercise 2.3.2, show that the tensors of energetic and quasienergetic equivalence, determined by (4.38) and (4.68), for the problem on tension of a beam have the form 4

.n/

.n/

E D QD 4

3 X

.n/

E ˛ˇ eN ˛ ˝ eN ˇ ˝ eN ˇ ˝ eN ˛ :

˛;ˇ D1 I

E ˛ˇ

II III IV V 2k˛ kˇ 1 2 D ; E ˛ˇ D ; E ˛ˇ D 1; E ˛ˇ D ; E ˛ˇ D k˛ kˇ : k˛ kˇ k˛ C kˇ k˛ C kˇ

194

4 Constitutive Equations Table 4.6 For Exercise 4.2.14 .n/

c ˛G

nDI

n D II

n D IV

nDV

.n/

a =2

1  U2 =

U0  1

0

a=2

U1 =

U1

a=2

1  U0 =

U2  1

a2 =2

c 0A

.n/

c 1A

.n/

2

c 2A

0

.n/

.1 C a /=2

U2 =

U0

1=2

c 1B

a=2

U1 =

U1

a=2

.n/

1=2

U0 =

U2

.1 C a2 /=2

0

1  U0 =

U2  1

a2 =2

c 0B

.n/

c 2B

.n/

c 0C

.n/

2

c 1C

a=2

U1 =

U1

a=2

.n/

c 2C

a =2

1  U2 =

U0  1

0

.n/

1=2

U0 =

U2

.1 C a2 /=2

c 1D

a=2

U1 =

U1

a=2

.n/

.1 C a2 /=2

U2 =

U0

1=2

c 0D

.n/

c 2D

2

4.2.14. Using the results of Exercises 2.2.2 and 2.3.3, show that for the problem on .n/ .n/ .n/

.n/

simple shear (see Example 2.2) the tensors C , G, A and g have the form .n/

.n/

.n/

.n/

C G D c 0G eN 21 C c 1G O3 C c 2G eN 22 ;

G D A; B; C; D;

.n/

where the coefficients c ˛G are determined by Table 4.6. Here  D U0 U2  U12 . 4.2.15. Using the results of Exercise 2.2.3, show that for the problem on rotation of .n/ .n/ .n/

.n/

a beam with tension (see Example 2.3) the tensors C , G, A and g have the forms .n/

CD

.n/

AD

3 3 .n/ 1 X nIII 1 X nIII .k˛  1/Ne˛ ˝ eN ˛ ; G D k eN ˛ ˝ eN ˛ ; n  III ˛D1 n  III ˛D1 ˛ 3 1 X nIII .k  1/p˛ ˝ p˛ ; n  III ˛D1 ˛

.n/

g D

n D I; II; IV; VI where p˛ D O0  eN ˛ .

3 1 X nIII k p˛ ˝ p˛ ; n  III ˛D1 ˛

4.2 Energetic and Quasienergetic Couples of Tensors

195

4.2.16. Show that if the tensors F and r ˝ v are diagonal (for example, as in the problem on tension of a beam): FD

3 X

k˛ eN ˛ ˝ eN ˛ ;

3 X

r ˝vDDD

˛D1

D˛ eN ˛ ˝ eN ˛ ;

˛D1 .n/

then expressions (4.102) for 4 X take the forms 4

I

X D .F1 ˝ F1T /.1432/ D

3 X

k˛1 kˇ1 ˛ˇ ;

˛;ˇ D1 4

II

X D .F1 ˝ E/.1432/ D

3 X

k˛1 ˛ˇ ;

˛;ˇ D1 4

III

X D III D

3 X

˛ˇ ;

4

IV

X D .F ˝ E/ T

˛;ˇ D1 4

3 X

D

.1432/

k˛ ˛ˇ ;

˛;ˇ D1 3 X

V

X D .FT ˝ F/.1432/ D

k˛ kˇ ˛ˇ ;

˛;ˇ D1

where ˛ˇ  eN ˛ ˝ eN ˇ ˝ eN ˇ ˝ eN ˛ . .n/

4.2.17. Prove that all the quasienergetic stress tensors S can be expressed in the form (4.66). 4.2.18. Show that for n D III; IV; V formulae (4.38) and (4.41) hold. 4.2.19. Prove formula (4.68) for different values of n. ı

4.2.20. Introducing components of the eigenvectors p˛ and p˛ with respect to the basis e ri : e kie rk ; pi D Q .n/

ı

ı e kie rk ; pi D Q

.n/

show that tensors 4 E (4.38) and 4 Q (4.68) in this basis have the components 4 .n/

.n/

.n/

e ijkle E DE rj ˝e rk ˝e rl ; ri ˝e

e ijkl D E

3 X ˛;ˇ D1

ı

ı

4

.n/

.n/

e ijkle QDQ rj ˝e rk ˝e rl ; ri ˝e

.n/

e i˛ Q ej Q ek el e ijkl D E˛ˇ Q ˇ ˇ Q˛ ; Q

3 X ˛;ˇ D1

e i˛ Q ej Q ek el E˛ˇ Q ˇ ˇ Q˛ :

196

4 Constitutive Equations .n/

.n/

4.2.21. Show that for n D I and V the tensors 4 Q and 4 E can be represented in the explicit form in terms of F and V: 4

I

E D .F1T ˝ F1 /.1432/ ;

4

I

Q D .V1 ˝ V1 /.1432/ ;

V

4

E D .F ˝ FT /.1432/ ;

4

Q D .V ˝ V/.1432/ :

V

4.2.22. Using Eqs. (2.283), (2.284), (2.311), and (2.321), prove formulae (4.98), (4.101) and (4.102) for G D A and B. 4.2.23. Using formulae (4.42), (4.43), (4.65), (4.71), and (2.143), show that the energetic and quasienergetic deformation tensors and measures can be represented in terms of the corresponding power of the deformation gradient: .n/

1 .FT  F/.nIII/=2 ; n  III

.n/

1 ..FT  F/.nIII/=2  E/; n  III

.n/

1 .F  FT /.nIII/=2 ; n  III

.n/

1 ..F  FT /.nIII/=2  E/; n  III

GD g D

CD AD

n D I; II; IV; V: 4.2.24. Prove that symmetric tensors commuting with each other are coaxial and vice versa (see Sect. 4.2.21).

4.3 The Principal Thermodynamic Identity 4.3.1 Different Forms of the Principal Thermodynamic Identity Let us return to the question on finding constitutive equations in continuum mechanics. Write out the thermodynamic laws (3.124) and (3.166):  .de=dt/ D T  .r ˝ v/T C qm  r  q;

(4.107)

 .d=dt/ D qm  r  q C w :

(4.108)

Eliminating r  q among these equations, we get 

de d   dt dt



 w.i / C w D 0:

(4.109)

4.3 The Principal Thermodynamic Identity

197

On substituting the expression (4.92) for the stress power w.i / into (4.109), we obtain .n/

de d CG d OT d .n/ e-form W     TG    SG   C w D 0; dt dt dt dt n D I; : : : ; VI

(4.110)

G D A; B; C; D:

This relationship is called the principal thermodynamic identity (PTI) written in the generalized e-form. Substitution of expressions (4.87)–(4.90) into (4.109) in place of w.i / gives the principal thermodynamic identity in the forms Aen , Bne , Cne and Dne : .n/

Aen

W

d .n/ d C de    T  C w D 0;  dt dt dt

W

de d .n/ d G     T  C w D 0; dt dt dt

(4.111)

.n/

Bne

(4.112)

.n/

Cne

W

de d .n/ d A ı d OT     S  S C w D 0; dt dt dt dt

W

de d ı d g .n/ d OT    S  S  C w D 0: dt dt dt dt

(4.113)

.n/

Dne

(4.114)

The remarkable property of the principal thermodynamic identity consists in the fact that the identity does not include the gradients with respect to coordinates (i.e. values of the type r  q), and, therefore, it may be considered as some relation .n/

.n/

.n/

connecting the changes in three main values: e,  and C (or e,  and G; or e, , A .n/

and OT ; or e, , g and OT ) at a local point of a continuum. The principal thermodynamic identity is a basis for derivation of different constitutive equations of continua. Using the expression (4.8) for the stress power w.i / , from (4.109) we get the relation dF ı de ı d ı  (4.115)   P C w D 0; dt dt dt ı

which is called the principal thermodynamic identity in the e-form in the material ı description and which contains the density  and the Piola–Kirchhoff tensor P deı

termined in K.

198

4 Constitutive Equations

4.3.2 The Clausius–Duhem Inequality Substituting the expression (3.167) for the dissipation function w into the principal thermodynamic identity (4.110) and taking into account that q  is always nonnegative due to the Planck inequality (3.158a), we obtain the relation .n/

d .n/ 1 de d CG d OT    TG    SG   C q  r  6 0;  dt dt dt dt 

(4.116)

which is called the Clausius–Duhem inequality in the e-form in the spatial description. In a similar way, from (4.115) and (3.175) we obtain the Clausius–Duhem inequality in the material description: ı de



dt

ı

 

d dF 1ı ı P C q  r  6 0: dt dt 

(4.117)

4.3.3 The Helmholtz Free Energy Introduce a new thermodynamic function follows:

called the Helmholtz free energy as

D e  :

(4.118)

It is clear that d dt

D

de d d   : dt dt dt

(4.119)

On determining the derivative de=dt from (4.119) and substituting its expression into (4.109), we obtain 

d d C   w.i / C w D 0: dt dt

(4.120)

Substitution of (4.119) into (4.110) yields .n/

d CG d OT d d .n/  C   TG    SG   C w D 0: dt dt dt dt

(4.120a)

This relation is called the principal thermodynamic identity in the generalized -form.

4.3 The Principal Thermodynamic Identity

199

Taking different values of index G D A; B; C or D, from (4.120) we obtain the following four -forms of the principal thermodynamic identity: .n/

d .n/ d C d C   T  C w D 0;  dt dt dt

(4.121)

.n/

d .n/ d G d C   T  C w D 0;  dt dt dt

(4.122)

.n/

d d .n/ d A ı d OT  C   S  S C w D 0; dt dt dt dt

(4.123)

.n/

ı d OT d d .n/ d g  C   S  S C w D 0I dt dt dt dt

(4.124)

where (4.121) is called the principal thermodynamic identity in the form An , (4.122) – in the form Bn , (4.123) – in the form Cn , and (4.124) – in the form Dn . In a similar way, from (4.115) and (4.119) we obtain the principal thermodyı

namic identity in the

-form in the material description: ıd



dt

ı

C 

d dF ı P C w D 0: dt dt

(4.125)

4.3.4 The Gibbs Free Energy Introduce one more thermodynamic function .n/

AD



1 .n/ .n/ T   C; 

n D I; : : : ; V;

(4.126)

called the Gibbs free energy in the form An , and determine its derivative with respect to time: 0 1 .n/

.n/

d d B T C .n/ T d .n/ d .n/ AD  @ A C    C: dt dt dt   dt

(4.127)

On determining the derivative d =dt from this equation, and then substituting its expression into (4.121), we obtain the principal thermodynamic identity in the form An : 0 1 .n/

.n/

.n/ d d BTC dA C  C  C   @ A C w D 0;  dt dt dt  .n/

n D I; : : : ; V: .n/

This identity connects changes in the functions  A ,  and T =.

(4.128)

200

4 Constitutive Equations

For the remaining forms Bn ; Cn and Dn , the Gibbs free energy is introduced as follows: in the form Bn W in the form Cn W in the form Dn W

.n/

B D

.n/

C D

.n/

DD



1 .n/ .n/ T   G; 

(4.129)



1 .n/ .n/ S   A; 

(4.130)



1 .n/ .n/ S  g: 

(4.131)

Introduce also the generalized Gibbs free energy

.n/

G

8 .n/ ˆ ˆ ˆ  A; ˆ ˆ ˆ ˆ .n/ ˆ ˆ < ; B D .n/ ˆ ˆ ˆ  C; ˆ ˆ ˆ ˆ ˆ ˆ :.n/  D;

if G D A; if G D B;

(4.132)

if G D C; if G D D:

Then, by simple manipulations similar to (4.127), from (4.122)–(4.124) we get the principal thermodynamic identity in the generalized -form: .n/

.n/

.n/ dG d TG d OT d  C  C CG   . /  SG   C w D 0; dt dt dt  dt

n D I; : : : ; VI

(4.133)

G D A; B; C; D:

For different values of index G, the principal thermodynamic identity takes the forms An , Bn , Cn and Dn . The Gibbs free energy can also be introduced formally with the help of the tensors P and F: 1  D  ı P   F; (4.134)  that is called the Gibbs free energy in the material description; and for the funcı

tion  we have the principal thermodynamic identity in the -form in the material description: dP ı d ı d ı  (4.135) C  CF C w D 0: dt dt dt

4.3 The Principal Thermodynamic Identity

201

4.3.5 Enthalpy Finally, introduce one more thermodynamic function .n/

i

G

De

.n/ 1 .n/ T G   CG; 

n D I; : : : ; VI

G D A; B; C; D;

(4.136) .n/

called the enthalpy. It follows from (4.126) and (4.129)–(4.131) that i

G

is con-

.n/

nected to  G by the relation .n/

.n/

i

G

D  G C :

(4.137)

Taking different values of index G in (4.136), we get the following four different forms of the enthalpy: .n/

i

A

.n/ iC

1 .n/ .n/ T   C;  1 .n/ .n/ D e  S   A; 

.n/

De

i

B

.n/ iD

1 .n/ .n/ T   G;  1 .n/ .n/ De S g: 

De

(4.138)

From (4.137) we find the rate .n/

.n/

diG d G d d D C C : dt dt dt dt

(4.139)

On substituting this expression into the principal thermodynamic identity in the form (4.133), we get .n/

0

.n/

1

diG d B T G C SG d OT d .n/ w  C CG   @  C D 0; A dt dt dt   dt  n D I; : : : ; V;

(4.140)

G D A; B; C; DI

that is the principal thermodynamic identity in the generalized i -form. Taking different values of index G, we obtain the principal thermodynamic identity in the forms Ain , Bni , Cni and Dni . Similarly to (4.136), introduce the enthalpy in the material description: ı

i D e  .1=/P   F D  C ;

(4.141)

202

4 Constitutive Equations

with the help of which from (4.135) and (4.141) we obtain the principal thermodynamic identity in the i -form in the material description: ı

di d F dP w  C ı  C ı D 0: dt dt dt  

(4.142)

4.3.6 Universal Form of the Principal Thermodynamic Identity We can write the following universal expression of the principal thermodynamic identity for all the four forms (4.110), (4.120), (4.133) and (4.140): d d d d OT w  C N KM C D 0: dt dt dt dt 

(4.143)

Here we have introduced scalar functions:  called the thermodynamic potential, and ; and also symmetric tensors N, K and M. Table 4.7 shows their values for different forms of the principal thermodynamic identity. The relationship (4.143) is called the universal form of the principal thermodynamic identity.

Table 4.7 Different variants of values of , , , N, K and M in the universal form of PTI (4.143) in the spatial description Generalized Forms N form of PTI of PTI   N K M 1

-form (4.120)

An , Bn , Cn , Dn





2

e-form (4.110)

Aen , Bne , Cne , Dne

e





3

e-form (4.110)

Aen , Bne , Cne , Dne



1=

e

4

-form (4.120)

An , Bn , Cn , Dn



1=

5

-form (4.133)

An , Bn ,   Cn , Dn

6

-form (4.133)

An , Bn ,   Cn , Dn

7

i -form (4.140)

Ain , Bni , Cni , Dni

8

i -form (4.140)

Ain , Bni , Cni , Dni









.n/

.n/

.n/

T G =

CG

.n/

.n/

T G =

CG

.n/

.n/

T G =

CG

.n/

.n/

T G =

CG

.n/

CG

G







1=

G

 C G =





CG

.n/

.n/

i 

G

1=

.n/

.n/

.n/

.n/

i

G

 C G =

SG = SG = SG = SG =

.n/

T G =

SG =

.n/

T G =

SG =

.n/

T G =

SG =

.n/

T G =

SG =

4.3 The Principal Thermodynamic Identity

203

Table 4.8 Different variants of values of , , , N, K and M in the universal form of PTI (4.143) in the material description N Form of PTI   N K M 1

ı

2

e-form (4.115)

3

e-form (4.115)

4 5 6 7 8

ı

ı

-form (4.125)

ı

-form (4.135) ı

-form (4.135) ı

i -form (4.142)

ı



-form (4.125)

ı

i -form (4.142)



ı

P= ı

e





P=



1=

e

P=

   i 

ı

1= 

P= 

1=  1=

ı

  i

F F= F F=

F

0

F

0

F

0

F

0 ı

P= ı

P= ı

P= ı

P=

0 0 0 0

It should be noted that for each of the four forms of PTI (4.110), (4.120), (4.133), and (4.140) there are two variants of a choice of the thermodynamic potential. For example, for the e-form (4.110) we may put either  D e,  D  and D , or  D ,  D e and D 1=. Both the variants are given in Table 4.7. The same holds for other forms of PTI as well. Thus, for the four forms of PTI, there are eight different variants of a choice of the thermodynamic potential. Notice that the choice of temperature  as a thermodynamic potential yields D 1=. But the thermodynamic laws do not exclude the situation when  ! 0; in this case D 1= ! C1, that should be taken into consideration when temperature  is chosen as the potential . All the four forms of PTI in the material description (4.115), (4.125), (4.135), and (4.142) can also be written in the single form (4.143). Values of the functions , , , and of the tensors N, K and M for these forms are given in Table 4.8. Notice that in the material description the tensor M is zero-tensor, and the tensors N and K are not symmetric. For each of the four forms of PTI in the material description, there are also two variants of a choice of the thermodynamic potential; thus, we have eight different sets of the functions , , , N, K and M in the universal form of PTI (4.143).

4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives .n/

.n/

.n/

.n/

Theorem 4.12. Let the tensors T and C (or the tensors S and A ) be coaxial in pairs for each n; then the principal thermodynamic identity (4.143) takes the form  P T C w D 0;  h C h  N   Kh  M   O 

h D f ; U; V; J; S g: (4.144)

204

4 Constitutive Equations

H Since  and  are scalar functions, their co-rotational derivatives coincide with the total derivative with respect to time: P D  h ; P D h : The second and third summands in (4.143) for the - and e-forms of PTI, according to the notation introduced in Table 4.7 and formula (4.92), give the generalized form of the stress power .n/

.n/

PT DNK P T: P CMO w.i / D T G   G G C SG   O

(4.145)

Then for w.i / Theorem 4.11 is valid; and, due to this theorem, we can really replace P by the co-rotational derivative Kh , if the tensors N and K, i.e. the total derivative K .n/

.n/

.n/

.n/

the tensors T and C (or the tensors S and A ), are coaxial. For the -form of PTI, the second and third summands in (4.143) do not coincide with w.i / , but they can be written with the help of Table 4.7: .n/

.n/

P T: P T D  C G   . T G =/ C SG   O P CMO NK .n/

.n/

.n/

(4.146)

.n/

If the tensors T and C (or the tensors S and A ) are coaxial in pairs for each .n/

.n/

n, then tensors C G and T G = are coaxial as well. Then, following the proof of Theorem 4.11, we get P D N   Kh ; NK (4.147) as was to be shown. N

Exercises for 4.3 .n/

.n/

4.3.1. Using formulae (4.43) and (4.70), show that the Gibbs free energies  A ,  B , .n/

.n/

 C and  D , determined by formulae (4.126), (4.129)–(4.131), are not coincident; and they are connected by the following relations: .n/

.n/

AD BC

.n/ 1 I1 . T /; n  III

.n/

.n/

C D DC

.n/ 1 I1 . S /: n  III .n/

.n/

4.3.2. Using the results of Exercises 4.2.3 and 4.2.8, show that  B and  D , and .n/

.n/

.n/

.n/

also  A and  C coincide in pairs:  A D  C ;

.n/

.n/

 B D  D ; n D I, II, IV, V: ı

4.3.3. Using the definition (3.56) of the tensor P, show that the Gibbs free energy  in the material description (4.134) can be written in the form D

 .1=/ I1 .T/:

4.4 Principles of Determinism, Equipresence and Local Action

205

4.4 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action 4.4.1 Active and Reactive Variables Let us consider now the principal thermodynamic identity in the form An (4.121). .n/

.n/

Split the functions , , , T =, C and w = included in (4.121) into the two groups: functions included in (4.121) through their derivatives, except , are called the reactive variables: .n/

R D f; C g;

(4.148)

the remaining functions are called the active variables: .n/

ƒ D f ; ; T =; w =g:

(4.149)

This separation has the following meaning: with the help of active variables we describe some generalized internal forces caused by external actions on a material point M of a continuum considered; and with the help of reactive variables we describe the response of a continuum to these external actions. The choice of reactive and active variables is not unique: considering the principal thermodynamic identity in the form Cn (4.123), as reactive variables we take the functions .n/

R D f; A ; Og;

(4.150)

and as active variables – the functions .n/

ı

ƒ D f ; ; S =; S=; w =g:

(4.151)

Similarly, for the principal thermodynamic identity in the form Bn (4.122) the variables R and ƒ are chosen as follows: .n/

.n/

ƒ D f ; ; T =; w =g;

R D f; Gg;

(4.152)

and for PTI in the form Dn (4.124): .n/

R D f; g ; Og;

.n/

ı

ƒ D f ; ; S =; S=; w =g:

(4.153)

We can write out the active and reactive variables for the universal form of PTI (4.136) in the spatial description as follows: R D f; K; Og;

ƒ D f; ; N; M; w =g;

(4.154)

where all the generalized thermodynamic functions are determined by Tables 4.7 and 4.8.

206

4 Constitutive Equations

4.4.2 The Principle of Thermodynamically Consistent Determinism Axiom 11 (The principle of thermodynamically consistent determinism). For every continuum, active variables ƒ are completely defined by reactive variables R; in other words, there is a mapping of the generalized space of reactive variables XR into the space of active variables Xƒ : fM W

XR ! Xƒ ;

(4.155)

which is called the operator relation (or the operator) and written in the form ƒ D fM.R/;

R 2 XR ; ƒ 2 Xƒ I

(4.156)

the relation is consistent with thermodynamics, i.e. satisfies identically the principal thermodynamic identity (4.143). A particular case of the operator (4.155) is the mapping determined by the ordinary function ƒ.t/ D f .R.t//; (4.157) where the active variables ƒ.t/ depend on the reactive ones R.t/ considered at the same times. Other examples of the operators will be given below. If PTI is considered in the form An (4.121), then the operator (4.156) with use of (4.148) and (4.149) takes the form 8 .n/ ˆ < D M ..n/ C; /;  D M . C ; /; (4.158) .n/ .n/ .n/ ˆ :T D F M . C ; /; w D wM  . C ; /: Here we have taken into consideration that density  can always be represented as a .n/

function of C (see Sect. 4.2.19). Relations (4.158) are constitutive equations of a continuum, they are also called the model An of a continuum. Notice that, according to Axiom 11, the operators (4.156) are not entirely arbitrary, because after their substitution into PTI (4.121) we get the equation .n/

.n/ d .n/ M / d  F M .C; /   d C C w . C ; / D 0;  M . C; / C . M C; dt dt dt

which should be satisfied identically for all processes of changing the arguments .n/

C . /, . /, 0 6 6 t. If PTI is considered in the form Bn (4.122), then the constitutive equations (4.158) with use of (4.152) are written as follows:

4.4 Principles of Determinism, Equipresence and Local Action

8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

207

.n/

D M . G; /; .n/

 D M . G; /;

.n/ .n/ ˆ ˆ M . G; /; ˆ T DF ˆ ˆ ˆ .n/ ˆ :  w D wM  . G; /; n D I; II; IV; V;

(4.159)

.n/

because the density  can be expressed as a function of G (see Sect. 4.2.19). The relations (4.159) are called the model Bn . Similarly, using PTI in the form Cn and the relation (4.156), we get the constitutive equations 8 .n/ ˆ ˆ D M . A; O; /; ˆ ˆ ˆ ˆ .n/ ˆ ˆ ˆ  D . M A ; O; /; ˆ < .n/ .n/ (4.160) M . A ; O; /; S DF ˆ ˆ ˆ .n/ ˆ ˆ ˆ ˆw D wM  . A ; O; /; ˆ ˆ ˆ .n/ :ı M 0 . A ; O; /; n D I; : : : ; V; SDF which are called the model Cn of a continuum. According to PTI in the form Dn , from (4.153) and (4.156) we obtain the constitutive equations 8 .n/ ˆ ˆ D M . g ; O; /; ˆ ˆ ˆ .n/ ˆ ˆ ˆ  D . M g ; O; /; ˆ < .n/ (4.161) M ..n/ S DF g ; O; /; ˆ ˆ ˆ .n/ ˆ  ˆ w D wM  . g ; O; /; ˆ ˆ ˆ ˆ :ı M 0 ..n/ SDF g ; O; /; which are called the model Dn of a continuum. Here we have taken into account that density  can be represented as a function .n/

.n/

of A or of g (see Sect. 4.2.19). M in all the relations (4.156)–(4.159) are Notice that the tensor operators F different. If PTI is considered in the universal form (4.143), then Axiom 11 gives the constitutive equations 8 ˆ M K; O/; ˆ ˆ D .; ˆ < D .; M K; O/; ˆN D N.; M K; O/; ˆ ˆ ˆ :   w D wM .; K; O/;

(4.162) M M D M.; K; O/:

208

4 Constitutive Equations

Considering different forms of PTI, by Tables 4.7 and 4.8, we obtain a large collection of models of continuum mechanics: 20 models for each of the -, e-, and -forms (and each of the forms has two representations), and also four forms: ı

ı

ı

ı

, e,  and  (each of the forms has two representations as well). We finally get 20  .4  2 C 4  2/ D 320 (!) models. It should be noted that the models An , Bn , Cn and Dn are most widely used in continuum mechanics.

4.4.3 The Principle of Equipresence Analyzing the constitutive equations (4.158)–(4.162), one can notice that for each model all the operator relations include the same reactive variables. This fact is expressed by the following axiom. Axiom 12 (The principle of equipresence). Constitutive equations must be operators of the same reactive variables. In particular, it follows from the principle that since the models An do not contain the rotation tensor O, the corresponding forms of constitutive equations (4.158) cannot include the tensor O as an argument. Another considerable corollary of the principle of equipresence is the following assertion: since the temperature gradient r  does not appear in PTI (4.143), the gradient r  cannot be involved in the set of active and reactive variables ƒ and R in constitutive equations (4.156). This conclusion will be used in Sect. 4.11.1.

4.4.4 The Principle of Local Action Notice that the general definition of operator (4.156) admits the situation when the process of changing the active variables ƒ.X i ; t/ at a material point X i depends on the process of changing the reactive variables R.X 0i ; / at other material points ı

X 0i ¤ X i , where X i ; X 0i 2 V . To prohibit such dependences, we introduce one more basic principle. Axiom 13 (The principle of local action). For any continuum, operators of constitutive equations in each of the forms (4.156) are such that active variables at every ı

material point X i 2 V depend only on reactive variables at the same point X i 8t > 0. Remark 1. The principle of local action allows us to construct constitutive equations describing adequately the behavior of most real continuous media. Nevertheless, the principle is not an universal physical law, because there are continua, which should be described not by local constitutive equations. t u

4.5 Definition of Ideal Continua

209

Continua satisfying the principle of local action are sometimes called simple. Remark 2. Notice that the principle of local action admits the dependence of constitutive equations (4.156) at a material point M on the M-point coordinate X i . This dependence has the explicit form ƒ D fM.R; X i /;

(4.163)

or, for example, for the models An : 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

.n/

D M . C ; ; X i /; .n/

 D . M C ; ; X i /;

.n/ .n/ ˆ ˆ M . C ; ; X i /; ˆ T DF ˆ ˆ ˆ .n/ ˆ :  w D wM  . C ; ; X i /:

(4.164)

Such dependence on X i is physically justified and can be caused either by the influence of nonmechanical factors (for example, electromagnetic field etc.): in this case the dependence on X i is continuous; or by nonhomogeneity of a continuum (such continua are, for example, composite materials, multiphase continua etc.): in this case the dependence (4.163) is a discontinuous (usually piecewise continuous) function of X i . t u A continuum, whose constitutive equations (4.156) are the same for all maı

terial points X i 2 V , is called homogeneous; otherwise, a continuum is called inhomogeneous.

4.5 Definition of Ideal Continua 4.5.1 Classification of Types of Continua Let us consider now the most widely used forms of the operators (4.156). Depending on the operator form, we distinguish the following main types of continua:    

ideal continua viscoelastic continua of the differential type viscoelastic continua of the integral type plastic continua

In Sects. 4.5–4.11 and Chap. 6 we will consider the most widely used type of continuous media, namely ideal continua. Chaps. 7 and 8 deal with viscoelastic continua

210

4 Constitutive Equations

of the differential and integral types, respectively; and finally, Chap. 9 investigates plastic continua. All the mentioned types of continua will be considered for the case of large deformations.

4.5.2 General Form of Constitutive Equations for Ideal Continua Definition 4.2. A continuum is called ideal, if its constitutive equations (4.156) are ordinary functions of active variables, i.e. the relations (4.157), where values of ƒ.t/ and R.t/ are determined at the same times t, hold. The model An of an ideal continuum is said to be considered, if the model An of a continuum has been chosen and the corresponding operator constitutive equations (4.158) are functions of the indicated arguments; in particular, the free energy .t/ D

.n/

. C .t/; .t//

(4.165)

.n/

is a scalar function of one tensor argument C and one scalar argument . Below, if the symbol M of operator is absent, then a relation of the form (4.165) will be considered as a function of current values of arguments at time t: .t/ D .n/

. C .t/; .t//, and the symbol t will be omitted. All the functions (4.157) are assumed to be continuously differentiable, then we can determine, for example, the total derivative of the free energy d =dt. Using the rule of differentiation of a scalar function with respect to tensor argument (see [12]), for models An of an ideal continuum we get .n/

dC @ @ d d  D C ; .n/ dt dt @ dt @C

(4.166)

.n/

where @ =@ C is the symmetric second-order tensor. Since the constitutive equations (4.156) must be consistent with thermodynamics, i.e. should satisfy PTI (4.121), so, substituting (4.166) into PTI (4.121) and collecting like terms, we obtain 0

.n/

1

.n/ TC B@ @ .n/  A   d C C  @C



 @ w C  d C dt D 0: @ 

(4.167)

4.5 Definition of Ideal Continua

211

.n/

The differentials d C , d and dt are independent, therefore the identity (4.167) holds if and only if coefficients of the differentials vanish, i.e. the following relations are valid: 8.n/ .n/ .n/ ˆ ˆ < T D .@ =@ C /  F . C ; /; (4.168)  D @ =@; ˆ ˆ :  w D 0: In other words, relations (4.168) are equivalent to the identity (4.167). The relations (4.168) are the constitutive equations for the model An of ideal continua, and they are the particular case of relations (4.156). Analyzing the relations (4.168), we conclude that: (1) Ideal continua are nondissipative, i.e. their dissipation function w is identically zero. (2) The model of an ideal continuum is specified by the only scalar function (4.165), and the remaining constitutive equations are determined by differentiation of this function, according to formulae (4.168). .n/

Remark. Notice that the tensor function F . C ; / in (4.168) is not potential. Since .n/

.n/

the density  is a scalar function of C (see Sect. 4.2.19), the function F . C; / would be potential if there were some scalar function .n/

.n/

0

D

0

.n/

.n/

. C; / such that T D

.n/

F . C ; / D @ 0 =@ C : Tensor function F . C; /, satisfying the relations (4.168), is called quasipotential. t u One can say that this is the model Bn of an ideal continuum, if some model Bn of a continuum has been chosen and the corresponding constitutive equations (4.157) are functions of the indicated arguments, in particular: D

.n/

. G; /:

(4.169)

Determine the derivative of (4.169) with respect to t, substitute the result into .n/

PTI (4.122) and then collect like terms. Since the differentials d G, d and dt are independent, we obtain that the relation (4.122) is equivalent to the following system of equations: 8.n/ .n/ .n/ ˆ ˆ < T D .@ =@ G/  F . G; /; (4.170)  D @ =@; ˆ ˆ :  w D 0; which are called the constitutive equations for the model Bn of an ideal continuum. The models Bn of ideal continua are also nondissipative and have the potential (4.169).

212

4 Constitutive Equations

If we apply the models Cn to an ideal continuum, then all the active variables ƒ.t/ in (4.160) depend on values of the reactive variables R.t/ at the same time; in particular, the free energy is a function in the form D

.n/

. A ; O; /:

(4.171)

Differentiating the function with respect to t, we obtain .n/

d dA @ @ d OT @ d  D C  C : .n/ dt dt @O dt @ dt @A

(4.172)

On substituting this expression into PTI (4.123), we get the identity 0

1

0 1 ı   @ w S C .n/ @ @ SA B@ T  C  d C dt D 0: (4.173)  d O C @ .n/  A d A C  @O  @  @A .n/

.n/

All the differentials d A, d OT , d and dt are independent, therefore the identity is equivalent to the following set of relations 8.n/ .n/ ˆ ˆ S D  .@ =@ A /; ˆ ˆ ˆ ˆ <ı S D  .@ =@O/; ˆ ˆ ˆ  D @ =@; ˆ ˆ ˆ : w D 0;

(4.174)

which are the constitutive equations for the model Cn of ideal continua. Models Cn of ideal continua, as well as models An , are nondissipative and potential; however, ı

for Cn there is an additional relation, connecting the rotation tensor of stresses S to . If we consider the models Dn of an ideal continuum, then all the relations (4.161) are functions of the indicated arguments; in particular, has the form D

.n/

. g ; O; /:

(4.175)

Substituting the function (4.175) into PTI (4.124), due to independence of the dif.n/

ferentials d g , d and dt, we get the following set of relations

4.6 The Principle of Material Symmetry

213

8.n/ .n/ ˆ ˆ S D  .@ =@ g /; ˆ ˆ ˆ <ı S D  .@ =@O/; ˆ ˆ  D @ =@; ˆ ˆ ˆ :  w D 0;

(4.176)

which are called the constitutive equations for the model Dn of ideal continua. Finally, if for an ideal continuum we consider PTI in the universal form (4.143), then all the relations (4.162) are functions of the indicated arguments; in particular, the thermodynamic potential  has the form  D .; K; O/:

(4.177)

Its derivative with respect to t is P D

@ dK @ d OT @ d  C  C : @K dt @O dt @ dt

(4.178)

By Axiom 11, the relations (4.162) should identically satisfy PTI in the form (4.143). Substituting (4.178) into (4.143) and collecting like terms, due to independence of the differentials d K, d O and d, we get the following system of constitutive equations: 8 ˆ ˆ ˆ D @=@; ˆ
4.6 The Principle of Material Symmetry 4.6.1 Different Reference Configurations Let us consider now the principle of material symmetry mentioned in Sect. 4.1. This principle introduces several significant concepts of continuum mechanics: anisotropy (and isotropy) of a continuum, invariants of tensors, and also indifferent tensor functions and functionals. The principle of material symmetry briefly consists in that the constitutive equations of a continuum (4.156) must remain unı

changed under certain transformations of a reference configuration K. Let us give a mathematical formulation of the principle. Constitutive equations (4.156) correspond to the motion of a continuum from ı



K to K. If we choose another reference configuration K, then the motion of the

214

4 Constitutive Equations 

continuum K ! K is described, in general, by other constitutive equations, which can be written in the form 



ƒ D fM .R/; 

(4.180)



where ƒ and R are active and reactive variables (4.148), (4.149) or (4.150), (4.151),  or (4.154) corresponding to the motion K ! K. The operator fM , in general, can 

differ from fM (4.156). The main problem is to find reference configurations K such  that the motion K ! K does not change the operator fM, i.e. in place of (4.180) we have the relation 



ƒ D fM.R/:

(4.181)



Let us find such configurations K.

ı



ı

Positions of the same point M in K and K are described by the radius-vectors x  and x, respectively (Fig. 4.1): ı

ı

x D x.X i /; ı





x D x.X i /;

(4.182)



where x and x are connected by the differentiable transformation ı

ı 

x D x.x i /:

(4.183) ı

For the same point M, we can introduce local bases and metric matrices in K 

and K:

Fig. 4.1 Positions of point M in actual and different reference configurations

4.6 The Principle of Material Symmetry

215 ı

@x ; @X i

ı

ri D ı

ı

ı

ı

ı

 ri

@x ; @X i 



g ij D ri  rj ; ı



D

(4.184)



g ij D ri  rj ; i

ri D g ij rj ;

(4.185)





r D gij rj ;

(4.186) ı



and also the deformation gradients at the transformations K ! K, K ! K and ı



K ! K:



ı

F D ri ˝ ri ;





F D ri ˝ ri ;

ı

H D ri ˝ ri : ı

(4.187)



The tensor H, describing the motion from K to K, connects the deformation 

gradients F and F:



F D F  H;



F D F  H1 :

(4.188)





Thus, each reference configuration K is assigned to its local basis ri and its tensor ı

H. And the transformation from one reference configuration K to another reference 

ı

configuration K is the passage from description of a continuum in the basis ri to  description in the basis ri .

4.6.2 H -indifferent and H -invariant Tensors With the help of the tensor H and relationship (4.188), we can connect any tensor ı

, which has a physical meaning and is defined for the transformation K ! K, and 



a corresponding tensor  defined for the transformation K ! K. Definition 4.3. A tensor  is called H -indifferent, if its components remain 

unchanged under the transformation from one reference configuration K to ı

another K:

ı



ı



i1 :::in .x/ D i1 :::in .x/: ı

(4.189) 



Here i1 :::in and i1 :::in are components of corresponding tensors  and  in ı



ı



configurations K and K with radius-vectors x and x, respectively: ı

ı

ı ı

ı

 



 



.x/ D i1 :::in .x/ri1 ˝ : : : ˝ rin ; .x/ D i1 :::in .x/ri1 ˝ : : : ˝ rin : (4.190)

216

4 Constitutive Equations

Definition 4.4. A tensor  is called H -invariant, if it remains unchanged under the ı



transformation from K to K, i.e.  

ı

.x/ D .x/:

(4.191) ı



Local bases vectors ri and ri remain unchanged under the transformation K !

K, therefore they are H -invariant.

ı

ı



Local basis vectors ri change in going from K to K, therefore they are not H  invariant but they can be resolved for their basis as well as the vectors ri : ı

 ri

ı

ri D ıij rj ;



D ıij rj ;

(4.192) ı

ı

i.e. components ıij of vectors ri remain unchanged under the transformation K ! 

ı

ı

K; hence, the vectors ri are H -indifferent (in a similar way, we can show that ri are H -indifferent vectors). Besides, due to formulae (4.184) and (4.187), we get the relations  ri

and also

ı

ı

D H  ri D ri  HT :

i

ı

(4.193)

ı

r D H1T  ri D ri  H1 :

(4.194)

Substituting (4.193) into (4.190) and taking (4.189) into account, we obtain that any H -indifferent tensor satisfies the relation  



 

ı



ı ı

ı

.x/ D i1 :::in .x/ri1 ˝ : : : ˝ rin D i1 :::in .x/ri1  HT ˝ : : : ˝ rin  HT : (4.195) For a scalar ', H -indifference means that 

'.x/ D '.x/;

(4.196)

for a vector a:  

ı

ı

a.x/ D a.x/  HT D H  a.x/;

(4.197)

and for a second-order tensor:  

ı

.x/ D H  .x/  HT :

(4.198)

Figure 4.2a shows an example of H -invariant vector a, which remains unchanged ı



under the transformation K ! K. Figure 4.2b shows an example of H -indifferent ı





vector b, which moves together with the basis ri : vector b in basis ri has the same ı coordinates as vector b in basis ri .

4.6 The Principle of Material Symmetry

217

a

b

Fig. 4.2 (a) – H -invariant vector a; (b) – H -indifferent vector b

ı



The metric tensor E is the same in all the configurations: K, K and K, i.e. the tensor is H -invariant. Indeed, according to formulae (4.193) and (4.194), we have 





ı

ı

E D ri ˝ ri D H  ri ˝ ri  H1 D H  E  H1 D H  H1 D E:

(4.199)

The deformation gradient F is neither H -invariant nor H -indifferent, because ı



during the passage from K to K the tensor is transformed by the law (4.188), which is different from both (4.191) and (4.198). .n/ .n/ .n/

.n/

Similarly to the tensors C , A, G and g describing the deformation of a local ı

point M under the transformation from K to K, we can define the deformation 

tensors and measures during the passage from K to K, for example: V



C D C D

V 1 T  1  .F  F  E/; G D FT  F: 2 2

.n/ .n/ .n/

(4.200)

.n/

But each of the tensors C , A, G and g as well as F is neither H -indifferent nor V

V

ı



H -invariant. For example, the tensors G and C during the passage from K to K are transformed as follows: V

G D

V 1 1  1 G D FT  F D H1T  FT  F  H1 D H1T  G  H1 ; 2 2 2 V V 1 C D H1T  C  H1 C .H1T  H1  E/: 2

218

4 Constitutive Equations I

Only the right Almansi deformation measure G1 (2.75) and hence G are H indifferent, because I  1 1 1 G D  G1 D  F1  F1T D  H  F1  F1T  HT 2 2 2 I 1 D  H  G1  HT D H  G  HT I 2

(4.201)

I

but C is neither H -indifferent nor H -invariant: I I I 1 1 C D G  E D H  C  HT  .H  HT  E/: 2 2

ı

(4.201a)

Since the actual configuration K remains unchanged under the transformation 

K ! K and the Cauchy stress tensor T is defined in K, so the tensor remains ı



unchanged under the transformation K ! K, i.e. T is H -invariant: 

T D T:

(4.202)

.n/ .n/

However, other stress tensors T , S and P are not H -invariant, because they depend .n/ .n/

not only on T but also on the deformation gradient F. Among the tensors S , T and V

P, only the tensor T is H -indifferent, because, according to (4.188) and (4.202), we have V







V

T D F1  T  F1T D H  F1  T  F1T  HT D H  T  HT I

(4.203) ı

I



but, for example, the tensor T is transformed during the passage from K to K as follows: I







I

T D FT  T  F D H1T  FT  T  F  H1 D H1T  T  H1 :

(4.203a) ı

Remark. The type of transformation (4.200a) and (4.203a) during the passage K ! 

K proves to be useful for continuum mechanics, although this type is neither H ı



indifferent nor H -invariant. Therefore tensors  transforming under K ! K by the type: 

 D H1T    H1 ; will be called H -pseudoindifferent.

(4.203b) t u

4.6 The Principle of Material Symmetry

219

4.6.3 Symmetry Groups of Continua Suppose that there exists a set of transformations of the reference configuration ı



H W K ! K and each of the transformations does not change the constitutive equations (4.156); i.e. if the relation (4.156) is valid, then relation (4.181) holds at these transformations as well. Such transformations of the reference configuration are called H -transformations, and each transformation is put in correspondence with a tensor H. ı  ı All H -transformations K ! K occur without changes in the density : 

ı

 D ;

(4.204)

ı

because the density  is the property of a continuum, and H -transformations, by definition, do not change properties of a continuum. ı



Density  in actual configuration K is independent of transformations K ! K, and hence the density  is both H -indifferent and H -invariant. ı



Theorem 4.13. The set of all H -transformations K ! K of the reference conı

figuration K of a continuum (or the set of H-tensors) with constitutive equations (4.156), for which the relations (4.181) remain valid, constitutes a group called the ı

symmetry group G s of the continuum. The group is sometimes called the group of equivalence. H Show that the set of all H -transformations actually has the properties of a group (see the definition of a group in [12]); i.e. in this set the operation of multiplication, which is superposition of H -transformations, is defined, and the set includes the unit and inverse elements. Indeed, superposition of two H -transformations is also H -transformation with the tensor H2  H1 . This assertion can be proved in the following way. Let there be ı   b with the deformation transformation H1 W K ! K and transformation H2 W K ! K gradients: 

F D F  H1 1 ;



b F D F  H1 2 :

(4.205)

Notice that since the energetic and quasienergetic deformation tensors are uniquely expressed in terms of the deformation gradient F (see Exercise 4.2.23), and the energetic and quasienergetic stress tensors – in terms of T and F by formulae (4.36) and (4.66), so the constitutive equations (4.156) can be written in the universal form ƒ D fM.R/; R D f; Fg; ƒ D f ; ; T=; w =g; for all the models An , Bn , Cn and Dn .

(4.206)

220

4 Constitutive Equations

By the definition of H -transformation, from the constitutive equations (4.206) after H1 -transformation we get the relation (4.181): 





ƒ D fM.R/; R D f  ; F g D f; F  H1 1 g; 

ƒDf







 

(4.207)



;  ; T =; .w / =g D f ; ; T=; w =g;

and from the constitutive equations (4.207) after H2 -transformation we obtain b b D fM.R/; ƒ

(4.208)

where b D fb R ; b Fg;

b D fb; b ƒ ; b T=; b w =g:

(4.209)

Thus, from the relations (4.206) after the transformation H2  H1 we get relations b and ƒ b (4.209), included in the relations, can be represented in (4.208), because R the form b D f; F  .H2  H1 /1 g; R

b D f ; ; T=; w =g: ƒ

(4.210)

Here we have taken into account that all the scalars under H -transformations remain   D b,  D  D b  etc., and the tensor T is H -invariant: unchanged: D  T. TDTDb The tensor H D E is the unit element of the set of H -transformations. Using the expressions (4.206) for reactive and active variables and choosing bDR H1 D H and H2 D H1 , from the relations (4.210) one can easily get that R b D ƒ. Thus, after the transformation H1  H2 D H  H1 we get constitutive and ƒ equations (4.156) with R; ƒ in the form (4.206). Hence, for any H -transformation, the transformation with the tensor H1 is H -transformation as well. N

4.6.4 The Statement of the Principle of Material Symmetry We have shown that if there is a set of H -transformations satisfying (4.181), then the set is a group. However, the question, whether the set always exists, remains open. The following axiom gives a positive answer to the question. Axiom 14 (The principle of material symmetry). For every continuum with the ı

constitutive equations (4.156) in the motion K ! K with an arbitrary reference ı

ı

configuration K, there exists a corresponding symmetry group G s being the group

4.7 Definition of Fluids and Solids

221 ı



of H -transformations of the reference configuration H W K ! K, which do not 

change the constitutive equations, i.e. relation (4.181) holds in any motion K ! K: 



ƒ D fM.R/:

(4.211)

4.7 Definition of Fluids and Solids 4.7.1 Fluids and Solids ı

Let us consider examples of symmetry groups G s . Theorem 4.14. Every H-tensor corresponding to H -transformation is unimodular, i.e. it satisfies the relation det H D ˙1: (4.212) H Indeed, the continuity equation (3.8) and relations (4.188) yield ı



    D jdet Fj D jdet .F  H/j D jdet F  det Hj D jdet Hj:  

(4.213)

Due to (4.204), from (4.213) we get Eq. (4.212). N Notice that H -transformations may be not continuous (such transformations cannot be represented as a sequence of transformations convergent to the identical one). An example of such transformation is the H -transformation of reflection with respect to some plane. For such H -transformations, in the continuity equation, unlike Eq. (3.8) written for continuous transformations, one should take the magnitude of ı det F in order that the expression = be positive. This has been made in (4.213). The set of all unimodular tensors constitutes the full unimodular group U [12]. Thus, we get the following theorem. ı

Theorem 4.15. A symmetry group G s of any continuum is a subgroup of the full unimodular group: ı

G s  U:

(4.214) ı

Definition 4.5. A continuum, which for any reference configuration K has a symı

metry group G s coinciding with the full unimodular group: ı

G s D U; is called a fluid.

(4.215)

222

4 Constitutive Equations ı

Consider H -transformations of a reference configuration K with orthogonal tensors H satisfying the condition H  HT D E:

(4.216)

The set of all tensors satisfying the condition (4.216) constitutes the full orthogonal group I [12]. b Definition 4.6. A continuum, for which there exists a reference configuration K b s is a subgroup of the full orthogonal group: such that its symmetry group G b s  I; G

(4.217)

is called a solid (a solid body). b of a solid, for which the condition (4.217) is satisfied, is The configuration K called undistorted.

4.7.2 Isomeric Symmetry Groups ı

Remind that a symmetry group G s of a continuum (a group of H -transformations) ı b is defined for some fixed configuration K. For another reference configuration K, b s as well, but the group is, in according to Axiom 14, there is a symmetry group G ı

general, not coincident with G s . ı

b s corTheorem 4.16 (of Noll). For every continuum, symmetry groups G s and G ı b are isomeric, i.e. they are responding to different reference configurations K and K connected by the relation ı b s D S G s S1 : G (4.218) Here S is the nonsingular deformation gradient corresponding to the transformation ı ı b which connects local bases rıi and b b K ! K, ri in K and K: ı

S Db ri ˝ ri ;

ı

b ri D S  r i ;

ı

b ri D ri  S1 :

(4.219)

H If the relationship (4.218) really holds, this means that we can put every transforı

mation tensor H of a symmetry group G s in correspondence with the tensor b D S  H  S1 H b s and vice versa. of a group G

(4.220)

4.7 Definition of Fluids and Solids

223

Fig. 4.3 The connection of different configurations

ı

b s are called isomeric. Figure 4.3 shows reference conSuch groups G s and G ı  b and their corresponding configurations K, K e after H - and H bfigurations K, K 

transformations. The deformation gradients F, b F, F and e F transform configurations ı  b K and K e into K, respectively. K, K, According to the representation (4.206) of constitutive equations, we see that for b -transformations the following relations hold: all H - and H ƒ D fM.; F/ D fM.; F  H1 / 8F;

(4.221)

M M b b b1 / 8b F: ƒDf .; b F/ D f .; b FH

(4.222)

M b Here the hat b in f means that the dependence of fM on F and b F may be different in different configurations. Since F and b F are connected by the tensor S: ı ı b F D ri ˝b ri D ri ˝ ri  rj ˝b rj D F  S1 ;

(4.223)

M M b b fM.; F/ D f .; b F/ D f .; F  S1 /:

(4.224)

we get the relation

Since this relation must be satisfied for any deformation gradient F, we can apply the relation to F  H1 with substitution of the product in place of F: M b fM.; F  H1 / D f .; F  H1  S1 /: ı

(4.225)

Since H  G s , so, by (4.221), two equalities (4.224) and (4.225) prove to be coincident, i.e. M M b b .F  H1  S1 /: (4.226) f .F  S1 / D f

224

4 Constitutive Equations

Returning from F to b F by the relation F D b F  S, from (4.226) we get M b M b b b ƒDf .F/ D f .F  S  H1  S1 /:

(4.227)

Thus, the transformation S  H1  S1 retains the form of the tensor function fM bs . without changes. This means that the tensor S  H1  S1 belongs to the group G 1 1 1 b b b Then there is a tensor H 2 G s such that H D S  H  S . From this relation we get formula (4.220). N b for bases rıi and b Let us resolve now tensors H and H ri : ı

ı

H D Hji ri ˝ rj ;

b ijb bDH H rj : ri ˝b

(4.228)

On substituting (4.228) into (4.220) and taking (4.219) into account, we get b ijb ri ˝b ri ˝b H rj D Hji S  ri ˝ rj  S1 D Hji b rj :

(4.229)

Thus, due to uniqueness of a tensor resolution for a dyadic basis, we find that b have the same components but with respect b ij D Hji . Hence, the tensors H and H H to different bases. Such tensors are called isomeric (see [12]). Theorem 4.16 as well as Definitions 4.5 and 4.6 have been suggested by Noll. ı

Remark 1. The conditions of Theorem 4.16 imply that as a symmetry group G s ı

corresponding to a configuration K, we take a subgroup of the unimodular group U , ı

which is maximum permissible by the constitutive equations (4.206); i.e. if for K ı b s . This there is one more symmetry group Gs0 , then Gs0  G s . The same holds for G condition is assumed to be always satisfied below. ı

Remark 2. Since during the passage from one reference configuration K to another ı b due to Theorem 4.16, we obtain, in general, different symmetry groups G s or K, b s , so the concept of a symmetry group of a continuum is relative and dependent G ı

on the choice of reference configuration. For example, if a group G s is orthogonal ı b the group may be no longer orthogonal. in K, then in configuration K Let us apply Theorem 4.16 to solids. According to Definition 4.6, an undistorted ı

configuration K is not uniquely defined. The following theorem establishes a relation between symmetry groups of two undistorted configurations. ı

b s of a solid, which correspond to two Theorem 4.17. Symmetry groups G s and G ı b are connected by the relation undistorted configurations K and K, ı

b s D O0 G s OT0 ; G

(4.230)

4.7 Definition of Fluids and Solids

225

where O0 is the orthogonal rotation tensor accompanying the transformation S W ı b K ! K. H Rewrite formula (4.220) in the form b  S D S  H; H

(4.231)

b2G b s and H 2 Gs are orthogonal tensors. where H Since the tensor S is nonsingular, we can write its polar decomposition S D O0  U0 D V0  O0 ;

(4.232)

where the tensors U0 and V0 are connected by the relation V0 D O0  U0  OT0 :

(4.233)

On substituting (4.232) and (4.233) into (4.231), we get b  O0  U0 D V0  O0  H: H

(4.234)

b  O0  U0 included in (4.234) is nonsingular, because det B D The tensor B  H b det Hdet S D ˙det S ¤ 0. Then the tensors U0 and V0 are symmetric and positiveb  O0 and O0  H are orthogonal. Thus, formula (4.234) is definite, and the tensors H a polar decomposition of the tensor B. But, due to Theorem 2.9, this decomposition b  O0 D O0  H is valid and can be rewritten in the is unique, therefore the relation H form b D O0  H  OT0 : H

(4.235)

This equation has proved formula (4.230). N Consider the relationship (4.234). In the equation, replace V0 by U0 with use of bT from the left. formula (4.233) and then multiply both sides of (4.234) by OT0  H As a result, we obtain bT  O0  U0  OT  O0  H D HT  U0  H: U0 D OT0  H 0

(4.236)

Here we have used formula (4.235). Thus, from (4.236) we establish that the tensors U0 and H are commutative: H  U0 D U 0  H

ı

8H 2 G s :

(4.237)

226

4 Constitutive Equations

4.7.3 Definition of Anisotropic Solids Consider a solid body. According to the principle of material symmetry, the solid b s in a corresponding undistorted conpossesses some orthogonal symmetry group G b bs . figuration K. Such solid body is called the anisotropic body of the class G b s in an undisDefinition 4.7. A solid is called isotropic, if its symmetry group G b torted configuration K coincides with I : b s D I: G

(4.238)

It should be noted that the condition (4.238) is satisfied, in general, only in some b Under the transformation to another reference configundistorted configuration K. ı

uration, for example to K, the solid may be no longer isotropic. b s of the full orthogonal group I (including the group There are 39 subgroups G b s (such I itself); wherein 32 subgroups contain a finite number of elements H 2 G groups are called point groups), and 7 subgroups include a continuous set of elements H (they are called continuous). The description of all elements of the groups one can find in [12]. bs  I , which are most Consider below only the following four subgroups G widely used in continuum mechanics:    

b s D E (point group, which consists of two elements). The triclinic group G bs D O (point group, which consists of eight elements). The orthotropy group G b s D T3 (continuous group). The transverse isotropy group G b s D I (continuous group). The isotropy group (which is full orthogonal) G ı

b of a solid its local basis ri (and As noted above, to each configuration K (and K) ı b s ) correspond. Orthogonal group G bs b ri ), and also the symmetry group G s (and G b is characterized by its orthonorcorresponding to the undistorted configuration K mal basis b ci , which is called the principal basis of anisotropy, and coordinate axes oriented along the vectorsb ci are called the principal axes of anisotropy. b and the reference configuration Figure 4.4 shows the undistorted configuration K ı

ı

K, bases ri and ri directed tangentially to the coordinate lines X i (dashed lines),

Fig. 4.4 Distorted reference configuration and undistorted configuration of a solid

4.7 Definition of Fluids and Solids

227

and also basis b ci directed along the curves (continuous lines) characterizing the geometric structure of a solid and its anisotropy. It should be noted that anisotropic properties of solids are caused, as a rule, by a specific geometric structure of the bodies, where not all directions have the same properties. Such materials are, for example, composites, rubber-cord materials and tires reinforced by fibres, metals and alloys, where a grained microstructure is oriented along the rolled direction, rocks and grounds having a layered structure, etc. Notice that the basis b ci can depend, in general, on coordinates x of a considered material point M, in this case a bs . solid is said to possess curvilinear anisotropy of the class G b s are represented in the principal If tensors of H -transformations from group G anisotropy basis: bDH b ij b b2G bs : H cj ; H (4.239) ci ˝b b i are orthogonal: then matrices of components of the tensors H j b ij H b j D ıi ; H k k

(4.240)

b i and H b j coincide). (in orthonormal bases the matrices H j i b ij bs , there are eight matrices of the standard form H For the orthotropy group G (see [12]): b ij 2 fE; C; R˛ ; D˛ g; ˛ D 1; 2; 3; H (4.241) where 0 1 1 1 0 0 100 E D @0 1 0A ; C D @ 0 1 0 A ; 0 0 1 001 0 1 0 1 0 1 1 0 0 1 0 0 10 0 R1 D @ 0 1 0A ; R2 D @0 1 0A ; R3 D @0 1 0 A ; 0 01 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 D1 D @0 1 0 A ; D2 D @ 0 1 0 A ; D3 D @ 0 1 0A : 0 0 1 0 0 1 0 0 1 0

(4.242)

Here E is the unit matrix, C is the matrix of centrally symmetric transformation, R˛ are the matrices of mirror image relative to the planes being orthogonal to vectors b c˛ , and D˛ are the matrices of rotation by angle  about the axes oriented along vectorsb c˛ . b s D E, there are only two matrices H b ij : For the triclinic group G b ij 2 fE; C g; H corresponding to the identical and centrally symmetric transformations.

(4.243)

228

4 Constitutive Equations

b s D T3 , there is one axis of rotation called For the transverse isotropy group G the axis of transverse isotropy; in this case it is the axis oriented along the vectorb c3 b ij have the form and matrices H b ij 2 fQ  ; D Q ; R˛ Q g; H 3 3 3

˛ D 1; 2; 3I D 1; 2:

(4.244)



Here Q3 is the matrix of rotation by angle about the axis oriented along the vector b c3 : 0 1 cos sin 0  Q3 D @ sin cos 0A ; 0 6 < 2: (4.245) 0 0 1 b s D T2 , the axes of transverse isotropy are b s D T1 and G For the groups G oriented along the vectorsb c1 and b c2 , respectively. ri of undistorted configuration The principal basis of anisotropyb ci and the basisb b b (in general, not K may be not coincident (see Fig. 4.4). If we introduce a tensor Q b b b ij b orthogonal), which connects these two bases: b ci D Q  b ri ; Q D Q rj ; then ri ˝ b b the tensors of H -transformations (4.239) from group G s can be represented in the basisb ri as well: b ki Q bj b b ij Q bDH b ijb cj D H rl : (4.246) H ci ˝b l rk ˝b

4.7.4 H -indifference and H -invariance of Tensors Describing the Motion of a Solid ı

Let us consider a solid continuum. A reference configuration K is assumed to be ı

undistorted, then a symmetry group G s of the continuum corresponding to the conı

figuration K is orthogonal; and H-tensors of transformations involved in the group ı

G s are orthogonal. According to formula (4.230), there exists an orthogonal tensor ı bs : O0 connecting elements of the groups G s and G b D O0  H  OT0 H

ı

b2G bs : 8H 2 G s ; H

(4.247)

ı

For undistorted configuration K, there is its principal anisotropy basis; the basis will be denoted below by b ci . Although this basis may differ from the basis introduced b below they will not be considered in Sect. 4.7.3 for undistorted configuration K, simultaneously, so there will be no confusion for the notationb ci . Notice that all constructions made in Sects. 4.7 and 4.8 can be written in a similar way for undistorted b But since for formulating the problems in continuum mechanics we configuration K. ı

need the constitutive equations in a reference configuration K and the passage from

4.7 Definition of Fluids and Solids

229

ı

b to K is described by the formulae of tensor transformation from Sect. 4.6.2, so the K final form of the constitutive equations can prove to be rather awkward. Therefore ı

we will consider below only the case of undistorted reference configuration K. ı

Since for undistorted configuration K all H-tensors are orthogonal, we introduce the notation  (4.248) HT D Q; 

where Q is the orthogonal tensor describing the transformation of a solid from ı



K to K. ı

Definition 4.8. A tensor  specified in configuration K is called H -indifferent rel

ı

ı



ative to group G s  I , if for any orthogonal tensor of transformation Q W K ! K from this group, the tensor  satisfies the condition 







 D QT    Q;

ı

8 Q 2 Gs:

(4.249)

In other words, there exist tensors, which are H -indifferent relative only to certain subgroups of the orthogonal group; tensors, which are H -indifferent relative to any H -transformation, are called absolutely H -indifferent. The definition (4.188) yields 



F D F  Q D F  HT ;

(4.250)

i.e. the deformation gradient is neither H -indifferent nor H -invariant even relative ı

to subgroups G s  I . The following combinations for F are neither H -indifferent nor H -invariant as well, because 









F1 D Q  F1 ;

FT D QT  FT ;



F1T D F1T  Q:

(4.251)

The metric tensor is both H -indifferent and H -invariant relative to any orthogoı

nal group G s  I , because the relation (4.199) yields 







E D QT  E  Q D QT  Q D E:

(4.252)

Notice that for arbitrary H -transformations (which are not orthogonal), the metric tensor is only H -invariant (see (4.199)).

230

4 Constitutive Equations .n/

Theorem 4.18. All the quasienergetic deformation tensors A and deformation .n/

.n/

measures g are H -invariant, and all the energetic deformation tensors C and deı

.n/

formation measures G are H -indifferent relative to any orthogonal group G s  I . H Prove that the Cauchy–Green deformation tensor C is H -indifferent, and the Almansi deformation tensor A is H -invariant. The relations (4.200) and (4.251) give V



C D C D

  V  1 T  1  .F  F  E/ D .QT  FT  F  Q  E/ D QT  C  Q; 2 2

(4.253)

  I   1 1 .E  F1T  F1 / D .E  F1T  Q  QT  F1 / D A: 2 2

(4.254)

and also I



A D A D

I

In a similar way, we can prove that the tensor C D ƒ is H -indifferent and the tensor V

A D J is H -invariant. Show that the right stretch tensor U is H -indifferent, the left stretch tensor V is H -invariant, and the rotation tensor O accompanying deformation is neither H -indifferent nor H -invariant. On substituting the polar decomposition of F into (4.250), we get 







F D V  .O  Q/ D V  O:

(4.255) 

Here, on the one hand, we have obtained the new orthogonal tensor O  Q, and, 

on the other hand, we have found the polar decomposition for F. Since the polar decomposition is unique, we get 



VDV

and



O D O  Q:

(4.256)

Formula (2.137) for the polar decomposition yields 













U D OT  F D QT  O  F  Q D QT  U  Q;

(4.257)

i.e. the tensor U is H -indifferent. .n/

In a similar way, one can prove the theorem for the remaining tensors C .n/

and A. N

4.7 Definition of Fluids and Solids

231 I

As shown in Sect. 4.6.2, only the deformation measure G is absolutely H -indifferent (i.e. H -indifferent relative to any but not only to orthogonal H -transformations). .n/

Theorem 4.19. All the energetic stress tensors T are H -indifferent, and all the .n/

quasienergetic stress tensors S are H -invariant relative to any orthogonal group ı

Gs  I . .n/

H The energetic stress tensors T are H -indifferent, because (I)













I

T D FT  T  F D QT  FT  T  F  Q D Q  T  Q;

(II)

   1 T   F  T  O C OT  T  F 2     II 1 T T Q  F  O  Q C QT  O  T  F  Q D Q  T  Q etc. (4.258) D 2

T D

.n/

Tensors S are H -invariant, because they are formed by the contraction of H invariant tensors T and V. N V

As shown in Sect. 4.6.2, the only tensor T is absolutely H -indifferent, and the III

only tensor S D T is absolutely H -invariant. ı

The rotation stress tensor S is neither H -invariant nor H -indifferent, and it is transformed similarly to the rotation tensor O: ı

S D

ı  1    1  1    .V  T  V  V  T  V/  O D S  O: 2

(4.259)

4.7.5 H -invariance of Rate Characteristics of a Solid ı



The velocity v is absolutely H -invariant (for any transformations K ! K), because this vector is defined in actual configuration K. For the same reason, the velocity gradient L D .r ˝ v/T D .@v=@X i / ˝ ri

(4.260)

is absolutely H -invariant as well. Therefore the tensors D and W are also absolutely H -invariant:   D D D; W D W: (4.261)

232

4 Constitutive Equations

P Oldroyd BOl , Theorem 4.20. The total derivative with respect to time B, CR d D Cotter–Rivlin B , mixed co-rotational B and B relative to the eigenbasis of the left stretch tensor BV , Jaumann BJ derivatives are absolutely H -invariant, if they are applied to an absolutely H -invariant tensor B. The same co-rotational derivatives, and also derivative in the eigenbasis BV and ı

the spin derivative BS are H -invariant relative to group G s  I , if they are applied ı

to a tensor B, which is H -invariant relative to the group G s  I . 

H Suppose that a tensor B is H -invariant: B D B. Then H -invariance of the derivaP BOl , BCR , Bd , BD and BJ follows from H -invariance of the tensors L and tives B, W, for example, P       .BJ / D B C B  W  W  B D BP C B  W  W  B D BJ :

(4.262)

ı

If a tensor B is H -invariant only relative to a subgroup G s of the full orthogonal 

group, then from (4.262) it follows that BJ is H -invariant relative to the subgroup ı

G s as well. We should consider particularly only BV . Since the left stretch tensor V is H 

ı

ı

invariant (see (4.256)) for all H D QT 2 G s (involved in orthogonal groups G s ), its eigenbasis pi and the total derivative of the eigenbasis pP i are H -invariant as well: P  pi D pP i :



pi D pi ;

(4.263)

Then the spin V and rotation tensor OV are H -invariant:  P   V D pi ˝ pi D pP i ˝ pi D V ; 

(4.264)



QV D pi ˝ eN i D pi ˝ eN i D OV :

(4.265)

From these relations we get that the tensor BV is H -invariant: P       .B/V D B  V  B C B  V D BP    B C B  V D BV : 

(4.266)

Let us consider now the spin tensor  D O  OT . According to the property (4.256) of transformation of the rotation tensor O, we get 









P  Q  QT  OT D O P  OT D ;  D O  OT D O ı

i.e. the spin  is H -invariant relative to any orthogonal subgroup G s  I .

4.7 Definition of Fluids and Solids

233 

Then the spin derivative (2.371) in configuration K has the form 











BS D B    B C B   D BP    B C B   D BS ;

(4.267)

ı

if a tensor B is H -invariant relative to G s  I . Notice that the derivatives BS and BV are not absolutely H -invariant even if a tensor B is absolutely H -invariant. N The theorem certainly remains valid for an arbitrary H -invariant vector a (see Exercise 4.7.3). A corollary of this theorem is H -invariance of the total and co-rotational derivative of the quasienergetic deformation tensors: .n/

.n/

. A  / D A 

.n/

.n/

. A  /V D A V :

and

(4.268)

Theorem 4.21. The total derivative BP of a H -indifferent tensor B with respect to time is H -indifferent as well. H Since a transformation tensor H is independent of time t, so, differentiating the indifference condition 

B D H  B  HT ;

(4.269)

we get 

.B/ D H  BP  HT :

N

(4.270)

The remaining co-rotational derivatives, mentioned in Theorem 4.20, are neither H -indifferent nor H -invariant, if they are applied to a H -indifferent tensor. A corollary of this theorem is absolute H -indifference of the total derivative of I

the energetic deformation measure G: I

I

.G / D H  G  HT ; .n/

(4.271) .n/

and also H -indifference of the derivatives G  and C  relative to any orthogonal ı

group G s : .n/



.n/



. G  / D QT  G   Q;



H D QT :

(4.271a)

Theorem 4.22. The co-rotational derivative in the eigenbasis of the right stretch tensor BU is H -indifferent if it is applied to a tensor B, which is H -indifferent ı

relative to group G s  I .

234

4 Constitutive Equations

H At first, let us consider how the tensors OU and U are modified in H -transformations. According to the relation (2.276) between O, OU and OV , and also formula (4.256) for the tensor O, we get 





O D OV  OTU ;

(4.272)

or 





O  Q D OV  OTU ;



O D OV  OTU  QT D OV  OTU :

(4.273)

Thus, the rotation tensor OU is neither H -indifferent nor H -invariant: 



OU D QT  OU :

(4.274)

Using the definition (2.269) of the tensor U , we obtain 













P U  OT  Q D QT  U  Q; U D OU  OTU D OT  O U

(4.275)

i.e. the spin of the right stretch tensor U is H -indifferent. Let a tensor B be H -indifferent. According to formula (2.359), we get 























BU D B  U  B C B  U D QT  B  Q  QT  U  Q  QT  B  Q 











COT  B  Q  QT  U  Q D QT  BU  Q: N

(4.276)

This theorem is valid for any H -indifferent vector as well (see Exercise 4.7.4). Remark. If a tensor B is H -invariant, then its derivative BU is neither H -invariant nor H -indifferent. t u Tables 4.9 and 4.10 give the information on H -indifference and H -invariance of basic vectors and tensors.

Exercises for 4.7 b can be obtained from configuration 4.7.1. Show that if undistorted configuration K ı

K with the help of volume expansion transformation, i.e. the transformation tensor S (4.219) is spherical: S D kE; ı

bs D G s : then G

k ¤ 0;

4.7 Definition of Fluids and Solids

235

Table 4.9 H -indifference and H -invariance of basic vectors and tensors H -indifference relative to orthogonal Tensors Absolute H -pseudo- subgroup Absolute H -invariance and vectors H -indifference indifference Gs  I ri ,ri   C

H -invariance relative to orthogonal subgroup Gs  I C

ı ıi ri , r

C



E



C

C

C

F















C





C



C









C







C

C













C

TDS







C

C

P













C

C









C





C



C





S , n D I; II; IV; V 







C

.n/

C, n D I; : : : ; V I

G II IV

G,G V

G .n/ .n/

A, g , n D I; : : : ; V III

I

T II III IV

T; T; T V

T .n/ ı

C



S











P  OT V , D O









C

U





C





v, L, D, W







C

C

ı



4.7.2. Show that if H -transformations H W K ! K are not orthogonal, then the tensor V is not H -invariant, and the tensor U is not H -indifferent. 4.7.3. Prove Theorem 4.20 for a H -invariant vector a. 4.7.4. Prove Theorems 4.21 and 4.22 for a H -indifferent vector a.

236

4 Constitutive Equations

Table 4.10 H -indifference and H -invariance of basic vectors and tensors H -indifference relative to orthogonal Tensors Absolute H -pseudo- subgroup Absolute H -invariance and vectors H -indifference indifference Gs  I Bh , h D f; Ol;    C CR; d; D; J; Sg if B is absolutely H -invariant Bh , h D f; Ol; CR; d; D; J; S; V g, if B is H -invariant

H -invariance relative to orthogonal subgroup Gs  I C









C

C



C





Bh , h D f; U g  if B is H -indifferent



C





ı

relative to G s  I P if B is absolutely B, H -indifferent

ı

relative to G s  I

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua Let us apply now the principle of material symmetry to ideal solids and consider corollaries of the principle. Ideal solids are usually called elastic continua.

4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids ı

Let there be an anisotropic ideal solid of class G s with undistorted reference conı

figuration K. Suppose that constitutive equations of the anisotropic ideal solid correspond to models An (4.165), (4.168). Apply the principle of material symmetry 

in the form (4.181) to these models. Then we obtain that for any Q-transformation ı



from K to K, the relations D .n/ 

.n/ 

.n/

. C  ; /; .n/ 

T D .@ =@ C /  F . C ; /;

(4.277)  D @ =@

(4.278)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations 

237

ı

hold at the motion K ! K, if for K ! K Eqs. (4.165), (4.168) are satisfied. Here 

we have taken into account that all scalar functions are H -invariants, i.e.  D , 

D



,  D . ı

.n/

.n/

Since the tensors T and C are H -indifferent relative to group G s , from (4.278) we get the following assertion: the principle of material symmetry holds if the constitutive equations of ideal models An of anisotropic continua satisfy the conditions 



.n/



.n/





ı

QT  F . C ; /  Q D F .QT  C  Q; /; 8Q 2 G s ; @ .n/ @  T .n/  . C ; / D .Q  C  Q; /; @ @ 

.n/

.n/

(4.279a) (4.279b)



D  . C ; / D  .QT  C  Q; /:

(4.279c)

Theorem 4.23. The principle of material symmetry holds for models An of ideal solids if and only if the condition (4.279c) for the potential is satisfied. H The condition (4.279c) is necessary, because if the principle of material symmetry holds, then all the three conditions (4.279) are satisfied. Prove that the condition (4.279c) is sufficient. Let only the condition (4.279c) be satisfied. On substituting the expression for the potential into (4.279a) and differentiating the scalar with respect to the composite tensor argument (see [12, 14]), we get 

QT 

@ .n/



.n/





.QT  C  Q; /  Q

@C 



D  QT  Q 

@ (n)



.n/







.QT  C  Q; /  QT  Q D 

@C

@ (n)

:

(4.280)

@C

In a similar way, one can prove that the condition (4.279b) is satisfied as well. N

4.8.2 Scalar Indifferent Functions of Tensor Argument Consider in detail the condition (4.279c) imposed on the function isfying the principle of material symmetry.

.n/

. C ; / for sat-

Definition 4.9. A scalar function './ of a tensor argument is called indifferent ı



ı

relative to symmetry group G s  I , if for each tensor Q 2 G s the following condition is satisfied:   './ D '.QT    Q/: (4.281)

238

4 Constitutive Equations

With the help of the concept of an indifferent function, the results of Sect. 4.8.1 can be formulated as the following theorem. Theorem 4.24. For the model An of an ideal solid continuum with symmetry group ı

ı

G s and undistorted configuration K, the principle of material symmetry holds if and only if the free energy condition (4.279c).

.n/

. C ; / is an indifferent scalar function, i.e. it satisfies the ı

ı

Consider a solid with undistorted configuration K and symmetry group G s . ı

According to Sect. 4.7.4, the principal anisotropy basis in the configuration K is 

ı

denoted by b ci , and an arbitrary transformation tensor Q 2 G s can be resolved for this basis (see (4.239)):  b ijb cj ˝b Q D HT D H ci ; (4.282) 

b ij are components of the transformation tensor H D OT (for example, where H of the forms (4.241) and (4.244)). Let there be a scalar function './, which is ı

indifferent relative to the same group G s , then the function can be written in the component form b ij /; ' D './ D '. (4.283) i.e. in terms of components of the tensor b ijb D cj : ci ˝b

(4.284)

The condition (4.281) of indifference of a scalar function './ in this case takes the form b ij / D '.0ij /; '.

b  b kl 8 Hji 2 G bi H bs : 0ij D H k l j

(4.285)

In other words, the form of a scalar indifferent function remains unchanged at reı b ij by 0ij for each matrix H ij of the group G s . placing  As an example, consider a scalar linear function of a symmetric tensor argument './ D M    C '0 ;

(4.286)

where M is a symmetric second-order tensor, which is independent of  and defining the function './, and '0 D const. If the function (4.286) is indifferent, then the condition (4.281) for the function has the form 



M    D M   .QT    Q/:

(4.287)

Using the rules (4.4) and (4.5) of rearrangement of multipliers in the scalar product and taking the symmetry of tensors  and M into account, we get

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations 

239



.M  QT  M  Q/    D 0:

(4.288)

Since this relation must be satisfied for any , so the relation holds if and only if the tensor M satisfies the condition 





ı

M D QT  M  Q 8 Q 2 G s :

(4.289)

Definition 4.10. The second-order tensor M satisfying the condition (4.289) for 

ı

ı

each orthogonal tensor O 2 G s is called indifferent relative to the group G s . Comparing formula (4.289) with (4.249) and (4.191), we obtain that the tensor ı

M is indifferent relative to group G s if and only if the tensor is both H -indifferent ı

and H -invariant relative the group G s : 





M D QT  M  Q

M D M;



ı

8Q 2 G s :

(4.290)

On comparing this condition with (4.249), we get that the linear scalar function ı

(4.286) is indifferent relative to group G s if and only if the tensor M defining this ı

function is indifferent relative to the same group G s . This example exhibits that indifferent scalar functions (and as will be shown below, not only scalar functions) are constructed with the help of indifferent tensors.

4.8.3 Producing Tensors of Groups Let us consider now the set S3.2/ of all symmetric second-order tensors M, which ı

are indifferent relative to some group G s  I .



Let b ci be the principal anisotropy basis, in which all orthogonal tensors Q 2 ı b ij are orthogonal matrices of the standard form G s has the form (4.282), where H (4.241), (4.244), etc. .2/ The set S3 originates a finite-dimensional space of dimension k (see [12]), where we can choose a basis being a collection of symmetric tensors Os. / ( D 1; : : : ; k) having the following properties: ı

 Each of the tensors Os. / is indifferent relative to the group G s . .2/

 Any other tensor M 2 S3

indifferent relative to the same group is a linear combination of the tensors Os. / : MD

k X  D1

where not all a are zero.

a Os. / ;

(4.291)

240

4 Constitutive Equations

Table 4.11 The producing tensors of groups

Symmetry group Orthotropy O Transverse isotropy T3 Isotropy I

Producing tensors Os. / , D 1; : : : ; k b c 2 ; D 1; 2; 3 b c32 ; E

Number k 3 2

E

1

ı

The tensors Os. / are called the producing tensors of the group G s . For the groups of transverse isotropy T3 , orthotropy O (see Sect. 4.7.3) and also for the full orthogonal group I , the tensors Os. / are given in Table 4.11 (see [12]). Here we have introduced the notation b c2 D b c ˝b c :

(4.292)

Then the representation (4.291) for the groups O, T3 and I takes the form MD

3 X

ab c2

ı

for G s D O;

 D1 ı

c23 C a2 E for G s D T3 ; M D a1b

(4.293)

ı

M D a E for G s D I; i.e. all second-order tensors indifferent relative to the full orthogonal group I are proportional to the metric tensor.

4.8.4 Scalar Invariants of a Second-Order Tensor Let us show how indifferent scalar functions (not only linear) are constructed with the help of indifferent tensors. At first, we give two definitions. Definition 4.11. Let there be a second-order tensor  having in the orthonormal b ijb ci ˝ b basis b ci the following form:  D  cj , and also some symmetry group 

ı

b s has the form (4.282), then G s  I , each transformation tensor of which Q 2 G any scalar function of the tensor  bij / I .s/ ./ D I .s/ . ı

being indifferent relative to the group G s is called a scalar invariant of the tensor ı

 relative to the group G s .

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

241

.s/

Definition 4.12. The system of r scalar invariants I ./ (where D 1; : : : ; r/ ı

of tensor  relative to transformation group G s is called the functional basis of ı

independent invariants of the tensor relative to the group G s , if  the system is functionally independent;  any other scalar invariant I .s/ ./ (not involved in the system) of the tensor  ı

relative to the same group G s can be represented as a function of the invariants .s/ I : (4.294) I .s/ ./ D f .I1.s/ ./; : : : ; Ir.s/ .//; where f .I1 ; : : : ; Ir / is a scalar function of r variables, i.e. f W Rr ! R1 . Below, unless otherwise stated, the function f is assumed to be continuously b ij / – continuously differentiable funcdifferentiable in Rr and the invariants I .s/ . .s/ 6 1 6 tions I W R ! R in R (the number of independent components of a symmetric second-order tensor  is equal to six). The superscript s of the invariants ı

means that for different groups G s invariants of the same tensor are different. The definitions of functional dependence and independence of the invariant system are given in [12]. In [12] one can find a proof of the following theorem on the number of elements in the functional basis. Theorem 4.24a. For any symmetric second-order tensor , its functional basis of ı

independent invariants relative to group G s  I consists of r elements, where ı

 r D 3 for the isotropy group G s D I ;

ı

 r D 5 for the transverse isotropy group G s D T3 ; ı

ı

 r D 6 for the orthotropy group G s D O and the triclinic group G s D E.

As the functional basis, we can choose different systems of invariants. In continuum mechanics, one often uses polynomial invariants I.s/ ./ formed by contracı

tion of the producing tensors Os. / of the corresponding group G s with the tensor  itself (linear invariants), or with its tensor powers ˝, ˝˝ (quadratic and cubic invariants, respectively). Functional bases constructed by this method for four considered symmetry groups are shown below. ı

The triclinic group G s D E: 1 cˇ Cb cˇ ˝b c˛ /;    .b c˛ ˝b 2 ˛; ˇ; D 1; 2; 3; ˛ ¤ ˇ ¤ ¤ ˛;

c2 ; I.E / D    b

or

.E / I3C D

b 11 ;  b 22 ;  b 33 ;  b 23 ;  b 13 ;  b 12 g: I.E / D f

(4.295)

242

4 Constitutive Equations ı

The orthotropy group G s D O: I4.O/ D .b c2  /   .b c3  /;

c2 ; D 1; 2; 3; I˛.O/ D    b I5.O/ D .b c1  /   .b c3  /; or

.O/

I6

D ..b c1  /  .b c2  //   .b c3  /;

b 13  b 23 g: b11 ;  b 22 ;  b 33 ;  b 223 ;  b213 ;  b 12  I.O/ D f

(4.296)

(4.296a)

ı

The transverse isotropy group G s D T3 : .3/

I1

D .E b c23 /   ; .3/

I3 .3/

I4

.3/

I2

D   b c23 ;

D ..E b c23 /  /   .b c23  /; .3/2

D 2   E  I2

.3/

.3/

 2I3 ;

I5

(4.297)

D det ;

or b 11 C  b 22 ;  b 33 ;  b 213 C  b 223 ;  b 211 C  b 222 C 2 b 212 ; det g: (4.297a) I.3/ D f ı

The isotropy group G s D I : I.I / D I ./;

D 1; 2; 3:

(4.298)

Here I ./ are the principal invariants of tensor , which are defined by formulae (4.48): I1 ./ D    E; I2 ./ D

1 2 .I ./  I1 .2 //; I3 ./ D det : 2 1

(4.299)

Let us prove that the principal invariants of a tensor are indifferent, for example, for I1 : 







I1 ./ D E    D .Q  E  QT /    D E   .QT    Q/ 



ı

b 2 G s D I: D I1 .QT    Q/ 8 Q (4.300) In a similar way, we can prove that I1 .2 / satisfies the condition (4.281); and hence I2 ./ satisfies the condition as well. For I3 ./, we have 











I3 .QT  Q/ D det .QT  Q/ D det QT det det Q D det  D I3 ./; (4.301) 

ı

because Q is an orthogonal tensor of the group G s D I .

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations .O/

243

.3/

One can find a proof of indifference of the invariants I˛ and I˛ in [12]. .s/ In braces, formulae (4.295)–(4.297) show component forms of the invariants I in the basisb ci . Notice once more that functional bases of invariants may be different; their choice, in general, is defined by a considered problem. So for the triclinic group ı

G s D E, besides (4.295), one can use another basis / e D I ./; I .E 

D 1; 2; 3;

.E / .E / e I 3C D I3C :

(4.302)

Here we have taken into account that the principal invariants I ./ are invariants ı

relative to any subgroup G s  I .

4.8.5 Representation of a Scalar Indifferent Function in Terms of Invariants According to Theorem 4.24a, any scalar function of tensor argument './, which ı

is indifferent relative to group G s , can be represented as a function of invariants I˛.s/ ./ of the tensor, i.e. './ D '.I1.s/ ./; : : : ; Ir.s/ .//:

(4.303)

Let us return now to models An of solids and apply this result to the free energy .n/

. C ; /. According to the principle of material symmetry, this function must be ı

indifferent relative to some group G s  I ; then the free energy can be represented .n/

as a function of the functional basis of independent invariants I˛.s/ . C / of the tensor .n/

C , i.e. .n/

. C ; / D

.n/

.n/

.I1.s/ . C /; : : : ; Ir.s/ . C /; /:

(4.304)

The function (4.304) indifferent relative to the orthotropy group O is called the orthotropic scalar function; the function depends on six invariants (r D 6), for example (4.296). The function (4.297) indifferent relative to the transverse isotropy group T3 is called the transversely isotropic scalar function; the function depends on five invariants .r D 5/, for example (4.303). The function (4.304) indifferent relative to the isotropy group I is called the isotropic scalar function; the function depends on three principal invariants: .n/

. C ; / D

.n/

.I˛ . C /; /;

˛ D 1; 2; 3:

(4.305)

244

4 Constitutive Equations

4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua Let us return now to the conditions (4.279) and consider in detail the first condition .n/

(4.279a) imposed on the tensor function of the tensor argument F . C ; /. Definition 4.13. The tensor function of tensor argument F ./ is called indifferent 

ı

ı

relative to a symmetry group G s  I , if each tensor Q 2 G s satisfies the condition 







QT  F ./  Q D F .QT    Q/:

(4.306)



ci (see (4.282)), and also the tensor  and tensor Resolving the tensor Q for basisb function F for the basis b ci : b ijb cj ; ci ˝b F ./ D F

bij D F b ij . b kl /; F

(4.307)

bklb D cl ; ck ˝b

(4.308)

we can write the component form of the condition (4.306) of indifference of the tensor function: b j1 j2 . b i 1 i 2 .H b i1 H b i2 F b k2  b l1 l2 /: b kl / D F b k1 H H j1 j2 l1 l2

(4.309)

.n/

The tensor function F . C ; / in the models An is quasipotential, i.e the function satisfies the first condition of (4.168): .n/

.n/

F . C ; / D .@ =@ C /:

(4.310) ı

.n/

where the scalar function . C ; / is indifferent relative to the group G s , and hence this function can always be represented in the form (4.304) of a function of ten.s/

.n/

.n/

sor invariants I˛ . C /. Then for the quasipotential function F . C; / we have the representation r X .n/ .n/ T D F . C ; / D ' I.s/ (4.311) C;  D1

where .n/

.n/

' D ' .I1.s/ . C /; : : : ; Ir.s/ . C /; / D .@ =@I.s/ /;

.n/

.s/ I.s/ C D @I =@ C : (4.312)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

245

.n/

Functions ' depend only on invariants I.s/ . C / and , because the density  .n/

is uniquely expressed in terms of three principal invariants of the tensor C (see Sect. 4.2.19). Formula (4.311) is called the representation of an indifferent quasipotential tensor function in the tensor basis or the invariant representation of constitutive equations for the model An of elastic continua. On substituting different collections (4.295)–(4.298) in place of invariants I.s/ into (4.311), we obtain the representations of tensor functions indifferent relative to the groups E, O, T3 and I . For the triclinic group E, the invariant representation (4.311) has the form .n/

T D

 3  X 1 'b c2 C ' C3 O ; 2  D1

(4.313a)

and the component form of this function for the basis b c is written as follows: .b n/

T

ij

 3  X 1 i j i j i j D ' ı ı C ' C3 .ı˛ ıˇ C ıˇ ı˛ / ; 2  D1

(4.313b)

where ' are scalar functions of invariants I.E / (4.295): .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ ' D ' . C 11 ; C 22 ; C 33 ; C 23 ; C 13 ; C 12 ; /:

(4.313c)

For the orthotropy group O, the invariant representation has the form .n/

T D

3 X

.n/ 1 'b c2 C .'4 O1 ˝ O1 C '5 O2 ˝ O2 C '7 O3 ˝ O3 /   C 2  D1 .n/

.n/

C3'6 6 Om     C ˝ C;

(4.314a)

and the corresponding component form of the function with respect to the basis b ci is written as follows: .b n/

T

ij

D

.b n/ .b n/ .b n/ 1 j j ' ıi ıj C .'4 C 23 C C 12 C 13 '6 /.ı2i ı3 C ı3i ı2 / 2  D1 3 X

.b n/ .b n/ .b n/ 1 C .'5 C 13 C '6 C 12 C 23 /.ı1i ı3j C ı3i ı1j / 2 .b n/ .b n/ .b n/ 1 C .'7 C 12 C '6 C 13 C 23 /.ı1i ı2j C ı2i ı1j /: 2

(4.314b)

246

4 Constitutive Equations

.n/ .b n/ .b n/ .n/ Here T ij and C ij are components of the tensors T and C with respect to the basis b ci , and O , D 1; 2; 3, and 6 Om are tensors indifferent relative to the group O, which are constructed with the help of the producing tensors Os. / of the group (see Table 4.11 and [12]):

O D b c˛ ˝b cˇ Cb cˇ ˝b c˛ ; 6

Om D

˛ ¤ ˇ ¤ ¤ ˛;

3 X

1 48

˛; ˇ; D 1; 2; 3;

.O˛ ˝ Oˇ C Oˇ ˝ O˛ / ˝ O :

(4.315) (4.316)

˛;ˇ; D1

˛¤ˇ¤ ¤˛

When deriving the expression (4.314a) from the representation (4.311), one should use the rules of differentiation of the invariants being scalar functions of a tensor argument (see [12]). In order for formula (4.314a) to be symmetric in indices 1, 2, .O/

and 3, we must complement the system of invariants I the seventh invariant .O/

I7

.n/

. C / ( D 1; : : : ; 6) with

.n/ .n/ .b n/ D .b c12  C/   .b c22  C / D C 12 ;

(4.317)

which is not independent. In this case, functions ' in (4.314a) depend on all these seven invariants b .b n/ .b n/ .n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ ' D ' . C 11 ; C 22 ; C 33 ; C 23 ; C 13 ; C 12 C 23 C 13 ; C 12 ; /:

(4.318)

The function (4.314a) is called the orthotropic tensor function. For the transverse isotropy group T3 , the representation (4.311) takes the form .n/

1 '3 .O1 ˝ O1 C O2 ˝ O2 / 2 !   .n/ .n/ 1 C2'4  .O1 ˝ O1 CO2 ˝ O2 /b c32 ˝b c32 '5 I1    CC'5 C 2 ; 2

T D .'1 C I2 '5 /.E b c23 / C .'2 C I2 '5 /b c23 C

(4.319) and the component form with respect to the basis b ci is written as follows: .b n/ ij

T

.b n/ j D .'1 C I2 '5 /ı ij C .'2  '1  2'4 C 33 /ı3i ı3 .b n/ .b n/   C.'3  2'4 / .ı2i ı3j C ı3i ı2j / C 23 C .ı1i ı3j C ı3i ı1j / C 13

b .b .b n/ .n/ n/ C.2'4  I1 '5 / C ij C '5 C i k C kj ;

(4.320)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

247

where b .b n/ n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .n/  .b ' D ' C 11 C C 22 ; C 33 ; C 13 2 C C 23 2 ; C 11 2 C C 22 2 C 2 C 12 2 ; det C;  : (4.321) Here  is the symmetric fourth-order unit tensor (2.22a). For the full orthogonal (isotropic) group I , the representation (4.311) takes the form .n/

T D

or

.b n/ ij

T

D

1ı

1E

ij

.n/

C

2C .b n/ ij

C

2C

.n/

C

3C

C

.b n/ .b n/ i kj ; 3C k C

2

;

(4.322)

where 1

D ' 1 C ' 2 I1 C ' 3 I2 ;

2

D .'2 C I1 '3 /;

.n/ .n/   .n/ ' D ' I1 . C/; I2 . C /; I3 . C /;  ;

D



3

D '3 ;

(4.322a)

.n/ .n/ .n/  I1 . C /; I2 . C /; I3 . C /;  :

When deriving formula (4.322), we used the rules of differentiation of the principal invariants of a tensor (see [12]): @I2 ./ D EI1 ./  T ; @

@I1 ./ D E; @

@I3 ./ D I3 ./1 D T2  I1 ./T C EI2 ./: @

(4.323)

b s , the indifferent quasipotential tensor functions (4.311) Remark 1. For all groups G can be represented in the single tensor form .n/

.n/

.n/

.n/

.n/

T D 4 M   C C  C 2 C 6 L     . C ˝ C/:

(4.324)

The tensor 4 M is called the quasilinear elasticity tensor,  is called the parameter of quadratic elasticity, and 6 L – the tensor of quadratic elasticity. For the isotropy group I , from (4.322) it follows that 4

MD

1 .n/

I1 . C /

E˝EC

2 ;

D

3;

6

L D 0:

(4.325a)

248

4 Constitutive Equations

For the transverse isotropy group T3 , from (4.319) it follows that 4

  '1 C I2 '5 '2  ' 1 MD E˝EC  2'4 b c23 c23 ˝b I1 I2  ' 3 C  '4 .O1 ˝ O1 C O2 ˝ O2 / C .2'4 C '5 /I1 ; 2 6  D '5 ; L D 0:

(4.325b)

For the orthotropy group O, from (4.314a) it follows that 4

MD

3 X '

b c2 .0/  I  D1 

˝b c2 C 6

'4 '5 '7 O1 ˝ O1 C O2 ˝ O2 C O3 ˝ O3 ; 2 2 2 (4.325c)

L D 3'6 6 Om ;

 D 0:

The main result of this section is the derivation of representations (4.311), (4.313a), (4.314a), (4.319), and (4.322) for indifferent tensor functions. These representations show that tensor nonlinearity for indifferent tensor functions cannot be .n/

.n/

.n/

.n/

higher than the second power, i.e. the tensor powers C 3 , C ˝ C ˝ C and higher powers are not involved in these functions; but the scalar functions (4.312), (4.318), (4.321), and (4.322a) of invariants are nonlinear in a more complicated form. The obtained representations of tensor functions are the consequences of applying the principle of material indifference to the models An of solids; and they demonstrate how this principle seeming not sufficiently analytical allows us to derive analytical forms of constitutive equations. Notice that the representations (4.313a), (4.314a), (4.319), and (4.322) reject attempts to construct indifferent tensor functions in the form of generalized tensor Taylor series. t u Remark 2. In continuum mechanics, one sometimes uses the concept of a natural state of a continuum, that means the configuration Ke where the deformation gradient F coincides with the metric tensor, the Cauchy stress tensor T take some values (for example, the tensor is spherical: Te D p e E, where p e is a constant having the sense of a pressure in the natural state), and the temperature and free energy are equal to some natural values: Ke W

T D Te ; F D E;

 D e;

D

e

:

(4.326) .n/

.n/

In the natural state, all the energetic and quasienergetic stress tensors T and S .n/

.n/

coincide with Te , all the deformation tensors A and C are zero-tensors, and the .n/

.n/

deformation measures g and G coincide with E to within the factor:

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

Ke W

.n/

.n/

T D S D Te ;

.n/

.n/

A D C D 0;

.n/

.n/

g DGD

1 E: n  III

249

(4.327)

ı

For solids, one often assumes that in the reference configuration K a continuum is at its natural unstressed state, i.e. Te D 0 and e D 0 ; then for the solid we have the relations ı

.n/

.n/

.n/

.n/

.n/

.n/

K W T D S D T D 0; A D C D 0; g D G D

1 E;  D 0 ; n  III

D

0:

(4.328) There is a class of problems (for example, problems on appearing technological or initial stresses in structures), for which the conditions (4.328) are replaced by (4.326), (4.327). ı

If in K a continuum is at an unstressed state, then in each reference configuration



ı

K obtained from K by H -transformation involved in the corresponding symmetry ı

group G s the continuum is at the unstressed state as well. In this case, we impose additional conditions on the constitutive equations (4.311) or (4.324), namely the ı

conditions (4.328) in K, that lead to certain constraints on the potential . For example, for the models An of isotropic ideal continua, from (4.322) it follows that in ı

K D Ke we have the relations '1 D '1 .0; 0; 0; 0 / D 0;

.0; 0; 0; 0 / D 0: 

(4.329)

4.8.7 Quasilinear and Linear Models An of Elastic Continua For the quasilinear models An of elastic continua, in formulae (4.324) we should assume that  D 0;

6

L D 0;

(4.330)

then we obtain the relation .n/

.n/

T D 4M   C :

(4.330a)

For an isotropic continuum, the relations (4.330) mean that the scalar function D '3 in (4.322) vanishes: '3 .I1 ; I2 ; I3 / D 0: This relation is the additional condition for the invariants I1 ; I2 ; I3 . On expressing one of the invariants from this relation, we can eliminate this invariant between arguments of the functions ' 3

.n/

(4.322a). In continuum mechanics, one usually expresses the cubic invariant I3 . C / in this way, in order that only linear and quadratic invariants remain as arguments

250

4 Constitutive Equations

of the functions (4.322a). Then relations (4.322) for isotropic quasilinear elastic continua take the form .n/

T D

1E

.n/

 '2 C ; .n/

1

.n/

D '1 C '2 I1 ; ' D ' .I1 . C /; I2 . C /; /; D 1; 2:

For the transverse isotropy group T3 , relations (4.330) lead to one additional condition '5 .I1.3/ ; : : : ; I5.3/ / D 0; which allows us to eliminate the cubic invariant I5.3/ among arguments of the functions ' (4.321); and the number of arguments of the functions ' reduces to four. For the orthotropy group O, relations (4.330) lead to the condition '6 .I1.O/ ; : : : ; I7.O/ / D 0; that also allows us to eliminate the cubic invariant I6.O/ between arguments of the scalar functions ' (4.312). For linear models An of solids, we assume that the free energy (4.304) is a .n/

quadratic function of invariants I.s/ . C/, i.e. the function is a quadratic form of linear invariants I.s/ ( D 1; : : : ; r1 ) and a linear form of quadratic invariants I.s/ ( D r1 C 1; : : : ; r2 ) relative to the corresponding group: ı



ı

ı

D  0 C N0 C

r1 X

m N  I.s/ C

 D1

r1 r2 X 1 X .s/ .s/ lˇ I Iˇ C l  I.s/ ; (4.331) 2  Dr C1 ;ˇ D1

1

.s/ where 0 , N 0 , m N  , lˇ are the constants; the cubic invariants I , D r2 C1; : : : ; r, are not involved in the expression (4.331), therefore for linear models An both the conditions (4.330) hold. Notice that the total number of different constants lˇ does not exceed 21, because the matrix .lˇ / consisting of the constants has a maximum size 6  6 (6 is the maximum number of invariants I.s/ ) and the matrix is symmetric; and a symmetric n  n matrix has n.n C 1/=2 different components. The functions ' have the forms

' D J

r1 X

.s/

lˇ Iˇ C J m ; D 1; : : : ; r1 ;

ˇ D1

(4.332) ı

' D J l  ; D r1 C 1; : : : ; r2 I J D =;

.s/

.n/

I C D @I.s/ =@ C ;

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

251

and the constitutive equations (4.311) and (4.324) for linear models take the form .n/

T DJ

r1 X

m N  O.s/  CJ

 D1

r1 X

r2 X

.s/

lˇ I.s/ Oˇ C J

;ˇ D1

 Dr1 C1

.s/

l   I C :

(4.333)

ı

If in K a continuum is at an unstressed state, then substituting (4.331) and (4.333) into the normalization equation (4.328), we find that the constants N 0 and m N  may be only zero: N 0 D 0; m N  D 0; D 1; : : : ; r1 : (4.334) Thus, for linear models An there are no purely linear summands. (1) For isotropic elastic continua, there is only one linear invariant and one quadratic invariant (r1 D 1, r2 D 2); therefore for isotropic linear models An the free energy (4.331) has the form ı



ı

D

0

C

1 2

 .n/ .n/ ı l1 C l2 I12 . C/  2l2 I2 . C / D 

0

C

.n/ l1 2 .n/ I1 . C / C l2 I1 . C 2 /; 2 (4.335)

where l1 and l2 are the constants (l11 D l1 C 2l2 , l22 D 2l2 ). For the isotropic linear model, the functions ' (4.312) and  (4.322a) have the forms '1 D J.l1 C 2l2 /I1 ; '2 D 2l2 J; '3 D 0; (4.336) ı J D =; 1 D l1 I1 J; 2 D 2l2 J; the tensor 4 M (4.325a) has only two independent components: 4

M D J.l1 E ˝ E C 2l2 /;

(4.337)

and the constitutive equations (4.322) are written as follows: .n/

.n/

.n/

T D J.l1 I1 . C /E C 2l2 C /:

(4.338)

Since the relations (4.338) contain the factor J , they are not linear; therefore the models (4.335) and (4.338) are often called semilinear. The semilinear model AIV is called John’s model, and the model AV – Murnaghan’s model. (2) For transversely isotropic continua, there are two linear invariants I.s/ and two quadratic invariants .r1 D 2; r2 D 4/; therefore, for transversely isotropic linear models An , the free energy (4.331) has the form ı



ı

D

0

1 C .l11 I1.3/2 C 2l12 I1.3/ I2.3/ C l22 I2.3/2 / C l33 I3.3/ C l44 I4.3/ ; (4.339) 2

and contains five constants lˇ .

252

4 Constitutive Equations

The functions ' (4.312) for this model have the forms .3/

.3/

'1 D J.l11 I1 C l12 I2 /; '3 D J l33 ;

.3/

.3/

'2 D J.l22 I2 C l12 I1 /;

'4 D J l44 ;

'5 D 0;

(4.340)

the tensor 4 M (4.325b) has five independent components: 4

 c23 ˝b M D J l11 E ˝ E C e l 22b c23 C .l12  l11 /.E ˝b c23 Cb c23 ˝ E/ C

l

33

2

   l44 .O1 ˝ O1 C O2 ˝ O2 / C 2l44  ;

(4.341)

e l 22 D l22  2l44  2l12 C l11 ; and the constitutive equations (4.319) are written as follows: .n/

T D J .l11 I1.3/ C l12 I2.3/ /.E b c23 / C ..l22  2l44 /I2.3/ C l12 I1.3/ /b c23 ! ! .n/ .n/ l33 C  l44 .O1 ˝ O1 C O2 ˝ O2 /   C C 2l44 C : 2

(4.342)

(3) For orthotropic continua, there are three linear invariants and three quadratic invariants .r1 D 3; r2 D r D 6/; therefore, for orthotropic linear models An , the free energy (4.331) has the form 3 3 X 1 X .O/ .O/ .O/ D 0C lˇ I Iˇ l3C;3C I3C 2  D1

ı

ı



(4.343)

;ˇ D1

and contains nine constants .l11 ; l22 ; : : : ; l66 ; l12 ; l13 ; l23 /, the tensor 4 M (4.325c) has nine independent components: 4

3 X

MDJ

c2 ˝b lˇb c2ˇ C J

;ˇ D1

3 X

l3C;3C O ˝ O ;

(4.344)

 D1

and the constitutive equations (4.314a) take the form .n/

T D

3 X ;ˇ D1

J lˇ Iˇ.O/b c2 C

3 X  D1

.n/

J l3C;3C O .O   C /:

(4.345)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

253

4.8.8 Constitutive Equations for Models Bn of Elastic Continua The theory stated in Sects. 4.8.1–4.8.6 is formally valid for models Bn with the only .n/

difference that the energetic deformation tensors C in formulae of these sections .n/

must be replaced by the corresponding energetic deformation measures G, because .n/

.n/

G as well as C are H -indifferent relative to the orthogonal group I . In particular, relations (4.304), (4.311), and (4.312) take the forms .n/

.s/

.n/

.n/

.I1 . G/; : : : ; Ir.s/ . G/; /;

. G; / D

r X

.n/

.n/

T D F . G; / D

.s/

'

 D1 .n/

@I

;

.n/

(4.346)

@G

.n/

' D ' .I1.s/ . G/; : : : ; Ir.s/ . G/; / D 

@ .s/

:

@I

In a similar way, we modify formulae (4.313), (4.319), and (4.322); for example, for the group I the representation (4.346) takes the form D

.n/

.n/

.I1 . G/; : : : ; I3 . G/; /;

D '1 C '2 I1 C '3 I2 ;

1

2

.n/

.n/

T D

1E

C

.n/

2G

.n/

C

D .'2 C '3 I1 /;

3G 3

2

;

D '3 ;

(4.347)

.n/

' D ' .I1 . G/; : : : ; I3 . G/; / D .@ =@I /: ı

If in K a continuum is at an unstressed state, then relations (4.347) must satisfy the conditions (4.328), which lead to the following conditions of normalization for functions ': '1 .a1 ; a2 ; a3 / C '2 .a1 ; a2 ; a3 /

2 1 D 0: (4.347a) C '3 .a1 ; a2 ; a3 / n  III .n  III/2

ı

Here we have used that in K .n/

.n/

I . C / D I .0/ D 0; I1 . G/ D .n/

3  a2 ; I2 . G/ D .n  III/2

.n/

3  a1 ; n  III

1 I3 . G/ D  a3 : .n  III/3

(4.347b)

254

4 Constitutive Equations

For linear models Bn , the free energy is also chosen in the form (4.331), where .n/

I D I . G/; in particular, for an isotropic linear continuum we have ı

ı

D





ı

0

C

0

C

 .n/ .n/ .n/ l1 C l2 I12 . G/  2l2 I2 . G/ C mI1 . G/; 2

(4.348)

and the constitutive equations take the form similar to (4.338): .n/

.n/

.n/

T D J.m C l1 I1 . G//E C 2J l2 G:

(4.349)

ı

If the continuum in K is at an unstressed state, then from (4.328) we obtain the following relations between the constants 0 , l1 , l2 and m: mD

3l1 C 2l2 ; n  III

0

D

3.3l1 C 2l2 / ı

2.n  III/2

;

hence relations (4.348) and (4.349) take the forms ı



ı

D

0

C

.n/ .n/ 3.3l1 C 2l2 / 3l1 C 2l2 .n/ l1  I1 . G/ C I12 . G/ C l2 I1 . G 2 /; 2 2.n  III/ n  III 2  .n/ .n/ .n/ 3l1 C 2l2 T D J .l1 I1 . G/  /E C 2l2 G : (4.350) n  III

The same relations can be derived from (4.335) and (4.338) by using the substitution .n/

.n/

C D G  .1=.n  III//E (see Exercise 4.8.11), i.e. the models Bn (4.350) and An (4.335), (4.338) are equivalent; they have only different forms. In Sect. 4.9 we will show that for some types of ideal continua there exist models An and Bn which are not equivalent.

4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua Establish corollaries to the principle of material symmetry for models Cn of solids. ı

Let there be a solid with undistorted reference configuration K and symmetry ı

group G s . If constitutive equations of the solid correspond to the models Cn (4.171), ı

(4.174) for each motion K ! K, then, according to the principle of material symmetry (4.181), the following relations must hold:

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations .n/ 

.n/

255



.n/

S D .@ =@ A  /  ˆ. A  ; O; /; D

.n/



. A  ; O; /;  D @ =@;

ı



ı .n/

(4.351)



S D .@ =@O/  ˆ. A  ; O; / 



ı



in the motion K ! K for each configuration K obtained from K by Q-transformation ı

involved in the group G s . .n/

.n/

Since S and A are H -invariant tensors relative to any orthogonal transformaı

tions (see Theorems 4.18 and 4.19) and the tensors S and O are transformed by the laws (4.256) and (4.259), so from (4.351) we obtain that the principle of material symmetry holds, if constitutive equations of the models Cn of solids satisfy the conditions .n/

.n/



ˆ. A ; O; / D ˆ. A; O  Q; /;

(4.352a)

 @ .n/ @ .n/ . A ; O; / D . A ; O  Q; /; @ @

(4.352b)

ı .n/



ı .n/



ˆ. A; O; /  Q D ˆ. A ; O  Q; /; .n/

. A ; O; / D



.n/



(4.352c) ı

. A ; O  Q; /; 8 Q 2 G s :

(4.352d)

Theorem 4.25. The principle of material symmetry holds for the models Cn of ideal solids if and only if the condition (4.352d) for the potential is satisfied. H The condition (4.352d) is necessary, because if the principle of material symmetry holds, then (4.352d) follows from (4.351). Prove that the condition (4.352d) is sufficient. The relations (4.352a) and (4.352b) can be proved in the same way as in Theorem 4.23, by differentiating the function .n/

in (4.352d) with respect to its arguments A and . To prove the relation (4.352c), one should differentiate the function (4.352d) with respect to O: ı .n/

ˆ. A ; O; / D

@ .n/ @ .n/  . A ; O; / D . A ; O; / @O @O D

.n/

@ 

@O





ı .n/





. A ; O; /  QT D ˆ. A ; O  Q; /  QT :

256

4 Constitutive Equations 

Here we have used the relation between derivatives of  @ @ D   QT ; @O @O



with respect to O and O: 

O D O  Q;

(4.353)

which follows from the differential form d0

D



@ 

  d OT D

@O



@

  .QT  d OT / D



@O



@ 

 QT   d O T D

@O

@   d OT ; @O

.n/

where d 0 is the partial differential at fixed A. N The condition (4.352d) imposed on the function differs considerably from the condition of indifference (4.279c), which has been obtained for models An . .n/

.n/

Definition 4.14. A scalar function P . A ; O/ of two tensor arguments A and O is ı

called a pseudoinvariant relative to group G s , if ı

(1) the function is rotary-indifferent relative to the symmetry group G s  I , i.e. for 

ı

.n/

each tensor Q 2 G s and for every A and O the following condition is satisfied: .n/



.n/

P . A ; O/ D P . A ; O  Q/;

(4.354)

.n/

(2) for every A and O the following identity holds: .n/

A

@P .n/

@A



@P .n/

.n/

AD

@A

@P  OT : @O

(4.355)

.n/

The function . A ; O; / satisfying the principle of material symmetry is a pseudoinvariant, because the function satisfies the condition (4.352d) of rotary indifference; and the function satisfies the relation (4.355), because from (4.174) and (4.76) it follows that 1 0 .n/ .n/ .n/ .n/ .n/ @ .n/ ı @ @ @ @   AAO D .A  S  S  A/O D  @A  : 0 D S .n/ .n/ @O @O @O @A @A (4.356) .n/

.n/

ı

Theorem 4.26. Let I˛.s/ . C / be an invariant of the tensor C relative to G s , then the scalar function .n/ .n/ P˛.s/ . A ; O/ D I˛.s/ .OT  A  O/ (4.357) ı

is a pseudoinvariant relative to the same group G s .

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

257 ı

.n/

H (1) Indeed, if I˛.s/ is an invariant of the tensor C relative to group G s , then, by ı

definition, the invariant satisfies the condition of H -indifference relative to G s : 

.n/



.n/



ı

I˛.s/ . C / D I˛.s/ .QT  C  Q/ 8Q 2 G s : .n/

(4.358)

.n/

But the tensors C can always be expressed in terms of A by the formula (see Exercise 4.2.8) .n/

.n/

C D OT  A  O:

(4.359)

Substitution of (4.359) into (4.358) yields 

.n/



.n/



ı

I˛.s/ .OT  A  O/ D I˛.s/ .QT  OT  A  O  Q/ 8Q 2 G s :

(4.360)

On defining the function P˛.s/ by formula (4.357), from (4.360) we obtain that the function satisfies the condition 

.n/



ı

P˛.s/ . A; O/ D P˛.s/ .A; O  Q/ 8Q 2 G s ;

(4.361) ı

i.e. the function P˛.s/ (4.357) is rotary-indifferent relative to the group G s . (2) Show that each scalar function (4.357) satisfies the condition (4.355). At first, .n/

determine the derivative @P˛ =@ A by using the concept of the partial differential at fixed O: d 0 P˛.s/ D

@P˛.s/ .n/

.n/

  d AT D

@A D

dP˛.s/ .n/

dP˛.s/ .n/

.n/

  d CT

dC

0

.n/ T

  .O  d A  O/ D @O  T

dP˛.s/

dC

.n/

1 O

TA

.n/

  d A:

dC

(4.362) Here we have taken into account that P˛.s/ , by (4.357) and (4.359), can be considered .n/

as a function of C . Comparing the second and last equalities, we get @P˛.s/ .n/

@A

DO

dP˛.s/ .n/

dC

 OT :

(4.363)

258

4 Constitutive Equations .n/

In a similar way, using the partial differential at fixed A, we obtain .n/ @P˛.s/ dP˛.s/   d CT   d OT D .n/ @O dC .s/ .n/ .n/ dP˛ D   .d OT  A  O C OT  A  d O/ .n/ dC .s/ .s/ .n/ .n/ dP˛ dP˛ D A O   OT C  OT  A   d O .n/ .n/ dC dC ! .n/ .n/ dP˛.s/ dP˛.s/ T D A O O  O  A  O  d OT : .n/ .n/ dC dC

d 0 P˛.s/ D

(4.364)

Here we have taken into account that d O D O  d OT  O. From (4.364) we get .n/ .n/ @P˛.s/ dP˛.s/ dP˛.s/ O  OT  A  O: D A O .n/ .n/ @O dC dC

(4.365)

On substituting (4.363) and (4.365) into (4.355), one can verify that the relation (4.355) is identically satisfied: .n/

.n/ @P˛.s/ dP˛.s/  OT  OT D A  O  .n/ .n/ .n/ @O @A @A dC .s/ .n/ .n/ dP˛.s/ dP dP˛.s/ ˛ O   OT  A  A  O   OT C O   OT  A  0: .n/ .n/ .n/ dC dC dC (4.366)

A

@P˛.s/



@P˛.s/

.n/

A

Thus, each scalar function (4.357) is a pseudoinvariant. N Theorem 4.26 gives the method of construction of pseudoinvariants P˛.s/ : they .s/

.n/

can be formed by the corresponding invariants I˛ . C / with the help of formula (4.357). The number z of elements in a functional basis of pseudoinvariants .n/

P˛.s/ . A ; O/ is determined by the following theorem. .n/

Theorem 4.27. A functional basis of pseudoinvariants P˛.s/ . A ; O/ relative to group ı

G s consists of z elements, where ı

 z D 3 for the isotropy group G s D I ,

ı

 z D 5 for the transverse isotropy group G s D T3 ,

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

259

ı

 z D 6 for the orthotropy group G s D O, ı

 z 6 6 for each subgroup G s  I . .n/

H (1) Since the symmetric tensor A has six independent components and the orthogonal tensor O has three independent components, the total number of func.s/ tional pseudoinvariants P˛ cannot exceed nine. However, due to the additional constraint (4.355) imposed on the functions P˛.s/ , the number of independent pseudoinvariants P˛.s/ diminishes. The left-hand side of (4.355) is a skew-symmetric tensor, therefore the set (4.355) contains only three independent relations. As a result, we find that the number z of independent pseudoinvariants does not exceed ı

9  3 D 6 for arbitrary orthogonal subgroup G s  I . (2) Show that for the isotropy group I the number of pseudoinvariants P˛.s/ is equal to three. Perform a proof in two stages. Stage 1. For isotropic continua, the condition (4.354) must be satisfied for the full 

orthogonal group, i.e. 8 Q 2 I . Since the tensor O is also orthogonal, the condition (4.354) in the basis b ci can be rewritten as follows: .b n/ .b n/ b kl / D P˛.s/ . A ij ; O b 0kl / P˛.s/ . A ij ; O

.n/

8O0 ; O 2 I; 8 A ;

(4.367)

.b n/  b kl / can be considered as where O0 D O  O 2 I . The component form P˛.s/ . A ij ; O a function of nine variables, which is given on the set R6 V0 , where V0  R3 is the b kl . Since in the condition (4.367) range of components of the orthogonal tensors O b kl and O b 0kl take all values of V0 (I is the full orthogonal group), the components O so from the condition (4.367) it follows that the function remains unchanged on the set V0 , i.e. the function is a constant; and within the domain of definition b kl , i.e. P˛.s/ are independent of the rotation R6  V0 the function is independent of O tensor O: .n/ .b n/ P˛.s/ D P˛.s/ . A/ or P˛.s/ D P˛.s/ . A ij /: (4.368)

Stage 2. Notice that although the indifference condition (4.281) has not been imposed on the function (4.368), the function P˛.s/ is nevertheless indifferent, however, possibly relative to not the whole group I but at least relative to the triclinic group 

ı

G s D E  I (this group contains only the identical Q D E and centrally symmet

ric Q D E transformations, and the condition (4.281) is always satisfied for each .s/

.n/

function P˛ . A /).

ı

The problem consists in finding a maximum subgroup G s , relative to which the function (4.368) is indifferent.

260

4 Constitutive Equations

We know that in the group E there exists the special functional basis of inde.n/

pendent invariants (4.302). Then we can apply formula (4.304) to P˛.s/ . A ; / and .n/

/ I .E consider P˛.s/ as a function of the functional basis e ˛ . A / (4.302): .n/

.n/

.n/

.n/

P˛.s/ D P˛.s/ .I1 . A/; : : : ; I3 . A/; e I 4 . A /; : : : e I 6 . A /; /:

(4.369)

Substituting the potential (4.369) into (4.355), we obtain 6 X

.n/

' A 

/ @e I .E 

 D1

.n/

@I.E /



.n/

@A

!

.n/

A

D 0;

(4.370)

@A

where ' D @P˛.s/ =@I.E / . .n/

According to formula (4.323) of differentiation of the principal invariants I1 . A / and the formula of differentiation of the remaining three invariants: / @e I .E  C3 .n/

.n/

b ; .A/ D O

˛; ˇ; D 1; 2; 3I ˛ ¤ ˇ ¤ ¤ ˛;

(4.371)

@A

b  1 .b O c˛ ˝b cˇ Cb cˇ ˝b c˛ /; 2

(4.372)

Eq. (4.370) takes the form 1E

D

C

.n/ 2

2A

1E C

.n/

C '3 A 3 C .n/

3 .n/ 1X ' C3 O  A 2  D1

.n/

2 3 2 A C '3 A C

3 .n/ 1X ' C3 A  O ; 2  D1

(4.373)

where 1 and 2 are determined by formulae (4.322a). Hence the following identity must hold: 3 X

.n/

.n/

' C3 .O  A  A  O / D 0:

(4.374)

 D1 .n/

Since the product of the tensors O and A is not commutative (one can check this .n/ .b n/ cj ), we obtain that the identity ci ˝b fact with multiplying O (4.315) by A D A ijb (4.374) holds if and only if the coefficients '3C ( D 1; 2; 3) identically vanish: / ' D @P˛.s/ =@e I .E  0; D 4; 5; 6: 

(4.375)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

261

Thus, for an isotropic continuum, each pseudoinvariant P (4.369) cannot depend on .n/

/ the invariants e I .E  . A /, D 4; 5; 6; but P depends only on three principal invari.n/

ants I . A/, i.e. for isotropic continua a basis of pseudoinvariants consists of three elements. (3) For a transversely isotropic continuum, a proof is analogous (see Exercise 4.8.7). N Consider several important corollaries of the theorem. For isotropic continua, the potential (see (4.171)) is independent of the tensor ı

O (because is a pseudoinvariant), and the rotation stress tensor S, due to this fact, is identically zero (with taking formula (4.176) into account): D

.n/

ı

.n/

. A ; /;

@ . A ; / D 0: SD @O

(4.376)

ı

Since for orthogonal groups G s D T3 ; O and I the number z of elements in .n/

the functional basis of pseudoinvariants P˛.s/ . A ; O/ is equal to the number r of .s/

ı

.n/

elements in the basis of invariants I˛ . C / of the corresponding group G s (see .n/

Theorems 4.24 and 4.27), and for each invariant I˛.s/ . C / we can construct a pseu.s/ doinvariant by formulae (4.357) and (4.359), so the basis of pseudoinvariants P˛ .n/

.s/

can be constructed only with the help of invariants I˛ . C / (4.357). Hence, for the models Cn , the potential can always be written in the form D

.n/

. A ; O; / D

.n/

.P˛.s/ . A ; O/; / D

.n/

.I˛.s/ .OT  A  O/; /:

(4.377)

On substituting (4.377) into (4.174), we obtain the general representation of constitutive equations for models Cn of ideal continua in the tensor basis (this is an analog of (4.311)): .n/

.n/

S D ˆ. A ; O; / D

r X

'

@I.s/ .n/

 D1

;

(4.378)

@A .n/

' D .@ =@I.s/ / D ' .I˛.s/ .OT  A  O/; /:

(4.379)

In particular, for an isotropic solid, the relations (4.378) and (4.379) of the model Cn have standard forms, which are similar to (4.322): .n/

S D

1

D ' 1 C ' 2 I1 C ' 3 I2 ; 

2

1E

C

.n/

2A

.n/

C '3 A 2 ;

D ' 2 C I 1 ' 3 ; ' D 

(4.380) @ ; @I

D

.n/

.I . A/; /:

262

4 Constitutive Equations

Let us establish a connection between the relation (4.378) of models Cn and the relation (4.311) of models An with taking into account that the quasienergetic stress tensors and quasienergetic deformation tensors are connected to the corresponding energetic tensors with the help of the rotation tensor O (see Exercise 4.2.3): .n/

.n/

S D O  T  OT :

(4.381)

Substitution of (4.381) and (4.359) into (4.378) yields .n/

O T O D T

r X

.s/

' O 

@I

 D1

.n/

 OT :

(4.382)

@C

Comparing (4.382) with (4.311), we conclude that the models An and Cn of ideal continua differ from each other only by multiplication of the constitutive equations by the rotation tensor O. Thus, we can consider these models as different forms of the same relations. However, we have shown that the form Cn (4.378) of constitutive equations is the only possible form with the rotation tensor O, which satisfies the principle of material symmetry. In addition, the forms An and Cn prove to be rather different, when they have been differentiated with respect to time (that will be shown in Sect. 4.8.12); therefore, we will distinguish the models An and Cn from each other. Notice that for nonideal continua a distinction between the models An and Cn is more considerable, that will be shown in Chapters 7 and 8. .n/

Since the quasienergetic measures g as well as the quasienergetic deformation .n/

tensors A are H -invariant relative to the group I , so all the formulae of Sect. 4.8.8 .n/

.n/

still hold for the models Dn of ideal solids (4.161) with substitution A ! g . In particular, the constitutive equation (4.380) for an isotropic solid in the models Dn takes the form .n/

S D

1E

C

.n/ 2

.n/

g C '3 g 2 ;

D

.n/

.I . g /; /:

(4.383)

4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua The constitutive equations (4.311), (4.346), and (4.378) for models An , Bn , Cn and .n/

Dn of elastic continua with the help of the generalized energetic tensors T G and .n/

C G (4.91) can be written in the unified form .n/

.n/

T G D F G . C G ; O; / D

r X  D1

' I.s/ G;

(4.384)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

263

where we have denoted the derivative tensors .n/

.s/ I.s/  G D @I G =@ C G ;

' D .@ =@I.s/ G /;

D

.n/

.I.s/ G ; /;

.n/

(4.385)

.s/ .s/ T I.s/ G D hG I . C G / C .1  hG /I .O  C G  O/;

G D A; B; C; DI

n D I; : : : ; V:

Here we have introduced the following functions, which are indicators of the model type: ( ( 1; G D A; B; 1; G D A; C; hG D (4.386) hN G D 0; G D C; D; 0; G D B; D: The constitutive equation (4.384) for all the models can always be reduced to the dependence of the Cauchy stress tensor T on the deformation gradient F. For the .n/

.n/

purpose we should use relations (4.37) and (4.67) expressing T in terms of T and S . Substituting the constitutive equations (4.311) of models An , (4.346) of models Bn , (4.378) of models Cn and Dn into relations (4.37) and (4.67), we obtain the general representation of constitutive equations for all the models of elastic continua: .n/

T D F G .F; /;

G D A; B; C; D;

(4.387)

where .n/

4

.n/

.n/

F G .F; /  E G   F G . C G ; / D

r X  D1

.n/

.s/

' 4 E G   I G ;

(4.388)

and 4

.n/

.n/

.n/

E G D hG 4 E C .1  hG /4 Q D

3 X

ı

ı

E˛ˇ p˛ ˝ pˇ ˝ .hG pˇ ˝ p˛ C .1  hG /pˇ ˝ p˛ /

(4.389)

˛;ˇ D1

are the generalized tensors of energetic equivalence. .n/

For the models at n D I and V, the tensors 4 E G can be written in the explicit form (see Exercise 4.2.21) in terms of F and V, therefore for models An , Bn , Cn and Dn (n D I and V) the tensor function (4.387) has the explicit form

264

4 Constitutive Equations I

I

F A D F1T  F .C; /  F1 ; V

V

F A D F  F .C; /  FT ; I

1

FC D V V

I

 ˆ.A; O; /  V

1

I

;

F C D V  ˆ.A; O; /  V;

V

V

F B D F  F .G; /  FT ; I

V

I

F B D F1T  F .G; /  F1 ;

1

FD D V V

(4.390)

I

 ˆ.g; O; /  V

1

;

V

F D D V  ˆ.g; O; /  V:

.n/

The generalized energetic tensors C G can be expressed in terms of F by formulae (4.41), (4.42), (4.65), and (4.71) (see Exercise 4.2.23). These formulae can be written in the unified form .n/

C G .F/ D

1 ..hG FT  F C .1  hG /F  FT /.nIII/=2  hN G E; n  III n D I; II; IV; V:

(4.391a)

For these models at n D III, in place of relations (4.391a) Eq. (4.387) is complemented with the differential relations (2.317) and (4.43b), (4.62), (4.71b) for III

determining the tensors CG in terms of F. These relations can also be written in the generalized form III

CG D

hG 1 P T P  U1 C 1  hG V1  .FP  FT C F  FP T /  V1 ; U  .F  F C FT  F/ 2 2 III

CG .0/ D 0; G D A; C I

III

CG .0/ D E; G D B; D:

(4.391b)

Fractional and negative powers of tensors of the type FT  F are determined with ı the help of the passage to the eigenbases pi and pi . According to the representation T T (2.140) of tensors F  F and F  F in the eigenbases, we can rewrite formulae (4.391a) in the equivalent form .n/

CG

3 hN G 1 X nIII ı ı D .˛ .hG p˛ ˝ p˛ C .1  hG /p˛ ˝ p˛ //  E; n  III ˛D1 n  III

(4.391c) n D I; II; IV; V: Remark. The constitutive equation (4.387) can be represented as a function of the inverse gradient F1 , that is sometimes convenient for formulation of elasticity .n/

.n/

problems. For the purpose, the tensors of energetic equivalence 4 E and 4 Q, and .n/

C G should be expressed in terms of F1 : .n/

C G .F1 / D

1 ..hG F1 F1T C.1hG /F1T F1 /.IIIn/=2  hN G E/; (4.392) n  III

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

265

ı

where eigenvalues ˛ and eigenvectors p˛ , p˛ must be expressed in terms of F1 , i.e. in place of (2.140) we must consider the equation F1T  F1 D

3 X

2 ˛ p˛ ˝ p˛

˛D1

to determine ˛ and p˛ . Then in place of (4.387) we have the constitutive equation .n/

T D F G .F1 ; /;

(4.393)

.n/

.n/

where F G is expressed by the same formulae (4.388), (4.389) and (4.385), and C G is considered as a function in the form (4.392). u t

4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases The constitutive equations (4.322), (4.347), (4.380), and (4.383) of models An , Bn , ı Cn , and Dn for isotropic elastic continua can be expressed in the eigenbases p˛ or .n/

.n/

p˛ . According to the results of Exercise 4.2.4, the tensors C and G can be resolved .n/

ı

.n/

for the basis p˛ , and the tensors A and g – for the basis p˛ . On substituting the resolutions into the constitutive equations (4.322), (4.347), (4.380) and (4.383), we .n/

.n/

ı

obtain that the tensors T and S at n D I, II, IV, V are diagonal in the bases p˛ and p˛ : 3 3 X X .n/ .n/ ı.n/ ı .n/ ı T D  ˛ p˛ ˝ p˛ ; S D  ˛ p˛ ˝ p˛ ; (4.394) ˛D1 ı.n/

˛D1

.n/

Here  ˛ and  ˛ are eigenvalues of the stress tensors, which for models Bn and Dn have the forms .n/

˛ D

ı.n/



˛

1

C

2

n  III

C nIII ˛

'3 2.nIII/ ; '˛ D  .n  III/2 ˛

@ .n/

;

@I˛ . G/ ' @ 3 D 1C C ˛2.nIII/ ; '˛ D  nIII ; ˛ 2 .n/ n  III .n  III/ @I˛ . g /

(4.395)

2

1

D '1 C '2 I1 C '3 I2 ;



2

D '2 C I1 '3 :

(4.396)

266

4 Constitutive Equations

For models An and Cn , analogous relations are given in Exercise 4.8.9. Since the .n/

.n/

principal invariants of the tensors G and g are coincident (see Exercise 4.8.2), i.e. .n/

.n/

.n/

ı.n/ ˛

.n/

I˛ . G/ D I˛ . g /, the eigenvalues  ˛ and  continua coincide as well: .n/ ˛

ı.n/ ˛;

D 

.n/

of the tensors S and T for isotropic

˛ D 1; 2; 3:

(4.397)

Theorem 4.28. For models An , Bn , Cn and Dn of isotropic elastic continua, eigen.n/

.n/

.n/

values  ˛ of the tensors T and S can always be represented in the form .n/

 ˛ D IVn ˛

and

@ ; ˛ D 1; 2; 3; n D I; II; IV; VI @˛

ı

 D 1 2 3 ;

(4.398)

can be considered as a function of ˛ : D

.1 ; 2 ; 3 ; /:

(4.399)

H Since, for models Bn and Dn of isotropic continua,

is a function of I˛ . G/ D

.n/

.n/

.n/

I˛ . g / and  and the principal invariants I˛ . G/ are uniquely expressed in terms of ˛ (see Exercise 4.8.3), so the potential can always be considered as a function in the form (4.399). To prove formula (4.398), we differentiate the composite function D .Iˇ ; / with respect to ˛ and multiply the result by IVn : ˛ IVn ˛

X @ @Iˇ X @Iˇ @ D IVn D IVn 'ˇ ˛ ˛ @˛ @Iˇ @˛ @˛ 3

3

ˇ D1

˛D1

 D IVn C '1 nIV ˛ ˛

'2 C nIII / nIV .nIII  ˇ n  III ˛  '3 nIV nIII  .  / C  ˇ ˛ .n  III/2

D '1 C

'2 '3 C nIII /C .ˇ  /nIII : .nIII  ˇ n  III .n  III/2

Here we have used the results of Exercise 4.8.4.

(4.400)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

267

On the other hand, substitution of (4.396) into (4.395) yields .n/

˛

!   2.nIII/ nIII ˛ nIII ˛ ˛ D ' 1 C ' 2 I1  C '3 I2  I1 C : (4.401) n  III n  III .n  III/2

According to the relations between I1 , I2 and ˛ (see Exercise 4.8.3), we see that the expressions (4.400) and (4.401) exactly coincide as was to be proved. A proof of the theorem for models An and Cn should be performed as Exercise 4.8.10. N Formulae (4.394), (4.398) and (4.399) are called the representation of the constitutive equations of isotropic continua in the eigenbasis. Let us consider now the case when n D III. Theorem 4.28a. For isotropic elastic continua, the tensor B defined by (2.317) is ı

energetically equivalent to the logarithmic deformation tensor H (2.167); the tenIII

sor G defined by (4.43a) is energetically equivalent to the logarithmic deformation ı Q the measure III measure H1 (2.167); the tensor Y (4.62) – to the tensor H; g (4.71a) – to the measure H1 , in the sense that the following relations hold: III

ı

III

III

III

III

III

P D S H P D T   G w.i / D T   BP D T   H D S   Y III

D T

ı

H1

III

III

III

P1 D D S  g D SH

3 III X ˛ P ˛ ;  ˛D1 ˛

(4.402)

III

III

where  ˛ D ˛ are eigenvalues of the tensors T and T, which are connected to eigenvalues ˛ of the tensors U and V by the relations III

˛ D  ˛ D ˛

@ ; @˛

D

.1 ; 2 ; 3 ; /:

(4.403)

H (a) For all the models Cn (n D I; : : : ; V) of isotropic continua, the rotation tensor ı

S is zero (see (4.376)), therefore, by the definition of the tensor (4.52), for these models including CIII , we obtain V  T  V1  V1  T  V D 0;

(4.404)

III

or V2  T  T  V2 D 0; i.e. the tensor T D S commutes with V2 , and hence the tensors are coaxial (see Sect. 4.2.21). Thus, T and V2 have a common eigenbasis p˛ and can be written in the form V2 D

3 X ˛D1

2˛ p˛ ˝ p˛ ;

III

TDSD

3 X III  ˛ p˛ ˝ p˛ : ˛D1

(4.405)

268

4 Constitutive Equations

P in the eigenbasis p˛ with use of its definition (4.62): Represent the derivative Y ! 3 3 X X 1 1 P D Y pˇ ˝ pˇ P ˛ p˛ ˝ p˛ C ˛ .pP ˛ ˝ p˛ C p˛ ˝ pP ˛ /  2 ˛D1 ˇ ˇ D1 ! 3 3 X 1 X 1 C p˛ ˝ p˛  .P ˇ pˇ ˝ pˇ C ˇ .pP ˇ ˝ pˇ C pˇ ˝ pP ˇ // 2 ˛D1 ˛ ˇ D1 ! ! 3 3 X ˇ P ˛ 1 X ˛ .pP ˛  pˇ / p˛ ˝ pˇ D ı˛ˇ C  ˛ 2 ˇ ˛ ˛D1 ˇ D1 ! 1 C .pP ˛ ˝ p˛ C p˛ ˝ pP ˛ / : (4.406) 2 Take the properties of orthonormal eigenbasis p˛ into account: p˛  pˇ D ı˛ˇ ; pP ˛  pˇ C p˛  pP ˇ D 0; 1 1d jp˛ j2 D 0; p˛  pP ˛ D .p˛  p˛ / D 2 2 dt

(4.407)

P (4.406): and contract the tensor T (4.405) with Y P D TY

3 X

˛

˛D1

P ˛ : ˛

(4.408)

Q (2.167), But the same result can be obtained if we consider the Hencky tensor H determine its derivative: Q D H

3 X

lg ˛ p˛ ˝ p˛ ;

˛D1

! 3 X P ˛ P Q p˛ ˝ p˛ C lg ˛ .pP ˛ ˝ p˛ C p˛ ˝ pP ˛ / ; HD ˛ ˛D1

(4.409)

and contract the derivative with T (4.405): PQ D w.i / D T   H

3 X

˛ P ˛ =˛ :

(4.410)

˛D1

Q Comparing (4.408) with (4.410), we get that for isotropic continua the tensors H P P Q and Y are energetically equivalent, and the contraction T   Y D T   H gives the ı

stress power, because for isotropic continua S D 0.

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

269

Since the logarithmic deformation measure H1 (2.167) satisfies the relation PQ and III P Eq. (4.402) still holds for the measures: P 1 D H, g  D Y, H III

III

III P 1: S   g D S   H

(b) Consider the model AIII for an isotropic continuum. For this case, the constitutive III

equation (4.322) holds. From the equation it follows that the tensors T and B are coaxial. To prove this fact it is sufficient to resolve the tensor B for its eigenbasis e p˛ III

and to substitute the result into (4.322); then we get that the tensor T is also diagonal in the same basis e p˛ : III

TD

3 X III  ˛e p˛ ; p˛ ˝ e

BD

˛D1

3 X

B˛e p˛ : p˛ ˝ e

(4.411)

˛D1

Unlike the model CIII , in this case there is no initial information that T is coaxial ı with U, and e p˛ , in general, may be not coincident with p˛ . However, for the model AIII , there is an implicit additional condition that Eq. (2.317) for the tensor BP must be P can be replaced by its energetically equivalent integrable integrable or the tensor B tensor. ı Take into account that the tensor BP in the eigenbasis p˛ has the same form as (4.406): BP D

3 X ˛D1

! ! 3 ˇ ı  ı P ˛ 1 X ˛ ı ı .p˛  pˇ / p˛ ˝ pˇ ı˛ˇ C  ˛ 2 ˇ ˛ ˇ D1

! 1 ı ı ı ı C .p˛ ˝ p˛ C p˛ ˝ p˛ / : 2

(4.412)

On the other hand, from (4.411) it follows that BP D

3 X

.BP˛e p˛ C B˛ .e pP ˛ ˝ e p˛ C e p˛ ˝ e pP ˛ //: p˛ ˝ e

(4.413)

˛D1 III

Contract the tensor BP with T and use both the expressions (4.412) and (4.413): 3 3 3 X X X P ˛ 2 III T   BP D ˛ p˛ C p˛ e p ˛ C ˛ˇ p˛ pˇ ˛ ˛D1  D1

III

ˇ D1

!! D

3 X III  ˛ BP ˛ ; ˛D1

(4.414)

270

4 Constitutive Equations

where ı

ı

e p ˛ D p˛  e p ;

p˛ D p˛  e p ;

˛ˇ D



˛

ˇ



ˇ  ı  ı .p˛  pˇ /: ˛

III

According to the relation (4.414), the coefficient of  ˛ must form the total derivative; that occurs if and only if p˛ D 0 at ˛ ¤ and p˛˛ D 1, i.e. the bases e p˛ and ı

ı

p˛ coincide: e p˛ D p˛ . In this case III

P D w.i / D T   B

3 3 X X III III  ˛ BP ˛ D  ˛ .lg ˛ / : ˛D1

(4.415)

˛D1

ı

III

The contraction of the tensor T with H gives the same expression, where ı

HD

3 X

ı

ı

lg ˛ p˛ ˝ p˛ ;

˛D1 ı



H D

3 X

ı

(4.415a)

ı

..lg ˛ / p˛ ˝ p˛ C lg

ı ˛ .p˛

ı

ı

˝ p˛ C p˛ ˝

ı p˛ //:

˛D1 ı

Hence, the tensor B can be replaced by its energetically equivalent tensor H. ı Since the logarithmic deformation measure H1 (2.167) satisfies the relation ı

ı

III

P so relationship (4.402) still holds for the measures: H1  D H , and G D B, III III III ıP T   G D T   H1 : (c) Since the left-hand sides of Eqs. (4.408) and (4.415) are coincident, the rightIII

hand sides must coincide as well; this means that eigenvalues of the tensors T and III

T are coincident: ˛ D  ˛ . Substitution of formulae (4.402) into (4.120) yields    3  X @ @  ˛ ˛ d˛ C   C  d D 0; @˛ @ ˛D1

(4.416)

because for an isotropic continuum, can always be considered as a function of ˛ and : D .˛ ; /. From (4.416) we get the constitutive equations (4.403). N Uniting the assertions of both theorems (4.398) and (4.403), we obtain that for all the models An , Bn , Cn and Dn (n D I; : : : ; V) constitutive equations of isotropic elastic continua take the form .n/ ˛

D ˛IIIn ˛ ; D

˛ D ˛ .˛ ; /;

@ ; n D I; : : : ; V; @˛ ı

 D 1 2 3 :

(4.417)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

271

The stress power w.i / defined by (4.2), with use of relations (4.87)–(4.90), (4.394), .n/

.n/

.n/

ı

.n/

(4.402), and expressions for C , G, A and g in terms of ˛ and p˛ (see Exercise 4.2.4) for all the models An , Bn , Cn and Dn , can be represented in the form 3 X .n/  ˛ nIV (4.417a) P ˛ ; n D I; : : : ; V: w.i / D ˛ ˛D1

Thus, for isotropic elastic continua, the generalized constitutive equations (4.387) for all the models An , Bn , Cn and Dn reduce to the single form T D F B .F; /; ı

F B .F; / D .1 2 3 /

3 X

˛

˛D1

@ p˛ ˝ p˛ ; @˛

(4.418a) D

.˛ ; /;

˛ ; p˛ k F:

If ˛ and p˛ are considered as functions of F1 , then the constitutive equations (4.418a) can be written in the form (4.393): T D F B .F1 ; /;

(4.418b)

where the tensor function F B .F1 ; / has the same form as in (4.418a). Since, for all the models An , Bn , Cn and Dn (n D I; : : : ; V) of isotropic elastic .n/ .n/

.n/

.n/ .n/

.n/

continua, all the tensors T , C and G, and also S , A and g are coaxial, so these tensors are commutative; then, according to Theorem 4.11, the stress power can be expressed with the help of the co-rotational derivative (see Sect. 4.2.21): .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

w.i / D T   C h D T   G h D S   A h D S   g h ; n D I; : : : ; V; III

ı

III

ı

III

(4.419)

h D f; U; V; J; S g: III

Q and g D H1 . Here C D H, G D H1 , A D H

4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua ‘in Rates’ For some problems in solid mechanics (usually for dynamic processes), it is convenient to represent the constitutive equations ‘in rates’.

272

4 Constitutive Equations

At first, consider the models An and Bn of elastic continua, which correspond to the constitutive equations (4.384) (at G D A; B) independent of the tensor O: .n/

.n/

.n/

T G D F G . C G ; / D

r X  D1

.s/

' I G :

(4.420)

Theorem 4.29. For the models An and Bn of ideal solids, the constitutive equations (4.420) at G D A; B can be represented ‘in rates’: .n/

.n/

.n/

.n/

P T  D 4 PG . C G ; /   C G C T G ;

G D A; B:

(4.421)

H Differentiating the relation (4.420) with respect to t and using the definition of the derivative tensors (4.385), we get .n/

r X



T D

@'

.s/ IˇG @Iˇ.s/

;ˇ D1



.n/

C G

! r X .n/ @I.s/ @' P .s/ G  I G C C '   C G : .n/ @  D1 @CG

(4.422)

.n/

Introducing the fourth-order tensor 4 PG and tensor T G as follows: 4

r X

PG D

.s/

.s/

.s/

.'ˇ I G ˝ IˇG C ' ıˇ 4 I G /;

.n/

T G D

;ˇ D1

r X @' .s/ I ; @  G

(4.423)

ˇ D1

where 'ˇ D

@' .s/

D

@Iˇ

@ .s/



@Iˇ

!

@ .s/

4 .s/ I G

;

@I

D

@I.s/ G .n/

@CG

.s/

D

@2 I .n/

.n/

;

(4.424)

@CG@CG

we get that (4.422) can be represented in the form (4.421). N Let us consider now the models Cn and Dn of ideal isotropic solids. As shown in Sect. 4.8.9, for isotropic continua the constitutive equations (4.384) are independent of the tensor O and can also be represented in the form (4.420) at G D C; D. Theorem 4.30. For the models Cn and Dn of isotropic ideal solids, the constitutive equations (4.420) can be represented ‘in rates’ as follows: .n/ h

.n/

.n/

.n/

P S D 4 PG . C G ; /   C hG C S G ;

G D C; D;

where we have introduced the notation h for the co-rotational derivatives h D fU; V; J; S g:

(4.425)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

273

H The co-rotational derivatives of a second-order tensor has the general form (formula (2.373)) .n/

.n/

.n/

.n/

S h D S   Zh  S  S  ZTh ;

(4.426)

Zh D fU ; V ; W; g; h D fU; V; J; S g: On substituting the right-hand side of (4.420) into (4.426), we obtain .n/ h

S D

3 X

.'P  I G C ' Ih G /;

(4.427)

 D1

because 'P  D ' h . Here we have taken into account that for isotropic continua the .n/

relation (4.420) contains only three principal invariants I . C G /. Let us show now that 'P can be represented with the help of co-rotational derivatives as follows: 'P ˛ D

3  X .n/ .n/ d @'˛ P  '˛ .I . C G /; / D  : '˛ I G   C hG C dt @  D1

(4.428)

Here we have used the following notation for derivative tensors of the principal .n/

.n/

invariants: I G D @I . C G /=@ C G :. We must show only that the total derivative in the double contractions can be replaced by the co-rotational one: .n/

.n/

I G   C G D I G   C hG :

(4.429)

Indeed, for isotropic continua, the derivative tensors I G have the form (4.323), and .n/

.n/

they are proportional to tensors E, C G and C 2G ; thus, all the tensors I G are coaxial .n/

.n/

with C G , and hence I G and C G are commutative. According to the representation (4.426), we obtain .n/

.n/

.n/

.n/

I G   C G h D I G   C G  I G   Zh  C G  I G   C G  ZTh : .n/

(4.430)

Since I G and C G are commutative, the second and third summands on the righthand side of (4.430) are canceled in pair; thus, formulae (4.429) and (4.428) really hold.

274

4 Constitutive Equations

Determine the co-rotational derivatives Ih G in relation (4.427) with use of for.n/

mula (4.426) (the formula still holds after substitution S ! I G ) and also the formula of differentiation of the principal invariants (4.323): D 0;

Ih1G

Ih2G

@I2

D

!h

.n/

.n/

.n/

D .EI1  C G /h D E ˝ E   C hG  C hG ;

.n/

@C G .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

Ih3G D . C 2G  I1 C G C EI2 /h D C hG  C G C C G  C hG  I1 C hG  IP1 C G C IP2 E .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

D C hG  C G C C G  C hG  I1 C hG  . C G ˝ E/   C hG C E ˝ .EI1  C G /   C hG : (4.431) Here we have applied the rule (4.429) of substitution of the co-rotational derivative in place of the total one. On substituting (4.428) and (4.431) into (4.427) and collecting like terms, we get the representation (4.425) of the constitutive equations, where the tensors 4 PG and .n/

S G have the forms 4

PG D

3 X

P ıˇ /; .'ˇ I G ˝ I G C ' 4e

.n/

S G D

;ˇ D1

3 X @' I G ; @  D1

(4.432)

where 4e P1

4e P3

D 0;

.n/

4e P2

D E ˝ E  ;

.n/

D .E ˝ C G /.3214/ C . C G ˝ E/.1423/ C I1 .E ˝ E  / .n/

.n/

.E ˝ C G C C G ˝ E/: N

(4.433)

Remark 1. As will be shown in Sect. 4.10.7, the use of co-rotational derivatives in relations (4.425) ensures their correctness, that cannot be achieved by applying the usual derivative with respect to time to models Cn and Dn . t u Remark 2. The relations (4.421) and (4.425) can be applied formally to problems of continuum mechanics without relations (4.420). In this case, there appears a question whether the relations satisfy the principle of material symmetry. This principle .n/

holds true for the models Cn and Dn in rates (4.425), because all the tensors S and .n/

.n/ .n/

C G D f A ; g g are H -invariant. One should prove that relations (4.421) ‘in rates’ for the models An and Bn satisfy the principle of material symmetry as well (Exercise 4.8.8). u t

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

275

Using the representations (4.380) and (4.383) of constitutive equations for the models Cn and Dn of isotropic continua and differentiating them, we get the relation ‘in rates’ .n/

.n/

.n/

.n/ h 2 CG

S h D P 1 E C P 2 C G C 'P3 C 2G C

.n/

which is another form of relations (4.425). Here 1

D '1 C '2 I1 C '3 I2 ;



2

.n/

.n/

.n/

C '3 . C hG  C G C C G  C hG /; (4.434) ˛

are expressed by the formulae

D '2 C I1 '3 ;

.n/

I D I . C G /:

(4.435)

Remark 3. The relations ‘in rates’ for the models An , Bn (4.420) and for the models Cn , Dn (4.425) can be written in the single form .n/

.n/

.n/

.n/

P G D A; B; C; D; T hG D 4 PG . C G ; /   C hG C T G ;

(4.436)

if for G D A; B the parameter h takes on the only value h D fg (the total derivative with respect to t), and for G D C; D the parameter h takes on the values h D fU; V; J; S g but only isotropic continua are considered. .n/

Then we can use the generalized relations (4.105) between C hG and L and rewrite Eqs. (4.436) as follows: .n/ T hG

.n/

.n/

D 4 P Gh .F; /   L C T G P ;

(4.437)

Where 4

.n/

.n/

P Gh D 4 PG   4 B Gh ;

(4.438)

.n/

and 4 B Gh are determined by (4.106). .n/

According to formula (2.373) for the co-rotational derivatives T hG of a symmetric second-order tensor, we can rewrite the relation (4.437) in the form .n/ T G

.n/

.n/

.n/

.n/

P  ZGh  T G  T G  ZTGh D 4 P Gh   L C T G :

Here we have introduced the notation ( ZGh D

l0;

(4.439)

G D A; B;

(4.440)

Zh ; G D C; D; .n/

.n/

and taken into account that for G D A; B, by definition, T hG D T G . The tensors Zh are defined by formulae (2.374).

276

4 Constitutive Equations .n/

According to the continuity equation, the total derivative T G takes the form (see formulae (3.20)) .n/

.n/ .n/ @ T G C r  .v ˝ T G / D  T G : @t On substituting (4.441) into (4.439), we obtain

(4.441)

.n/

.n/ .n/ .n/ .n/ .n/ @ T G P C r  .v ˝ T G / D 4 P Gh   L C ZGh  T G C  T G  ZTGh C  T G ; @t G D A; B; C; D; h D f; U; V; J; S g; n D I; : : : ; V; (4.442) – the generalized divergence form of the constitutive equations ‘in rates’ for ideal solids. t u

4.8.13 Application of the Principle of Material Symmetry to Fluids Up to now in this section we considered only solids. Applying the principle of material symmetry to fluids, we obtain the following important theorem. Theorem 4.31. For the models An (n D I; II; IV; V) of ideal fluids, the constitutive equations (4.165), (4.168), satisfying the principle of material symmetry, can be written in the form p .n/1 T D G ; n  III  1=.IIIn/ ı .III  n/; p D p.I3 ; / D I3 .@ =@I3 / .n  III/3 I3 .n/

 D @ =@; D

.n/

.I3 . G/; /;

(4.443a) (4.443b) (4.443c) (4.443d)

where T D pE; 2

(4.444a)

p D p.; / D  .@ =@/; D  .; /;

(4.444b) (4.444c)

 D @ =@:

(4.444d)

The function p introduced by formulae (4.443b) and (4.444b) is called the pressure. The tensor T proportional to the unit tensor (see relation (4.444a)) is called spherical.

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

277 ı

H According to Definition 4.5, in each reference configuration, including K, a fluid ı

has the unimodular symmetry group G s D U . Therefore, the relations (4.278) must ı



ı

hold for each H-transformation from K to K, where H 2 G s D U (i.e. det H D 1). .n/

.n/

Notice that the tensors T and C are not H -indifferent relative to the unimodular I

group (except T, which is absolutely H -indifferent), therefore relations (4.279) do not hold for a fluid, and for the function in place of the indifference condition (4.279c) we have a more general condition .n/

ı

.n/

. C  ; / 8H 2 G s D U:

(4.445)

The unique solution of Eq. (4.445) is the scalar function

depending on the

. C ; / D

.n/

determinant of the corresponding energetic deformation measure G, i.e. .n/

1 det . C C E/;  n  III

D

! D

.n/

.I3 . G/; /:

(4.446) .n/

Indeed, one can immediately verify that the third principal invariant I3 . G/ always satisfies the condition (4.445): .n/

.n/

I3 . G  / D I3 . G/

8H 2 U;

(4.447)

because at n D I, II, IV, V this invariant can be expressed in terms of the densities ı ratio = (see formulae (4.80)): .n/

I3 . G/ D

1 ı .=/IIIn ; 2 .n  III/

(4.448)

ı

and densities  and  are always absolutely H -indifferent (see Sect. 4.6.3). Moreover, the unimodular H -transformations have been introduced as transformations of ı a reference configuration, which hold the only scalar function – density , therefore .n/

the functions I3 . G/ are unique solutions of Eq. (4.445). All other solutions are only .n/

functions of I3 . G/, in particular the function (4.446). Thus, we have shown that for a fluid the free energy really has the form (4.446). On substituting (4.448) into (4.446), we obtain that for all the models An can be expressed only in terms of  and : D

.n/

.I3 . G/; / D



1 ı .=/IIIn ;  3 .n  III/

   .; /:

(4.449)

278

4 Constitutive Equations .n/

Substituting (4.446) into (4.168), we determine the tensors T .n/

T D

@ .n/

D

@C

@ @I3 : @I3 .n/ @G

(4.450)

Here we have taken into account that, by formula (4.43), from the derivative with .n/

.n/

.n/

.n/

respect to C one can pass to the derivative with respect to G, because @ G=@ C D . .n/

According to formula (4.323) for @I3 =@ G: .n/

.n/

.n/ .n/

1

@I3 . G/=@ G D I3 . G/ G

;

(4.451)

we can rewrite the constitutive equations (4.450) in the form .n/ @ I3 G 1 : @I3

(4.452)

.n/ @ I3 . G/.III  n/ @I3

(4.453)

.n/

T D

Introducing the notation for pressure: pD

.n/

and substituting the expression (4.448) for  in terms of I3 . G/ into (4.453), from (4.452) we obtain the exact representations (4.443a) and (4.443b) of constitutive equations for a fluid. Let us show that all four relations (4.443a) can be represented in the form .n/

(4.444a). To see this, one should rewrite the expression for G 1 with the help of formulae (4.31) and (4.47): I

G1 D 2G D 2FT  F; IV

G1 D U1 D

II 1 G1 D U D  .FT  O C OT  F/; 2

1 1 .F  O C OT  F1T /; 2

(4.454)

V

G1 D 2G1 D 2F1  F1T ; .n/

and then substitute these expressions and the expressions for energetic tensors T (see Table 4.1) into relations (4.443a). As a result, we obtain a linear equation with respect to the tensor T for each n: nDIW

n D II W n D IV W

FT  T  F D pFT  F; 1 1 T (4.455) .F  T  O C OT  T  F/ D  p.FT  O C OT  F/; 2 2 1 1 1 .F  T  O C OT  T  F1T / D  p.F1T  O C OT  F1T /; 2 2 nDVW F1  T  F1T D pF1  F1T :

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

279

A solution of each of the linear equations is the exact expression (4.444a), and this solution is unique (indeed, each equation in (4.455) can be rewritten in components, for example, with respect to the basis ri ; then we get a linear nonsingular equation for components of the tensor T ij and the equation has a unique solution). Formula (4.444b) for pressure follows from (4.453), if we express invariant .n/

I3 . G/ in terms of  according to (4.448): @ .; / @ I3 .III  n/ @ @I3 ı 1 1 @  I3III  n .n  III/3=.IIIn/ I3 .III  n/ D 2 .@ =@/: (4.456) D @ .III  n/

pD

Finally, we should show that the derived constitutive equations (4.443a) satisfy the material symmetry principle, i.e. from (4.443a) we can get the relationship .n/ 

T D

p .n/1 G n  III

8H 2 U:

(4.457) .n/

Notice that the pressure p, according to (4.443b), is a function only of I3 . G/ and , and hence p is always H -indifferent: p  D p. And since the stress tensor T is also absolutely H -indifferent, so for all the models An the relation (4.444a) holds 

in any configuration K: ı



T D p  E



8H 2 U; H W K ! K:

(4.458) 

On substituting (4.458) into relations (4.455) written in an arbitrary configuration K: 









FT  T  F D p  FT  F; (4.459)        1 T   1 n D II W .F  T  O C OT  T  F/ D  p.FT  O C OT  F/ etc. 8H 2 U; 2 2 nDIW

we find that the relations are identically satisfied (independently of transformations ı



of the tensors F and O during the passage from K to K!). Relations (4.459) are exactly equivalent to (4.457), hence the relations (4.457) ı



really hold for any unimodular H -transformations H W K ! K. N .n/

Remark 1. Theorem 4.31 shows that in place of energetic deformation tensors C .n/

for fluids we use only the corresponding measures G; in other words, the models An and Bn for fluids coincide. For solids, the models An and Bn prove to be considerably distinct. t u

280

4 Constitutive Equations

Theorem 4.32. For the models Cn (n D I; II; IV; V) of ideal fluids, the constitutive equations (4.165), (4.168) satisfying the principle of material symmetry can be written in the form 8 .n/ ˆ p .n/1 ˆ ˆ ˆ S D  n  III g ; ˆ ˆ 1=.IIIn/ ı < @  p D p.I3 ; / D I3 .III  n/; .n  III/3 I3 @I3 ˆ ˆ  D @ =@; ˆ ˆ ˆ ˆ .n/ : D .I3 . g /; /;

.4:460a/ .4:460b/ .4:460c/ .4:460d/

where expressions of the tensor T for all the models are the same, and they have the form (4.444). .n/

.n/

The third invariant in (4.460) is determined by tensor g : I3 D I3 . g /. H A proof of the theorem is similar to the proof of the preceding theorem. According to the principle of material indifference (4.211) for the potential the following relation must hold for the models Cn of an ideal fluid: .n/

. A ; O; / D



.n/

. A  ; O; /

,

ı

8H 2 G s D U:

(4.461)

.n/

Tensors A are not H -invariant under unimodular transformations, therefore relation (4.352d) does not hold. The unique solution of Eq. (4.461) is the scalar function depending on the third .n/

principal invariant of the quasienergetic deformation measure g , which is con.n/

nected to A by relation (4.71): D

.n/

.I3 . g /; /:

(4.462)

.n/

From (4.448) it follows that the invariant I3 . g / can be expressed in terms of the ı densities ratio =: .n/

.n/

I3 . g / D I3 . G/ D

1 ı .=/IIIn : 3 .n  III/

ı

(4.463) .n/

Since the densities  and  are always H -indifferent, the function I3 . g / is also H -indifferent under any unimodular transformations, i.e. .n/

.n/

I3 . g  / D I3 . g /

ı

8H 2 G s D U;

(4.464)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

281

hence, the function (4.462) is a solution of Eq. (4.461). Uniqueness of the solution (4.462) follows from formula (4.463), because unimodular H -transformations have been defined as transformations retaining the only scalar function being the density . On substituting the function (4.462) into the constitutive equations (4.174), we ı

.n/

find the quasienergetic stress tensors S and tensor S: .n/

S D

.n/

@ .n/

D

@A

@ @I3 . g / ; .n/ @I3 @g

(4.465)

ı

S D .@ =@O/  0: .n/

(4.466)

.n/

Here we have taken into account that @ g =@ A D . With use of formula (4.323) for the derivative of the tensor determinant, we can rewrite the constitutive equations (4.465) in the form .n/

S D

.n/ @ I3 g 1 : @I3

(4.467)

Introducing in formula (4.467) the notation for pressure e pD

.n/ @ I3 . g /.III  n/ D p; @I3

(4.468)

we obtain the representation (4.460a) of constitutive equations for the models Cn of an ideal fluid. .n/

.n/

Since, by (4.463), the invariants I3 . g / and I3 . G/ are coincident, the pressure p introduced by (4.453) coincides with e p introduced by (4.468). On substituting the .n/

expression (4.463) for  in terms of I3 . g / into (4.468), we get formula (4.460c). Let us show now that all four constitutive equations (4.460a) are really equivalent to the representation (4.444a) for the Cauchy stress tensor T. According to the .n/

definitions of quasienergetic tensors S (see Table 4.2) and quasienergetic measures .n/

g (4.71), constitutive equations (4.460a) can be rewritten in the form nDIW n D II W n D IV W nDVW

V  T  V D pV  V; 1 .V  T C T  V/ D pV; 2 1 1 .V  T C T  V1 / D pV1 ; 2 V1  T  V1 D pV1  V1 :

(4.469)

One can verify that an unique solution of each of the linear equations for the tensor T is the expression (4.444a).

282

4 Constitutive Equations ı

Since the rotation tensor of stresses S for an ideal fluid is identically zero (see ı

(4.466)), so, by the definition (4.52) of the tensor S, the following relation must be satisfied for an ideal fluid: V  T  V1 D V1  T  V:

(4.470)

But the relation is always satisfied, because we have shown that the stress tensor for an ideal fluid is spherical (see (4.444a)). It remains to show that the derived constitutive equations (4.460a) satisfy the principle of material symmetry, i.e., according to (4.460a), the following relations must hold: .n/ p  .n/1 S D  g 8H 2 U; (4.471) n  III Or at n D I W at n D II W









V  T  V D p  V2 ;    1   .V  T C T  V/ D p  V etc. 2

(4.472)



But the relations (4.472) are really satisfied, because the tensor T, by (4.458), is also ı



spherical under any H -transformations H W K ! K. N Remark 2. As follows from Theorem 4.32, the models Cn and Dn for ideal fluids .n/

coincide, because the energetic deformation tensors A do not appear in the consti.n/

tutive equations (they are included only in the corresponding measure g ).

t u

As a result of the section, it should be noted that  all the models An , Bn , Cn and Dn of ideal fluids are equivalent and can be written

in the form (4.444);  nevertheless, for a fluid, the constitutive equations (4.443) and (4.460) can also

be applied to different problems of continuum mechanics. Remark 3. Proofs of Theorems 4.31 and 4.32 show that in order for the principal of material symmetry to hold, it is not necessary that the tensors involved in the constitutive equations be either H -indifferent or H -invariant; the principal requirement consists in that they must be transformed in the consistent way during the passage ı



from K to K (see formulae (4.459) and (4.472)).

t u

Let us note another property. The principle of material symmetry has allowed us to find explicit forms of the constitutive equations for ideal solids and fluids; these are formulae (4.444), (4.311), (4.313a), (4.314a), (4.322), and (4.319), as distinct from the implicit forms (4.168) and (4.174).

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

283

4.8.14 Functional Energetic Couples of Tensors Let us return now to the representation (4.13) of energetic couples of tensors and explain a method of construction of new energetic couples with the help of the theory of invariants suggested above. Consider an arbitrary tensor function of tensor U .e/

.e/

C D C .U/;

(4.473)

such that: .e/

.e/

(a) it forms a symmetric second-order tensor C D C T , (b) it is indifferent relative to the full orthogonal group I (i.e. it is an isotropic function), (c) it is differentiable. For such tensor functions, we have the theorem (see, for example [12]) that the functions can be represented in terms of two tensor powers U and U2 in the form .e/

C D !1 E C !2 U C !2 U2 ;

(4.474)

where !˛ are scalar functions of the principal invariants !˛ D !˛ .I1 .U/; I2 .U/; I3 .U//;

(4.475)

being differentiable with respect to I˛ within the whole domain of definition. In a particular case, functions !˛ may be the derivatives of one scalar function: !˛ D @I@˛ !.I1 ; I2 ; I3 /, but this is not necessary. Determine the derivative of (4.474) with respect to t: .e/ 

.e/

P C D 4 C U   U:

(4.476) .e/

.e/

Here 4 C U is a fourth-order tensor being the derivative tensor of C with respect to the tensor argument U: 4

.e/

C U D !2 C!3 .U˝ECE˝U/C

3 X

.!1˛ EC!2˛ UC!3˛ U2 /˝I˛U ; (4.477)

˛D1

where !ˇ ˛ D

!ˇ ; @I˛

I˛U D

@I˛U ; ˛ D 1; 2; 3: @U

(4.478)

284

4 Constitutive Equations .e/

.e/

The stress tensor T being coupled to C must satisfy the equation .e/

.e/

IV

P T   C  D T   U:

(4.479) .e/

Substituting (4.476) into (4.479), we obtain the equation for T : .e/

IV

.e/

T D 4 CU   T:

(4.480) .e/ .e/

Thus, we have found the family of new energetic couples of tensors . T ; C/, they will be called the functional energetic couples. .n/

.n/

The couples . T ; C/ when n > IV are particular cases of the functional energetic couples, because according to the Hamilton–Cayley theorem (see [12]) U3 D I1 .U/U2  I2 .U/U C I3 .U/E;

(4.481)

we can express any integer power Un in terms of only the first two tensor powers: E, U and U2 in the form (4.474), where the coefficients !˛ are functions of the principal invariants (4.475). .e/

.e/

In a similar way, we can introduce the functional energetic couples . C ; T / as follows: .e/

.e/

C D C .U1 /;

II

.e/

TD4 C

.e/

U1

 T ;

(4.482)

.e/

.e/

where 4 C U1 is the derivative tensor of C with respect to the tensor argument U1 in the form (4.477). .n/

.n/

The energetic couples . T ; C/ when n 6 II are particular cases of the functional .e/

.e/

couples . T ; C /. We will say that functional energetic couples are of the A-type, if the following initial conditions hold: .e/

C.E/ D 0;

.e/

.e/

C .E/ D 0

.e/

(4.483) .n/

(the tensors C and C are generalizations of energetic deformation tensors C ). If the initial conditions .e/

C.E/ D E;

.e/

C .E/ D E

(4.484)

4.8 Corollaries of the Principle of Material Symmetry and Constitutive Equations

285

are satisfied, then we will say that the functional couples are of the B-type (such .e/

.e/

.n/

tensors C and C are generalizations of energetic deformation measures G). We can also introduce functional quasienergetic couples of tensors of the C -type as follows: .e/

.e/

.e/

.e/

.e/

A.E/ D 0; .e/

IV

.e/

.e/

.n/

4

.e/

SD4 A

.e/

A

(4.485)

A .E/ D 0;

II

S D 4 AV   S ;

The derivative tensors 4 A V and C ! A and U ! V.

.e/

A D A .V1 /;

A D A.V/;

V1

.e/

V1

 S :

have the form (4.477) with substitutions

.n/

Quasienergetic couples . S ; A / when n > IV are particular cases of the functional quasienergetic couples. Functional quasienergetic tensor couples of the D-type differ from ones of the C -type by the initial conditions .e/

A.E/ D E;

.e/

A .E/ D E:

(4.486)

The use of the functional couples of tensors considerably extends the scope for constructions of advanced models of continuous media. However, methods of construction of these models are the same as the ones considered above for models An , Bn , Cn , and Dn . Therefore we will not consider functional couples of tensors in this book, but they are kept in mind for future investigations.

Exercises for 4.8 4.8.1. Using the relation (2.31) between the tensors U and V, show that principal invariants of the tensors always coincide: I˛ .U/ D I˛ .V/;

˛ D 1; 2; 3:

4.8.2. Using the results of Exercise 4.8.1, show that principal invariants of the ten.n/

.n/

sors G and g coincide: .n/

.n/

I˛ . G/ D I˛ . g /;

˛ D 1; 2; 3; n D I, II, IV, V;

286

4 Constitutive Equations

where .n/

I3 . G/ D

1 .I3 .U//nIII : .n  III/3 .n/

.n/

Show that principal invariants of the tensors A and C are coincident as well: .n/

.n/

I˛ . A / D I˛ . C/;

˛ D 1; 2; 3:

4.8.3. Using the results of Exercise 4.2.4, show that principal invariants of the ten.n/

.n/

sors G and g can be expressed in terms of eigenvalues ˛ of the tensors U and V as follows: .n/

.n/

I1 . G/ D I1 . g / D .n/

3 1 X nIII  ; n  III ˛D1 ˛ .n/

I2 . G/ D I2 . g / D

.n/

.n/

I3 . G/ D I3 . g / D

1 .1 2 3 /nIII ; .n  III/3

1 ..1 2 /nIII C .2 3 /nIII C .1 3 /nIII /: .n  III/2

4.8.4. Using the results of Exercise 4.8.3, show that .n/

@I1 . G/ D nIV ; ˛ @˛

.n/

@I2 . G/ 1 D C nIII /; nIV .nIII  ˇ @˛ n  III ˛

.n/

@I3 . G/ 1 D nIV .ˇ  /nIII ; ˛ ¤ ˇ ¤ ¤ ˛: @˛ n  III ˛ 4.8.5. Show that for quasilinear models An and Bn (see Sect. 4.8.7) the tensors PG (G D A; B) coincide with the quasilinear elasticity tensor 4 M introduced in Sect. 4.8.6: 4

4

PA D 4 PB D 4 M;

and for linear models An the tensor 4 M has the form 4

e M D J 4 M;

ı

J D =;

where the constant-tensor 4 M depends only on producing tensors of the corresponding symmetry group Gs . 4.8.6. Using the results (4.444) of Theorems 4.31 and 4.32, show that the Gibbs .n/

.n/

free energy  G (4.126) and enthalpy i

G

(4.137) of an ideal fluid have the forms

4.9 Incompressible Continua .n/

.n/

287

! .n/ p I1 . G 1 / AD C 1 ; .n  III/ n  III ! .n/ p I1 . g 1 / C 1 : .n  III/ n  III

p C ; .n  III/

DD B D

.n/

C D

.n/

4.8.7. Prove Theorem 4.27 for a transversely isotropic continuum. .n/ .n/ .n/ .n/

.n/

4.8.8. Using properties of H -indifference of the tensors C, T , G, T  , C  and .n/

G  (see Sect. 4.7.4), prove that the relations (4.421) satisfy the principle of material symmetry. 4.8.9. Show that for the models An and Cn the relations (4.395) and (4.396) beı.n/ .n/ ˛,  ˛

tween eigenvalues of the stress tensors  .n/

ı.n/ ˛

˛ D 

'3 2 ..nIII/  1/2 ; .nIII  1/ C n  III ˛ .n  III/2 ˛ D '1 C '2 I1 C '3 I2 ;  2 D '2 C I1 '3 ;

D 1

and ˛ have the forms

1

C

.n/

.n/

where '˛ D .@ =@I˛ . C // for the models An and '˛ D .@ =@I˛ . A // for the models Cn : 4.8.10. Using the results of Exercise 4.8.9, prove Theorem 4.28 for the models An and Cn of isotropic elastic continua. 4.8.11. Show that relationships (4.350) for the models Bn can be derived from relations (4.335) and (4.338) for the models An after substitution of formula (4.43) into the relations.

4.9 Incompressible Continua 4.9.1 Definition of Incompressible Continua Consider a particular case of continua, that is frequently used in practice. Definition 4.15. A continuum is called incompressible, if in each actual configuraı tion K its density  coincides with the density  in a reference configuration: ı

 D  D const 8K 8t > 0:

(4.487)

288

4 Constitutive Equations

There are both incompressible fluids and incompressible solids. An incompressible fluid is often called simply a fluid; and if a fluid is not incompressible, then it is called a compressible fluid or gas. As follows from the continuity equation in Lagrangian variables (3.8), for incompressible continua, a volume of each elementary domain dV remains unchanged: ı

D

H)

ı

dV D d V ;

(4.488)

and, hence, a volume jV .t/j of the domain occupied by an incompressible continı

uum, remains unchanged as well: jV .t/j D jV j D const. It follows from the continuity equation (3.8) that for an incompressible continuum there is an additional condition imposed on the deformation gradient: det F D 1:

(4.489)

According to the polar decomposition (2.137), we find that for an incompressible continuum the stretch tensors always have the unit determinant: det V D det F  det OT D det F D 1;

det U D 1:

(4.490)

Substituting the formulae into (4.42) and (4.71), we get that determinants of all the energetic and quasienergetic deformation measures are always equal to a constant value: 

.n/

det G D det

1 UnIII n  III 

.n/

det g D det



1 VnIII n  III

D

1 1 det UnIII D ; .n  III/3 .n  III/3

 D

1 ; .n  III/3

(4.491)

n D I, II, IV, V:

(4.492)

4.9.2 The Principal Thermodynamic Identity for Incompressible Continua .n/

.n/

For incompressible continua, not all components of the tensors F, U, V, C and A are independent, because there are additional conditions (4.489)–(4.492) imposed on the tensors. Thus, for incompressible continua, the constitutive equations (4.168) and (4.174) do not hold, because their derivation was based on independence of the .n/

.n/

differentials d C or d A. In order to find a general form of constitutive equations for incompressible continua we should transform formulae (4.121) and (4.123) of the principal

4.9 Incompressible Continua

289

thermodynamic identity. Introduce two additional scalar functions, values of which are identically zero: .n/

.n/

. G/ D det G 

1 D 0; .n  III/3

.n/

.n/

. g / D det g 

1 D 0; .n  III/3

(4.493)

and, according to formulae (4.491) and (4.492), derivatives of the functions with respect to their tensor arguments have the forms @ .n/

.n/ .n/

D det G G 1 D

@G

.n/

.n/ 1 G 1 ; 3 .n  III/

@ .n/

D

@g

.n/1 1 g : 3 .n  III/

(4.494)

.n/

Since . G/ and . g / are identically zero, their total differentials are identically zero as well: d D 0. Rewrite the principal thermodynamic identity in forms (4.121) and (4.123) with adding the zero summand e p d : d

.n/

.n/

e p d  T   d C C  d C w dt D 0;

(4.495)

and d

.n/

.n/

ı

e p d  S   d A  S   d OT C  d C w dt D 0;

(4.496)

where .n/

.n/

.n/

.n/

d D .@ =@ G/   d G D .@ =@ g /   d g D 0:

(4.497)

Here we have introduced the non-zero function e p called the indeterminate Lagrange multiplier.

4.9.3 Constitutive Equations for Ideal Incompressible Continua If we consider the model An of ideal continua, then (4.165):

is a function in the form

.n/

D

. C ; /:

(4.498)

Determining the differential of the function , substituting the result into PTI (4.495) and taking formula (4.497) into account, we get 0

.n/

1

.n/ e p @ C T B@  dC C A @ .n/    .n/ @C @G



 w @ C  d C dt D 0: @ 

(4.499)

290

4 Constitutive Equations .n/

Due to the choice of Lagrange multiplier e p , increments of the tensors C may .n/

be considered again as independent ones and the expressions at d C, d and dt in (4.499) can be equated to zero; then we get the constitutive equations for the models An of incompressible ideal continua 8 .n/ .n/ .n/ ı ˆ 1 ˆ ˆ < T D .p=.n  III// G C .@ =@ C /;  D @ =@; ˆ ˆ .n/ ˆ :  D . C ; /; w D 0;

(4.500)

where p De p =.n  III/2 : .n/

(4.500a)

.n/

Replacing the tensors C by G in (4.499), we obtain constitutive equations for the models Bn of incompressible ideal continua: 8 .n/ .n/ .n/ ı ˆ 1 ˆ ˆ < T D .p=.n  III// G C .@ =@ G/;  D @ =@; ˆ ˆ .n/ ˆ :  D . G; /: w D 0; If we consider the models Cn of ideal continua, then (4.171): D

(4.501)

is a function in the form

.n/

. A ; O; /;

(4.502)

and PTI (4.496) takes the form ! ! ı! .n/ @ @ p @ w S e S T    dO C  C  d C dt D 0:  d A C .n/   .n/ @O  @  @g @A (4.503) @

.n/

.n/

The introduction of Lagrange multiplier e p allows us to consider the increments d A, d O, d and dt as independent; then from (4.503) we get the constitutive equations for the models Cn of incompressible ideal continua 8 .n/ .n/ .n/ ı ˆ ˆ S D .p=.n  III// g 1 C .@ =@ A /; ˆ ˆ ˆ ˆ ı <ı S D .@ =@O/; ˆ ˆ  D @ =@; ˆ ˆ ˆ .n/ ˆ :  w D 0; D . A ; O; /:

(4.504)

4.9 Incompressible Continua

291 .n/

.n/

In a similar way, the substitution A ! g yields the constitutive equations for the models Dn of incompressible ideal continua: 8 .n/ .n/ .n/ ˆ ı ˆ ˆ S D .p=.n  III// g 1 C .@ =@ g /; ˆ ˆ ˆ <ı ı S D .@ =@O/; ˆ ˆ  D @ =@; ˆ ˆ ˆ ˆ .n/ :w D 0; D . g ; O; /:

(4.504a)

Just as for a fluid (see formulae (4.443)), the function p is called the hydrostatic pressure. Notice that for incompressible continua the pressure p as well as the indeterminate Lagrange multiplier e p is an additional unknown function that cannot be .n/

.n/ .n/ .n/

expressed in terms of  and C, or A , G, g .

4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids Relations (4.500) and (4.504) hold for both incompressible solids and fluids. Let us consider incompressible fluids. Theorem 4.33. For the models An (n D I; II; IV; V) of ideal incompressible fluids, the constitutive equations (4.500) satisfying the principle of material symmetry can be written in the form 8.n/ .n/ ˆ ˆ < T D .p=.n  III// G 1 ;  D @ =@; ˆ ˆ : D ./;

(4.505)

and the equation for the Cauchy stress tensor for all the models An has the form T D pE;

(4.506)

where p is an unknown function being independent of . H Since a considered continuum is a fluid, so, as shown in the proof of Theorem 4.31, its free energy (4.446)):

.n/

can depend only on the third invariant I3 . G/ and  (formula D

.n/

. C ; / D

.n/

.I3 . G/; /:

(4.507)

292

4 Constitutive Equations

This assertion holds for all fluids: compressible and incompressible. But for incom.n/

pressible fluids the invariant I3 . G/ is always a constant (formula (4.491)); thus, in this case may change only with changing temperature, i.e. the function (4.507) can be written in the form D

./:

(4.508)

.n/

Then @ =@ C  0, and relations (4.500) for incompressible fluids really take the form (4.505). Just as in Theorem 4.31, one can prove that formula (4.506) follows from (4.505). Notice that for incompressible fluids, formula (4.444b) connecting the pressure p to the density  does not hold, and p is an unknown function. N Theorem 4.34. For the models Cn (n D I; II; IV; V) of ideal incompressible fluids, the constitutive equations (4.504) satisfying the principle of material symmetry can be written in the form 8.n/ .n/ ˆ ˆ S D .p=.n  III// g 1 ; < (4.509)  D @ =@; ˆ ˆ : D ./; where equations for the tensor T for all the models are the same and have the form (4.506), and the pressure p is an unknown function. H A proof of the theorem is analogous to the proof of Theorem 4.33. N

4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases Let us consider now an incompressible ideal solid, which is called an incompressible elastic continuum. According to the principle of material symmetry, such a continb As before, b s in undistorted configuration K. uum must have some symmetry group G ı

a reference configuration K is assumed to be undistorted. If we consider the models An of incompressible solids (4.500), then the free energy .s/

.n/

.n/

. C ; /, according to .n/

formula (4.304), is a function of the invariants I˛ . C / of the tensor C relative to ı

the group G s : D

.n/

.n/

.I1.s/ . C /; : : : ; Ir.s/ . C /; /:

(4.510)

4.9 Incompressible Continua

293 ı

However, for incompressible solids, the density  does not vary . D /, and hence, .n/

according to formula (4.81), the principal invariants I˛ . C / are connected by one relation .n/

.n/

.n/ 1 I1 . C / I2 . C / C I3 . C / C D 0; n  III .n  III/2

(4.511)

.n/

i.e. among three invariants I˛ . C / (˛ D 1; 2; 3), only two ones are independent. .n/

Since, as shown in Sect. 4.8.4, the principal invariants I˛ . C/ can always be included .s/

ı

.n/

in the functional basis I˛ . C / of any orthogonal group G s , so for incompressible continua the functional basis contains only .r 1/ invariants and the function (4.510) has the form D

.n/

.n/

.s/ .I1.s/ . C /; : : : ; Ir1 . C/; /:

(4.512)

For incompressible isotropic and transversely isotropic continua, the third principal .n/

.n/

invariant I3 . C / is usually eliminated from the complete functional basis I˛ . C / (see formulae (4.297) and (4.298)), and for orthotropic incompressible continua, .O/

.n/

the invariant I6 . C / is excluded from the set (4.296). On substituting the free energy (4.512) into (4.500), we obtain the following general representation of the constitutive equations for incompressible elastic continua in the tensor basis: r1 .n/ .n/ @I.s/ p .n/1 X .s/ T D G C ' .I1.s/ . C /; : : : ; Ir1 . C/; / ; .n/ n  III  D1 @C

.n/

(4.513)

where ı

' D .@ =@I.s/ /;

(4.514)

which is similar to the representation (4.311), (4.312). In particular, the constitutive equations for an isotropic incompressible elastic continuum has the form (compare with (4.322)): .n/

T D 1

ı

p .n/1 G C n  III

1E

D '1 C '2 I1 ;

' D .@ =@I /; D 1; 2I

2

D



.n/

2 C;

D '2 ; .n/

(4.515) .n/

.I1 . C /; I2 . C /; /:

294

4 Constitutive Equations

If we assume the normalization conditions (4.326) and (4.327), where Te D p e , then the potential is subject to the additional conditions (compare them with (4.329)): '1 .0; 0; e / D p0  p e ;

.0; 0; e / D 0;

(4.515a)

where p0 is a value of the hydrostatic pressure p in a natural state. The models Bn of incompressible elastic continua give constitutive equations, .n/

.n/

which are similar to (4.513) and differ from them only by the substitution C ! G: .n/

T D .n/

r1 .s/ p .n/1 X @I G C ' ; .n/ n  III  D1 @G .n/

(4.516)

ı

.s/ . G/; / D .@ =@I.s/ /: ' .I1.s/ . G/; : : : ; Ir1

In particular, for the models Bn of isotropic incompressible elastic continua, we have .n/

T D

p .n/1 G C n  III

1E

.n/

 '2 G;

ı

1

D '1 C '2 I1 ; ' D .@ =@I / D ' .I1 ; I2 ; /;

D

.n/

.n/

.I1 . G/; I2 . G/; /: (4.517)

If we assume the normalization conditions (4.326) and (4.327), then the potential is subject to the additional conditions (compare with (4.347a)) '1 .a1 ; a2 ; e / C .2=.n  III//'2 .a1 ; a2 ; e / D p0  pe ;

.a1 ; a2 ; e / D 0: (4.518)

Here we have used the notation (4.347b). For the models Cn of incompressible ideal solids, we can use the theorems of Sect. 4.8.9 and represent the free energy in the form (4.378), then we get the following representation of the constitutive equations (4.504) in the tensor basis: .n/

S D

r1 p .n/1 X @I.s/ ; g C ' .n/ n  III  D1 @A

.n/

ı

' D ' .I˛.s/ .OT  A  O/; / D 

@ .s/

(4.519)

; ˛ D 1; : : : ; r  1:

@I

ı

In this case the relation (4.504) for the rotation tensor S is automatically satisfied. If we consider the models Cn of incompressible isotropic continua, then, according to (4.380), the free energy can be represented as a function of three principal

4.9 Incompressible Continua

295

.n/

invariants I˛ . A /. However, for incompressible continua, these three invariants are also connected by the relation (4.511): .n/

I3 . A / C

.n/ .n/ 1 1 I . A/ D 0; I2 . A / C 1 n  III .n  III/2

i.e. only two ones of them are independent, therefore only on two invariants; for example, on I1 and I2 : D

.n/

can be considered to depend

.n/

.I1 . A/; I2 . A /; /:

(4.520)

Then the constitutive equations (4.509) take the form .n/

S D

p .n/1 g C n  III .n/

1E



.n/

2 A;

(4.521) .n/

' D ' .I1 . A /; I2 . A /; /: Just as for compressible continua, the constitutive equations for the models An (4.515) and for the models Cn (4.521) can be connected only by the rotation tensor .n/

.n/

O, because I˛ . C / D I˛ . A / (see Exercise 4.8.2). The constitutive equations for the models Dn (4.504a) in the tensor basis are written as follows: .n/

S D

r1 .s/ p .n/1 X @I g C ' ; .n/ n  III  D1 @g

.n/

ı

' D ' .I˛.s/ .OT  G  O/; / D 

@ .s/

(4.522)

; ˛ D 1; : : : ; r  1:

@I

These equations are equivalent to relations (4.516) and can be obtained from them with the help of multiplication by the tensor O (see (4.382)).

4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids Just as in Sect. 4.8.10, constitutive equations (4.513) for the models An , (4.516) for the models Bn , (4.519) for the models Cn and (4.522) for the models Dn can be

296

4 Constitutive Equations .n/

written in a general form – with the help of the generalized energetic tensors T G , .n/

.n/

C G and G G (4.91): .n/

TG D 

r1 p .n/1 X .s/ GG C '  I G : n  III  D1

(4.523)

Here we have used the notation (4.385) and (4.386). These relations can be reduced to the form (4.387) by expressing the Cauchy stress tensor in the explicit form: .n/

e G .F; /; T D pE C F

(4.524)

where .n/

e G .F; /  F

r1 X

.n/

' H  G :

(4.525)

 D1

The remaining notation has been introduced in Sect. 4.8.10.

4.9.7 Linear Models of Ideal Incompressible Elastic Continua As an example, let us consider linear models of ideal elastic isotropic continua, for which the free energy (4.515) or (4.516) is a quadratic function of the correspond.n/

.n/

ing invariants I . C / or I . A /. For isotropic linear models An of incompressible continua, has the same form (4.335) as well as for corresponding compressible isotropic continua; but constitutive equations change due to the appearance of hydrostatic pressure p; ı



ı

D

ı

0

C N0 C

.n/ .n/ l1 C 2l2 2 .n/ N 1 . C /; I1 . C /  2l2 I2 . C / C mI 2

.n/ .n/ p .n/1 T D G C .m N C l1 I1 . C //E C 2l2 C : n  III

.n/

(4.526)

ı

Since we consider incompressible continua, so J D = D 1. In order that the constitutive equations satisfy the conditions (4.326), (4.327) in the natural configuration Ke at Te D p e , in Ke the relations (4.515a) must hold; in this case they take the form p0 D p e C m; N where p0 is the value of the function p in Ke .

N 0 D 0;

(4.527)

4.9 Incompressible Continua

297

For isotropic linear models Bn of incompressible continua, the free energy has the form (4.348) as well as for compressible continua, and the constitutive equations are similar to (4.526): ı



ı

D

ı

0

C

0

C

.n/ .n/ l1 C 2l2 2 .n/ I1 . G/  2l2 I2 . G/ C mI1 . G/; 2

(4.528)

.n/ .n/ p .n/1 T D G C .m C l1 I1 . G//E C 2l2 G: n  III

.n/

The normalization conditions (4.326) and (4.327) lead to the restrictions (4.518) on the form of the potential , which in this case yield relations between 0 , m, l1 , l2 and p0 : p0 D p e C m C

3l1 C 2l2 ; n  III

0

D

3.3l1 C 2l2 / ı

2.n  III/2



3m ı

.n  III/

:

(4.529)

Relations (4.526), (4.527) for models An and (4.528), (4.529) for models Bn are completely equivalent (just as for compressible continua), and the models can be .n/

.n/

obtained one from another with the help of substitution C D G  .1=.n  III//E (see Exercise 4.9.1). Each of the models includes three independent constants m, l1 and l2 that is by one constant more than for the corresponding models An (4.335) and Bn (4.350) of compressible continua. Considering for the models (4.526) and (4.528) special cases of values of the constants, for example, assuming m D 0 and m N D 0, from the normalization conditions (4.527) and (4.529) we find new values of p0 , N 0 and 0 , while the models An and Bn are not equivalent. The models Bn (see (4.528)) are most widely used in practice; in these models we assume that l1 C 2l2 D 0;

l2 D . =2/.1  ˇ/.n  III/2 ;

m D .1 C ˇ/.n  III/; (4.530)

where and ˇ are two new independent constants; and from the normalization conditions (4.529) we have 0

ı

D 6 =;

p0 D p e C .3  ˇ/.n  III/2 :

(4.531)

Relations (4.528) in this case can be written in the following form (see Exercise 4.9.2):   .n/ .n/ ı ı  D  0 C .1 C ˇ/..n  III/I1 . G/  3/ C .1  ˇ/..n  III/2 I2 . G/  3/ ; ! ! .n/ .n/ .n/ p .n/1 1Cˇ 2 T D G C .n  III/ C .1  ˇ/I1 . G/ E  .1  ˇ/ G : n  III n  III (4.532)

298

4 Constitutive Equations

The set of relations (4.532) at n D IV is called the model of K.F. Chernykh, which in the particular case when ˇ D 1 reduces to the Bartenev–Hazanovich model. The set of relations (4.532) at n D V is called Mooney’s model, which in the special case when ˇ D 1 reduces to Treloar’s model.

4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis For incompressible continua, three eigenvalues i are not independent, because they are connected by the relations (4.490): D I3 .U/  1 D 1 2 3  1 D 0:

(4.533)

As shown in Sect. 4.9.2, we can introduce the Lagrange multiplier p and rewrite the principal thermodynamic identity (4.495) in the form d

 p d 

3 X .n/  ˛ nIV d˛ C  d D 0; ˛

(4.534)

˛D1

where d D

3 X @ d˛ D 0: @˛ ˛D1

(4.535)

Here we have used the expression (4.417a) for the stress power w.i / . Then d˛ (˛ D 1; 2; 3) can be considered as independent. Determining the derivatives of functions (4.533): @ 1 D ˇ  D ; ˛ D 1; 2; 3; ˛ ¤ ˇ ¤ ¤ ˛; @˛ ˛

(4.536)

and substituting them into (4.534), we obtain    3  X ı.n/ p @ @ nIV    ˛ ˛  d˛ C  C  d D 0: @˛ ˛ @ ˛D1

(4.537)

As a result, we find the following constitutive equations for incompressible continua (the models Bn and Dn ): .n/

ı

 ˛ D p˛IIIn C IVn ˛

@ ; @˛

˛ D 1; 2; 3; n D I; : : : ; V;

(4.538)

4.9 Incompressible Continua

299

or with taking (4.417) into account: ı

˛ D p C ˛

@ ; @˛

.n/ ˛

D ˛IIIn ˛ :

(4.539)

Thus, we have proved the following theorem. Theorem 4.35. The constitutive equations (4.515) and (4.521) for incompressible isotropic continua can be written in the form of three scalar relations (4.539) between ˛ and ˛ . As an example of constitutive equations in the eigenbasis, consider the potentials (4.532) for incompressible continua and pass from I˛ to ˛ by the formula from Exercise 4.8.3: BI ; BV W BII ; BIV W

D

D

2 2 C .1 C ˇ/.2 1 C 2 C 3  3/ C .1  ˇ/.21 C 22 C 23  3/;

(4.540)

1 1 C .1 C ˇ/.1 1 C 2 C 3  3/ C .1  ˇ/.1 C 2 C 3  3/:

(4.541)

0

0

Here we have used the incompressibility condition 1 2 3 D 1. The second invariant I2 can be expressed as follows: I

2 2 2 2 2 2 2 2 4I2 .G/ D 2 1 2 C 2 3 C 1 3 D 1 C 2 C 3 :

(4.542)

On substituting (4.540) and (4.541) into (4.539), we get the corresponding constitutive equations in the eigenbasis: 2 2 BI ; BV W ˛ D p C 2 ..1  ˇ/.21 C 22 C 23 /  .1 C ˇ/.2 1 C 2 C 3 //; (4.543) 1 1 BII ; BIV W ˛ D p C ..1  ˇ/.1 C 2 C 3 /  .1  ˇ/.1 1 C 2 C 3 //: (4.544)

Exercises for 4.9 4.9.1. Show that the linear models An (4.526), (4.527) and Bn (4.528), (4.529) of incompressible continua are equivalent. 4.9.2. Prove that relations (4.532) follow from (4.528) under the assumptions of the model (4.530).

300

4 Constitutive Equations

4.10 The Principle of Material Indifference 4.10.1 Rigid Motion Let us consider now one more basic principle mentioned in Sect. 4.1, namely the principle of material indifference. This principle consists in the assertion that constitutive equations of a continuum cannot depend on the choice of a frame of reference, where they are considered. The principle of material symmetry supposes a change of reference configuraı



tion: K ! K, but, according to the principle of material indifference, we consider special motions of actual configuration: K ! K0 , namely rigid motions. Define the rigid motion of a continuum. Take in configuration K (Fig. 4.5) an ı arbitrary point M0 with its radius-vector x0 (it is not to be confused with x) and b which differs from K only by rotation as a rigid whole pass to configuration K, about the point M0 . Then an arbitrary vector x  x0 of K becomes the vector b x  x0 b and in the configuration K; b x  x0 D .x  x0 /  Q.t/:

(4.545)

b respectively, and Q Here x and b x are coordinates of the same point M in K and K, is the orthogonal rotation tensor: Q D Q ij eN i ˝ eN j ;

QT D Q1 ; QT  Q D E;

(4.546)

where Q ij is the orthogonal matrix of rotation. b to the following configuration K0 , which differs from K b only by a Pass from K parallel carry with vector a, then a radius-vector of the point M in K0 is determined as follows: x0 D b x C a D x0 C a C .x  x0 /  Q.t/; (4.547)

Fig. 4.5 The rigid motion of a continuum

4.10 The Principle of Material Indifference

or

  x0 .X i ; t/ D x00 .X0i ; t/ C x.X i ; t/  x0 .X0i ; t/  Q.t/;

301

(4.548)

where X i and X0j are Lagrangian coordinates of the points M and M0 , respectively, and x00 D x0 C a is the radius-vector of the point M0 in K0 , which is the ı

instantaneous center of rotation. The rotation tensor Q.t/ and radius-vectors x.t/ ı and x0 .t/ depend on time t. The motion, satisfying Eq. (4.548), is called the rigid motion of a continuum.

4.10.2 R-indifferent and R-invariant Tensors Let an arbitrary tensor A.x/ be given at a point x in configuration K: A.x; t/ D Ai1 :::in ri1 ˝ : : : ˝ rin ;

(4.549)

where ri is the local basis in K. A tensor A is called indifferent relative to the rigid motion (4.548) or Rindifferent, if under the transformation from K to K0 the corresponding tensor A0 is determined by the same rules as the tensor A, but only in K0 : A0 .x0 ; t/ D A0i1 :::in r0i1 ˝ : : : ˝ r0in I for A0 we have Ai1 :::in .x; t/ D A0i1 :::in .x0 ; t/;

(4.550)

where r0i is the local basis in K0 . Components of R-indifferent tensor A move together with the local basis vectors, i.e. they are ’frozen’ into the continuum. A tensor A is called invariant relative to the rigid motion (4.548) or Rinvariant, if (4.551) A0 .x0 ; t/ D A.x; t/: The concept of invariance of a tensor relative to the rigid motion, does not coincide with the definition of a tensor as an invariant object, because under the transformation K ! K0 we obtain, in general, different tensors A and A0 , and their coincidence is a certain condition imposed on the tensor A. Differentiation (4.548) with respect to X i yields @x @x0 D r0i D  Q D ri  Q: i @X @X i

(4.552)

302

4 Constitutive Equations

Thus, in the rigid motion, vectors of a local basis are transformed by the law (4.552), and the definition of R-indifferent tensor (4.550) takes the form A0 .x0 ; t/ D A0i1 :::in .x0 ; t/r0i1 ˝ : : : ˝ r0in D Ai1 :::in .x; t/ri1  Q ˝ : : : rin  Q: (4.553) For a scalar ', R-indifference means that

for a vector b:

' 0 .x0 ; t/ D '.x; t/;

(4.554)

b0 .x0 ; t/ D b.x; t/  Q D QT  b.x; t/;

(4.555)

and for a second-order tensor A: A0 .x0 ; t/ D A0i1 i2 r0i1 ˝ r0i2 D Ai1 i2 ri1  Q ˝ ri2  Q D QT  Ai1 i2 ri1 ˝ ri2  Q D QT  A  Q: (4.556) According to (4.548), the vector e x D x  x0 is R-indifferent. Consider other examples.

4.10.3 Density and Deformation Gradient in Rigid Motion Density  is a R-indifferent scalar. Indeed, in going from K to K0 Eq. (3.6) yields  dV D 0 dV 0 ;

(4.557)

where 0 and dV 0 are the density and elementary volume in the configuration K0 ; but in the rigid motion dV 0 D dV and, hence, 0 .x0 ; t/ D .x; t/:

(4.558)

Local basis vectors ri are R-indifferent due to (4.552). The deformation gradient F0 in K0 is determined as well as F in K: ı

ı

F0 D r0i ˝ ri 0 D QT  r0i ˝ ri D QT  F:

(4.559)

The deformation gradient is not a R-indifferent tensor, because in the rigid motion ı ı a reference configuration remains unchanged: r0i D ri . The transpose deformation gradient is not R-indifferent as well, because ı

ı

FT0 D ri 0 ˝ r0i D ri ˝ ri   Q D FT  Q:

(4.560)

4.10 The Principle of Material Indifference

303

The inverse gradients are also not R-indifferent: F1T0 D .FT0 /1 D .FT  Q/1 D Q1  FT1 D QT  FT1 ; 10

F

0 1

D .F /

D .Q  F/ T

1

1

DF

Q

T1

DF

1

 Q:

(4.561a) (4.561b)

Local vectors of a reciprocal basis are R-indifferent, because ı

ı

ı

ı

ri 0 D rj 0 ıji D rj 0 ˝ r0j  ri 0 D F1T0  ri 0 D QT  F1T  ri D QT  ri :

(4.562)

The unit tensor E D QT  E  Q D Q T  Q D E is both R-indifferent and R-invariant. With the help of (4.562) we can determine the nabla-operator r 0 in K0 : r 0 D r0i .@=@X i /:

(4.563)

It follows from (4.562) and (4.563) that r 0 D QT  r ;

r D ri

@ @ D eN i i : @X i @x

(4.563a)

4.10.4 Deformation Tensors in Rigid Motion The left Almansi deformation tensor is R-indifferent, because A0 D

1 0 1 .E  F1T0  F10 / D .E0  QT  F1T  F1  Q/ D QT  A  Q: (4.564) 2 2

The right Almansi deformation tensor is R-invariant: ƒ0 D

1 0 1 .E  F10  F1T0 / D .E  F1  Q  QT  F1T / D ƒ: 2 2

(4.565)

The left Cauchy–Green deformation tensor is R-invariant, because C0 D

1 T0 0 1 .F  F  E0 / D .FT  Q  QT  F  E/ D C: 2 2

(4.566)

The right Cauchy–Green deformation tensor is R-indifferent: J0 D

1 0 T0 1 .F  F  E0 / D .QT  F  FT  O  QT  Q/ D QT  J  Q: 2 2

(4.567)

304

4 Constitutive Equations

The right stretch tensor U is R-invariant. Indeed, by the definition of the polar decomposition, formula (4.560) of transformation of F has the form F0 D QT  F D QT  O  U D O0  U0 I but since QT  O is orthogonal and the polar decomposition is unique, so U D U0

and O0 D QT  O:

(4.568)

Thus, the tensor U is R-invariant. The rotation tensor O accompanying deformation is neither R-indifferent nor R-invariant. In a similar way, using the definition of V and the connection between O0 and Q, we have F0 D QT  F D QT  V  O D V0  O0 D V0  QT  O; i.e. we obtain that QT  V D V0  QT : Therefore, the tensor V is R-indifferent: V0 D QT  V  Q:

(4.569)

From (4.568) and (4.569) we derive the following theorem. .n/

Theorem 4.36. All the energetic deformation tensors C and energetic deforma.n/

tion measures G, which are proportional to UnIII , are R-invariant, and all the .n/

quasienergetic deformation tensors A and quasienergetic deformation measures .n/

g , which are proportional to VnIII , are R-indifferent (n D I; II; IV; V). III

III

The tensors C and A will be considered in Sect. 4.10.7.

4.10.5 Stress Tensors in Rigid Motion The Cauchy stress tensor is R-indifferent. To prove the assertion, use the Cauchy relation (3.47): tn D n  T: By definition, the force vector tn is ‘tracking’, i.e. the vector moves together with the corresponding surface element d† in any motion; and, hence, tn is R-indifferent. The vector n also moves with the surface element; thus, n is R-indifferent as well. Then t0n D n0  T0 D n  Q  T0 D tn  Q D n  T  Q;

(4.570)

4.10 The Principle of Material Indifference

305

and we finally get Q  T0 D T  Q

and T0 D QT  T  Q;

(4.571)

i.e. T is R-indifferent. The Piola–Kirchhoff tensor P is neither R-indifferent nor R-invariant, because q q ı 10 ı 0 P D g=g F  T D g=g F1  Q  QT  T0  Q D P  Q: 0

(4.572)

.n/

Theorem 4.37. All the principal energetic stress tensors T are R-invariant, and .n/

all the principal quasienergetic stress tensors S are R-indifferent: .n/

.n/

T0 D T;

.n/ 0

.n/

S D QT  S  Q;

n D I; : : : ; V:

(4.573)

.n/

H By the definition of the tensors T (see Table 4.1), we have I

I

T0 D FT0  T0  F0 D FT  Q  QT  T  Q  QT  F D T;  II 1  T0 0 T0 D F  T  O0 C OT0  T0  F0 2  II 1 T F  Q  QT  T  Q  QT  O C OT  Q  QT  T  Q  QT  F D T; (4.574) D 2 III

III

T0 D OT0  T0  O0 D OT  Q  QT  T  Q  QT  O D T etc., .n/

and, by the definition of the tensors S (see Table 4.3), we get I

I

S0 D V0  T0  V0 D QT  V  Q  QT  T  Q  QT  V  Q D QT  S  Q;  II 1 0 0 S0 D V  T C T0  V0 2  II 1 T Q  V  Q  QT  T  Q C QT  T  Q  QT  V  Q D QT  S  Q; (4.575) D 2 III

III

S 0 D T0 D QT  S  Q etc. N

306

4 Constitutive Equations

4.10.6 The Velocity in Rigid Motion Theorem 4.38. The velocity v is not R-indifferent. H According to the definition (2.190) and formula (4.548), we get v0 D

d x0 d x00 P T  .x  x0 /; D C .v  v0 /  Q C Q dt dt

(4.576)

where v D d x=dt, and v0 D d x0 =dt is the velocity of the point M0 (which is independent of X i ). It follows from (4.576) that the vector v is not R-indifferent. N

4.10.7 The Deformation Rate Tensor and the Vorticity Tensor in Rigid Motion Theorem 4.38a. The deformation rate tensor D is R-indifferent, and the vorticity tensor W is neither R-indifferent nor R-invariant. H Due to (4.576), we get @v0 L D .r ˝ v/ D r ˝ D QT  r i ˝ @X i 0

T0

i0



@v P T  ri QCQ @X i



P D QT  r ˝ v  Q C QT  Q; P (4.577) D QT  r ˝ v  Q C QT  ri ˝ ri  Q   @v0 @v P T  ri ˝ r i  Q L0 D .r ˝ v/T0 D ˝ ri 0 D Q T ˝ CQ i @X @X i P T  Q: D QT  .r ˝ v/T  Q C Q

(4.578)

On summing (4.577) and (4.578), we find D0 D

1 .r ˝ v/0 C .r ˝ v/T0 D QT  D  Q: 2

(4.579)

The vorticity tensor is transformed as follows: W0 D

1 PTQ ..r 0 ˝ v0 /T  r 0 ˝ v0 / D QT  W  Q C Q 2 P N D QT  W  Q  QT  Q:

(4.580)

4.10 The Principle of Material Indifference

307

4.10.8 Co-rotational Derivatives in Rigid Motion Theorem 4.39. The total derivative of any R-invariant tensor with respect to time is R-invariant. H Let a tensor  be R-invariant, then P 0 D d : N and  dt

0 D 

(4.581) .n/ .n/

Hence, the total derivatives of the energetic deformation tensors C , G and of the .n/

energetic stress tensors T with respect to time are R-invariant. Theorem 4.40. The derivative of any R-indifferent tensor  with respect to time is neither R-indifferent nor R-invariant. H For the tensor  we have P T    Q C QT   P Q .0 / D .QT    Q/ D Q P  Q: N C QT    Q ¤ QT  

(4.582)

III

P are R-invariant, the tensor C D B, defined by relations Since the tensors U and U III

(2.317), is also R-invariant. But the tensor A D Y, defined by (2.318), is neither III

P which is not RR-invariant nor R-indifferent, because A depends on the tensor V, III

indifferent. In place of the tensor A D Y, for isotropic continua one usually use the III

Hencky tensor H (2.167), which is energetically equivalent to A D Y (see Theorem 4.28a)) and which is R-indifferent. Indeed, the eigenvalues ˛ of the tensor U do not change in the motion K ! K0 , because the tensor U remains unchanged itself; but since ˛ are the eigenvalues of the tensor V as well, so V can be represented as follows: V0 D

3 X

˛ p0˛ ˝ p0˛ D QT 

˛D1

3 X ˛D1

From this equation it follows that p0˛ D QT  p˛ ; and the eigenvectors p˛ are R-indifferent.

˛ p˛ ˝ p˛  Q:

308

4 Constitutive Equations

The Hencky tensor (2.167) in K0 has the following form: H0 D

3 X

lg ˛ p0˛ ˝ p0˛ D QT 

˛D1

3 X

lg ˛ p˛ ˝ p˛  Q D QT  H  Q:

˛D1

Thus, the tensor H is really R-indifferent. ı

The Hencky tensor H (2.167) is clear to be R-invariant. To obtain R-indifferent rate characteristics of a R-indifferent tensor, one should use the co-rotational (objective) derivatives with respect to time, which have been introduced in Sect. 2.5. Theorem 4.41. The Oldroyd derivatives of any R-indifferent vector a and of any R-indifferent tensor T are R-indifferent as well. H From (2.350) we get P .a0 /Ol D aP 0  a0  LT0 D .a  Q/  a  Q  QT  L  Q  a  Q  QT  Q P aLQaQ P D QT  .Pa  LT  a/ D QT  aOl ; D QT  aP C a  Q (4.583) and .T0 /Ol D TP 0  T0  LT0  L0  T0 D .QT  T  Q/  QT  T  Q  QT  LT  Q P  QT  L  Q  Q T  T  Q  Q P T  Q  QT  T  Q QT  T  Q  QT  Q P T  T  Q C QT  T  Q P D QT  TP  Q  QT  T  LT  Q  QT  L  T  Q C Q P Q P  T  QDQT  .TT P  LT L  T/  QDQT  TOl  Q: N QT  T  Q (4.584) The Oldroyd derivatives of any R-invariant vector a and of any R-invariant tensor … are neither R-invariant nor R-indifferent. Theorem 4.42. The Cotter—Rivlin derivatives of any R-indifferent vector a and of any R-indifferent tensor T are R-indifferent as well. H From (2.351) we obtain P  QT  a .a0 /CR D aP 0 C LT0  a0 D .aT  Q/ C QT  LT  Q  QT  a C QT  Q P T  a D aT  .Pa C LT  a/ D QT  aCR ; P T  a C QT  aP C QT  LT  a  Q DQ (4.585)

4.10 The Principle of Material Indifference

309

and from (2.352): PTTQ .T0 /CR D TP 0 C LT0  T0 C T0  L0 D QT  TP  Q C Q P C QT  LT  Q  QT  T  Q C QT  Q P  QT  T  Q C QT  T  Q C Q T  T  Q  Q T  L  Q C QT  T  Q   Q T  Q D QT  .TP C LT  T C T  L/  Q D QT  TCR  Q: N

(4.586)

The Cotter–Rivlin derivative of any R-invariant vector (or of any R-invariant tensor) is neither R-invariant nor R-indifferent. Theorem 4.43. The mixed left and right co-rotational derivatives of any Rindifferent tensor T are R-indifferent. H From (2.355) we get P T  T  Q C QT  T  Q P .T0 /d D TP 0  L0  T0 C T0  L0 D QT  TP  Q C Q P T  Q  Q T  T  Q C QT  T  Q  Q T  L  Q QT  L  Q  QT  T  Q  Q P T  Q D QT  .TP  L  T C T  L/  Q D QT  Td  Q CQT  T  Q  Q (4.587) and .T0 /D D TP 0 C L0T  T0  T0  L0T D QT  TP  Q C QT  LT  Q  QT  Q P  QT  T  Q  QT  T  Q  QT  LT  Q  QT  T  Q  QT  Q P CQT  Q T T P P T  T  QCQT  T  QDQ P  .TCL  TT  LT /  QDQT  TD  Q: N CQ (4.588)

The mixed co-rotational derivatives of any R-invariant tensor are neither Rinvariant nor R-indifferent. Theorem 4.44. The rotation tensor OU and the spin of the right stretch tensor U defined by formulae (2.269) are R-invariant. H Since the right stretch tensor U is R-invariant, i.e. it remains unchanged in the ı rigid motion, its eigenvectors pi remain without changes as well: ı

ı

p0i D pi :

(4.589) ı

Due to (4.581), the total derivative of the R-invariant vector p˛ with respect to time is R-invariant as well: P ı ıP p0i D pi : (4.590)

310

4 Constitutive Equations

Thus, we get P ı ı ıP ı 0U D p0i ˝ p0i D pi ˝ pi D U ;

ı

ı

O0U D p0i ˝ eN i D pi ˝ eN i D OU : N (4.591)

Theorem 4.45. The derivative relative to the eigenbasis of the right stretch tensor, applied to any R-invariant vector a or tensor …, is R-invariant. H From (2.357) and (2.358) we get .a0 /U D aP 0 C a0  0U D aP C a  U D aU ; P 0  0  …0 C …0  0 D … P  U  … C …  U D …U : N .…0 /U D … U U (4.592) Theorem 4.46. The rotation tensor OV and the spin of the left stretch tensor V , defined by formulae (2.271), are neither R-indifferent nor R-invariant. H From the relation (2.276) between O, OU and OV we get O0 D O0V  OT0 U:

(4.593)

Since the rotation tensor O is transformed by formula (4.568), we find that QT  O D O0V  OTU ;

O D Q  O0V  OTU D OV  OTU :

Thus, the rotation tensor OV is neither R-indifferent nor R-invariant: O0V D QT  OV :

(4.594)

The spin of the left stretch tensor V , defined by formula (2.271) with the help of the rotation tensor OV , is transformed in the rigid motion as follows: P 0  OT0 D QT  O P V  OT  Q C Q P T  OV  OT  Q 0V D O V V V V P T  Q: D QT  V  Q C Q

(4.595)

Thus, 0V is neither R-indifferent nor R-invariant. N Theorem 4.47. The derivative of any R-indifferent vector a or of any R-indifferent tensor T relative to the eigenbasis of the left stretch tensor is R-indifferent. H Due to (2.361), we get .a0 /V D .a  Q/ C a  Q  0V P C a  Q  Q T  V  Q  a  Q  Q T  Q P D aP  Q C a  Q D .Pa C a  V /  Q D aV  Q;

(4.596)

4.10 The Principle of Material Indifference

311

and PTTQ .T0 /V D TP 0  0V  T0 C T0  0V D QT  TP  Q C Q P  QT  V  Q  Q T  T  Q  Q P T  Q  QT  T  Q C QT  Q  Q PTQ C QT  T  Q  QT  V  Q C QT  T  Q  Q D QT  .TP  V  T C T  V /  Q D QT  TV  Q: N

(4.597)

P  OT accompanying deformation is neither RTheorem 4.48. The spin  D O indifferent nor R-invariant. H According to (4.568), we obtain P 0  O0T D Q P T  O  OT  Q C QT  O P  OT  Q 0 D O T T P  Q: N D Q QCQ

(4.598)

Theorem 4.49. The spin derivative, defined by formulae (2.368) and (2.369), of any R-indifferent object is R-indifferent. H For any R-indifferent vector, from (2.370) we get P C a  Q  QT    Q a0S D aP 0 C a0  0 D aP  Q C a  Q T P  Q D .Pa C a  /  Q D aS  Q: Ca  Q  Q (4.599) In a similar way, for any R-indifferent tensor from (2.371) we obtain TS0 D QT  TS  Q: N

(4.600)

Theorem 4.50. The Jaumann derivative is R-indifferent, if it is applied to any Rindifferent vector or tensor. H For any R-indifferent vector, from (2.364) we get P .a0 /J D .a  Q/J D .a  Q/ C a  Q  W0 D aP  Q C a  Q T T Ca  Q  .Q  W  Q  Q  Q/ D .Pa C a  W/  Q D aJ  Q D .aJ /0 : (4.601) In a similar way, we can verify that the derivative TJ (2.365) is R-indifferent: .T0 /J D QT  TJ  Q: N

(4.602)

Table 4.12 gives summary data on R-indifference and R-invariance of basic vectors and tensors.

312

4 Constitutive Equations Table 4.12 List of basic R-invariant and R-indifferent tensors Tensors and vectors R-indifference R-invariance ı

ı

ri , ri



C

ri , ri

C



E

C

C

F







C

.n/ .n/

C, G, n D I; : : : ; V

.n/ .n/

A, g , n D I; : : : ; V

C



T

C



P







C

S , n D I; : : : ; V

C



.n/

T , n D I; : : : ; V

.n/

v, L, W





D

C



Bh , h D f; U g



C

C



if B is R-invariant Bh , h D fOl; CR; d; D; V; S; J g, if B is R-indifferent OU , U



C

OV , V , 





4.10.9 The Statement of the Principle of Material Indifference Axiom 15 (The principle of material indifference (or the principle of objectivity)). In the rigid motion, all the universal laws of continuum mechanics in the differential form (3.303) and also the constitutive equations (4.156), connecting the active ƒ and reactive R variables, remain unchanged. In other words, if the relations (3.303) and the constitutive equations ƒ D fM.R/

(4.603)

hold in configuration K, then in K0 the following relations must hold: 0

d 0 AN0˛ D r 0  BN ˛0 C 0 C˛0 ; dt

˛ D 1; : : : ; 6;

(4.604)

and also ƒ0 D fM.R0 /; where fM is the same operator.

(4.605)

4.10 The Principle of Material Indifference

313

4.10.10 Material Indifference of the Continuity Equation Let us establish what constraints are imposed by the principle of material indifference on the continuity equation (see the system (3.303) at ˛ D 1). Due to (4.558), the density  is always R-invariant. Thus, by Theorem 4.39, we obtain d d (4.606) 0 .1=0 / D  .1=/: dt dt According to the results of Exercise 4.10.1, r 0  v0 D r  v:

(4.607)

Thus, we have proved the following theorem. Theorem 4.51. The principle of material indifference always remains valid for the continuity equation, i.e. from (3.303) at ˛ D 1 we always get that 0

d dt



1 0



D r 0  v0 :

(4.608)

4.10.11 Material Indifference for the Momentum Balance Equation Consider the momentum balance equation (3.303) at ˛ D 2. At first, prove the following auxiliary assertion. Theorem 4.52. The acceleration vector d v=dt D d 2 x=dt 2 D xR of a material point is neither R-indifferent nor R-invariant, and in the rigid motion it is transformed as follows: 0

P  02 /  .x0  x0 / C 20  .Px0  xP 0 /; xR 0 D QT  .Rx  vP 0 / C xR 00 C . s s 0 s 0

(4.609)

where P T  Q: 0s D Q

(4.610)

H From formula (4.548), for the rigid motion we get x  x0 D .x0  x00 /  QT :

(4.611)

Then formula (4.576) for v0 with taking (4.611) into account yields v0 D .v  v0 /  Q C xP 00 C 0s  .x0  x00 /;

(4.612)

314

or

4 Constitutive Equations

v D v0 C .v0  xP 00 /  QT  Q  0s  .x0  x00 /:

(4.613)

Differentiation of (4.612) with respect to time t gives 0

P C xR 0 C 0  .Px0  xP 0 / C  P  .x0  x0 /: (4.614) xR 0 D vP 0 D .Rx  vP 0 /  Q C .v  v0 /  Q 0 s 0 0 s Going from v to v0 by formula (4.613), we find that P Q P T  Q  0s  .x0  x00 / xR 0 D .Rx  vP 0 /  Q C .v0  xP 00 /  QT  Q P 0  .x0  x0 / D .Rx  vP 0 /  Q C xR 0 C xR 0 C 0  .Px0  xP 0 / C  0

s

0

0

s

P 0  02 /  .x0  x0 / C 20  .Px0  xP 0 /: C . s s 0 s 0

0

(4.615)

The presence of the last three summands in the formula means that the acceleration vector xR is not R-indifferent. It is clear that xR is not R-invariant as well. N Let us consider how the momentum balance equation (3.72) is transformed during the passage from K to K0 . Theorem 4.53. The momentum balance equation in Eulerian variables (3.72), written in K in the form a D r  T C f; (4.616) where (1) the vector a is defined as follows: in K it coincides with the total acceleration xR , and during the passage to K0 it is transformed as a R-indifferent vector: KW

a D xR I

K0 W

a0 D xR  QI

(4.617)

(2) during the passage from K to K0 the vector f is assumed to be R-indifferent: f0 D f  Q;

(4.618)

satisfies the principle of material indifference, i.e. in K0 the momentum balance equation has the form 0 a0 D r 0  T0 C 0 f0 : (4.619) H Indeed, let the vectors f and a be transformed during the passage from K to K0 by formulae (4.617) and (4.618), then we get 0 a0  r 0  T0  0 f0 D 0 xR  Q  r 0  T0  0 f  Q: Since the tensor T is R-indifferent, so, according to the results of Exercise 4.10.2, the divergence r  T is also a R-indifferent vector. Then, taking the equality  D 0 and formula (4.616) into account, we get 0 a0  r 0  T0  0 f0 D .a  r  T  f/  Q D 0:

4.10 The Principle of Material Indifference

315

Thus, formula (4.619) really follows from (4.616). N Remark 1. If one uses the traditional form of the momentum balance equation (3.72), then, according to the principle of material indifference, during the passage to K0 the equation should be written in the form 0 .d v0 =dt/ D r 0  T0 C 0 f0 I but since the velocity v and the acceleration d v=dt are not R-indifferent vectors, this equation does not hold. This fact is one of the paradoxes of continuum mechanics: the traditional form of the momentum balance equation (3.72) does not satisfy the principle of objectivity, i.e. this depends on the choice of a coordinate system. To eliminate this paradox we should define the acceleration vector a in the special Rindifferent way (4.617). The definition (4.617) includes the axiomatic assumption on the existence of an inertial system O eN i such that in configuration K the acceleration a is the second derivative of radius-vector xR . The existence of the inertial system O eN i is guaranteed by Axiom 5 (see Sect. 3.2.1). t u Remark 2. The expression for acceleration a0 in K0 with taking formula (4.615) into account can be written in the form 0 0 0 P 0s  02 a0 D xR 0 C v0  Q  xR 00  . x0  xP 00 /:  s /  .x  x0 /  2s  .P

(4.617a)

Taking the conclusion of Theorem 4.53 and Remark 1 for the theorem into account, we should return to Axiom 5 (see Sect. 3.2.1) and reformulate this axiom in the following way. Axiom 5a. (1) For every continuum B 8t > 0 there exists a vector F .i / .B; t/, called the body inertia force, which in the inertial system O eN i for configuration K has the form KW

F .i / D d I=dt;

(4.620a)

where I is the momentum vector of the body (3.26), and the vector F .i / is transformed during the passage to K0 in the R-indifferent way: K0 W

F 0.i / D F .i /  Q:

(4.620b)

(2) For any pair of bodies B and B1 8t > 0 there exists a vector of bodies interaction force F .B; B1 ; t/, which may be zero-valued and has the following properties:  the vector is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 , B1 D B10 [ B100 ;

316

4 Constitutive Equations

 the vector is R-indifferent during the passage to K0 :

F 0 D F  QI  the vector is inertial, i.e. the summarized vector of external forces acting on

the body B, F .B; t/ D F .B; B e ; t/ (where B e D U n B is the surroundings of the body B) in K is equal to the body inertia force vector: K W F .i / D F :

(4.621)

A corollary of this axiom is the following theorem. Theorem 4.53a. If Axiom 5a is assumed to be valid, then under the transformation from configuration K to K0 the momentum balance law retains its form: K0 W F 0.i / D F 0 :

(4.621a)

This theorem is an integral analog of Theorem 4.53. Axiom 5a ensures that the conditions involved in statements of Theorems 4.53 and 4.53a be satisfied. Thus, due to the acceptance of Axiom 5a, the main purpose of this section (to ensure that the momentum balance equations satisfy the principle of material indifference) has been achieved. Axiom 5 introduced in Sect. 3.2.1 is entirely contained in the generalized Axiom 5a; therefore, all the conclusions, obtained above according to Axiom 5, remain valid.

4.10.12 Material Indifference of the Thermodynamic Laws The following axiom asserts material indifference of the thermodynamic functions. Axiom 7a (Objectivity of thermodynamic functions). The thermodynamic functions U , Q, H , QN  and temperature  are R-indifferent scalars: U 0 D U;

Q0 D Q;

H 0 D H;

QN 0 D QN  ;

 0 D :

(4.622)

It follows from (4.622) that the specific internal energy and specific entropy are R-indifferent scalars: 0 D I (4.622a) e 0 D e; and from (4.622), (3.106), and (3.159) we get that the heat fluxes from mass and surface sources, and also the specific internal entropy production are R-indifferent scalars: 0 qm D qm ;

0 q† D q† ;

q 0 D q  ;

(4.622b)

4.10 The Principle of Material Indifference

317

because the mass m is independent of the rigid motion and d† is always defined in the R-indifferent way. The normal n and relations (3.111) are R-indifferent (see Sect. 4.10.5), then we obtain that the heat flux vector q is R-indifferent: q0 D q  Q;

(4.623)

because the following relation holds: 0 D n0  q0 D n  Q  q0 D n  q D q† : q†

Theorem 4.54. The principle of material indifference holds for the first thermodynamic law, i.e. from the energy balance equation (3.118), written for K in the form 

 de dt

 C v  a  r  .T  v/ C r  q  qm  f  v D 0;

(4.624)

we get the following energy balance equation in K0 : 0

 de 0 dt

 0  0 f0  v0 D 0; C v0  a0  r 0  .T0  v0 / C r 0  q0  0 qm

(4.625)

where a0 is defined by (4.617). H Consider transformations of separate summands on the left-hand side of equation (4.625). Due to Axiom 5a, the momentum balance equation (4.619) holds in K0 ; then, multiplying the left and right sides of this equation by v0 , we get 0 v0  a0 D v0  r 0  T0 C 0 f0  v0 ;

(4.626a)

or 0 v0  a0  r 0  .T0  v0 / C T0   .r 0 ˝ v0 /T  0 f0  v0 D 0:

(4.626b)

Due to Axiom 7a and the results of Exercises 4.10.2 and 4.10.5, we have r 0  q0 D r  q;

0

de 0 de D : dt dt

(4.627)

Since the tensor T0 is symmetric (see Exercise 4.10.6), the results of Exercise 4.10.4 and Theorem 4.38 yield T0   .r 0 ˝ v0 /T D T0   D0 D T   D D T   .r ˝ v/T :

(4.628)

318

4 Constitutive Equations

Modify the left-side expression in (4.625) with taking formulae (4.626b), (4.627) and (4.628) into account: 0

 de 0 dt

 0 C v0  a0  r 0  .T0  v0 / C r 0  q0  0 f0  v0  0 qm

D

de 0  T0   .r 0 ˝ v0 /T C r 0  q0  qm dt

D

de  T   .r ˝ v/T C r  q  qm : dt

(4.629)

Since Eq. (4.624) is satisfied, the heat influx equation (3.124) being its consequence holds as well. On comparing (4.629) with (3.124), we conclude that the expression (4.629) must vanish, and therefore the equation (4.625) really holds. N Theorem 4.55. The entropy balance equation (the second thermodynamic law) satisfies the principle of material indifference, i.e. from (3.303) at ˛ D 4 we get the equation  0 0 q 0 0 0 d 0 .q C q 0 / D 0: (4.630)  r   dt 0 0 m H A proof of the theorem should be performed as Exercise 4.10.7. N

4.10.13 Material Indifference of the Compatibility Equations Let us verify whether equations (3.303) at ˛ D 5 and 6 always satisfy the principle of material indifference. Theorem 4.56. The kinematic equation (3.303) at ˛ D 5 and the dynamic compatibility equation (3.303) at ˛ D 6 always satisfy the principle of material indifference. ı

H Since u0 D x0  x, we have d u0 =dt D d x0 =dt D v0 ; and hence the kinematic equation is always satisfied in K0 : 0 .d u0 =dt/ D 0 v0 :

(4.631)

Let the compatibility equation (3.303) at ˛ D 6 be satisfied in K: 

d FT  r  .F ˝ v/ D 0: dt

(4.632)

4.10 The Principle of Material Indifference

319

Consider the similar expression in K0 : 0

d FT0  r 0  .0 F0 ˝ v0 / dt d FT P  0 FT0  r 0 ˝ v0  .r 0  0 F0 / ˝ v0 : (4.633) D 0  Q C FT  Q dt

Here we have used formula (4.559). With taking this equation and (4.578) into account, the third summand takes the form P 0 FT0  r 0 ˝ v0 D FT  Q  QT  r ˝ v  Q C FT  Q  QT  Q T T P D F  r ˝ v  Q C F  Q:

(4.634)

The fourth summand in (4.633) is zero, because r 0  0 F0 D r0i 

@0 F0 @F D ri  Q  Q T  D r  F D 0: i @X @X i

(4.635)

The last equality in (4.635) is valid due to Theorem 3.24. On substituting (4.634) and (4.635) into (4.633), we finally get   d FT T  r  . F ˝ v / D   F  r ˝ v  .r  F/ ˝ v  Q  dt dt   d FT D   r  .F ˝ v/  Q D 0; (4.636) dt 0 dF

T0

0

0 0

0

because Eq. (4.632) is satisfied by condition. N

4.10.14 Material Indifference of Models An and Bn of Ideal Continua Consider the constitutive equations (4.165) and (4.167) of models An of ideal continua (both solids and fluids) in actual configuration K. Then in configuration K0 these equations, according to the principle of material indifference (4.603)–(4.605), must be written in the form 8 .n/ .n/ .n/ ˆ ˆ ˆ T 0 D .@ =@ C /  F . C 0 ;  0 /; < .n/ (4.637) D . C 0 ;  0 /; ˆ ˆ ˆ :  D @ =@; where

.n/

and F . C 0 ;  0 / are the same tensor functions as in (4.165) and (4.167).

320

4 Constitutive Equations

But Eqs. (4.637) really hold, if the constitutive equations (4.165) and (4.167) are .n/

.n/

.n/

.n/ .n/

.n/

satisfied, because all the tensors T and C are R-invariant, i.e. T 0 D T , C 0 D C ; and the temperature  0 , due to (4.622), is also R-invariant:  0 D . For models Bn , the situation is analogous. Thus, we can formulate the following theorem. Theorem 4.57. For models An and Bn of ideal continua (both solids and fluids), the principle of material indifference is always identically satisfied for any potentials .n/

. C ; / and

.n/

. G; /.

4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua The situation is different when the principle of material indifference is applied to models Cn of ideal continua. Let in configuration K the relations (4.171) and (4.174) be satisfied, then in configuration K0 , according to the principle of material indifference (4.603)–(4.605), the following equations must hold: 8.n/ .n/ .n/ ˆ ˆ S 0 D .@ =@ A 0 /  ˆ. A 0 ; O0 ; /; ˆ ˆ ˆ ˆ .n/ < D . A 0 ; O0 ; /; ˆ ˆ D @ =@; ˆ ˆ ˆ ˆı ı .n/ : S0 D @ =@O0  ˆ. A 0 ; O0 ; /;

(4.638)

ı

where ˆ, ˆ and

are the same tensor functions as in (4.171) and (4.174). .n/

.n/

Since the tensors S and A are R-indifferent (see Theorems 4.36 and 4.37) and the tensor O is transformed during the passage from K to K0 by formula (4.568), ı

so the relations (4.638) hold if and only if the functions ˆ, ˆ, and following equations for any orthogonal tensor Q: 8 .n/ .n/ ˆ ˆQT  ˆ. A ; O; /  Q D ˆ.QT  A  Q; QT  O; /; ˆ ˆ ˆ ˆ ı .n/ ı .n/ ˆ <.1=2/.QT  ˆ. A ; O; / C ˆ. A ; O; /  Q/ .n/ ı ˆ ˆ ˆ D ˆ.QT  A  Q; QT  O; /; ˆ ˆ ˆ .n/ ˆ : .n/ . A ; O; / D .QT  A  Q; QT  O; / 8Q:

satisfy the

.4:639a/

.4:639b/ .4:639c/

4.10 The Principle of Material Indifference

321

Theorem 4.58. For models Cn of ideal continua (both solids and fluids), the principle of material indifference remains valid if and only if the scalar function .n/

. A ; O; / satisfies the condition (4.639c) for any orthogonal tensor Q. H If the principle of material indifference is valid, then all the relations (4.639), including (4.639c), hold. Prove the converse assertion. If the condition (4.639c) is satisfied, then, differentiating the function

.n/

.n/

. A ; O; / with respect to tensor arguments A and O and .n/

ı

taking the relations ˆ D @ =@ A and ˆ D @ =@O into account, we get that the two remaining equations (4.639a) and (4.639b) hold as well; and hence the equations (4.638) remain valid. N On comparing the condition (4.639c) with conditions (4.281) and (4.354), we obtain that for satisfaction of the principle of material indifference, the function .n/

. A ; O; / must be both indifferent under the full group of orthogonal transforma.n/

tions Gs D I with respect to the first argument A and rotation-indifferent under the same group I with respect to the second argument O. Theorem 4.59. For models Cn of ideal solids, the principle of material indifference is always satisfied. H For models Cn , the potential

can always be represented in the form (4.377):

.n/

. A ; O; / D

.n/

.I˛.s/ .OT  A  O/; /:

(4.640a)

To verify that the condition (4.639c) is satisfied, we should substitute the tensor .n/

.n/

QT  A  Q in place of A and the tensor QT  O in place of O on the right-hand side of the relation. After these substitutions we get .n/

.QT  A  Q; QT  O; / D D

.n/

.I˛.s/ .OT  Q  QT  A  Q  QT  O/; / .n/

.I˛.s/ .OT  A  O/; /:

(4.640b)

On comparing the expressions (4.640a) with (4.640b), we find that they coincide; hence the relations (4.639c) hold for any orthogonal tensor Q. N Theorem 4.60. Constitutive equations in the form of models Cn (4.460) for ideal fluid always satisfy the principle of material indifference. .n/

.n/

.n/

H Tensors S are R-indifferent, and tensors g 1 are R-indifferent as well (since g is R-indifferent, the following relation holds: .n/0 1

g

.n/

.n/

.n/

D .QT  g  Q/1 D Q1  g 1  Q1T D QT  g 1  Q:

322

4 Constitutive Equations

Then, multiplying equation (4.460a) by QT from the left and by Q from the right, we get the following relation in configuration K0 : S0 D  The function

p .n/0 1 g : n  III

(4.641)

.n/

.I3 . g /; / (4.460d) always satisfies the condition (4.639c), be-

.n/

cause I3 . g / is an invariant under any orthogonal transformation; therefore the pressure p (4.460b) is a R-invariant function: p 0 D p, and hence the constitutive equations (4.460) satisfy the principle of material indifference. N For models Bn , the situation is analogous. Thus, the constitutive equations for models An , Bn , Cn and Dn of ideal continua, which have been derived in Sect. 4.8, satisfy the principle of material indifference.

4.10.16 Material Indifference for Incompressible Continua All the constitutive equations derived above for incompressible fluids (4.505), (4.509) and incompressible solids (4.513), (4.521) satisfy the principle of material indifference. Indeed, the constitutive equations for incompressible continua differ from the corresponding constitutive equations for compressible continua by the ad.n/

.n/

ditional summand .p=.n  III// G 1 or .p=.n  III// g 1 . But this summand, similarly to the model of ideal compressible fluid (see Theorem 4.59), is really consistent with the principle of material indifference.

4.10.17 Material Indifference for Models of Solids ‘in Rates’ Theorem 4.61. The models An and Bn (4.421) of ideal solids ‘in rates’ automatically satisfy the principle of material indifference, and the models Cn and Dn (4.425) ‘in rates’ for isotropic continua satisfy the principle only if h is a corotational derivative from the following set: h D fV; J; S g:

(4.642)

The principle of material indifference does not hold for the derivative h D U in eigenbasis of the right stretch tensor and for the total derivative with respect to time d=dt. .n/

.n/

.n/

.n/

H (1) Since all the tensors T and C G D f C; Gg are R-invariant, the relations (4.421) remain unchanged in rigid motions K ! K0 and really satisfy the principle of material indifference.

4.10 The Principle of Material Indifference

323

(2) Consider constitutive equations (4.425) for models Cn and Dn in the form (4.434). Under the transformation from K to K0 , these equations take the form .n/

.n/

.n/

.n/

. S h /0 D '1h 0 E C '2h 0 C 0G C '3h 0 C 0G2 C '20 . C hG /0 .n/

.n/

.n/

.n/

C '30 .. C hG /0  C 0G C C 0G  . C hG /0 /; .n/

.n/

.n/

.n/

C G D f A ; g g: (4.643)

.n/

Tensors S and C G are R-indifferent (see Theorems 4.36 and 4.37), then, due to .n/

.n/

Theorems 4.43, 4.49, and 4.50, the co-rotational derivatives S h and C hG are also R-indifferent, where h is included in the set (4.642): .n/

.n/

. S h /0 D QT  S h  Q;

.n/

.n/

. C hG /0 D QT  C hG  Q:

(4.644)

.n/

The functions '˛ depend on invariants I˛ . C G / (see (4.380)), and, according to .n/

the results of Exercise 4.10.5, invariants I˛ . C G / are R-invariant scalar functions. Therefore the functions '˛ are also R-invariant, i.e. '˛0 D '˛ ; and, by Theorem 4.39, we get .'P ˛ /0 D .'˛h /0 D '˛h : (4.645) Substitution of (4.644) into (4.434) yields  .n/ .n/ .n/ QT  S h  Q D QT '1h E C '2h C G C '3h C 2G  .n/ .n/ .n/ .n/ .n/ C'2 C hG C '3 . C hG  C G C C G  C hG /  Q:

(4.646)

Thus, the relations (4.644) hold in K0 . For the co-rotational derivative h D U and the total derivative with respect to time, the relations (4.644) and hence (4.646) are not valid, and the principle of material indifference does not hold. N

Exercises for 4.10 4.10.1. Using formulae (4.578) and (4.560), show that the divergence of the velocity is both R-indifferent and R-invariant: r 0  v0 D r0i 

@v0 D E   r 0 ˝ v0 D r  v: @X i

324

4 Constitutive Equations

4.10.2. Show that the divergence of any R-indifferent second-order tensor B (and of any R-indifferent vector b) is R-indifferent: r 0  B0 D r0i 

@B0 D QT  r  B; @X i

r 0  b0 D r  b:

4.10.3. Show that for any R-indifferent vector b, its curl is also R-indifferent: r 0  b0 D QT  r  b: 4.10.4. Prove that the contraction of any two R-indifferent tensors A and B gives a scalar being both R-indifferent and R-invariant: A0   B0 D A   B: 4.10.5. Show that the principal invariants of any symmetric R-indifferent tensor A are both R-indifferent and R-invariant: I˛0 .A0 / D I˛ .A0 / D I˛ .A/;

A0 D QT  A  Q:

4.10.6. Show that if a second-order tensor B is symmetric and R-indifferent, then the tensor B0 is symmetric as well. 4.10.7. Prove Theorem 4.55 with use of the method given in the proof of Theorem 4.54. 4.10.8. Show that from (4.622b) we get R-indifference of the dissipation rate w (3.167): w0 D w : 4.10.9. Let there be a scalar function of a tensor argument: D  .A/ and A0 D OT  A  O; where O is a second-order tensor being independent of A. Prove that @ .A/ @ .A/  O: D OT  0 @A @A

4.11 Relationships in a Moving System 4.11.1 A Moving Reference System In continuum mechanics, one frequently considers not only a rigid motion of an actual configuration, investigated in Sect. 4.10, but also a rigid motion of a

4.11 Relationships in a Moving System

325

Fig. 4.6 The moving reference system O 0 eN0i for K

coordinate system, methods of description for which are analogous. Let us consider such a motion. In Sect. 2.1.7 we have introduced a moving system O 0 eN 0i (2.39), which is characterized by the radius-vector x0 .t/ of the point O 0 (the center of rotation) and the orthogonal tensor Q.t/ of rotation of the system O 0 eN 0i about a fixed (immovable) system O eN i (Fig. 4.6), where eN 0i D Q  eN i D Qij eN j :

(4.647)

The position of a point M in the system O eN i is defined by its radius-vector x, and in the system O 0 eN 0i – by the radius-vector e x D x  x0 D e x i eN 0i ;

(4.648)

e x 0 D QT  e x;

(4.649)

(see (2.26)). Introduce the vector x in whose coordinates e x in the basis eN i are the same as coordinates of the vector e the moving basis eN 0i : i

e xQDe x i eN 0i  Q D e x i eN i D e x 0i eN i ; x0 De

(4.650)

e x 0i D e xi :

(4.651)

i.e.

Equations (4.647) and (4.648) yield (see (2.27)) j e x D x  x0 D .x i  x0i /Nei D e x i eN 0i D e x i Qi eN j ;

xj ; x i  x0i .t/ D Qji .t/e

@x i D Qji .t/: @e xj

(4.652)

326

4 Constitutive Equations

4.11.2 The Euler Formula Let us differentiate formula (4.648): e xP D xP  xP 0 :

(4.653)

Since xP D v and xP 0 D v0 are velocities of the point M and the point O 0 , respectively, relative to the fixed system O eN i , according to formula (4.649) fore x we obtain v D v0 C .e x 0  QT / :

(4.654)

Differentiating the producte x 0  Q and using formula (4.649) once more, we find that P  QT /: v D v0 C e xP 0  QT e x  .Q

(4.655)

P  QT is skew-symmetric (see Sect. 2.4.10) Since the tensor Q is orthogonal, then Q and it can be represented in terms of the corresponding vector !s in the form P  QT D !s  E D Q  Q P T: s  Q

(4.656)

The vector !s is in general not coincident with the vorticity vector ! introduced by ı

formula (2.227); they coincide in the case when during the passage from K to K a continuum executes only a rigid motion, whose rotation tensor coincides with the tensor Q of a moving system. According to (4.656), formula (4.655) takes the form v D v0 C e xP 0  QT e x  s I

(4.656a)

and the following relations hold: e x  s D Ts  e x D s  e x D .!s  E/  e x D !s e x:

(4.657)

According to formulae (4.650), we can introduce the vector xP 0  QT D e xP i eN i  QT D e xP i eN 0i ; vr  e

(4.658)

which is called the relative velocity of a point M in the moving system O 0 eN 0i . Components of the vector vr in basis eN 0i are the derivatives e xP i of components e x i of the 0 radius-vector e x in the same moving basis eN i ; in other words, vr is determined by the same rules as the velocity v D xP D xP i eN i in a fixed system O eN i . According to (4.656)–(4.658), formula (4.655) takes the form x C vr : v D v0 C !s e

(4.659)

4.11 Relationships in a Moving System

327

The velocity v is called the total velocity of a point M, and ve D v0 C !s  e x – the translational velocity. Formula (4.659) called the Euler formula means that the total velocity of a material point is equal to a sum of the translational and relative velocities.

4.11.3 The Coriolis Formula Similarly to the acceleration vector of a material point M in a fixed system O eN i : xR D xR i eN i ;

(4.660)

let us introduce the relative acceleration vector of a point M in the moving system O 0 eN 0i : xR i eN 0i : (4.661) ar D e By (4.647) and (4.650), the vector ar can be written as follows: ar D e xR 0  QT : xR i Q  eN i D e

(4.662)

Theorem 4.62 (of Coriolis). The acceleration xR of a material point M in a fixed coordinate system and the relative acceleration ar in a moving system are connected by the Coriolis formula xR D ar C ae C ac ; (4.663) where ae D vP 0 C ! P s e x C !s  .!s e x/;

(4.664)

ac D 2!s  vr D 2!s  .e xP 0  QT /;

(4.665)

ae is called the translational acceleration, ac – the Coriolis acceleration, and xR – the total acceleration. H Consider formula (4.655) for the velocity v and differentiate this expression with respect to t: PTC P s e P xR D vP D vP 0 C e xR 0  QT C e xP 0  Q x C s  e x:

(4.666)

Here we have used formula (4.656) for the spin s . Modify the third summand on the right-hand side P T De P T D .Q P  QT /  .e e xP 0  QT  Q  Q xP 0  QT / D s  vr D !s  vr ; (4.667) xP 0  Q

328

4 Constitutive Equations

and the fourth summand according to (4.656) P s e  x D .!s  E/  e xD! P s  E e xD! P s e x:

(4.668)

The last summand on the right-hand side of (4.666) can be transformed according to (4.649) and (4.658): P T/ s  e xP D s  .e x 0  QT / D s  . e xP 0  QT / C s  .e x0  Q P T // D !s  vr C s  .s  e x  .Q  Q x/ D s  vr C s  .e x /: D !s  vr C !s  .!s e

(4.669)

On substituting formulae (4.667)–(4.669) into (4.666), we obtain Eq. (4.663). N Remark 1. Due to the definition (2.336) of co-rotational derivatives ah , expressions (4.658) and (4.662) for the relative velocity vr and the relative acceleration ar can be represented as co-rotational derivatives aQ of the vector e x De x i eN i 0 in the or0 thonormal moving basis hi D eN i : xQ e xP i eN 0i ; vr D e

ar D e x QQ D vQ xR i eN 0i : r e

(4.670)

In Sect. 2.1.7 we have derived the relations (2.50) and (2.53) between the moving basis eN 0i and basese ri and e ri : e je eN 0i D P i rj ;

e ij eN 0j ; e ri D P

e ij D @e ej ; Q x i =@X

e j eN 0j ; e ri D Q i

(4.671)

e ij D @X e i =@e P xj ;

e ij , according to (2.44), are independent of t; then fore ij and P where the matrices Q mulae (4.670) can be written in the basis e ri as well: vr D e vire vrie ri D e ri ; e vir D

d j ei .e x P j /; dt

ar D e aire arie ri D e ri ;

e air D

d 2 j ei de vri .e x Pj/ D : dt 2 dt

(4.672)

These formulae are exactly analogous to representations of the velocity and accelı eration in the fixed basis eN i and in the basis ri of Lagrangian coordinates: d 2x dx ı ı ı ı D xR i eN i D aN i eN i D ai ri ; (4.673) D xP i eN i D vN i eN i D vi ri ; a D dt dt 2 d d  i  @X j  ı i d2 i d ıi d 2  i  @X j  ı x vi D x i D a D x D v D ; x :  dt dt @x i dt 2 dt dt 2 @x i vD

4.11 Relationships in a Moving System

329

Remark 2. For moving bases of the type eN 0i , which are connected to eN i with the help of the orthogonal tensor Q by formula (4.647), in Sect. 2.4.10 we have derived formula (2.266); this formula applied to eN 0i takes the form ePN 0i D !s  eN 0i :

(4.674)

Resolving the vector !s for the basis eN 0i : !s D !s0i eN 0i , we get ! P s D .!s0i eN 0i / D !P s0i eN 0i C !s0i ePN 0i D !s  .!s0i eN 0i /:

(4.675)

Since !s  !s D 0, we obtain ! P s D !Q s ;

!Q P s0i eN 0i ; s !

(4.676)

N 0i . where !Q s is the co-rotational derivative (2.336) in the basis e

t u

4.11.4 The Nabla-Operator in a Moving System e in a In Sect. 2.1.7 we have proved Theorem 2.2 that the nabla-operators r and r moving coordinate system are coincident: e; r Dr

@ e De r ri ; ei @X

r D ri

@ ; @X i

(4.677)

e i are curvilinear spatial coordinates (2.45), and e ri and e ri are local bases in where X i e K relative to the coordinates X introduced by (2.48) and (2.53). According to relations (4.671), the nabla-operator r can be represented in bases eN i and eN 0i as follows: r D eN i

 xj  @ @ ik m 0s @e N e D ı ı P ms k @x i @x i @e xj @ @ @ j D ı i k ıms Pkm Pi eN 0s j D ısj eN 0s j D eN 0j j : @e x @e x @e x

(4.678)

Here we have used the formulae of passage from one orthonormal basis to another: eN i D ı i k eN k ;

eN 0m D ıms eN 0s ;

(4.679)

and taken into account that both the matrices Qji and Pji are orthogonal, and Pki Plj ı kl D ı ij :

(4.680)

330

4 Constitutive Equations

As follows from (4.678) and (4.672), (4.673), in a moving system O 0 eN 0i its orthonormal basis eN 0i plays the same role as the basis eN i in a fixed system, and local bases e ri i and e ri are similar to bases r and ri relative to the nabla-operator, because r D eN i

@ @ @ e D eN 0i @ D e D ri Dr ri ; e @x i @X i @e xi @X i ı

(4.681)

ı

and they are analogous to bases ri and ri relative to the derivative with respect to time.

4.11.5 The Velocity Gradient in a Moving System Applying the relations (4.681) to formula (4.656a), we obtain e ˝ v r  .r e ˝e r ˝vDr x/  s :

(4.682)

Here we have taken into account that v0 and s are independent of coordinates. Since @ e ˝e x j eN 0j / D eN 0i ˝ eN 0i D E (4.683) r x D eN 0i i ˝ .e @e x (similarly to r ˝ x D E), formula (4.682) takes the form e ˝ v r  E  !s : r ˝vDr

(4.684)

In particular, since the tensor s is skew-symmetric, so e  vr : r vDr

(4.685)

4.11.6 The Continuity Equation in a Moving System On substituting the relation (4.685) into the continuity equation in Eulerian variables (3.15), we obtain e  vr I d=dt D r (4.686) P so or, since the co-rotational derivative Q of a scalar coincides with , 1  

!Q e  vr Dr

(4.687)

4.11 Relationships in a Moving System

331

is the continuity equation in a moving system O 0 eN 0i , which has the same form as the corresponding Eq. (3.15) in a fixed system O eN i if all the values are considered e. relative the system O 0 eN 0i , i.e. with substitutions v ! vr and r ! r

4.11.7 The Momentum Balance Equation in a Moving System For the momentum balance equation, the situation is different: substitution of the Coriolis formula (4.663) and the relation (4.681) into (3.72) yields e  T C f: .e xR 0  QT C ae C ac / D r

(4.688)

Here we have taken into account that the stress tensor T is defined in actual configuration K and is independent of the choice of a coordinate system (but components of eij with respect to the basis eN 0 : T D T eij eN 0 ˝ eN 0 depend on this choice). the tensor T i i j According to representation (4.670) for the acceleration ar , we can rewrite equation (4.688) in the form e  T C e e x QQ D r f; (4.689) where e f D f  ae  ac D f  vP 0  ! P s e x  !s  .!s e x/  2!s  vr

(4.690)

is the specific mass force in a moving system, which includes the translational and Coriolis accelerations. If v0 and !s are assumed to be known, then the translational acceleration ae is a known value, but the Coriolis acceleration, due to the presence of the velocity vr , is an unknown vector. According to (4.670), the momentum balance equation (4.689) takes the form e e vQ r D r  T C  f:

(4.691)

This equation is similar to the momentum balance equation (3.72) in a fixed system e, the co-rotational O eN i ; the distinction consists only in that the nabla-operator r Q derivative vr and the velocity vr are considered in a moving system O 0 eN 0i , and the specific mass force e f involves additional terms, namely the translational ae and Coriolis ac accelerations.

4.11.8 The Thermodynamic Laws in a Moving System The scalar thermodynamic functions e, , , and qm are independent of the choice a coordinate system, because they were introduced in an actual configuration. The heat flux vector q, as well as T, is also defined in K; therefore, it remains unchanged during the passage to the system O 0 eN 0i , and only its components with respect to basis eN 0i become different. The specific kinetic energy determined by the velocity

332

4 Constitutive Equations

magnitude squared also changes; indeed, according to the Euler formula (4.659) we have 1 2 1 1 xj/ jvj D v  v D .jvr j2 C jv0 j2 C j!s e 2 2 2 x/ C v0  vr C vr  .!s e x/: Cv0  .!s e However, if the heat influx equation (3.124) is considered as a statement of the first thermodynamic law, then it does not include .1=2/jvj2; and, thus, we obtain the heat influx equation in a moving system O 0 eN 0i : 

de eq e ˝ vr /T C qm  r D T   .r dt

(4.692)

Here we have used formula (4.684) and also the fact that T   s D 0, because the tensor T is symmetric, and s D E  !s is skew-symmetric. In order to find an analog of the energy balance law in the form (3.118), one should consider the contraction vQ xR i eN 0i /  .e xP j eN 0j / D e xR i e xP j ıij D r  vr D .e

1 i j 1 .e xP e xP ıij / D .vr  vr / D jvr j =2: 2 2 (4.693)

Multiplying Eq. (4.691) by vr , from (4.693) we obtain the kinetic energy balance equation in a moving system 

d jvr j e  T C e vr e f: D vr  r dt 2

(4.694)

On summing (4.694) and (4.692), we finally get the energy balance law in a moving coordinate system jvr j2 d eC  dt 2

! e  .T  vr  q/ C e f  vr C qm : Dr

(4.695)

The second thermodynamic law (3.166) during the passage to O 0 eN 0i retains its form d e  q C qm C w :  (4.696) D r dt

4.11.9 The Equation of Deformation Compatibility in a Moving System Let us consider now the dynamic compatibility equation in the form (3.277): d FT D FT  r ˝ v: dt

(4.697)

4.11 Relationships in a Moving System

333 ı

ı

By analogy with the Cartesian coordinates x i for K in the system O eN i , we introduce ı

ı

Cartesian coordinates e x i in the moving system O 0 eN 0i also for the configuration K (according to (4.652)): ı

ı

ı

j e x i D P ij .x j  x00 /; ı

(4.698)

ı

ı

j where P ij , Qij , x00 are values of Pji , Qji and x0j at t D 0. Since x i and x i are ı

connected by x i D x i .x j ; t/, so from (4.652) and (4.698) we find the relation ı

between e x i and e xj : ı

e x i D Pji .x j  x0j / D Pji .x j .x k ; t/  x0j / ı

ı

ı

k D Pji .x j .x00 C Qkme x m ; t/  x0j /  e x i .e x m ; t/: (4.699) ı

Introduce the basis eN 0i D eN 0i .0/ that is in general not coincident with eN i ; for this basis, according to (4.647), we have ! ı ı ı ı ı @x i ı 0 i x j /Nei D Q  eN i D Q  QT  eN 0i ; eN j D Qj eN i D eN i D .@x i =@e j @e x t D0 (4.700) ı ! ı ı ı ı @e x i ı0j 0j T 0i i i 0j eN D Q  Q  eN ; eN D P j eN D eN ; ı @x j ı

ı

ı

where Q D Qij eN i ˝ eN j is the value of the orthogonal tensor Q at t D 0 (if eN 0i D eN i , ı

then Q D E). Remind the representation of the deformation gradient F in the Cartesian basis eN i (see Exercise 2.1.5) and find its representation in the basis eN 0i : ı

ı ı x k @e xm @x i @e @e xk 0 j e km eN 0 ˝ eN 0m ; N N ˝ e D F D ı eN i ˝ eN D k ı e eN k ˝ eN 0m  F i k ı ı @e x @x j @e x m @x j @e xm (4.701)

@x i

j

ı

e km D .@e x k =@e x m /: F e km but with respect Introduce two more tensors, which have the same components F to other dyadic bases: e e km eN 0 ˝ eN 0m ; FDF k

e km eN k ˝ eN m : F0 D F

(4.702)

F0 D QT  e F  Q:

(4.703)

According to (4.700) and (4.647), we get ı

F De F  Q  QT ;

334

4 Constitutive Equations

Using the representation (4.701), we can determine the derivative with respect to time e km ı ı d FT dF e km eN 0m ˝ ePN 0 D eN 0m ˝ eN 0k C F k dt dt k e ı ı dF m 0m e km eN 0m ˝ eN 0  !s : (4.704) D Q  QT  eN ˝ eN 0k  Q  QT  F k dt ı

Here we have taken into account that eN 0m is independent of t and used formulae (4.700) and (4.674) for ePN 0k . If in the first summand we replace the basis eN 0i by ei according to (4.647) and e km =dt/Nek ˝ eN m , then we obtain take into account that d F0 =dt D .d F ! ı d F0T d FT T T e (4.705) DQQ  Q  Q  F  !s : dt dt Rearrange the right-hand side of the compatibility equation (4.697) with the help of (4.703) and (4.684): ı

e ˝ vr  e FT  r ˝ v D Q  QT  .e FT  r FT  !s /:

(4.706)

On substituting (4.705) and (4.706) into (4.697), we finally get the dynamic compatibility equation in a moving coordinate system O 0 eN 0i : d F0T e ˝ vr :  QT D e FT  r dt

(4.707)

e km dF d F0T FTQ  QT D eN 0m ˝ eN 0k D e dt dt

(4.708)

Q Since Q

(here we have used the definition (2.338) of the co-rotational derivative in the moving orthonormal basis eN 0i ), so Eq. (4.707) can be written in the equivalent form e e ˝ vr ; FTQ D e FT  r

(4.709)

which is a formal analog of the compatibility equation (4.695) in a fixed coordinate system. Equation (4.707) can also be written in the form similar to (3.303), ˛ D 5. To do this, one should determine the divergence of the tensor e F according to formulae (4.701) and (3.272): ı

ı

ı

e  .e r F/ D r  .F  Q  QT / D .r  .F/  Q  QT C FT   r ˝ .Q  QT / ı

D .r  F/  Q  QT D 0:

(4.710) ı

Here we have taken into account that the rotation tensors Q and QT are independent of coordinates x i .

4.11 Relationships in a Moving System

335

e  e It follows from (4.710) that .r F/ ˝ vr D 0. Multiplying (4.709) by  and adding the result to the last equation, we obtain e  .e e FTQ D r F ˝e vr /:

(4.711)

4.11.10 The Kinematic Equation in a Moving System An analog of the kinematic equation (3.303) (˛ D 6) in a fixed system O eN i : uP D v;

(4.712)

during the passage to a moving system O 0 eN 0i is the first relation of (4.670) for the relative velocity vr , which is expressed in terms of the co-rotational derivative.

4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System Equations (4.686), (4.691), (4.695), (4.696), (4.711), and (4.670) can be written in the single form similar to (3.303): e N ANQ ˛ D r  B˛ C C˛ ;

˛ D 1; : : : ; 6;

(4.713)

where we have introduced the generalized vectors 0

0 1 1 1= vr B B C C T vr B B C C B B C C 2 T  v e C .jv j =2/  q B B C C r r AN˛ D B C ; BN˛ D B C; B B q= C C  B B C C @ @ A A e x 0 T e e F  F ˝ vr

0

1 0 B C e f B C Be C B f  vr C qm C C˛ D B C;  B.qm C q /= C B C @ A vr 0 (4.714) which are similar to the corresponding generalized vectors of (3.304).

4.11.12 Constitutive Equations in a Moving System ı

Consider the case when eN 0i D eN i , i.e. the moving basis eN 0i at t D 0 coincides with eN i , ı

then Q D E, and from (4.703) we get e F D F  QT :

(4.715)

336

4 Constitutive Equations

Comparing (4.715) with (4.188), we can establish an analogy between the transformation of coordinate system O eN i ! O 0 eN 0i described by the rotation tensor ı



Q D eN 0i ˝ eN i and the orthogonal transformation of reference configuration K ! K, 

ı



which, according to Sect. 4.6, is determined by the tensor H D ri ˝ ri D QT . The 

tensor Q is an analog of the tensor QT D H. Drawing on the analogy, we can notice that all the results of Sect. 4.6 for the ten.n/ .n/

.n/ .n/

sors C , A and T , S under orthogonal H -transformations still hold for orthogonal transformations of a coordinate system. Thus, according to Theorems 4.18 and 4.19, .n/

.n/

all the energetic tensors C and T are H -invariant, i.e. .n/

.n/

.n/

.n/

e C D Q  C  QT ;

e T D Q  T  QT ;

(4.716)

.n/

.n/

T and e where e C are the tensors determined in O 0 eN 0i . .n/

.n/

Tensors A and S are H -invariant, and the tensor O is transformed by the formula similar to (4.256) .n/

.n/

e A D A;

.n/

.n/

e S D S;

e D O  QT : O

(4.717)

Therefore, the constitutive equations for models An of ideal solids (4.311) during the passage to O 0 eN 0i take the form .n/

QT  e T QD

r X  D1

' D 

@

.s/

; I C D .s/

@I

@I.s/ .n/

D QT 

@I.s/ .n/

.s/

' I  C ;

(4.718) .n/

.n/

 Q; I.s/ . C/ D I.s/ .QT  e C  Q/:

C @e

@C

(4.719) .n/

.n/

e ci D Q  b ci have the same C in basis b As follows from (4.716), the tensors e T and e .n/ .b n/ .b n/ .n/ components T ij , C ij as the tensors T and C in basis b ci of principal anisotropy e axes of a body. The basis b ci is orthonormal, and it is called the basis of moving e ci the invariants anisotropy axes. As follows from (4.718) and (4.719), in the basis b .n/

.n/

s/ e e s D QT  Gs  Q have the same form I.e C relative to the group G . C/ of the tensor e .s/

.n/

as the invariants I . C / relative to the group Gs while they are written in basisb ci : .n/ .n/ .n/ .b n/ .b n/ s/ e s/ . C / D I.e . C ij / D I.s/ . C ij / D I.s/ . C/ D I.s/ .QT  e C  Q/; I.e

(4.720)

4.11 Relationships in a Moving System

337

e s with (the index .e s/ means that the invariant is considered relative to the group G e the basis of moving anisotropy axes e ci ). According to (4.719) and (4.720), the constitutive equations (4.718) take the form .n/

e T D

r X  D1

s/ ' I .e e;

.n/

s/ .e s/ e I .e e D @I =@ C :

s/ ' D .@ =@I.e /;

C

C

(4.721)

.n/

For isotropic continua (Gs D I ), the principal invariants I . C / coincide with .n/

C /; therefore the constitutive equations (4.721) exactly coincide with the corI . e responding relations in a fixed system O eN i and have the form (4.322) .n/

e T D

1E

C

.n/

.n/ e2

eC

2C

3C

;

˛

D

.n/

e

˛ .I . C /;

/:

(4.722)

For transversely isotropic continua with the group Gs D T3 , whose invariants .n/ .b n/ .3/ .3/ I . C / D I . C ij / are chosen in the form (4.297), the corresponding invariants .n/

e C / in basis b e ci have the same form, and they are determined by the rule (4.720): I.3/ . e .n/

.n/

e e C; c23 /   e I1.3/ D .E  b e I3.3/

.n/

3/ e I2.e C; Db c23   e .n/

e I4.3/

e e D ..E  b c23 /  e C /   .b c23  e C /;

.n/

3/ I5.e C; D det e

.n/ e2

D C E

e I2.3/2



(4.723)

e 2I3.3/ :

In a similar way, we can determine invariants for other groups Gs  I . For models Bn of ideal solids, the constitutive equations (4.346) are transformed during the passage to O 0 eN 0i in a similar way and take the form .n/

e T D

r X  D1

' D 

.n/

@ s/ @I.e

s/ ' I .e e;

.e s/ s/ e Ie D @I.e =@ G;

;

G

G

.n/

.n/

(4.724)

s/ e e  Q/: I.e . G/ D I.s/ .QT  G

For models Cn of ideal solids, the constitutive equations (4.378) and (4.379) in a moving system O 0 eN 0i with the help of (4.717) are written as follows: .n/

e S D

r X  D1

s/ ' I .e e; A

.n/

' D 

.e s/ @I s/ ; ; I .e D A e .n/ s/ @I.e e @A

@

.n/

s/ e T e e I.e .O  A  O/ D I.s/ .QT  OT  e A  O  Q/:

(4.725)

338

4 Constitutive Equations

For ideal fluids, the constitutive equations (4.444) retain their form during the passage to O 0 eN 0i , because the tensor T is H -invariant, and  and  are independent of orthogonal transformations of a reference configuration.

4.11.13 General Remarks The above reasoning allows us to make the following conclusion: during the passage to a moving coordinate system the balance equations (4.713) and (4.714) and the constitutive equations (4.721) are written in the same form as for a fixed system except that: (1) the additional terms ae and ac should be included in the expression for the mass force f, and (2) the derivatives with respect to time should be replaced by the co-rotational derivatives. The presence of the translational ae and Coriolis ac accelerations in the set (4.713) is usually quite considerable. However, for the special case of the accelerationless rectilinear motion of a moving system O 0 eN 0i , for example, along the axis O eN 1 when Q D E; v0 D vN 10 eN 1 ; vN 10 D const; (4.726) we have !s D 0;

ae D 0;

ac D 0;

(4.727)

i.e. there are no translational and Coriolis accelerations; and, thus, Eqs. (4.713) exactly coincide with Eqs. (3.303) in a fixed coordinate system. In this case all processes and phenomena described by the equation sets will be coincident as well. Therefore, an observer moving with such a moving coordinate system cannot recognize whether he moves rectilinearly and without acceleration or he is at rest. This conclusion in a less general formulation was made by Galilei and Newton when they were studying only kinematics of nondeformable bodies; therefore, the motion (4.726) is called Galilei’s motion.

Exercises for 4.11 4.11.1. Show that if a moving system O 0 eN 0i rotates with a constant angular speed ! about the fixed axis O eN 3 and O 0 D O, then for the Coriolis formula (4.663) we have v0  0; ! P s  0; !s D ! eN 3 ; e x3  x 1 eN 1 C x 2 eN 2 D rer ; ae D fcp  ! 2e x3 D ! 2 rer ; and expression (4.690) for e f takes the form e f D f C fcp  ac ;

4.12 The Onsager Principle

339

where the vector fcp is called the specific centripetal force, the direction of which coincides with the direction of vector er of the cylindrical coordinate system Oer e' eN 3 with the axis O eN 3 .

4.12 The Onsager Principle 4.12.1 The Onsager Principle and the Fourier Law Let us pay attention to the fact that the principal thermodynamic identity forms (4.121)–(4.124), which were the foundation of derivation of constitutive equations for the models An , Bn , Cn and Dn , do not include the heat flux vector q. However, the vector q appears in the set of balance laws (3.303), and without some relations for q this system is not closed even with taking account of constitutive equations (4.158), (4.159), (4.160) or (4.161). To derive the relations for q, one should use the Onsager principle [45, 46]. Formula (3.169) for specific internal entropy production yields q  D w 

q  r  > 0: 

(4.728)

The value q  is nonnegative due to the Planck inequality (3.158a). Axiom 16 (The Onsager Principle). In order for the Planck inequality to hold, the specific internal entropy production q  must be representable in the quadratic form q  D

X

Qˇ Xˇ D

ˇ D1

X

Lˇ Xˇ X > 0:

(4.729)

ˇ; D1

Here Qˇ D

X

Lˇ X

(4.730)

ˇ D1

are functions called thermodynamic fluxes, and Xˇ are functions called thermodynamic forces. As follows from (4.729), the matrix Lˇ is symmetric: Lˇ D Lˇ :

(4.730a)

Remark. As shown in Sect. 4.4.3, the principles of thermodynamically consistent determinism and equipresence result in the fact that all the active variables ƒ included in constitutive equations (4.158)–(4.161) are independent of the temperature gradient r . Hence, the dissipation function w being among the active variables is independent of r  as well. Thus, if we assume r   0 8M 2 V and 8t > 0, then

340

4 Constitutive Equations

the value of w remains unchanged and the summand q  r  vanishes; and from (4.728) we obtain w > 0:

(4.731)

This inequality is known (see (3.171)) as the dissipation inequality. The reasoning has proved the fact that the dissipation inequality is a consequence of the Planck inequality (4.728) and principles of thermodynamically consistent determinism and equipresence. For ideal continua, w  0, then the Planck inequality (4.728) yields q  r  6 0;

(4.732)

this relation is called the Fourier inequality. For nonideal continua, the Fourier inequality (4.732) does not follow from the Planck inequality (4.728) and it should be assumed to be an independent axiom (see Axiom 10 in Sect. 3.5.2). t u  The Onsager principle P is applied as follows: expression (4.728) for q is represented as the sum ˇ Qˇ Xˇ ; for example, for ideal continua we choose the temperature gradient X1 D r  as a thermodynamic force X1 and the value Q1 D .1=/q as a corresponding thermodynamic flux. Then, according to the Onsager principle, the thermodynamic flux must be a linear function of X1 , i.e.

1  q D Q1 D L11 X1 D L11 r : 

(4.733)

The coefficient L11 in this case is a second-order tensor, and the product L11 D  is called the heat conductivity tensor; then relation (4.733) takes the form q D   r ;

(4.734)

which is called the Fourier law in the spatial description. Due to the Fourier inequality, the heat conductivity tensor  is positive-definite: r     r  > 0:

(4.735)

The Fourier law can be formulated in the material description as well. To do this, ı one should replace the vector q in (4.734) by the vector q according to formula ı

(3.128) and the gradient r  by r  according to the formula similar to formulae ı

(2.23): r  D F1T  r . Then we obtain the relation ı

ı

ı

q D   r ;

(4.736)

4.12 The Onsager Principle

341

ı

where  is the heat conductivity tensor in a reference configuration: q ı  D g=g F1    F1T : ı

(4.737)

The Fourier law (4.734) is included in the general system of constitutive equations (4.158), (4.159), (4.160) or (4.161) of a continuum. This law can also be written in the form of operator relation (4.156), where the vector q is an active variable and the vector r  is a reactive one. Then, depending on the considered model An , Bn , Cn or Dn , the heat conducı

tivity tensors  and  should be assumed to be operators of the corresponding set of reactive variables R: .n/

M C ; /; models An W  D . .n/

M G; /; models Bn W  D . .n/

M A ; O; /; models Cn W  D . .n/

M g ; O; /; models Dn W  D .

ı ıM .n/  D . C; ı ıM .n/  D . G; M .n/ ı ı  D . A ; O; ı ıM .n/  D . g ; O;

/; /; /; /:

(4.737a)

4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law ı

Consider the Fourier law (4.736) in a reference configuration K with use of one of the models An ; : : : ; Dn ; for definiteness, we choose the model Bn . Since the Fourier law as well as the constitutive equations (4.158)–(4.161) describes some physical properties of a continuum, so this law must satisfy the principle of material ı

ı

symmetry (4.180)–(4.181) if the vector q is considered as ƒ and the vector r  – as ı

R. In other words, if Eq. (4.736) holds in a configuration K, then there exists a symı

ı



metry group G s such that for each H -transformation from this group (H W K ! K) the form of function (4.736) remains unchanged: ı

ı .n/

ı

q D . G; /  r :

(4.738a)

and 

ı .n/



q D . G  ; /  r :

(4.738b)

342

4 Constitutive Equations

Theorem 4.63. The principle of material symmetry (4.738a), (4.738b) holds for the ı

Fourier law in the material description if and only if in K there exists a symmetry ı

group G s such that the heat conductivity tensor is transformed as follows: ı .n/

. G; / D

q ı .n/ ı  ı g=g H1  . G  ; /  H1T 8H 2 G s :

(4.739)

H Since the vector q and temperature  are defined in K, they are H -invariant. Then ı

ı

q and r  under H -transformations change as follows: 

q

qD 

  1

g=g F 

r  D ri

q q ı   1  q D g=g H  F  q D g=g H  q;

ı @ i ı ı k @ 1T D r ˝ r  r D H  r : i @X i @X k

(4.740)

Let the principle of material symmetry be valid, then relations (4.738a) and (4.738b) hold. On substituting (4.740) into (4.738b), we obtain

or

q ı ı ı ı  g=g H  q D   H1T  r ;

(4.741)

q ı ı  ı q D  g=g .H1    H1T /  r :

(4.742)

ı

Comparing (4.738a) with (4.742), we see that in this case the relation (4.739) ı

must hold for the heat conductivity tensor . Prove the converse assertion. Let the condition (4.739) be satisfied. Substituting the condition into (4.736), we obtain formula (4.742). Performing the transformations in the reverse order, we get formulae (4.738a) and (4.738b). N Theorem 4.63 still holds for the remaining models An , Cn and Dn . For them the relation (4.739) has the forms q ı .n/ ı  g=g H  . C ; /  HT ; q ı .n/ ı .n/ ı  . A  ; O ; / D g=g H  . A ; O; /  HT ; q ı .n/ ı .n/ ı    . g ; O ; / D g=g H  . g ; O; /  HT : ı .n/

. C  ; / D

(4.743)

Theorem 4.64. The Fourier law (4.734) in the spatial description identically satisfies the principle of material symmetry for each symmetry group Gs . ı



H Indeed, for any H -transformation of a reference configuration K ! K, since the vectors q and r  are always H -invariant, relation (4.734) retains its form with the same heat conductivity tensor  for each tensor H. N

4.12 The Onsager Principle

343 ı

Certainly, there is no discrepancy between relation (4.739) for the tensor  durı



ing the passage from K to K and invariability of the tensor , because formula (4.739) together with (4.737) and formula (4.188) of transformation of the deformation gradient F ensure that the tensor  be independent of the choice of reference 

configuration K: D

q q q ı ı      ı ı  g=g F    FT D g=g F  H    HT  FT D g=g F    FT D ./ : 

Here ./ is the tensor  defined in configuration K, which, as it was expected, 

coincides with the tensor  itself (it is not to be confused with , which is the ı



tensor  defined in configuration K).

4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law ı

ı

The heat flux vector q and temperature gradient r  are defined in a reference conı

figuration K, therefore they remain unchanged at rigid motions from K to K0 (i.e. ı

ı

ı

ı

q0 D q, .r /0 D r ). Hence, the Fourier law (4.736) in the material description also remains unchanged under the transformation from K to K0 : ı

ı

ı

q0 D   .r /0

(4.744)

for any orthogonal rotation tensor Q. This means that relation (4.736) of the Fourier law satisfies the principle of material indifference (4.603), (4.605) with the same ı

heat conductivity tensor , i.e.

ı

ı

0 D :

(4.745)

For the Fourier law (4.734) in the spatial description, the situation is different. Theorem 4.65. The Fourier law (4.734) in the spatial description satisfies the principle of material indifference (4.603), (4.605) (i.e. the relation q0 D   .r /0

(4.746)

follows from (4.734) for any rigid motion K ! K0 ) if and only if the heat conductivity tensor is both R-invariant and R-indifferent: 0 D ;

0 D QT    Q:

(4.747)

344

4 Constitutive Equations

Remind that tensors satisfying the condition (4.747) for any orthogonal tensor Q, according to (4.290), are called indifferent relative to the full orthogonal group. H The heat flux vector q is R-indifferent (just as the tensor T). To prove the fact, one should use relation (3.111) and take into account that the scalar function of heat 0 influx due to surface sources q† is R-invariant: q† D q† , and the normal vector n is R-indifferent, then 0 q† D q0  n0 D q0  QT  n D q† D q  n:

(4.748)

q0 D QT  q:

(4.749)

Thus, we obtain

The gradient of a scalar is also R-indifferent: .r /0 D ri 0

@ @ D Q T  ri D QT  r : @X i @X i

(4.750)

Multiplying the relation (4.734) by QT , we obtain QT  q D QT    r  D QT    Q  QT  r ;

(4.751)

q0 D 0  r 0 ;

(4.752)

0 D QT    Q:

(4.753)

or

As follows from (4.752), the principle of material indifference holds if and only if the tensor  is indifferent, i.e. 0 D . N Notice that relations (4.747) are consistent with (4.745): substituting formulae (4.747), (4.559), and (4.560) into (4.737), we obtain q q ı ı 10 ı 0 0 1T0  D g=g F    F D g=g F1  Q  0  QT  F1T q ı ı D g=g F1    F1T D ; (4.754) that is exactly the relation (4.745).

4.12.4 The Fourier Law for Fluids For fluids, one usually use the Fourier law in the spatial description (4.734). According to Theorem 4.64, the relation (4.734) for any tensor  satisfies idenı

tically the principle of material symmetry for each symmetry group G s , including ı

the unimodular group G s D U corresponding to a fluid.

4.12 The Onsager Principle

345

To satisfy the principle of material indifference, the tensor  in (4.734) must be indifferent relative to the full orthogonal group Gs D I , i.e. the relations (4.747) must hold. As shown in Sect. 4.8.3 (see (4.293)), Eq. (4.747) for the group I has an unique solution being the spherical tensor :  D E;

 > 0;

(4.755)

where  is the constant called the heat conductivity coefficient. Substitution of (4.755) into (4.734) gives the Fourier law for fluids q D r :

(4.756) ı

With the help of Eq. (4.737) we can write the heat conductivity tensor  for fluids as follows: q q ı ı ı  D  g=g F1  F1T D  g=g G1 : (4.757) This tensor is not spherical, but it satisfies Eq. (4.739). The Fourier law (4.736) in the material description for fluids has the form q ı ı ı q D  g=g G1  r :

(4.758)

4.12.5 The Fourier Law for Solids For solids, one usually apply the Fourier law in the material description (4.736). ı

As noted in Sect. 4.12.3, for any tensor  this relation satisfies identically the principle of material indifference. To satisfy the principle of material symmetry, by ı

ı

Theorem 4.63, the tensor  must be indifferent relative to a group G s corresponding to the symmetry group of a considered solid, i.e. the relation (4.739) must hold. According to the results of Sect. 4.8.3 (see formulae (4.293)), we find that: ı

ı

 if G s D I (an isotropic solid), then the tensor  has the only independent comı

ponent :

ı

ı

 D E;

(4.759)

ı

b3 (a transversely isotropic solid with the transverse isotropy vectorb  if G s D T c3 ), ı

ı

ı

then the tensor  has two independent components – 1 and 2 : ı

ı

ı

c23 C 2 E;  D 1b

(4.760)

346

4 Constitutive Equations ı

b (an orthotropic solid with the principal orthotropy basis b c ), then the  if G s D O ı

ı

tensor  has three independent components –  : ı

D

3 X

ı

c2 : b

(4.761)

 D1

To find the tensor  for solids one should substitute formulae (4.759)–(4.761) into (4.737). For example, for an isotropic continuum the tensor  has the form q D

q ı ı ı ı g=g F    FT D  g=g F  FT :

(4.762)

On substituting expressions (4.759)–(4.761) into (4.736), we get the Fourier law for isotropic solids ı

ı

q D E;

(4.763)

the Fourier law for transversely isotropic solids ı

ı

ı

q D .1b c23 C 2 E/;

(4.764)

and the Fourier law for orthotropic solids ı

qD

3 X  D1

ı

b c2 :

(4.765)

Chapter 5

Relations at Singular Surfaces

5.1 Relations at a Singular Surface in the Material Description 5.1.1 Singular Surfaces Up to now we considered the case when all functions appearing in the balance laws: , u, v, T, F, f etc. are continuously differentiable functions of coordinates X i (or of x i ) and time t. However, in practice one often meets with problems, where this condition is violated. For example, for the phenomena of impact, explosion, combustion etc., a part of the indicated functions in a domain V considered can suffer a jump discontinuity across some surface S (Fig. 5.1). As a singular surface we can consider an interface S between two contacting media (Fig. 5.2). Besides differential equations in a domain V , mathematical formulation of problems in continuum mechanics should involve conditions on the surface bounding the domain V (boundary conditions). This boundary can also be considered as a singular surface for the functions. The integral balance laws still hold for a volume V containing a singular surface S ; but differential equations of the corresponding laws are not valid across such surface S , because in deriving the equations we have essentially used the condition of continuous differentiability of the functions inside V . Therefore, we need to establish appropriate consequences of the balance laws for the case when there is a singular surface for the functions. This is a purpose of the chapter.

5.1.2 The First Classification of Singular Surfaces Consider a continuum occupying a volume V in K. We assume that there is a surface S dividing the volume V into two subdomains VC and V such that inside VC and V all functions (, v, F etc.) considered are continuously differentiable. However, during the passage from VC to V across S some of the functions can suffer a jump discontinuity. Consider the set Mt of material points belonging to the singular Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 5, c Springer Science+Business Media B.V. 2011 

347

348

5 Relations at Singular Surfaces

Fig. 5.1 The singular surface S in domain V

Fig. 5.2 The interface S between two media VC and V

Fig. 5.3 There is no transition of material points across a singular surface ı

surface S.t/ at a time t > 0. This set Mt in a reference configuration K belongs to ı

ı

a surface S.t/, which, in general, depends on t and divides V into two subdomains VC and V . At different times t (t1 6 t 6 t2 ), positions of the singular surface S.t/ in the actual configuration may be distinct, and there is an alternative:  Sets Mt , corresponding to surfaces S.t/ 8t 2 Œt1 ; t2 , coincide  Not all sets Mt , corresponding to surfaces S.t/ at considered times t (t 2 Œt1 ; t2 ),

are coincident In the first case, one can say that material points do not cross the singular surface S.t/ 8t 2 Œt1 ; t2 ; in the second case, there occurs a transition of material points across the singular surface. If there is no transition of material points across the interface S.t/, 8t 2 Œt1 ; t2 , ı

ı

then the position of the surface S .t/ in K 8t 2 Œt1 ; t2  remains unchanged (Fig. 5.3). While material points cross a singular surface at t 2 Œt1 ; t2 , the location of the ı

surface S.t/ in the reference configuration varies (Fig. 5.4). Let us give now a classification of singular surfaces (Fig. 5.5).

5.1 Relations at a Singular Surface in the Material Description

349

Fig. 5.4 Material points cross a singular surface

Fig. 5.5 Classification of singular surfaces

Definition 5.1. A singular surface S.t/, across which unknown functions , u, v, T, F etc. suffer jump discontinuities, is called a surface of a strong discontinuity. If these functions remain continuous across S.t/ and only their first derivatives with respect to time t and coordinates (r , r ˝u, r ˝v etc.) suffer jump discontinuities, then S.t/ is called a surface of a weak discontinuity. The transition of material points can occur only across surfaces of a strong discontinuity. Definition 5.2. A surface of a strong discontinuity S.t/, across which there is no transition of material points 8t 2 Œt1 ; t2 , is called a surface of a contact discontinuity if constitutive equations in domains VC .t/ and V .t/ are the same at all times t considered; otherwise, it is called a surface of contact. Definition 5.3. A surface of a strong discontinuity S.t/, across which there is a transition of material points at t 2 Œt1 ; t2 , is called a surface of a shock wave if constitutive equations are the same at all times t considered; otherwise, S.t/ is called a surface of phase transformation. Besides the classification introduced above, singular surfaces can be split into coherent and incoherent ones.

350

5 Relations at Singular Surfaces

Definition 5.4. A singular surface S.t/, across which the radius-vectors x of material points M 2 S.t/ suffer jump discontinuities, is called incoherent; otherwise, the surface is called coherent. For an incoherent surface S.t/, a local neighborhood dV of every point M, belonging to the surface S.t/ at some time t, at the time t C t is divided into two half-neighborhoods dVC and dV , moving apart one from another, because if ı

ı

in K the radius-vector x.M/ of the point M is continuous, then the radius-vector x.M; t C t/ of this point is no longer continuous: ı

Œx D 0;

Œx ¤ 0:

(5.1)

For a coherent surface S.t/, local half-neighborhoods dVC and dV of its every point M 2 S.t/ at any time t do not move apart one from another, and the jump of ı the radius-vector x is zero if Œx D 0: ı

Œx D 0;

Œx D 0:

(5.2)

The example of an incoherent surface is a surface of contact (an interface) between an ideal fluid and a solid: an ideal fluid slips along a solid without grip. If at time t we choose two contacting points MC and M of an incoherent surface S.t/, then their radius-vectors xC and x coincide but the radius-vectors ı

ı

ı

xC and x in K may be not coincident, i.e. instead of (5.1) the following relations may hold ı

Œx ¤ 0;

Œx D 0:

It is evident that an incoherent surface is a surface of a strong discontinuity.

5.1.3 Axiom on the Class of Functions across a Singular Surface ı

ı

Let us consider now a surface of a strong discontinuity S .t/ dividing a volume V in ı

ı

ı

K into two subdomains V C and V  . ı

ı

Definition 5.5. Let there be a function A.x; t/ defined in a domain V which conı

tains a singular surface S . The value ˇ ˇ ŒA D AC ˇ ı  A ˇ ı ; S

S

(5.3)

5.1 Relations at a Singular Surface in the Material Description

where

ˇ A˙ ˇ ı D S

ı

351

ı

A˙ .x; t/;

lim ı

x!x† ı ı ı ı x2V ˙ ; x † 2S ı

is called the jump of the function across the surface S . If a surface S.t/ is coherent, then from (5.2) it follows that the functions ı

x i .X i ; t/ do not suffer jump discontinuities across the surface S .t/: ˇ ˇ x i .X j ˇ ı ; t/ D x i .X j ˇ ı ; t/: SC

S

ı

Then in (5.3) we can always pass from coordinates x i to x i and consider the jump of a function across a corresponding singular surface S.t/ in K: ˇ ˇ ŒA D AC ˇS  A ˇS ; where

ˇ A˙ ˇ S D

lim

i x i !x† x i 2V˙ ; x i 2S.t / †

(5.4)

A.x i ; t/:

ı

ı

Axiom 17. For a continuum containing in K a singular surface S.t/, which divides ı

ı

ı

ı

ı

ı

the volume V into two parts V C and V  , the functions , A˛ and B ˛ appearing in ı

ı

the balance equation system (3.310) are assumed to be smooth in V C and V  : ı

A˛ D

8ı
ı

ı

˛C ;

x 2 V C;

˛ ;

x 2 V ;

ı :A

ı

B˛ D

ı

ı

8ı
ı

ı

ı

ı

˛C ;

x 2 V C;

˛ ;

x 2 V ;

(5.5)

ı

and across the singular surface S .t/ they can suffer a finite jump discontinuity, i.e. ı

ı

ŒA˛  < C1;

ŒB ˛  < C1:

(5.6)

The functions C˛ included in the system (3.310) are also assumed to be smooth in ı

ı

V C and V  , and across the singular surface they can suffer an unbounded jump discontinuity of the type as the ı-function has: ı

ı

ı

ı

ı

e ˛ C C ˛† ı.x†  x/; C˛ D C

e˛ D C

8
ı

ı

ı

ı

˛C ;

x 2 V C;

˛ ;

x 2 V :

(5.7)

352

5 Relations at Singular Surfaces ı

ı

ı

Here C ˛† .x† ; t/ is a finite, smooth across S .t/ function satisfying the condition Z

ı

ı

Z

ı

ı

ı

C ˛† ı.x†  x/ d V D ı

ı

C ˛† d †:

(5.8)

ı

V

S

ı

The function C 4† consists of the two summands: the jump of entropy production by ı

ı

external sources CN 4† and the jump of entropy production by internal sources C 4† , which is assumed to be always non-negative: ı

ı

ı

ı

C 4† D CN 4† C C 4† ; C 4† > 0:

(5.8a)

To apply Axiom 17 to a continuum containing not one but several singular surı

ı

ı

faces S ˛ , one should divide the whole domain V into subdomains V ˛ (˛ D ı

1; 2; : : :), each of which contains only one singular surface S ˛ .

5.1.4 The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface Let us establish a rule of differentiation of a volume integral for the case when the ı

ı

ıı

ı

integration domain V contains a singular surface S .t/ for the functions A˛ .x i ; t/ ı

defined in V . ı According to Axiom 17, determine this derivative for each of the volumes V C .t/ ı

and V  .t/: @ @t

Z

ıı

@ A˛ d V D @t

ı

Z

ı

V

ı

ı

 A˛ ı

Z

ı

@ dV C @t

V  .t /

ı

ı

ı

C A˛C d V :

(5.9)

ı

V C .t /

Let us use the definition of the derivative of a scalar function with respect to a scalar argument (time): @ @t

Z

ı

ı

ı

ı

 A˛ .x; t / d V ı

V  .t/

D lim

t!0

0 1 B B t @

Z

ı

ı

ı

ı

Z

 A˛ .x; t C t / d V  ı

ı

V  CV

ı

V

1 ıC ı ı  A˛ .x; t / d V C A

5.1 Relations at a Singular Surface in the Material Description

0

Z

B D lim B t!0 @

ı

ı

ı

353

ı

A˛ .x; t C t /  A˛ .x; t / ı 1 dV C  t t ı

ı

V

Z

1 ıC ı ı ı  A˛ .x; t C t /d VC A

ı

V

ı Z @A˛ ı 1 ı ı ı ı dV C lim  A˛ .x; t C t / dV:  t!0 @t t

Z

ı

D ı

(5.10)

ı

V  .t/

V

ı

ı

Here V is the change in the volume V  due to transition of material points across ı

ı

ıı

ı

ı

the singular surface in K: V D x†  nS , and x† is the change in the radiusı

ı

vector x of points belonging to the surface S .

ı

The corresponding elementary volume d V is determined as follows (Fig. 5.6): ı

ı

ı

ı

d V D d x†  n d †;

(5.11)

ı ı ˇ ı ı ı where n is the normal directed from V  to V C , and d x† D d x.X i ˇ ı ; t/ is the †

ı

infinitesimal increment of the radius-vector x of points belonging to the singular surface. Then the vector ı

ı

c D d x† =dt

(5.12)

ı

is the velocity of the singular surface S . ı Due to the construction, the vector c is different from zero if and only if material ı

points cross the singular surface S .t/ (and hence S.t/). According to (5.11) and (5.12), the second summand in the last line of formula (5.10) can be written as follows: Z t !0

ı

ı

ı

ı

 A˛ .x; t C t/

lim

ı

V

Z

D lim

t !0

ı

dV t ı

d x† ı ı  A˛ .x; t C t/  n d† D t ı

ı

V

Fig. 5.6 The infinitesimal change in a volume due to transition of material points across a singular surface

ı

Z

ı

ı

ı

ı

ı

 A˛ v†  n d †: ı

†

(5.13)

354

5 Relations at Singular Surfaces

Then the expression for the derivative of the first integral in (5.9) takes the form @ @t

Z

ı

ı

Z

ı

ı

ı

@A˛ ı  dV C @t ı

 A˛ .x; t/ d V D ı

ı

V

V

Z

ı

ı

ı

ı

ı

 A˛ c  n d †:

(5.14)

ı

S

The derivative of the second integral in (5.9) is determined in a similar way: @ @t

Z

ı

ı

ı

ı

C A˛C .x; t/ d V ı

V C .t /

0 1 B B t !0 t @

Z

ı

ı

ı

ı

ı

ı

ı

V C V ı

@A˛C ı C dV  @t ı

D ı

ıC ı ı ı C A˛C .x; t/ d V C A

C A˛C .x; t C t/ d V 

D lim

Z

1

Z

VC

VC

Z

ı

ı

ı

ı

ı

C A˛C c  n d †:

(5.15)

ı

S ı

ı

Here we have taken into account that normals n to the surface S for the volumes ı ı V C and V  are opposite in direction. On substituting (5.14) and (5.15) into (5.9), we obtain the following theorem. ı

ı

Theorem 5.1. For any functions A˛ defined in a domain V , which contains a sinı

gular surface S, and satisfying Axiom 17, the rule of differentiation of a volume integral holds: @ @t

Z

ıı

ı

Z

A˛ d V D ı

V

ı ı @A˛

 ı

@t

ı

Z

ı

dV 

ı

ı

ı

ı

ı

.C A˛C   A˛ /D d †;

(5.16)

ı

V

S

where ı

ı

ı

D D cn is the normal speed of propagation of the singular surface.

(5.16a)

5.1 Relations at a Singular Surface in the Material Description

355

ı

According to Definition 5.4 and the fact that in K the total and partial derivatives with respect to t coincide, Eq. (5.16) can be rewritten in the form d dt

Z

ıı

Z

ı

A˛ d V D ı

ı ı d A˛



dt

ı

V

Z

ı

dV 

ıı

ı

ı

ŒA˛ D d †:

(5.17)

ı

V

S

ı

5.1.5 Relations at a Coherent Singular Surface in K ı

Let a singular surface S.t/ be coherent. Consider its image in K, being the singular ı

ı

ı

surface S and a point M 2 S. Let us construct a special domain V h called the neighborhood of the singular surface and containing the point M (Fig. 5.7). ı

ı

ı

To construct the domain V h we choose the part †h of the surface S so that ı

M 2 †h and introduce curvilinear coordinates X i so that the curves X 1 ; X 2 belong ı

ı

to the surface S and the curves X 3 be oriented normally to S . ı

ı

ı

The domain V h is bounded by the surface @V h consisting of the lateral surfaces ı

eh˙ (their equations are †h˙ (their equations are X 3 D ˙h=2), the end surfaces † 3 3 I 0 < X < h=2 and h=2 < X < 0, and X , I D 1; 2, satisfy the equation of the ı

ı

contour L† bounding the surface †h : X I D X I† .s/, 0 6 s 6 s0 ). Let us use the integral form of the balance laws (3.313), which hold for an arbiı

trary finite volume, including V h : @ @t

Z

ıı

ı

Z

A˛ d V D ı

Vh

Fig. 5.7 The domain for derivation of conditions at a singular surface

ı

ı

Z

C˛ d V C ı

Vh

ı

ı

ı

n  B ˛ d †: ı

@V h

(5.18)

356

5 Relations at Singular Surfaces

According to theorem (5.16) and conditions (5.7) and (5.8) for the functions C˛ , the balance equations (5.18) can be written in the form Z

ı ı @A˛

 ı

@t

Z

ı

ıı

dV 

ı

ı

Vh

Z

ı

ı

ı

ŒA˛ D d † 

Z

ı

ı

†h

ı

ı

e˛ d V  C

n  B˛ d †  ı

@V h

Z

ı

ı

C ˛† d † D 0:

ı

Vh

†h

ıı

(5.19)

ı

Here ŒA˛  is the jump of the functions at the singular surface †h : ıı

ı

ı

ı

ı

ŒA˛  D .C A˛C   A˛ / ı :

(5.20)

†h

ı

Notice that the domain V h has been constructed in the special way, so we can consider the one-parametric family of such domains, numbering them by the parameter ı

ı

ı

ı

h 2 .0; h0 /. Then, since, according to Axiom 17, the functions .@A˛ =@t/, n  B ˛ ı

and C˛ are continuous in domains V h˙ and can suffer only a finite discontinuı

ı

ı

ity across †h (a field of the normal n.X i / at every point X i of the domain V h˙ ı

ıı

ı

ı

is defined as follows: n.X i / D ˙n.X I /), and the functions ŒA˛ D and C ˛† are ı

ı

continuous across †h , we can pass to the limit as h ! 0 (i.e. the domain V h shrinks to a point). So we have the equations lim

h!0

lim

h!0

ı @A˛



ı

@t

j †h j ı

ı

d V D 0;

(5.21)

ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

n  B ˛ d † D nC  B ˛C C n  B ˛ D n  .B ˛C  B ˛ / D n  ŒB ˛ ;

ı

j †h j

Z Vh

Z

1

1

ı

@V h

(5.22) lim

h!0

lim

h!0

1

Z

ı

j†h j ı

1

Z

ı

j†h j ı

ı

ı

ı

†h

ı

ı

e ˛ d V D 0; C

ı

ı

ŒA˛ D d † D ŒA˛ D;

lim

h!0

Vh

1

Z

ı

j †h j ı

ı

ı

(5.23)

ı

C ˛† d † D C ˛† :

(5.24)

†h

ı

Dividing each summand in (5.19) by j†h j and passing to the limit as h ! 0, with the help of formulae (5.21)–(5.24) we obtain the following theorem.

5.1 Relations at a Singular Surface in the Material Description ı

ı

357

ı

Theorem 5.2. For the functions , A˛ , B ˛ and C˛ : ı

ı

 Defined in a domain V which contains a coherent singular surface S,  Satisfying Axiom 17,  Satisfying the integral balance laws (3.313), ı

the following relations at the singular surface S .t/ hold: ıı

ı

ı

ı

ı

ŒA˛ D D n  ŒB ˛  C C ˛† ;

˛ D 1; : : : ; 6:

(5.25)

5.1.6 Relation Between Velocities of a Singular Surface ı

in K and K Define the velocity of a singular surface S.t/ in K as follows:   ˇ d x† Xiˇı ; t ; S.t / dt

cD

x† 2 S.t/:

(5.26) ı

Even if there is no transition of material points across a singular surface (i.e. S is independent of t), the vector c may be different from zero due to the motion of ı

material points M of the singular surface under the transformation from K to K. ı Let us establish now a relation between the velocities c and c of a singular surface ı

in K and K, respectively, defined by Eqs. (5.12) and (5.26). Consider the relations (5.2) at a coherent surface with use of the motion law (2.4) for material points of the ı

ı

surface S.t/. During the passage from coordinates X i to x i this law takes the form  x†

    ˇ ˇ ıi ıi jˇ jˇ X ı ; t D x x † .X ı ; t/; t D x x † .X ı ; t/; t ; 

ˇ

iˇ

S

ı

SC

(5.27)

S

ı

ı

ı

where x† D x i† eN i are the radius-vectors of points of the singular surface S in K, depending not only on X jˇ but also on ˇ t. Here we have used the condition of coherence of the surface S.t/: X i ˇ ı D X i ˇ ı . SC

S

On substituting (5.27) into the definition of c (5.26), we find that cD

@x† @t



ˇ ı x i† .X j ˇ ı ; SC



ı

@xC @x i† ı ı t/; t C ı D v†C CFC  c D v† CF  c: (5.28) @t i @x

358

5 Relations at Singular Surfaces

Here we have used the definition (2.181) of the velocity of a material point M at the singular surface:   ˇ @x† iˇ v†˙ D X ı ; t ; S˙ @t ı

and also the definition (5.12) of the velocity of a singular surface in K: ı

ı

@x i @x† cD D † eN i ; @t @t ı

ı

and introduced the notation for the deformation gradient at the singular surface S .t/:  F˙ D

@x m ı

@x i

ˇ ˇ ˇ ı eN m ˝ eN i :

(5.29)

S˙

Thus, we have proved the following theorem. ı

Theorem 5.3. Velocities c and c of a coherent singular surface S.t/ in reference and actual configurations, which are determined by (5.12) and (5.26), respectively, are connected by the relation ı

c D v†˙ C F˙  c;

(5.30)

c D F1 ˙  .c  v†˙ /:

(5.31)

or ı

5.2 Relations at a Singular Surface in the Spatial Description 5.2.1 Relations at a Coherent Singular Surface in the Spatial Description Let us establish now relations between jumps of the functions ˛ , A˛ , B˛ and C˛ ı

ı

at a coherent singular surface S.t/ corresponding to the singular surface S .t/ in K. ı

To do this, multiply Eq. (5.25) by the surface element d †: ıı

ı

ı

ı

ı

ı

ı

ı

Œ A˛ c  n d † D n  ŒB˛  d † C C ˛† d †:

(5.32)

ı

Theorem 5.4. The functions B ˛ and B˛ defined by formulae (3.308) and (3.311) satisfy the relations ı

ı

ı

n  ŒB ˛  d † D n  ŒB˛  d†;

˛ D 1; : : : ; 6:

(5.33)

5.2 Relations at a Singular Surface in the Spatial Description

359

ı

H When ˛ D 2 we have B2 D T and B 2 D P; then, according to (3.55), we get ı

ı

n  ŒP d † D n  ŒT d†:

(5.34)

When ˛ D 3 we have ı

ı

B3 D T  v  q; B 3 D P  v  q; and hence ı

ı

ı

n  ŒP  v  q d † D n  ŒT  v  q d†: ı

(5.35)

ı

When ˛ D 4 we have B4 D q=; and B 4 D q=. Therefore, according to (3.129), the following relation holds: ı

ı

ı

n  Œq= d † D n  Œq= d†:

(5.36)

When ˛ D 6 we have ı

ı

B6 D F ˝ v; B 6 D E ˝ v: According to formula (2.116), we get  q  ı ı ı ı ı n  ŒF ˝ v d† D n   g=gE ˝ v d † D n  ŒE ˝ v d †; ı

(5.37)

as was to be shown. N To modify the left-hand side of (5.32), we use Eq. (5.31); then we find the relations ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

ı

C D d † D C c  n d † D C n  F1 C  .c  v†C / d † ı

D  n  F1   .c  v† / d †:

(5.38)

Using the property of transformation of oriented surface elements (see (2.116)) applied to the ones on the singular surface: ı

ı

ı

˙ n  F1 ˙ d † D ˙ n d†;

(5.39)

we get ı

ı

ı

˙ D d † D ˙ n  c d†  ˙ n  v†˙ d† D ˙ .D  n  v†˙ / d†:

(5.40)

360

5 Relations at Singular Surfaces

Here we have denoted the normal speed of the singular surface in K by D D c  n:

(5.41)

On substituting (5.40) and (5.33) into (5.32), we obtain the following theorem. ı

ı

Theorem 5.5. Equations (5.25) at a coherent singular surface S in K are equivalent to the relations ŒA˛ D C n  Œv ˝ A˛   n  ŒB˛  D C˛† ;

(5.42)

which hold at the corresponding singular surface S.t/ in K. Here the jumps of functions across the singular surface S , according to (5.2), are determined by the equations ŒA˛  D .C A˛C   A˛ /S :

(5.43) ı

In relations (5.42) we have introduced singular sources in K connected to C ˛† by the relations ı

ı

C˛† d† D C ˛† d †:

(5.44)

5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface Let us derive now a rule of differentiation of an integral over a moving volume V .t/. This rule is a generalization of formula (3.12) for the case when the volume V .t/ contains a singular surface S.t/. ı

ı

ı

Consider the function A˛ .x i ; t/ (where x i 2 V ) satisfying Axiom 17. Going ı

ı

from K to K, according to the motion law x i D x i .x j ; t/, we find that the funcı ı tion A˛ .x j ; t/ D A˛ .x i .x j ; t/; t/ D A˛ .x j ; t/ is continuously differentiable within the volumes VC and V and can suffer a finite jump discontinuity across the interface S.t/ between the volumes, i.e. ( A˛ D where

A˛C ; x i 2 VC ; A˛ ;

x i 2 V ;

ŒA˛  < C1;

ˇ ˇ ŒA˛  D AC ˇS  A ˇS :

(5.45)

(5.46)

5.2 Relations at a Singular Surface in the Spatial Description

361

According to Theorem 5.1, the differentiation rule (5.17) holds for the function ı

A˛ in K. Transform the integral on the left-hand side of (5.17) as follows: d dt

Z ı

Z

ı

d A˛ d V D dt ı

V

ı

VC

D

d dt

Z

ı

d A˛C C d V C dt ı

ı

ı

A˛  d V ı

V

Z

A˛C C dV C VC

d dt

Z

A˛  dV D V

d dt

Z A˛ dV: V

(5.47) ı

ı

Here we have used the fact that inside the volumes V C and V  , and also inside VC ı and V , the functions ˙ and A˛˙ are smooth (differentiable) and satisfy the contiı

nuity equation (3.6) allowing us to go from integrals over V ˙ to integrals over V˙ . In a similar way, modify the first integral on the right-hand side of formula (5.17): Z

ı dA˛

 ı

dt

ı

Z

dV D

 V

V

dA˛ dV: dt

(5.48)

The second integral on the right-hand side of formula (5.17) can be transformed by ı

using the relation (5.40) between surface elements d † and d†: Z

ı

Z

ı

ŒA˛ D d† D ı

ı

ı

ı

ı

ı

ı

.C AC D d †   A D d †/ ı

S

S

Z

D

.C AC .D  n  vC / d†   A .D  n  v // d† S

Z ŒA.D  n  v/ d†:

D

(5.49)

S

On substituting formulae (5.47)–(5.49) into relation (5.17), we obtain the following theorem. Theorem 5.6. For any function A˛ , defined in a volume V .t/, which contains a coherent singular surface S.t/, and satisfying the corollary (5.45) of Axiom 17, the following rule of differentiation of an integral over a moving volume, which contains a singular surface, holds: d dt

Z

Z A˛ dV D

V

 V

dA˛ dV  dt

Z ŒA˛ .D  vn / d†: S

(5.50)

362

5 Relations at Singular Surfaces

Formula (5.50) may be written in another form. Splitting the first integral on the right-hand side of (5.50) into integrals over volumes VC and V and using formula (3.20), we get   Z  Z  dA˛ @ dV D dV:  .A˛ / C r  .v ˝ A˛ / dt ˙ @t ˙

V˙

(5.51)

V˙

On substituting (5.51) into (5.50), we obtain two more representations of the rule of differentiation of an integral over a moving volume, which contains a singular surface S.t/: d dt

Z 

Z A˛ dV D V

V

@A˛ C r  .v ˝ A˛ / @t



Z dV 

ŒA˛ .D  vn / d† S

(5.52)

and d dt

Z

Z A˛ dV D V

V

@A˛ dV C @t

Z

Z n  .v ˝ A˛ / d†  †

ŒA˛ .D  vn / d†: S

(5.53)

5.3 Explicit Form of Relations at a Singular Surface 5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration ı

ı

Let us write now relations (5.25) in the explicit form. Using the notations A˛ , B ˛ and C˛ (see (3.304), (3.308), (3.311)) and substituting them into (5.25), we obtain the explicit form of the relations between jumps of the functions at a surface of a strong discontinuity in K: ı ı

ŒD D 0; ı

ı

ı

ı

ŒvD C n  ŒP C C 2† D 0; ı

ı

ı

ı

ı

Œ.e C v2 =2/D C n  ŒP  v  q C C 3† D 0; ı

ı

ı

ı

ı

ŒD  n  Œq= C C 4† D 0; ı

ı

ı

ı

ı

ŒuD C C 5† D 0; ı

ı

ı

ŒFT D C n ˝ Œv C C 6† D 0:

(5.54)

5.3 Explicit Form of Relations at a Singular Surface

363

5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration Substitution of the notations A˛ , B˛ and C˛ (see (3.304) and (3.308)) into relations (5.42) gives their explicit forms Œ.D  vn / D 0; Œv.D  vn / C n  ŒT C C2† D 0; Œ.e C v2 =2/.D  vn / C n  ŒT  v  q C C3† D 0; Œ.D  vn /  n  Œq= C C4† D 0; Œu.D  vn / C C5† D 0; ŒFT .D  vn / C n  ŒF ˝ v C C6† D 0;

(5.55)

vn˙ D v†˙  n:

(5.56)

where ı

The functions C ˛† and C˛† in these relations, in general, are independent. Expressions for them can be given with the help of additional assumptions on singular ı

surfaces S and S .

5.3.3 Mass Rate of Propagation of a Singular Surface ı

Denote a mass rate of propagation of a singular surface in K by ı

ı ı

M D D C ;

(5.57)

then the first relation of (5.54) yields ı

ı

ı

ı

ı

C D D  D D M :

(5.58)

The first equation of (5.55) gives C .D  v†C  n/ D  .D  v†  n/  M;

(5.59)

where M is the mass rate of propagation of the singular surface in actual configuration K.

364

5 Relations at Singular Surfaces ı

Theorem 5.7. Mass rates M and M are connected by the relation M D

kC C ı

C

ı

1=2 M ; k˙ D .n  g1 D .n  F˙  FT˙  n/1=2 ; ˙  n/

where

ı

(5.60)

ı

kC C =C D k  = ; ı

i.e. Œk= D 0

(5.61)

ı ı

or Œ=.k / D 0:

H Substitution of expression (5.41) and formula (5.30) into the definition (5.59) yields ı

M D C .D  v†C  n/ D C .c  v†˙ /  n D C n  FC  c:

(5.62)

ı

Apply formula (2.124), connecting the normals n and n, to material points at the singular surface: ı ı or n  F D k n: (5.63) n  FC D kC n; On substituting (5.63) into (5.62), we finally obtain ı

ı

M D k C C c  n D

kC C ı

ı

M:

(5.64)

C In a similar way, we can show that M D

k  ı



ı

M: N

(5.65)

One frequently introduces also the speed of propagation of a singular surface D0 D D  v†C  n;

(5.66) ı

which, according to (5.30) and (5.41), is connected to c by the relation ı

D0 D n  FC  c:

(5.67) ı

If there is no transition of material points across a singular surface, then c D 0 and hence ı ı D D D0 D 0; M D M D 0: (5.68) Relation (5.59) with use of (5.66) can be written in the form ŒD0 D  Œvn :

(5.69)

5.3 Explicit Form of Relations at a Singular Surface

365

ı

The introduction of the mass rate M allows us to rewrite the relations (5.25) for jumps as follows: ı

ı

ı

ı

M ŒA˛  C n  ŒB ˛  C C ˛† D 0;

˛ D 2; : : : ; 6;

(5.70)

or in the explicit form ı ı

ŒD D 0; ı

(5.70a)

ı

ı

M Œv C n  ŒP C C2† D 0;   ı ı v2 ı ı M eC C n  ŒP  v  q C C 3† D 0; 2 ı

ı

(5.70b) (5.70c)

ı

ı

M Œ  n  Œq= C C 4† D 0;

(5.70d)

ı

ı

M Œu C C5† D 0; ı

ı

ı

(5.70e)

ı

M ŒFT  C n ˝ Œv C C6† D 0:

(5.70f)

In a similar way, relations (5.42) with use of (5.59) can be rewritten as follows: M ŒA˛  C n  ŒB˛  C C˛† D 0;

˛ D 2; : : : ; 6;

(5.71)

or in the explicit form Œ.D  v  n/ D 0; M Œv C n  ŒT C C2† D 0;   v2 C n  ŒT  v  q C C3† D 0; M eC 2 M Œ  n  Œq= C C4† D 0; M Œu C C5† D 0;

(5.72) (5.73)

M ŒFT  C n  ŒF ˝ v C C6† D 0:

(5.77)

(5.74) (5.75) (5.76)

5.3.4 Relations at a Singular Surface Without Transition of Material Points ı

If there is no transition of material points across a singular surface, i.e. M D M D 0, ı

then from (5.71) it follows that at S and S the following relations hold: ı

ı

ı

ı

ı

n  ŒB ˛  C C ˛† D 0; x 2 S ; n  ŒB˛  C C˛† D 0; x 2 S; ˛ D 2; : : : ; 6I

(5.78) (5.79)

366

5 Relations at Singular Surfaces

or

8 ı ı ˆ ˆn  ŒP C C2† D 0; ˆ ˆ ˆ ı ˆ ı ı ˆ ˆ n  ŒP  v  q C C ˆ 3† D 0; < ı ı n  Œq= C C 4† D 0; ˆ ˆ ı ˆ ˆ ˆ C 5† D 0; ˆ ˆ ˆ ı ˆ ı ı :n ˝ Œv C C6† D 0;

and also

8 ˆ ˆn  ŒT C C2† D 0; ˆ ˆ ˆ ˆ ˆ
(5.80) ı

ı

x 2 SI

(5.81) x 2 S:

5.4 The Main Types of Singular Surfaces ı

Let us analyze now a possible form of the surface functions C˛† and C ˛† for a singular surface S.t/, which, in general, may be incoherent.

5.4.1 Jump of Density ı

At any singular surface S.t/ there is no jump of mass (as C1 D C 1  0), hence ı

C 1† D C1† D 0:

(5.82)

ı

As follows from (5.70a), the density jump Œ may be different from zero only if ı there is no transition of material points across the singular surface (the velocity c ı

and hence the speed D are zero): ı

D D 0;

ı

Œ ¤ 0:

(5.83) ı

If material points cross a singular surface, then the density jump  is always zero: ı

D ¤ 0;

ı

ΠD 0:

Notice that the density jump Πin this case is non-zero (see formula (5.72)).

(5.84)

5.4 The Main Types of Singular Surfaces

367

5.4.2 Jumps of Radius-Vector and Displacement Vector The made assumptions on differentiability of all functions ˙ , A˛˙ and B˛˙ in domains V˙ up to a boundary S mean the absence of discontinuities within a medium in an actual configuration, i.e. the impossibility of the motion of two media so as it is shown in Fig. 5.8. This fact can be described mathematically as follows: for all points of a surface S in K their radius-vectors x on the both sides of the surface are coincident: x†C  x† D 0

or Œx D 0:

(5.85)

However, the displacement vector u may suffer a jump discontinuity across S . A surface † is called a surface without singular displacements, if the following conditions are satisfied: ı C 5† D C5† D 0: (5.86) At a surface without singular displacements, from (5.72)–(5.77) we have: ı

 If there is no transition of material points across S (D0 D D D 0 and M D ı

M D 0), then a jump of the displacement vector may be nonzero: M D 0;

Œu ¤ 0:

(5.87)

 If material points cross the surface S (D0 ¤ 0 and M ¤ 0), then there is no

jump of the displacement vector: M ¤ 0;

Œu D 0:

(5.88) ı

For coherent surfaces S (see Definition 5.4), besides x, the radius-vector x in a reference configuration is also continuous: ı

Œx.x; t/ D 0:

Fig. 5.8 Such motion of a continuum is impossible under the made assumptions on differentiability of all functions ˙ , A˛˙ and B˛˙

(5.89)

368

5 Relations at Singular Surfaces

Examples of incoherent surfaces are a surface of interaction of a solid and an ideal gas continua, on which there is slip of gas, and also a surface S with singular displacements, on which the following condition holds: C5† ¤ 0:

(5.90)

At an incoherent surface, material points MC and M of a continuum, being neighboring on different sides of the surface in a reference configuration, can move apart within an arbitrary distance from one another without leaving the surface, so there will be no discontinuities at the surface S (Fig. 5.9). The displacement vector in this case suffers a jump discontinuity M ¤ 0;

Œu ¤ 0:

Figure 5.10 shows a logical scheme of coherent and incoherent surfaces.

Fig. 5.9 Displacements of neighboring points at an incoherent surface

Fig. 5.10 Logical scheme of singular surfaces

(5.91)

5.4 The Main Types of Singular Surfaces

a

369

b

c

Fig. 5.11 Scheme of a singular surface S.t / in K: (a) coherent, (b) semicoherent, and (c) incoheri ent. Here X i and X are Lagrangian coordinates in subdomains VC and V

5.4.3 Semicoherent and Completely Incoherent Singular Surfaces An incoherent singular surface S.t/ is called semicoherent, if the following conditions are satisfied at this surface: ı

C 6† D 0;

C6† D 0:

(5.92)

If at an incoherent surface S.t/ the conditions ı

C 6† ¤ 0;

C6† ¤ 0

(5.93)

are satisfied, then the surface is called completely incoherent. There are no singular jumps of the deformation gradient F across semicoherent surfaces (Fig. 5.11).

5.4.4 Nondissipative and Homothermal Singular Surfaces A singular surface is nondissipative, if there are no singular sources of entropy production at the surface: ı C 4† D 0; C4† D 0: (5.94) Surfaces of melting, solidification, sublimation and precipitation of gas on solids, and also many surfaces of phase transformations are usually nondissipative. Surfaces of shock waves, surfaces of chemical reactions etc. are examples of dissipative surfaces. Notice that the function C4† , according to (5.8a), is the jump of entropy production by external heat sources and by internal sources. If there are no external heat

370

5 Relations at Singular Surfaces

sources (such processes of motion are called adiabatic), then q D 0, qm D 0 and e 4 D 0, and C4† is non-negative: C  C4† D C4† > 0;

(5.94a)

The vector of surface forces C2† and the energy of surface forces C3† are usually connected with the phenomenon of surface tension:  C2† D P† n;

P† D †

1 1 C R1 R2

 :

(5.95)

Here R1 and R2 are the principal curvature radii of the surface S.t/ (see [12]), and † is the coefficient of surface tension being a characteristic of the medium surface, which is determined in experiments. A corresponding value of the surface forces energy is determined as follows: C3† D .1=2/C2†  .v†C C v† /:

(5.96)

When there are electromagnetic effects, the functions C2† and C3† are connected also with surface electromagnetic forces and energy. When there are no electromagnetic effects and surface tension, we have ı

C2† D C2† D 0

ı

and C 3† D C3† D 0:

(5.97)

A surface † is called homothermal, if there is no jump of temperature across this surface: Œ D 0 at x 2 †: (5.98) This condition is used in many problems on phase transformations, chemical reactions, shock waves etc.

5.4.5 Surfaces with Ideal Contact If a surface S.t/, across which there is no transition of material points, is homothermal, coherent and nondissipative, then it is called a surface with ideal contact. For this surface, ı

C ˛† D 0; and

ı

Œx D 0;

M D 0;

C˛† D 0; Œ D 0;

˛ D 1; : : : ; 6; Œu D 0;

Œv D 0:

(5.99) (5.100)

Since material points do not cross a surface with ideal contact, the velocity of the surface S.t/ coincides with the velocity v of material points M of S.t/; and the jump of the velocity is zero that follows from (5.59) or (5.70f).

5.4 The Main Types of Singular Surfaces

371

Relations (5.99) and (5.100) should be complemented with relations (5.81), which in this case have the forms n  ŒT D 0;

(5.101)

n  ŒT  v D n  Œq;

(5.102)

n  Œq= D 0;

(5.103)

n  ŒF ˝ v D 0:

(5.104)

As follows from (5.100) and (5.103), the jump of the normal component of the heat flux vector is zero: n  Œq D 0; (5.105) and relation (5.102) is not independent. This relation is identically satisfied if equations (5.100) and (5.103) hold. Since Œv D 0, Eq. (5.104) is equivalent to the relation n  ŒF D 0: According to (2.116), this relation is a geometric condition for the normal vector ı

in K: ı

ı

ı

0 D Œnd † D Œn  Fd†=: Thus, in the system (5.100)–(5.105) the following relations are independent: 8 ˆ ˆ ˆŒu D 0; ˆ
(5.106)

These equations are called the relations at a surface with ideal contact. ı

In a similar way, in K from (5.100) and (5.80) we obtain the following independent relations at a surface with ideal contact: 8 ˆ Œu D 0; ˆ ˆ ˆ
(5.107)

372

5 Relations at Singular Surfaces

5.4.6 On Boundary Conditions Before we have considered the case when in both subdomains VC and V the functions ˙ , A˛˙ and B˛˙ are unknown and should be found. However, Eq. (5.71) can also be used for another case: when in one of the subdomains, for example in VC , the functions C , A˛C and B˛C are given, and in the other subdomain (i.e. in V ) the functions  , A˛ and B˛ are unknown. In this case, relations (5.71) at the interface S between VC and V serve to obtain boundary conditions at S for formulation of initial boundary-value problems. Notice that, as a rule, not all the functions C , A˛C and B˛C in the domain VC can be given in an arbitrary way; in formulating a problem of continuum mechanics, some of the functions should usually remain unknown and be determined in its solving.

ı

5.4.7 Equation of a Singular Surface in K ı

An equation of any smooth surface, including a singular surface S .t/, can be given ı implicitly, namely with the help of one scalar equation in coordinates x i : ı ı

ı

f .x i ; t/ D 0:

SW

(5.108)

ı

The argument t shows that the surface function f may vary with time. ı Introducing curvilinear coordinates XQ I (I D 1; 2) on the surface S .t/ [12], we ı

ı

can give the surface S in the parametric form in coordinates x i : ı

ı

x i† D x i .XQ I ; t/; ı

(5.109) ı

ı

where x† D x i† eN i is the radius-vector of a point of the surface S . ı

Introduce vectors tangent to the surface S and the unit normal vector being orthogonal to them: ı

@x† I D ; @XQ I ı

ı

nD

ı

ı

ı

ı

1  2 j 1   2 j

ı

ı

n  I D 0; I D 1; 2:

;

(5.110)

ı

ı

By tangent vectors I , we can always construct orthonormal tangent vectors  I ; ı

ı

ı

therefore, at surface S there is an orthonormal basis  I ; n (I D 1; 2) such that ı

ı

ı

ı

n   I D 0;  1   2 D 0; I D 1; 2:

(5.110a)

5.4 The Main Types of Singular Surfaces

373

By substituting (5.109) into (5.108), the equation of a singular surface can be represented as a composite function of curvilinear coordinates on the surface: ı ı f .x i .XQ I ; t/; t/ D 0. Differentiating this equation with respect to XQ I , we get ı

ı

@f @x i† D 0: ı I @x i @XQ ı ı

Introducing the vector r f D

ı

@f ı

@x i

(5.111)

eN i called the surface gradient and using formulae ı ı

ı

(5.110), we can represent Eq. (5.111) as the scalar product r f  I D 0: In other ı ı

words, the surface gradient r f is orthogonal to the tangent vectors; hence, the ı ı

ı ı

ı

vector r f is collinear to the vector n. Then, normalizing the vector r f , we find the following important formula: ı

ı ı

ı ı

n D r f =jr f j;

(5.112) ı

ı

which allows us to calculate the normal n to the surface S given implicitly by equation (5.108). Determine the total differential of Eq. (5.108) with respect to t: ı

ı

ı

@f ı @f dt C ı d x i† D 0: df D @t @x i This equation can be rewritten as follows: 0 ı 1 ı @f @f C @ ı eN i A  @t @x i

ıj

@x † eN j @t

! D 0:

(5.113)

ı

According to the definitions (5.12) of the velocity c of a singular surface and the surface gradient, we get ı

@f ı ı ı C c  r f D 0: @t

(5.114) ı

With the help of definition (5.16a) for the normal speed D of a singular surface and formula (5.112) for the normal vector, we can rewrite Eq. (5.114) as follows: ı

ı ı ı @f C Djr f j D 0: @t

(5.115)

374

5 Relations at Singular Surfaces ı

This equation is called the differential equation of the singular surface S .t/; it ı

ı

allows us to find Eq. (5.108) of the surface S .t/ if only the normal speed D and ı

initial position of the surface S.0/ are given: ı

S .0/ W

ı ı

f .x i ; 0/ D 0:

(5.116)

5.4.8 Equation of a Singular Surface in K Let us derive now an equation of a singular surface S.t/ in an actual configuration K. This equation has the implicit form similar to (5.108) SW

f .x i ; t/ D 0I

(5.117)

and the parametric equation of a singular surface is analogous to (5.109): i x† D x i .XQ I ; t/;

I D 1; 2:

(5.118)

Here x† D x i† eN i is the radius-vector of a point of the surface S . Differentiating this radius-vector with respect to curvilinear coordinates on the surface, we find the expression for tangent vectors to S.t/ and for the normal vector: I D

@x† @XQ I

nD

1  2 ; n  I D 0; j1  2 j

I D 1; 2:

(5.119)

By tangent vectors I , we can always construct orthonormal tangent vectors  I ; therefore, at the surface S there is an orthonormal basis  I ; n (I D 1; 2) such that n   I D 0;  1   2 D 0; I D 1; 2:

(5.119a)

Substituting the function (5.118) into (5.117), we represent the equation of a singular surface as a composite function of curvilinear coordinates on the surface: f .x i .XQ I ; t/; t/ D 0: Differentiation of this equation with respect to X I yields i @f @x† D 0: @x i @XQ I

(5.120)

Introducing the surface gradient r f D .@f =@x i /Nei and using formulae (5.119), we can rewrite Eq. (5.120) in the form r f  I D 0. On comparing this equation with formula (5.119) of mutual orthogonality of tangent vectors and a normal vector,

5.4 The Main Types of Singular Surfaces

375

we obtain the following connection between the unit normal vector n and the surface gradient r f : rf nD ; (5.121) jr f j being an analog of formula (5.112). Determine the total differential of Eq. (5.117) with respect to t: df D

@f @f i D 0: dt C i dx† @t @x

According to (5.26), this equation takes the form @f C c  r f D 0: @t

(5.122)

With use of the definition (5.41) of the normal speed D of a singular surface S.t/ in K, Eq. (5.122) becomes @f C Djr f j D 0: (5.123) @t This equation is called the differential equation of the singular surface S.t/ in K. Passing from D to the speed D0 determined by (5.66), Eq. (5.123) can be written in the form @f (5.124) C .D0 C v†C  n/jr f j D 0: @t This equation allows us to find an equation of the surface S.t/ in the form (5.117) if ı

only the normal speed D and initial position of the surface S.0/ D S .0/ are given: S.0/ W

ı

f .x i ; 0/ D f .x i ; 0/ D 0:

(5.125)

Exercises for 5.4 5.4.1. Show that Eq. (5.124) of propagation of a singular surface can be rewritten in the form @f C D0 jr f j C v†C  r f D 0; @t and when there is no transition of material points across the singular surface, the equation reduces to @f C v†C  r f D 0: @t

Chapter 6

Elastic Continua at Large Deformations

6.1 Closed Systems in the Spatial Description 6.1.1 RU VF -system of Thermoelasticity Let us formulate the problem for the general case of elastic (i.e. ideal) solids at large deformations; their constitutive equations have been derived in Sects. 4.5–4.9. The general system of balance laws (3.307) in the spatial (Eulerian) description consists of the continuity equation .˛ D 1/, the momentum balance equation .˛ D 2/, the energy balance equation .˛ D 3/, the dynamic compatibility equation .˛ D 5/ and the kinematic equation .˛ D 6/: @ C r  v D 0; @t

(6.1a)

@v C r  v ˝ v D r  T C f; @t

(6.1b)

@" C r  .v" C q/ D r  .T  v/ C f  v C qm ; @t

(6.1c)

@FT C r  .v ˝ FT  F ˝ v/ D 0; @t @u C r  .v ˝ u/ D v: @t

(6.1d) (6.1e)

As noted above, the balance laws system (3.307) consists of six groups of equations .˛ D 1; : : : 6/; the total number of scalar equations is 1 C 3 C 1 C 1 C 9 C 3 D 18. However, after complementing the system with the constitutive equations, we should exclude one thermodynamic equation from the set. Indeed, remind (see Sect. 4.4) that the part of the constitutive equations is equivalent to the principal thermodynamic identity, which has been obtained by summation of two thermodynamic equations (see Sect. 4.3); therefore, the principal thermodynamic identity can replace one of them. One usually eliminates the entropy balance equation (3.307)

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 6, c Springer Science+Business Media B.V. 2011 

377

378

6 Elastic Continua at Large Deformations

at ˛ D 4. In this case the system (6.1a)–(6.1e) consists of 17 scalar equations and contains 27 scalar unknowns: ; v; u; T; e; ; q; F k x; t:

(6.1f)

Each of the vector fields v, u and q is equivalent to three scalar functions, the tensor field T – to six scalar functions, and the field F – to nine scalar functions. To close the system we should complement it with constitutive equations. A part of the equations has the same form for all ideal solids, namely the Fourier law (4.734) and the expressions for specific total energy " and internal energy e (see (4.118) and (4.168)):

"DeC

q D   r ;

(6.2a)

jvj2 D 2

(6.2b)



@ v2 C : @ 2

The remaining constitutive equations for ideal solids depend on the choice of a continuum model. Forms of the constitutive equations have been considered in Sect. 4.8. Remind the general representation of constitutive equations for elastic continua in the form (4.387) (see Sect. 4.8.10): .n/

T D F G .F; /; .s/

.n/

.I G . C G .F//; / 

D

(6.3a) .F; /;

(6.3b)

.n/

where the tensor functions F G and the potential depend on the choice of model An , Bn , Cn or Dn and are defined by relations (4.385)–(4.391): .n/

.n/

.n/

.n/

F G .F; / D 4 E G   T G D 4 E G   F G .F; / D

r X  D1

.n/

.s/ .s/ I G D @I G =@ C G ;

.n/

.s/

' 4 E G   I G ;

' D .@'=@I.s/ G /;

.n/

.n/

.s/ .s/ T I.s/ G D hG I . C G / C .1  hG /I .O  C G  O/; .n/

CG D

4

.n/

3 X

hN G 1 ı ı .nIII .hG p˛ ˝ p˛ C .1  hG /p˛ ˝ p˛ //  E; ˛ n  III ˛D1 n  III

EG D

3 X ˛;ˇ D1

ı

ı

E˛ˇ p˛ ˝ pˇ ˝ .hG pˇ ˝ p˛ C .1  hG /pˇ ˝ p˛ /:

(6.4)

6.1 Closed Systems in the Spatial Description

379 ı

.n/

Here I.s/ are invariants of the tensors C G relative to the symmetry group G s of a considered elastic continuum; and ı

˛ ; p˛ ; p˛ k F

(6.5)

are eigenvalues and eigenvectors, which are functions of the tensor F; the coeffiıı

cients E˛ˇ are expressed in terms only of ˛ (see (4.39)); O D pı ˝ p is the rotation tensor accompanying the deformation, which also depends only on F; and hG and hN G are function-indicators of the model type: ( hG D

1;

G D A; B;

0;

G D C; D;

(

1; hN G D 0;

G D A; C; G D B; D:

(6.4a)

The internal energy e can be considered as a function of F and , i.e. eD



@ D e.F; /: @

(6.6)

Then we obtain the system of 10 scalar constitutive equations (the vector relation (6.2a) is equivalent to three scalar equations, relation (6.2b) where (6.3b) has been substituted is one scalar equation, and (6.3a) is equivalent to six scalar equations), which together with (6.1) constitute the closed system of 27 scalar equations for 27 scalar unknown functions (6.1f). The number of equations and unknowns in the system can be decreased by substituting the constitutive equations (6.2) and (6.3) into (6.1b) and (6.1c); as a result, we obtain the system of 17 scalar equations for 17 unknowns: ; ; ; v; F k x; t:

(6.7)

The set (6.1) is called the RU VF -system of thermoelasticity. .n/ .n/ Pr .s/ Remark 1. Choosing the function T G D F G . C G ; / D  D1 ' I G for the models An , Bn , Cn or Dn in a certain form (one of the forms considered in Sect. 4.8), we get special representations of the tensor functions (6.4). So if the model An is considered, then the general representation of the function has the form (see (4.324)): .n/

.n/

.n/

.n/

.n/

F A . C; / D 4 M   C C  C 2 C 6 L     . C ˝ C /;

(6.8)

where 4 M is the quasilinear elasticity tensor,  is the parameter of quadratic elasticity, 6 L is the tensor of quadratic elasticity; they depend, in general, on the

380

6 Elastic Continua at Large Deformations .n/

invariants I.s/ . C /. For quasilinear models An , we assume (see (4.330)) that  D 0, 6 L D 0: .n/

.n/

F A . C ; / D 4 M   CI

(6.8a)

ı

ı

and for linear models, in addition, 4 M D J 4 M, where 4 M is the tensor of elastic .s/

.n/

moduli, which is independent of I . C/. On substituting the given expression for F A into (6.4), we obtain the correspond.n/

ing representation of the tensor function F A : .n/

.n/

.n/

T D F A .F; / D E   F A D E   .n/

CD

4

4

3 1 X nIII ı ı  p˛ ˝ p˛ ; n  III ˛D1 ˛

4

4

.n/

.n/

.n/

.n/

!

M   C C  C C L     .C ˝ C/ ;

.n/

E D

3 X

2

6

ı

ı

E˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ :

˛;ˇ D1

t u Remark 2. In place of the energy balance equation (6.1c), we can use its equivalent – the heat influx equation (3.125): @ e C r  .ve C q/ D T   r ˝ vT C qm ; @t

(6.9a)

or the entropy balance equation (3.307) at ˛ D 4, which can be written in the form (3.166): d  (6.9b) D r  q C qm : dt The constitutive equations (6.2b) in this case are replaced by  D @ =@ D .F; /:

(6.9c)

Equations (6.9a) or (6.9b) sometimes prove to be more convenient than the energy balance equation (6.1c), because they contain a smaller number of summands. However, in numerical solving Eq. (6.1c), having a completely divergence form (Eq. (6.9a) has a divergence form up to the summand T  r ˝ vT ), is frequently more preferable. The entropy balance equation (6.9b) can be rewritten by using the principal thermodynamic identity in the form (4.120): d 1 d C  T  r ˝ vT D 0: dt dt 

(6.10)

Here we have used the definition (4.2) for the stress power w.i / and taken into account that for elastic media w D 0.

6.1 Closed Systems in the Spatial Description

381

Differentiating (6.10) with respect to  and taking into account that  and r ˝ vT D FP  F1 are expressed only in terms of F and are independent of , and F and  are independent variables, we obtain @ d 1 @T d @ D C  r ˝ vT : dt @ @ dt  @ On substituting the relation (6.9c) into the last equation, we find the following expression for the rate of changing specific entropy: @2 d 1 @T d D 2    r ˝ vT : dt @ dt  @

(6.10a)

Substitution of (6.2a), (6.3a), and (6.10a) into (6.9b) gives the equation  c"

@ C v  r @t

 D r  .  r / C

@ .n/ F .F; /   r vT C qm ; @

(6.11)

which is called the heat conduction equation for an elastic continuum in the spatial description. Here we have introduced the function @2 ; (6.11a) @ 2 that is called the heat capacity of an elastic continuum at fixed deformations. t u c" .F; / D 

6.1.2 RVF -, RU V -, and U V -Systems of Dynamic Equations of Thermoelasticity Notice that the displacement vector u formally appears only in the kinematic equation (6.5), and this equation can be excluded from the general system. In this case the system is considered to be the RVF -system of thermoelasticity. However, due to boundary conditions, Eq. (6.5) cannot always be excluded from the general system (see Sect. 6.3). The number of equations and unknowns in the system (6.1)–(6.3) can be reduced by eliminating the deformation gradient F between the unknowns (6.1f). To do this, one should recall that the inverse gradient has a vector potential (see (2.77)): F1 D E  r ˝ uT :

(6.12)

This relation is, in essence, a compatibility equation: the dynamic compatibility equation (6.1d) has been derived from this equation (see Sect. 3.7.1). Therefore, relation (6.12) can be considered in place of Eq. (6.1d). Then, substituting (6.12)

382

6 Elastic Continua at Large Deformations

into the constitutive equations (6.3), we represent them as functions of the inverse gradient (see Sect. 4.8.10, formulae (4.392) and (4.393)): .n/

T D F G .F1 ; /; .n/

1 1 .I.s/ G . C G .F //; /  .F ; /;

D

eD



(6.13a)

@ D e.F1 ; /: @

.n/

ı

Here the tensor function F G has the same form (6.4) but ˛ , p˛ and p˛ are considered to depend on F1 : ı (6.13b) ˛ ; p˛ ; p˛ k F1 : Substituting the constitutive equations (6.13a) and (6.2a) into Eqs. (6.1b) and (6.1c), we obtain the RU V -system of thermoelasticity @ C r  v D 0; @t

(6.14a)

.n/ @v C r  v ˝ v D r  F G .E  r ˝ uT ; / C f; @t

(6.14b)

.n/ @" C r  v" D r  .  r  C F G .E  r ˝ uT ; /  v/ C f  v C qm ; @t (6.14c) @u C r  .v ˝ u/ D v; (6.14d) @t v2 " D e.E  r ˝ uT ; / C ; (6.14e) 2

which consists of eight scalar equations for eight scalar unknowns: ; ; u; v k x; t: .n/

(6.15)

.n/

Remark 3. The dependence of tensors 4 E G and C G on F1 (see Eqs. (6.4) and (6.13b)) is sufficiently complex; and, in general, it even hasn’t got an explicit anı alytic expression (eigenvalues ˛ and eigenvectors p˛ and p˛ , in general, cannot be expressed in terms of F1 analytically, but only in the form of a computing algorithm (see Sect. 2.3.2)). Due to this, the dependence of the tensor T on the displacement gradient r ˝ u in the theory of finite elastic deformations has a rather complex form. Even for linear models (see Sect. 4.8.7 and (6.8)), where the ten.n/

.n/

.n/

.n/

sors T and C G are linearly connected, the dependences C G .F1 / and 4 E G .F1 / lead to a complicated nonlinear connection between T and r ˝ u. However, the

6.1 Closed Systems in the Spatial Description

383 I

I

I

I

models AI and BI are the exceptions; for them the tensors GA D C D ƒ, CB D G I

I

I

and 4 EA D 4 EB D 4 E can be expressed in terms of F1 in a sufficiently simI

ple form (see Exercise 4.2.21 and (2.73a)), and the tensor dependences CG .F1 / I

and 4 EG .F1 / are quadratic. The RU V -system of dynamic equations of elasticity theory (6.14) for these models is essentially simplified and has the form (for the model AI ) 8 ˆ @=@t C r  v D 0; ˆ ˆ ˆ <@v=@t C r  v ˝ v D r  T C f; ˆ@"=@t C r  v" D r  .  r  C T  v/ C f  v C qm ; ˆ ˆ ˆ : @u=@t C r  .v ˝ u/ D v; 8 P .s/ ˆ T D r D1 ' F1T  I  F1 ; ˆ ˆ ˆ ˆ .s/ .s/ .s/ ˆ D .I.s/ .ƒ/; /; ˆ
(6.16a)

(6.16b)

I

The substitution ƒ ! G gives a similar system for the model BI . There is no need to find eigenbases and eigenvalues ˛ for the models AI and BI , that discerns these models among the remaining ones and explains their wide application to the spatial description. The models AV and BV are of a higher level of complexity. For the model AV , constitutive equations (6.16b) are replaced by 8 P T ˆ T D r D1 ' F  I.s/  F ; ˆ ˆ ˆ .s/
(6.16c)

For this model, there is no need to determine eigenvectors and eigenvalues as well; but, unlike the models AI and BI , one should invert the inverse deformation gradient in order to restore the tensor F. The models AII , BII and AIV , BIV are the most complicated for numerical solving the problems with arbitrary geometry of the domain V ; for them we need to calcuı late ˛ , p˛ and p˛ . t u Remark 4. Notice that the density  in the systems (6.14) and (6.16) is considered to be an independent unknown value, therefore one should retain the multiplier ı J D = in the constitutive equations (see Sect. 4.8.7) in its form not expressing

384

6 Elastic Continua at Large Deformations .n/ .n/ .n/

.n/

the factor in terms of the tensors C, A , G or g by formulae (4.81) and (4.82). But if such substitution has been performed, then we obtain that  can be expressed in terms of F, and the relation  D .F/ is the well-known continuity equation in Lagrangian variables (3.8): ı

ı

 D  det F1 D  det .E  r ˝ uT /:

(6.17)

Since this equation is exactly equivalent to the continuity equation (6.14a) (see Sect. 3.1), so in this case the continuity equation should be excluded from the general system (6.14) or (6.16), and Eq. (6.17) should be substituted into the remaining equations of the systems including the constitutive equations. Thus, we obtain the U V -system (6.14b)–(6.14e), (6.17) consisting of seven equations for the seven unknowns ; u; v k x; t:  (6.18) Remark 5. Reduction of the number of equations and unknown functions usually makes the system more complicated. For example, the RU VF -system (6.1) involves the first-order derivatives with respect to x and t, but the U V -system is of the mixed order: for v – the first order, and for u – the second order with respect to x and the first order with respect to t. Therefore, often it is more convenient to consider the RU VF -system; although it contains more equations, all the equations are of the same type. t u

6.1.3 T RU VF -system of Dynamic Equations of Thermoelasticity Sometimes in solving special problems for solids it is more convenient to increase the number of equations even in comparison with the RU VF -system (6.1) by in.n/

troducing the tensor T or the generalized energetic stress tensor T G (see (4.91)) as an additional unknown tensor. To do this, consider the constitutive equations ‘in rates’ (4.442). Expressing the .n/

tensor T in terms of the energetic stress tensors T G by relations (4.37), (4.67), and (4.387): .n/

.n/

.n/

.n/

.n/

T D 4 E   T D 4 Q   S D 4 E G   T G;

(6.19)

we obtain the T RU VF -system of thermoelasticity: @ C r  v D 0; @t

(6.20)

.n/ .n/ @v C r  v ˝ v D r  .4 E G   T G / C f; @t

(6.21)

6.1 Closed Systems in the Spatial Description

385

.n/ .n/ @" C r  v" D r  .  r  C .4 E G   T G /  v/ C f  v C qm ; @t @u C r  .v ˝ u/ D v; @t

(6.22) (6.23)

.n/

.n/ .n/ @ T G C r  .v ˝ T G / D 4 P Gh .F; /   r ˝ vT @t .n/

.n/

.n/

P CZGh  T G C  T G  ZGh C  T G ; (6.24) @FT C r  .v ˝ FT  F ˝ v/ D 0; (6.25) @t which consists of 23 equations for 6 C 1 C 1 C 3 C 3 C 9 D 23 scalar unknowns: .n/

T G ; ; ; u; v; F k x; t:

(6.26)

An advantage of the system is that all its equations have the same divergence form. The tensors appearing in Eq. (6.24) have been determined in Sect. 4.8.12: the .n/

tensor 4 P Gh – by formulae (4.423), (4.438), (4.106), (4.99), and (2.375), the tensor ZGh – by (4.440) and (2.374), and the tensor TG – by (4.423). All the tensors are functions of F and r ˝ v. Remark 6. Equation (6.24) has been derived in Sect. 4.8.12 for elastic continua models not containing the rotation tensor O, i.e. for models An , Bn , and also for isotropic materials described by models Cn and Dn . Therefore, for these models the T RU VF -system (6.20)–(6.25) holds. t u

6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description To rewrite the equation systems given above in components, one usually uses fixed bases e ri and e r i (see Sect. 2.1.7). Local bases ri and ri are rarely applied for the reason mentioned in Sect. 3.1.2. The RU VF -system (6.1) in basise ri has the component form @ e v j / D 0; C r j .e @t @e vi ei; ej .e e ji / D f Cr v ie vj  T @t

386

6 Elastic Continua at Large Deformations

@" e i e ie e jke v ."ıij  T g ki / C e q j / D f vje g ij C qm ; C r j .e @t e ij @F ek .e e ij  F e kj e (6.27) vkF v i / D 0; Cr @t @e ui ej e v je u i D e vi: Cr @t The constitutive equations (6.2a) and (6.2b) in components are written as follows: ej ; e q i D e ij r

"D



1 i j @ C e ve v e g ij : @ 2

(6.28)

The constitutive equations (6.3) and (6.4) have the component forms .n/

e ij .F e k ; /; eij D F T l G D

.n/

e ij e k .I.s/ G .F G .F l //; /  .n/ e ij F G

D

3 X  D1

.n/

.s/ e ijkl e ' E G I  G;kl ;

.n/

e e kl D @I.s/ I .s/ G =@ C G ;  G;kl I.s/ G D .n/ e ij C G

D

.n/ e kl hG I.s/ . C G/

e k ; /; .F l

' D .@ =@I.s/ G /;

C .1 

(6.29)

.n/ e ij O e e hG /I.s/ . C G i k O jl /;

3 ı ı hN G 1 X nIII ei Q e i˛ Q e j˛ C .1  hG /Q ej ˛ .hG Q e g ij ; ˛ ˛/  n  III  D1 n  III

eijkl D E G

3 X

ı

ı

e i˛ Q e j˛ .hG Q ek Q el ek el E˛ˇ Q ˇ ˛ C .1  hG /Qˇ Q ˛ /:

˛;ˇ D1 ı

e i˛ of eigenvectors pı ˛ , p˛ (see e i˛ , Q Here eigenvalues ˛ and Jacobian matrices Q e i of the deforExercise 4.2.20), according to (6.5), are functions of components F j mation gradient F in the basis e ri : ı

e i˛ ; Q e i˛ k F e ij : ˛ ; Q

(6.30)

To rewrite the RU V -system (6.14) in components, one should only exclude the dynamic compatibility equation from (6.27) and replace it by the component form of relations (6.13a): ele e 1 /k D .F e k /1 D ı k  r .F uk : l l l

(6.31)

6.1 Closed Systems in the Spatial Description

387

As a result, we obtain the system @ e vj / D 0; C r j .e @t @e vi ei; ej .e v ie vj  e T j i / D f Cr @t @" e i j e ie ej .e ei / C f ej m // D r v ."ıi  e gmi T ij r vj e g ij C qm ; C r j .e @t @e ui ej e vje u i D e vi; Cr @t @ 1 i j "D  v e g ij ; C e ve @ 2 .n/

.n/

ele e ij .ı k  r e ij D F uk ; /; T G l

D

(6.32)

ele .ıkl  r uk ; /:

Here, just as in (6.13b), we have turned to the inverse gradient F1 in the constitutive equations (6.29). The component form of the system (6.16) coincides with (6.32), and the ten.n/

e ij .F k / corresponding to this system (the model AI ) is given in sor function F l Exercise 6.1.3. The T RU VF -system has the component form @ e vj / D 0; C r j .e @t .n/ .n/ @e vi ei; ej .e e ijkl T e Gkl / D f v ie vj  E Cr G @t .n/ .n/ @" e i e ie ei / C f ej .e e kj ml T e Gml e v ."ıij  E gki // D r ij r vj e g ij C qm ; C r j .e G @t e ij @F ek .e e ij  F e kj e vk F v i / D 0; Cr @t @e ui ej .e vj e u i / D e vi; Cr @t (6.33) .n/

.n/ e Gij @ T ek .e Cr T Gij / vk e @t .n/

.n/

.n/

.n/

@ e vl C  Z e k T e k e e k r e e ; D P Ghij l ke Ghi Gkj C  Z Gh j T Gi k C  T Gij @t .n/

.n/

e k are components of the tensors P Gh and ZGh with e k and Z where P Ghij l Ghi respect to the basise ri .

388

6 Elastic Continua at Large Deformations

6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations Continuum mechanics widely uses the concept of models of processes (i.e. motions of media under some assumptions). One of them is the model of quasistatic proı

cesses (i.e., in a certain sense, very slow motions from K to K). Definition 6.1. One can say that this is the model of quasistatic processes in a solid, if the terms containing the velocity v can be neglected in Eqs. (6.1b) and (6.1c) of the RU VF -system (6.1) in comparison with the other terms; i.e. in these equations one assumes that v  0: (6.34) Figure 6.1 shows the schematic graph of varying all known and unknown functions  with time for quasistatic and dynamic processes (for the latter motions we cannot assume the model of quasistatic processes). Due to (6.34), the momentum balance equation (6.1b) and the heat conduction equation (6.11) used in place of the energy balance equation (6.1c) take the forms r  T C f D 0; c"

@ D r  .  r / C qm : @t

(6.35)

Complementing this system with constitutive equations (6.12) and (6.13): .n/

T D F G .E  r ˝ uT ; /;

(6.36)

we obtain the closed system of four scalar equations for four scalar functions: ; u k x; t:

Fig. 6.1 The typical graph of functions of time for dynamic (1), quasistatic (2), and static (3) processes

(6.37)

6.1 Closed Systems in the Spatial Description

389

This system is called the quasistatic equation system of thermoelasticity in the spatial description. The density  in the system is assumed to be expressed in terms of displacements by the continuity equation (6.17). The remaining equations of the system (6.1) for the model of quasistatic processes hold in general only approximately, and, therefore, they are not considered. However, if one considers the equation system (6.35) for static processes, for which starting from a certain time t0 the functions  and u are independent of t (see Fig. 6.1), then Eqs. (6.1a), (6.1d), and (6.1e) take the forms @=@t D 0;

@F=@t D 0;

@u=@t D 0;

(6.38)

and hold exactly at t > 0; and their solution is written as follows:  D .x/;

F D F.x/;

u D u.x/:

(6.39)

Here the right-hand sides of the relations can be found after solving the system (6.35), (6.36).

Exercises for 6.1 6.1.1. Show that for the quasilinear models AI (see Remark 1 or Sect. 4.8.7) the constitutive equations (6.8a) have the explicit analytic form similar to (6.16b): T D F1T  .4 M   ƒ/  F1 ; ƒD

1 .r ˝ u C r ˝ uT  r ˝ uT  r ˝ u/; 2

F1 D E  r ˝ uT ;

or  T D .E  r ˝ u/ 

4

M

   1 E  r ˝ uT  r ˝ u  .E  r ˝ uT /: 2

6.1.2. Using formula (4.322), show that for isotropic continua the relations (6.9a) can be written as follows: T D F1T  . 1

C 2ƒ C D '1 C '2 I1 C '3 I2 ;  3

D '3 ;

/  F1 ; D '2 C I1 '3 ;

3ƒ

1E

2

2

I˛ D I˛ .ƒ/; ˛ D 1; 2; 3:

Using formula (4.337), show that for the linear model AI of isotropic continua this relation takes the form T D J.l1 I1 .ƒ/F1T  F1 C 2l2 F1T  ƒ  F1 /;

390

6 Elastic Continua at Large Deformations

or

   1 T T D J l1 r  u  r ˝ u   r ˝ u .E  r ˝ u/  .E  r ˝ uT / 2  1 C 2l2 .E  r ˝ u/  .E  r ˝ uT /  r ˝ u  .E  r ˝ u/T ; 2 where l1 and l2 are the constants. 6.1.3. Show that constitutive equations (6.16b) for the model AI in basise ri have the component form eij D T

r X  D1

e I .s/  ms

D

ems ; @I.s/ =@ƒ

.s/ e 1 /m .F e 1 /se ' e g i ke g jl .F k l I  ms ;

' D .@ =@I.s/ /;

D

e ms /; /; .I.s/ .ƒ

ese eke eke eke eme e 1 /m D ı m  r e ms D 1 .r us C r um  r um r us /; .F um : ƒ k k 2 Using the result of Exercise 6.1.1, show that for the quasilinear model AI this component representation takes the form e e 1 /p .F e 1 /q f ems ; T ij D e g i ke gjl .F k l M pqms ƒ and for the model AI of isotropic continua – the form e 1 /p .F e 1 /q eij De g i ke g jl .F T k l



g pq C 1e

e ms C g pme g qs ƒ 2e

 mu esv e e g e g e g : ƒ ƒ 3 pm qs uv

6.2 Closed Systems in the Material Description 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description For the material description, we should consider the balance law system (3.310) having the explicit form ı

 D  det F1 ; ı

ı

(6.40a) ı

.@v=@t/ D r  P C f; ı

ı

ı

ı

.@=@t/ D r  q C qm ;

(6.40b) (6.40c)

ı

@FT =@t D r ˝ v;

(6.40d)

@u=@t D v:

(6.40e)

6.2 Closed Systems in the Material Description

391

Here we have used the entropy balance equation (3.174) in place of the energy balance equation. The system (6.40) as well as (6.1) becomes closed after complementing it with the constitutive equations consisting of the two equations ı

ı

ı

q D   r ;

 D @ =@;

(6.41)

and the universal constitutive equations (6.3) and (6.4) written for the Piola– Kirchhoff tensor: .n/

P D F ıG .F; /;

.s/

D

.n/

.I G . C G .F//; / 

.F; /:

(6.42)

Here we have introduced the tensor function .n/ F ıG .F; /

.n/

ı

 .=/F1  F G .F; /:

(6.43)

The system (6.40)–(6.42) contains 16 scalar unknowns (the density  is assumed to be expressed in terms of F): ; u; v; F k X i ; t;

(6.44)

which are functions of Lagrangian coordinates X i and time t, and consists of 16 scalar equations (after substitution of (6.41) and (6.42) into (6.40)). This system is called the U VF -system of thermoelasticity in the material description. For solids, this system proves to be more preferable, because a domain of defı

inition of the functions (6.44) is known: this is V  Œ0; tmax . The exceptions are ı

problems with phase transformations, where V varies with time and is determined in the process of solving. However, in this case the system (6.40)–(6.42) also proves to be more preferable than the corresponding system (6.1)–(6.6) in the spatial description. .n/

Remark 1. Although the definition (6.43) of the tensor function F ıG includes F1 , this function can be represented as a function of F. To do this, we should use the representation of F1 in the eigenbasis (see (2.171)) and the expression (6.4) of the .n/

function F G ; then we obtain .n/

P D F ıG .F; / D

r X  D1

4

.n/

.n/

E ıG D F1  4 E G D

3 X

ı

ı

ı

.n/

'  4 E ıG   I.s/ G; ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ .hG pˇ ˝ p˛ C .1  hG /pˇ ˝ p˛ /;

˛;ˇ D1 ı

(6.45)

E ˛ˇ D E˛ˇ =˛ ;

ı

ı

'  D .=/' ; G D A; B; C; D:

392

6 Elastic Continua at Large Deformations .n/

.s/

The form of the tensors I G coincides with (6.9c). The tensors 4 E ıG are called Lagrangian tensors of energetic equivalence; they connect the Piola–Kirchhoff .n/

stress tensor to T G :

.n/

.n/

P D 4 E ıG   T G :

(6.46) .n/

.n/

For example, choosing the tensor functions T G D F G . C G ; / for the models An .n/

in the form (6.8), we obtain tensor function F ıG in the form .n/

.n/

.n/

.n/

.n/

P D 4 E ı   .4 M   C C  C 2 C 6 L     . C ˝ C //; .n/

(6.47)

.n/

where 4 E ı D F1  4 E G . Notice that although the equation system (6.40) contains the Piola–Kirchhoff stress tensor P, the final (after solving the system (6.40) with boundary and initial conditions) analysis of a stress field in a solid usually needs the Cauchy stress tensor T to be known; therefore, besides constitutive equations (6.45), the material description involves the relations (6.4). t u Remark 2. Although the tensor function (6.45) has a rather complex form, which in general cannot be expressed analytically, this form may be simplified for two exceptional models AV and BV , because ı

PD

ı

ı

V  1  V F  T D F1  .F  T  FT / D T  FT :   

(6.48)

V

On substituting the expression (6.4) for the tensor function T D F A .F; / into (6.48), we obtain the following tensor function having the explicit analytical representation: V

P D F ıA .F; / D

r X

T I.s/  F ;

(6.49)

 D1 ı

ı

'  D .@ =@I.s/ /;

D

.s/ .I.s/ .C/; /; I.s/  D @I =@C;

C D .1=2/.FT  F  E/: V

V

The substitution C ! G in (6.49) yields the tensor function F ıB .F; / for the model BV . For the models AV and BV , there is no need to determine eigenbases and eigenvalues in constructing the functions (6.49). The mentioned advantages discern the models AV and BV among others while using the material description as well as the

6.2 Closed Systems in the Material Description

393

models AI and BI while using the spatial description (see Remark 3 in Sect. 6.1). These models are the most frequently applied to computations of problems of elasticity theory at large deformations. The models AI and BI are of a higher level of complexity in the material description. For the model AI , constitutive equations (6.49) are replaced by the relations I

P D F ıA .F; / D ı

ı

r X

ı

1 '  F1  F1T  I.s/  F ;

(6.50)

 D1

'  D .@ =@I.s/ /;

.s/ D .I.s/ .ƒ/; /; I.s/  D @I =@ƒ; 1 ƒ D .E  F1  F1T /: 2

For this model, although there is no need to determine eigenvalues and eigenvectors, however, in comparison with the models AV and BV , one should invert the tensor F1 . Just as in the spatial description, the models AII , BII and AIV , BIV are the most complicated for computations. t u The entropy balance equation in the system (6.40) can be modified just as in Sect. 6.1.1 by using the principal thermodynamic identity in the material description (4.125): ı ıd ı d  (6.51) C   P   r ˝ vT D 0: dt dt Differentiating the identity with respect to  and using the second relation of (6.41), we obtain the following expression for the rate of changing the specific entropy in the material description: ı @



@t

2 ı@

D 

@ 2

@P ı @    r ˝ vT : @t @

(6.52)

On substituting (6.52) and (6.41), (6.42) into the entropy balance equation of the system (6.40), we get the desired heat conduction equation for an elastic continuum in the material description: ı

c"

ı

.n/ ı ı ı @  .n/ ı D r  .  r / C .4 E ıG   T G /   r ˝ vT C qm ; @t 

(6.53)

where the heat capacity c" is determined by (6.11a). Here we have taken into account that equations (6.45) give the following expression for the derivative with respect to : ı

ı r X .n/ .n/ @'  .s/ @ .n/ı  .n/ @P D I G D 4 E ıG   T G ; F G .F; / D 4 E ıG   @ @ @   D1

(6.53a)

394

6 Elastic Continua at Large Deformations .n/

where the tensor T G has been introduced by formula (4.423): .n/

T G D

r X @' .s/ I : @  G  D1

(6.53b)

6.2.2 U V - and U -systems of Thermoelasticity in the Material Description Return to the general system (6.40)–(6.42) and notice that the deformation gradient F can be excluded from the system with the help of Eq. (2.77): ı

F D E C r ˝ uT :

(6.54)

Then we obtain the U V -system of thermoelasticity in the material description: ı @v

 ı

c"

@t

ı

ı

.n/

ı

D r  F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı ı F G .E C r ˝ uT ; /   r ˝ vT C qm ; D r  .  r / C @t @ @u=@t D v;

(6.55)

consisting of seven equations for seven scalar unknowns: ; u; v k X i ; t:

(6.56)

The density  does not appear in this system explicitly, and it can always be evaluated with the help of the continuity equation (6.17): ı

ı

 D  det .E C r ˝ uT /:

(6.57)

Since in the material description the velocity v is connected to the displacement vector u by an explicit kinematic relation, we can eliminate the velocity v between the unknowns. As a result, we obtain the U -system of thermoelasticity in the material description: 2 ı@ u



@t 2

ı

.n/

ı

ı

D r  F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı @u ı D r  .  r / C C qm ; c" F G ..E C r ˝ uT /; /   r ˝ @t @ @t ı

(6.58)

6.2 Closed Systems in the Material Description

395

for four scalar unknowns: ; u k X i ; t:

(6.59)

In particular, for the exceptional model AV , the U -system (6.58) with account of (6.49) and (2.78) is written in the form 2 ı@ u

ı

ı

D r  P C f; 0 1 r X ı ı ı ı ı @ ı T T A c" ' I.s/ D r  .  r / C @  C   .F  r ˝ v / C qm ; @t  D1 

@t 2

@2

ı

' D  PD

r X  D1

CD

ı

@@I.s/

.s/

'  I C  F T ;

D

;

ı

ı

' D 

@ @I.s/

(6.60)

;

.s/

.I.s/ .C/; /;

I C D @I.s/ =@C;

ı ı ı 1 ı .r ˝ u C r ˝ uT C r ˝ u  r ˝ uT /; 2

ı

F D E C r ˝ uT :

6.2.3 T U VF -system of Thermoelasticity in the Material Description .n/

Using the relations (6.46) between the tensors P and T G and complementing the system (6.40) with constitutive equations (4.442) ‘in rates’, we obtain the T U VF system of thermoelasticity in the material description: ı @v

 ı

c"

@t

ı

.n/

.n/

ı

D r  .4 E ıG   T G / C f;

.n/ .n/ ı ı ı ı @ ı ı D r  .  r / C .=/.4 E ıG   T G /   r ˝ vT C qm ; @t

@u=@t D v;

ı

(6.61)

@F=@t D r ˝ vT ;

.n/

.n/ .n/ .n/ .n/ ı @ @TG D 4 P ıGh .F; /   r ˝ vT C ZGh  T G C T G  ZGh C T G ; @t @t

for 22 scalar unknowns: .n/

T G ; ; u; v; F k X i ; t:

(6.62)

396

6 Elastic Continua at Large Deformations

Here we have introduced the notation for the fourth-order tensor 4

.n/ P ıGh

.n/

 .4 P .1243/  F1T /.1243/ : Gh

(6.63)

The remaining notations are the same as in Sect. 6.1.3. The heat conduction equation in (6.61) has been written with the help of (6.53). Component forms of equation systems of elasticity theory in the material deı

ı

scription are usually used in the basis ri of a reference configuration K (see Exercise 6.2.2 and 6.2.3).

6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description The model of quasistatic processes introduced in Sect. 6.1.5 can be considered for the material description as well. In this case the equation system (6.40)–(6.42) with account of the assumption (6.34) takes the form ı

ı

r  P C f D 0; ı

c"

ı

@ ı D r  .  r / C qm ; @t .n/

(6.64)

ı

(6.65)

ı

P D F ıG .E C r ˝ u; /:

(6.66)

The system consists of four scalar equations for four scalar unknown functions: ; u k X i ; t;

(6.67)

and it is called the quasistatic equation system of thermoelasticity in the material description. In this case the dynamic compatibility equation and the kinematic equation in system (6.40) are satisfied only approximately, and they are not considered. These equations are satisfied exactly for static processes starting from some time t0 .

Exercises for 6.2 6.2.1. Show that constitutive equations (6.49) for the quasilinear model AV can be written in the form ı

P D .=/.4 M   C/  FT ;

T D F  .4 M   C/  FT ;

6.2 Closed Systems in the Material Description

397

or in terms of the displacement gradient: 

ı

 PD 

4

   ı ı 1ı T M   E C r ˝ u  r ˝ u  .E C r ˝ u/: 2

6.2.2. Using the results of Exercise 6.1.1, show that for isotropic media the relations (6.49) have the forms PD.

ı 1E ı

ı

C

2C ı

1

C

ı 3C

2

ı

/  FT ;

T D JF  .

ı

D ' 1 C ' 2 I1 C ' 3 I2 ;  ı

ı

' ˛ D .@ =@I˛ /;

ı

ı 1

ı 2C

ı

ı

2

C

D ' 2 C I1 ' 3 ;

I˛ D I˛ .C/;

C

ı

ı 3C

2

/  FT ;

ı

D '3;

3

˛ D 1; 2; 3:

Show that for the linear model AV of isotropic continua these relations take the forms P D l1 I1 .C/FT C 2l2 C  FT ;

T D J.l1 I1 .C/F  FT C 2l2 F  C  FT /;

or in terms of the displacement gradient:   ı ı ı 1ı T P D l1 r  u C r ˝ u   r ˝ u .E C r ˝ u/ 2 ı

ı

ı

ı

ı

Cl2 .r ˝ u C r ˝ uT C r ˝ u  r ˝ uT /  .E C r ˝ u/: 6.2.3. Using the result of Exercise 6.2.1, show that the constitutive equations for ı the quasilinear model AV in basis ri have the component form ı

ı

j

ı

ı

ı

P i D .=/M jkms "ms F lk g li I and for the model AV of isotropic media: ı

P ji D .

ı

ı jk 1g

T ij D

ı

C

ı ij 1g

ı

ı j m ı ks g "ms 2g

C

ı

C

ı i m ı js 2 g g "ms

ı

ı ı ı j m ı ks ı uv g g "mu "sv /F lk g li ; 3g

C

ı

ı i m ı js ı uv 3 g g g "mu "sv : ı

6.2.4. Show that the U VF -system (6.40)–(6.42) and (6.45) in basis ri in the material description has the component form

398

6 Elastic Continua at Large Deformations

8 ı ı ı ı ıı ˆ .@vi =@t/ D r j P j i C f i ; ˆ ˆ ˆ ˆ ı ı ı ı ı ı ı ı <ı .@e=@t/ D r j .j i r i / C P ij r i vk g kj C qm ; ı ı ˆ ˆ@F i =@t D r j uı i ; ˆ ˆ j ˆ : ıi ı @u =@t D vi ; 8 ı ˆ ıij
G

l .n/ ık .s/ ıij .I G . C G .F l //;

D

/ 

ı

.F kl ; /;

8 .n/ Pr ı .n/ıijkl ı .s/ ˆ ˆ ˆ F ıij D ' I  Gkl ; ˆ  EG  D1 G ˆ ˆ .n/ ˆ ı ı ı ˆ .s/ .s/ .s/ ˆ ˆ I D @I G =@ C ıkl '  D .@ =@I G /; ˆ G ; <  Gkl .n/ .n/ ı ı .s/ .s/ .s/ ıij / C .1  hG /I . C G O i k O jl /; I G D hG I . C ıkl G ˆ ˆ ˆ .n/ ı ı ˆ j j ˆ hG ı ij 1 P3 nIII .h Q i i ˆ C ıkl ˆ G ˛ Q˛ C .1  hG /Q˛ Q˛ /  n  III g ; ˛D1 ˛ G D n  III ˆ ˆ ˆ.n/ ı ı ı ˆ j : E ıijkl D P3 k k l i l ˛;ˇ D1 E ˛ˇ Q˛ Qˇ .hG Q ˇ Q˛ C .1  hG /Qˇ Q ˛ /: G ı

ı

ı

Here p˛ D Qi˛ ri , p˛ D Q˛i pi . ı

6.2.5. Show that the U -system (6.60) for the model AV in basis ri has the component form 8 ı ı ı ıı ı j ˆ .@2 ui =@t 2 / D r j P i C f i ; ˆ ˆ ˆ ˆ ı ı ı ı ı ı ı ı ı ˆ ˆ ˆ.@e=@t/ D r j .j i r i / C P ji r j .@uk =@t/g i k C qm ; ˆ ˆ ı ı ˆ P ı <ıj .s/ P i D r D1 '  .@I G =@"jk /F lk g li ; ı ı ˆ ˆ'  D .@ =@I.s/ /; D .I.s/ ."jk /; /; e D  .@ =@/; ˆ ˆ ˆ ı ı ı ı ı ı ı ˆ ı ı ˆ ˆ "jk D .1=2/.r j uk C r k uj C r j um r k ui g mi /; ˆ ˆı ˆ ı ı ı : i F k D ıki C r k um g mi : ı

ı

ı

Here "jk D C jk are components of the deformation tensor C D "ij ri ˝ rj (see (2.68)). V

ı

6.2.6. Using the fact that components of the tensor T with respect to basis ri coincide with components T ij of the Cauchy tensor T with respect to basis ri (see Exercise 4.2.1), show that for the model AV components T ij can be evaluated with the help of constitutive equations (6.3a): T

ij

D

FAij

D

r X  D1

'

@I.s/ G @"kl

ı

ı

g i k g jl :

6.3 Statements of Problems for Elastic Continua at Large Deformations

399

6.3 Statements of Problems for Elastic Continua at Large Deformations 6.3.1 Boundary Conditions in the Spatial Description Remind that the closed systems of differential equations given in Sects. 6.1.1–6.1.4 hold only in a domain V not containing singular surfaces S.t/. In the presence of such surfaces, in place of these systems one should consider the relations for the jumps (5.72)–(5.77). (1) For a homothermal coherent surface S1 .t/, the relations have the form 8 ˆ Œ.D  v  n/ D 0; ˆ ˆ ˆ ˆ ˆ M Œv C n  ŒT C C2† D 0; ˆ ˆ ˆ <M Œe C v2 =2 C n  ŒT  v  q C C D 0; 3† (6.68) ˆ Œ D 0; ˆ ˆ ˆ ˆ ˆ Œu D 0; ˆ ˆ ˆ : M ŒFT C n  ŒF ˝ v D 0: Here Œv D v0  v00 are the jumps of functions across the singular surface S1 .t/ (see Fig. 6.2), and M D 0 .D  v0  n/ D 00 .D  v00  n/ is the mass rate of propagation of the singular surface (see (5.59)). For quasistatic processes in an elastic continuum (see Sect. 6.1.5), according to the definition (6.34), we should assume that v D 0 in relations (6.68), then in place of (6.68) we have 8 ˆ Œ D 0; ˆ ˆ ˆ ˆ ˆ ˆ
Œu D 0; ˆ ˆ ˆ ˆ M Œe  n  Œq C C3† D 0; ˆ ˆ ˆ :Œ D 0:

Fig. 6.2 Boundary conditions for solids in configuration K

(6.68a)

400

6 Elastic Continua at Large Deformations

One can neglect the last equation in (6.68) as well as the deformation compatibility equation in the quasistatic equation system. (2) In the special case when a surface S2 .t/ is a surface with ideal contact, the relations (5.106) hold at this surface (Fig. 6.2): 8 ˆ Œu D 0; ˆ ˆ ˆ
(6.69)

The boundary † of a domain V is also a singular surface; therefore, we should formulate boundary conditions at † by using the relations for jumps (6.68). Let us specify several main special cases of boundary conditions. (3) When material points cross the surface part †1 , boundary conditions at †1 for a solid have the form M Œv C n  ŒT C C2† D 0;   v2 C n  ŒT  v  q C C3† D 0: M eC 2

(6.70)

Here Œv D ve  v etc., where functions without subscripts correspond to a domain V considered and are unknown, and functions with subscript e correspond to the surroundings of the domain V and are assumed to be known. The remaining relations for jumps in the system (6.68) take no part in formulating boundary conditions, they only impose constraints on parameters of the surroundings. If the model of quasistatic processes is considered for an elastic continuum, then, according to the definition (6.34), we assume that v  0 in the boundary conditions (6.70). In this case the conditions take the form ( n  ŒT C C2† D 0; (6.70a) M Œe  n  Œq C C3† D 0: (4) If the surroundings on the part †2 is an ideal fluid, then for the fluid the Cauchy stress tensor is spherical: Te D pe (see (4.444a)), and relations (6.70) take the form (see Exercise 6.3.5) 8 ˆ ˆ <M Œvn  Tn D pe  Cn† ; (6.71) M ŒvI  TI D 0; I D 1; 2; ˆ ˆ :M Œe C v2 =2  T v D p v  C 0 e ; n n e ne n 3†

e 03† C

2 1X D C3†  .vI C vI e /2  Œqn : 2 I D1

6.3 Statements of Problems for Elastic Continua at Large Deformations

401

Here e, Tn D n  T  n, vn D n  v and TI D  I  T  n are unknown functions referred to the solid; e0 , pe , vI e and vne , qne are given functions referred to the fluid. For quasistatic processes, the relations (6.71) take the form 8 ˆ ˆ
TI D 0; I D 1; 2; ˆ ˆ :M Œe  Œq D C : n 3†

(6.71a)

(5) Across the surface part †3 of a domain V , the transition of material points may be absent (M D 0). If this surface is in addition coherent, then the following boundary conditions hold at the surface: n  T De tne ;

nq De q ne ;

(6.72)

where e tne D n  Te C C2† ; e q ne D n  qe  C3† are given values. (6) If M D 0 and the surrounding medium is an ideal fluid, then although such a surface †4 is not coherent (see Sects. 5.1.2 and 5.4.2), the boundary conditions (6.72) hold at the surface; and the stress vectore tne is collinear to the normal vector: e tne D pe n: (6.73) The boundary condition (6.73) is frequently called the tracking loading, because it follows the change of the normal to the surface †4 in the actual configuration. On the contrary, the boundary condition (6.72) admits the existence of the stresses vector e tne with its fixed value and direction in any actual configuration; such a boundary condition is called the fixed loading (Fig. 6.3). (7) When M D 0, at the surface part †5 instead of the stress vector tn D n  T we can give the displacement vector (or the radius-vector x) of a material point M and temperature : u D ue ;  D e :

Fig. 6.3 Fixed loading of the crane boom under the action of the gravity force of a suspended freight

(6.74)

402

6 Elastic Continua at Large Deformations

In this case the relations for jumps (5.81) take no part in formulation of boundary conditions, they only impose constraints on parameters of the surroundings. (8), (9) In addition, two more combinations of boundary conditions at the surface parts †6 and †7 of a domain V are possible: (

(

u D ue ; n  q D qne ;

n  T D tne ;

(6.75)

 D e :

(10) For solids, boundary conditions can be given at artificial boundaries. The symmetry conditions are the most widely used conditions of this type; they remain valid when: (a) there is a fixed symmetry plane …† in the domain V , which divides the domain into two parts V 0 and V 00 (b) at outer boundaries †0 and †00 of the domains V 0 and V 00 boundary conditions are symmetric with respect to …† , and mass forces and heat sources are symmetric too (c) constitutive equations of a solid and values of the unknown functions at the initial time are symmetric with respect to …† In this case, fields of all the unknown functions must remain symmetric with respect to …† at any time t > 0. This means that the derivatives of all the scalar unknowns with respect to the normal to …† must vanish, the motion of points along the normal to …† must be absent (otherwise, the symmetry is violated), and the stress vector tn must be collinear to the normal vector n to …† , i.e. the following relations at …† must hold: @=@n D 0;

u  n D 0;

v  n D 0;

n  T   I D TI D 0;

q  n D 0;

(6.76)

I D 1; 2:

6.3.2 Boundary Conditions in the Material Description All the considered types of boundary conditions correspond to an actual configuraı

tion K; however, they can be rewritten in terms of a reference configuration K and ı

then used to formulate problems in K. ı

ı

ı

(1) For a singular surface S 1 .t/ inside a domain V , when the surface S 1 .t/ is homothermal and coherent, the conditions (5.70) should be satisfied:

6.3 Statements of Problems for Elastic Continua at Large Deformations

8 ı ı ˆ ˆ Œ D D 0; ˆ ˆ ˆ ı ı ˆ ı ˆM ˆ Œv C n  P C C 2† D 0; ˆ ˆ ˆ ı ı < ı ı  M Œ  n  Œq C  C 4† D 0; ˆ ˆ Œu D 0; ˆ ˆ ˆ ˆı T ı ı ˆ ˆ M ŒF C n ˝ Œv D 0; ˆ ˆ ˆ : Œ D 0: ı ı

ı

403

(6.77)

ı

ı

Here M D 0 D D 00 D is the mass rate of propagation of the singular ı

surface in K (see (5.58)). Notice that the set (6.77) includes the condition for the entropy jump (the third relation), because the balance equation system (6.40) contains just the entropy balance equation. If instead of this equation the system (6.40) involved the energy balance equation, then the set (6.77) should contain a relation for the energy jump (the relation (5.70c)). For quasistatic processes, according to (6.34), the system (6.77) takes the form ı Œ D 0; ı

ı

n  P C C2† D 0; Œu D 0; ı

ı

(6.77a)

ı

ı

 M Œ  n  Œq C  C 4† D 0; Œ D 0: (For the quasistatic statement, the deformation compatibility equation is not considered.) ı

ı

(2) If the surface S 2 .t/ inside the domain V is a surface with ideal contact, then the relations (5.107) hold at the surface: ı

ı

ı

Œu D 0; n  ŒP D 0; Œ D 0; n  Œq D 0: ı

(6.78)

ı

(3) At the outer surface part †1 of the domain V (corresponding to the surı

face †1 ), where M ¤ 0 and M ¤ 0, the following conditions should be satisfied: 8ı ı <M Œv C nı  ŒP C C 2† D 0; (6.79) ı ı ı :M Œ  n  Œq= C Cı D 0: 4†

Here Œv D ve v etc., where functions without subscripts correspond to the considered domain V , and functions with subscript e – to the surroundings (given values).

404

6 Elastic Continua at Large Deformations

For quasistatic processes, these relations take the forms ı

ı

n  ŒP C C2† D 0; ı

ı

(6.79a)

ı

ı

M Œ  n  Œq= C C 4† D 0: ı

ı

(4) At the part †2 , corresponding to the surface part †2 , where M ¤ 0 and M ¤ 0, and where the surrounding medium is an ideal fluid, boundary conditions have the same form (6.79), but the Piola–Kirchhoff stress vector ı

ı

in the surroundings n  Pe D tne is determined by the formula ı tne

ı

ı

D pe .=/ n  F1 :

(6.80)

Here pe is the pressure in the fluid (referred to actual configuration K), and ı ,  and F1 are parameters of the solid. ı

(5) At the part †3 , corresponding to the homothermal coherent surface †3 , ı

where M D 0 and M D 0, the boundary conditions (6.79) become ı

ı

ı

n  P D tne ; ı

ı

ı

n  q D q ne ;

(6.81)

ı

where tne and q ne are given values. ı

(6) At the part †4 , corresponding to the surface part †4 , where parameters ı

of the ideal fluid are given and M D 0, M D 0, we have the conditions ı

ı

(6.81). In these conditions the stress vector tne has the form (6.80), and q ne is determined by qne : ı ı

ı

ı

q ne D qe  n.d†=d †/ D qne k.=/; ı

ı

where k is evaluated by formula (2.121). Therefore, at †4 the following boundary conditions must be satisfied: ı

ı

ı

n  P D pe .=/n  F1 ; ı

ı ı

ı

n  q D qne k.=/; ı

ı

(6.82)

ı

k D .n  G1  n/1=2 : ı

(7) At the part †5 , corresponding to the surface †5 , the boundary conditions (6.74) formally remain unchanged.

6.3 Statements of Problems for Elastic Continua at Large Deformations

405

ı

(8), (9) At the part †6 , corresponding to the surface †6 , the boundary conditions (6.75) become (

u D ue ; ı

ı

(ı ı n  P D tne ;

ı

n  q D q ne ;

(6.83)

 D e :

(10) The symmetry conditions (6.76), defined in K at the plane …† which is ı

fixed at any time t > 0, during the passage to K take the form ı

u  n D 0;

ı

ı

ı

v  n D 0;

q  n D 0;

ı

ı

ı

ı

n  P   I D 0;

(6.84)

because at …† : n D n and  I D  I .

6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description Let us complement now each of the closed systems formulated in Sect. 6.1 with the boundary conditions given in Sect. 6.3.1 and also with initial conditions (the number of which is equal to the number of derivatives with respect to t in the closed system) and Eq. (5.124) of the moving surface † of a body V considered. Then we obtain statements of the corresponding initial boundary-value problems for elastic continua at large deformations. Thus, the statement of the dynamic RU VF -problem of thermoelasticity in the spatial description consists of:  Equation system (6.1) defined in V  .0; tmax /.  Constitutive equations (6.2)–(6.4) given in V  .0; tmax /.  Equation (5.124) of the moving surface † of a body V :

@f =@t C D0 jrf j C v  r f D 0:

(6.85)

 Boundary conditions (6.68)–(6.75) given at †  .0; tmax /.  The initial conditions given in V ı

t D 0 W  D ; v D v0 ;  D 0 ; F D E; u D 0; f D f 0 :

(6.86)

The statement of the dynamic RU V -problem of thermoelasticity consists of:  Equation system (6.14) defined in V  .0; tmax /.  Equation (6.85) of the moving surface † of a body V .  Boundary conditions (6.68)–(6.75) at †  .0; tmax /, where the constitutive

equations (6.2)–(6.4) have been substituted.

406

6 Elastic Continua at Large Deformations

 The initial conditions in V ı

t D 0 W  D ; v D v0 ;  D 0 ; u D 0; f D f 0 :

(6.87)

In a similar way, we can give statements of the dynamic U V -problem of thermoelasticity and the dynamic U -problem. The statement of the dynamic T RU VF -problem of thermoelasticity consists of:  Equation system (6.20)–(6.25) defined in V  .0; tmax /.  Equation (6.85) of the moving surface † of a body V .  Boundary conditions (6.68)–(6.75) at †  .0; tmax /, where relations (6.2)–(6.4)

and (6.19) have been substituted.  The initial conditions in V ı

t D 0 W T D T0 ;  D 0 ;  D ; u D 0; v D v0 ; F D E; f D f 0 : (6.88) By a given value of the Cauchy stress tensor T0 and F D E at t D 0, one can .n/

evaluate T G at t D 0. The statement of the quasistatic problem of thermoelasticity in the spatial description consists of: Equation system (6.35) defined in V  .0; tmax /. Constitutive equations (6.36) in V  .0; tmax/. Equation (6.85) of the moving surface of a body. Boundary conditions (6.68a), (6.69), (6.70a), (6.71a), (6.72)–(6.75) (the velocity v does not appear in the boundary conditions (6.75) for this statement).  The initial conditions    

t D0W

 D 0 ;

f D f 0:

Remark 1. For all the statements considered, the constitutive equations hold both inside the domain V and at its boundary †. t u Remark 2. Equation (6.85) of the moving surface † of a body V holds for all its parts †˛ mentioned in Sect. 6.3.1, where at the parts †1 and †2 : M ¤ 0, and at the remaining parts M D 0. Moreover, Eq. (6.85) can be written for singular surfaces S1 .t/ and S2 .t/ moving inside the volume V . For surfaces †˛ , across which there is no transition of material points .M D 0/, Eq. (6.85) may be replaced by formula (2.76): ˇ ˇ ı (6.89) x D x.x i ˇ†˛ ; t/ C u.x i ˇ†˛ ; t/; ˇ where x i ˇ†˛ are coordinates of points at the surface †˛ .

t u

6.3 Statements of Problems for Elastic Continua at Large Deformations

407

Remark 3. In all the problem statements considered above, the rate D0 of phase transformation is assumed to be a given function in the form D0 D D0 .F; ; e /;

(6.90)

where e D fe ; Fe ; e ; ve g are parameters of the surroundings.

t u

Remark 4. All the problem statements considered above are, in general, coupled; i.e. the heat conduction problem consisting of the energy balance equation (6.1c) with the corresponding boundary conditions (6.68)–(6.75) for qn or  and the initial condition (6.86) for  cannot be solved separately from the elasticity problem at large deformations. Indeed, if we consider, for example, the dynamic RU VF problem of thermoelasticity and if in place of the energy equation (6.1c) we use the heat conduction equation (6.11), then there are six causes of the connection: (1) Equation (6.11) involves the convective term v  r  containing the velocity; .n/

@ (2) this equation also includes the term @ F   r ˝ vT depending on F and r ˝ v; (3) the heat conductivity tensor for a solid is usually defined in a reference configı

ı

uration K (this is the tensor ), and in K the tensor is determined by formula ı

ı

(4.737)  D .=/F    FT ; and hence  depends upon F; (4) the domain V of integration of the system (6.1) is unknown and determined by solving Eq. (6.85) or by formula (6.87), which contain the mechanical unknowns: the velocity v and the displacement vector u; (5) the rate D0 of phase transformation (6.90), in general, may depend on the mechanical unknowns F and T; (6) in turn, constitutive equations of elasticity (6.3a) depend upon the temperature . .n/

@ F   r ˝ vT to the heat conduction equation can A contribution of the term @ often be neglected, the rate D0 is usually assumed to depend only on the temperature: D0 D D0 .; e /. However, the remaining causes 1, 3, and 4, ensuring the system (6.1a), (6.1b), (6.11), (6.12), and (6.19) to be coupled, in general, cannot be neglected. So the statement of the coupled dynamic RU VF -problem of thermoelasticity at large deformations differs fundamentally from the thermoelasticity problem under infinitesimal deformations, where the connection, as a rule, can be neglected. Due to this, problems of thermoelasticity at infinitesimal deformations are called weakly coupled, and the RU VF -problem is called strongly coupled. The dynamic RU V -, U V - and T RU VF -problems of thermoelasticity in the spatial description are also strongly coupled. t u

408

6 Elastic Continua at Large Deformations

6.3.4 Statements of Thermoelasticity Problems in the Material Description Statements of all the problems in the material description, when there are no phase ı

ı

transformations .M D 0/, are formulated in the domain V of a reference configuration, which is known; this is their main advantage over the problem statements in the spatial description. ı

ı

ı

If a part of the surface † of domain V is moving due to phase transformations ı

.M ¤ 0/, then the domain V becomes unknown too, and the system (6.40) of balance laws should be complemented with Eq. (5.115) for determining a shape of ı

ı

the surface †.t/ of the domain V . The statement of the dynamic U VF -problem of thermoelasticity in the material description consists of: ı

 Equation system (6.40) defined in V  .0; tmax /. ı

 Constitutive equations (6.41)–(6.43) in V  .0; tmax /. ı

 Equation (5.115) of the motion of a surface † due to phase transformations: ı

ı

ı ı

@f =@t C Djr f j D 0:

(6.91) ı

 Boundary conditions (6.77)–(6.84) (a part of which may be absent) at † 

.0; tmax /, where the constitutive equations (6.41) and (6.42) have been substituted.  The initial conditions in V t D 0 W  D 0 ; u D 0; v D v0 ;

ı

F D E; f D 0:

(6.92)

The statement of the dynamic U V -problem of thermoelasticity in the material description consists of: ı

 Equation system (6.55) defined in V  .0; tmax /. ı

 Equation (6.91) of the moving surface † of a body. ı

 Boundary conditions (6.77)–(6.84) at †  .0; tmax /, where the constitutive equa-

tions (6.41), (6.42), and (6.54) have been substituted. ı

 The initial conditions in V

t D0W

ı

 D 0 ; u D 0; v D v0 ; f D 0:

In a similar way, we can give a statement of the dynamic U -problem.

(6.93)

6.3 Statements of Problems for Elastic Continua at Large Deformations

409

The statement of the dynamic T U VF -problem of thermoelasticity in the material description consists of: ı

 Equation system (6.61) defined in V  .0; tmax /. ı

 Equation (6.91) of the moving surface † of a body. ı

 Boundary conditions (6.77)–(6.84) at †  .0; tmax /, where the constitutive

equations (6.41) and (6.46) have been substituted. ı

 The initial conditions in V

t D0W

.n/

ı

.n/

T G D T G0 ;  D 0 ; u D 0; v D v0 ; F D E; f D 0: (6.94)

The statement of the quasistatic problem of thermoelasticity in the material description consists of: ı

 Equation system (6.64), (6.65) defined in V  .0; tmax /. ı

 Constitutive equations (6.66) in V  .0; tmax /.

ı

 Boundary conditions (6.77a), (6.78), (6.79a), and (6.80)–(6.84) at † (the equation

of (6.84) for v does not appear in the statement). ı

 Equation (6.91) of the moving surface † of a body.  The initial conditions ı

t D 0 W  D 0 ;

f D 0:

(6.95)

ı

The speed D in all the statements mentioned above is assumed to be a given function in the form ı ı D D D.F; ; e /; (6.96) where e D fe ; Fe ; e ; ve g are parameters of the surroundings. Remark 5. The statements of thermoelasticity problems in the material description, which have been formulated above, are weakly coupled when there are no phase transformations or when the rate of phase transformations depends only on temı

ı

perature: D D D.; e /, because in this case the heat conduction problem (for example, in the U VF -statement this is Eq. (6.53) with the corresponding heat boundary and initial conditions from (6.77)–(6.84), (6.86), and also (6.85)) and the mechanics problem (the remaining equations in the U VF -problem) are connected .n/

ı

.n/

ı

only by the entropy term .=/.4 E ıG  T G /r ˝vT in (6.53) and by the dependence of the entropy jump Œ upon F. These effects in most problems of thermoelasticity can be neglected; as a result, the problems of heat conduction and mechanics turn out to be uncoupled. Usually, one first solves the problem of heat conduction and ı

ı

finds the temperature field .x; t/ in V , and with the known temperature field one then solves the problem of mechanics.

410

6 Elastic Continua at Large Deformations ı

If the phase transformation rate D essentially depends on the temperature  and on the mechanical unknowns F and T (in some problems on solids with phase transformations there is such a situation), then problems of thermoelasticity in the material description turn out to be strongly coupled (i.e. the connection cannot be neglected). t u

6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations An important particular case of the motion of elastic continua, occurring in practice, is the case when the temperature is a constant:  D 0 D const

8t > 0:

In this case, one can say that this is the model of isothermal processes in solids. For this model, the energy balance equation (or the entropy balance equation) is eliminated among the balance equations of the general system and the temperature  is eliminated between the unknowns for all the statements considered in Sects. 6.3.3 and 6.3.4. The corresponding problems are called problems of elasticity at large deformations. For example, the statement of the quasistatic problem of elasticity in the spatial description has the form 8 ˆ r  T C f D 0 in V; ˆ ˆ ˆ .n/ ˆ ˆ ˆ
(6.97)

and the statement of the quasistatic problem of elasticity in the material description is written as follows: 8ı ı ı ˆ ˆ r  P C f D 0 in V ; ˆ ˆ ˆ ı .n/ ı ˆ ˆ ˆP D F ı .E C r ˝ uT / in V ; ˆ < G ı ı ı ı ı (6.98) n  P D tne ; at †1 ; : : : ; †4 ; †7 ; ˆ ˆ ı ı ˆ ˆ ˆ u D ue ; at †5 ; †6 ; ˆ ˆ ˆ ı ˆ ı ı ı : u  n D 0; n  P   I D 0 at †8 :

6.3 Statements of Problems for Elastic Continua at Large Deformations

411

Notice that the boundary conditions (6.70) and (6.71) at the surface parts †1 and †2 , across which there is a transition of material points .M ¤ 0/, for quasistatic processes coincide with the conditions (6.72) and (6.73) at surfaces without such transition, because under these conditions the term M Œv can be neglected in comparison with n  ŒT . For incompressible continua, a statement of the quasistatic problem of elasticity theory in the spatial description includes the constitutive equations from Sect. 4.9.6: 8 ˆ r  T C f D 0 in V; ˆ ˆ ˆ ˆ ˆ det F1 D 1 in V; ˆ ˆ ˆ ˆ .n/ ˆ ˆ ˆ e G .F1 / in V ;
in V ; F1 D E  r ˝ uT ˆ ˆ ˆ ˆ ˆ n  T De tne ; at †1 ; : : : ; †4 ; †7 ; ˆ ˆ ˆ ˆ ˆ u D ue ; at †5 ; †6 ; ˆ ˆ ˆ : u  n D 0; n  T   I D 0 at †8 : The second equation in this system is the equation of incompressibility. .n/

e G .F1 ; / has the same form as the corresponding function The tensor function F .n/

F G .F1 ; / in (6.13a) and (6.4), and differs from the last one only by the smaller .n/

number .r  1/ of invariants I.s/ . C G / in the potential (6.3b) due to the incompressibility condition. The problem (6.99) is solved for the unknown functions

D

.n/

.I.s/ G . C G /; /

u; p k x i : The corresponding statement of the quasistatic problem of elasticity for an incompressible continuum in the material description has the form 8ı ı ı ˆ ˆ r  P C f D 0 in V ; ˆ ˆ ˆ ı ˆ ˆ ˆdet F D 1 in V ; ˆ ˆ ˆ ˆ .n/ ı ˆ ˆ ˆ e ı .F/ in V ; ˆP D pF1 C F < G ı

ı

in V ; FDECr ˝u ˆ ˆ ˆ ı ı ı ı ˆ ı ˆ ˆn  P D tne ; at †1 ; : : : ; †4 ; †7 ; ˆ ˆ ˆ ı ı ˆ ˆ ˆ u D u ; at † ; † ˆ e 5 6; ˆ ˆ ı ˆ :u  nı D 0; nı  P  ı D 0 at † ˛ 8;

(6.100)

412

6 Elastic Continua at Large Deformations

and is solved for the same unknown functions but depending on the coordinates X i : u; p k X i : .n/

.n/

e ı .F; / differs from F ı (6.45) only by the number of inThe tensor function F G G variants being smaller by 1 (this number is equal to .r  1/). The inverse gradient in (6.100) is determined by the formula (see [12]) F1 D F2  I1 .F/F C I2 .F/E: Remark 1. Since in actual configuration K the solution domain V is unknown, the systems (6.97) and (6.99) must be complemented with relations (6.89), which allow us to find the unknown geometry of the domain V . For the problems (6.98) and ı

t u

(6.100) in the material description, the solution domain V is known. .n/

.n/

Remark 2. A form of the tensor functions F G and F ıG in the statements (6.30)–(6.33) is determined by Eqs. (6.4), (6.13a) and (6.45), and in general, as noted in Remark 1 of Sect. 6.1.1, it is rather complicated and has no explicit analytical representation. The exceptions are the models AI and BI for the problems (6.30) and (6.32) in the spatial description and the models AV and BV for the problems (6.31) and (6.33) in the material description; they admit an explicit analytical representation of the tensor functions mentioned above (see (6.16b), (6.16c) and Exercises 6.1.1 and 6.1.2). If we consider a special model of an elastic continuum, for example, the quasi.n/

.n/

linear model An , then as the tensor functions F G and F ıG in (6.30) and (6.31) we can use the representations (6.8a) and (6.47) under the assumption that  D 0 and 6 e where 4 M e L  0. For linear models An , the tensor 4 M has the form 4 M D J 4 M, ı

is the elastic moduli tensor, which for different symmetry groups G s of an elastic continuum is chosen in accordance with the representations from Sect. 4.8.7. t u

6.3.6 Conditions on External Forces in Quasistatic Problems Remind (see Sect. 3.3.1) that besides the motion equations (or the equilibrium equations) having a local character, in continuum mechanics there are integral laws: the momentum balance law (3.32) and the angular momentum balance law (3.82). For quasistatic processes, inertial forces in these laws can be neglected; thus, we obtain the equations Z

Z f dV C

V

†

e tn d† D 0;

Z

Z x  f dV C V

†

x e tn d† D 0:

(6.101)

6.3 Statements of Problems for Elastic Continua at Large Deformations

413

These equations are additional conditions on external forces f and e tne and displacements ue in the quasistatic problem (6.97). Using the boundary conditions of (6.97), we can rewrite the conditions (6.101) in the form Z X Z X Z e f dV C tn d† D 0; tne d† C V

Z

˛D1;:::;4;7 †˛

X

x  f dV C V

Z

˛D1;:::;4;7 †˛

˛D5;6 †˛

x e tne d† C

X Z

˛D5;6 †˛

ı

.x C ue /  tn d† D 0: (6.102)

These conditions can be expressed in terms of a reference configuration as follows: Z

ı

ı

f d V C

V

Z

ı

ı

V

X

ı

Z ı

˛D1;:::;4;7 †˛ ı

x  f d V C

X

Z

ı

˛D1;:::;4;7 †˛

X Z ı ı ı e ı tne d † C tn d † D 0; ı ˛D5;6 †˛

X Z ı ı e ı ı ı x  tne d † C ı .x C ue /  t n d † D 0: ˛D5;6 †˛

(6.103) e ı As a result, we find conditions on the vectors tne , f and ue for the quasistatic problem in the material description.

6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description In solving problems of continuum mechanics at large deformations, variational statements of the problems are widely used. To state these problems we introduce the concept of a kinematically admissible vector field u.x/, which is defined in a domain V [ †, two times continuously differentiable in this domain and satisfies the boundary conditions of the problem (6.97) at the surface parts †u D †5 [ †6 and †8 , where displacements are given: ˇ uˇ†u D ue ;

ˇ u  nˇ† D 0: 8

(6.104)

If the vector field u.x/ in addition satisfies all the remaining equations of the system (6.97), then it is called real. So a real vector field u is the desired solution of the problem (6.97). Introduce the concept of the variation ıu of a vector field as the difference of two kinematically admissible fields. The variation ıu satisfies the zero boundary conditions ˇ ˇ ıuˇ†u D 0; ıu  nˇ† D 0: (6.105) 8

414

6 Elastic Continua at Large Deformations

Introduce integral characteristics of the motion of a continuum: Ae is the work done by external forces acting over the displacements u, which consists of the work Ae† done by external surface forces and the work Aem done by external mass forces: Z A D e

Ae†

C

Aem ;

Ae†

D †

e tne  u d†;

Z Aem

D

f  u dV:

(6.106)

V

Here † D †1 [ †2 [ †3 [ †4 [ †7 is the part of the body surface, where the stresses vectore tne included in system (6.97) is given. In addition, introduce the potential energy … of a body as follows: Z …D



dV;

(6.107)

V .n/

. C G ; O; / is the free energy (potential) (6.3b) in isothermal processes,

where

.n/

which depends on the generalized energetic deformation tensor C G and the rotation tensor O in accordance with the models An , Bn , Cn and Dn . Write the functional L.u/ D …  Ae ;

(6.107a)

called Lagrangian, and introduce the concept of the variation of functionals ıL, ı… and dAe over displacements u, which can be calculated by the same rules as the differential df of function f .t/. Then for ı… we obtain the expression Z ı… D ı

Z 

dV D ı

V

ı

ı



ı

Z

dV D

V

ı

ı

ı

V

ı

Z

dV D

ı

dV:

(6.108)

V

Here we have used the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2). Let us apply now formula (4.178) of differentiation of the scalar function .n/

. C G .t/; O.t/; / with respect to the argument t to the case of isothermal processes (when  D const/ and ideal continua (when w D 0/. According to this formula, we can determine the variation ı : .n/

ı . C G ; O; / D 

ı .n/

@CG

.n/

  ıCG C 

.n/ .n/ ı   ıOT D T G   ı C G C SG   ıOT : @O

(6.109)

Since formula (6.109) holds for all n D I; : : : ; V and G D A; B; C; D, then, choosing n D V and G D A, we obtain V

ı

D T   ıC D

1V T   ı.FT  F C FT  ıF/ 2

6.3 Statements of Problems for Elastic Continua at Large Deformations

415

1 V T F  T  F  .F1T  ıFT C ıF  F1 / 2 ı ı 1 D T   .F1T  r ˝ ıu C .r ˝ uT /  F1 / D T  ".ıu/: 2 (6.110) D

Here we have used the relation (2.73) between C and F, the relation (4.25) between V

T and T, and also formulae (2.79) and (2.37) which yield the relations ı

ıFT D r ˝ ıu;

F1T  ıFT D r ˝ ıu:

(6.111)

In Eqs. (6.110), we have denoted the linear deformation tensor " and its variation: ".u/ D

1 1 .r ˝ u C r ˝ uT /; ı".u/ D ".ıu/ D .r ˝ ıu C r ˝ ıuT /: (6.112) 2 2

Equating the right-hand sides of (6.109) and (6.110), we find that .n/

.n/

T G   ı C G C SG   ıOT D T   ".ıu/:

(6.113)

This equation is an analog of formulae (4.92) and (4.9) for the stress power when the tensor T is symmetric. The tensor ".ıu/ is an analog of the tensor Ddt D ".vdt/ in (6.113). Substituting (6.109) and (6.113) into (6.108) and evaluating the variation ıAe , we find the expression for the variation of Lagrangian Z

Z

ıL D

T   ".ıu/ dV  V

Z f  ıu dV 

V

†

e tne  ıu d†:

(6.114)

Let us formulate now the main theorem. Theorem 6.1 (Lagrange’s variational principle). Among all kinematically admissible fields u.x/, a real field is distinguished by the fact that its and only its Lagrangian L has a stationary value: ıL D 0:

(6.115)

H Let the field u satisfy Eq. (6.114). Then, substituting (6.114) into (6.115), we obtain Z Z Z e T   ".ıu/ dV  f  ıu dV  (6.116) tne  ıu d† D 0: V

V

†

416

6 Elastic Continua at Large Deformations

According to (6.112) and the properties of the product of the divergence of a tensor and a vector (see [12]), we can represent the first integral of (6.116) in the form Z

Z

Z

T   ".ıu/ dV D V

T   r ˝ ıuT dV D V

V

Z D

n  T  ıu d† C †

Z r  .T  u/ dV 

2 Z X

ıu  r  T dV V

Z

˛D1 †8

n  T   ˛ ıu˛ d† 

V

ıu  r  T dV: (6.117)

Here we have taken into account that the tensor T is symmetric and applied the Gauss–Ostrogradskii formula, and also used that ıu D 0 at †u , and at the part †8 the following relationship holds:

n  T  ıu D tn  ıu D tnn n C

2 X

1 ! 0 2 X tn˛  ˛  @ıun n C ıu˛  ˛ A ˇ D1

˛D1

D tnn ıun C

2 X ˛D1

tn˛ ıu˛ D

2 X

tn˛ ıu˛ ;

˛D1

as ıun D n  ıu D 0 at †8 . Here n,  ˛ is the orthonormal basis; and tnn , tn˛ and un , u˛ are projections of the vectors tn and u onto the basis vectors. On substituting (6.117) into (6.116), we obtain 2 Z X ˛D1 †8

Z .n T  ˛ /ıu˛ d†C

†

.n Te tne / ıu d†

Z ıu .r  Tf/ dV D 0: V

(6.118)

Since functions ıu are arbitrary, from Eq. (6.118) it follows that all expressions in parentheses must vanish. Thus, the equilibrium equations in the system (6.97) actually hold, and the boundary conditions at † and †8 are satisfied. At the remaining surface part †u , the boundary conditions are satisfied due to the choice of functions u. Thus, u is a solution of the problem (6.97). Conversely, let u.x/ be a real field of displacements, which satisfies the system (6.97). Then, multiplying the equilibrium equations and the boundary conditions at † and †8 by ıu, we get Eq. (6.118). Following in the reverse order all manipulations from (6.117) to (6.114), we verify that Eq. (6.114) actually holds. N Equation (6.116) is called the variational equation, and the variational statement of the quasistatic problem (6.97) serves for finding a kinematically admissible displacement field u.x/ satisfying the variational equation (6.116).

6.3 Statements of Problems for Elastic Continua at Large Deformations

417

6.3.8 Variational Statement of Quasistatic Problem in the Material Description In a similar way, we can formulate a variational statement of the quasistatic problem (6.98). With the help of formulae (6.110), (2.37), and (4.5) we can transform the variation ı as follows: ı

1 T .T   r ˝ ıu C T   r ˝ ıuT / 2 ı ı 1 D .TT  F1T   r ˝ ıu C F1  T   r ˝ ıuT / 2 ı ı ı   D ı .PT   r ˝ ıu C P   r ˝ ıuT / D ı P   r ˝ ıuT : 2  D

(6.119)

ı

Then using formulae (3.55) of passage from K to K and substituting (6.119) into ı

(6.116), we obtain the variational equation in K Z

ı

ı

ı

Z

P   r ˝ ıuT d V 

V

ı

ı

ı

Z

f  ıu d V 

V

ı

†

ı tne

ı

 ıu d † D 0:

(6.120)

6.3.9 Variational Statement for Incompressible Continua in the Material Description Let us formulate now a variational statement of the quasistatic problem (6.99) of elasticity for incompressible materials. Introduce a field of possible pressures p.x/ being an arbitrary scalar field defined and continuously differentiable in V [ †, a real field of pressures p.x/ satisfying (together with a real field of displacements u.x/) the system (6.99), and also the variation of pressures ıp being the difference of two possible pressures. The potential energy for an incompressible continuum is introduced as follows: Z …D

.

e p / dV:

(6.121)

V

Here e p is the indeterminate Lagrange multiplier: e p D p.n  III/2 (see (4.500a)), .n/

and is a scalar function of the tensor G G , whose value is zero (see (4.493)): .n/

.n/

. G G / D det G G 

1 D 0: .1  III/3

(6.122)

418

6 Elastic Continua at Large Deformations

The variation of the function is calculated by the same rules as the differential d

(see (4.494)): ı D

@

.n/

.n/

  ı GG D

@G G

.n/ .n/ 1 G 1 G  ı G G : 3 .n  III/

(6.123)

Since constitutive equations for an incompressible elastic continuum can be written in the form (4.523) .n/

TG D 

p .n/1 e G C FG; n  III G

r1 X eG D  @ F D ' I.s/ G; .n/  D1 @GG

(6.124)

the variation ı , by analogy with formula (6.109), can be represented as follows: .n/

.n/

e G   ı C G C SG   ıOT : ı . C G ; O; / D F

(6.125)

According to (6.123) and (6.125), we find Z

Z  eG  e p ı  ıe p / dV D F

.n/ p .n/1  G G   ı G G dV n  III V V Z Z  .n/  1 det G G  ıe p dV C SG   ıOT dV  .n  III/3 V V Z .n/ Z  .n/  .n/ 1 D . T G   ı C G C SG   ıOT / dV  det G G  ıe p dV .n  III/3 V V Z Z 1 T   ".ıu/ dV  ..det F/nIII  1/ıp dV: (6.126) D n  III V V

ı… D

.ı

Here we have used formulae (6.124) and (6.113), which hold for incompressible materials too, and taken into account that .n/

1 1 det UnIII D .det U/nIII .n  III/3 .n  III/3 .n/ 1 D .det F/nIII D det g .n  III/3

det G D

and det F D det U D det V. According to constitutive equations for incompressible elastic continua in the form (6.99), we finally obtain Z ı… D V

.n/

e G  pE/   ".ıu/ dV  .F

1 n  III

Z ..det F/nIII  1/ıp dV: (6.127) V

6.3 Statements of Problems for Elastic Continua at Large Deformations

419

Writing Lagrangian L.u; p/ according to formulae (6.107a), (6.121) and (6.106), we can formulate Lagrange’s variational principle for incompressible continua. Theorem 6.2. Among all kinematically admissible fields u.x/ and possible pressures p, real fields of displacements and pressures for incompressible continua are distinguished by the fact that their and only their Lagrangian L.u; p/ has a stationary value: ıL.u; p/ D 0: (6.128) Substituting the expressions (6.107a), (6.121) and (6.106) into (6.128) and taking into account that the variations ıu and ıp are mutually independent, we obtain the following system of variational equations for incompressible elastic continua: 8
.n/ R R e G  pE/   ".ıu/ dV  e F . V V f  ıu dV  † tne  ıu d† D 0; :R V .det .E  r ˝ u/  1/ıp dV D 0:

(6.129) A proof of the theorem is similar to the proof of Theorem 6.1 (see Exercise 6.3.6). A variational statement of the quasistatic problem (6.100) of elasticity for an incompressible continuum can be obtained from (6.129) with the help of transformations (6.119): 8 .n/ ı ı ı ı ı R ı R e ˆ
Exercises for 6.3 6.3.1. Show that the statement (6.99) of the quasistatic problem of elasticity theory in the spatial description for the model AI of incompressible media has the form (by using the relation (6.16b)): r  T C f D 0; det F1 D 1 in V; 8 Pr1 1T 1 ˆ  I.s/  F ; ˆ ˆT D pE C  D1 ' F ˆ
†8

420

6 Elastic Continua at Large Deformations

6.3.2. Show that the quasistatic problem statement (6.97) of elasticity theory for the models Bn and Dn of isotropic media can be written in the form (by using the constitutive equations (4.418a)): ı

r  T C f D 0;

T D 1 2 3

3 X

˛

˛D1 1

p˛ k F ; ˇ uˇ† D ue ;

˛ ; ˇ ˇ e n  T † D tne ;

u

1

F ˇ uˇ

@ .ˇ / p˛ ˝ p˛ ; @˛

D E  r ˝ uT ;

†8

ˇ n  Tˇ†   I D 0:

 n D 0;

8

Here for incompressible continua, constitutive equations take the form 8
˛D1

:det F1 D 1:

˛

@ .ˇ / p˛ ˝ p˛ ; @˛

6.3.3. Show that the quasistatic problem statement (6.98) of elasticity theory in the material description for the models Bn and Dn of isotropic media can be written in the form (by using the results of Exercise 6.3.2): ı

ı

ı

r  P C f D 0;

PD

3 X @ .ˇ / ı p ˛ ˝ p˛ ; @˛

˛D1

ı

ı

ˇ ı n  Pˇ ı

†

˛ ; p˛ ; p˛ k F1 ; F1 D E  r ˝ uT ; ˇ ˇ ˇ ı ı ı ı D tne ; uˇ ı D ue ; uˇ ı  n D 0; n  Pˇ ı   I D 0; †u

†8

†8

where for incompressible continua, constitutive equations take the form 8

ı

r  P C f D 0; det F D 1 in V; 8 Pr1 ı .s/ T 1 ˆ '  I  F ; ˆ ˆP D pF C  D1 ˆ ı ı ˆ .s/ .s/
ı

ı

ı

ˆ C D .1=2/.r ˝ u C r ˝ uT C r ˝ u  r ˝ uT /; ˆ ˆ ˆ ı ˆ : F D E C r ˝ uT in V ; ˇ ˇ ˇ ˇ ı ı ı ı ı n  Pˇ ı D tne ; uˇ ı D ue ; uˇ ı  n D 0; n  Pˇ ı   I D 0: †

†u

†8

†8

6.4 The Problem on an Elastic Beam in Tension

421

ı

Show that the problem for basis ri can be written in components: ık

ıı

ı ı

r j P ji C f i D 0; det .F l / D 1; 8ı ı ı ı P ı j j ˆ P i D p.F 1 /i C r1 '  .@I G =@"jk /F lk g i l ; ˆ  D1 ˆ ˆ ı ı ˆ ˆ ˆ ' D .@ =@I.s/ /; D .I.s/ ."jk /; /; ˆ <  ı ı ı ı ı ı ı ı ı "jk D .1=2/.r j uk C r k uj C r j um r k ui g mi /; ˆ ˆ ı ı ı ı ˆ ˆ ˆ F ik D ıki C r k um gmi ; ˆ ˆ ˆ ı ı : ı 1 j .F /i D .1=2/j mk i qs F qm F sk ; ı ı ı jˇ ı ı ıj ˇ ı ı ˇ ı ı ˇ nj P i ˇ ı D t nei ; ui ˇ ı D uei ; ui ˇ ı ni D 0; nj P k ˇ ı kI D 0: †

†u

†8

ı

†8

j

(One should derive an expression for .F 1 /i by using the relation (2.125) between components of a matrix and its inverse.) 6.3.5. Prove relations (6.71). 6.3.6. Prove Theorem 6.2.

6.4 The Problem on an Elastic Beam in Tension 6.4.1 Semi-Inverse Method Consider several classical examples of solving the elasticity problems (6.97) and (6.98). Practically all main analytical solutions of quasistatic problems of elasticity theory at large deformations can be obtained by using the semi-inverse method, where proceeding from the geometric shape of a body, in a reference configuration one selects (guesses) a law of its motion x.X i ; t/, according to which the deformation gradient field F.X i ; t/ is calculated. Then one can evaluate the stress tensor fields T and P by constitutive equations. Further, one should verify that the tensor fields P.X i ; t/ and T.X i ; t/ obtained satisfy the equilibrium equations and boundary conditions of the systems (6.97) and (6.98). If the motion law is guessed right, then these equations and conditions are satisfied identically.

6.4.2 Deformation of a Beam in Tension Consider the problem on a beam in tension; then the motion law has the form (2.6) x ˛ D k˛ .t/ X ˛ :

(6.131)

422

6 Elastic Continua at Large Deformations

Table 6.1 Sequential calculation of the deformation tensors Tension (2.6)

Shear (2.7)

Rotation with tension (2.8)

Local basis vectors and metric matrices

Exercise 2.1.1

Exercise 2.1.2

Exercise 2.1.3

Deformation gradient F, C; G; A; J

Exercise 2.2.1

Exercise 2.2.2

Exercise 2.2.3

Tensors U; V; O

Exercise 2.3.2

Exercise 2.3.3

Exercise 2.3.4

Exercise 4.2.13

Exercise 4.2.14

Exercise 4.2.15

The motion law

.n/

.n/ .n/

.n/

Tensors C; A; G; g

The deformation gradient F for this problem becomes (see Exercise 2.2.1) FD

3 X

k˛ eN ˛ ˝ eN ˛ :

(6.132)

˛D1

The deformation tensors for this problem were calculated above (this procedure is shown in Table 6.1). .n/

.n/

In particular, the tensors C and A have the form .n/

.n/

CDAD

3 .n/ X C ˛ eN ˛ ˝ eN ˛ ;

(6.133)

˛D1 .n/

C˛ D

1 .k nIII  1/; n D I; II; IV; VI n  III ˛

.n/

C ˛ D ln k˛ ; n D III: (6.134)

6.4.3 Stresses in a Beam Assume that a beam is made of elastic (ideal, solid) isotropic material described by linear models An (4.338): .n/

.n/

.n/

T D J.l1 I1 . C /E C 2l2 C /;

ı

J D =:

(6.135)

Let us calculate stresses in the beam. On substituting the expressions (6.133) and .n/

(6.134) for the energetic deformation tensors C into (6.135), we find components .n/

T ˛˛ of the energetic stress tensors

6.4 The Problem on an Elastic Beam in Tension .n/

T D

423

3 .n/ X T ˛˛ eN ˛ ˝ eN ˛ ; ˛D1

0 0

.n/

T ˛˛

1 1 3 X 1 @l1 @ D kˇnIII  3A C 2l2 .k˛nIII  1/A ; .n  III/k1 k2 k3

(6.136)

ˇ D1

n D I; II; IV; V; III

T ˛˛

1 D k1 k2 k3

l1

3 X

!

lg k˛ C 2l2 lg k˛ :

˛D1

Since in this problem the rotation tensor O is the unit tensor (see Exercise 2.3.2), .n/

.n/

.n/

.n/

the quasienergetic stress tensors S coincide with T (see Exercise 4.2.3): S D T . To determine the Cauchy stress tensor T we use the tensors of energetic equiva.n/

lence 4 E and formulae (4.37) having the following forms in this problem: I

nDI W

T DF1T  T  F1 ;

nDII W

T D.1=2/.F1T  T C T  F1 /;

nDIII W

T DT;

II

II

III

(6.137) IV

IV

nDIV W

T D.1=2/.F  T C T  FT /;

nDV W

T DF  T  FT :

V

On substituting the expression (6.132) for F and formulae (6.133) into (6.137), we find components ˛˛ of the Cauchy stress tensor T with respect to the Cartesian basis for different models An : TD

3 X

˛˛ eN ˛ ˝ eN ˛ ;

.n/

˛˛ D k˛nIII T ˛˛ ;

(6.138)

˛D1

˛˛

0 0 1 1 3 X k˛nIII @l1 @ D kˇnIII  3A C 2l2 .k˛nIII  1/A ; .n  III/k1 k2 k3 ˇ D1

n D I; II; IV; V; III

˛˛ D T ˛ ;

n D III:

(6.139a) (6.139b)

Since the expression for T obtained is independent of the coordinates X i , the tensor T (6.138) automatically satisfies the equilibrium equations of the system (6.97): r  T D 0 when there are no mass forces (f D 0).

424

6 Elastic Continua at Large Deformations

6.4.4 The Boundary Conditions Boundary conditions of the system (6.97) applying to the problem on a beam in tension become x1 D 0 W

n  u D 0;

1

nuD x D h1 W ˛ x D ˙h˛ =2 W

n  T   ˛ D 0;

˛ D 2; 3I

u1e ;

n  T   ˛ D 0; ˛ D 2; 3I n  T D 0; ˛ D 2; 3:

(6.140)

In other words, according to the problem statement from Example 2.1, at one end of the beam .x 1 D 0/ the symmetry condition is given, at the other end .x 1 D h1 / – the displacement u1e , and the lateral surfaces .x I D ˙hI =2/ are free of loads. Since at the surface x 1 D 0 we have n D Ne1 ,  ˛ D eN ˛ (˛ D 2; 3) and at the lateral surface x ˛ D ˙h˛ =2: n D eN ˛ , the boundary conditions (6.140) take the form x1 D 0 W

u1 D 0;

x D h1 W

u D

1

1

˛

u1e ;

x D ˙h˛ =2 W

eN 1  T  eN ˛ D 0;

˛ D 2; 3;

eN 1  T  eN ˛ D 0;

˛˛ D 0;

˛ D 2; 3I ˛ D 2; 3:

(6.141a) (6.141b) (6.141c)

As follows from formula (6.138), there are no tangential stresses in this problem; therefore, the conditions eN 1 T eN ˛ D 0 are automatically satisfied in the whole beam. The boundary condition u1 D 0 at x 1 D 0 is satisfied due to (6.131), and from the boundary condition at x 1 D h1 we express k1 in terms of u1e : u1e D h1  h01 D .k1  1/h01 ;

k1 D 1 C u1e = h01 :

(6.142)

6.4.5 Resolving Relation 1  k1 On substituting (6.139a) into the boundary condition (6.141c), we find that the components 22 and 33 of the stress tensor T in this problem must be zero, and only the component 11 is nonzero. As a result, uniting (6.139b) and (6.141c), we obtain three equations 1 1 0 0 3 nIII X k1 @l1 @

11 D kˇnIII  3A C 2l2 .k1nIII  1/A ; .n  III/k1 k2 k3 ˇ D1

0 0 D l1 @

3 X

n D I; II; IV; V; 1 kˇnIII  3A C 2l2 .k˛nIII  1/;

(6.143) ˛ D 2; 3;

ˇ D1

for the three unknowns: k1 , k2 and 11 (k1 is determined by (6.142)).

(6.144)

6.4 The Problem on an Elastic Beam in Tension

425

Notice that the system (6.143)–(6.144) admits the solution k2 D k3 ; then with the help of (6.144) we can express k2 and k3 in terms of k1 : k˛nIII  1 D .k1nIII  1/; where D

˛ D 2; 3;

(6.145)

l1 2.l1 C l2 /

(6.146) .n/

.n/

is the Poisson ratio. Calculating the ratio of components C 2 and C 1 of the energetic deformation tensors, for which Eqs. (6.134) hold, and using formula (6.145), we find that Poisson’s ratio is the ratio of these components with the opposite sign: .n/

.n/

.n/

.n/

C 2 = C 1 D C 3 = C 1 D :

(6.147)

This relationship holds for all models An . For most real isotropic elastic materials, values of the Poisson ratio are within the interval 0 <  < 0:5. Therefore, for a .n/

beam in longitudinal tension along the axis Ox 1 when C 1 > 0, the transverse de.n/

.n/

formations are negative: C 2 D C 3 < 0, i.e. there occurs a transverse compression of the beam. This phenomenon is called the Poisson effect. On substituting the expression (6.145) into (6.143), we find the resolving relation for the problem: the dependence of the stress 11 on the elongation ratio k1 :

11 D 2l2 .1 C /

k1nIII1 nIII  1/.1  .k1nIII  1//2=.nIII/ ; .k n  III 1 n D I; II; IV; V:

(6.148)

By the same method, we obtain the resolving relation for the model AIII (see Exercise 6.4.1): k2 D k3 D 1=k1 ;

11 D 2l2 .1 C /

lg k1 : k112

(6.149)

6.4.6 Comparative Analysis of Different Models An Compare the dependences k2 .k1 / and 11 .k1 / obtained for different models An . Notice that for all models An , the dependences k2 .k1 / (6.145) and (6.149) are monotonically decreasing, i.e. the extension of the beam along the axis Ox 1 is accompanied by its compression along the axes Ox 2 and Ox 3 (remind that the values k˛ > 1 correspond to extension along the axis Ox ˛ , and the values 0 < k˛ < 1 – to compression along Ox ˛ ). This property is called the Poisson effect.

426

6 Elastic Continua at Large Deformations

a

b

c

Fig. 6.4 Dependence k2 .k1 / for different models An : (a) – n D I and n D II, (b) – n D III, (c) – n D IV and n D V

However, for the models AI and AII , the function k2 .k1 / has horizontal and vertical asymptotes (Fig. 6.4): k1 D

k1

 

 1C

1=.IIIn/

and k2 D

k21

 

1 1C

1=.IIIn/

; n D I; II;

and for the models AIV and AV there are no asymptotes, but there is ultimate value k1 D .1 C =/1=.nIII/ , at which the solution degenerates with loss of the physical meaning. For the model AIII , according to (6.149), the function k2 .k1 / also has horizontal and vertical asymptotes which are coincident with the abscissa and ordinate axes, respectively (Fig. 6.4). Comparing graphs of the functions 11 .k1 / (6.148) and (6.149), we can see that for all models An (n D I; : : : ; V), values of the stresses 11 are positive for the beam in tension along the axis Ox 1 .k1 > 1/ and negative for the beam in compression along the axis Ox 1 .0 < k1 < 1/, that agrees with actual stresses observed in practice. Notice that values of k1 can be found in experiments by displacements u1e according to formula (6.142), and stresses 1 – according to formula (3.64), which for the beam takes the form b11 D F11 =†1 :

11 D T

(6.150)

Here F11 is the component referred to the axis Ox 1 of the external force vector F 1 D F11 eN 1 applied to the beam side †1 orthogonal to Ox 1 . However, for the models AI and AII , the function 11 .k1 / in tension (when k1 > 1) first grows and then diminishes, and in compression (when k1 < 1) the magnitude j 11 .k1 /j also first increases and then decreases, while 11 .k1 / D 0 (see Fig. 6.5, where we have denoted the dimensionless stress N 11 D 11 =.2l2 .1 C //). The monotone increase of the function 11 .k1 /, that is usually observed in experiments, occurs only within the interval k1 < k1min 6 k1 6 k1max . The extremum valmax ues k1min and ˇ k1 , at which monotonicity of the function 11 .k1 / is violated (i.e. @

.k /ˇ D 0), are sometimes interpreted as ultimate values of loss @k1 11 1 k1 Dk1min ;k1max of stability of the material in compression or in tension, respectively. However, this

6.4 The Problem on an Elastic Beam in Tension

427

Fig. 6.5 The function

11 .k1 / for the models AI and AII ,  D 0.3

Fig. 6.6 The function

11 .k1 / for the models AIII , AIV and AV ,  D 0.3

effect cannot be identified with the phenomenon of loss of stability for structures of solids in compression, that is connected, as a rule, not with non-linearity of the constitutive equations but with the nonlinear dependence of deformations upon displacements. For the models AIII and AIV , the function 11 .k1 / is monotonically increasing at all values of k1 . For the model AIV , the function is defined only within the interval 0 < k1 < k1 , and 11 ! C1 as k1 ! k1 . For the model AV , the function 11 .k1 / in tension (k1 > 1) behaves as well as it does in the model AIV ; and in compression the function has an extremum point (k1 D kmin , see Fig. 6.6). The model AIII is the only one of models An , for which the function 11 .k1 / exists and is monotone along the whole semiaxis k > 0, and 11 .k/ ! 1 as k ! 0. Thus, the problem on a beam in tension shows that models An at different n yield distinct (even in kind) dependences of the stress on the elongation ratio 11 .k/. The question what model An should be applied to a specific elastic continuum is solved by comparing experimental data with computed results determined by different models; in particular, by comparing the functions 11 D 11 .k1 / (6.148) and (6.149) with corresponding experimental dependences. In experimental investigations, instead of k1 one usually uses the relative elongation u1 h1  h01 ı1 D k1  1 D e0 D ; (6.151) h1 h01 and the dependence 11 D 11 .ı1 / is called the diagram of deforming.

428

6 Elastic Continua at Large Deformations

Notice also that in experiment one frequently determines the Piola–Kirchhoff stress P11 , that is the ratio of component F11 of the external force vector (measured ı

in experiments) to the undistorted surface element †1 : ı

P11 D F11 =†1 :

(6.152)

ı

In the problem on tension, the areas †1 and †1 are connected by the relation ı

†1 =†1 D k2 k3 D k22 ;

(6.153)

therefore, P11 and 11 are connected by

11 D P11 =k22 :

(6.154)

According to (6.148), (6.149) and (6.154), we find the following expressions for the function P11 .k1 /: k1nIII1 nIII  1/; n D I; II; IV; V; .k n  III 1 lg k1 D 2l2 .1 C / ; n D III: k1

P11 D 2l2 .1 C / P11

(6.155)

Notice that the functions P11 .k1 / are independent of the Poisson ratio. Thus, if we have the experimental diagram P11 .k1 /, then we cannot calculate values of  by (6.155); they may be determined by changing the thickness of a specimen in tension or in compression according to formula (6.146). .ex/ .ı1 / obtained Figure 6.7 shows the experimental diagram of deforming P11 by the method mentioned above for filled rubber, where ı1 D k1  1, and also

.ex/

Fig. 6.7 Experimental diagram of deforming P11 .ı1 / for filled rubber in tension and its approximation by linear models An at  D 0.4

6.4 The Problem on an Elastic Beam in Tension

429

approximation of this diagram with the help of models An by formulae (6.155). A value of the constant l2 for these models was determined by minimizing the meansquare distance ı1.i / between computed and experimental curves at N points: D

N P11 .ı1.i / / ˇˇ2 1 X ˇˇ ˇ1  .ex/ ˇ N P11 .ı1.i / / i D1

!1=2 ! min:

(6.156)

The Poisson ratio for filled rubber proved to be close to  D 0.4. Values of the constant l2 , obtained by the method mentioned above, are shown in Table 6.2; this table gives also the minimum mean-square deviations corresponding to these optimal values of l2 . The values found allow us to conclude that the model AIII gives the best approximation to the experimental diagram of deforming (see Fig. 6.7). The models .ex/ .ı1 /, when ı1 ranges from AI and AII satisfactorily approximate the diagram P11 0 to 22 and 45%, respectively. At higher values of ı1 the condition k1 > kmax holds, and functions P11 .ı1 / for these models give the non-physical effect of lowering the stresses P11 . .ex/ Figure 6.8 shows diagrams of deforming 11 .ı1 / for the same rubber, which .ex/ were recalculated from the experimental diagram P11 .ı1 / by formulae (6.154), where k2 were determined by (6.145) and (6.149). Since the functions k2 .k1 / are .ex/ distinct for different models An , the diagrams 11 .ı1 / are also distinct for these models. Table 6.2 Values of the constant l2 for rubber in different models An

Fig. 6.8 Experimental diagram of deforming in tension for filled rubber, recalculated in terms of the Cauchy stresses by models An when  D 0.4

n I II III IV V

l2 , MPa 29.6 27.1 16.4 7.5 3.0

, % 37 17 9 42 58

430

6 Elastic Continua at Large Deformations

Fig. 6.9 Diagrams of deforming for filled rubber in tension, computed in terms of the Cauchy stresses by models An when  D 0.4

Figure 6.9 exhibits diagrams 11 .ı1 / for models An , computed by formulae (6.148) and (6.149), where the constant l2 was found with the help of approxi.ex/ .ı1 / by functions (6.155) for rubber (see mation to the experimental diagram P11 Table 6.2). .ex/ The situation is also possible when in experiment one determines not P11 .ı1 / .ex/ but the diagram of deforming 11 .ı1 /. Then the constant l2 can be found from .ex/ the condition of the best approximation to the function 11 .ı1 / by the dependence (6.148) or (6.149), while the coefficient  is usually determined by Eq. (6.147). For the considered class of filled rubbers, the qualitative relative position of the functions .ex/ .ı1 / is the same as for the Piola–Kirchhoff stresses; in this case the

11 .ı1 / and 11 model AIII gives the best results too.

Exercises for 6.4 6.4.1. Prove the relations (6.148) for the linear model AIII .

6.5 Tension of an Incompressible Beam 6.5.1 Deformation of an Incompressible Elastic Beam Consider the problem on tension of a beam (see Example 2.1 and Sect. 6.4.2), but the beam is assumed to be an incompressible elastic (ideal) solid. Let boundary conditions for the beam be the same conditions (6.140), and a law of the beam motion will be sought in the form (6.131) too. Then the deformation gradient F is expressed by formula (6.132), and the deformation measures and tensors have the same forms as the ones for a compressible continuum and are determined

6.5 Tension of an Incompressible Beam

431 .n/

by Table 6.1. In particular, the energetic deformation measures G have the form (see Exercise 4.2.13): .n/

.n/

GD g D

3 1 X nIII k eN ˛ ˝ eN ˛ ; n D I; II; IV; V: n  III ˛D1 ˛

(6.157)

Since the solid was assumed to be incompressible, the condition of incompressibility (4.489) must be satisfied. Then substituting the expression (6.132) for F into (4.489), we obtain the relation 1 D det F D k1 k2 k3 ;

(6.158)

from which we can express k2 and k3 in terms of k1 (in this problem k2 D k3 ): p k3 D k2 D 1= k1 :

(6.159)

This simple relation is an analog of relations (6.145) and (6.149) for a compressible material; from (6.159) it follows that for incompressible continua the Poisson effect also occurs (see Sect. 6.4.5) but the dependence of k2 and k3 upon k1 has an universal character: k2 and k3 are independent of the constants  and ˇ of an incompressible continuum (for a compressible medium, the situation is different). Since in the problem on tension of a beam k1 is immediately evaluated from the boundary condition (6.142), values of k2 and k3 can be found by formula (6.159) without use of the constitutive equations too.

6.5.2 Stresses in an Incompressible Beam for Models Bn Let constitutive equations of an elastic incompressible beam correspond to linear models Bn (4.532): p .n/1 T D G C .n  III/2 n  III

.n/



!  .n/ .n/ 1Cˇ C .1  ˇ/I1 . G/ E  .1  ˇ/ G ; n  III (6.160)

which contain two constants:  and ˇ. .n/

On substituting relation (6.157) into this expression, we find that the tensors T in this problem, just as for compressible media, do not contain tangential stresses: .n/

T D

3 .n/ X T ˛˛ eN ˛ ˝ eN ˛ ; ˛D1

(6.161)

432

6 Elastic Continua at Large Deformations

X  3  T ˛˛ D pk˛IIIn C .n  III/ 1 C ˇ C .1  ˇ/ knIII  k˛nIII ;

.n/

 D1

˛ D 1; 2; 3;

n D I; II; IV; V:

(6.162) Substitution of these expressions into (6.137) yields a representation of the Cauchy stress tensor T for models Bn : TD

3 X

˛˛ eN ˛ ˝ eN ˛ ;

(6.163)

˛D1

 X  3

˛˛ D p C .n  III/k˛nIII 1 C ˇ C .1  ˇ/ knIII  k˛nIII ;  D1

˛ D 1; 2; 3: (6.164)

6.5.3 Resolving Relation 1 .k1 / Since k2 and k3 are coincident, in (6.164) there are only two independent relations: when ˛ D 1 and ˛ D 2, and from the boundary condition (6.141c) it follows that

22 D 0. As a result, we have the two equations

11 D p C .n  III/k1nIII .1 C ˇ C 2.1  ˇ/k2nIII /; 0 D p C .n  III/k2nIII .1 C ˇ C .1  ˇ/.k1nIII C k2nIII //

(6.165)

for the two unknowns: 1 and p. Eliminating p and using Eq. (6.159), we find the resolving relation between the stress 11 and the longitudinal elongation ratio k1 :  

11 D .n  III/ .1 C ˇ/ k1nIII   C.1  ˇ/ k1.nIII/=2 

1



.nIII/=2

k1 1  ; n D I; II; IV; V: nIII

k1

(6.166)

6.5.4 Comparative Analysis of Models Bn 11 .k / determined by formula (6.166) Figure 6.10 exhibits graphs of the functions  1 for different values of the parameter ˇ and different n. For the models BII and BIV , all the functions are convex upwards at all values of the constant ˇ within the in11 .k / have a point terval 1 6 ˇ 6 1. For the models BI and BV , the functions  1

6.5 Tension of an Incompressible Beam

433

Fig. 6.10 The functions 11 = determined by formula (6.166) for different values of ˇ and n

of inflection at ˇ < 1 and ˇ > 1, respectively: in the case of compression (when 0 < k < 1/ these functions are convex upwards, and in the case of tension (when k > 1) they are convex downwards. For the model BI with ˇ D 1 and the model 11 .k/ have no point BV with ˇ D 1 (Treloar’s model), the graphs of the functions  of inflection, they are convex upwards. 11 .k / have the physical When ˇ > 1 and ˇ < 1, graphs of the function  1   meaning only within some interval kmin < k1 < kmax , because for other values of k1 monotonicity of the function is violated. .ex/ Figures 6.11 and 6.13 show experimental diagrams of deforming 11 .ı1 / for rubber and polyurethane elastomer and also the dependences 11 .ı1 / (6.166) approximating these experimental curves. For each model Bn , the constants  and ˇ have been chosen in the optimal way by minimizing the mean-square distance between the computed and experimental diagrams of deforming in tension, calculated at N points: 0 !2 11=2 N

11 .ı1.i / / 1 X A 1  .ex/ ! min: (6.167) D@ N

.ı1.i / / i D1

11

434

6 Elastic Continua at Large Deformations

Fig. 6.11 Experimental diagram of deforming for rubber in tension and its approximation by models Bn and Treloar’s model

Fig. 6.12 Experimental diagram of deforming for rubber in compression and its approximation by models Bn and Treloar’s model

Fig. 6.13 Experimental diagram of deforming for polyurethane in tension and its approximation by models Bn

.ex/

Values of the stresses 11 .ı1 / have been determined by experimental values of .ex/ the Piola–Kirchhoff stresses P11 .ı1 / with the help of formula (6.154) for incompressible continua: .ex/

.ex/

11 D P11 k1 :

(6.168)

The models BIV and BV with ˇ D 1 were also considered (the Bartenev– Hazanovich and Treloar’s models, see Sect. 4.9.7).

6.5 Tension of an Incompressible Beam

435

Table 6.3 Values of the constants  and ˇ for rubber and polyurethane Rubber Polyurethane n , MPa ˇ , % n , MPa ˇ , % I 5.145 0.13 7.8 I 3.15 0.616 11.3 II 19.11 1 8.7 II 11.5 0.45 14.5 IV 19.11 1 8.7 IV 11.56 0.45 14.5 V 5.145 0.13 8.7 V 3.15 0.616 11.3 4.41 1 11 V

Fig. 6.14 Experimental and computed diagrams of deforming for polyurethane in compression

Table 6.3 shows values of the constants  and ˇ, calculated by the method mentioned above. This table gives also values of the mean-square distance  between experimental and computed curves. All the models Bn exhibit practically the same accuracy of approximation to the experimental curves; the models BI and BV are some more accurate. Figures 6.12 and 6.14 show computed and experimental diagrams of deforming for rubber and polyurethane in compression, where the computed diagrams were obtained by formulae (6.166), in which the constants  and ˇ were predetermined by diagrams of deforming in tension. For the materials considered, the models BII and BIV forecast more accurately a behavior of polyurethane in compression than the models BI and BV . It should be noted that experimental results in compression depend considerably on the conditions of testing, in particular, on the shape of a specimen and the method of its fastening. For example, in compression of specimens with a cylindrical shape the diagrams 11 .ı1 / depend essentially on the ratio of their initial diameter and thickness. Moreover, at a certain value of ı1 there usually occurs a loss of stability of a shape of the specimen; for example, the cylindrical shape often becomes the barrel-type one. The circumstances indicated appear to be main causes of the deviation of the computed diagrams 11 .ı1 / from the experimental .ex/ one 11 .ı1 / in compression.

436

6 Elastic Continua at Large Deformations

6.5.5 Stresses in an Incompressible Beam for Models An To describe the process of deforming of an incompressible elastic beam, we can choose other models stated in Sect. 4.9. For comparison, let us consider the linear models An (4.526) .n/

T D

.n/ .n/ p .n/1 G C .m N C l1 I1 . C //E C 2l2 C; n  III

n D I; II; IV; V;

(6.169)

which contain three constants: m, N l1 and l2 . The algorithm of determining the diagram of deforming 11 .ı1 / for these models .n/

is the same as the one for models Bn . The stresses T ˛˛ in the problem on tension have the form .n/

T ˛˛ D pk˛IIIn C m N C

IIIn l1 2l2 .k1nIII C 2k1 2  3/ C .k nIII  1/; n  III n  III ˛ ˛ D 1; 2; 3: (6.170)

Cartesian components ˛˛ of the Cauchy stress tensor are determined by formulae .n/

(6.138): ˛˛ D k˛nIII T ˛˛ ; as a result, we obtain the equations

11

 D p C m N C

22

   IIIn l1 2l2 nIII nIII 2 C 2k1 3 C  1/ k1nIII ; k .k n  III 1 n  III 1 (6.171)   IIIn IIIn IIIn l1 2l2 D 0 D pC mC N .k1nIII C2k1 2 3/C .k1 2 1/ k1 2 : nIII nIII

Eliminating p among these equations, we find the desired function 11 .k1 /, which can be written in the form .n/

.n/

.n/

11 D m N Q.k1 / C l1 M .k1 / C l2 N .k1 /;

(6.172)

where .n/

IIIn 2

Q D k1nIII  k1

.n/

;

IIIn IIIn 1 .k1nIII C 2k1 2  3/.k1nIII  k1 2 /; n  III .n/ IIIn IIIn 2 N D ..k1nIII  1/k1nIII  .k1 2  1/k1 2 /: n  III

M D

(6.173)

Figures 6.15 and 6.16 exhibit results of approximation to the experimental .ex/ diagrams of deforming 11 .ı1 / for rubber and polyurethane with the help of the

6.5 Tension of an Incompressible Beam

437

Fig. 6.15 Experimental diagram of deforming for rubber and its approximation by models An of incompressible continua

Fig. 6.16 Experimental diagram of deforming for polyurethane and its approximation by models An of incompressible continua

Table 6.4 Values of the constants m, N l1 , and l2 for rubber and polyurethane Rubber Polyurethane n m, N MPa l1 , MPa l2 , MPa , % n m, N MPa l1 , MPa l2 , MPa I 0.2 0.2 17.6 21.9 I 0.2 0.2 10.6 II 2.8 0.2 19.8 16.1 II 25 0.2 29.4 IV 19.8 0.2 2.4 28 IV 0.2 0.2 9 V 10.6 0.2 0.2 31 V 5.4 0.2 0.2

, % 21 9 34 38

functions (6.172) for different n DI, II, IV and V. The constants m, N l1 and l2 were calculated by minimizing the functional of the mean-square distance between the .ex/ .ı1 / at N points: functions 11 .ı1 / and 11 0

N 1 X

11 .ıi / D@ 1  .ex/ N

11 .ıi / i D1

!2 11=2 A

! min:

(6.174)

Values of the constants m, N l1 , and l2 calculated by this method are shown in Table 6.4.

438

6 Elastic Continua at Large Deformations

The model AII demonstrates the best approximation to the experimental diagram of deforming for the two considered types of materials (see Figs. 6.15 and 6.16 and Table 6.4); values of  for this model do not exceed 16.1% for rubber and 9% for polyurethane. Notice that for the model AII , unlike other models An and Bn considered above, the function 11 .ı1 / has a point of inflection as well as the .ex/ .ı1 /. experimental diagram 11 For other solids, the situation may be different. Consideration of the whole complex of energetic (or quasienergetic) models is appropriate, because, on performing calculations by all the models, one can choose the model giving the best results for a specific elastic continuum.

Exercises for 6.5 6.5.1. Consider the problem on an incompressible elastic beam in tension with the help of the linear models An (4.526) when m N D p0 D 0: .n/

T D

.n/ .n/ p G1 C l1 I1 . C /E C 2l2 C: n  III .n/

Show that in this case the final expression for the component 1 of the Cauchy stress tensor has the form .n/

1 D

l1 .k nIII C 2k1.IIIn/=2  3/.k1nIII  k1.IIIn/=2 / n  III 1 C

2l2 .IIIn/=2 .IIIn/=2 .k1 1//; .k nIII .k1nIII 1/k1 nIII 1

n D I; II; IV; V:

6.6 Simple Shear 6.6.1 Deformations in Simple Shear Consider the problem on an elastic solid body, whose shape is a parallelepiped, in simple shear. The motion of the body is determined by Eq. (2.7) (see Example 2.2 in Sect. 2.1.1): x i D X i C aı1i X 2 ;

i D 1; 2; 3:

(6.175)

The deformation tensors for this problem have been calculated above (see Table 6.1).

6.6 Simple Shear

439

6.6.2 Stresses in the Problem on Shear Assume that the body considered is isotropic and complies with the linear model .n/

.n/

An (4.338). Calculate the stress tensors T by substituting the tensors C from Exercise 4.2.14 into Eq. (4.338) or (6.135): .n/

T D

3 .n/ X .n/ T ˛˛ eN 2˛ C T 12 O3 ;

(6.176)

˛D1 .n/

.n/

.n/

.n/

T 11 D .l1 C 2l2 / c 0A C l1 c 2A ;

.n/

.n/

.n/

T 22 D .l1 C 2l2 / c 2A C l1 c 0A ;

.n/

T 12 D 2l2 c 1A ;

.n/

.n/

.n/

T 33 D l1 . c 0A C c 2A /;

n D I; II; IV; V: .n/

.n/

.n/

Here c 0A , c 1A and c 2A are determined by Table 4.6 from Exercise 4.2.14. .n/

.n/

Since the shear stresses T 12 are nonzero, the tensors T in this problem are not diagonal: each of them has four nonzero components. To evaluate the Cauchy stress tensor T, one should use formulae (4.37) and (4.38). Since in the problem on shear the rotation tensor O is different from E, calculating the tensors of energetic equiv.n/

alence 4 E is more complicated than in the problem on tension, because we need relations (4.37) to be represented in the eigenbases: .n/

.n/

T D 4E  T D

3 X

1 3 .n/ X .n/ E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛   @ T   eN 2 C T 12 O3 A 0

.n/

3 X ˛;ˇ D1

ı

 D1

˛;ˇ D1

D

ı

.p/ T˛ˇ p˛ ˝ pˇ

(6.177)

Here we have denoted components of the Cauchy stress tensor with respect to the eigenbasis p˛ by 1 3 .n/ X .n/ ı ı ı ı ı ı @ T   pN ˇ pN ˛ C T 12 .pN ˛1 pN ˇ 2 C pN ˛2 pN ˇ1 /A ; 0 .n/

.p/ D E ˛ˇ T˛ˇ

 D1

(6.178)

440

6 Elastic Continua at Large Deformations ı

ı

and also introduced components pN ˛ of resolution of the eigenvectors p˛ for the basis eN  : 1 0 s1 s1 b1 0 ı ı ı ı (6.179) p˛ D pN ˛i eN i ; pN ˛ D p˛  eN  D @ 0 s2 b2 0A ; 0 0 1 p a s˛ D .1 C b˛2 /1=2 ; b˛ D  .1/˛ 1 C a2 =4: 2 ı

Here we have used the expression for p˛ from Exercise 2.3.3: ı

ı

p˛ D s˛ .Ne1 C b˛ eN 2 /; ˛ D 1; 2I pN 3 D 1: With the help of (6.179) we can rewrite formulae (6.178) in the explicit form .p/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

T11 D E 11 . T 11 s12 C T 22 s12 b12 C 2 T 12 s12 b1 /; .p/ T22 D E 22 . T 11 s22 C T 22 s22 b22 C 2 T 12 s22 b2 /; .p/ T12

.n/

.n/

.n/

.n/

(6.180)

D E 12 . T 11 C T 22 b1 b2 C T 12 .b1 C b2 //s1 s2 ; .n/

.p/ .p/ .p/ D T23 D 0; T33 D T 33 : T13 .n/

The matrices E ˛ˇ are determined by formulae (4.39); and eigenvalues ˛ appearing in these formulae, according to the result of Exercise 2.3.3, have the forms 1 D

p 1 C b1 jaj;

2 D 1 1 ; 3 D 1:

(6.181)

Using the expression for the eigenvectors (see Exercise 2.3.3) p˛ D .1/˛C1 s˛ .b˛ eN 1 C eN 2 /; from (6.177) we obtain the following representation for the Cauchy stress tensor: TD

3 X

˛˛ eN 2˛ C 12 O3 :

(6.182)

˛D1

Here ˛ˇ are components of the Cauchy stress tensor with respect to the Cartesian basis eN i , which in the problem on shear have the forms .p/ 2 2 .p/ .p/ 2 2 s1 b1  2T12 s1 s2 b1 b2 C T22 s 2 b2 ;

11 D T11 .p/

.p/

.p/

22 D T11 s12  2T12 s1 s2 C T22 s22 ; .p/ 2 .p/ .p/ 2 .p/

12 D T11 s1 b1  T12 s1 s2 .b1 C b2 / C T22 s2 b2 ; 33 D T33 :

(6.183)

6.6 Simple Shear

441

Thus, the Cauchy stress tensor (6.182) in the problem on simple shear also proves to be independent of coordinates; therefore, the tensor automatically satisfies the equilibrium equations in a Cartesian basis: r TD

3 X

.@ ˛ˇ =@x ˇ /Ne˛ D 0:

(6.184)

˛;ˇ D1

6.6.3 Boundary Conditions in the Problem on Shear In the problem on simple shear, boundary conditions corresponding to the motion law (6.175) have the form x 2 D h2 W 2

x D0W

u1 D u1e ; u2 D 0; u3 D 0; ˛

u D 0; ˛ D 1; 2; 3;

(6.185a) (6.185b)

x D 0; h W u D 0; eN 1  T  eN 3 D 0; eN 2  T  eN 3 D 0;

(6.185c)

x D 0; h W eN 1  T  eN 3 D 0; u D 0; u D

(6.185d)

3

1

3

1

3

2

2

u1e X 2 = h2 :

Thus, the side x 2 D 0 of the parallelepiped remains fixed, and at the opposite side x 2 D h2 we give the displacement u1e shifting the parallelepiped towards the axis Ox 1 , at the sides x 3 D 0 and x 3 D h3 there is no displacement along the axis Ox 3 and both tangential stresses. At the inclined sides x 1 D 0; h1 we give the zero tangential stress 13 D 0, the zero displacement u D 0 and the longitudinal displacement u1 varying by a linear law along the coordinate X 2 . We can verify that the motion law (6.175) and the stress tensor T (6.182) automatically satisfy the boundary conditions (6.185a)–(6.185c), because Eqs. (6.175) and (6.182) yield ui D x i  X i D aı1i X 2 ; ˛ D 1; 2; 3;

˛3 D eN ˛  T  eN 3  0; ˛ D 1; 2:

(6.186a) (6.186b)

From (6.185d) and (6.186a) we can find values of the function a through u1e : a D u1e = h2 :

(6.187)

According to (6.187), the first boundary condition in (6.185d) is satisfied. The second and third boundary conditions in (6.185d) are satisfied identically due to formulae (6.186) and (6.187). The problem on a simple shear with the boundary conditions (6.185) is realized approximately in the experiment on longitudinal shear of a thin strip of considered material (for example, of rubber), placed between two rigid (for example, steel) sheets displaced one parallel to another (Fig. 6.17).

442

6 Elastic Continua at Large Deformations

Fig. 6.17 The experimental scheme, where the solution of the problem on a simple shear is approximately realized

Fig. 6.18 Diagrams of deforming in shear for linear models An and  D 0.4

At some distance from the ends x 1 D 0; h1 and x 3 D 0; h3 in such a plate the motion law (6.175) holds and there is a uniform stress state.

6.6.4 Comparative Analysis of Different Models An for the Problem on Shear Notice that in the problem on simple shear, besides the shear stresses 12 , all normal stresses 11 , 22 and 33 are nonzero; this phenomenon, which is typical for large deformations, is called the Poynting effect. The dependence 12 .a/ expressed by formulae (6.183) is called the diagram of deforming in simple shear (remind that a D tan ˛ D u1e = h2 is the tangent of shear angle for the parallelepiped). Figures 6.18 and 6.19 exhibit diagrams of deforming in shear for different linear models An and different values of the Poisson ratio (the stress 12 is referred to l2 , and the constant l1 is expressed in terms of  by formula (6.146)). The increase in value of  from 0 to 0.5 leads to displacing the diagram 12 .a/ into the domain of higher values. A peculiarity of the problem on a simple shear is the fact that the diagrams 12 .a/ coincide for the models AI and AV , and also for the models AII and AIV (Fig. 6.18). However, corresponding diagrams of the normal stresses 11 .a/ and

22 .a/ determined by (6.183) are essentially distinct: for the models AI and AII the stresses 11 and 22 are negative, and for the models AIV and AV they are positive (Figs. 6.20 and 6.21).

6.6 Simple Shear

443

Fig. 6.19 Diagrams of deforming in shear for the linear model AI with different values of the Poisson ratio

Fig. 6.20 Dependence of the stress 11 on the shear angle a in the problem on a simple shear for linear models An and  D 0.4

Fig. 6.21 Dependence of the stress 22 on the shear angle a in the problem on a simple shear for linear models An and  D 0.4

6.6.5 Shear of an Incompressible Elastic Continuum Consider a simple shear for linear models Bn of incompressible continua (4.532). Notice that the motion law (6.175) even for compressible continua conserves their volume, because det F D 1 (see Exercise 4.2.14). Therefore, for incompressible continua a simple shear is also described by the law (6.175) and the boundary conditions (6.185a), (6.185b) and (6.185d), and in place of conditions (6.185c) one should consider the case of free ends: x 3 D 0; h3 W

33 D 0; 13 D 0; 23 D 0:

(6.188)

444

6 Elastic Continua at Large Deformations .n/

On substituting the expression for the tensors G in the problem on shear (see Exercise 4.2.14) into (4.532), we find 3 p .n/1 X .n/ 2 .n/ G C T ˛˛ eN ˛ C T 12 O3 ; n  III ˛D1    .n/ .n/ 1 1Cˇ 2 T 11 D .n  III/ ; C .1  ˇ/ c 2B C n  III n  III    .n/ .n/ 1 1Cˇ 2 T 22 D .n  III/ ; C .1  ˇ/ c 0B C n  III n  III   .n/ .n/ .n/ 1Cˇ 2 T 33 D .n  III/ C .1  ˇ/. c 0B C c 2B / ; n  III .n/

T D

.n/

(6.189)

.n/

T 12 D .n  III/2 c 1B ;

.n/

.n/

.n/

where c 0B , c 1B and c 2B are determined by Table 4.6 from Exercise 4.2.14. .n/

With the help of the tensors of energetic equivalence 4 E by analogy with (6.177) we obtain the expression for T: T D pE C

3 X ˛;ˇ D1

.p/ T˛ˇ p˛

˝ pˇ D

3 X

˛˛ eN 2˛ C 12 O3 ;

(6.190)

˛D1

.p/ where T˛ˇ are determined by the same formulae (6.180). Here we have used that .n/

.n/

E   G 1 D .n  III/E. Cartesian components of the Cauchy stress tensor ˛ˇ can be determined with the help of expressions for the eigenvectors p˛ in the Cartesian basis eN i ; as a result, we obtain relations similar to (6.183): 4

.p/ 2 2 .p/ .p/ 2 2

11 D p C T11 s1 b1  2T12 s1 s2 b1 b2 C T22 s2 b2 ; .p/

.p/

.p/

22 D p C T11 s12  2T12 s1 s2 C T22 s22 ; .p/ .p/ .p/ .p/

12 D T11 s12 b1  T12 s1 s2 .b1 C b2 / C T22 s22 b2 ; 33 D p C T33 :

(6.191) Just as for a compressible material, in the problem on a simple shear 13 D 23 D 0. Then, substituting the formula for 33 into the boundary condition (6.188), we find that   .n/ .n/ .n/ 1Cˇ .p/ C .1  ˇ/. c 0B C c 2B / : (6.192) p D T33 D T 33 D .n  III/2 n  III

6.6 Simple Shear

445

Here we have used formulae (6.189). Substitution of (6.192) into (6.191) yields a final expression for the normal stresses 11 and 22 , and the stress 33 in this case proves to be zero in the whole body: 33 D 0. Figures 6.22–6.25 exhibit diagrams of deforming 12 .a/ in shear, and also the functions 11 .a/ and 22 .a/ for linear models Bn of incompressible materials. For incompressible continua, not only the diagrams of deforming 12 .a/ for the models BI and BV , and also for the models BII and BIV are coincident, but also the normal stresses 11 .a/ and 22 .a/ (Figs. 6.22–6.24); here for all the models Bn 11 > 0, while 22 < 0. With growing the parameter ˇ from 1 to 1, all the diagrams j 11 .a/j and j 22 .a/j are displaced into the domain of smaller values (Fig. 6.25). The diagrams of deforming 12 .a/ are independent of the parameter ˇ.

Fig. 6.22 Diagrams of deforming in shear for models Bn of incompressible continua

Fig. 6.23 Dependence of the stress 11 on the shear angle a in the problem on a simple shear for models Bn of incompressible continua

Fig. 6.24 Dependence of the stress 22 on the shear angle a in the problem on a simple shear for models Bn of incompressible continua

446

6 Elastic Continua at Large Deformations

Fig. 6.25 Dependence of the stress 11 on the shear angle a in the problem on a simple shear for the model BI of incompressible continua with different values of the coefficient ˇ

Exercises for 6.6 6.6.1. Consider the problem on shear for the models AI and AV with replacing the general equations (4.37) by the explicit relations between the Cauchy stress tensor and energetic stress tensors: I

V

T D F1T  T  F1 D F  T  FT : Show that in this case formulae (6.183) for stresses ˛ˇ take the form I

I

I

I

I

I

˛˛ D T ˛˛ ; ˛ D 1; 3; 22 D T 22  2aT 12 C a2 T 11 ; 12 D T 12  aT 11 for the model AI , and V

V

V

V

11 D T 11 C 2aT 12 C a2 T 22 ; 22 D T 22 ;

V

V

V

33 D T 33 ; 12 D T 12 C aT 22

for the model AV .

6.7 The Lam´e Problem 6.7.1 The Motion Law for a Pipe in the Lam´e Problem In the problems on tension and simple shear considered above, a stress tensor field T.X i / was uniform, that ensured the equilibrium equations to be satisfied. With the help of the semi-inverse method we can find an analytical solution also for the problem on a pipe under external and internal pressures (the Lam´e problem), where the field T.X i / is not uniform.

6.7 The Lam´e Problem

447

Fig. 6.26 For the Lam´e problem

ı

In this problem, a body B in K and K is a pipe (a thick-walled cylinder) of finite length h3 (Fig. 6.26), which under the action of pressures pe2 and pe1 on the outer r D r2 and inner r D r1 surfaces, respectively, has the same symmetry axis Ox 3 D Oz, i.e. remains self-similar. At the end z D h3 of the cylinder the displacement ue3 is given, and at the second end z D 0 the symmetry condition is assumed to be satisfied. Let us introduce cylindrical coordinates r; '; z (Fig. 6.26); and as Lagrangian coı ı ı ordinates X i we choose X 1 D r, X 2 D ' and X 3 D z, which are the values of ı

cylindrical coordinates of material points at the initial time t D 0 in K. Then the ı

ı

radius-vector x of a material point M in K can be resolved for basis vectors of the cylindrical coordinate system er ; e' ; ez (i.e. for the physical basis) as follows: ı

ı

ı

x D rer C zez :

(6.193)

Since the pipe retains its axial symmetry in K under the action of pressures and axial displacement, a law of the motion is sought in the form x D rer C zez :

(6.194)

Here r and z are functions in the forms ı

r D f .r ; t/;

ı

z D k.t/z;

(6.195)

k.0/ D 1:

(6.196)

where ı

ı

f .r ; 0/ D r;

448

6 Elastic Continua at Large Deformations

6.7.2 The Deformation Gradient and Deformation Tensors in the Lam´e Problem ı

Determine the local basis vectors ri and ri by using formulae (2.10), (6.194), and (6.195): @x @ @f ı ı ı D ı .f .r; t/er .'/ C zkez / D f 0 er ; f 0  ı ; 1 @X @r @r @x @ @er ı ı D ı .f er .'/ C zkez / D f ı D f e' ; r2 D @X 2 @' @'

r1 D

(6.197)

@x @ ı D ı .f er C zkez / D kez ; r3 D 3 @X @z ı

ı

r1 D er ;

ı

ı

r2 D re' ;

r3 D ez :

Here we have taken into account the formulae of differentiation of physical basis vectors of a cylindrical coordinate system (see [12]). ı ı Let us form the metric matrices gij , g ij and g ij , g ij : 0

0 02 1 1 0 0 f 02 0 0 f gij D ri  rj D @ 0 f 2 0 A ; gij D @ 0 f 2 0 A ; 0 0 k2 0 0 k 2 1 1 0 0 1 0 0 1 0 0 ı ı ı ı C B ı C B ı g ij D ri  rj D @0 r 2 0A ; g ij D @0 r 2 0A ; 0 0 1 0 0 1 ı

g D .f 0 f k/2 ;

(6.198)

ı

g D r2

and find local vectors of the reciprocal bases by formulae (2.13): r1 D

er ; f0

e' 1 ; r3 D ez ; f k e' ı2 ı3 r D ı ; r D ez : r

r2 D

ı

r 1 D er ;

(6.199)

According to (6.198) and (6.199), we calculate the deformation gradient ı

F D r i ˝ r i D f 0 e r ˝ er C ı

f ı

r

e' ˝ e' C kez ˝ ez D FT ; ı

F1 D ri ˝ ri D

r 1 1 er ˝ er C e' ˝ e' C ez ˝ ez ; f0 f k ı

J D = D det F1 D

ı

r ; f 0f k

(6.200)

6.7 The Lam´e Problem

449

which, just as for the problem on a beam in tension, is a diagonal tensor but its components depend on Lagrangian coordinates. Since f D r is the radius and k is the elongation ratio of a cylinder along its axis, values of f and k are always positive; and the last relation of (6.200) yields that f 0 is positive too. Thus, there are constraints on signs of f 0 , f and k: f 0 > 0; f > 0; k > 0:

(6.201)

Since F is diagonal, one can readily find the stretch tensors U and V and the rotation tensor O: U D V D F; O D E: (6.202) With the help of formulae (4.25) and (4.42) we find the energetic deformation ten.n/

.n/

sors C and deformation measures G: .n/

CD

1  ..f 0 /nIII  1/er ˝ er n  III

 ı C..f =r/nIII  1/e' ˝ e' C .k nIII  1/ez ˝ ez ; (6.203)

.n/

GD

1  0nIII ı f er ˝ er C .f =r/nIII e' ˝ e' C k nIII ez ˝ ez : n  III

6.7.3 Stresses in the Lam´e Problem for Models An Assume that constitutive equations of the cylinder correspond to the linear model An (6.135) of an isotropic elastic continuum. Then substituting the expression (6.203) into (6.135), we find the energetic stress tensors .n/

.n/

.n/

.n/

T D T r er ˝ er C T ' e' ˝ e' C T z ez ˝ ez ; ı

.n/

r Tr D .l1 I1 C 2l2 .f 0nIII  1//; .n  III/f 0 f k ı

.n/

T' D .n/

r ı .l1 I1 C 2l2 ..f =r/nIII  1//; .n  III/f 0 f k ı

TzD

r .l1 I1 C 2l2 .k nIII  1//; .n  III/f 0 f k ı

I1 D .f 0 /nIII C .f =r/nIII C k nIII  3:

(6.204)

450

6 Elastic Continua at Large Deformations .n/

Since the tensors F and T are diagonal, from Eqs. (6.137), (6.200), and (6.204) we obtain the expression for the Cauchy stress tensor .n/

T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez D FnIII  T ;

(6.205)

ı

r D

' D

r .f 0 /nIII1 .l1 I1 C 2l2 .f 0nIII  1//; .n  III/f k

1 ı ı .f =r/nIII1 .l1 I1 C 2l2 ..f =r/nIII  1//; .n  III/f 0 k ı

z D

rk nIII1 .l1 I1 C 2l2 .k nIII  1//: .n  III/f 0 f

Components of the stress tensor in this problem prove to depend on the coordinate ı r and time t, therefore, the equilibrium equations are not automatically satisfied. In this case, as a rule, it is convenient to use the equilibrium equations (6.100) in the material description. To do this, we should calculate the Piola–Kirchhoff stress tensor by formulae (3.56): PD

.n/ 1 1 1 ı ı ı F  T D FnIII1  T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez ; (6.206) J0 J

where ı

r D

.f 0 /nIII1 ı ..l1 C 2l2 /f 0nIII C l1 .f =r/nIII  .3l1 C 2l2 / C l1 k nIII /; n  III ı

' D

ı

z D

1 ı ı .f =r/nIII1 ..l1 C 2l2 /.f =r/nIII n  III Cl1 f 0nIII  .3l1 C 2l2 / C l1 k nIII /;

(6.207)

k nIII1 ı ..l1 C 2l2 /k nIII C l1 .f 0nIII C .1=r/nIII /  .3l1 C 2l2 //: n  III

6.7.4 Equation for the Function f ı

Writing components of the divergence r  P with respect to the physical basis er ; e' ; ez (see [12]), we can represent the equilibrium equation projected onto the axis Oer as follows: ı

@ r ı

@r

ı

C

ı

r  ' ı

r

D 0:

(6.208)

6.7 The Lam´e Problem

451 ı

ı

ı

ı

ı

Here we have taken into account that r , ' and z are independent of ' and z. The other two projections of the equilibrium equation onto the axes Oe' and Oez are satisfied identically. On substituting the expressions (6.207) into (6.208), we obtain an ordinary difı ı ferential equation of the second order for the function f .r ; t/. The function f .r; t/ is determined up to the two constants of integration C1 .t/ and C2 .t/ being functions only of time; to evaluate them one should use boundary conditions.

6.7.5 Boundary Conditions of the Weak Type Boundary conditions at inner and outer surfaces of the cylinder have the form (6.80) (gas or fluid pressures pe1 and pe2 are given). According to formula (6.206) and ı ı the fact that at the surfaces r D r ˛ (˛ D 1; 2) the normal vectors have the form ı n D er , the boundary condition (6.80) for this problem becomes ı

ı

r D r˛ W

ı

ı

r D pe˛

f .r ˛ ; t/k ı

;

˛ D 1; 2:

(6.209)

r˛ ı

Boundary conditions at the end surfaces z D 0; h3 serve for determining the function k. For example, if at the surface z D 0 the symmetry conditions are given, ı and at z D h03 – the pressure pe3 , then we have ı ˇ uz D .z  z/ˇızD0 D 0;

ı

zD0W ı

ı

z D h3 W

z D pe3 :

(6.210) (6.211)

On substituting the expression (6.194) and (6.205) into (6.210) and (6.211), we verify that the first boundary condition is satisfied identically, and the second one is an additional differential equation for the function f and the constant k. The solution found above cannot satisfy this boundary condition completely, therefore we should consider other variants of boundary conditions. One of such variants weakens the boundary condition (6.211) by replacing it with the integral one ı

ı

z D h3 W

Z

r2 r1

1

z rdr D  pe3 .r22  r12 /: 2

(6.212)

One can say that such a condition is of the weak type. The method of replacing exact boundary conditions by the integral ones is called the Saint-Venant method.

452

6 Elastic Continua at Large Deformations ı

According to (6.195), we have rdr D f df D ff 0 d r. Then, substituting the expression (6.205) for z into (6.212) and passing to the reference configuration, we rewrite the condition (6.212) as follows: k nIII1 2l1 ı n  III r 2  rı 2 2 1

Z



ı

r2

0 nIII

.f /

ı

C

nIII ! ı ı r dr

f ı

r

r1

!

C.l1 C 2l2 /k

nIII

ı

 .3l1 C 2l2 / D pe3

ı

f 2 .r 2 /  f 2 .r 1 / ı

ı

r 22  r 21

! :

(6.213) Expressing the function k by Eq. (6.213) and substituting the result into (6.207) and then formulae (6.207) into (6.208) and (6.209), we obtain an integro-differential ı equation for determining the function f .r; t/. For the case when n D IV (John’s model AIV ), Eqs. (6.208) and (6.209) have a simple analytical solution: ı

ı

f .r; t/ D C1 .t/r C

C2 .t/ ı

:

(6.214)

r

Indeed, in this case Eqs. (6.207) become ı

r D 2.l1 C l2 /C1  2l2 ı

' D 2.l1 C l2 /C1 C 2l2

C2 ı

r2 C2 ı

r2

C l1 k  .3l1 C 2l2 /; (6.215) C l1 k  .3l1 C 2l2 /;

and relations (6.205) take the forms ı

r D

r2 ı

l1 .2C1 C k/ C 2l2 C1 

r2

C2

.C1 r 2 C C2 /k ı

' D

l1 .2C1 C k/ C 2l2 C1 C

ı

.C1 r 2  C2 /k

ı

r2

C2 ı

r2 !

!

!

 .3l1 C 2l2 / ; !

 .3l1 C 2l2 / ; (6.216)

ı4

z D

r ı

.C12 r 4  C22 /

.l1 .2C1 C k/ C 2l2 k  .3l1 C 2l2 // :

Substituting the expressions (6.215) into (6.208), one can readily verify that the equilibrium equation is identically satisfied.

6.7 The Lam´e Problem

453

Substitution of (6.215) into the boundary conditions (6.209) and (6.211) yields 2.l1 C l2 /C1 

2l2

2.l1 C l2 /C1 

2l2

ı

r 21 ı

r 22

C2 C l1 k  .3l1 C 2l2 / D pe1 C1 C

C2

C2 C l1 k  .3l1 C 2l2 / D pe2 C1 C

C2

! ;

ı

r 21 ı

r 22

!

(6.217) :

After substitution of (6.214), the boundary condition (6.213) for the model AIV takes the form ! C22 2 2C1 l1 C .l1 C 2l2 /k  .3l1 C 2l2 / D pe3 C1  ı ı : (6.218) r 21 r 22 Solving the system of three algebraic equations (6.217), (6.218), we find C1 , C2 , and k.

6.7.6 Boundary Conditions of the Rigid Type In place of (6.212) we can consider another boundary condition, namely the condition of the rigid type when displacements uez along the axis Oz are given: ı

ı

z D h3 W

ı ˇ uz D .z  z/ˇı

ı

zDh3

D uez :

(6.219)

Then we obtain the following simple expression for k: ı

k D 1 C .uez =h3 /;

(6.220)

which is similar to the corresponding expression of (6.142) in the problem on a beam in tension. For the model AIV , this relation takes the place of the condition (6.218) and the system (6.217), (6.218) for C1 , C2 , and k becomes linear; its solution has the form C1 D C2 ;

ı2

C2 D r 1

1  2.k  1/ ; .1 C .1  2/e p1 /  .1  2/.1  e p1 /

e p1  e p 2 ˇ02  1 C ˇ02

D ; e p2  e p1

ı

ı

ˇ0 D r 1 = r 2 :

Here e p ˛ D pe˛ =2l2, and the Poisson ratio (6.146) has been introduced.

(6.221)

454

6 Elastic Continua at Large Deformations

Expressions for outer r2 and inner r1 radii of the cylinder in K can be found with the help of formulae (6.194) and (6.214): ı

ı

ı

ı

r˛ =r ˛ D f .r ˛ ; t/=r ˛ D C1 C .C2 =r 2˛ /;

˛ D 1; 2:

(6.222)

For other models An , we can also consider the boundary condition (6.219), which yields the expression (6.220) for k. Then, substituting the expressions (6.207) into (6.208) and (6.209), we obtain one nonlinear differential second-order equation for the function f with two boundary conditions.

6.8 The Lam´e Problem for an Incompressible Continuum 6.8.1 Equation for the Function f Consider the Lam´e problem on a cylindrical pipe under internal and external pressures (see Sect. 6.7), but let the pipe be made of isotropic incompressible material described by linear models Bn (see (4.532) or (6.160)). In this case the motion law for the pipe is also sought by the semi-inverse method in the form (6.194) and (6.195); therefore, all the relations (6.197)–(6.203) hold. From the condition of incompressibility of a material considered det F D 1 and from (6.201) it follows that ı the function f .r; t/ must satisfy the equation ı

f 0 f k D r: ı

(6.223) ı

Rewriting this equation in the form f df D k1 r d r ; one can easily find its solution (choosing a positive root) ı r2 f2 D C C; (6.224) k where C is the constant of integration.

6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum On substituting Eq. (6.223) into (6.203), we find 1 0  nIII ı !nIII 1 @ r f GD er ˝ er C ı e' ˝ e' C k nIII ez ˝ ez A : n  III fk r

.n/

(6.225)

6.8 The Lam´e Problem for an Incompressible Continuum

455

Having substituted this expression int the constitutive equation (6.160), we obtain .n/

that the energetic stress tensors T have the diagonal form (6.204) in the case of an incompressible continuum too, but their components are different from the ones for compressible materials: 

.n/

T r D p

fk ı

T ' D p

 C .n  III/ 1 C ˇ C .1  ˇ/

r f

T z D pk

nIII

! Ck

nIII

;

r !nIII

 C .n  III/ 1 C ˇ C .1  ˇ/ 

.n/

f ı

r ı

.n/

nIII

nIII

C .n  III/ 1 C ˇ C .1  ˇ/

ı

r fk

ı

r fk

nIII

nIII

 C

! C k nIII f

;

nIII ! ;

ı

r

n D I; II; IV; V: (6.226) For the Cauchy stress tensor, relations (6.205) hold as well, and their components are written as follows:

r ; ' D p C e

' ; z D p C e

z;

r D p C e

(6.227)

where ı

r e

r D .n  III/ fk  e

' D .n  III/

f

!nIII

 1 C ˇ C .1  ˇ/

f

nIII

ı

! Ck

nIII

;

r 

nIII

1 C ˇ C .1  ˇ/

ı

r

 e

z D .n  III/k nIII 1 C ˇ C .1  ˇ/

ı

r fk

ı

r fk

nIII

nIII

 C

! C k nIII f ı

;

nIII ! :

r

(6.228) With the help of the relation P D F tensor: ı

r D r

fk ı

r

;

1

T, we find components of the Piola–Kirchhoff ı

' D

ı

r

' ; f

ı

z D

z : k

(6.229)

456

6 Elastic Continua at Large Deformations

6.8.3 Equation for Hydrostatic Pressure p Substitution of the Piola–Kirchhoff tensor components (6.229) into the equilibrium equation (6.208) yields an ordinary differential equation for unknown function ı p.r; t/ being the hydrostatic pressure in an incompressible continuum: 

pf k

0

ı

C

ı

p.f 2 k  r 2 /

D

ı

r 2f

r

ı

e h

@ r e h D ı .f ke

r/  e

': f @r

; ı r

(6.230)

Rearranging the left-hand side of Eq. (6.230) with the help of (6.223) and (6.224), we reduce the equation to the form dp f k ı

dr

ı

r

D

e h ı

:

(6.231)

r

Integration of this equation yields

Z ı h ı 1 r e p D p0 C (6.232) d r: ı k r1 f Here p0 is the constant of integration, which together with C can be found from the boundary conditions (6.209) after substitution of expressions (6.229) and (6.232) into them: 1 p0 C k

Z

ı

r2 ı

r1

e h ı ı ı

r .r 2 /; p0 D pe1 C e

r .r 1 /: d r D pe2 C e f

(6.233)

For the constant C , from (6.233) we obtain the nonlinear algebraic equation: 1 F .C /  e

r .r 2 /  e

r .r 1 /  k ı

ı

Z

ı

r2 ı

r1

ı e h dr  p D 0; p D pe1  pe2 ; (6.234) f

into which the expressions (6.224), (6.228), and (6.230) for e

r , f and e h should be substituted.

6.8.4 Analysis of the Problem Solution Figure 6.27 shows the graph of the function F .C /. In the general case, for all n and ˇ the equation F .C / D 0 may have two roots C1 and C2 , among which one should choose the least root C1 , because just the one satisfies the normalization condition F .C1 / D 0 at k D 1 and p D 0 when there is no loading and deformations of the cylinder. For this case

6.8 The Lam´e Problem for an Incompressible Continuum

C D 0;

457 ı

f D r;

' D e

z  0 D .n  III/.3  ˇ/ D const; e

r D e p D p0 D const; p0 D pe1 C 0 ; ı

ı

(6.235)

ı

r D ' D z D p0 C 0 D pe1 D const: The stresses r , ' and z in the cylinder are zero when pe1 D pe2 D 0; and they are equal to each other but are nonzero when pe1 D pe2 ¤ 0. When k D 1 and p > 0 (excess internal pressure), the least root C1 of the function F .C / is positive; and when p < 0 (excess external pressure) the root C1 is negative (Fig. 6.27). With growing the value k > 1 (longitudinal extension), the root C1 is displaced into the domain of negative values (there occurs transverse compression of the cylinder); and with decreasing the value k < 1 (longitudinal compression), on the contrary, the root C1 is displaced into the domain of positive values (there occurs lateral dilatation of the cylinder). For the function F .C /, a peculiarity of interest is the existence of the ultimate value p : while p > p , there are no roots of the function F .C / (see Fig. 6.27). This means that at such values of p the nonlinear Lam´e problem has no solution (unlike the Lam´e problem in linear elasticity theory, which has a solution at all values of p). If we consider the process of monotone increasing the pressure difference p from 0 up to p , then the cylinder radius also monotonically grows; and when p > p a solution does not exist. This effect is called the loss of stability of a material in tension. ı ı Figure 6.28 shows the dependence of the dilatation coefficient yR D .r1  r 1 /=r 1 of the cylinder upon the dimensionless pressure difference p= (when p > 0)

Fig. 6.27 The function F .C /

Fig. 6.28 Dependence of the coefficient of relative dilatation of a thin-walled cylinder on internal excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

458

6 Elastic Continua at Large Deformations ı ı

ı

ı

ı

for the cylinder with a very thin wall: h=r 1 D 0.01, where h D r 2  r 1 is the initial thickness of the wall. The function p.yR / is nonlinear and exists only if ı h.r/ on p 6 p  . A fact of interest is a very weak dependence of the function e the coefficient ˇ when k D 1; due to this, the functions F .C / and p.yR / are also practically independent of the values of ˇ within the interval Œ1; 1 . Moreover, values of the function p.yR / at n D I and V, and also at n D II and IV are practically not distinguishable in pairs. Therefore, when k D 1 there are only two essentially distinct functions p.yR / at n D I and II (Fig. 6.28). Their corresponding ultimate ı ı

values p  = are 0.04 and 0.0072 while h=r 1 D 0.01. Ultimate magnitudes of the  dilatation coefficient yR are 218 and 41%. If we consider the cylinder with a thicker ı

 prove to be smaller. wall h, then the ultimate values yR In the case of compression when p < 0, the root C1 is negative, and from Eq. (6.224) it follows that there exists a limiting value ı

C1 D r 21 =k;

(6.236)

such that the root C1 cannot take on values smaller than C1 . Hence there also exists a limiting negative value of the pressure difference p such that when p < p  there is no solution of the Lam´e problem. Graphs of the function p.yR / in compression are shown in Fig. 6.29. Just as in tension, for the case when k D 1, the functions p.yR / at different values of the parameter ˇ are practically coincident, and they are distinct only for the models n D I; V and n D II; IV. ı ı

In the case when h=r 1 D 0.01, we found the limiting value p  = D 3.2 for n D I; V, and for n D II; IV: p  = D 0.33. It should be noted that for real thin-walled structures at essentially smaller values of pressure in compression such that jpj=  jp  j=, there occurs a loss of stability of the structure itself; thus, the values p  = are usually not realized. Figures 6.30 and 6.31 exhibit distributions of the stresses r = and ' = versus the cylinder thickness at different values of the parameter ˇ and different n when ı ı pe2 D 0. The radial stress r .Nr /, where rN D r=r 1 , depends weakly on ˇ and

Fig. 6.29 Dependence of the coefficient of relative compression of a thin-walled cylinder on external excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

6.8 The Lam´e Problem for an Incompressible Continuum

a

459

b

Fig. 6.30 Distribution of the radial stresses versus the thickness of a thick-walled cylinder for the model BI at different values of ˇ (a) and for different models of incompressible material .ˇ D 1/ (b)

Fig. 6.31 Distribution of the tangential stresses versus the thickness of a thick-walled cylinder for different models of incompressible material and different values of the parameter ˇ

n; it monotonically decreases from the value pe1 to 0. The tangential stress ' .r/ depends considerably more on ˇ and n especially for thin-walled cylinders. So when r2 =r1 D 2, for the model BI and ˇ D 1 the stress ' is positive everywhere, and it reaches its minimum value at the interior point r=r1  1.2 of the cylinder. When ˇ 6 0.6, the stress ' on the inner surface of the cylinder becomes negative, and when ˇ D 1, the stress ' is negative in the whole cylinder. For the models BII , BIV and BV , the stress ' is always positive. For the model BII , when ˇ D 1,

460

6 Elastic Continua at Large Deformations

Fig. 6.32 Distributions of radial and tangential stresses versus the thickness of a thin-walled cylinder for different models of incompressible material .ˇ D 1/

the stress ' reaches its maximum on the outer surface of the cylinder; and for the models BIV and BV , at all ˇ the function ' .Nr / is always monotonically decreasing and reaches its maximum on the outer surface. For thin-walled cylinders, the stress ' is positive and practically constant versus the cylinder thickness; it is almost independent of ˇ and n, and its value is close to the value determined by the theory of thin linear-elastic shells at small deformations: ı

ı

'  p r 1 =h (Fig. 6.32).

Chapter 7

Continua of the Differential Type

7.1 Models An and Bn of Continua of the Differential Type 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type Let us consider now nonideal continua. Practically all real bodies are nonideal media, and they can be considered as ideal ones in a certain approximation. According to the general theory of constitutive equations stated in Sect. 4.4.2, a continuum is nonideal if its operator constitutive equations (4.156) include the dissipation function w being nonzero. Models of nonideal materials, which are widely used in practice, are models of continua of the differential type. Definition 7.1. A continuum is called a continuum of the differential type, if corresponding operator relations (4.156) are usual functions of active variables R.t/ and P their derivatives R.t/, i.e.   P ƒ.t/ D f R.t/; R.t/ :

(7.1)

Functions (7.1) are assumed to be continuously differentiable. One can say that this is the model An of a continuum of the differential type, if some model An of a continuum has been chosen and its corresponding operator constitutive equations (4.156) are simply functions of the arguments indicated above and their rates. In particular, the Helmholtz free energy has the form .t/ D

.n/

.n/

. C .t/; C  .t/; .t//:

(7.2)

(In continuum mechanics, the set of arguments of the function (7.2) does not involve the derivative P .)

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 7, c Springer Science+Business Media B.V. 2011 

461

462

7 Continua of the Differential Type

The total derivative of

with respect to time has the form .n/

.n/

d2 C @ d d dC @ @  C  D C : 2 .n/ .n/ dt dt @ dt dt  @C @C .n/

(7.3)

.n/

The partial derivatives @ =@ C and @ =@ C  are symmetric second-order tensors. The remaining constitutive equations (4.158), connecting the active variables to the reactive ones, for the model An of a continuum of the differential type have the forms .n/ .n/ .n/

.n/

T D T . C; C  ; /;

.n/ .n/

 D . C ; C  ; /;

.n/ .n/

w D w . C ; C  ; /:

(7.4)

By analogy with the tensor function connecting the energetic tensors of stresses and deformations, introduce the two tensor functions .n/

.n/

.n/ .n/

T e . C ; / D T . C; 0; /;

(7.5)

and .n/

.n/

.n/

T v D T  T e;

(7.6)

called the function of equilibrium stresses and the function of viscous stresses, respectively. Substituting the expressions (7.3) and (7.6) into PTI (4.121) and collecting like terms, we get 0

1

  .n/ .n/ .n/ .n/ Te C @ @ 1 B@  C  d C .w  T v   C  / dt D 0: @ .n/  A   d C C .n/   d C C  @  @C @C (7.7) .n/

.n/

.n/

Since the differentials d C , d C  , d and dt are mutually independent, the identity (7.7) is equivalent to the equation system 8 .n/ .n/ .n/ ˆ ˆ ˆ T e D .@ =@ C/  F . C ; /; ˆ ˆ ˆ ˆ .n/ ˆ < @ =@ C  D 0; ˆ ˆ ˆ  D @ =@; ˆ ˆ ˆ ˆ .n/ ˆ :  .n/ w D T v   C :

.7:8a/ .7:8b/ .7:8c/ .7:8d/

7.1 Models An and Bn of Continua of the Differential Type

463 .n/

Just the fact that terms within the first parentheses of (7.7) are independent of C  .n/

.n/

ensures that the differentials d C and d C  be mutually independent. In turn, the .n/

.n/

presence of T e instead of T within the first parentheses of (7.7) and the validity of .n/

Eq. (7.8b) ensure that terms within these parentheses be independent of C  . Thus, continua of the differential type have the following properties: (1) They are dissipative (i.e. nonideal), because for them the dissipation function w is not identically zero. .n/

.n/

(2) The tensor function of equilibrium stresses T e (but not T ) is quasipotential (see Sect. 4.5.2). (3) The quasipotential

.n/

depends only on C and : D

.n/

. C ; /;

(7.9)

.n/ .n/

.n/

the remaining functions , T v , w depend on C , C  and . According to the property (2), for models of the differential type it is not sufficient to specify only one function (7.9); in addition we need the viscous stresses function (7.6) to be given: .n/ .n/

.n/

T v D F v . C ; C  ; /:

(7.10)

According to formulae (7.4)–(7.6), this function depends also on the deformation .n/

tensor rates C  . Relationships (7.8)–(7.10) are constitutive equations for models An of continua of the differential type.

7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type The tensor function of viscous stresses (7.10) is not quite arbitrary: it satisfies the conditions (7.5) and (7.6), i.e. .n/

F v . C ; 0; / D 0:

(7.11)

In addition, the function must satisfy the Onsager principle (see Sect. 4.12.1, Axiom 16). This principle is applied to continua of the differential type as follows.

464

7 Continua of the Differential Type

First, we should form the scalar function (4.728) being the specific internal entropy production: q  D w 

.n/ .n/ .n/ q q  r  D F v . C ; C  ; /   C    r  > 0:  

(7.12)

Here we have used expression (7.8d) for the dissipation function w of continua of the differential type. Then we represent the expression (7.12) in the form (4.729) with thermodynamic forces .n/

X2 D C 

X1 D r ;

(7.13)

(where X1 is a vector, and X2 is a second-order tensor) and thermodynamic fluxes .n/ .n/

.n/

1 Q1 D  q; 

Q2 D T v D F v . C ; C  ; /:

(7.14)

According to the Onsager principle, thermodynamic fluxes Qˇ must be linear (tensor-linear) functions of Xˇ in the form (4.730), i.e. 8 ˆ <.1=/q D L

11  r  .n/ .n/ 

.n/

C L12 C  ;

.n/ .n/ ˆ : T D F . C ; C ; / D L r  C L   C  : v v 12 22

.7:15/ .7:16/

Here L11 is a second-order tensor, L12 is a third-order tensor, and L22 is a fourthorder tensor. Notice that since the viscous stresses function F v must depend only .n/

.n/

on C  and C and may not depend upon the temperature gradient r  (because the opposite conflicts with the principle of equipresence), so in (7.16) we have L12  0; and hence we obtain the relationships q D   r ; .n/ .n/

  L11 ; .n/

F v . C ; C  ; / D 4 Lv   C  ;

4

Lv  L22 :

(7.17) (7.18)

Equation (7.17) is the known Fourier law, which exactly coincides with the Fourier law (4.734) for ideal continua; and Eq. (7.18) is the quasilinear Stokes law for viscous stresses. The fourth-order tensor 4 Lv is called the viscosity tensor. .n/

Since q is independent of C  and F v is independent of r , the Planck inequality (7.12) is equivalent to the two inequalities q  r  D r    r  > 0; .n/

.n/

.n/

w D F v   C  D C    4 Lv   C  > 0;

(7.19) (7.20)

7.1 Models An and Bn of Continua of the Differential Type

465

which are the Fourier inequality and the dissipation inequality for continua of the differential type, respectively. From these inequalities it follows that the tensors  and 4 Lv are nonnegative-definite. In addition, from (7.20) it follows that the dissipation function w for continua of the differential type is a quadratic scalar function .n/

of C  , and the viscosity tensor 4 Lv has the following symmetry in components: 4

b Lijkl D b Lijlk ;

Lv D b Lijklb cj ˝b ck ˝b cl ; ci ˝b

b Lijkl D b Lj i kl ;

b Lijkl D b Lklij ;

(7.21)

whereb ci is an orthonormal basis. The total number of components of the tensor 4 Lv is 81, and the number of independent components, according to Eqs. (7.21), does not exceed 21 (see [12]). Notice that the viscosity tensor 4 Lv and its components, in general, depend on .n/

.n/

the tensor C and also on C  (the Onsager principle (7.18) does not prohibit this dependence, it only requires that F v be a quasilinear tensor function (see Sect. 4.8.7 and [12])). .n/

.n/

If the tensor 4 Lv is independent of C  and depends only on C , then the function of viscous stresses F v is pseudopotential, i.e. satisfies the relation .n/

.n/ .n/

T v D F v . C; C  ; / D

.n/ 1 .@w =@ C  /: 2

(7.22)

Such function F v is also called potential with respect to the second tensor argument. Equation (7.22) can readily be verified by differentiating the scalar function w .n/

(7.20) with respect to the argument C  .

7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type Notice that the principle of material symmetry (Axiom 14), and also Definitions 4.5 and 4.6 of fluids and solids, respectively, have been formulated for arbitrary operator constitutive equations (4.156); i.e. they can be applied not only to ideal materials but also to continua of the differential type and other types of nonideal materials. According to this principle, models An of continua of the differential type with bs constitutive equations (7.8)–(7.10) in some undistorted reference configuration K ı

ı

(as this configuration we will choose K) have a symmetry group G s such that ı

ı



for each transformation tensor H 2 G s (H W K ! K/ the constitutive equations 

(7.8)–(7.10) in the reference configuration K have the form

466

7 Continua of the Differential Type

8 .n/ .n/ .n/ ˆ   ˆ T D F . C ; /  .@ =@ C  /; ˆ e ˆ ˆ ˆ ˆ ˆ .n/ ˆ ˆ ˆ D . C  ; /; ˆ ˆ <  D @ =@; ˆ ˆ ˆ ˆ .n/ .n/ ˆ ˆ    ˆ / D T   C  ; .w ˆ v ˆ ˆ ˆ ˆ .n/ .n/ ˆ.n/ : T v D F v . C  ; C  ; /:

.7:23a/ .7:23b/ .7:23c/ .7:23d/ .7:23e/

Let us consider first the models An of solids of the differential type, whose symı

metry group G s is a subgroup of the full orthogonal group I . For these materials, the further construction is the same as in Sect. 4.8 for ideal solids: since the tensors ı

.n/

.n/

.n/

.n/

T and C are H -indifferent relative to the group G s , tensors T e (7.5), T v (7.6) .n/

and C  are also H -indifferent, according to Theorem 4.21. Then Eqs. (7.23) are equivalent to the relations 



.n/



.n/



QT  F . C; /  Q D F .QT  C  Q; /; 

.n/

. C ; / D 



.n/

.QT  C  Q; /;



.n/ .n/

(7.24a)



.n/





(7.24b)

.n/



QT  F v . C ; C  ; /  Q D F v .QT  C  Q; QT  C   Q; /; 

(7.24c)

ı

for any Q 2 G s . Since .n/

.n/



.n/





.n/



.n/

.n/

.w / D T v   C  D QT  T v  Q   QT  C   Q D T v   C  D w ;

(7.25)

Eq. (7.23d) is always satisfied. Theorem 7.1. For models An of solids of the differential type, the principle of material symmetry (7.24) holds if and only if the two conditions (7.24b) and (7.24c) for the function and the viscous stresses function F v are satisfied. H The conditions (7.24b) and (7.24c) are necessary, because if the principle of material symmetry holds, then all conditions (7.24) are satisfied. Prove that the conditions (7.24b) and (7.24c) are sufficient. If the condition (7.24b) is satisfied, then (7.24a) follows from (7.24b); a proof of this fact is similar to the proof of formula (4.280). N

7.1 Models An and Bn of Continua of the Differential Type

467

7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases In comparison with ideal continua, for solids of the differential type not only the condition (7.24b), being the condition of indifference of the scalar function ı

.n/

. C ; / relative to some group G s (see Definition 4.9), must be satisfied but also .n/ .n/

the condition (7.24c) for the tensor function F v . C ; C  ; / of two tensor arguments. .n/

For the scalar function . C ; /, the same representations as were derived in Sect. 4.8 remain valid; in particular, these are representations (4.304) as functions ı

.n/

of invariants I.s/ . C / relative to group G s : D

.n/

.n/

.I1.s/ . C/; : : : ; Ir.s/ . C /; /;

(7.26) .n/

and also representations (4.311) and (4.324) for the tensor function F . C ; / (7.8a). For example, for an isotropic solid of the differential type, from (4.322) we obtain the expression for equilibrium stresses .n/

Te D

1E

C

.n/

2C

.n/

C

3C

2

;

(7.27)

where  are determined by formulae (4.322a). By analogy with Definition 4.13 (see Sect. 4.8.6) for a function of one tensor ar.n/ .n/

gument, the function F v . C ; C  ; / of two tensor arguments, which satisfies the ı

condition (7.24c), is called indifferent relative to the group G s (see [12]). The following theorem gives representations of this function in tensor bases. Theorem 7.2. Any tensor function of viscous stresses (7.10), which is ı

 indifferent relative to some orthogonal group G s  I , i.e. satisfies the condition

(7.24c), .n/

 quasilinear in the second argument C  , i.e. satisfies the conditions (7.18) and

(7.20), ı

can be represented in the tensor basis of the corresponding group G s : .n/

.n/ .n/

.n/

T v D F v . C ; C  ; / D .1 E ˝ E C 22 /   C  ;

(7.28)

or .n/

.n/

.n/

T v D 1 I1 . C  /E C 2 C  ;

(7.29)

468

7 Continua of the Differential Type ı

— for an isotropic continuum of the differential type .G s D I /, .n/

T v D .1 E ˝ E C 2b c3 C 3 .E ˝b c23 Cb c23 ˝ E/ c23 ˝b .n/

C 4 .O1 ˝ O1 C O2 ˝ O2 / C 25 /   C 

(7.30)

ı

— for a transversely isotropic continuum of the differential type .G s D T3 /, .n/

Tv D

3 X

.˛b c˛ C 3C˛ .b c2ˇ ˝b c2 Cb c2 ˝b c2ˇ / c2˛ ˝b

˛D1 .n/

C6C˛ O˛ ˝ O˛ /   C  ; ˛ ¤ ˇ ¤  ¤ ˛; ˛; ˇ;  D 1; 2; 3; (7.31) ı

— for an orthotropic continuum of the differential type .G s D O/. The tensorsb c2˛ and O˛ are determined by formulae (4.292) and (4.315). H Substituting relation (7.8) of quasilinearity of the viscous stresses function F v into (7.24c), we obtain that the following condition must be satisfied: 



.n/

.n/ 



.n/

QT  .4 Lv   C  /  Q D 4 Lv  .QT  C  Q/ 8 C  :

(7.32)

Hence the viscosity tensor 4 Lv must satisfy the equation (see Exercise 7.1.1) 4











Lv D 4 Lv     .Q ˝ Q ˝ Q ˝ Q/.57312468/

ı

8Q 2 G s :

(7.32a)

This condition is an analog of the condition (4.289) for fourth-order tensors. Therefore, a fourth-order tensor 4 Lv satisfying Eq. (7.32a) is called indifferent relative to ı

the group G s . All indifferent fourth-order tensors are formed with the help of the operations of tensor and scalar products by producing tensors of a corresponding group ı

G s (see Sect. 4.8.3) and can be resolved for the tensor basis 4 Os. / by analogy with P the resolution (4.291) of second-order tensors: 4 Lv D k D1 a 4 Os. / . The number k of elements of this basis coincides with the number of independent components of ı

the tensor 4 Lv ; it is equal to 2, 5, and 9 for the groups G s D I; T3 ; O, respectively (see [12]). The elastic moduli tensors 4 M defined by formulae (4.337), (4.341), and ı

(4.344) are also indifferent relative to the groups G s D I; T3 ; O, respectively; and the representations (4.337), (4.341), and (4.344) are the desired resolutions for the tensor bases, being the same for each group [12]. Hence, the viscosity tensor can alı

ways be represented in the similar form for the corresponding group G s D I; T3 ; O too. Thus, the representations (7.28)–(7.31) actually hold. N

7.1 Models An and Bn of Continua of the Differential Type

469

Remark. The coefficients  in representations (7.28)–(7.31) are, in general, scalar .n/

.n/

functions of corresponding invariants of the tensors C and C  : .n/ .n/

 D  . C ; C  ; /:

(7.33)

But since we have assumed everywhere that  remain unchanged under H transformations in a corresponding group, so the coefficients  must satisfy the relations 

.n/ .n/

.n/







.n/



ı

 . C; C  ; / D  .QT  C  Q; QT  C   Q; / 8Q 2 G s :

(7.34)

Such scalar functions are called simultaneous invariants of two tensor arguments ı

relative to a group G s considered. According to the theorem proved in [12], for ı

each group G s there is a functional basis of independent simultaneous invariants .s/

.n/ .n/

J . C; C  / ( D 1; : : : ; z), where  z D 9 for the full orthogonal group I ,  z D 11 for the transverse isotropy group T3 ,  z D 12 for the orthotropy group O. .n/

.n/

As a functional basis of simultaneous invariants J.s/ . C ; C  / of two tensors we can choose the following sets [12]: ı

for the isotropy group G s D I : .n/

.n/

.I / J3C˛ D I˛ . C  /; ˛ D 1; 2; 3;

J˛.I / D I˛ . C /; .n/

.n/

J7.I / D C   C  ;

.n/

.n/

J8.I / D C 2   C  ;

.n/

.n/

J9.I / D C   . C  /2 I

(7.35)

ı

for the group G s D T3 : .n/

J˛.3/ D I˛.3/ . C /; ˛ D 1; : : : ; 5; .n/

.n/

.n/

J5Cˇ D Iˇ . C  /; ˇ D 1; : : : ; 4; .3/

.3/

.n/

.n/

c23 /  C /   .b c23  C  /; J11 D C   C   2J10  J2 J8 I J10 D ..E b (7.36) .3/

.3/

.3/

.3/

.3/

ı

for the group G s D O: .n/

J˛.O/ D I˛.O/ . C /; ˛ D 1; : : : ; 6I .n/

.n/

.O/ D .b c22  C/   .b c23  C  /; J10

.n/

.O/ J6Cˇ D Iˇ.O/ . C  /; ˇ D 1; 2; 3; 6I .n/

.n/

.O/ J11 D .b c21  C /   .b c23  C  /:

(7.37)

470

7 Continua of the Differential Type .s/

Invariants I˛ of corresponding groups are determined by formulae (4.295)–(4.297). Then the viscous coefficients can always be represented as functions of simultaneous invariants: .n/ .n/

.n/ .n/

 D  .J1 . C ; C  /; : : : ; Jz.s/ . C ; C  /; /: .s/

(7.38)

The dissipation function (7.20) for an isotropic medium of the differential type, according to (7.29), has the form .n/

.n/

w D 1 I12 . C  / C 2 I1 . C 2 /: 

(7.39)

7.1.5 Models Bn of Solids of the Differential Type For solids of the differential type, models Bn can be obtained formally from corre.n/

.n/

sponding models An by replacing the tensors C with the measures G. In particular, constitutive equations (7.8)–(7.10) become 8 .n/ .n/ .n/ ˆ ˆ T e D F . G; / D .@ =@ G/; ˆ ˆ ˆ ˆ ˆ .n/ ˆ < D . G; /; .n/ ˆ ˆw D .n/ ˆ T v   G ; ˆ ˆ ˆ ˆ .n/ .n/ ˆ :.n/ T v D F v . G; G  ; /:

(7.40)

For models Bn of isotropic continua of the differential type, the tensors of equilibrium and viscous stresses have the forms .n/

Te D



.n/

C

.n/

2

(7.41)

T v D 1 I1 . G  /E C 22 G  ;

(7.42)

.n/

.n/

where the coefficients of invariants:

C

;

1E

2G

3G

.n/

are expressed by formulae (4.322), and  are functions .n/ .n/

.n/ .n/

 D  .J1 . G; G  ; /; : : : ; J10 . G; G  ; //:

(7.43)

In a similar way, from (7.28) and (7.29) we get equations for models Bn of transversely isotropic and orthotropic materials of the differential type.

7.1 Models An and Bn of Continua of the Differential Type

471

7.1.6 Models Bn of Incompressible Continua of the Differential Type Similarly to models Bn for elastic incompressible continua (see Sect. 4.9), we can introduce models Bn for incompressible continua of the differential type. As shown in Sect. 4.9, for these models the potential in (7.40) depends only on r  1 linear .n/

and quadratic invariants I˛.s/ . G/. In particular, for an isotropic incompressible continuum of the differential type,

.n/

.n/

.n/

depends only on I1 . G/ and I2 . G/, and T e is a

.n/

quasilinear function of G: .n/

T D

D

.n/

p .n/1 G C n  III

.n/

.I1 . G/; I2 . G/; /;

1

1E

.n/

C

.n/

.n/

2G

C 1 I1 . G  /E C 22 G  ;

2

D '2 ; '˛ D .@ =@I˛ /:

ı

D ' 1 C ' 2 I1 ;

(7.44) For the simplest model Bn of isotropic incompressible continua of the differential type, we assume that the viscous coefficients are connected by the relation 2 1 D  2 : 3

(7.45)

Then the constitutive equation (7.44) takes the form .n/

T D

p .n/1 G C n  III

1E

C

.n/

2G

.n/

C 22 dev G  ;

(7.46)

where .n/ .n/ 1 .n/ (7.47) dev G D G  I1 . G/E 3 is the deviator of the tensor (the more detailed information on deviators can be found in Sect. 8.2.13 and [12]). For the simplest models Bn , the first principal invariants of the stress tensor .n/

.n/

T and the deformation measures G are connected by the elastic relations (see Exercise 7.1.4) .n/

I1 . T / D 

.n/ p I1 . G 1 / C 3 n  III

being independent of the deformation rates.

1

C

.n/

2 I1 . G/;

(7.48)

472

7 Continua of the Differential Type

When solving problems in practice, one usually applies models Bn of incompressible continua of the differential type with steady creep, where the potential .n/

is independent of the invariants I˛ . G/: .n/

T D

.n/ p .n/1 G C 22 dev G  ; n  III

D

./;

(7.49)

This model describes the phenomenon of creep being a change in deformation of a body with time at constant stresses (see Sect. 7.4). Remark. The consistency conditions (4.328), which must be satisfied by constitutive equations at a natural unstressed state, for materials of the differential type should be complemented by the requirement that the rates of the deformation ten

sors and measures vanish in K: ı

KW

.n/

.n/

T D S D T D 0; .n/

.n/

A  D C  D 0;

.n/

.n/

A D C D 0;

.n/

.n/

.n/

g DGD

.n/

g D G  D 0;  D 0 ;

1 E; n  III D

0:

(7.50)

All the constitutive equations (7.29), (7.30), (7.31), (7.41), (7.42), (7.44), (7.46), and (7.49) derived above satisfy these conditions. t u

7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant, the constitutive equations (7.8)–(7.10) and also (7.28)–(7.31) for solids of the differential type are the same in actual configurations K and K0 obtained one from another by a rigid motion. Therefore, the principle of material indifference for models An and Bn of solids of the differential type (as well as for elastic continua) is satisfied identically.

Exercises for 7.1 7.1.1. Show equivalence of relations (7.32) and (7.32a).

7.2 Models An and Bn of Fluids of the Differential Type

473

7.1.2. Show that the component representations of functional bases of simultaneous invariants (7.35)–(7.37) in the basis b c˛ have the forms for the transverse isotropy ı

group G s D T3 : b b .b n/ .b n/ .n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.3/ D f C 11 C C 22 ; C 33 ; C 213 C C 223 ; C 211 C C 222 C 2 C 212 ; det C; .b n/

.b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ C 11 C C 22 ; C 33 ; . C 13 /2 C . C 23 /2 ; . C 11 /2 C . C 22 /2 C 2. C  12 /2 ;

b b b .n/ .b n/ .n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ C 13 C 13 C C 23 C 23 ; C 11 C 11 C C 22 C 22 C 2 C 12 C 12 gI

.b n/

ı

for the orthotropy group G s D O: b .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.O/ D f C 11 ; C 22 ; C 33 ; C 223 ; C 213 ; C 12 C 13 C 23 ; C 11 ; C 22 ; C 33 ; b b .b .n/ .b n/ .b n/ .b n/ .n/ n/ C 23 C 23 ; C 13 C 13 ; C 12 C 13 C 23 g:

.b n/

7.1.3. Show that for the simplest model Bn of isotropic incompressible continua of the differential type (7.46), the relation (7.48) between the first principal invariants holds.

7.2 Models An and Bn of Fluids of the Differential Type 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type Let us use the principle of material symmetry (7.23) to derive constitutive equations for fluids of the differential type. ı

For fluids, the symmetry group G s , relative to which Eqs. (7.23) hold, is the ı

.n/ .n/

.n/

unimodular group G s D U . Tensors T , C and C  are no longer H -indifferent V

I

I

relative to this group, except T and the tensor G D C  .1=2/E (see Table 4.9); ı

therefore, relations (7.24) for G s D U do not hold and one should use their general form (7.23).

474

7 Continua of the Differential Type .n/

Due to Theorem 4.31, from these relations it follows that the function F . C; / has the form (4.443a) .n/

.n/

T e D F . C; / D 

p .n/1 G ; n  III

(7.51)

.n/

p D p.I3 . G/; /;

(7.52)

where p is the pressure being a scalar function only of the third invariant of the ten.n/

.n/

sor G. Thus, for fluids, the tensor C appears in models An only in the combination .n/

.n/

1 G D C C nIII E; i.e. in fact, models An and Bn are coincident. For the tensor of viscous stresses, the situation is analogous.

7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type I

I

Let us consider further only the models AI and AV . The tensors T and G are trans

formed during the passage to configuration K according to formulae (4.201) and (4.203a). On substituting these formulae into (7.23e), we obtain that for the group ı

G s D U the following equation must hold: I

I

I

I

H1T  F v .G; G ; /  H1 D F v .H  G  HT ; H  G  HT ; / 8H 2 U: (7.53) I

I

I

Here we have gone from the argument C to G, because just the measure G satisfies 

I

the relation (7.53), and the tensor C is transformed during the passage to K in another way, namely according to (4.201a), and its expression involves the additional term: .1=2/.H  HT  E/. I

I

The tensor function F v .G; G ; / satisfying the condition (7.53) is called AI -unimodular or AI -indifferent relative to the group U . Notice that the group U contains a subgroup being the full orthogonal group I ; therefore, if H D OT and Q 2 I , then formula (7.53) becomes (7.24c). In other words, if a tensor function is AI -unimodular, then the function is isotropic (i.e. indifferent relative to the group I ). Then, since the Onsager principle holds for all I

I

groups and the function F v .G; G ; / is quasilinear: I

I

I

I

Tv D F v .G; G ; / D 4 Lv   G ;

(7.54)

7.2 Models An and Bn of Fluids of the Differential Type

475

so for this function one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. But such a representation may be not AI -unimodular; in particular, the representation (7.29) does not satisfy the condition (7.53). The following isotropic tensor function proves to be appropriate: I

Tv D F v D

I I I I 1 .1 I1 G1 C 22 G1  G  G1 /; 4

(7.55)

where I

I

I

I

I1 .G1  G / D G1   G :

(7.56)

We can verify that the function (7.55) is AI -unimodular, i.e. satisfies the Eq. (7.53): I

I

I

I

I

I

I

I

4F.G ; G ; / D 1 .G1   G /G1 C 22 G1  G  G1 I

I

I

D 1 .H1T G1  H1   H  G  HT /H1T  G1  H1 I

I

I

C 22 H1T  G1  H1  H  G  HT  H1T  G1  H1 I

I

I

D H1T  .1 J1 G1 C 22 G1  G  G1 /  H1 I

I

D 4H1T  F .G; G ; /  H1 :

(7.57)

As follows from (7.57), the function (7.55) satisfies also the condition (7.24c) for Gs D I , i.e. it is isotropic. Thus, the isotropic tensor function (7.55) constructed is AI -unimodular. Thus, we obtain the following theorem. Theorem 7.3. Any quasilinear AI -unimodular tensor function of viscous stresses (7.54) can be represented in the form (7.55) or, with the help of the Cauchy stress tensor, in the form (7.58) Tv D 1 I1 .D/E C 22 D; where D is the tensor of deformation rates (2.225). H The first assertion of the theorem (formula (7.55)) has been proved above. Let us prove that the representations (7.55) and (7.58) are equivalent. I

Go from the energetic tensor of viscous stresses Tv to the tensor of viscous stresses T: I

Tv D F1T  Tv  F1 :

(7.59)

Substitution of (7.55) into (7.59) yields I I I I 1 1T F  .1 G1 J1 C 22 G1  G  G1 /  F1 4 1 P 1  FT /: D .21 J1 E  42 F  G 4

Tv D

(7.60)

476

7 Continua of the Differential Type I

I

Here we have taken into account that G D  12 G1 D  12 F1  F1T and G1 D I

2G D 2FT  F: According to formula (2.284), we can express the tensor G in terms of the deformation rate tensor D: I

P 1 D 2F1  D  F1T : 2G D G Taking into account that I

I

I

I

P 1 I1 .G1  G / D G1   G D G   G D 2FT  F   F1  D  F1T D 2D   E D 2I1 .D/; (7.61) from (7.60) we really get formula (7.58). N

7.2.3 Simultaneous Invariants for Fluids of the Differential Type I

I

Notice that the coefficients 1 and 2 in formula (7.55) may depend on G and G : I

I

 D  .G; G ; /:

(7.62)

However, they cannot vary under unimodular transformations: I

I

I

I

 .G; G ; / D  .H  G  HT ; H  G  HT ; /;

8H 2 U;

(7.63)

i.e.  must be scalar AI -unimodular functions of two tensor arguments. Such functions are called simultaneous AI -invariants relative to the group U . Theorem 7.4. A functional basis of independent simultaneous AI -invariants .U /

I

I

I

I

J .G; G / of the tensors G and G relative to the unimodular group U consists of not more than five elements, which can be chosen as follows: I

I

I

I

J.U / D I .G1  G /;  D 1; 2; 3I J4.U / D I3 .G/; J5.U / D I3 .G /: (7.64) I

I

.U / H Each simultaneous invariant J .G; G / is an AI -unimodular function of two tensor arguments, i.e. it satisfies Eq. (7.63). Then the invariant is an isotropic tensor function of two arguments if H D QT and Q 2 I  U ; i.e. it is a simultaneous invariant relative to the group I .

7.2 Models An and Bn of Fluids of the Differential Type

477

In the group I , a functional basis of simultaneous invariants can be formed by I

I

I

I

contractions of powers of the tensors: G1  G , .G1 /2   G etc. However, among all the contractions, only contractions of the tensor I

I

G D G1  G

(7.65)

are invariants relative to the group U (see Exercise 7.2.1). The number r of independent invariants of this tensor cannot exceed three (invariants of the tensor relative to the group U are its invariants relative to the group I too, and for the group I : r D 3). As these invariants we can choose I .G /,  D 1; 2; 3. Moreover, each I

I

of the tensors G and G has one unimodular invariant (see Theorem 4.31), these I

I

are det G and det G , respectively. There are no other independent simultaneous I

I

invariants of the tensors G and G relative to this group. N Remark 1. Notice that in the theorem the number r does not exceed 5; but the invariI

I

ant I3 .G / is not independent: it may be expressed in terms of I3 .G/ and I3 .G /. Let us show this fact. .U / Consider the representation (7.58), then simultaneous AI -invariants J (7.64) can be written as functions of the principal invariants of the tensor D and of the density : I

I

J1.U / .G; G / D I1 .G / D 2I1 .D/; .U /

J2

I

1 2 .I .G /  I1 .2G // D 4I2 .D/; 2 1

.U /

P 1 D 8I3 .D/; .G; G / D det G D det G  det G 1 ı ı .U / J4.U / D  .=/2 ; J5 D .=/2 I3 .D/: 8

I

.G; G / D I2 .G / D J3

.U /

Hence J5

I

I

.U /

is not independent, because it is expressed in terms of J3 .U /

J5

.U /

D J3

.U /

J3

:

(7.66) .U /

and J4

:

(7.67)

Due to Theorem 7.4, the viscous coefficients  (7.62) can be represented as functions of the simultaneous invariants (7.64):  D  .J1.U / ; : : : ; J4.U / ; /;

(7.68)

or in the form  D  .I .D/; ; /;

 D 1; 2; 3: 

(7.69)

478

7 Continua of the Differential Type

7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type Let us consider now the model AV .



V

V

The tensors T and G are transformed during the passage to K by formulae (4.200a) and (4.203). On substituting these formulae into (7.23e), we obtain that the function of viscous stresses must satisfy the relation V

V

V

V

H  F v .G; G ; /  HT D F v .H1T  G  H1 ; H1T  G  H1 ; / V

(7.70)

V

8H 2 U . Such a tensor function F v .G; G ; / is called AV -unimodular or AV -indifferent relative to the group U . According to the Onsager principle, this function must be quasilinear: V

V

V

V

Tv D F v .G; G ; / D 4 Lv  G :

(7.71)

The condition for AV -unimodularity (7.70) goes into the condition for the isotropy of a tensor function when H D QT and Q 2 I  U ; therefore, for the function (7.71) one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. The following isotropic tensor function is appropriate: V

Tv D F v D V

V V V V 1 .1 I1 G1 C 2 G1  G  G1 /; 4

(7.72)

V

where I1 D I1 .G1  G /: We can immediately verify that this function is AV -unimodular, i.e. satisfies the relation (7.70) (Exercise 7.2.3). Thus, the following theorem holds. Theorem 7.5. Any quasilinear AV -unimodular tensor function of viscous stresses (7.71) can be represented in the form (7.72) or, with the help of the Cauchy stress tensor, in the form (7.58). H The first part of the theorem has been proved above, therefore we will show only equivalence of representations (7.72) and (7.58). V

Going from Tv to Tv by the formula V

Tv D F  Tv  FT ;

(7.73)

7.2 Models An and Bn of Fluids of the Differential Type

479

from (7.72) we obtain V V V V 1 F  .1 G1 I1 C 22 G1  G  G1 /  FT 4 1 P  F1 /: D .21 I1 E C 42 F1T  G 4

Tv D

V

(7.74)

V

Here we have taken into account that G D 12 G D 12 FT  F and G1 D 2G1 D P in terms of D: 2F1  F1T : With the help of formula (2.284) we can express G V

P D 2G D 2FT  D  F: G

(7.75)

Using the relations V

V

P D 2F1  F1T   FT  D  F D 2I1 .D/; I1 .G1  G / D G1   G

(7.76)

from (7.74) and (7.75) we actually obtain representation (7.58). N

7.2.5 Viscous Coefficients in Model AV of a Fluid of the Differential Type V

V

Coefficients  in (7.72) are functions of G and G : V

V

 D  .G; G ; /

(7.77)

and remain unchanged under unimodular transformations: V

V

V

V

 .G; G ; / D  .H1T  G  H1 ; H1T  G  H1 ; /;

8H 2 U I (7.78)

i.e. they are scalar AV -unimodular functions. In other words, they are simultaneous V

V

AV -invariants of the tensors G and G relative to the group U . For these invariants, Theorem 7.4 and Remark 1 still remain valid. Therefore,  (7.77) can be represented as functions  D  .J1.U / ; : : : ; J4.U / ; /

(7.79)

of simultaneous AV -invariants formed by formulae (7.64): V

V

J.U / D I .G1  G /;

 D 1; 2; 3;

V

J4.U / D det G:

(7.80)

480

7 Continua of the Differential Type

These invariants can be expressed in terms of the principal invariants of the tensor D and : V

V

J1.U / .G1  G / D 2I1 .D/; J2.U / D 4I2 .D/; .U /

J3

.U /

D 8I3 .D/; J4

D

1 ı .=/2 ; 8

(7.81)

hence the viscous coefficients (7.79) can be written in the form (7.69) as well.

7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type Since for both the models AI and AV constitutive equations can be written in the single generalized form (7.58) and (7.68), it is appropriate to give a common name for the models of fluids of the differential type. Definition 7.2. The models AI and AV (and also BI and BV coincident with them) of fluids of the differential type, whose constitutive equations (7.51), (7.55), and (7.72) can be represented in the single form (7.58), (7.68), i.e. 8 ˆ T D Te C Tv ; ˆ ˆ ˆ ˆ
(7.82)

are called the model of a viscous fluid. According to (7.20), the dissipation function w for a viscous fluid takes the form .n/

.n/

w D T v   C  D Tv   D D 1 I12 .D/ C 22 I1 .D2 /:

(7.83)

The Fourier law (7.17) for a viscous fluid has the same form as the one for an ideal fluid: q D  r ;

 D E:

(7.84)

If the viscous coefficients  are independent of the invariants I .D/ and the density , then the model (7.82) is called the model of a linear-viscous fluid or the model of a Newtonian fluid.

7.2 Models An and Bn of Fluids of the Differential Type

481

7.2.7 Constitutive Equations for Incompressible Viscous Fluids Similarly to Sect. 4.9.4 where the model of incompressible ideal fluids has been stated, we can introduce a model of an incompressible viscous fluid. Theorem 7.6. The models AI and AV (and also BI and BV ) of incompressible fluids of the differential type (7.51), (7.55), (7.68) and (7.72), (7.79) can be represented in the single form 8 ˆ ˆ ˆT D Te C Tv ; ˆ ˆ
(7.85)

where p is a function independent of . H A proof of the theorem is similar to the proof of Theorem 4.33. N The models AI and AV (and also BI and BV ) of an incompressible fluid of the differential type, represented in the form (7.85), are called the model of an incompressible viscous fluid; and if  are independent of I .D/, then – the model of a linear-viscous incompressible fluid.

7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant, the constitutive equations (7.51), (7.55), and (7.72) for fluids of the differential type are the same in actual configurations K and K0 differing from one another by a rigid motion. Therefore, the principle of material indifference for models An and Bn of fluids of the differential type (as well as for ideal fluids) is identically satisfied. We should consider separately only representation (7.85) for fluids, which in configuration K0 takes the form T0v D 1 I1 .D0 /E C 22 D0 ;  D  .I1 .D0 /; : : : ; I3 .D0 /; ; /:

(7.86)

This relation actually holds, because both the tensors D and Tv are R-indifferent (see Table 4.12), and the invariants I .D/ and I .D0 / are coincident.

482

7 Continua of the Differential Type

Exercises for 7.2 7.2.1. Prove that contractions of the tensor (7.65) give scalar invariants relative to the group U . 7.2.2. Show that tensor 4 Lv for the AI -unimodular function (7.54) has the form 4

Lv D

I I I I 1 .1 G1 ˝ G1 C 22 .G1 ˝ G1 /.1432/ /; 4

and for the AV -unimodular function (7.71) – the form 4

Lv D

V V V V 1 .1 G1 ˝ G1 C 22 .G1 ˝ G1 /.1432/ /: 4

7.2.3. Show that the tensor function (7.72) satisfies Eq. (7.70). 7.2.4. Show that the dissipation function (7.40) for models BI and BV of a fluid of the differential type has the form w D 1 I12 .D/ C 22 I1 .D2 /:

(7.87)

7.3 Models Cn and Dn of Continua of the Differential Type 7.3.1 Models Cn of Continua of the Differential Type According to models Cn of continua of the differential type, the free energy and the quasienergetic stress tensors are functions in the form D .n/

.n/ .n/

P /; . A ; A  ; O; O;

.n/ .n/ .n/

P /; S D S . A; A  ; O; O;

ı

(7.88)

ı .n/ .n/

P /: S D S. A; A  ; O; O;

.n/ .n/

P /: w D w . A ; A  ; O; O; .n/

ı

(7.89) (7.90) .n/

ı

Introducing the tensors of equilibrium stresses S e , Se and viscous stresses S v , Sv : .n/

.n/

.n/

S v D S  S e;

.n/

.n/ .n/

S e D S . A ; 0; O; 0; /;

ı

ı

ı

Sv D S  Se ; ı

ı .n/

Se D S. A ; 0; O; 0; /;

(7.91) (7.92)

7.3 Models Cn and Dn of Continua of the Differential Type

483

and substituting them into PTI (4.123), we obtain the identity 0

1

0 1 ı .n/ .n/ @ @ @ @ S S eC e B  PT  A   d OT C   dO A   d A C .n/   d A C @ @ .n/  P  @O  @ O @A @A .n/

 C

 .n/ .n/ ı @ 1 P dt D 0: C  d C .w  S v   A   Sv   O/ @  .n/

(7.93)

.n/

P T , d and dt are independent, the identity Since all differentials d A, d A  , d OT , d O (7.93) is equivalent to the equation system 8 .n/ .n/ .n/ ˆ ˆ S D .@ =@ A /  ˆ. A; O; /; ˆ e ˆ ˆ ˆ ˆ .n/ ı ˆı ˆ <Se D .@ =@O/ D ˆ. A ; O; /; .n/ .n/ ı ˆ ˆ  ˆ P T; D S   A  C Sv   O  D @ =@; w ˆ v ˆ ˆ ˆ ˆ .n/ .n/ ˆ : P D 0; @ =@ A  D 0; D . A ; O; /; @ =@O

(7.94)

.n/

P Just as in the i.e. , just as for models An , cannot depend on the rates A  and O. case of models An , Eqs. (7.94) should be complemented by relations connecting the .n/

ı

tensors of viscous quasienergetic stresses S v , Sv to the reactive variables .n/

ı

.n/ .n/

P /; S v D ˆ v . A; A  ; O; O;

ı

.n/ .n/

P /: Sv D ˆ v . A ; A  ; O; O;

(7.95)

The relations (7.94) and (7.95) are constitutive equations for models Cn of continua of the differential type. However, these constitutive equations prove to be not correct: they do not satisfy .n/ .n/

.n/

the principle of material indifference. Indeed, since the tensors S , S e , S v and .n/

.n/

A are R-indifferent and the tensors A  are not R-indifferent, relations (7.95) are transformed during the passage from K to K0 as follows (even if there is no tensor O being not R-indifferent in these relations): .n/

.n/ .n/

S 0v D QT  ˆ v . A ; A  ; /  Q .n/

.n/

.n/

.n/

P T  A  Q C QT  A  Q; P /: D ˆ v .QT  A  Q; QT  A   Q C Q

(7.96)

484

7 Continua of the Differential Type

One can show that for isotropic continua of the differential type satisfying the Onsager principle, the tensor of viscous stresses ˆ v has the form .n/

.n/

ˆ v D 1 I1 . A  /E C 22 A  :

(7.97)

Derivation of this equation is performed by the same method as the one used for models An (see Sect. 7.1.2). On substituting (7.97) into (7.96), we obtain that this equation does not hold. Therefore, models Cn of continua of the differential type cannot describe adequately a behavior of materials.

7.3.2 Models Cnh of Solids with Co-rotational Derivatives In order to obtain a correct model Cn , we should replace the total derivative by some co-rotational one in Eqs. (7.88)–(7.89). One can say that this is the model Cnh of continua with co-rotational derivatives, .n/ ı

if active variables ƒ D f ; S ; Sg depend on the following reactive ones: D

.n/

.n/

P /; . A ; A h ; O; O;

.n/

.n/ .n/ .n/

ı

ı .n/ .n/

P /; S D S . A ; A h ; O; O;

(7.98) (7.99)

P /; S D S. A; A h ; O; O;

(7.100)

h D f V; S; J g:

(7.101)

where Substitution of (7.98)–(7.100) into PTI (4.123) yields 0

1

0 1 ı .n/ .n/ @ @ S S @ eC e B PT  AO A   A  C .n/   A h C @ @ .n/   @O  @A @Ah .n/

C

@ P @O

RTC O



 .n/ .n/ ı @ 1 P T / D 0: (7.102) C  P C .w  S v   A   Sv   O @ 

7.3 Models Cn and Dn of Continua of the Differential Type .n/

485

.n/

P T, O R T and P are independent, the identity (7.102) Since all derivatives A  , A h , O is equivalent to the equation system 8 .n/ ˆ ˆ A ; O; /; D . ˆ ˆ ˆ ˆ .n/ ˆ ˆ P D 0; @ =@ A h D 0; ˆ @ =@O ˆ < .n/ .n/ .n/ S e D .@ =@ A /  ˆ. A ; O; /; ˆ ˆ ˆ ı .n/ ˆı ˆ ˆ S D .@ =@O/ D ˆ. A ; O; /; ˆ e ˆ ˆ ˆ .n/ ı :  .n/ P T: w D S v   A  C Sv   O

.7:103a/ .7:103b/ .7:103c/ .7:103d/ .7:103e/

For viscous stresses, according to the principle of equipresence, we should give the functions .n/

.n/ .n/

P /; S v D ˆ v . A ; A h ; O; O; ı

ı

.n/ .n/ h

P /: Sv D ˆ v . A ; A ; O; O;

(7.104) (7.105)

Notice that the expression (7.103e) for the dissipation function, which has been derived formally, conflicts with the principle of equipresence, because w depends on the following arguments: .n/ .n/

.n/

P /; w D w . A; A  ; A h ; O; O;

(7.106)

that is inconsistent with the dependences (7.98)–(7.100). However, we will show below that this inconsistence may be avoided.

7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids Consider the principle of material symmetry for models Cnh of solids. According to the principle, the following relations must hold during the passage to reference 

configuration K: .n/ S e

.n/



ı .n/



.n/



D ˆ. A  ; O; /;

ı

Se D ˆ. A  ; O; /; D

. A  ; O; /;

(7.107a) (7.107b) (7.107c)

486

7 Continua of the Differential Type .n/

.n/



.n/

.n/



.n/

P  ; /; S v D ˆ v . A  ; A h ; O; O ı

ı

(7.107d)

P  ; /; Sv D ˆ v . A  ; A h ; O; O .n/

(7.107e)

ı

.n/

P ; w D S v   A  C Sv   O 

(7.107f)

ı

for each configuration K obtained from K with the help of H -transformation ı



ı

(K ! K) from group G s . From Eqs. (7.103e) and (7.107f) it follows that the dissipation function (7.106) must satisfy the relation .n/

.n/



.n/

.n/

.n/

ı

.n/

P  ; / D w . A ; A  ; A h ; O; O; P / 8H 2 G s : w . A  ; A  ; A h ; O; O (7.108) ı

Below, we will consider solids of the differential type, i.e. the case when G s  I . ı

.n/

.n/

Since for all orthogonal groups G s the tensors A and A h are H -invariant (see Tables 4.9 and 4.10) and the tensor O is transformed by formula (4.256) during the ı



passage from K to K, relation (7.108) can be written in the form .n/

.n/



.n/



.n/

.n/



.n/

ı

P  Q; / D w . A ; A  ; A h ; O; O; P / 8Q 2 G s  I: w . A ; A  ; A h ; O  Q; O (7.109) The similar relation follows from (7.103a) and (7.107c) for the Helmholtz free energy : .n/



. A ; O  Q; / D



.n/

. A ; O; /

ı

8Q 2 G s  I:

(7.110)

In other words, according to Definition 4.14, the dissipation function (7.106) and ı

the free energy (7.103a) must be rotation-indifferent relative to G s  I . Theorem 7.7. For models Cnh of isotropic continua, the dissipation function (7.106) and the free energy (7.103a), which satisfy the principle of material symmetry (i.e. P i.e. Eqs. (7.109) and (7.110)), are independent of the tensors O and O; D

.n/

. A ; /;

.n/ .n/

.n/

w D w . A ; A  ; A h ; /:

(7.111) (7.112)

H A proof of the theorem is similar to the proof of the first stage of Theorem 4.27 (see Exercise 7.3.3). N

7.3 Models Cn and Dn of Continua of the Differential Type

487

.n/

P are independent, from (7.112) and (7.103e) we obtain Since the tensors A  and O that for isotropic solids the following relations must hold: ı

ı

Sv  ˆ v  0; .n/

.n/

(7.113a)

.n/

S v D ˆ v . A; A h ; /I

(7.113b)

i.e. the rotation tensor of viscous stresses is identically zero, and the tensor of vis.n/

P cous stresses S v is independent of O and O. Moreover, since for isotropic continua is independent of O, from (7.103d) it follows that the rotation tensor of equilibrium stresses is identically zero too: ı

ı

Se D ˆ  0: ı

(7.114)

ı

Since both the rotation tensors Sv and Se are identically zero, so, according to .n/

.n/

Theorem 4.9 of Sect. 4.2.18, the corresponding tensors S v and S e for isotropic .n/

continua are commutative with A : .n/

.n/

.n/ .n/

.n/

.n/

.n/ .n/

S v  A D A  S v;

(7.115)

S e  A D A  S e:

(7.116)

Then, according to Theorem 4.11, the stress power w.i / and hence the viscous stress power can be represented in terms of the co-rotational derivatives: .n/

.n/

.n/

.n/

wv  S v   A  D S v   A h D w ;

h D fU; V; S; J g:

(7.117)

.n/ .n/

Since the tensor of viscous stresses depends only on A , A h and , from (7.117) and (7.103e) it follows that the dissipation function (7.112) for isotropic continua is .n/

independent of A  , i.e. .n/ .n/

w D w . A ; A h ; /; that ensures that the principle of equipresence be satisfied.

(7.118)

488

7 Continua of the Differential Type

7.3.4 Viscosity Tensor in Models Cnh Let us apply now the Onsager principle to models Cnh of isotropic continua. Due to this principle, the function of viscous stresses (7.113b) is quasilinear with respect .n/

to A h : .n/

.n/

.n/

.n/

S v D ˆ v . A ; A h ; / D 4 Lv   A h :

(7.119)

Here 4 Lv is the viscosity tensor being positive-definite: .n/

.n/

.n/

.n/

w D S v   A h D A h   4 Lv   A h > 0;

(7.120)

and having the symmetry (7.21). Theorem 7.8. For models Cn of isotropic solids, the viscosity tensor 4 Lv has the form 4

Lv D 1 E ˝ E C 22 ;

(7.121)

and the function of viscous stresses becomes .n/

.n/

.n/

S v D ˆ v D 1 I1 . A h /E C 22 A h ;

(7.122)

where  are the viscous coefficients. .n/

H Notice that the quasilinear function (7.119) of the argument A h is indifferent relative to the triclinic group E, i.e. satisfies the condition 

.n/





.n/

QT  .4 Lv   A h /  Q D 4 Lv   .QT  A h  Q/

(7.123)



when Q 2 fE; Eg. Just as in Theorem 4.27, we want to find a maximum orthogonal subgroup ı

G s  I , relative to which the tensor function (7.119) is indifferent. But in the triclinic group there is a tensor basis consisting of six tensors (see [12]), .n/

.n/

for which we can resolve any tensor function (7.119); this is the set fE; A h ; A h2 ; O1 ; O2 ; O3 g; i.e. .n/

.n/ h

.n/

.n/

S v D Lv   A D '1 E C '2 A C '3 A 4

h

h2

C

3 X  D1

where ' are the resolution coefficients.

'3C O ;

(7.124)

7.3 Models Cn and Dn of Continua of the Differential Type

489

As shown above (see (7.115)), for isotropic solids of the differential type the ı

.n/

rotation viscosity tensor Sv commutes with the tensor A . Substitution of (7.124) into (7.115) yields 3 X

.n/

.n/

'3C .O  A  A  O / D 0:

(7.125)

 D1

Hence the coefficients '3C ( D 1; 2; 3) in the representation (7.124) must always vanish. .n/

Moreover, since the function (7.119) is quasilinear in A h , so '3 D 0; and then from (7.124) we obtain that the function (7.119) for an isotropic continuum can always be written in the form .n/

.n/

.n/

S v D '1 E C '2 A h D 4 Lv   A h :

(7.126)

Introducing the notation .n/

'1 D 1 I1 . A h /;

'2 D 22 ;

(7.127)

from (7.126) and (7.127) we really get representations (7.121) and (7.122). N

7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids Let us establish now a final form of constitutive equations for the models Cnh of isotropic solids. Theorem 7.9. For models Cnh of isotropic solids, constitutive equations (7.111) and (7.103c) for equilibrium stresses can be represented as functions of three principal .n/

invariants I˛ . A /: .n/

. A ; / D .n/

.n/

.n/

S e D ˆ  .@ =@ A / D

where

˛

˛ D 1; 2; 3;

.I˛ . A /; /; 1E

C

.n/

2A

C

.n/ 2

3A

(7.128) ;

are expressed by formulae (4.336).

H A proof of the theorem is similar to the proof of Theorem 4.27. N

(7.129)

490

7 Continua of the Differential Type

Formulae (7.122) and (7.129) give the possibility to derive a final representation of constitutive equations for the model Cnh of an isotropic solid: .n/

S D

1E

.n/

C

2A

.n/

C

3A

2

.n/

.n/

C 1 I1 . A h /E C 22 A h :

(7.130)

The relation obtained is correct, because it satisfies the principle of material indifference. Indeed, in the rigid motion from actual configuration K to K0 the relationship (7.130) is transformed as follows: 1E

C

.n/ 0

2A

D QT  .

.n/

C

1E

3A

C

02

.n/

2A

.n/

.n/

C 1 I1 . A h 0 /E C 22 A h 0 C

.n/

3A

2

.n/

.n/

C 1 I1 . A h /E C 22 A h /  Q:

(7.131)

Remark 1. Notice that although the representation (7.117) for the viscous stress power still holds for the co-rotational derivatives included in (7.117), however, for the final result (7.130) we need to return to the list (7.101) with the excluded right derivative in the eigenbasis h D U , because this derivative, according to Table 4.12, .n/

does not maintain R-indifference of the derivative A h for the R-indifferent ten.n/

sor A . But all co-rotational derivatives occurring in the list (7.101) satisfy this requirement, that has been used in deriving the formula (7.131). t u Remark 2. The coefficients



occurring in (7.130) are functions of the invariants

.n/

.n/ .n/

I . A / (see formulae (4.336)), and the viscous coefficients depend on A , A h and : .n/ .n/

 D  . A; A h ; /:

(7.132)

Since the functions (7.132) cannot vary in rigid motions of actual configuration K ! K0 , they must satisfy the relation .n/

.n/

.n/

.n/

 . A ; A h ; / D  .QT  A  Q; QT  A h  Q; / .n/

8Q 2 I

(7.133)

.n/

(due to R-indifference of A and A h ). But this means that  must be H -indifferent relative to the full orthogonal group. Then, according to the results of Sect. 7.1.4, the functions (7.132) must depend .n/

.n/

on simultaneous invariants J˛ of the tensors A and A h relative to the group I :  D  .J1 ; : : : ; J10 ; /;  D 1; 2I

.n/

.n/

J˛ D J˛ . A ; A h /;

˛ D 1; : : : ; 10: (7.134)

Simultaneous invariants J˛ satisfy the condition (7.133) and can be formed as shown in (7.35).

7.3 Models Cn and Dn of Continua of the Differential Type .n/

491

.n/

Due to H -invariance of the tensors A and A h , under orthogonal transformations ı



of reference configuration K ! K the viscous coefficients (7.132) always remain unchanged, that has been used in proving the formula (7.131). t u

7.3.6 Models Dnh of Isotropic Solids Models Dnh of solids of the differential type can be obtained from Eqs. (7.98)– .n/

.n/

(7.100) for models Cnh by replacing the deformation tensors A with the measures g . .n/

Due to H -invariance of the tensors g , all manipulations performed for models remain valid for models Dnh too. In particular, the final relations (7.130) for isotropic solids in the model Dnh take the form Cnh

.n/

S D

1E

C

.n/ 2

g C

3

.n/2

.n/

.n/

g C 1 I1 . g h /E C 22 g h ;

(7.135)

.n/

where the coefficients  are functions of the invariants I . g / (see formulae (4.380)), and the viscous coefficients  depend on simultaneous invariants of the .n/

.n/

tensors g and g h :  D  .J1 ; : : : ; J10 ; /;  D 1; 2I

.n/ .n/

J˛ D J˛ . g ; g h /; ˛ D 1; : : : ; 10: (7.136)

Relations (7.135) satisfy the principles of material symmetry (since all the tensors .n/ .n/

.n/

S , g and g h are H -invariant under orthogonal transformations) and material in.n/ .n/

.n/

difference (since all the tensors S , g , and g h are R-indifferent).

Exercises for 7.3 7.3.1. Prove Theorem 7.9. 7.3.2. Prove that the Onsager principle yields representation (7.97) for the functions ˆ v . 7.3.3. Prove Theorem 7.7. 7.3.4. Using relations (7.120) and (7.122), show that for models of isotropic continua of the differential type the dissipation function has the form .n/

.n/

w D 1 I12 . A h / C 22 I1 .. A h /2 /:

492

7 Continua of the Differential Type

7.4 The Problem on a Beam in Tension 7.4.1 Rate Characteristics of a Beam Consider the problem on a beam in tension, which has been investigated in Sect. 6.4.1, but the beam is made of material described by the model Bn of incompressible continua of the differential type with steady creep (see (7.49)). The .n/

motion law and the tensors F, C in this case retain their forms (6.131)–(6.134), and the velocity v and the velocity gradient L, according to (2.223), become 3 X

vD

kP˛ X ˛ eN 2˛ ;

˛D1

LD

3 P X @v k˛ 2 i ˝ r D eN : i @X k˛ ˛ ˛D1

(7.137)

Here we have used that r˛ D eN ˛ =k˛ (see Exercise 2.1.1). .n/

The rates C  of the deformation tensors and their deviators can be found by differentiating the expression (6.133): 3 X

.n/ 

C D

k˛nIII1 kP˛ eN 2˛ ;

.n/

dev G  D

˛D1 .n/

f 1 .k1 / D

3 .n/ X f ˛  .k1 /Ne2˛ ;

(7.138)

˛D1

.n/ .n/ nIII 2 1 .n/ .k1nIII  k1 2 /; f 2 D f 3 D  f 1 : 3.n  III/ 2

(7.139)

Here p we have taken into account that for incompressible materials k2 D k3 D 1= k1 (see Sect. 6.5.1).

7.4.2 Stresses in the Beam .n/

On substituting the expression (7.138) for dev C  into constitutive equations (7.49), .n/

we find components T ˛˛ of the energetic stress tensors .n/

T D

3 .n/ .n/ X .n/ T ˛˛ eN 2˛ ; T ˛˛ D pk˛IIIn C 22 f ˛  : ˛D1

(7.140)

7.4 The Problem on a Beam in Tension

493

The Cauchy stress tensor T is calculated by the same relations (6.138) just as for elastic materials: 3 X .n/ TD ˛˛ eN 2˛ ; ˛˛ D k˛nIII T ˛˛ ; (7.141) ˛D1 .n/

˛˛ D p C 22 f

˛

 nIII k˛ :

(7.142)

Just as in the case of an elastic beam, the Cauchy stress tensor (7.141) is independent of coordinates; therefore, the equilibrium equations for the beam, when there are no mass forces, are identically satisfied.

7.4.3 Resolving Relation  .k1 ; kP 1 / On substituting (7.142) into the boundary condition (6.141c), we find that, just as for an elastic beam, the components 22 and 33 must be zero, and the only component 11 is nonzero. As a result, we obtain two relations .n/

11 D p C 22 f

1

 nIII k1

.n/

IIIn 2

and 0 D p C 2 f 1  k1

;

(7.143)

for the three unknowns: k1 , p and 11 . Excluding p from this system, we get the following resolving relation for the problem: 11 D

22 .n/2 L .k1 /kP1 ; 3k1

(7.144)

where .n/

1 IIIn L .k1 / D k1nIII C k1 2 : 2

(7.145)

7.4.4 Comparative Analysis of Creep Curves for Different Models Bn Consider a stepwise process of loading (Fig. 7.1) when the stress 11 .t/ is given in the form (7.146) 11 .t/ D  0 h.t/; where  0 is the constant, and h.t/ is the Heaviside function ( 0; t 6 0; h.t/ D 1; t > 0:

(7.147)

The elongation function ı1 .t/ D k1 .t/  1 (measured in experiments) for a beam of material of the differential type under such loading is usually called the creep

494

7 Continua of the Differential Type

Fig. 7.1 The stepwise process of loading (a) and the typical curve of creep (b)

a

b

Fig. 7.2 The process of loading–unloading (a) and creep curve in this process (b)

a

b

curve. A typical creep curve for materials, whose properties depend on the deformation rate, is shown in Fig. 7.1. There are three typical sections along this curve: (1) the initial section, (2) the section of steady creep, and (3) the section of unsteady creep. The value of elongation ı1 .0C / at t ! 0C is called the instantly elastic elongation. The creep effect is observed for metals and alloys under high temperatures. If at the time t1 unloading occurs (Fig. 7.2): 11 .t/ D  0 .h.t/  h.t  t1 //;

(7.148)

then the elongation ı1 .t/ decreases, and for materials of the differential type the elongation does not return to its initial zero value as t ! 1 (Fig. 7.2), i.e. there appears a residual deformation of creep ı1 .1/. If the process of loading 11 .t/ is given, then Eq. (7.144) is a nonlinear ordinary differential equation for the functions k1 . Solving the equation, we obtain k1 .t/ D H

1

Z

t 0

! 11 . / d ; 2

2 H.k1 /  3

Z

k1 1

.n/ 2

L .k/ d k: k

(7.149)

On substituting (7.146) into (7.149), we find that under a constant load the elongation of the beam grows with time: ı1 .t/ D H 1 . 0 t =2 /  1;

(7.150)

and under unloading the elongation remains constant and does not decrease: ı1 .t/ D ı1 .t1 / at t > t1 .

7.4 The Problem on a Beam in Tension

495

Fig. 7.3 Creep curves for nickel alloy (solid curves are computations, dashed lines are experimental data)

Figure 7.3 shows computed and experimental creep curves ı1 .t/ for nickel alloy at temperature 1100ıC in compression for different values of the stress  0 6 0. Theoretical curves have been computed according to formula (7.149) for different models Bn , and the creep curves ı1p .t/ are the difference in elongation between the experimental values ı1 .t/ and the initial value: ı1p .t/ D ı1 .t/  ı1 .0C /. One of the experimental creep curves ı1p .t/ at the smallest value  0 was used for determining the constant 2 , that was calculated by minimizing the mean-square distance between computed and experimental results at N points being times ti : !1=2 N ı1 .ti / 2 1 X ! min: (7.151) 1

D N ı1p .ti / i D1

Table 7.1 shows values of the constant 2 for nickel alloy, computed by the method mentioned above for different models Bn . The model BII exhibits the best approximation to the experimental data for the case considered (Fig. 7.3). It should be noted that for many metals, values of the high-temperature creep elongation ı1p .t/ at sufficiently great t considerably exceed instantly elastic values ı1 .0C /, therefore the last ones are often neglected in computations of creep problems. The considered models Bn of continua of the differential type with steady creep fall into this class of models.

496

7 Continua of the Differential Type Table 7.1 Values of the constant 2 and the relative error ı of approximation to the creep curves for models Bn of continua of the differential type for nickel alloy at temperature 1100ı C n I II IV V 2 , GPas 30 30 32 35 ı, % 18 17 19 20

Fig. 7.4 Diagrams of deforming for nickel in compression

7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type Consider one more regime of deforming, when the deforming process of a beam is given by the law ( bt 2 =2t1 ; t < t1 ; k1 .t/ D 1 C (7.152) b.t  t1 =2/; t > t1 : where b D const is the rate of deforming, and t1 is the beginning of deforming with a constant rate (the initial interval 0 6 t 6 t1 is necessary for the consistency conditions (7.50) to be satisfied). The rate kP1 is determined by the expression kP1 D and

p

.2k1 b=t1 /

kP1 D b

when t < t1

when t > t1 :

Substituting (7.152) into (7.144), we obtain the relation 11 .k1 / called the diagram of deforming for the beam considered. Figure 7.4 exhibits the experimental diagram of deforming for nickel in compression at 1100 ı C and b D 0:00025 s1 and computed diagrams, for which the viscosity coefficient 2 has been evaluated with the help of the creep curves (see Sect. 7.4.4 and Table 7.1). The model BII gives the best approximation to the experimental data.

Chapter 8

Viscoelastic Continua at Large Deformations

8.1 Viscoelastic Continua of the Integral Type 8.1.1 Definition of Viscoelastic Continua Besides models of the differential type considered in Chap. 7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues. Further, viscoelastic continua of the differential type will be called continua of the differential type, and viscoelastic continua of the integral type – simply viscoelastic materials. Definition 8.1. A medium is called a viscoelastic continuum of the integral type (or simply a viscoelastic continuum), if any of the models An , Bn , Cn or Dn is assumed for the medium, and corresponding operator constitutive equations (4.156) or (4.158)–(4.161) are functionals of time t: Dt

ƒ.t/ D f .R.t/; Rt .//;

(8.1)

D0

i.e. values of active variables ƒ.t/ depend not only on values of reactive variables R.t/ at the same instant of time but also on their prehistory Rt ./  R.t  /, i.e. on their values at all preceding instants of time 0 <  6 t, starting from initial one  D 0. Due to such specific dependence, viscoelastic continua are also called continua with memory. For viscoelastic continua: 1. “The present can depend only on the past but not on the future”; therefore, all the functionals (8.1) depend only on their previous history (prehistory) R.t  /, 0 <  6 t. Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 8, c Springer Science+Business Media B.V. 2011 

497

498

8 Viscoelastic Continua at Large Deformations

2. “The past is not infinite”, i.e. the times  > t do not affect the functionals (8.1). This means that R./  0 at  < 0I

Rt ./ D R.t  /  0 at t > :

(8.2)

8.1.2 Tensor Functional Space To perform operations with functionals (8.1), we need some additional knowledge of functional analysis. Consider the set of previous histories of k-order tensors k Tt ./ D k T.t  / (0 <  6 t), and define for two arbitrary prehistories k Tt1 and k Tt2 their scalar product as follows: Zt .

k

Tt1 ; k Tt2 /t

k

D

Tt1 ./ „ƒ‚…  : : :  .k Tt2 .//.k:::1/  2 ./ d :

(8.3)

k

0

The function ./ is called the function of memory. This function is positive, continuous, monotonically decreasing, defined within the interval Œ0; C1/ and squared-integrable, i.e. Z1  2 ./ d  D 02 < C1: (8.4) 0

Since the memory function is monotonically decreasing, quantities of the prehistory k Tt ./ D k T.t  / at small values of  give a greater contribution to the scalar product (8.3) than quantities of k Tt ./ at large values of . In other words, a continuum better remembers events having occurred at times closer to the current instant t than the ones at more remote times. The functionals (8.1) are assumed to have this property too; therefore, viscoelastic continua of the integral type are also called continua with fading memory. Let us consider now the set k Ht of processes of changing the tensor k T./ (0 <  6 t) and assign to each process the pair .k T.t/; k T.// consisting of values of the tensor k T.t/ at time t and its prehistory k T./ D k T.t  / (0 <  6 t). Then we can introduce the scalar product of processes k T1 ./ and k T2 ./ included in k Ht : 

where



k

k

T1 ; k T2

 t

D



k

   T.t/; k T.t/ C k Tt1 ; k Tt2 ; t

 .k ::: 1/  T1 .t/; k T2 .t/ D k T1 .t/ „ƒ‚…  : : :  k T2 .t/ k

is the scalar product of k-order tensors.

(8.5)

(8.6)

8.1 Viscoelastic Continua of the Integral Type

499

The set k Ht of all processes of changing the tensor k T./ (0 6  6 t), for which the scalar product (8.5) exists and which at each fixed  are elements of the tensor space T3k .E3 / with the operations of addition and multiplication by a number, is called the tensor functional space k Ht . The space k Ht is a Hilbert space, because we can always go into a Cartesian basis, where the components TN i1 :::ik ./ of tensors included in k Ht are squared.m/ integrable functions, i.e. they belong to the function space L2 Œ0; t (where m D 3k ), which is known as a Hilbert space. Due to the property (8.2), the scalar product of processes included in k Ht (8.5) can be written in the form Z1 k

.

Tt1 ; k Tt2 /t

D

k

Tt1 ./ „ƒ‚…  : : :  .Tt2 .//.k:::1/  2 ./ d  < C1;

(8.7)

k

0

which often proves to be convenient for analysis of the models of a viscoelastic continuum. In the space k Ht , there is a natural norm of the process k T./: k k T kD .k T; k T/1=2 t ;

(8.8)

where .  /t is the scalar product (8.5).

8.1.3 Continuous and Differentiable Functionals Using the norm (8.8), we can introduce the concept of a continuous functional in the form (8.1) m

t

S D m F .k T.t/; k Tt .//; D0

(8.9)

which is considered as the mapping of the domain U contained in the space k Ht into the domain V of the space m Ht : m

FW

U  k Ht ! V  m Ht :

(8.10)

Definition 8.2. The functional (8.10) is called continuous in a domain U  k Ht , if for each process k T 2 U the following condition is satisfied: 8" > 0 9ı > 0 such T, where .k T C k e T/ 2 U and that for every process k e T kt < ı; k ke

(8.11)

500

8 Viscoelastic Continua at Large Deformations

values of the operators in the norm (8.8) of the space k Ht are sufficiently close: t

kmF



k

D0

 t T.t/; k Tt ./ C k e Tt ./  m F .k T.t/; k Tt .// kt < ": T.t/ C k e D0

(8.12)

The functional (8.10) is called linear if it satisfies the two conditions 

t

F

k

D0

T1 .t/ C k T2 .t/; k T1 ./ C k T2 ./ t

D F

D0

t

F

D0



k



  t  T1 .t/; k T1 ./ C F k T2 .t/; k T2 ./ ; D0

   t  s k T.t/; s k T./ D s F k T.t/; k T./ ; D0

(8.13)

(8.14)

for all processes k T1 ./ and k T2 ./ included in k Ht and for every real number s. In space k Ht we can use Riesz’s theorem that any linear functional (8.9) can be represented as the scalar product of a fixed element from k Ht and an arbitrary process k T./ from k Ht ; so a scalar linear functional in 2 Ht has the form e.t; t/   TT .t/ C f .T.t/; T .// D 

Zt

t

e .t; t  /   TT .t  / 2 ./ d ; (8.15a)

0

e.t; t/ is the instantaneous value at  D t of where e .t; t  / is the prehistory and  e the fixed process  .t; / (0 6  6 t) for the given functional f (the appearance of one more argument t for the process e .t; / means that this process can vary with changing the time interval considered). Replacing the variable t   D y and introducing the notation e  T .t; y/ 2 ./ D T .t; y/,  0 D  .t; t/, after the reverse substitution y !  we obtain another representation of the linear scalar functional Zt f D  0   T.t/ C

.t; /   T./ d ;

(8.15b)

0

called Volterra’s representation. For viscoelastic continua, it is convenient to use the Dirac ı-function ı.t/ having the following main property: (

Zt B.t; /ı.t0  / d  D 0

for any continuous tensor process B.t; /.

B.t; t0 /; t0 2 Œ0; t; 0;

t0 … Œ0; t;

(8.16)

8.1 Viscoelastic Continua of the Integral Type

501

According to (8.16), the linear functional (8.15b) can be written in the form Zt A.t; /   T./ d ;

f D

(8.15c)

0

A.t; / D  0 ı.t  / C .t; /:

(8.15d)

Definition 8.3. The functional (8.9) is called Fr´echet–differentiable at point m T 2 U of a domain U  m Ht , if there exist two functionals @F and ıF having the following properties:  they are defined over the Cartesian product of the space Ht

@m F W

k

Ht  k Ht ! m Ht I

ımF W

k

Ht  k Ht ! m Ht ;

(8.17)

 they can be written in the form similar to (8.9) m

t ˇ T.t// P1 D @m F .k T.t/; k Tt ./ˇk e D0

D

@

t

F .k T.t/; k Tt .//  : : :  k T.k:::1/ .t/;

@k T.t/ D0

m

t ˇ t T .//; P2 D ı m F .k T.t/; k Tt ./ˇk e

(8.18) (8.19)

D0

(the vertical line separates two different arguments of the process),  they are linear and continuous in the second argument,  they satisfy the condition: 8" > 0 9ı such that for any process .k e T.t/; k e Tt .//,

for which .k T1 .t/; k Tt1 .//  U and k k T kt < ı;

(8.20)

k m F kt 6 " k k e T kt ;

(8.21)

the following inequality holds:

where t

t

m F D m F .k T1 .t/; k Tt1 .//  m F .k T.t/; k Tt .// D0

D0

t ˇ ˇ t T.t//  ı F .k T.t/; k Tt ./ˇk e T .//; @m F .k T.t/; k Tt ./ˇk e t

D0

D0

(8.22) k

. T1 .t/;

k

Tt1 .//

ke

k et

 . T.t/ C T.t/; T ./ C T .//: k

k

t

(8.23)

502

8 Viscoelastic Continua at Large Deformations

The operator (8.19) is called the Fr´echet–derivative; and the right-hand side of expression (8.18) is the partial derivative of F (considered as a tensor function of T.t/) with respect to the tensor argument T.t/. If the functional (8.9) is Fr´echet–differentiable, then it is continuous (see Exercise 8.1.2). The set of processes k T.t/ 2 k Ht 0 having the first and second continuous derivaP R and k T.t/, which belong to k Ht 0 , will be denoted tives with respect to time t: k T.t/ by Ut 0 . Theorem 8.1. Let the functional (8.9) be Fr´echet–differentiable in k Ht 0 , then there exists such t (t 2 .0; t 0 /) that for all processes k T./ 2 Ut the process m S.t/ is differentiable with respect to t and the following rule of differentiation of the functional with respect to time holds: d dt

m

S.t/ D

@ @

t

F .k T.t/; k Tt .//  : : : 

k T.t/ D0

d k .k:::1/ T .t/ dt

t ˇ C ı F .k T.t/; k Tt ./ˇk TP t .//: D0

Here k

TP t D

(8.24)

d d k T.t  / D  k Tt ./: d.t  / dt

H A proof of the theorem can be found in [31]. N Remark. The theorem gives the possibility to calculate the Fr´echet–derivatives of the operators (8.9) by evaluating the ordinary derivative of the functions S.t/ with respect to t according to formula (8.24). t u Example 8.1. Determine the Fr´echet–derivatives of the linear operator (8.15b) for the case when .t; / D .t  / and .t; t/ D  .0/. According to formula (8.24), we calculate the ordinary derivative with respect to time t by the rule of differentiation of an integral with a varying upper limit: d dT f D 0   .t/ C  .0/   T.t/ C dt dt

Zt

 0 .t  /   T./ d ;

(8.25)

0

where  0 .y/ D @.y/=@y. Comparing (8.25) with (8.24), we find the partial derivative and the Fr´echet–derivative: d @F T.t/; D  0; dt @T Zt ıF D .0/   T.t/ C  0 .t  /   T./ d :  @F D  0  

0

(8.26)

8.1 Viscoelastic Continua of the Integral Type

503

8.1.4 Axiom of Fading Memory For viscoelastic continua, in addition we assume the following axiom. Axiom 18 (of fading memory). The functionals (8.1) occurring in constitutive equations of viscoelastic continua are Fr´echet–differentiable and hence satisfy the rule of differentiation with respect to time (8.24): t t ˇ t d @ d P .//: (8.27) f .R.t/; Rt .// R.t/ C ı f .R.t/; Rt ./ˇR ƒ.t/ D dt @R.t/ D0 dt D0

The interconnection between Fr´echet–differentiability and fading memory of functionals is clarified by the theorem on relaxation. Let there be a process R./ (0 6  6 t), which is arbitrary up to some time t0 < t, and when t0 >  > t it remains constant: R./ D R.t0 /;

t0 >  > t:

(8.28)

Such process R./ is called a process with the constant extension (Fig. 8.1). In addition, consider a static process 

R./ D R.t0 / D const;

0 6  6 t:

(8.29)

Then the following theorem can be formulated. Theorem 8.2. Let the functional f (8.1) be Fr´echet–differentiable, then its partial derivative @f and Fr´echet–derivative ıf , and also the function ƒ.t/ for any process R./ with the constant extension at fixed time t0 have the limits as t ! C1, which

Fig. 8.1 For Theorem 8.2

504

8 Viscoelastic Continua at Large Deformations 



coincide with values of the derivatives @f , ı f and ƒ , respectively, in correspond

ing static processes R./: lim



t0

t



f .R.t/; Rt .// D ƒ  f .R.t0 /; Rt0 .//;

t !C1 D0

D0

t0    ˇ ˇ P D @f  @ f .R.t0 /; Rt0 ./ˇ0/; lim @ f .R.t/; Rt ./ˇR.t// t

t !C1 D0

D0

t0    ˇ ˇ lim ı f .R.t/; Rt ./ˇRP t .// D ı f  ı f .R.t0 /; Rt0 ./ˇ0/: t

t !C1 D0

(8.30)

D0

In simple words, for a process R./ such that starting from some time t0 the process reaches a constant level, in a certain time interval a viscoelastic continuum forgets the process R./ up to the time t0 , because the continuum response, expressed by the functionals f , @f and ıf , at sufficiently great values of t differs little from its response to a static process. H Consider the process 

e R./ D R./  R./;

0 6  6 t:

(8.31)

Since e R./ 0

at t0 >  > t;

(8.32)

so we have e kR

k2t

e2

Zt

e2

Zt

R .t  / ./ d  D

D R .t/ C

2

0

Zt0 D

e 2 ./ 2 .t  / d  R

0

e 2 ./ 2 .t  / d  6 c R

Zt  2 ./ d :

(8.33)

t t0

0

Due to the property (8.4) and monotone decreasing the function ./, we find that e kt ! 0 at t ! C1. kR Since a Fr´echet–differentiable functional f is continuous as well as @f and ıf , the condition (8.12) yields the inequality t





t





e e t .//  f .R.t/; Rt .// kt < "I Rt ./ C R k f .R.t/ C R.t/; D0

(8.34)

D0

this means that the first limit of (8.30) exists. In a similar way, we can prove that the second and third limits of (8.30) exist too. N

8.1 Viscoelastic Continua of the Integral Type

505

8.1.5 Models An of Viscoelastic Continua For models An of viscoelastic continua, the free energy is a functional in the form (8.1), and as reactive variables one should choose the set (4.148): D

.n/

t

!

.n/ t

t

C .t/; .t/; C ./;  ./ :

(8.35)

D0

According to the rule (8.24) of differentiation of a functional, we obtain the expression for the total derivative of with respect to t: .n/

d dC @ @ d  D C Cı : .n/ dt dt @ dt @C

(8.36)

Substituting this expression into PTI (4.121) and collecting like terms, we get 0

.n/

1

.n/ TC B@ @ .n/  A   d C C  @C



    @ w C  d C Cı dt D 0: @ 

.n/

(8.37)

.n/

When the prehistories C t ,  t and the current values C .t/ and .t/ are fixed, the .n/

increments d C , d and dt can vary arbitrarily, therefore the identity (8.37) holds when and only when the coefficients of these increments vanish. As a result, we obtain the equation system 8 .n/ .n/ .n/ t .n/ ˆ ˆ ˆ T D .@ =@ C / D F . C .t/; .t/; C t ./;  t .//; < D0 (8.38)  D @ =@; ˆ ˆ ˆ :  w D ı ; that together with (8.35) is a system of constitutive equations for models An of viscoelastic continua. Just as for ideal continua, for viscoelastic materials it is sufficient to give only the free energy functional (8.35), then the remaining relations are determined by its differentiation according to formulae (8.38). Notice that although relations (8.38) are formally similar to the corresponding relations (4.168) for models An of ideal continua, they essentially differ by the fact .n/

that in (8.38) there is a functional dependence on C and . Moreover, viscoelastic continua are dissipative: for them the dissipation function w is not identically zero.

506

8 Viscoelastic Continua at Large Deformations

8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua According to the principle of material symmetry (Axiom 14), for each viscoelastic b Just as in continuum there exists an undistorted reference configuration K. ı

Sect. 4.7.4, for simplicity, let the reference configuration K be undistorted. Then ı

ı

for K there is a subgroup G s  U of the unimodular group U such that for each ı

ı



transformation tensor H 2 G s (H W K ! K) the constitutive equations (8.38) writı



ten for K are transformed during the passage to K as follows: 8.n/ .n/ .n/ t .n/ ˆ   t t ˆ T D F . C ; ; C ;  / D .@ =@ C  /; ˆ ˆ D0 ˆ ˆ ˆ .n/ t .n/ < . C  ; ; C t  ;  t /   ; D D0 ˆ ˆ ˆ D @  =@; ˆ ˆ ˆ ˆ ı :   .w / D .ı / ; 8H 2 G s :

(8.39)

.n/ .n/

For viscoelastic solids, due to H -indifference of all the tensors T , C .t/ and .n/

.n/

C t ./ D C.t  / (see Sect. 4.7.4), relations (8.39) take the forms 

.n/

t



.n/

t



.n/





.n/



QT  F . C ; ; C t ;  t /  Q D F .QT  C  Q; ; QT  C t  Q;  t /; D0

t

D0

.n/

D0



t

. C ; / D

.n/





.n/





(8.40a)

ı

.QT  C  Q; ; QT  C t  Q;  t / 8Q 2 G s ;

(8.40b)

D0

ı

Dı



:

(8.40c)

Theorem 8.3. The principle of material symmetry in the form (8.40) holds for models An of viscoelastic solids if and only if the condition (8.40b) for is satisfied. H It is evident that the condition (8.40b) is necessary. Prove that this condition is sufficient. Let the condition (8.40b) be satisfied. Then, since the functional F is a tensor function being the derivative of with respect to .n/

C .t/, so with the help of the method used in the proof of Theorem 4.23 we can prove that the condition (8.40b) yields (8.40a). To prove Eq. (8.40c) we should use 

and rewrite it in K:

formula (8.36) for ı

.n/

ı



d  1 .n/ d C  @  d  T   : D dt  dt @ dt

(8.41)

8.1 Viscoelastic Continua of the Integral Type

Since

D



507

ı





and the passage from K to K is independent of t, so d =dt D d .n/

.n/

.n/ 

=dt.

.n/ 

Due to H -indifference of the tensors T and C , we have the relation T  .d C =dt/D .n/

.n/

T   .d C =dt/, and also @  =@ D @ =@. Thus, the right-hand side of Eq. (8.41) coincides with ı , and hence .ı / D ı ; i.e. Eq. (8.40c) actually holds. N

8.1.7 General Representation of Functional of Free Energy in Models An Scalar functional

(8.35) satisfying the condition (8.40b) is called functionally ı

indifferent relative to the group G s . Let us find a general representation of such a ı

.n/

functional in terms of invariants of tensor C of the corresponding group G s . In Sect. 8.1.3 we have derived a general representation of a linear scalar functional in the form (8.15d). Similarly to (8.15d), define a quadratic scalar functional as a double integral in the form Zt Zt 2

4

D 0

.n/

.n/

A.t; 1 ; 2 /     . C .1 / ˝ C.2 //.4321/ d 1 d 2 ;

(8.42)

0

where 4 A.t; 1 ; 2 / is a fixed fourth-order tensor called the core of the functional. We also define a n-fold scalar functional: Zt m

D

Zt :::

0

e m .t; 1 ; : : : ; m / d 1 : : : d m ;

(8.43)

0

where .n/

.n/

e m .t; 1 ; : : : ; m / D 2n A.t; 1 ; : : : ; m /  : : :  . C .1 /˝: : :˝ C .m //.m;m1;:::;2;1/ : „ƒ‚… 2m

(8.44)

Here 2n A.t; 1 ; : : : ; m / is the core of this functional (it is a fixed tensor of order .2n/ depending on m C 1 arguments). Theorem 8.4 (Stone–Weierstrass). Any continuous scalar functional (8.35) in space Ht can be uniformly approximated by n-fold scalar functionals (8.43): D

t

.n/

.n/ t

. C .t/; .t/; C ./;  .// D

D0

t

1 X

m;

(8.45)

mD1

where the equality means that the partial sum uniformly converges in the norm (8.8).

508

8 Viscoelastic Continua at Large Deformations

H A proof of the theorem for the space Ht can be found in [9]. N Let us consider the integrand n .t; 1 ; : : : ; n / of the n-fold functional (8.43). At any fixed set of values t; 1 ; : : : ; m , this expression is a scalar function (but not .n/

.n/

a functional!) of n tensor arguments C.i /  C i (i D 1; : : : ; m): em .t; 1 ; : : : ; m / D em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm /:

(8.46)

Here while values of t; 1 ; : : : ; m vary, the number and the form of tensor arguments of the functions remain unchanged. On substituting the representation (8.45) into the condition (8.40b) of functional indifference of , we find that the functions e m (8.44) at each fixed value t; 1 ; : : : ; m must satisfy the condition em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm / 







D em .t; 1 ; : : : ; m ; QT  C1  Q; : : : ; QT  Cm  Q/;

(8.47)

ı

i.e. they must be H -indifferent scalar functions relative to the group G s . Theorem 8.5. Every scalar function (8.46) of m tensor arguments C1 ; : : : ; Cm , ı

which is indifferent relative to a group G s , can be represented as a function of finite number z (z 6 6m) of simultaneous invariants J.s/ D J.s/ .C1 ; : : : ; Cm /;

 D 1; : : : ; z;

(8.48)

ı

relative to the group G s in the form em D e m .t; 1 ; : : : ; m ; J.s/ .C1 ; : : : ; Cm //:

(8.49)

H By analogy with simultaneous invariants of two tensors, which have been considered in Sect. 7.1.4, we now introduce simultaneous invariants of m tensors relative ı

to a group G s . The simultaneous invariants are scalar functions J (8.48), which are ı

H -indifferent relative to the group G s ; i.e. they satisfy the relations 









J.s/ .C1 ; : : : ; Cm / D J.s/ .QT  C1  Q; : : : ; QT  Cm  Q/ 8Q 2 Gs :

(8.50)

Their functional basis consists of z simultaneous invariants, where z is a finite number, which cannot exceed the total number of components of all the tensors, i.e. z 6 6m. Moreover, since J.s/ form a basis, any other H -indifferent scalar funcı

tion relative to the same group G s can be expressed in this basis. But the function e (8.40) is just such a function due to (8.47), therefore the relation (8.49) actually holds. N

8.1 Viscoelastic Continua of the Integral Type

509

Substitution of the expression (8.49) into (8.43) and then into (8.45) yields the following general representation of the continuous functional (8.35), which is funcı

tionally indifferent relative to a group G s : D

1 Z X mD1 0

Zt

t

:::

.n/

.n/

e m .t; 1 ; : : : ; m ; J . C .1 /; : : : ; C.m /// d 1 : : : d m :

0

(8.51)

In deriving this formula we have used representation (8.15c) of linear functionals with the help of ı-function. Let us perform now the inverse operation: segregate the ı-type constituent from the cores e m , that allows us to separate the instantaneous .n/

value of the tensor C .t/ from its prehistory. Formula (8.15d) can be generalized for functions of m C 1 arguments t; 1 ; : : : ; m as follows: m X

e m .t; 1 ; : : : ; m ; J.s/ / D

ı.t  1 / : : : ı.t  k /

kC1 ; : : : ; m ; J.s/ /:

mk .t;

kD0

(8.51a)

We assume that at k D m the argument kC1 D mC1 does not appear among .s/ arguments of the function mk : mm D mk .t; J /. On substituting (8.51a) into (8.51), we get D

1 X m Z X

Zt

t

.n/

::: „ƒ‚…

mD1 kD0 0

mk

mk 0

.n/

.n/

.n/

t; kC1 ; : : : ; m ; J.s/ C.t/; : : : ; C .t/; ƒ‚ … „ k

!!

C.kC1 /; : : : ; C.m /

d kC1 : : : d m :

(8.52)

Rearrange summands in the expression (8.52) and take into account that simulta.n/

neous invariants J.s/ of m tensors, among which there are k tensors C.t/, can always be expressed in terms of simultaneous invariants of .m  k C 1/ ten.n/

.n/

.n/

sors C .kC1 /; : : : C.m / and C.t/. Then we finally obtain the general form of the functional (8.35) .n/

D '0 .t; J.s/ . C .t/// C

1 Z X mD1 0

Zt

t

::: „ƒ‚… m

 .n/ 'm t; 1 ; : : : ; m ; J.s/ . C .t/;

0

 .n/ C.1 /; : : : ; C.m // d 1 : : : d m ;

.n/

(8.53)

510

8 Viscoelastic Continua at Large Deformations ı

which is functionally indifferent relative to the group G s . Here we have introduced the notation: '0 – the instantly elastic part and 'm – cores of the functional: 1 X

'0 D

mm .t;

J.s/ .C.t///;

mD1

'm D

1 X kD0

mCk;k .t;

.n/

.n/

1 ; : : : ; m ; J.s/ . C.t/ : : : C .t/; C.1 /; : : : ; C.m ///: ƒ‚ … „ k

.s/

(8.54)

.n/

.s/

.n/

Simultaneous invariants J . C .t// of one tensor are simply invariants I . C.t// ı

of the tensor relative to the same group G s . The expression (8.53) is the desired general representation of the functional (8.35) for models An .

8.1.8 Model An of Stable Viscoelastic Continua Definition 8.4. One can say that this is the model An of a stable viscoelastic continuum, if the functional of the model is invariant relative to a shift of the pro.n/

cess of deforming and heating in time; i.e. if there are two processes . C ./; .// .n/

C ./; e .//, 0 6  6 t1 , such that they are different from each other only by a and e shift in time: 8 < .n/ .n/ e e . C ./; .// D . C .  t0 /; .  t0 //; t0 <  6 t1 ; (8.55) :.0;  /; 0 6  6 t0 I 0 then the corresponding values of the functionals and e are different only by a shift in time as well (Fig. 8.2): ( .t  t0 /; t0 < t 6 t1 ; eD (8.56) .0/; 0 6 t 6 t0 : Remark. Since the functional

for stable continua is invariant relative to a shift .n/

in time, its partial derivatives with respect to C .t/ and .t/ and also the Fr´echet– derivative ı have this property; hence the constitutive equations (8.38) are invariant relative to a shift in time too. Due to the property of invariance, stable continua are also called non-aging, and their constitutive equations do not change with time themselves when there are no deformations and variations of temperature. t u Let us consider two important models of stable continua.

8.1 Viscoelastic Continua of the Integral Type

511

Fig. 8.2 For the definition of a stable viscoelastic continuum

8.1.9 Model An of a Viscoelastic Continuum with Difference Cores One can say that this is the model An of a viscoelastic continuum with difference cores, if in the general representation of the functional (8.53) there is no explicit dependence of cores 'm on the times t and i , but there is a dependence only on their difference t  i or on the temperatures .t/ and .i /: .n/

.n/

.n/

'm D 'm .t1 ; : : : ; tm ; .t/; .1 /; : : : ; .m /; J.s/ . C .t/; C.1 /; : : : ; C.m ///; (8.57) and '0 does not depend explicitly on temperature: .n/

'0 D '0 .I.s/ . C .t//; .t//:

(8.58)

Functions 'm are assumed to satisfy the following conditions of normalization and symmetry with respect to any permutations of the first m arguments: 'm .t  1 ; : : : ; t  m ; 0 ; : : : ; 0 ; 0; : : : ; 0/ D 0;

(8.59)

'm .y1 ; : : : ; yn ; : : : ; yl ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ / D 'm .y1 ; : : : ; yl ; : : : ; yn ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ /; where 0 D .0/, n D .n /, yn D t  n , and 1 6 n; l 6 m. Simultaneous invariants J.s/ can always be chosen to satisfy the normalization conditions J.s/ .0; : : : ; 0/ D 0;

 D 1; : : : ; z:

(8.60)

512

8 Viscoelastic Continua at Large Deformations

For the model with difference cores, the functional (8.53) has the form .t/ D

.n/ '0 .I.s/ . C .t//;

1 Z X

.t// C

Zt

t

:::

mD1 0

'm d 1 : : : d m ;

(8.61)

0

where 'm are determined by formula (8.57). Theorem 8.6. The model An of a viscoelastic continuum with difference cores is stable. .n/

H Let the first process be of the form C ./, 0 6  6 t, then the corresponding .n/

C ./ when  6 t0 functional .t/ has the form (8.61). Since the second process e is identically zero, so, due to the normalization conditions (8.59) and (8.60), we have 'm  0 when 0 6 i 6 t0 (i D 1; : : : ; m); therefore, the lower limits of the integrals in the expression (8.61) for functional e.t/ can be determined as t0 :   .n/ e e.t/ D '0 I.s/ . e C .t//;  .t/ C

1 Z X mD1 t

Zt

t

:::

 ' m t  1 ; : : : ; t   m ; e .t/; e .1 /; : : : ; e .m /;

t0

0

.n/ .n/ .n/  J.s/ . e C .t/; e C.1 /; : : : ; e C.m // d 1 : : : d m : (8.62) .n/

.n/

C .i / by e C.i  t0 / when i > t0 and then substituting i  t0 D e  i , we Replacing e obtain (when t > t0 ) .n/

e.t/ D '0 .I.s/ . C .t  t0 //; .t  t0 // tZt0 1 tZt0  X ::: 'm t  t0  e  1 ; : : : ; t  t0  e  m ; .t  t0 /; C mD1 0

0

.n/  .n/ .n/ .e  1 /; : : : ; .e C .t  t0 /; C .e  m /; J.s/ . e  1 /; : : : ; C.e  m // de  1 : : : de  m:

(8.63) Comparing (8.61) and (8.63), we verify that e.t/ and the time-shift t ! .t  t0 /, because at t D t0 e.t0 / D '0 .0; .0// D i.e. the relation (8.56) holds. N

.0/;

.t/ are connected only by

8.1 Viscoelastic Continua of the Integral Type

513

On substituting the functional (8.61) into (8.38), we get the general form of constitutive equations for stable continua: .n/

T D

z X  D1

.s/ @'0 @I

@I.s/  D

w D '10 C 

.n/

C

1 Zt X

1 Zt X

@'0 C @ mD1

:::

0

:::

0

Zt

.s/ @'m @J

@J.s/

.n/

! d 1 : : : d m ;

@ C .t/

@'m d 1 : : : d  m ; @.t/

0

Zt 

Zt

mD1 0

:::

mD1 0

@ C .t/

1 X

Zt

@'m 0 C 'mC1 @t

 d 1 : : : d m :

(8.64)

0

Here the cores 'm are functions in the form (8.57), and @'m =@t is the partial derivative of the function when its first arguments .t  1 /; : : : ; .m / vary and the arguments J.s/ are fixed (i.e. there is no differentiation with respect to J.s/ ). We have introduced the following notation for a value of the function 'mC1 (8.57) at mC1 D t:  0 D 'mC1 t  1 ; : : : ; t  m ; 0; .t/; .1 /; : : : ; .m /; .t/; 'mC1  .n/ .n/ .n/ .n/ J.s/ . C .t/; C.1 /; : : : ; C.m //; C .t/ : In deriving the expression for the dissipation function we have used the conditions (8.59).

8.1.10 Model An of a Thermoviscoelastic Continuum Let us consider the most widely used method to take the dependence of cores 'm (8.57) on temperature  into account. For the model An of a thermoviscoelastic continuum with difference cores, temperature  appears in simultaneous invariants J.s/ , i.e. .n/

.n/

.n/

'm D 'm t  1 ; : : : ; t  m ; J.s/ C  .t/; C  .1 /; : : : ; C  .m / where .n/

.n/

ı

C  ./ D C./  "./;

Z./ "./ D ˛.e /d e : ı

0

!! ; (8.65)

(8.66)

514

8 Viscoelastic Continua at Large Deformations ı

The tensor " is called the tensor of heat deformation, and ˛ – the tensor of heat expansion. Both the tensors are symmetric and H -indifferent relative to a considered ı

group G s :





QT  ˛  Q D ˛



ı

8Q 2 G s ;

(8.67) ı

.n/

therefore, the functions J.s/ of C  are also H -indifferent relative to the group G s . Taking the dependence of constitutive equations upon temperature as the difference between the deformation tensor and the heat deformation tensor (8.66) is called the Duhamel–Neumann model. In a similar way, the Duhamel–Neumann model describes the dependence of the function '0 on temperature: ! .n/

'0 D '0 .I.s/ C  .t/ ; .t//:

(8.68)

.n/

Since @ C  =@ D ˛, the derivatives with respect to .t/ in (8.64) for this model have the form z .s/ X @0 '0 @'0 @'0 @I ˛C D ; @ @I .n/ @.t/  D1 @C

z .s/ X @'m @'m @J   ˛..t//; D @.t/ @J .n/  D1 @ C  .t/ (8.68a)

where @0 =@ means the derivative with respect to the second argument in formula (8.68). .n/

.n/

.n/

.n/

Taking into account that @I.s/ =@ C  D @I.s/ =@ C and @J.s/ =@ C  D @J.s/ =@ C and substituting (8.68a) into (8.64), we obtain the expression for the specific entropy D

.n/ @0 '0 1 C ˛   T: @.t/ 

(8.69)

8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum One can say that this is the model An of a thermorheologically simple viscoelastic continuum, if the cores 'm (8.57) in constitutive equations (8.53) and (8.64) depend on temperature  in the functional way through the so-called reduced time:  .n/ .n/ 0 ; J.s/ . C .t/; C.1 /; : : : ; 'm D 'm t 0  10 ; : : : ; t 0  m  C.m // a ..1 // : : : a ..m //;

.n/

(8.70)

8.1 Viscoelastic Continua of the Integral Type

515

where 0

Zt

t D

Zi

i0

a ..e // de ;

D

0

a ..e // de 

(8.71)

0

is the reduced time being a functional of the function a ./ called the function of the temperature-time shift. The functions 'm (8.70) and a satisfy the normalization conditions     0 'm 0; : : : ; 0; J.s/ D 0; 'm t 0  10 ; : : : ; t 0  m ; 0 D 0; a .0 / D 1: (8.72) .n/

If the process C ./ is considered with respect to the reduced time 1 0  Z .n/ e  A D C ./; C. 0 / D e C @ a de

.n/

.n/

0

then, since d i0 D a ..i // d i , the functional (8.53) with the core (8.70) can be written with respect to the reduced time 0

.t / D '0

.n/ I.s/ . C .t 0 //;

!

t0

1 Z X

0

.t / C

Zt 0 :::

mD1 0 .n/

.n/

.n/

0 // J.s/ . C .t 0 /; C.10 /; : : : ; C.m

0 'm t 0  10 ; : : : ; t 0  m ;

0

!

0 d 10 : : : d m :

(8.73)

Substituting the functional (8.73) into (8.38) and using the differentiation rule (8.27), we obtain constitutive equations for a thermorheologically simple viscoelastic continuum 1 0 Zt 0 1 Zt .s/ X @I.s/ @'m @J @' 0 0 0 @ T D ::: C d 1 : : : d m A ; .s/ .s/ .n/ .n/ @I @J 0    D1 mD1 0 @C @ C .t / 0 z X

.n/

0

.n/

 D .@0 '0 =@/ C .1=/˛   T ; 

w D

a '10

C a

0 1 Zt X

mD1 0

Zt 0  ::: 0

Here we have used that @=@t D a .@=@t 0 /.

 @'m 0 0 C 'mC1 d 10 : : : d m : @t 0

(8.74)

516

8 Viscoelastic Continua at Large Deformations

Notice that with the help of (8.38) and (8.74) the dissipation function w can be represented in another equivalent form .n/

w D T  

 0  .n/ d @ '0 d .n/ d C  ˛ T ; C  dt dt @ dt

(8.75)

which proves to be useful for cyclic loading. Theorem 8.7. A thermorheologically simple continuum is stable. H The reduced time (8.71) for the shifted process of heating e ./ D .  t0 / with use of the normalization condition (8.72) can be represented in the form 0

Zt0

t D

Zt a d  C

0

Zt   e a ./ d  D t0 C a ..  t0 // d 

t0

t0 tZt0

D t0 C

a ..e  // de ; 0

0

t Z 0

 D t0 C

a ..e  // de ;

(8.76)

0

when t0 <  < t. The further proof is the same as the one in Theorem 8.6 (see Exercise 8.1.1). N

Exercises for 8.1 8.1.1. Complete the proof of Theorem 8.7. 8.1.2. Using Definitions 8.2 and 8.3, prove that if a functional is Fr´echet– differentiable, then it is continuous.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua 8.2.1 Principal Models An of Viscoelastic Continua Constitutive equations containing multiple integrals of the type (8.53), (8.61) or (8.73) are very awkward, and their application in practice is considerably difficult. Therefore, special models of viscoelastic materials, in which one may retain a finite number of integrals, are widely used.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

517

For the principal model An of a thermoviscoelastic continuum with difference cores, the sum (8.61) contains only one integral .m D 1/, i.e. in this model has the form D

.n/ '0 .I.s/ . C  /;

Zt

.n/

.n/

'1 .t  ; J.s/ . C  .t/; C  .// d :

/ 

(8.77)

0

Here '0 and '1 are functions of the arguments indicated, and the function '1 is chosen with the negative sign that can always be done by simple renaming the functions. Constitutive equations for the continuum considered have the form (8.64), where m should be assumed to be equal to 1: .n/

T D

z  X  D1

'0 J.s/ C

Zt 

 '1 J.s/ d  : C

(8.78)

0

Here we have denoted the partial derivatives of '0 and '1 .n/

'0 .J˛.s/ . C  .t/// D .@'0 =@I.s/ /; '1 .t  ;

.n/ .n/ J˛.s/ . C  .t/; C  .///

D

 D 1; : : : ; r;

.@'1 =@J.s/ /;

(8.79)

 D 1; : : : ; z; .n/

and also the partial derivative tensors of J.s/ with respect to C  .t/: .n/

.n/

.s/ .s/ J.s/ C D @J =@ C  .t/ D @J =@ C .t/; .s/

.n/

 D 1; : : : ; z:

(8.80)

.n/

The simultaneous invariants J . C  .t/; C  .// can be chosen so that the first r .n/

ı

ones form a functional basis of invariants I.s/ . C  .t// in the same group G s . We will assume below that J.s/ are ordered in such a way; then the following equations hold: .s/

.n/

J C D @I.s/ =@ C  .t/;  D 1; : : : ; rI

'0  0;  D r C 1; : : : ; z:

(8.81)

These equations have been used for deriving the relation (8.78). According to (8.64) and (8.69), the dissipation function w and the specific entropy  for the principal models An have the forms .n/

.n/

w D '1 .0; J.s/ . C  .t/; C  .t//C

Zt

.n/ .n/ @ '1 .t  ; J.s/ . C  .t/; C  .// d  > 0; @t

0

D

0

.n/ 1 @ '0 C ˛   T: @ 

(8.82)

518

8 Viscoelastic Continua at Large Deformations

Here @=@t is the partial derivative of '1 with respect to the first argument, and @0 =@ is the partial derivative of '0 with respect to the second argument. Simultaneous invariants J.s/ of two tensors can be written by analogy with the ones for continua of the differential type (see Sect. 7.1.4).

8.2.2 Principal Model An of an Isotropic Thermoviscoelastic Continuum For the principal model An of an isotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants J.I / consists of 9 invariants, which can be chosen as follows (see (7.35)): .n/

.n/

.I / D I˛ . C  .//; ˛ D 1; 2; 3; J˛.I / D I˛ . C  .t//; J˛C3 .I /

J7

.n/

.n/

D C  ./   C  .t/;

.I /

.n/

.n/

.I /

J8 D C 2 ./   C  .t/; r D 3 and z D 9:

J9

(8.83)

.n/

.n/

D C  ./   C 2 .t/;

The derivative tensors of these invariants are calculated by the formulae (see [12]) .I / J1C D E;

.n/

.I /

.n/

J2C D EI1 . C  .t//  C  .t/; .I /

.I /

.n/

.n/

.I /

J C3;C D 0;  D 1; 2; 3I

.I /

.n/

.n/

J8C D C 2 ./;

J7C D C  ./; .n/

.n/

.n/

.I /

J3C D C 2 .t/  I1 C  .t/ C EI2 ;

.n/

J9C D C  ./  C  .t/ C C  .t/  C  ./:

(8.84)

Substituting these expressions into (8.77) and collecting terms with the same tensor powers, we obtain constitutive equations for the principal model An of an isotropic thermoviscoelastic continuum: .n/

.n/

.n/

T D 'M 1 E C 'M 2 C  C 'M3 C 2 :

(8.85)

Here we have denoted the functionals Zt 'M1  '01 C '02 I1 .t/ C '03 I2 .t/ 

.'11 C '12 I1 .t/ C '13 I2 .t// d ; 0

.n/

.n/

Zt

.n/

.n/

'M2 C   .'02 C '03 I1 .t// C  .t/  ..'12 C '13 I1 .t// C  .t/  '17 C  .// d ; 0

(8.86)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .n/

519

.n/

'M3 C 2  '03 C 2 .t/ Z t  .n/  .n/ .n/ .n/ .n/ .n/ '13 C 2 .t/C'18 C 2 ./C'19 . C  .t/  C  ./C C  ./  C  .t// d :  0

Relation (8.85) is formally similar to the corresponding relation (4.322) for an isotropic elastic continuum, but in (8.85) 'M1 , 'M2 and 'M3 are no longer functions .n/

of invariants of the tensor C ; they are functionals in the form (8.86).

8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum For the principal model An of a transversely isotropic (relative to the group T3 ) thermoviscoelastic continuum with difference cores, the functional basis of simul.3/ taneous invariants J consists of 11 invariants, which can be chosen as follows (see (7.36)): .n/  .n/ .3/ J.3/ D I.3/ C  .t/ ;  D 1; : : : ; 5I J5C D I.3/ . C  .//;  D 1; : : : ; 4I   .n/ .n/ .3/ D ..E b c23 /  C  .t//   b c23  C  ./ ; J10 .n/

.n/

.3/ .3/ J11 D C  .t/   C  ./  2J10  J2.3/ J7.3/ ;

r D 5;

z D 11:

(8.87) Here the invariants I.3/ are determined by formulae (4.297). The partial derivatives J.3/ C of these invariants have the forms (see [12]) .n/ 1 .O1 ˝ O1 C O2 ˝ O2 /   C  .t/; 2 .n/ 1 D 24 O3   C  .t/; 4 O3    .O1 ˝ O1 C O2 ˝ O2 / b c23 ˝b c23 ; 2

.3/ .3/ .3/ D E b c23 ; J2C Db c23 ; J3C D J1C .3/ J4C

.3/

.n/

.n/

.3/

.3/

.3/

.3/

J5C D C 2 .t/  I1 C  .t/ C EI2 ; J6C D J7C D J8C D J9C D 0; .3/ J10C D

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 /   C  ./; 4

.n/

.3/ J11C D 4 O3   C  ./:

(8.88)

Substituting these expressions into (8.77) and rearranging the summands, we obtain constitutive equations for the principal model An of a transversely isotropic thermoviscoelastic continuum: .n/

.n/

.n/

.n/

c23 C .O1 ˝ O1 C O2 ˝ O2 /   'M3 C  C 'M4 C  C 'M5 C 2 : (8.89) T D 'M1 E C 'M 2b

520

8 Viscoelastic Continua at Large Deformations

Here we have denoted the functionals Zt 'M 1  '01 C '05 I2 

.'11 C '15 I2 .t// d ; 0

'M 2  '02  '01 

2'04 I2.3/

Zt 

.'12  '11  2'14 I2.3/ .t/  2'1;11 I2.3/ .// d ;

0 .n/

.n/ 1 1 'M 3 C   .'03 2'04 / C   2 2

!  .n/ .n/ Zt  '1;10 '13 '14 C  .t/C '1;11 C  ./ d ; 2 2 0

Zt   .n/ .n/ .2'14  '15 I1 .t// C  .t/ C '1;11 C  ./ d ; 'M4 C   .2'04  '05 I1 / C   .n/

.n/

0 .n/

'M5 C 2 

!

Zt

.n/

'15 d  C 2 .t/:

'05 

(8.90)

0

8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum For the principal model An of an orthotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants consists of 12 invariants, which can be chosen as follows (see (7.37)): .n/

J.O/ D I.O/ . C  .t//;  D 1; : : : ; 6I

.n/

.O/

J C6 D I.O/ . C  .//;  D 1; 2; 3; 6I

        .n/ .n/ .n/ .n/ .O/ .O/ c22  C  .t/   b c23  C  ./ ; J11 D b c21  C  .t/   b c23  C  ./ ; J10 D b r D 6;

z D 12:

(8.91) This set should be complemented by two more invariants (being dependent) in order to obtain relations symmetric with respect to the vectors b c2˛ : .n/

.n/

.O/ J13 D .b c21  C  .t//   .b c22  C  .//; .O/ J14

D

.n/ I7.O/ . C  .t//

.n/

(8.92) .n/

D .b c21  C  .t//   .b c22  C  .t//:

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

521

The partial derivative tensors of these invariants have the forms (see [12]) J.O/ c2 ;  D 1; 2; 3I C Db .n/

.O/

.n/ 1 .O ˝ O /   C  .t/;  D 1; 2I 2

.O/

J C3;C D .n/

.O/

J6C D 3 6 Om     C  .t/ ˝ C  .t/; .O/ J10C .O/ J13C

.O/

J C6;C D J12C D 0;  D 1; 2; 3I

.n/ 1 D .O1 ˝ O1 /   C  ./; 4 .n/ 1 D .O3 ˝ O3 /   C  ./; 4

.n/ 1 D .O2 ˝ O2 /   C  ./; 4 .n/ 1 .O/ J14C D .O3 ˝ O3 /   C  .t/; 2

(8.93)

.O/ J11C

where the tensor 6 Om is determined by formula (4.316). Substituting these expressions into (8.77) and grouping like terms, we obtain constitutive equations for the principal model An of an orthotropic thermoviscoelastic continuum: .n/

T D

3 X

.n/

.n/

.n/

.'M b c2 C O ˝ O   'M3C C  / C 'M 7 6 Om     C  ˝ C  :

(8.94)

 D1

Here we have denoted the functionals Zt 'M   '0 

  '1 t  ; J˛.O/ d ;  D 1; 2; 3;

0

.n/

'M3C C  

Zt 

.n/ 1 1 '0; 3C C   2 2

 .n/ 1 '1; 9C t  ; J˛.O/ C  ./ 2

0



C'1; 3C t  ;

J˛.O/

Zt 'M 7  3'06  3

 .n/ C  .t/ d ;  D 1; 2;

  '16 t  ; J˛.O/ d ;

0 .n/

.n/ 1 1 'M 6 C   '0;14 C   2 2

Zt 

 .n/ 1 '1;13 t  ; J˛.O/ C  ./ 2

0

  .n/ C'1;14 t  ; J˛.O/ C  .t/ d :

(8.95)

522

8 Viscoelastic Continua at Large Deformations

8.2.5 Quadratic Models An of Thermoviscoelastic Continua For the quadratic model An of a thermoviscoelastic continuum with difference cores, we retain two integrals in the sum (8.61), i.e. m D 1; 2. A form of constitutive equaı

tions for specific symmetry groups G s becomes considerably more complicated, because there appear double integrals and we need to consider simultaneous invariants J.s/ of three tensors. Therefore, one usually considers the particular case of the quadratic model when m D 1 and 2, but simultaneous invariants J.s/ of only two tensors appear there just as in the principal model: .n/ I.s/ . C  .t//; 

D '0

!

Zt 

'1 .t 

; J.s/

.n/

C

.n/

.n/

d

!

'2 .t  1 ; t  2 ; J.s/ C  .1 /; C  .2 / 0

!

C  .t/; C  .1 /

0

Zt Zt

.n/

d 1 d  2 :

(8.96)

0

Here '0 , '1 and '2 are functions of the arguments indicated. Since the core '2 in .n/

.n/

this model is independent of C  .t/, so @'2 =@ C  .t/  0, and constitutive equations prove to be coincident with (8.78); and hence they coincide with (8.85), (8.89) and (8.94) too. The distinction between the principal and quadratic models consists only in the forms of the functional and the dissipation function w . Such a situation is typical for viscoelastic continua when distinct functionals of the free energy correspond to the same relations between the tensors of stresses and deformations. Comparing the principal model (8.77) with the quadratic one (8.96), we can also notice that the principal model has only one core '1 appearing also in relation (8.78), and the quadratic model has two cores '1 and '2 , one of which is not included in relation (8.78) between stresses and deformations. Thus, for the principal model we can restore the functional of the free energy by Eqs. (8.78) up to the entropy term @'0 =@ and the constant '0 .0; 0 / D 0 . Models of viscoelastic continua having such a property are called mechanically determinate. The quadratic model is not mechanically determinate: due to the presence of the core '2 we cannot restore the form of by relations (8.78) between stresses and deformations. Nevertheless, this model is also used in practice due to its quadratic structure being typical for thermodynamic potentials.

8.2.6 Linear Models An of Viscoelastic Continua The quadratic model An of a thermoviscoelastic continuum with difference cores (8.96), where the functions '0 , '1 and '2 depend linearly upon the quadratic

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .s/

523 .s/

invariants J and quadratically upon the linear invariants J , is called linear (cubic invariants do not occur in this model): '0 D 1 X

0C

r1 1 X

lˇ I.s/ .t/Iˇ.s/ .t/ C

ı

2

;ˇ D1

r1

'1 D

ı

 ;ˇ D1

.s/

qˇ .t  /I.s/ .t/Iˇ ./ C

'2 D

r1 1 X ı

2 ;ˇ D1 1 Cı 

2

1

r2 X

ı



X

l  I.s/ .t/;

 Dr1 C1

r2

ı

  Dr1 C1

q  .t  /J.s/ .t; /; (8.97)

pˇ .t  1 ; t  2 /I.s/ .1 /Iˇ.s/ .2 /

r2 X

p  .t  1 ; t  2 /J.s/ .1 ; 2 /:

 Dr1 C1

Here lˇ , l  are constants; qˇ .t  /, q  .t  / are one-instant cores (they are functions of one argument, being symmetric in  and ˇ); pˇ .t  1 ; t  2 / and p  .t  1 ; t  2 / are two-instant cores (they are functions of two variables, being symmetric in  and ˇ, and also in t 1 and t 2 ). We have introduced the notation ! ! .n/

I.s/ ./ D I.s/ C  ./ ;

J.s/ .1 ; 2 / D J.s/

.n/

.n/

C  .1 /; C  .2 / ;

(8.98)

ı

.n/

where r1 is the number of linear invariants I.s/ . C  .t// in group G s .r1 6 r/, and .s/ .r2  r1 / is the number of the quadratic simultaneous invariants J .1 ; 2 / in this group, where r2 6 z. Since not all simultaneous invariants contained in the full basis .n/

.n/

J . C  .1 /; C  .2 // appear in the expression for the functional in linear