Viscoelastic Structures Mechanics of Growth and Aging
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Viscoelastic Structures Mechanics of Growth and Aging
Aleksey D. Drozdov Institute for Industrial Mathematics Ben-Gurion University of the Negev Be'ersheba, Israel
ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto
This book is printed on acid free paper. Copyright © 1998 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com Academic Press Limited 24-28 Oval Road, London NW1 7DX, UK http//www.hbuk.co.uk/ap/
Library of Congress Cataloging-in-Publication Data Drozdov, Aleksey D. Viscoelastic structures : mechanics of growth and aging / Aleksey D. Drozdov p. cm. Includes bibliographical references and index. ISBN 0-12-222280-6 (alk. paper) 1. Viscoelastic materialsmMechanical properties. 2. Polymersm ViscositymMathematical models. 3. ViscoelasticitymMathematical models. TA418.2.D763 1998 620.1 ~06---dc21 97-29072 CIP Printed in the United States of America 97 98 99 00 01 EB 9 8 7
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To my wife Lena
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Preface The book is concerned with constitutive equations for thermoviscoelastic media (at finite and small strains), mathematical models for the description of manufacturing polymeric articles and polymeric composites, and optimal design of structural members and processes of their fabrication. Its objective is to familiarize the reader with new mathematical models for advanced materials and processes and to demonstrate the effects of material, structural, and technological parameters on the characteristic features of viscoelastic structures. In the recent years, the viscoelasticity theory has attracted essential attention owing to • The study of new physical phenomena (e.g., physical aging, double yield, and anomalous temperature dependencies in semicrystalline polymers). • New spheres of applications (e.g., flow of short-fiber suspensions, processing of fiber-resin composites, and filament winding.). • New mathematical techniques for the description of technological processes (e.g., hyperbolic-parabolic partial integro-differential equations). Problems in the mechanics of viscoelastic media are analyzed in publications scattered among a number of joumals, from purely mathematical to application-oriented. The book aims to present the state of the art in the mathematical models and methods for the analysis of polymeric structures and processes of their manufacturing. The book is directed to applied mathematicians and specialists in the mechanical engineering. However, it may be also of interest to specialists in polymer science, as well as to engineers exploring advanced technological processes. The first part of the book (Chapters 1 to 5) can be used as a supplementary material to a course on the advanced strength analysis for graduate students in mechanical engineering. The exposition is based on two concepts. The first is a concept of adaptive links, which allows the viscoelastic response in polymeric media to be modeled as the behavior of a transient network of elastic springs (links) that arise and break due to micro-Brownian motion. The idea of adaptive links goes back to the Tobolsky model of a temporary network suggested in 1940s, but it has been widely used to derive constitutive equations for glassy polymers only recently. vii
viii
Preface
The other idea is the successive use of three basic configurations (reference, natural, and actual) for the accretion processes in viscoelastic media. A model of continuous accretion based on this concept allows us to apply the same mathematical technique to describe such different processes as erection of dams, formation of selfgravitating planets, winding of composite pressure vessels, and growth of biological tissues. The book consists of two parts. The first part (Chapters 1 to 5) is focused on constitutive relations for the thermoviscoelastic behavior of polymers. Chapter 1 provides a brief introduction to the kinematics of viscoelastic media with finite strains. Chapter 2 deals with linear constitutive models at small strains. We discuss differential, fractional differential, and integral constitutive equations, and introduce the concept of adaptive links. A brief survey is presented of creep and relaxation kernels and their properties. We introduce thermodynamic potentials for viscoelastic media and formulate basic variational principles. A model for an aging viscoelastic medium is derived and verified by comparison with experimental data. Chapter 3 is concerned with nonlinear constitutive models with small strains. After a survey of nonlinear differential and integral models, two constitutive models for crosslinked and noncrosslinked polymers are derived based on the concept of adaptive links. To validate these models, results of numerical simulation are compared with experimental data. Nonlinear constitutive relations with finite strains are studied in Chapter 4. We provide a survey of differential and integral constitutive equations in finite viscoelasticity and suggest two new approaches to the design of constitutive models. The first is based on the theory of fractional differentiation. We propose a fractional differential operator, which maps an objective tensor function into an objective tensor, and introduce several analogs of standard differential constitutive equations with fractional derivatives. The other is based on the concept of adaptive links. Combining a model of adaptive links with the Lagrange variational principle, we derive constitutive relations that extend the B KZ-type equations. A constitutive model is determined by a series of functions Xm(t, T) that characterize the reformation process for adaptive links and by a series of strain energy densities Wm. We discuss the choice of strain energy densities and demonstrate fair agreement between the models' prediction and experimental data. Chapter 5 is concerned with linear integral equations for thermoviscoelastic media with small strains. We provide a brief surveys of constitutive relations that account for the effect of temperature on the viscoelastic response, introduce two models based on the concept of adaptive links, and compare results of numerical simulation with experimental data. Finally, we extend the models to nonisothermal loading and calculate residual stresses built up in a polymeric cylindrical pressure vessel cooling on a metal mandrel. The other part of the book (Chapters 6 to 8) deals with growing viscoelastic media, the mass of which increases under loading owing to material supplied to a
Preface
ix
part of the boundary (surface accretion) or to a part of the volume (volumetric growth). The theory of growing (accreted) bodies reflects such diverse processes as growth of biological tissues, dusting-up, freezing, sol = gel, solid-liquid and solid-solid phase transitions, crystal growth, polymerization of adhesives, snowfalls, winding of fibers and magnetic tapes, manufacturing of large engineering structures (e.g., dams and embankments), etc. We concentrate on mathematical models for these processes at finite and small strains, and on the mechanical phenomena observed in their analysis. Chapter 6 is concerned with accretion at large deformations. Linear and nonlinear applied problems with small strains are studied in Chapter 7. Chapter 8 deals with optimization problems for accreted viscoelastic media. We analyze optimal choice of the rate of manufacturing for polymeric articles, optimal design of growing beams, optimization of the preload distribution for wound pressure vessels and pipes, and optimal choice of the cooling rate for polymeric vessels solidified in molds. The exposition is characterized by the following features: • We successively employ the model of an aging viscoelastic material, and demonstrate the effect of aging (both physical and chemical) on stresses and displacements in growing viscoelastic media. • We choose such problems for the analysis as allow an explicit (or at least, semianalytical) solution to be derived. Numerical techniques are employed only to demonstrate the effects of material and structural parameters on the obtained solutions. • For any problem under consideration, some engineering recommendations are formulated that may be used to simplify applied problems and to reduce the number of parameters by neglecting those whose effects are not significant. Financial support by the Israel Ministry of Science (grant 9641-1-96) is gratefully acknowledged. Aleksey D. Drozdov
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Contents Kinematics of Continua 1.1 Basic Definitions and Formulas 1.1.1 Description of Motion 1.1.2 Tangent Vectors 1.1.3 The Nabla Operator 1.1.4 Deformation Gradient 1.1.5 Deformation Tensors and Strain Tensors 1.1.6 Stretch Tensors 1.1.7 Relative Deformation Tensors 1.1.8 Rigid Motion 1.1.9 Generalized Strain Tensors 1.1.10 Volume Deformation 1.1.11 Deformation of the Surface Element 1.1.12 Objective Tensors 1.1.13 Velocity Vector and Its Gradient 1.1.14 Corotational Derivatives 1.1.15 The Rivlin-Ericksen tensors Bibliography
1 1 1 3 5 6 7 10 11 12 13 14 15 16 18 20 22 23
Constitutive Models in Linear Viscoelasticity 2.1 Differential Constitutive Models 2.1.1 Differential Constitutive Models 2.1.2 Fractional Differential Models 2.2 Integral Constitutive Models 2.2.1 Boltzmann's Superposition Principle 2.2.2 Connections Between Creep and Relaxation Measures 2.2.3 A Model of Adaptive Links 2.2.4 Spectral Presentation of the Function X(t, ~') 2.2.5 Three-Dimensional Loading 2.3 Creep and Relaxation Kernels 2.3.1 Creep and Relaxation Kernels for Nonaging Media 2.3.2 Creep and Relaxation Kernels for Aging Media 2.3.3 Properties of Creep and Relaxation Measures
25 25 26 28 34 35 39 41 44 48 54 54 59 66
xi
xii
Contents
2.4
Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity 2.4.1 Thermodynamic Potentials of Aging Viscoelastic Media 2.4.2 Variational Principles in Viscoelasticity 2.4.3 Gibbs' Principle and the Second Law of Thermodynamics 2.4.4 Thermodynamic Inequalities in Linear Viscoelasticity 2.5 A Model of Adaptive Links for Aging Viscoelastic Media 2.5.1 A Model of Adaptive Links 2.5.2 Validation of the Model 2.5.3 Prediction of Stress-Strain Curves for Time-Varying Loads Bibliography
71 72 73 77 79 80 81 89 93 97
Nonlinear Constitutive Models with Small Strains 3.1 Nonlinear Differential Models 3.2 Nonlinear Integral Models 3.2.1 Uniaxial Loading 3.2.2 Three-Dimensional Loading 3.3 A Model for Crosslinked Polymers 3.3.1 A Model of Adaptive Links 3.3.2 Determination of Adjustable Parameters 3.3.3 Constitutive Equations for Three-Dimensional Loading 3.3.4 Correspondence Principles in Nonlinear Viscoelasticity 3.4 A Model for Non-Crosslinked Polymers 3.4.1 A Model of Adaptive Links 3.4.2 A Generalized Model of Adaptive Links 3.4.3 Validation of the Model Bibliography
107 107 117 117 126 130 131 136 140 143 145 146 149 153 161
Nonlinear Constitutive Models with Finite Strains 4.1 Differential Constitutive Models 4.1.1 The Rivlin-Ericksen Model 4.1.2 The Kelvin-Voigt Model 4.1.3 The Maxwell Model 4.1.4 The Standard Viscoelastic Solid 4.2 Fractional Differential Models 4.2.1 Fractional Differential Operators with Finite Strains 4.2.2 Fractional Differential Models 4.2.3 Uniaxial Extension of an Incompressible Bar 4.2.4 Radial Deformation of a Spherical Shell 4.2.5 Uniaxial Extension of a Compressible Bar 4.2.6 Simple Shear of a Compressible Medium 4.3 Integral Constitutive Models 4.3.1 Linear Constitutive Equations 4.3.2 Constitutive Equations in the Form of Taylor Series
171 171 172 173 174 176 177 178 180 182 188 195 198 203 203 205
Contents
xiii
4.3.3 BKZ-Type Constitutive Equations 4.3.4 Semilinear Constitutive Equations 4.4 A Model of Adaptive Links 4.4.1 A Model of Adaptive Links 4.4.2 The Lagrange Variational Principle 4.4.3 Thermodynamic Stability of a Viscoelastic Medium 4.4.4 Constitutive Equations for Incompressible Media 4.4.5 Extension of a Viscoelastic Bar 4.5 A Constitutive Model in Finite Viscoelasticity 4.5.1 A Model of Adaptive Links 4.5.2 Uniaxial Extension of a Viscoelastic Bar 4.5.3 Biaxial Extension of a Viscoelastic Sheet 4.5.4 Torsion of a Viscoelastic Cylinder Bibliography
206 210 212 212 213 219 221 223 226 227 231 236 248 255
Constitutive Relations for Thermoviscoelastic Media 5.1 Constitutive Models in Thermoviscoelasticity 5.1.1 Thermorheologically Simple Media 5.1.2 The Proportionality Hypothesis 5.1.3 The McCrum Model 5.2 A Model of Adaptive Links in Thermoviscoelasticity 5.2.1 Governing Equations 5.2.2 A Refined Model of Adaptive Links 5.3 Constitutive Models for the Nonisothermal Behavior 5.3.1 Constitutive Equations for Isothermal Loading 5.3.2 Constitutive Equations for Nonisothermal Loading 5.3.3 Three-Dimensional Loading 5.3.4 The Standard Thermoviscoelastic Solid 5.3.5 Cooling of a Cylindrical Pressure Vessel Bibliography
262 262 262 270 272 275 275 284 294 297 302 306 307 313 328
Accretion of Aging Viscoelastic Media with Finite Strains 6.1 Continuous Accretion of Aging Viscoelastic Media 6.1.1 A Model for Continuous Accretion 6.1.2 Continuous Accretion of a Viscoelastic Cylinder 6.1.3 Continuous Accretion of an Elastoplastic Bar 6.2 Winding of a Cylindrical Pressure Vessel 6.2.1 The Lame Problem for an Accreted Cylinder 6.3 Winding of a Composite Cylinder with Account for Resin Flow 6.3.1 Kinematics of Deformation 6.3.2 Governing Equations 6.3.3 Accretion on a Rigid Mandrel 6.3.4 Accretion with Small Strains
337 337 338 347 353 371 375 393 394 398 404 406
xiv
Contents
6.4
Volumetric Growth of a Viscoelastic Tissue 6.4.1 A Brief Historical Survey 6.4.2 Constitutive Equations 6.4.3 Compression of a Growing Bar 6.4.4 The Lame Problem for a Growing Cylinder Bibliography
413 414 417 423 430 436
Accretion of Viscoelastic Media with Small Strains 7.1 Accretion of a Viscoelastic Conic Pipe 7.1.1 Formulation of the Problem 7.1.2 Kinematics of Accretion 7.1.3 Constitutive Equations 7.1.4 Governing Equations (Model 1) 7.1.5 Governing Equations (Model 2) 7.1.6 Numerical Analysis 7.2 Accretion of a Viscoelastic Spherical Dome 7.2.1 Formulation of the Problem 7.2.2 Governing Equations 7.2.3 Determination of Preload 7.2.4 Displacements in an Accreted Dome 7.2.5 Numerical Analysis 7.3 Debonding of Accreted Viscoelastic Beams 7.3.1 Accretion of a Two-Layered Beam 7.3.2 Accretion of an Elastic Beam on a Nonlinear Winkler Foundation 7.4 Torsion of an Accreted Elastoplastic Cylinder 7.4.1 Formulation of the Problem 7.4.2 Stresses and Strains in a Growing Cylinder 7.4.3 Accretion of an Elastic Cylinder 7.4.4 An Elastoplastic Cylinder with One Plastic Region 7.4.5 An Elastoplastic Cylinder with Two Plastic Regions Bibliography
446 446 446 447 450 451 455 458 464 465 467 472 474 475 480 480
Optimization Problems for Growing Viscoelastic Media 8.1 An Optimal Rate of Accretion for Viscoelastic Solids 8.1.1 Torsion of an Accreted Viscoelastic Cylinder With Small Strains 8.1.2 Extension of an Accreted Elastic Bar with Finite Strains 8.2 Optimal Accretion of an Elastic Column 8.2.1 Formulation of the Problem and Governing Equations 8.2.2 Optimal Regime of Loading 8.2.3 Optimal Regime of Accretion 8.3 Preload Optimization for a Wound Cylindrical Pressure Vessel 8.3.1 Formulation of the Problem and Governing Equations
511 511
489 499 499 501 502 505 509 510
512 523 532 532 536 538 542 542
Contents
8.3.2 Winding of a Nonaging Cylindrical Pressure Vessel 8.3.3 Winding of an Aging Cylindrical Pressure Vessel 8.4 Optimal Design of Growing Beams 8.4.1 Formulation of the Problem and Governing Equations 8.4.2 Optimal Thickness of a Nonaging Elastic Beam 8.4.3 Optimal Thickness of an Aging Elastic Beam 8.5 Optimal Solidification of a Spherical Pressure Vessel 8.5.1 Formulation of the Problem 8.5.2 Temperature Distribution 8.5.3 Stresses and Displacements 8.5.4 Stresses in a Pressure Vessel after Cooling 8.5.5 Numerical Analysis Bibliography Index
xv 548 550 554 555 558 562 569 570 572 573 579 582 589 593
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Viscoelastic Structures Mechanics of Growth and Aging
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Chapter I
Kinematics of Continua This chapter is concerned with kinematic concepts in the nonlinear mechanics of continua. We discuss Eulerian and Lagrangian coordinate frames, derive expressions for tangent and dual vectors, and introduce operators of covariant differentiation in curvilinear coordinates. Explicit formulas are developed for the main strain and deformation tensors, as well as for the volume and surface elements in an arbitrary configuration. Finally, we introduce corotational derivatives of objective tensors and discuss their properties. A more detailed exposition of these issues with the use of direct tensor notation, can be found, e.g., in Drozdov (1996).
1.1
Basic Definitions and Formulas
1.1.1
Description of Motion
In the nonlinear mechanics of continua, two different kinds of coordinate frames are distinguished. The first is the Eulerian (spatial) coordinate frame, which is fixed and immobile in space. Points of a moving medium change their positions in space with respect to the Eulerian coordinates. As common practice, Cartesian coordinates {xl,x2, x3}, cylindrical coordinates {r, O,z}, and spherical coordinates {r, 0, th} are employed as Eulerian coordinates. For cylindrical coordinates r -- V / ( x l ) 2 -k- ( x 2 ) 2,
x2
0 -- tan -1 ~--i-' Z -" X 3,
and for spherical coordinates r = V/(xl) 2 + (x2) 2 + (x3) 2,
0 = tan-1 V/(xl)2 + (X2)2 X3
,
1 _x _2 4) = tan- x l .
We denote unit vectors of Cartesian coordinates a s ~'1, ~'2, and 6'3, unit vectors of cylindrical coordinates as ~r, ~0, and ~z, and unit vectors of spherical coordinates as
2
Chapter 1. Kinematics of Continua
~'r, 6'0 and ~6, respectively. The following formulas are fulfilled for the derivatives of the unit vectors for a cylindrical coordinate frame:
Or
Or
o~ e r
o36
Or
tg e. ch _
t96'r -- 0,
o~ e z
~(1)
-- 6'~b,
-- 6'r,
O~6'th -- 0,
Oz
O3(]) -- 0, 06'z -- 0
Oz
Oz
(1.1.1)
and for a spherical coordinate frame: Oe.r
_ O,
O~e'O -- O,
Or
00
°3e'dP -- O,
Or --
6'0,
04) - 06 sin 0,
00
Or --
--6'r,
00
--
0,
&h - 06 cos 0,
0~b
--(6'r sin 0 + ~0 cos 0).
(1.1.2) Lagrangian (material) coordinates ~ = {~1, ~2, ~3} provide the other kind of coordinate frames, which are frozen into a moving medium. Position of any point with respect to the Lagrangian coordinates remains unchanged in time, while the frame moves together with the medium. As common practice, Lagrangian coordinates coincide with Eulerian coordinates at the initial instant, when the motion starts. Position of a point M with respect to an immobile spatial coordinate frame is determined by its radius vector ?. Two radius vectors are distinguished: the initial ?0(~) and the current ?(t, ~), where t stands for time (see Figure 1.1.1). To set a motion with respect to a Lagrangian frame means to establish a law = ?(t, ?o)
(1.1.3)
for any point ~ and for any instant t. Introducing the displacement vector fi(t, ~), we write Eq. (1.1.3) as
?(t, ~) = ?0(~) + fi(t, ~).
(1.1.4)
Some generally accepted requirements are imposed on admissible displacement fields: (i) The map ? = ?(t, ?0) is twice continuously differentiable. (ii) The map ? = ?(t, ~0) is globally one-to-one and it preserves orientation. Condition (i) is introduced for convenience and simplicity of exposition, and it may be violated for the analysis of crack propagation and shock waves in deformable media.
1.1. Basic Definitions and Formulas
3
Actual configuration
Initial configuration
M
fi
M
A
~_-'
O
Figure 1.1.1: The radius vectors and the displacement vector.
The first part of restriction (ii) means that two distinct material points cannot occupy the same position simultaneously, which implies that the map ~(t, ?0) is globally invertible. This assertion excludes such phenomena as, e.g., collapse of a cavity and attachment of strips. The other part of this restriction means that orientation of any three noncoplanar vectors does not change.
1.1.2
T a n g e n t Vectors
Let ?0(~) and ?(t, ~) be the radius vectors of a point M with Lagrangian coordinates = {~i} in the initial and actual configurations. We fix the coordinates ~2 and ~3 and consider a line drawn by the radius vector, when only the coordinate ~1 changes. This line is called the coordinate line ~ 1. Similarly, the coordinate lines ~2 and ~3 are introduced as shown in Figure 1.1.2. The vectors 07 gi -- o~i
(1.1.5)
are linearly independent, tangent to the coordinate lines ~i, and they form a basis. Volume V of a parallelepiped constructed on the tangent vectors gi is calculated as V = gl "(,~2 X g3) - g2"(,~3 X ,~1) - g3"(,~l X g2),
(1.1.6)
where the dot denotes the inner product, and x stands for the vector product (see Figure 1.1.3).
4
Chapter 1. Kinematics of Continua
~3
g'3 g'2
... --
~2 .
"''-.....°. o ,,,
Figure 1.1.2: Tangent vectors for a Lagrangian coordinate frame.
The dual vectors ~i are orthogonal to the tangent vectors gi, ~i . ~j __ ~j,i
where 8ji are the Kronecker indices
/ ~J :
{1°
i=, i =/= j.
g3
M
Figure 1.1.3: The elementary volume.
(1.1.7)
1.1. Basic Definitions and Formulas
5
Vectors gi and ~i are connected by the formulas ~1 __ g2 X g3
V
~2 __ g3 X gl
'
V
gl = Vg 2 X ~3,
~3 __ gl X g2
'
g2 = Vg 3 X ~1,
V g3 = Vg 1 X ~2.
(1.1.8)
Any vector g/can be expanded in tangent vectors gi and in dual vectors ~i, gt = q'g,i
:
(1.1.9)
qig,',
where qi are covariant components and qi are contravariant components of g/. Summation is assumed with respect to repeating indices, which occupy alternately the upper and the lower position. It follows from Eqs. (1.1.7) and (1.1.9) that qi = ~ . ~i,
qi = q " gi.
(1.1.10)
Let us calculate the differential of the radius vector °~P d ~ i = g i d ~ i. d F -- --2-~;
(1.1.11)
Multiplying Eq. (1.1.11) by itself, we find the square of the arc element ds d s 2 = d? " d? = ~ i d ~ i " $jd(; j = (gi " g j ) d ~ i d ~ j = g i j d ~ i d ~ j,
(1.1.12)
where the quantities (1.1.13)
gij = gi " g j
are covariant components of the metric tensor. Contravariant components g'J of the metric tensor are elements of the matrix inverse to the metric matrix [gij]. For any integers i and j we have gtk• gkj = ~j.i
It can be shown that gij = ~ i . ~j,
~i _ g i j ~ j ,
gi = gijg, j.
(1.1.14)
Equations (1.1.14) imply that covariant and contravariant components of the metric tensor allow the indices of tangent vectors to be raised and lowered.
1.1.3
The Nabla Operator
We multiply Eq. (1.1.11) by ~J, use Eq. (1.1.7), and find that d~i = ~i . d r .
(1.1.15)
6
Chapter 1. Kinematics of Continua
Differentiation of a smooth scalar function f(~) with the use of Eq. (1.1.15) yields Of = ~i O f df = --~ d~ i --~ " dr.
(1.1.16)
The Hamilton operator (the nabla operator) is introduced according to the formula ~7 -- ~i O~
03~i.
(1.1.17)
Combining Eqs. (1.1.16) and (1.1.17), we obtain (1.1.18)
d f = fT f . d?.
Equations (1.1.7) and (1.1.17) imply that
0
o~i -- gi" ~r.
(1.1.19)
By analogy with Eq. (1.1.16), we find that for a smooth vector function g/(~) dO =
d~' = d? • ~i 0~/
~T
(1.1.20)
where the tensor ~r~/is called covariant derivative of the vector field ?/(~), and T stands for transpose.
1.1.4
D e f o r m a t i o n Gradient
We now differentiate Eq. (1.1.4) with respect to ~i and find that 07 gi -- o~i
07o Off Off o~i + ~ = goi + a~ i ,
(1.1.21)
where g0i and gi are tangent vectors in the initial and actual configurations. Equations (1.1.19) and (1.1.21) imply that gi -- g0i + g0i" V0 ~ -- g0i" (I + ~0 ~) = (I + ~r0uT)'g0i, goi : gi -- gi " ~ ~l : gi " ( I -- ~ ~l) : ( I -- v ~lT) " gi,
(1.1.22)
where i is the unit tensor. Denote by ~(~) and ~,i(t, ~) the dual vectors, and by ~'0r(~) and ~r?0(t, ~) the deformation gradients -
0?
VO ~ -- g , ~ - ~
-- ~tO~i,
~7~0 = ~i -¢~0 • i. ~ = ~t~O
(1.1.23)
1.1. Basic Definitions and Formulas
The tensors ~'o? and ~r ?o are "gradients" of the map ?(t, ?o), which characterize it in a small vicinity of any point. In particular, if a map ?(t, ?0) preserves orientation, then det ~'07 > 0.
(1.1.24)
Let us discuss properties of the deformation gradients. It follows from Eqs. (1.1.23) that ~'o?o = ~' ? = i
(1.1.25)
and
¢0 ~T -- ~i~io,
V E~ -= ~Oi~i.
(1.1.26)
Substitution of expressions (1.1.22) into Eqs. (1.1.23) yields
Vo~ = i + Vo~,
V ~o = i -
~ ~.
(1.1.27)
Multiplying Eqs. (1.1.23) and using Eq. (1.1.7), we find that
Vor" Vro = i. It follows from this equality that ~r0? = ~r ?o 1.
(1.1.28)
We multiply the first equality in Eq. (1.1.23) by ~'0i, the other equality by ~,~, and use Eq. (1.1.7). As a result, we obtain g0i" ~707 -- gi,
~rr0. g0 -- ~i.
(1.1.29)
It follows from Eqs. (1.1.17), (1.1.28), and (1.1.29) that ~7 = ~ i ~-~ O3 = ¢ r0" g0i ~¢9 = ~7r0" ¢0,
~70 = ~7~O 1 . ~7 = VOP . ~7. (1.1.30)
Let us consider a vector d Po = ~oid~ i in the initial configuration and its image d ? = ~id,~ i in the actual configuration. Equations (1.1.20) and (1.1.30) imply that dP = dPo" VoP = Vo~ T" d~o,
1.1.5
dPo = d ? . V~o = VPff" dP.
(1.1.31)
Deformation Tensors and Strain Tensors
Denote by
dso
and
ds
the arc elements in the initial and actual configurations d s 2 = d?o " d?o,
ds 2 = d? . d?.
(1.1.32)
Substitution of expressions (1.1.31) into the second formula (1.1.32) yields ds 2 = d? • d? = d?o"
~ro? • ~'0 ?T • d?o = d?o" ~" d?o,
(1.1.33)
Chapter 1. Kinematics of Continua where ~' = ~'o~ • ~7o?r
(1.1.34)
is the Cauchy deformation tensor. It follows from Eqs. (1.1.27) and (1.1.34) that = ~? + 2~o(fi) + V0fi" Vofir,
(1.1.35)
where 1
&o(~) - ~(Vo~ + Vo~ T)
(1.1.36)
is the first (Cauchy) infinitesimal strain tensor. To obtain a reciprocal deformation tensor, we substitute expressions (1.1.31 ) into the first formula in Eq. (1.1.32) and find that
ds~ = dPo" dPo = d ~ . ~r~o-~'?~. dP = dP" ~'o" dP,
(1.1.37)
where ~'o = V?o" V?~
(1.1.38)
is the Almansi deformation tensor. According to Eqs. (1.1.27) and (1.1.38), ~o = i - 2~(fi) + V ft. V fir,
(1.1.39)
where 1
~(fi) = ~(~,fi + ¢fir)
(1.1.40)
is the second (Swainger) infinitesimal strain tensor. It follows from Eqs. (1.1.33) and (1.1.37) that the Cauchy and Almansi deformation tensors indicate changes in the arc element for transition from the initial to actual configuration. Substitution of expressions (1.1.23) into Eqs. (1.1.34) and (1.1.38) yields -/-j
g, = g i j g o g o ,
,,
go
• ..
= goijg, tg, J
(1.1.41)
Multiplying the deformation gradients ~'o? and V ?o, we may construct four symmetrical tensors. The Finger deformation tensor is determined as P = Vo ?r" ~'o? = i + 2~o(fi) + Vofir " ~7ofi,
(1.1.42)
and the Piola deformation tensor equals F0 = ¢ ~ "
~'~0 = i -
2~(fi) + ~,fir. eft.
(1.1.43)
It follows from Eqs. (1.1.28), (1.1.34), (1.1.38), (1.1.42), and (1.1.43) that P = g o 1,
/~0 = ~ - 1 .
(1.1.44)
1.1. Basic D e f i n i t i o n s a n d F o r m u l a s
9
Substitution of expressions (1.1.23) into Eqs. (1.1.42) and (1.1.43) implies that p
ij-= go gigj,
" "
FO -- g'Jgoigoj.
It follows from Eqs. (1.1.34), (1.1.38), (1.1.42), and (1.1.43) that Ik(P) = Ik(~),
Ik(P0) = I~(~0),
(1.1.45)
where I~ (k = 1, 2, 3) stands for the principal invariant of a tensor. Other deformation tensors can be presented as functions of the Cauchy and Finger tensors. For example, the Hencky deformation tensor is defined as 1
/:/= ~ lnF
(1.1.46)
[see Fitzgerald (1980).] In general, to construct the tensor/t we should find the eigenvalues and eigenvectors of the tensor F, which requires cumbersome calculations. It follows from Eqs. (1.1.33) and (1.1.37) that ds 2 -
d s 2 = d ?o " ~ " d ro -
ds 2 -
ds 2
d ?o " i . d ~o =
2d?o" ¢~" d?o,
= d~ • i " d? - d? "g0" d~ = 2 d ? • A" d~,
(1.1.47)
C-- ~(g--I) : E0(U) + ~~70~" ~70uT
(1.1.48)
where
is the Cauchy strain tensor and ^
1
A = ~(i-
1
~o) = ~(fi) - ~ ' f i " ~fir
(1.1.49)
is the Almansi strain tensor. Substitution of expressions (1.1.41) into Eqs. (1.1.48) and (1.1.49) implies that 1
-i -j
= -~(gij -- goij)gogo,
^
1
•
A = -~(gij - goij)g, ig, J,
which means that the Cauchy and Almansi strain tensors have the same covariant components, but in different bases. Obviously, their contravariant and mixed components may differ from each other. Similar to Eqs. (1.1.48) and (1.1.49), we define the Finger strain tensor ^ EF
1
= ~(I3(g0)/2" -- I)
(1.1.50)
and the Piola strain tensor /~F0 = ~1( I^- I3(~)P0) .
(1.1.51)
10
Chapter 1. Kinematics of Continua
Several constitutive equations for viscoelastic media employ the so-called difference histories of strains [see, e.g., Coleman and Noll (1961).] The difference history of the Cauchy strain Cd(t, T) equals Cd(t, "r) = C ( t ) - C ( t - "r),
(1.1.52)
where t~(t) is the Cauchy strain tensor. It follows from Eqs. (1.1.47) and (1.1.52) that the difference history of the Cauchy strain characterizes changes in the arc element for transition from the actual configuration at instant t - ~-to the actual configuration at instant t ds2 ( t ) - ds2 ( t - ~-) = 2d?0-Cd(t, ~') " d ?o .
The terminology used in the nonlinear mechanics has not yet been fixed. The deformation gradient is also called the distortion tensor. The Cauchy strain tensor is also called the Cauchy-Green strain tensor and the Green strain tensor. The Cauchy deformation tensor is also called the left Cauchy tensor, whereas the Almansi deformation tensor is called the Green tensor, the right Cauchy tensor, and the Euler strain tensor.
1.1.6
Stretch Tensors
According to the polar decomposition theorem, any nonsingular tensor can be presented as a product of a symmetrical positive definite tensor and an orthogonal tensor. Applying this assertion to the deformation gradient V0?, we arrive at the left polar decomposition formula ~ro? = O'l" O,
(1.1.53)
where tit is a symmetrical positive definite left stretch tensor, C]~ = ~]l, and O is an orthogonal rotation tensor, 0 T = 0 -1. Substitution of expression (1.1.53) into Eq. (1.1.34) implies that = Ol " O " O - '
" Ol =
It follows from this equality that
~Jl = ~1/2.
(1.1.54)
Another important relation is derived by using the right polar decomposition of the deformation gradient Vo? = O. ~-/r,
(1.1.55)
where Clr is a symmetrical positive definite right stretch tensor, Of = Clr, and O is an orthogonal rotation tensor, O r = 0 -1. It follows from Eqs. (1.1.42) and (1.1.55)
1.1. Basic Definitions and Formulas
11
that ~)rr = p 1/2.
(1.1.56)
The eigenvalues Vl, v2, v3 of the stretch tensors UI and Or coincide. These eigenvalues are called principal stretches. Equation (1.1.56) implies that I1(/~) = v 2 + v 2 + v 2,
/2(/~) =
v2v 2 + v2v 2 + v2v 2,
I3(F)=
v 2 V~2 V32 .
(1.1.57) Other deformation tensors can be also expressed in terms of the left and right stretch tensors. For example, substituting expression (1.1.56) into Eq. (1.1.46), we obtain the formula for the Hencky deformation tensor /~ = In Ur.
1.1.7
(1.1.58)
Relative D e f o r m a t i o n Tensors
The deformation tensors describe transformations from the initial (at instant t = 0) to the actual (at the current instant t) configuration. For the analysis of the viscoelastic behavior, it is convenient to use relative deformation tensors, which characterize the entire history of deformations in the interval [0, t]. Let us consider transition from the actual configuration at instant ~"to the actual configuration at instant t -> ~-. The corresponding deformation gradients
fTr?(t) = ~7~?o" fTo?(t) = gi(T)~,i(t), (Ttr(~') = (Ttro" (7or('r)= gi(t)~oi(r)
(1.1.59)
are called relative deformation gradients. It follows from Eqs. (1.1.59) that for any 0_<,r_
(1.1.60)
Relative deformation tensors are introduced with the use of Eqs. (1.1.34), (1.1.38), (1.1.42), (1.1.43), and (1.1.59). For example, the Cauchy and Finger tensors are determined as follows: ~°(t, 1-) = ~'TP(t) • ~rTpT(t) = ~'¢P0 " ~(t) • ~'TPff,
P c ( t , r) = fTr?T(t) • fTr?(t) = fTO?r(t) " ~ - l ( r ) " ~'0?(t).
(1.1.61)
Similar formulas can also be obtained for the relative strains tensors, for example, for the relative Cauchy strain tensor
1
C'<>(t, r) = ~ [~<>(t, r) - i]
(1.1.62)
12
Chapter 1. Kinematics of Continua
and for the relative Almansi strain tensor
1
-1] .
A<>(t, ~-) = ~[i - (F<>(t, ~'))
(1.1.63)
It follows from Eqs. (1.1.61) that for any t -> 0, ~<>(t, t) = i,
F<>(t, t) = L
(1.1.64)
Equations (1.1.44) show that only two deformation tensors (e.g., the Cauchy and Finger tensors) are independent, and the other two tensors are inverse to them. Equations (1.1.61) imply that for any 0 -< ~" - t < ~, F°(t, 1-) = [~°(~-,t)]-l,
(1.1.65)
which means that only one relative deformation tensor is independent, and the others may be expressed in terms of it.
1.1.8
Rigid Motion
A motion is called rigid if for any instant t - 0 the distance between two arbitrary points ~1 and ~2 remains unchanged ]P(t, ~1) - ?(t, ~2)1 =
Ir0(~l)
-
P0(~2)l.
For any rigid motion, there are a constant vector k~, a vector function R* (t) and an orthogonal tensor function O(t) such that for any point ~ and for any instant t, ?(t, ~) = R*(t) + [?0(~) - k~]-0(t).
(1.1.66)
Differentiation of Eq. (1.1.66) with respect to ~i with the use of Eq. (1.1.5) yields gi(t) = goi " O(t).
Combining this equality with Eqs. (1.1.23) and (1.1.27), we obtain C7o~ = O,
v ~o = 0 T,
CTor~= 0 - i,
~ r~ = i -
O r.
Substitution of these expressions into Eqs. (1.1.34), (1.1.36), (1.1.39), (1.1.40), (1.1.42), (1.1.43), (1.1.48) to (1.1.51) implies that
~,=b = k = P o =i,
~ = ~ = b ~ =b~0 =o,
~0= ~ ( o + 0 T ) - - i # 0 ,
~=i--~(0+0T)#0.
(1.1.67)
It follows from Eqs. (1.1.67) that the strain tensors ~, 6",/~F, and/~Fo adequately describe deformation of continua, since they vanish for any rigid motion. The infinitesimal strain tensors ~0 and ~ do not vanish for rigid motions with nonzero rotations.
13
1.1. Basic Definitions and Formulas
1.1.9
G e n e r a l i z e d Strain Tensors
Four strain tensors A, ~7, EF, and EF0 are expressed in terms of the deformation tensors ~,, ~'0, F, and F0. It is natural to generalize this approach and to treat any admissible tensor function Z of the deformation tensors as a generalized measure of strains. Several generalized strain tensors are introduced by Seth (1964). Conditions of admissibility for generalized strain tensors are formulated by Hill (1968): 1. Generalized strain tensors L coincide for two motions which differ from each other by a rigid motion. 2. Generalized strain tensors Z vanish for the identical transformation from the initial to the actual configuration, L(0) = 0. 3. L is a tensor of the second rank. 4. Z is an isotropic invertible tensor function of either the Cauchy deformation tensor ~, or the Finger deformation tensor F. 5. The derivative of a generalized strain tensor Z with respect to the strain tensor = 0. vanishes in the initial configuration,
Lele=0
6. For infinitesimal strains, a generalized strain tensor Z coincides with the first and second infinitesimal strain tensors. ^
7. The eigenvalues of a generalized strain tensor L are positive, provided the corresponding principal stretches are greater than unity. Two generalized strain tensors are widely used in applications: 1. The Eulerian m-tensor of strains e(em) =
1 ( D i n ~ 2 ~), m
1
= ln~,, 2
m 4: 0, (1.1.68) m = O,
~
2. The Lagrangian m-tensor of strains E(Lm) =
1j_
p-m/2),
m4:0, (1.1.69)
~lnP,
m--O.
The following formulas for the fractional powers of the Cauchy and Finger tensors are valid: 3 ~2 _ (I1 -- v2)g + 13 Vk21 ~m/2 = ~ V~ 2 4 2 k=l Ulc -- ll v[c + 13 Vk 2 3 m p2 -- (I1 - v ~ ) F2+ "I 3 v E 2 i pm/2 = ~ Vi~ 2 4 2 k=l V~: -- Il V~ + 13 Vk 2
(1.1.70)
14
Chapter 1. Kinematics of Continua
where vk are the principal stretches and Ik are the principal invariants of the Cauchy and Finger deformation tensors [see Morman (1986)].
1.1.10
Volume Deformation
It follows from Eq. (1.1.41) that ,,
-i -j
kj-i -
g = gijgogo = gikgo gogo j,
which implies that det[gik] g I3(g') -- det ~ = det [gikg koj] -- det[gik] det [golj] -- det[g0 kj] _ go'
(1.1.71)
where go = det[g0 ik] and g = det[gik]. Volume V of a parallelepiped erected on the tangent vectors gi is calculated using the formula for the triple product of vectors and Eq. (1.1.6)
[glxl .
V = det |g2xl I_g3x 1
glx 2
glx3] (1.1.72)
g2x2 g2x3 , g3x 2
g3x 3
where gixJ are projections of the tangent vectors gi on Cartesian axes x j. Multiplying Eq. (1.1.72) by itself, we obtain
V2 = det
rax [g2x1
,ix] [glx r, xl2 , xl 3xl]
g2x 2
g2x 3
g2x 2
g3x 2
[.g3x~ g3x2 g3x3
I_glx 3 gzx 3
g3x 3
[gl" gl gl'g2 gl" g3] [gll g12 g13] -- det g2 gl g2"g2 g2"g3 = det /gEl g22 g23 = g . g3 gl g3 "g2 g3 "g3 [.g31 g32 g33 It follows from this equality that V = ~.
(1.1.73)
Denote by dVo and d V the volume elements (volumes of the elementary parallelepipeds erected on the tangent vectors g0i and gi with sides d~ i) in the initial and actual configurations. It follows from Eq. (1.1.73) that dVo = v ~ d ~
1 d~ 2 d~ 3,
d V = v ~ d ~ 1 d~ 2 d~ 3.
(1.1.74)
Substitution of expression (1.1.71) into Eqs. (1.1.74) with the use of Eq. (1.1.45) implies that
dv _
dVo
U-go
1175
15
1.1. Basic Definitions and Formulas
A medium is called incompressible provided its volume element remains unchanged under deformation. According to Eq. (1.1.75), the incompressibility condition reads I3(F) = 1.
1.1.11
(1.1.76)
Deformation of the Surface Element
Let us consider a rectangular surface element in the initial configuration erected on vectors d ?d and d ?ff (see Figure 1.1.4). Denote by h0 the unit normal vector to the surface element and by dSo its area (1.1.77)
ho dSo = d ?d × d ?dI.
In the actual configuration, vectors d ?d, d ?if, and ho are transformed into d 7/, d ?", and h, and Eq. (1.1.77) is written as (1.1.78)
h d S = d? I × d? n.
The surface elements in the initial and actual configurations obey the equality [see, e.g., Drozdov and Kolmanovskii (1994)], (1.1.79) We multiply Eq. (1.1.79) by itself, use the condition h • h = 1, and obtain dS 2 = g ~o" ¢ ~ " ¢ ~o" nodS~. go
This equality together with Eqs. (1.1.43) and (1.1.44) implies that dS
• h0)l/2.
dSo
hdS
y
d? n dS
.//
d? I
Figure 1.1.4: The surface element.
(1.1.80)
Chapter 1. Kinematics of Continua
16
To derive another expression for this ratio, we present Eq. (1.1.79) as
hodSo =
~r0? • hdS =
n. fTo~rdS.
(1.1.81)
We multiply Eq. (1.1.81) by itself, use Eqs. (1.1.42), (1.1.44), and the condition h0 • h0 = 1, and find that
(h. F. n) 1/2 --
dS
(n. go 1. ~)1/2
(1.1.82)
Equations (1.1.47), (1.1.80), and (1.1.82) demonstrate the geometrical meaning of deformation tensors. The Cauchy tensor ~, and the Almansi tensor ~0 determine changes in the arc element, whereas the Finger tensor P and the Piola tensor F0 characterize changes in the area element for transition from the initial to the actual configuration.
1.1.12
Objective Tensors
Let us consider two motions of a continuum. At the current instant t, the radius-vector of a point ~ equals = R(t, ~)
(1.1.83)
F = R'(t, ~)
(1.1.84)
in the first motion, and equals
in the other motion. We choose an arbitrary point P as a pole and denote by/~0 and k~ radius vectors of the point P in the first and second motions, respectively (see Figure 1.1.5). The relative radius vectors (with respect to the pole) are denoted as/~ and/~/: = R0 + R,
R ' = k~ + R'.
(1.1.85)
The motions R and R~ differ from each other by a rigid motion, provided that for any point ~ and for any instant t ]/~(t)] = ] R(t)[.
(1.1.86)
Equality (1.1.86) implies that there exists an orthogonal tensor function t) = O(t) such that for any point ~ and for any instant t, k ~ = / ~ . 0.
(1.1.87)
Substitution of expression (1.1.87) into Eq. (1.1.85) yields Rt(t, ~) = k~(t) + [R(t, ~) - R0(t)]" O(t).
(1.1.88)
1.1. Basic Definitions and Formulas
17
79
M
M
79
O
Figure 1.1.5: Kinematics of two rigid motions.
Equation (1.1.88) determines a rigid motion superimposed on the motion (1.1.83). We differentiate Eq. (1.1.88) with respect to ~i to obtain the formula for the tangent vectors ~ ( t , ~) = gi(t, ~) " O(t) = 0 T (t) " gi(t, ~).
(1.1.89)
The dual vectors ~i and ~i! satisfy the equality ~it(t ' ~) = ~i(t, ~) " O(t) = 0 T(t) " ~i(t, ~).
(1.1.90)
A vector field 77 = qi gi,
~, = q i , ~ ,
(1.1.91)
frozen into a deformable medium, is called objective (indifferent with respect to rigid motions) if qit = qi.
(1.1.92)
It follows from Eqs. (1.1.89) and (1.1.92) that an objective vector satisfies the condition ~! = qi(~)T " gi) -- o T " q = q" O.
(1.1.93)
By analogy with Eq. (1.1.92), a tensor field of the second rank ~_~ = QiJ~i~j,
Ot = Q ijl-I-I gig j,
(1.1.94)
18
Chapter 1. Kinematics of Continua
frozen into a deformable medium, is called objective (indifferent with respect to rigid motions) if (1.1.95)
Qijt = Qij.
Using Eq. (1.1.89), formula (1.1.95) can be also presented in the invariant form
01= aiJ (OT . ~i) (~j . O) "- 0 T" O" O.
(1.1.96)
It follows from Eq. (1.1.96) that the product P • (~ of two objective tensors P and (~ is an objective tensor as well. 1.1.13
V e l o c i t y V e c t o r a n d Its G r a d i e n t
The velocity vector ~ is defined as the derivative of the radius vector ~ with respect to time 0 = at(t,~).
(1.1.97)
Applying the Hamilton operator to the velocity vector ~, we obtain the velocity gradient ~7~. This tensor is expanded into the symmetrical and skew-symmetrical components ~r~ = b -
t',
(1.1.98)
which are called the rate-of-strain tensor 1 D = -~(f7 9 T + f7 ~),
(1.1.99)
1 ~" = ~(~79T -- ~'9).
(1.1.100)
and the vorticity tensor
The rate-of-strain tensor [ is connected with the relative Finger deformation tensor F<>(t, r) by the formula [see, e.g., Drozdov (1996)], 10
D(t) = -~
F<>(t, T)
(1.1.101)
. T=t
Denote by ~ and ~/the velocity vectors corresponding to the motions (1.1.83) and (1.1.84) -
at'
v =
O---t-"
(1.1.102)
We transform the latter formula in Eq. (1.1.102) with the use of Eq. (1.1.88) 0/
-
d/~ ( d/~o~ + ~ • 0 + (R-/~o)" dt \ - --~-/
dO -dt"
(1.1 103) "
19
1.1. Basic Definitions and Formulas
For any orthogonal tensor O, dO dt
-
dO T dt
-O.
(1.1.104)
• O.
It follows from Eqs. (1.1.88), (1.1.103), and (1.1.104) that _ , _ dk~ +
dt
_ ddt[~
+
~-
dko~
-Yi-)
(dido) fp---~
"0-
( R - [ ~ o ) "O"
"0- (
,,
-k~)"
dO T dt " 0
dO T dt
"O"
(1..1 105)
The skew-symmetrical tensor (1.1.106)
fi = dOT • 0 dt
is called spin tensor. According to Eqs. (1.1.104) and (1.1.106), tensors 0 and 0 T satisfy the differential equations dO . dt
0 .
~, .
dOT dt
.
f t . 0 T.
(1.1.107)
It follows from Eqs. (1.1.105) and (1.1.106) that +
dt -
dt
( dko) ( dko) -~-J
+
a -
--d~ j
• 0 + 1~.
(R'-/?~).
(1.1
108)
To calculate the covariant derivative of the vector 9 ~, we use Eqs. (1.1.5), (1.1.7), (1.1.89), (1.1.90), and (1.1.108) fT ' TJ' = g-i t a ~l l - O T " g, i -a~~) t -- O T " g, i ( -O- ~ " 0 - : OT . (~, i -~- )~~ ) " 0 -- ~t~,[. " ) fi0
__
a ~t~ ~ it . f i
T • (V~) " O -- g, tg, " i • O" fi )
= O T. fT~. 0 - 0 T ' i " (9. fi = 0 T. fT~. O - ft.
(1.1.109)
It follows from Eqs. (1.1.108) and (1.1.109) that the velocity vector 9 and its gradient ~ are not objective. We calculate the rate-of-strain tensor b and the vorticity tensor Y according to Eqs. (1.1.99), (1.1.100), and (1.1.109) and find that D' -- O r . D . O,
Y' - 0 T . ~" . 0 + 19,.
(1.1.110)
20
Chapter 1. Kinematics of Continua
It follows from Eq. (1.1.110) that the rate-of-strain tensor/3 is objective, whereas the vorticity tensor t" is not indifferent with respect to rigid motions. Resolving the latter equation in Eq. (1.1.110) with respect to the spin tensor ~, we obtain
h= ?'-br.?.b.
(1.1.111)
1.1.14 Corotational Derivatives In the constitutive theory for inelastic media, corotational derivatives serve as analogs of material time derivatives (which are not indifferent with respect to rigid motions). Corotational derivatives are defined as linear operators, which transform an objective tensor into an objective tensor of the same range.
The J a u m a n n
Derivative Let gt(t, {~) be an objective vector field that satisfies Eq. (1.1.93). The derivative of ?:/with respect to time at fixed Lagrangian coordinates {~ is called the material derivative. Differentiation of Eq. (1.1.93) with the use of Eq. (1.1.107) yields
aq' at
= ~a q . o + q . dO a?::/ O - q . b . f ~ .,, . . . at
dt
at
Substitution of expressions (1.1.93) and (1.1.111) into this equality implies that
-
at
at
0-
0.
0
o)
= a~.O_q.O.?,+q.?.O at
= a~.o-O,.?,+q.?.O. at
It follows from this equality that
~?/~+?/~.~,~= (a?/ a-7
-a-7 + 0. ?
)
.0,
which means that the vector Z/~ = aZ/ + ~/" ~, at
(1.1.112)
is indifferent with respect to rigid motions. The vector g/<> is called the Jaumann derivative of an objective vector g/. Employing similar reasoning, the Jaumann derivative may be defined for an objective tensor of the second rank (~. Using Eqs. (1.1.96), (1.1.107), and (1.1.111),
1.1. Basic Definitions and Formulas
21
and repeating the preceeding calculations, we find that
aO.' at
db~ aO_. db • 0_" b + b ~ • ~ . b + b ~" (2" dt at dt
= h . b ~. O. b + b ~. aa~. b - U . O b . fi at
= (?'-
U.
? . b)" b ~" (2" b + b ~" a/_OO,b
at
- b ~ " O_ " b" (?' - b ~ " ? . b ) = ?"O_'-U'?'O_'O+O
~. aO..b
at
-O'.~'+b~.O.~.b. Therefore
aQ~-Y~'Q~+Q~'~'~=br'at
( aO- - ~ " O" +
. t ' ) . b,
which implies that the Jaumann derivative of an objective tensor
0< > _ aQ + Q" }" - ~'" 0 at
(1.1.113)
is indifferent with respect to superposed rigid motions.
The Oldroyd Derivatives
It follows from Eqs. (1.1.98) to (1.1.100) that
~" = b - ~7~,
t" = - D + ~7~T.
(1.1.114)
Let ~/be an objective vector field with the Jaumann derivative (1.1.112). Substitution of the first expression (1.1.114) into Eq. (1.1.112) yields
q+ = a~ + ~ . b - q . # 9 at
= qa + q . b ,
(1.1.115)
where F/A _ OF:/ _ Z/. ~'~. at
(1.1.116)
It follows from Eq. (1.1.115) that qA is an objective tensor, since ~/, D, and ?/<> are objective vectors and tensors. It is called contravariant (upper) convected derivative of a vector q [see Oldroyd (1950)].
Chapter 1. Kinematics of Continua
22
Let us now consider an objective tensor field (~ with the Jaumann derivative (1.1.113). Substitution of the first expression (1.1.114) into Eq. (1.1.113) yields 0 o _ 0(~ + (~. b - D - 0 0t
- (~" ~7~ + ¢ ~ . (~.
(1.1.117)
It follows from Eq. (1.1.99) that V0 = 2 / 3 - V0 r. Substituting this expression into Eq. (1.1.117), we obtain
0_<>_ OQ ot
+O.b-O.eo+b.O.-eor.o
=0 A + O . b + b . O ,
(1.1.118)
where the objective tensor ^
OA _ 0Q _ Q" V0 - V0 T. 0
Ot
(1.1.119)
is the contravariant (upper) convected derivative of an objective tensor Q. Using the latter expression in Eq. (1.1.114) and repeating similar transformations, we introduce the covariant (lower) convected derivatives of an objective vector g:/and an objective tensor (~ [see Oldroyd (1950)], g/v = Og/ + ~/" ¢~,r, 0t
~)v _ 0 0 + 0 " l~'vr + ~'0" Q. 0t
(1.1.120)
Spriggs et al. (1966) introduced the generalized corotational derivative of an objective tensor Q of the second rank 0 D = 0 ° + al(Q" b + D . (2) + a2(Q"/3)i + a3Ii(0)D,
(1.1.121)
where ak (k = 1, 2, 3) are arbitrary constants. For any tensor Q, the Jaumann and the Oldroyd corotational derivatives are particular cases of the generalized corotational derivative (1.1.121). The Jaumann derivative corresponds to the case al = a2 = a3 = 0; the upper Oldroyd derivative corresponds to the case al = - 1 , a2 = a3 -- 0; and the lower Oldroyd derivative corresponds to the case a l = 1, a2 = a3 = 0. ^
1.1.15
The Rivlin-Ericksen
Tensors
The Oldroyd covariant derivatives of the rate-of-strain tensor b were introduced by Oldroyd (1950) and used by Rivlin and Ericksen (1955) to construct constitutive models for viscoelastic fluids. The tensors, known as the Rivlin-Ericksen tensors, are
Bibliography
23
determined by the formulas A0 = i, ^
an+l = Anv -
~
= 2b = Vo + Vo T,
aAn + An" ~,oT + ~,~. ,21n Ot
(n = 1,2 .... ). (1.1.122)
Instead of the Oldroyd covariant derivatives, other corotational derivatives may be used. For example, employing the Oldroyd contravariant derivatives, we arrive at the White-Metzner tensors [see White and Metzner (1963)],
/~0 = - L ^
Bn+l =/TnA --
/~1 = 2D = ~r ~ + ~ oT, a/~n
at
--/Tn" ~'~ -- ~roT. /Tn
(n = 1,2 .... ). (1.1.123)
The Rivlin-Ericksen tensors (1.1.122) were used by Coleman et al. (1966), the WhiteMetzner tensors (1.1.123) were employed by Astarita and Marrucci (1974) and Huilgol (1979) to develop constitutive equations in the nonlinear mechanics of continua.
Bibliography [1] Astarita, G. and Marrucci, G. (1974). Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London. [2] Coleman, B. D., Markowitz, H., and Noll, W. (1966). Viscometric Flows of Non-Newtonian Fluids. Springer-Verlag, New York. [3] Coleman, B. D. and Noll, W. (1961). Foundations of linear viscoelasticity. Rev. Modem Phys. 33, 239-249. [4] Drozdov, A. D. (1996). Finite Elasticity and Viscoelasticity. World Scientific, Singapore. [5] Drozdov, A. D. and Kolmanovskii, V.B. (1994). Stability in Viscoelasticity. North-Holland, Amsterdam. [6] Fitzgerald, J. E. (1980). Tensorial Hencky measure of strain and strain rate for finite deformations. J. Appl. Phys. 51, 5111-5115. [7] Hill, R. (1968). On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229-242. [8] Huilgol, R. R. (1979). Viscoelastic fluid theories based on the left Cauchy-Green tensor history. Rheol. Acta 18,451-455. [9] Morman, K. N. (1986). The generalized strain measure with application to nonhomogeneous deformations in rubber-like solids. Trans. ASME J. Appl. Mech. 53,726-728.
24
Chapter 1. Kinematics of Continua
[10] Oldroyd, J. G. (1950). On the formulation of rheological equations of state. Proc. Roy. Soc. London A200, 523-541. [ 11] Rivlin, R. S. and Ericksen, J. L. (1955). Stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4, 323-425. [12] Seth, B. R. (1964). Generalized strain measure with applications to physical problems. In Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics. (M. Reiner, and D. Abir, eds.), pp. 162-172. Pergamon Press, Oxford. [13] Spriggs, T. W., Huppler, J. D., and Bird, R. B. (1966). An experimental appraisal of viscoelastic models. Trans. Soc. Rheol. 10, 191-213. [14] White, J. L. and Metzner, A. B. (1963). Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci. 7, 1867-1889.
Chapter 2
Constitutive Models in Linear Viscoelasticity
This chapter is concerned with constitutive relations for linear viscoelastic media with infinitesimal strains. The viscoelastic behavior is typical of a number of materials that are extremely important for applications: polymers and plastics [see, e.g., Aklonis et al. (1972), Bicerano (1993), Ferry (1980), Struik (1978), Tschoegl (1989), Vinogradov and Malkin (1980), and Ward (1971)], composites [see, e.g., Alberola et al. (1995)], metals and alloys at elevated temperatures [see, e.g., Skrzypek (1993) and Szczepinski (1990)], concrete [see, e.g., Bazant (1988) and Neville et al. (1983)], soils [see, e.g., Adeyeri et al. (1970)], road construction materials [see, e.g., Maccarrone and Tiu (1988) and Vinogradov et al. (1977)], building materials [see, e.g., Papo (1988)], biological tissues [see, e.g., Deligianni et al. (1994)], food-stuffs [see, e.g., Robert and Sherman (1988) and Struik (1980)], etc. Section 2.1 deals with linear differential and fractional differential models. In section 2.2, we discuss integral models and introduce the concept of adaptive links. Section 2.3 is concerned with properties of creep and relaxation measures. Basic thermodynamic potentials for an aging viscoelastic medium are derived in Section 2.4. A new constitutive models for an aging viscoelastic material is developed in Section 2.5.
2.1
Differential Constitutive Models
In this section, a brief survey is provided of differential and fractional differential models in linear viscoelasticity. For simplicity, we confine ourselves to uniaxial loading. 25
26
2.1.1
Chapter 2. Constitutive Models in Linear Viscoelasticity
Differential Constitutive Models
We begin with differential constitutive models, where the stresses and strains at the current instant t are connected by linear differential equations with constant coefficients. Any differential model for the viscoelastic behavior reflects some rheological model, where elastic elements (springs) and viscous elements (dashpots) are connected in series and in parallel. A linear spring obeys Hooke's law o- = EE,
(2.1.1)
where o" is the stress, E is the strain, and E is Young's modulus. A linear dashpot obeys Newton's law dE ~r = r/~-,
(2.1.2)
where r/is the Newtonian viscosity. To construct more complicated rheological models, the basic elements are connected in series or in parallel. To derive differential equations for a rheological model, the following rules are employed: • For elements connected in parallel, their strains coincide, and the total stress equals a sum of stresses in individual elements. • For elements connected in series, their stresses coincide, and the total strain equals a sum of strains in individual elements. The Maxwell model consists of a spring and a dashpot connected in series (see Figure 2.1.1). The constitutive equation of the Maxwell model reads dE 1 doo+ --. dt E dt rl
I I i
<
> ->
Figure 2.1.1: The Maxwell model.
(2.1.3)
2.1. Differential Constitutive Models
27
% iII Figure 2.1.2: The Kelvin-Voigt model. The Kelvin-Voigt model consists of a spring and a dashpot connected in parallel (see Figure 2.1.2). This model obeys the constitutive relation dE
cr = EE + r/-w.
(2.1.4)
at
To generalize these models, we combine springs and dashpots, as well as the Maxwell and Kelvin-Voigt elements, in parallel and in series. Several Maxwell elements connected in parallel provide the generalized Maxwell model (the MaxwellWeichert model). Several Kelvin-Voigt elements connected in series provide the generalized Voigt model. A model consisting of a Maxwell element in parallel with a spring is called the standard viscoelastic solid (the Zener model) (see Figure 2.1.3). Its constitutive equation reads
1
E~ at
+-or
~
=
(1+
~
-- +--e.
dt
(2.1.5)
Here E1 is Young's modulus of the spring and E2 and r/are Young's modulus and the Newtonian viscosity of the Maxwell element.
Figure 2.1.3: The standard viscoelastic solid.
28
Chapter 2. Constitutive Models in Linear Viscoelasticity
The differential constitutive equation (2.1.5) with the zero initial conditions is equivalent to the Volterra integral constitutive equation or(t) = E
[
e(t) +
/ot
Qo(t - T)E(r) d
.
(2.1.6)
Here E=EI
+E2
is Young's modulus and Qo(t) = - x
t [ 1 - exp ( - T ) I
(2.1.7)
is the relaxation measure with the parameters X=
E2 El + E 2 '
T-
~/ E2
Combining basic rheological elements, an arbitrary number of differential constitutive models can be designed. Some of these models are equivalent to each other, the others are independent [see Giesekus (1994)]. The constitutive equation of a rheological model with an arbitrary number of springs and dashpots reads do" dno " dE dmE ao~r + al dt + "'" + a n - - ~ -- boe + bl--d-[ + "'" + bm dt---W ,
(2.1.8)
where m and n are positive integers and ak and bl are adjustable parameters.
2.1.2
Fractional Differential Models
Fractional differential models (that are "transient" between differential and integral models) obey linear differential equations with constant coefficients and fractional derivatives. A fractional derivative is a Volterra operator with the Abel kernel
t o~ J~ (t) = F(1 + a)"
(2.1.9)
Here a is a real parameter, and F(z) is the Euler gamma-function of a complex variable z F(z) = fo c° t z- 1 exp(- t) dt.
(2.1.10)
Fractional differential constitutive models in linear viscoelasticity were proposed by Bagley and Torvik (1983a,b, 1986) and Rogers (1983), and were discussed in detail by Bagley (1987), Ffiedfich (1991a,b), Glockle and Nonnenmacher (1991, 1994), Heymans and Bauwens (1994), Koeller (1984), Nonnenmacher and Glockle (1991),
29
2.1. Differential Constitutive Models
Suarez and Shokooh (1995), and Tschoegl (1989). For a detailed exposition of the theory of fractional derivatives see, e.g., Miller and Ross (1994), Oldham and Spanier (1974), and Srivastava and Menocha (1984). Fractional differential models • Describe adequately the creep and relaxation processes in polymeric materials by means of simple relationships between stresses and strains with relatively small number of adjustable parameters [see, e.g., Bagley and Torvik (1983a) and Heymans and Bauwens (1994)]. • Reflect correctly the effect of vibrational frequency on the viscoelastic response in damping materials [see, e.g., Bagley and Torvik (1983b), Nashif et al. (1985), and Suarez and Shokooh (1995)]. • Lead to the non-Debay relaxation behavior [see, e.g., Glockle and Nonnenmacher (1993), Ngai (1987), and Nonnenmacher and Glockle (1991)]. • Take into account stochastic (micro-Brownian) motion of chain molecules at the microlevel in the phenomenological description of viscoelastic media [see Bagley and Torvik (1983a)], and provide a basis for application of fractal rheological models in viscoelasticity [see Heymans and Bauwens (1994)]. • Allow some interesting phenomena to be analyzed arising in the theory of wave propagation in media described by hyperbolic-parabolic equations [see e.g., Choi and MacCamy (1989) and Hrusa and Renardy (1985)]. Let f ( t ) be a sufficiently smooth function that vanishes in the interval ( - ~ , 0] and that is integrable in [0, T] for an arbitrary T > 0. The primitive Fa(t) for the function f ( t ) equals Fl(t) =
f0t f ( s ) d s .
The second primitive Fz(t) of f ( t ) is calculated as F2(t) =
d~" • f ( s ) ds = /0 t(t /0t/0
s ) f ( s ) ds.
Similarly, the nth primitive of f ( t ) reads
1 /0t (t
F , ( t ) = (n - 1)!
-- S) n- 1 f ( s )
ds,
(2.1.11)
where n! = 1 . 2 . • • n. Since F(n) = (n - 1)! for any positive integer n, Eq. (2.1.11) is presented as t
Fn(t) =
fo
Jn-1 (t - s ) f ( s ) ds.
30
Chapter 2. Constitutive Models in Linear Viscoelasticity
According to Eq. (2.1.11), the fractional operator F~(t) =
f0t J a - l ( t
(2.1.12)
- s)f(s)ds
is reduced to the standard operator of integration for a positive integer c~. It is convenient to rewrite Eq. (2.1.12) as F~(t) =
/0tJ ~ - l ( s ) f ( t
- s)ds =
/0 J ~ - l ( s ) f ( t
- s)ds.
(2.1.13)
Equation (2.1.12) determines the fractional operator F~(t) for an arbitrary a > 0. For a E (0, 1), the special notation is used [see, e.g., Glockle and Nonnenmacher (1991, 1994) and VanArsdale (1985)], D - ~ f ( t ) = F~(t) =
J~-l(t - s)f(s)ds.
(2.1.14)
To define the function F~ (t) for an arbitrary negative c~, we employ the formula dn dtnJ~+n(t) = J~(t),
(2.1.15)
which is satisfied for any real a and for any positive integer n. Let us consider the functional la(th) =
(2.1.16)
Ja_l(S)dp(s)ds.
For ct > 0, the integral in Eq. (2.1.16) converges for any continuous function th(t) such that
4~(0) = 0,
14~(t)l dt <
~.
To define the functional I,~(~b) for c~ E ( - n , - n + 1), we assume that ~b(t) has n continuous derivatives which vanish at t = 0. In this case, Eq. (2.1.16) is integrated n times by parts with the use of Eq. (2.1.15) to obtain I~(dp) =
J~-l(S)dp(s)ds
= J~(s)4)(s)lo -
ds ~_.
/o (s)
J~(s
s) ds
(s) ds
* * ,
= ( - 1 ) ~ o/0"~ J ~ + n - l ( S ) - d~d~ -~(s)as. a s ,~
(2.1.17)
2.1. Differential Constitutive Models
31
Since the integral in the right side of Eq. (2.1.17) converges, the functional I~(th) can be defined for a E ( - n , - n + 1) according to formula (2.1.17). Returning to the function fit), which vanishes at t = 0 with its derivatives, we set [see Eq. (2.1.13)]
F~(t)
=
fo
Ja+n-l(S)f(n)(t -- s)ds
=
fo'
Ja+n-l(S)f(n)(t -- s)ds,
(2.1.18)
provided a E ( - n , - n + 1). Here n is a positive integer and
dnf f(n)(t) = - ~ ( t ) . The function F~ (t) is defined now for any real a except for nonpositive integers. To define it on the whole real axis, we set F-n(t) =
lim
Fa(t).
t2g.--.* - - n - - O
Substitution of expressions (2.1.9) and (2.1.18) into this equality implies that
F-n(t) =
lim
a--,-n-O
f0t Ja+n(S)f (n+l)(t -
s) ds
lim foot sa+n f(n+l)(t _ S) ds a---*-n-0 F(1 + c~ + n)
fo
t f(n+ 1)(t
s) ds
f(n)(t).
(2.1.19)
It follows from Eq. (2.1.19) that for any positive integer n, function F-n(t) determines the nth derivative of f(t). This concept may be extended to an arbitrary negative real c~. In particular, the fractional derivative of the order a ~ [0, 1) is defined as follows:
daf
f{a}(t) = d---~(t) = F_a(t)
t
=
fO
=
/o'
=
df
J_~(s)-d- i (t
J _ ~ ( t - s)
F(1
_ S) ds (s) ds
1_ a) fOOt (t - s)- °~-c(s)ds. dfdt
(2.1.20)
It is of interest to establish a correspondence between the fractional derivative f{~}(t) and the fractional operator D -~ f(t). For this purpose, we integrate expression (2.1.14) by parts and use the initial condition f(0) = 0. As a result, we find that
32
D-(1-~)f(t)
Chapter 2. Constitutive Models in Linear Viscoelasticity
1
= F(1 - a)
1
(1 - a ) F ( 1
fOt (t f(s) - s) ~ ds
[
+ - a) -(t - s)l-af(s) s=t s=0
1 - a) (1 - a)F(1
/0 t(t
-- S) 1-a
/0t(t -- S) 1-a -jid f (s) & ] (2.1.21)
(s)ds.
Differentiation of Eq. (2.1.21) with respect to time implies that d (1 1 fot d t D - -~) f ( t ) = F(1 - a ) (t-
s)-
adf
-d-~(s)ds.
Finally, combining this equality with Eq. (2.1.20), we obtain d~f dt ~ (t) -
d (1 -~V-~) f ( t ) .
(2.1.22)
Equation (2.1.2) determines a Newtonian dashpot, where the stress is proportional to the first derivative of the strain. A natural generalization of this rheological element is a fractional dashpot with the constitutive equation
daE (2.1.23)
or = rl dt ~ ,
which is characterized by two material parameters: a E (0, 1) and ~. The limiting cases correspond to the Hookean spring (a = 0) and to the Newtonian dashpot ( a = 1).
Using the fractional dashpot, we can construct analogs of the Maxwell model (2.1.3): d~or
E d~e + -or = E~
dt ~
rl
dt ~ '
(2.1.24)
of the Kelvin-Voigt model (2.1.4):
daE
(2.1.25)
or = E e + ~ dt ~ ,
and of the Zener model (2.1.5): d '~or ~- 1 dt ~
-
--or = T
E d'~e dt ~
E1
+ ~e. T
(2.1.26)
Equation (2.1.26) is easily generalized by introducing derivatives of different fractional orders. For example, the following equation may be proposed for the standard viscoelastic solid: d ~ tr dt a
1
d/3a
E1
+ ~or= Ed-~ + Te,
(2.1.27)
33
2.1. Differential Constitutive Models
where a , / 3 , E, El, and T are adjustable parameters. However, not all extensions of the Zener model are thermodynamically admissible [see a discussion of this question in Friedrich (1991a)]. Models with fractional springs and dashpots permit experimental data in dynamic tests to be predicted adequately for a number of polymeric materials. To demonstrate fair agreement between results provided by the Zener model (2.1.26) and observations, we consider steady uniaxial oscillations of a viscoelastic specimen. In accordance with Burton (1983), to derive an equation for steady oscillations, we have to replace zero as the lower limit of integration in Eq. (2.1.20) by - ~ . As a result, we obtain the constitutive equation 1
[;
(t -
F(1 - a ) J _ o~ =
E F(1
I
og)
/:
~d° + or(t) s ) - ~ dt ( S ) d s T
(t - s) -'~ de
E1 e(t)
(2.1.28)
-:-(s)ds + ~ . tit-T
o~
As common practice, we seek solutions of Eq. (2.1.28) in the form o-(t) = o0 exp(~wt),
E(t) = E0 exp(w~t),
(2.1.29)
where o'0 and e0 are the amplitudes of oscillations to be found, ~o is the frequency of oscillations, and ~ = ~ 1. Substituting expressions (2.1.29) into Eq. (2.1.28) and introducing the new variable ~- = t - s, we find that
o0
E
too F(1 - a )
/0
= eo F(1 - a )
~'-'~ exp(-~to~')d~" +
1
~'-'~ e x p ( - ~or) d~- +
.
Calculation of the integrals with the use of Eq. (2.1.10) implies that E*(w)-
oo _ E1 + E T ( r w ) '~ ~0 1 + T(~w) '~ '
(2.1.30)
where E*(og) = E'(w) + tE"(w)
(2.1.31)
is the complex Young's modulus. Combining Eqs. (2.1.30) and (2.1.31), we obtain expressions for the storage modulus E'(oJ) and the loss modulus E'(o~) E'(to) =
[El + Eo9a cos(Tra/2)][1 + to'~ cos(ara/2)] + Eto 2a sin2(ara/2) [1 + toa cos('n'a/2)] 2 + o)2c~ sin2(Tra/2)
E'(to) =
(E - E1)o9c~sin(ara/2) [1 + to c~cos('n'a/2)] 2 + ~2c~ sin2(,n.c~/2) •
(2.1.32)
34
Chapter 2. Constitutive Models in Linear Viscoelasticity 4.0
0
0
0
logE ~ logE"
1.0
I
I
I
I
I
I
-2.0
I
log to
I
I
7.0
Figure 2.1.4: The storage modulus E ~(MPa) and the loss modulus E" (MPa) versus frequency to (Hz) of steady oscillations for poly(methyl methacrylate) (PM MA). Circles show experimental data obtained by Rogers (1983): unfilled circles: E'; filled circles: E'. Solid lines show prediction of the fractional Zener model (2.1.26) with c~ = 0.1946, E = 6354.0 MPa, E1 = 2130.0 MPa, and T = 0.8234 sec ~. Experimental data for PMMA and the dynamic moduli calculated according to Eq. (2.1.32) are plotted in Figure 2.1.4, which demonstrates fair agreement between experimental data in dynamic tests and their theoretical prediction. Calculation of the material response in static tests requires a more sophisticated analysis, since even for the simplest programs of loading (creep, relaxation, recovery, etc.) the behavior of the model (2.1.26) is expressed in terms of special functions (either the generalized Mittag-Leffler functions or the Wright functions).
2.2
Integral Constitutive Models
In this section, some integral constitutive models are discussed for linear viscoelastic media. We begin with Boltzmann's superposition principle and derive integral equa-
35
2.2. Integral Constitutive Models
tions for the creep and relaxation measures of aging viscoelastic media. Afterward, the concept of adaptive links is introduced, and a balance law is developed for the number of links. Finally, constitutive equations for uniaxial deformation are extended to three-dimensional loading.
2.2.1 Boltzmann's Superposition Principle Let us consider a specimen in the form of a rectilinear rod, which is in its natural (stress-free) state. At the initial instant t = 0, tensile forces are applied to the ends of the rod. Boltzmann's superposition principle states that the stress cr at the current instant t depends on the entire history of strains e in the interval [0, t]. Assuming this functional to be linear and applying Riesz's theorem, we find that
o(t) =
/0
X(t, ~')de(~'),
(2.2.1)
where X(t, T) is a function integrable in ~"for any fixed t >- 0. Equation (2.2.1) provides the general presentation of the stress-strain dependence in linear viscoelasticity. We suppose that the stress o- and the strain e are sufficiently smooth functions of time that satisfy the conditions o-(0) = 0,
e(0) = 0.
(2.2.2)
Integration of Eq. (2.2.1) by parts with the use of Eq. (2.2.2) implies that
OX (t, r)e(r) dr. or(t) = X(t, t)e(t) - foot -~r
(2.2.3)
It is convenient to present the relaxation function X(t, r) in the form
X(t, r) = E(r) + Q(t, r),
(2.2.4)
E(I") = X(~', 1")
(2.2.5)
where
is the current Young's modulus, and
Q(t, r) = x(t, r) - x ( r , r)
(2.2.6)
is the relaxation measure. It follows from Eq. (2.2.6) that for any t -> 0 (2.2.7)
Q(t, t) = O. The relaxation kernel R(t, r) is determined as
R(t, ~') -
10X
E(t) ar
(t, ~').
(2.2.8)
36
Chapter 2. Constitutive Models in Linear Viscoelasticity
Substituting expressions (2.2.5) and (2.2.8) into Eq. (2.2.3), we obtain the constitutive equation of a linear viscoelastic medium or(t) = E(t)
E /0' e(t) -
l
R(t, ~')e(~') d~"
.
(2.2.9)
Equations (2.2.3) and (2.2.9) describe the viscoelastic response in aging viscoelastic media, mechanical properties of which depend explicitly on time. For aging materials, the function X(t, ~') depends on two variables, t and ~-. Typical examples of aging media are polymers, concrete, and soils [see, e.g., experimental data presented in Arutyunyan et al. (1987) and Struik (1978)]. Aging elastic media provide the simplest example of aging viscoelastic materials. For an aging elastic solid, Young's modulus E(t) depends on time, whereas the relaxation function vanishes, Q(t, ~-) = 0.
(2.2.10)
Combining Eqs. (2.2.4) and (2.2.10), we find that X(t, 1") = E(~').
Substitution of this expression into Eq. (2.2.3) implies that or(t) = E ( t ) E ( t ) -
fot~T
(T)E(I")dI".
(2.2.11)
Differentiation of Eq. (2.2.11) with respect to time yields the differential constitutive equation with a time-dependent Young's modulus dodt
dE - E(t) d---t"
(2.2.12)
The mechanical response in nonaging viscoelastic media is time-independent, which means that the function X depends on the difference t - T only X(t, r) = Xo(t - r).
(2.2.13)
It follows from Eqs. (2.2.5), (2.2.6), and (2.2.13) that Young's modulus E of a nonaging viscoelastic medium is time-independent, E = X0(0),
(2.2.14)
and the relaxation function Q depends on the difference t - ~-, Q = EQo(t - ~).
(2.2.15)
Substituting expressions (2.2.14) and (2.2.15) into Eqs. (2.2.3) and (2.2.9), we obtain the constitutive equation of a nonaging viscoelastic material
2.2. Integral Constitutive Models
37
[
or(t) = E e(t) +
/0t
Qo(t - r)e(r) dr
1
=E[e(t)-ftR(t-r)e(r)dr],
(2.2.16)
where
dQo R(t) = - ~ ( t ) , dt
(2.2.17)
and the superimposed dot denotes differentiation. Another formulation of Boltzmann's superposition principle states that the strain e at the current instant t is a functional of the entire history of stresses. Assuming this functional to be linear and applying Riesz's theorem, we arrive at the constitutive equation similar to Eq. (2.2.1) e(t) =
J0
Y(t, r) do'(r),
(2.2.18)
where Y(t, r) is a function integrable in r for any fixed t >-- 0. We suppose that the stress o" and the strain e are sufficiently smooth functions of time, integrate Eq. (2.2.18) by parts, and use Eq. (2.2.2). As a result, we obtain the constitutive equation of an aging, linear, viscoelastic medium
e(t) = Y ( t , t ) o ( t ) - foot -~-~T(t, OY r)cr(r)dr.
(2.2.19)
The function Y (t, r) is presented in the form
Y(t, r) -
1
E(r)
+ C(t, r),
(2.2.20)
where
E(r) -
1
Y(r, r)
(2.2.21)
is the current Young's modulus, and C(t, z) = Y(t, r) - Y(r, r)
(2.2.22)
is the creep measure, which satisfies the condition
C(t, t) - O.
(2.2.23)
Substitution of expressions (2.2.20) and (2.2.21) into Eq. (2.2.19) yields
e(t) -
o-(t) E(t)
~
+ C(t, r) or(r) dr.
(2.2.24)
Chapter 2. Constitutive Models in Linear Viscoelasticity
38 Introducing the creep kernel
K(t, r) = - E ( t )
(2.2.25)
~ - ~ + C(t, r) ,
we rewrite Eq. (2.2.24) as e(t) = - ~
or(t) +
K(t, r ) ~ ( r ) d
.
(2.2.26)
An aging elastic medium is characterized by the condition
C(t, r) = 0. This equality together with Eqs. (2.2.20) and (2.2.24) implies that 1
Y(t, r) .-
E(r)
and
o-(t) ~'d~ ( ~ 1 )
e(t) - E(t) -
~r(r) dr.
(2.2.27)
Differentiating Eq. (2.2.27) with respect to time, we obtain the constitutive Eq. (2.2.12). For non-aging viscoelastic media, the function Y depends on the difference t - r only. According to Eqs. (2.2.21) and (2.2.22), this means that Young's modulus is constant, E(t) = E,
and the creep function depends on the difference t - r, 1 C = -~Co(t - r).
(2.2.28)
Substitution of expression (2.2.28) into Eqs. (2.2.24) and (2.2.25) implies the constitutive relation for a nonaging viscoelastic material
1[ /0t
e(t) = ff~ or(t) +
K(t-
I l o t(?0(t -
1 tr(t) + E
rlor(rld
r)o'(r)dr
1
(2.2.29)
where K(t) = -dCo ~ (t).
(2.2.30)
2.2. Integral Constitutive Models
39
The constitutive Eqs. (2.2.9) and (2.2.26) describe homogeneous viscoelastic media. For a viscoelastic solid with an arbitrary nonhomogeneity, Young's modulus, and the creep and relaxation kernels depend explicitly on Lagrangian coordinates {~,
1[
e(t, ~) - E(t, ~)
or(t, ~) +
[
or(t, ~) = E(t, ~) e(t, ~) -
/ot K(t, r, ~)or(r, !~) d r 1 ,
/0t
R(t, r, ~)e('r, ~) dr
]
(2.2.31)
.
For nonhomogeneously aging media, we assume that different portions were manufactured at different instants that preceded the initial instant t = 0 [see Arutyunyan et al. (1987)]. To describe the manufacturing process, we introduce a piecewise continuous and bounded function K(~), which equals the material age at a point ~ at the initial instant t = 0. Since the material response is characterized by the internal time t + K(~), the constitutive equations of a nonhomogeneously aging viscoelastic medium read e(t,~) =
1
E(t + K(~))
[cr(t,~) + 7ot K(t
or(t, ~) = E(t + K(~)) e(t, ~ ) -
]
+ K(~), r + K(~))o-(r, ~)dr ,
R(t + K(~), r + K(~))e(r, ~)dr .
(2.2.32)
Three approaches may be distinguished: (i) K(~) is a prescribed function, which characterizes the age distribution in a medium. (ii) K(~) is a control function, which is chosen to ensure optimal properties of a structure. (iii) K({~)describes environmental dependent aging caused by temperature [see, e.g., Stouffer and Wineman (1971) and Struik (1978)], by humidity [see Aniskevich et al. (1992), Knauss and Kenner (1980), Makhmutov et al. (1983), Morgan et al. (1980), Panasyuk et al. (1987), Shen and Springer (1977)], and by radiation [see McHerron and Wilkes (1993) and Sharafutdinov (1984)].
2.2.2
Connections Between Creep and Relaxation Measures
Let us derive an integral equation which expresses creep and relaxation measures of an aging viscoelastic medium in terms of each other. For this purpose, we substitute expression (2.2.19) into Eq. (2.2.3), take into account Eqs. (2.2.5) and (2.2.21), and obtain
,~(t) = E(t) [E(t) [,~(t) _ fo' -g-ss or" (t' s)cr(s) cls] -
t oqX
fO
[ or(s)
Ts (t's~ Y(-;f
s OY
fo
]
-~r (s, r)o'(r) dr ds
Chapter 2. Constitutive Models in Linear Viscoelasticity
40
= or(t)-
log
]
fOt [E(t)aY(t,s)+ E(s) --~s(t' s) or(s)ds ! Os
OX s) ds fo ~ -~T(s, OY r)or(r) dr. + fot --~s(t, This equality implies that
OY l_~ OX(t ' s) = E(t)--~s (t,s) + E(s) Os
fs t ~ ( t ,
OY(Z, s) dr. T)-~s
(2.2.33)
Substitution of expressions (2.2.4) and (2.2.20) into Eq. (2.2.33) yields
E(t)~
+C(t,s) +
1
0
E(s) Os
= fs t ~0 [E(T) + Q(t, ~')]~
[E(s) + Q(t, s)]
+ C(~',s) d~'.
(2.2.34)
Integrating Eq. (2.2.34) from T to t, we obtain
E(t) [(E-~t)+ C(t, t)) - ( E~T) + C(t, T))
=
fT t ds fs t ~0 [E(T) + Q(t, ~')]~
+ fT t ~ 1 ~0 [E(s) + Q(t, s)] ds
+ C(~',s) d~'.
(2.2.35)
We change the order of integration in the right-hand side of Eq. (2.2.35) and find that
ds
~[E(~') + Q(t, ~')]~ss ~
= fr t ~0-~T[E(T) + Q(t, r)]dT
+ C(T,s) dr
Os -E~ + c(r,s) ds.
We calculate the integral with the use of Eq. (2.2.23) and obtain f r t ~0 [E(T) + Q(t, ~')]aT fT ~ 0
-k~+C(~,s)
] ds
0 = f r t ~-~T[E(r) + Q(t, ~')]
+C(r,r)-
('
1
1
E(T)
1 - E(T) 1 - C(T, T) 1 d~'. 0 [E(~') + Q(t, r)] E(~') = fr t ~-~r
+ c(T,r)/] d~
2.2. Integral Constitutive Models
41
Substitution of this expression into Eq. (2.2.35) yields 1
1 - E(t)
E(T)
E1
+ C(t, T) = -
~-~r[E(~')+ Q(t, 1")] E(T) + C(~', T)
]
dr.
(2.2.36) Integration of the right side of Eq. (2.2.36) by parts with the use of Eqs. (2.2.7) and (2.2.23) implies that
f
[1
]
t 0 ~-~r[E(T) + Q(t, ~')] E(T) + C(T, T) dT
I1
= [E(t) + Q(t, t)] E(T) + C(t, T)
-
~
t
]-
[1
[E(T) + Q(t, T)] E(T) + C(T, T)
]
0C
[E(r) + Q(t, r)]-~r (r, T l d r
Q(t,T) E(T)
=E(t) IE~T ) + C ( t , T ) ] - 1
fTt [E(I")+
Q(t,
°3C(~', T) d~'. ~')]-~r
Substitution of this expression into Eq. (2.2.36) results in
Q(t,s) + E(s)
f
t
OC
[E(I-)+ Q(t, ~-)]-~r (1-, s) d~- = 0.
(2.2.37)
Equation (2.2.37) is a linear Volterra equation for the relaxation measure Q(t, s) provided that the creep measure C(t, s) and Young's modulus E(t) are given. Introducing the notation
M(t, s) = 1 + E(s)C(t, s),
(2.2.38)
we rewrite Eq. (2.2.37) in the form
Q(t, s) +
f tO--~rM(l-, s)Q(t, r) dr = - ft~T E(r)
(r, s) dr.
(2.2.39)
Equation (2.2.39) can be solved using the standard numerical methods for linear Volterra equations [see, e.g., Brunner and van der Houwen (1986) and Linz (1985)].
2.2.3
A M o d e l of A d a p t i v e L i n k s
Our objective now is to demonstrate that the response in an aging viscoelastic medium may be described by a network containing only elastic elements (without dashpots) provided the springs replace each other according to a given law. For this purpose,
Chapter 2. Constitutive Models in Linear Viscoelasticity
42
we transform Eq. (2.2.3) as follows:
or(t) = X(t, t)e(t) -
fo t -~r O~x(t, r)e(t)dr
+
fOt -~r O~x(t, r)[e(t)
- e(r)] d r
= X(t, 0)e(t) 4- foot -~-T OX (t, r)[e(t) - e('r)] d r
OX (t, r)eO(t, r ) d r , = X(t, 0)e(t) 4- ~o t -~-
(2.2.40)
where e¢(t, r) = e(t) - e(r) is the relative strain for transition from the actual configuration at instant r to the actual configuration at instant t. For definiteness, an interpretation of Eq. (2.2.40) is provided for polymeric materials. However, the proposed concept may be applied to an arbitrary viscoelastic medium. Let us consider a system of parallel elastic springs (which model links between chain molecules). At the initial instant t = 0, the system consists of X.(0, 0) links in the natural (stress4ree) state. Rigidity of any spring equals c. Within the interval [ r , r + dr], aX, ~(t, 8r
r)]t=~ d r
new links merge with the system. These links are connected in parallel to the initial links, and they are stress-free at the instant of their appearence. The latter means that the natural configuration of links arising at instant r coincides with the actual configuration of the system at that instant. The strain at instant t in links arising at instant r equals e <>(t, r). Due to the breakage process, some links annihilate. The number of initial links existing at instant t equals X,(t, 0), whereas the amount 8X,
~(t,
r) dr
determines the number of links arising within the interval [r, r + dr] and existing at instant t. To calculate the response in a network of parallel links, stresses in all the links should be added
o'(t) = o'o(t) +
do'(t, "r).
(2.2.41)
Here o0(t) is the stress at instant t in the initial links, and do'(t, r) is the stress at instant t in links joining the system at instant r. It follows from Hooke's law that
~ro(t) = cX,(t, 0)e(t), tgX, "r)[e(t) - e(r)] d'r. dtr(t, "r) = c -tgX, ~ z ( t , r)e~(t, 7")d'r = c-~r(t,
2.2. Integral Constitutive Models
43
Substitution of these expressions into Eq. (2.2.41) yields
{
fOt°gx*
or(t) = c X,(t, O)e(t) +
--~r (t, ~')[e(t)- e(T)] aT
}
t °~X
= X(t, O)e(t) +
fO
-~r (t, ~')[e(t) - e(r)] dr,
(2.2.42)
where
X(t, T) = cX.(t, ~').
(2.2.43)
Since expressions (2.2.40) and (2.2.42) coincide, the behavior of this system of adaptive links coincides with the behavior of an aging linear viscoelastic medium, which means that a system of adaptive links may model the mechanical response in a linear viscoelastic material. The reason for this assertion lies deeper than a simple coincidence of equations. A polymeric material may be treated as a network of long molecules mutually linked by chemical and physical crosslinks and entanglements. The chains move relatively to each other (micro-Brownian motion). When the relative displacement of two portions connected by a link reaches some ultimate value, the link breaks, and chains acquire "free edges" that are ready to create new links. These links emerge when appropriate free edges are located sufficiently close to each other owing to random wandering. After their onset, new links oppose the displacements of chains relative to their positions at the instant when the links arise. This scenario for the interaction of polymeric molecules coincides with the preceding scenario for a system of elastic springs, provided crosslinks and entanglements are treated as appropriate springs. The function X(t, T) is an average (deterministic) characteristic of random motion of chains at the microlevel. The quantity X(t, T) is proportional to the number of links arising before instant ~"and existing at instant t. The derivative
OX ~ ( t , "r) determines the rate of creation (at instant ~-) of new links which have not been broken before instant t. To determine potential energy of a network of parallel elastic springs, we add together the mechanical energies of individual links. The potential energy of the initial links existing at instant t equals C
-~X.(t, 0)EZ(t). The potential energy (at instant t) of links joining the system at instant ~"is calculated as
c OX,(t ' ,r)[e~(t ' T)]2 dT. 2 0~-
44
Chapter 2. Constitutive Models in Linear Viscoelasticity
Summing up these expressions, we obtain the potential energy of the entire network (strain energy density of an aging viscoelastic medium)
W(t) = -~ X,(t, O)e2(t) + 1 2
{
X(t,o)eZ(t) +
---~r (t, r)[e<>(t, r)] 2 dr
f0t
~(t,
r)[e(t) -- e(r)] 2 dr
}
.
(2.2.44)
Dafermos (1970) employed an expression similar to Eq. (2.2.44) as a Lyapunov functional for an aging, linear viscoelastic medium. A rheological model of elastic links between polymeric chains was suggested by Green and Tobolsky (1946). In that work, one-dimensional constitutive equations were proposed for elongation and shear of non-aging polymers with the exponential relaxation kernel. Yamamoto (1956) generalized the Green-Tobolsky concept and developed a statistical theory that permits relaxation kernels to be determined under some assumptions regarding breakage of polymeric chains. As a result, integro-differential equations were derived for a chain-distribution function, a chain-reformation function, and a chain-breakage function. For a comprehensive exposition of statistical models in viscoelasticity, see, e.g., Lodge (1989). Two shortcomings of the Yamamoto approach may be mentioned: (i) it is too cumbersome for engineering applications, since it requires integro-differential equations for chain-distribution functions to be solved, and (ii) no experimental confirmation exists for relations between chain-distribution and chain-reformation functions.
2.2.4
Spectral Presentation of the Function X(t, r)
Our purpose now is to derive an integral equation for the function X.(t, r) and to solve it. We suppose that adaptive links are divided into two types: the links of type I are not involved in the process of replacement, whereas the links of type II take part in this process. Denote by X E [0, 1] concentration of links of type I, and by 1 - X concentration of links of type II. Let g(t - r, r) be the relative number of links which have arisen at instant r and have lost before instant t. We can write
X.(t, 0) = X.(0, 0){X + (1 - X)[1 - g(t, 0)1}, OX, ~(t, Or
"r) = ~(r)[1 -- g(t -- r, r)],
(2.2.45)
where
OX,
• (r) = --~--(t, r){t=~
(2.2.46)
45
2.2. Integral Constitutive Models
is the rate of creation for new links. The total number of links at instant t is calculated
as f0 t -~T t~X,(t, ~') d~'.
X , ( t , t) = X , ( t , O) +
(2.2.47)
Substitution of expressions (2.2.45) into Eq. (2.2.47) with the use of Eqs. (2.2.5) and (2.2.43) implies that E(t) = E(0){X + (1 - X)[1 - g(t, 0)]} + c
~('r)[1 - g(t - % "r)] d-r.
(2.2.48)
Setting E,(t) -
E(t)
E(0)
-
X , ( t , t)
dO, U) -
X,(0, 0)'
c~(t)
E(0)
1
OX, - - (t, t), X,(0, 0) &-
-
we rewrite Eq. (2.2.48) as E , ( t ) = X + (1 - X)[1 - g(t, 0)] +
~,('r)[1 - g(t - %-r)] d-r.
(2.2.49)
For a given dimensionless Young's modulus E , ( t ) , Eq. (2.2.49) imposes restrictions on the functions ~ , ( t ) and g(t - ~', "r). In the general case, these functions cannot be found uniquely from Eq. (2.2.49). However, for non-aging media this equation allows us to derive explicit expressions for ~,(~') and g(t - "r, ~'). Indeed, for a non-aging material, E , ( t ) = 1,
dO, U) = alp,,
g(t - ~', ~') = go(t - ~').
(2.2.50)
Substitution of expressions (2.2.50) into Eq. (2.2.49) yields (1 - X)go(t) = ~ ,
/o t[ 1 -
go(t - "r)] d r = ~ ,
/o
[ 1 - g0(r)] dr. (2.2.51)
Differentiation of Eq. (2.2.51) with respect to time implies that dgo ~, - ~ (1 - go), dt 1 - X
The unique solution of Eq. (2.2.52) is go(t) = 1 - exp
g0(0) = 0.
(°,) - 1 - Xt
.
(2.2.52)
(2.2.53)
To find the function X(t, q-), we substitute expressions (2.2.50) and (2.2.53) into the equality t
X(t, T) = X(t, t) -
OX --~s (t, s) ds
* ft)t -~ x * --z--(t, l a s s) ds = c [ x (t,
(2.2.54)
46
Chapter 2. Constitutive Models in Linear Viscoelasticity
and obtain with the use of Eqs. (2.2.45) and (2.2.53) X ( t , ~') = E(0)
1 - ~,
[1 - go(t - s)] ds
= E(0) {1 - (1 - X) [1 - exp ( - 1 _ X Equation (2.2.55) expresses the relaxation function of a non-aging viscoelastic medium in terms of the rate of reformation q~, and the breakage function go(t). It follows from Eqs. (2.2.4) and (2.2.55) that the only relaxation measure for a non-aging viscoelastic material coincides with the relaxation measure of the standard viscoelastic solid [see Eq. (2.1.7)], Q0(t) = - ( 1 - X) [1 - exp ( - 1~*t _ X)1.
(2.2.56)
The constitutive relations (2.2.42) and (2.2.55) describe a network with only one kind of links. Observations demonstrate that several different kinds of links may be distinguished, such as "elastically active long chains" "elastically active slide and entanglement chains," and "elastically active short chains" [see He and Song (1993)]. Drozdov (1992, 1993) proposed a version of the model of adaptive links with M different kinds of links. Any kind of links is characterized by its strain energy density and relaxation measure. Links of different kinds arise and break independently of one another. Denote by ~m concentration of the mth kind of links (the ratio of the number of links of the mth kind to the total number of links), by ~m(~') and gm(t - r, T) the rates of creation and breakage for these links, and by Xm concentration of nonreplacing links (m = 1. . . . . M). The parameters T~mare assumed to be time-independent. The balance law for mth kind of links states that the total number of links of the mth kind at instant t rlmX,(t, t)
equals the sum of the number of initial links existing at instant t 'l~mX,(0 , 0){Xm "q- (l -- Xm)[ 1 - gm(t, 0)]}
and the number of links arising within the interval (0, t] and existing at instant t. The latter quantity is calculated as follows. Within the interval D', ~" + d~'], T/mX, (0 , 0)(I)m, (T) d~-
new links of the mth kind appear. At instant t, their number reduces to
~mX,(O,0)(I)m,(T)[1 -
gm(t - ~', "r)] dr.
(2.2.57)
47
2.2. Integral C o n s t i t u t i v e Models
Summing up these amounts for various intervals, we obtain rlmX,(O, 0) f 0 t @m,('r)[1 - gm(t -- T, ~')] d~'.
As a result, we arrive at the integral equations
/0 t ~m,('r)[1
E , ( t ) = Xm + (1 - Xm)[1 - gm(t, 0)] +
- gm(t -- 7, "r)] d'r, (2.2.58)
which should be satisfied for m = 1 . . . . . M. It follows from Eqs. (2.2.45) and (2.2.57) that olX,
M
O---~(t, 1") = X,(0, 0) Z
~mCI)m*('r')[1 -- gm(t -- "r, 1")].
(2.2.59)
m=l
Substitution of expression (2.2.59) into Eq. (2.2.54) implies that X ( t , T) = E(O)
{
st
E,(t) -
(I)m,(S)[1 - gm(t - s, s)] d s
"l~m
/
.
(2.2.60)
m=l
For a non-aging medium (2.2.50), Eq. (2.2.58) is solved explicitly
~m,t ) gm 0(t) = 1 - exp
- 1 - Xm
(2.2.61)
"
Substitution of expressions (2.2.50) and (2.2.61) into Eq. (2.2.60) yields X(t,T)=E(O)
{£ 1-
"rlm(1--Xm )
(I)m, -1-Xm
1-exp
m=l
It follows from Eqs. (2.2.4) and (2.2.62) that Qo(t) = - Z
]'Lm 1 - exp
-~m
'
(2.2.63)
m=l
where ].Lm = rim(1 -- Xm),
Tm -
1-
~ .
Xm
dPm,
(2.2.64)
Equation (2.2.63) implies that the relaxation measure of an arbitrary non-aging viscoelastic material equals a sum of exponential functions with positive coefficients. Differentiating Eq. (2.2.63) with respect to time and using Eq. (2.2.17), we find the relaxation kernel M
m,m
exp
m-
1
-
Chapter 2. Constitutive Models in Linear Viscoelasticity
48 Assuming that M ---, oo and
]£m ~ ]£(Tm)(Tm+m - Tm), we arrive at the presentation of the relaxation kernel for a non-aging viscoelastic medium R ( t ) = j0 "°°/z(T) T exp ( t-)- ~
dT
(2.2.65)
with a nonnegative relaxation spectrum/x(T). The nonnegativity condition for the relaxation spectrum was discussed in details by Beris and Edwards (1993) and Pipkin (1972).
2.2.5
Three-Dimensional Loading
It follows from Eqs. (2.2.9) and (2.2.26) that in order to construct a constitutive equation in linear viscoelasticity it suffices to replace Young's modulus in Hooke's law by an appropriate Volterra operator. In Eq. (2.2.9), Young's modulus E is replaced by the relaxation operator E(I - R), and in Eq. (2.2.26), the elastic compliance E -1 is replaced by the creep operator E-I(I + K). Here I is the unit operator, and for an arbitrary smooth function f(t),
K f = fOOt K(t, r ) f (r) dr,
Rf =
~0t R(t, r)f(r) dr,
(2.2.66)
where K(t, r) and R(t, r) are the creep and relaxation kemels. It is natural to suppose that the same procedure (replacement of elastic moduli by Volterra operators) may be carried out for three-dimensional loading as well. However, even for an isotropic elastic medium, several different versions of constitutive equations exist, and any version is determined by at least two elastic moduli. When we replace these moduli (or only one of them) by integral operators, we obtain different versions of constitutive equations in viscoelasticity, and we are faced with the problem of choosing appropriate constitutive relations. We confine ourselves to two different versions of constitutive equations for an isotropic elastic medium. According to the first, we write E (5+ v j) 6"- 1 + v 1 - 2------~
'
~=
1 ~7[(1 + v ) 8 -
vor?],
(2.2.67)
where 6" is the stress tensor, 5 is the strain tensor, o- = 11(6") and e = 11(5) are the first invariants of these tensors, and v is Poisson's ratio. Replacing Young's modulus E by an appropriate integral operator and assuming Poisson's ratio to be constant, we obtain the following constitutive equations in linear viscoelasticity: E (I-R)(~+ & - 1+ v
v d) 1 - 2-------~ '
~__ 1 b7(I + K)[(1 + v)& - vo-]]. (2.2.68)
2.2. Integral Constitutive Models
49
Experimental data show that Poisson's ratio of viscoelastic materials can change in time [see, e.g., Bertilsson et al. (1993), Ladizecky and Ward (1971), Nielsen (1965), Popov and Khadzhov (1980), Powers and Caddell (1972), Shamov (1965), Stokes and Nied (1988), and Theocaris (1979)]. To account for a dependence of Poisson's ratio u on time, we employ the other version of constitutive equations for an isotropic linear elastic medium o" = 3Ke,
~ = 2GO.
(2.2.69)
Here ~, ~ are the deviatoric parts of the strain and stress tensors, 1
1 ^
= ~o-7 + ~,
~ = ~I
+ ~,
and K , G are bulk and shear elastic moduli, which are connected with Young's modulus E and Poisson's ratio v by the formulas K =
E 3(1 - 2v)'
G =
E 2(1 + v)'
9KG 3 K + G'
E=
v=
3K-
2G
2 ( 3 K + G)" (2.2.70)
Replacing K and G in Eqs. (2.2.69) by appropriate Volterra operators, we obtain the following constitutive equations for a linear viscoelastic medium: o- = 3 K ( I -
Rb)~,
~ = 2 G ( I - Rs)~,
(2.2.71)
where Rb and Rs are bulk and shear relaxation operators with kernels Rb(t, ~) and Rs(t, T). Since a number of viscoelastic materials demonstrate purely elastic bulk response, Eqs. (2.2.71) can be simplified by setting Ro = 0 and Rs = R, o- = 3Ke,
~ = 2 G ( I - R)~,.
(2.2.72)
It is of essential interest to compare constitutive equations (2.2.68) and (2.2.72) with experimental data. For this purpose, we study two regimes of loading of a viscoelastic medium: pure shear and uniaxial extension. We consider a non-aging viscoelastic specimen in the form of rectilinear rod and introduce Cartesian coordinates {Xl, x2, x3}. For pure shear in the plane (Xl, x2), the only nonzero component of the stress tensor 6" is o'12, and the only nonzero component of the strain tensor ~ is e12. Constitutive Eqs. (2.2.68) and (2.2.72) imply that
/o /ot
1
R(t - "r)elz('r) d'r ,
E v elz(t) o'12(t)- 1 +
[
o'12(t) = 2G el2(t) -
R(t
-
-
'r)el2('i" ) d
In relaxation tests with ~12(t) -- ~0,
Leo,
t < 0 t > O,
.
(2.2.73)
50
Chapter 2. Constitutive Models in Linear Viscoelasticity
the drop in tangential stress is characterized by the function 0"12(t)
rz(t) = 1 -
0"12(0 ) "
Both equalities (2.2.73) yield the formula r2(t) = f 0 t R(T) dT.
(2.2.74)
For uniaxial extension in the Xl-direction, the only nonzero component of the stress tensor 6- is o'11, while the strain tensor ~ has three nonzero components •11 and E22 - - E 3 3 . According to the model (2.2.68), these quantities are expressed in terms of the strain •11 as (2.2.75)
• 22 "- E33 -" -- 1"•11,
with a time-independent Poisson's ratio v. It follows from Eq. (2.2.75) that • =
(1 -
2/~,)•11.
Substituting this expression into Eq. (2.2.68), we find that O'll(t) = E
I
E l l ( t ) --
/otR(t --
7") El l ('r) d
.
(2.2.76)
In relaxation tests with Ell
(t) = ~0' Le0,
t < 0 t>0,
(2.2.77)
fOOt R(~') d~',
(2.2.78)
Equation (2.2.76) implies that rl (t) = where rl (t) = 1 --
0-11(t) 0-11(0)
is a function that characterizes the drop in tensile stress. It follows from Eqs. (2.2.74) and (2.2.78) that the constitutive Eqs. (2.2.68) lead to the Coincidence of the functions rl(t) and r2(t) for tensile and shear tests. Let us now return to the constitutive Eqs. (2.2.72). We calculate the first invariant of the strain tensor - Ell + 2E22,
51
2.2. Integral Constitutive Models
and the nonzero components of the deviatioric part of the strain tensor 2
ell ---- ~(ell -- E22),
1
e22 ----e33 -- --~(Ell -- E22),
and substitute these expressions into Eq. (2.2.72). As a result, we obtain the first invariant and the nonzero components of the deviatoric part of the stress tensor 0"(t) = 39([ell(t) + 2ezz(t)], Sll(t) = ~4 G
{ [ell(t)
- e22(t)] -
f0 t R(t
- r)[ell(r) - e22(r)]d~" } ,
$22(t) = $33(t) [ell(t)- e22(t)] -- f0 t R(t- T)[Ell(T ) -- e22('r)] d'r / .
2{ = -sG
It follows from these equalities that the nonzero components of the stress tensor are Sll(t) = 9([ell(t) + 2Ezz(t)] + ~4 {G
[ell(t) - e22(t)] - ~0"t R ( t -
1")[e11(1-) - e22(r)]dr } , (2.2.79)
S22(t) = S33(t) = 9([Ell(t)+ 2E22(t)] 2G 3
-
{ [ela(t)
- e22(t)] -
f0 t R(t
- ~')[ell('r) - e22(~')]d'r } .
(2.2.80) Equating 0"22 and 0"33 to zero, we obtain from Eq. (2.2.80) 2
( 9(
+ ~1G )
=-I(9(-2G)
2 ~0"tR(t -- '1")E22('1)"d r E22(/) -- -~G 2 t ell(t)+-~GfoR(t
_
T)E11(T) d~"1 .
(2.2.81)
In this case Eqs. (2.2.79) and (2.2.80) imply that 0-11(t) = 2G
(
[ell(t)
-- Ezz(t)] -- f0 t R(t- 'r)[Ell('r) - Ezz('l')]d'l-} .
(2.2.82)
Equations (2.2.81) and (2.2.82) establish stress-strain relations for an arbitrary regime of loading. For the relaxation tests (2.2.77), Eq. (2.2.81) reads
Chapter 2. Constitutive Models in Linear Viscoelasticity
52
ezz(t) - 39( G+ G fot R(t - "r)E22('r)d'r
[ 2(35(; 3 K - 2 G++G) G f o o39( + G
t R(~')dr ] co.
(2.2.83)
Setting E22(t)
-- E33(t)
=
(2.2.84)
--~(t)eo,
and using Eq. (2.2.70), we find from Eq. (2.2.83) that ~ ( t ) = v + 1 - 2 V f o t R(t - r)[1 + ¢(r)]dr.
(2.2.85)
In the general case, the constant parameter v in Eq. (2.2.85) does not coincide with the time-dependent Poisson's ratio sr. We substitute expressions (2.2.77) and (2.2.84) into Eq. (2.2.82) and obtain after simple algebra rl (t) = 1 --
1 + ~ ( t ) - f o R ( t - 'r)[1 + ~'('r)] d'r 1 + ~'(0)
Combining this equality with Eq. (2.2.85), we find that
rl(t)= 1 -
(l+v)
1-2v
I1-2~(t) 1
1+ ~(0----~ "
(2.2.86)
Setting fro(t) = 1 + ~(t), we present Eqs. (2.2.85) and (2.2.86) as follows: ~'o(t)
_ 1 - 2v ~t
R ( t - r)sro(r)dr = 1 + v,
(2.2.87)
~ v
rl(t) = 1 --
1 -- 2v
~'0(0)
"
To validate the constitutive models (2.2.68) and (2.2.72), we plot in Figures 2.2.1 and 2.2.2 experimental data for the functions rl (t) and r2(t) obtained in tensile and torsional tests for polyethylene and poly(vinyl chloride). These data demonstrate significant discrepancies between the functions rl (t) and rE(t), which implies that the model (2.2.68) does not adequately predict the material response. To determine adjustable parameters of the model (2.2.72), we approximate the relaxation measure Qo(t) by the truncated Prony series [see Eq. (2.2.63)], M
Qo(t) = - Z/Xm[1 -- exp(--')'mt)], m=l
M
R(t) = E ]d'm]Imexp(-3'mt), m=l
(2.2.89)
2.2. Integral Constitutive Models
53
0.6 iiiii
.z
;
:f
O
o
0.2
I
10 -1
I
I
I
I
I
I
t
I
I
104
Figure 2.2.1: The dimensionless parameters rl and rE versus time t (hr). Circles show experimental data for polyethylene obtained by Popov and Khadzhov (1980). Curve 1 (torsion): fit of experimental data using the relaxation measure (2.2.89) with jb[, 1 - 0.178, /x2 = 0.399, 3'1 = 0.168, and 3'2 = 18.050 (dotted line shows the results of numerical simulation). Curve 2 (extension): prediction of the material response in tensile test with v = 0.13.
and determine parameters /d,m and ~/m, which ensure the best fit of experimental data for the function r2(t). Because the number of experimental data is rather small (less than 10), we confine ourselves to M = 2. Afterward, we solve Eq. (2.2.87) numerically with an arbitrary v and calculate the function rl(t) according to Eq. (2.2.88). The adjustable parameters v is chosen to ensure the best fit of experimental data for the function rl(t). Results of numerical simulation demonstrate that the constitutive model (2.2.72) correctly predicts the viscoelastic response at pure shear and uniaxial extension.
Chapter 2. Constitutive Models in Linear Viscoelasticity
54 0.25
O
I
I
I
I
I
I
10 -1
I
I
t
I 103
Figure 2.2.2: The dimensionless parameters rl and r2 versus time t (hr). Circles show experimental data for PVC obtained by Popov and Khadzhov (1980). Curve 1 (torsion): fit of experimental data with the use of the relaxation measure (2.2.89) with/xl = 0.094, /x2 = 0.103, 3'1 = 0.38, and 3'2 = 18.00 (dotted line shows the results of numerical simulation). Curve 2 (extension): prediction of the material response in tensile test with v = 0.09.
2.3
Creep and Relaxation Kernels
This section deals with creep and relaxation operators for linear viscoelastic media. We provide several examples of creep and relaxation measures and compare theoretical results with experimental data. Afterward, general features of creep and relaxation measures are discussed. For simplicity, we confine ourselves to uniaxial deformations.
2.3.1
Creep
and Relaxation
Kernels
for Nonaging
Media
Two types of relaxation measures are distinguished: regular and singular. A measure Q is called regular, provided it is twice continuously differentiable. If a measure Q
2.3. Creep and Relaxation Kernels
55
is only differentiable, and its derivative, the relaxation kernel R, has an integrable singularity, then the measure Q is called weakly singular.
Regular Measures We begin with regular relaxation measures for non-aging viscoelastic media. The simplest measure corresponds to the standard viscoelastic solid, see Eq. (2.1.7), Qo(t) = - x [ 1 -
exp ( - T )
(2.3.1)
],
where X is a material viscosity, and T is the characteristic time of relaxation. Differentiation of Eq. (2.3.1) with the use of Eq. (2.2.17) implies the formula for the relaxation kernel
x
R(t) = ~ e x p
( -t~)
(2.3.2)
.
The following advantages of the model (2.3.1) may be mentioned: 1. Equation (2.3.1) has a simple mechanical interpretation: it reflects the response in a rheological system consisting of two springs and a dashpot. 2. The creep kernel for the relaxation kernel (2.3.2) can be found explicitly. 3. Equation (2.3.1) describes qualitatively the material response observed in creep and relaxation tests. An important drawback of Eq. (2.3.1) is poor agreement with experimental data. To obtain a more sophisticated expression for the relaxation measure, truncated Prony series (finite sums of exponential functions) are used [see Soussou et al. (1970)], M
a°(t)=-ZXm [1-exp(--~m)
(2.3.3)
m=l
where Xm are material viscosities and Tm are the characteristic times of relaxation. The creep measure Co(t) corresponding to the relaxation measure (2.3.3) is also presented as a truncated Prony series, Co(t)=Z/3m m=l
E (')] 1-exp
-~
,
(2.3.4)
Tm
where ~3m a r e material viscosities, and ~'m are the characteristic times of retardation. As common practice, the characteristic times Tm and the characteristic viscosities Xm increase in m, while the parameters ]3m depend on m nonmonotonically [see, e.g., Kochetkov and Maksimov (1990)]. Equations (2.3.3) and (2.3.4) predict correctly experimental data for a number of viscoelastic materials, when the integer M is of the range from 5 to 15. The only drawback of these equations is a large number of adjustable parameters [see, e.g., Koltunov (1966)].
Chapter 2. ConstitutiveModels in Linear Viscoelasticity
56
Differentiation of Eqs. (2.3.3) and (2.3.4) with respect to time implies the formulas for the relaxation and creep kernels
zXrn (-~m) M
R(t) =
~m exp
m=l
K(t)=Z~exp
m={ "rm
-
,
(t)
--. "rm
(2.3.5)
A natural generalization of Eqs. (2.3.5) is an infinite sum of exponential functions, which may be presented in the integral form
R(t) = fo~ X(TT)exp ( - T ) dT, K ( t ) = f0 °~/3(T) T exp ( t-)~
dT,
(2.3.6)
where x(T) is the relaxation spectrum, and/3(T) is the retardation spectrum. Determination of relaxation and retardation spectra based on data in the standard creep and relaxation tests is an ill-posed problem. Some approaches to solving this problem are presented in Kaschta and Schwarzl (1994a, b) and Tschoegl (1989). The relaxation function Xo(t) = 1 + Qo(t) of the standard viscoelastic medium is calculated as
Xo(t) = where
~2 =
~ 2 -k- (1 -
~2) exp -- ~- ,
(2.3.7)
1 - X. The Laplace transform Xo(p) of the function Xo(t) equals
Xo(p) = P
+ 1
'
(2.3.8)
where p is the dual variable. To extend expression (2.3.8), Achenbach and Chao (1962) suggested the following formula for the Laplace transform of the relaxation function: "~°(P)= p-1 (P~T+6)2+l It is easy to check that the function
(2.3.9)
2.3. Creepand Relaxation Kernels
57
provides the inverse Laplace transform for the function (2.3.9). The corresponding relaxation measure
Qo(t)=-x
{ [ (l lx>t] 1-
1-
(~)
1 + v/l-
t}
exp(-~)
(2.3.10)
X
is characterized by two adjustable parameters X and T. As common practice, relaxation measures of non-aging viscoelastic materials are positive, decreasing, and concave. The Achenbach-Chao relaxation measure (2.3.10) is neither strictly decreasing nor concave. Moreover, it becomes negative for sufficiently small 6. These features make applications of Eq. (2.3.10) questionable. Another generalization of formula (2.3.1) is provided by the KohlrauschWilliam-Watts (stretched exponential) relaxation measure
Qo(t)=-x
1-exp
,
-
(2.3.11)
which is characterized by three adjustable parameters a, X, and T. Equation (2.3.11) is widely used to fit experimental data [see, e.g., Dean et al. (1995), Garbarski (1992), and Scanlan and Janzen (1992)]. A significant drawback of the relaxation measure (2.3.11) is that the corresponding creep measure Co(t) cannot be expressed in terms of elementary functions. Wortmann and Schulz (1994a, b) suggested employing the cumulative lognormal distribution as the relaxation measure
Qo(t) =
x/~[3
gt exp - ~
/3
(2.3.12)
where c~ and/3 are adjustable parameters. The model (2.3.12) correctly describes the mechanical response in several semicrystalline polymers. Askadskii (1987) and Askadskii and Valetskii (1990) proposed considering a viscoelastic medium as a system of interacting oscillators (microvoids). That model implies the following expression for the relaxation measure:
Qo(t) =-a
fo'Elr~21 + (f('r) -
1
j
a)ln(f(-r) - a) + (1 - f(-r) + a)ln(1 - f(-r) + a) d'r, (2.3.13)
where
f (t) = (1 + bt) -~, and a, b, a, and/3 are adjustable parameters. Equation (2.3.13) provides fair agreement with experimental data in relaxation tests for polyoxadiazole and polyamide.
58
Chapter 2. Constitutive Models in Linear Viscoelasticity
Weakly Singular Measures Findley et al. (1989) and Rabotnov (1969) suggested employing the power-law relaxation measure Qo(t) = -
,
(2.3.14)
where ~ E (0, 1) and T > 0 are adjustable parameters. Differentiation of Eq. (2.3.14) implies that R(t) = ~c~_l(t),
(2.3.15)
where = a F ( a ) T -~, J~(t) is the Abel kernel (2.1.9), and F(z) is the Euler gamma-function (2.1.10). The creep kernel K(t), corresponding to the relaxation kernel (2.3.15), reads K(t) = ~Za_ 1(t, ~),
(2.3.16)
where the fractional-exponential function Z~(t, A) is determined as the resolvent kernel for the kernel J~(t). The latter means that the unique solution x(t) of the Volterra equation t (t-
x(t)- A
fO
S) a
F(1 + a) x(s)ds = f(t)
(2.3.17)
is presented in the form x(t) = f ( t ) + X
I'
Z~ (t - s, X)f(s) ds.
(2.3.18)
For ~ E (0, 1), the function Z~(t, I ) cannot be expressed in terms of elementary functions. By analogy with Eq. (2.3.16), Rabomov (1969) proposed to present the creep kernel in the form g(t) = 71Z~_1(t,-n),
(2.3.19)
where c~ E (0, 1) and r/ > 0 are adjustable parameters, and Z~ is the fractionalexponential function. A serious drawback of Eq. (2.3.19) is the necessity to express both the creep and relaxation kernels in terms of special functions. By analogy with Eq. (2.3.14), it is natural to assume the creep measure to be a power-law function C0(t) =
,
(2.3.20)
where c~ E (0, 1) and T > 0 are adjustable parameters. The corresponding creep kernel reads K(t) = ~lJ~-1 (t),
(2.3.21)
2.3. Creepand Relaxation Kernels
59
and the relaxation kernel is written as
R(t) = rl 1/s sin( Tra fo °° Ig2sus+exp(-rll/sut)2u s cos(Tra)du+ 1"
(2.3.22)
Rzhanitsyn (1968) proposed a refined version of the creep kernel (2.3.21), the so-called generalized fractional-exponential function
K(t) = r/Js-1 (t) exp(-/3t),
(2.3.23)
where a E (0, 1), 13 > 0, and r / > 0 are adjustable parameters. The corresponding relaxation kernel reads
R(t) = ~/1/s exp(-/3t) sin(Trc~) ~~ 7/"
uS exp(-rll/sut)du U2 s
(2.3.24)
+ 2u s cos(Tra) + 1"
Garbarski (1992) demonstrated that Eq. (2.3.23) does not provide essential improvement in fitting experimental data compared to the kernel (2.3.21). To construct new creep and relaxation kernels, Garbarski (1992) suggested prescribing explicit expressions for the Laplace transforms of creep and relaxation kernels, which permit integral presentations to be derived for these kernels. As an example, the so-called root function is proposed /((p) =
l+av~
(2.3.25)
l + av/-fi + bp'
where a and b are material parameters. Equation (2.3.25) implies the following integral formulas for the creep and relaxation kernels:
K(t)
ab f ~ U3/2 exp(-ut) du 7r Jo (bu- 1)2 + a 2 u '
ac f ~ u3/2 exp(-ut) du R(t) = ~ (cu- 1)2 + a z u
(
c=
1) b .
(2.3.26)
Equations (2.3.26) demonstrate fair agreement with experimental data for several industrial polymers. However, the use of expressions (2.3.26) in engineering is questionable, because of the complicated expressions for creep and relaxation kernels.
2.3.2
Creep and Relaxation Kernels for Aging Media
The material response in an aging, linear, viscoelastic medium is characterized by the functions of two variables X(t, T) and Y (t, ~-). To determine these functions by fitting experimental data in the standard tests, a huge number of observations is necessary. To reduce this number, additional hypotheses are introduced, which permit these functions to be presented as superpositions of several functions of one variable. Two main approaches may be distinguished to constitutive equations for aging viscoelastic media. The first goes back to Struik (1978), who suggested that the
60
Chapter 2. Constitutive Models in Linear Viscoelasticity
material aging can be treated in the framework of the constitutive models for nonaging media by introducing an intemal time, similar to the intemal (pseudo) time in thermoviscoelasticity. In short-term creep tests, Struik (1978) demonstrated that creep curves for amorphous polymers corresponding to different elapsed times coincide after shift along the logarithmic time axis ("horizontal" shift). This assertion was checked in a number of studies [see, e.g., Espinoza and Aklonis (1993), McKenna (1989), Plazek et al. (1984), and Struik (1978, 1987a, b), to mention a few]. It has been shown that the Struik hypothesis is confirmed by experimental data for amorphous polymeric materials in short-term tests. Its extrapolation to long-term tests for amorphous polymers [see, e.g., Brinson and Gates (1995), Dean et al. (1995), and Matsumoto (1984)], as well as to semicrystalline polymers [see Struik (1987a, b)], may lead to significant discrepancies between experimental data and their prediction. To reduce these discrepancies, some "vertical" shift is introduced for creep and relaxation curves. Arutyunyan (1952) introduced simple phenomenological assumptions, which permit the function of two variables X(t, ~) to be reduced to several functions of one variable. According to the Arutyunyan model, two functions are employed instead of the function X(t, ~'): the current Young's modulus E(t) and the relaxation measure Q(t, ~). It is assumed that the current Young's modulus E(t) is positive, increases monotonically in time, and tends to some limiting elastic modulus E(oo) as time tends to infinity. The derivative
dE ---(t) dt is nonnegative for t --> 0, and it vanishes as time approaches infinity. To provide an interpretation of these assumptions in the framework of the model of adaptive links, we recall that the current Young's modulus E(t) is proportional to the number of adaptive links existing at instant t. The Arutyunyan hypotheses regarding the Young's modulus mean that • At any instant t - 0, the number of adaptive links is positive. • This number tends to some limiting value as time tends to infinity. • The rate of increase in the number of adaptive links is positive and vanishes with the growth of time. The dependence E(t) is approximated either by the exponential function F
I
t \l
E ( t ) - E ( 0 ) + [ E ( ~ ) - E(0)] [ 1 - exp ~ - ~ ) A ,
(2.3.27)
or by the stretched exponential function [see, e.g., Gul et al. (1992)],
E(t)=E(O)+[E(~)-E(O)]{1-exp[-(T)~]}.
(2.3.28)
Here E(0) is the initial elastic modulus, E(~) is the equilibrium elastic modulus, T is the characteristic time of aging, and 7 E (0, 1) is an adjustable parameter.
2.3. Creep and Relaxation Kernels
61
Expressions (2.3.27) and (2.3.28) provide fair agreement with experimental data for polypropylene, polyethylene, isobutylene, and nitride rubber. Arutyunyan (1952) proposed the following expression for the creep measure C(t, T):
C(t, I-) = th(~){1 - exp[-~/(t - ~-)]},
(2.3.29)
where th(~') is an aging function, and 3/is the characteristic rate of creep. It is assumed that the function th(~') is positive, decreases monotonically in time, and tends to some positive limiting value as ~"---* oo. Two expressions are employed for the aging function th(~'). The first was suggested by Arutyunyan (1952), N
d~(T) = ao + Z n=l
(2.3.30)
an T-~-T n
and the other was proposed by Prokhopovich (1963), N
dp(7")=ao+Zanexp(-'r) n=l
(2.3.31) Tn
where an and Tn are adjustable parameters. An important advantage of the creep measure (2.3.29) is that an appropriate relaxation measure Q(t, T) may be found explicitly. For this purpose, we differentiate twice the constitutive Eq. (2.2.24), use Eq. (2.3.29), and arrive at the ordinary differential equation
with the initial conditions o-(0) = 0,
dodE d-T(O) = E(O)~-~-(O).
Integration of Eq. (2.3.32) implies the constitutive Eq. (2.2.3) with the relaxation measure
Q(t, "r) = -'yE(~')th(l")
i tE(s)exp [-3t i s(1 + E(~)dp(~))d~
ds.
(2.3.33)
Despite the presence of explicit expressions for the creep and relaxation measures, applications of Eq. (2.3.29) are rather limited because of poor agreement between experimental data and the model predictions [see Drozdov (1996a)]. To refine the Arutyunyan model, we replace Eq. (2.3.29) by the equality
C(t, r) = ch(~')F(t - T),
(2.3.34)
Chapter 2. Constitutive Models in Linear Viscoelasticity
62
where th(~') is an aging function, and F(t) is some creep measure for a non-aging material. It follows from Eq. (2.3.34) that for any elapsed time te >- 0
C(t + te, te) = +(te)F(t),
(2.3.35)
which means that the creep measures corresponding to different elapsed times te should be proportional to each other
C(t + t~e,tie) _ th(t~e) C(t + te, re)
(~(te)"
Equation (2.3.35) implies that graphs of the creep measure C plotted versus time t in bilogarithmic coordinates may be obtained from each other by shift along the vertical axis. This assertion is in fair agreement with observations for a number of
log C
o o © ©
o
o
o
6
8
8
°
o
o
o
0 0 0 0
0 0 0
0 0 0
0 0
©
-D
-2
I
0
I
I
I
I
I
I
log t
I
I
4
2.3.1: The creep measure C(t + te, t¢) (GPa -1) versus time t (min) for tensile creep in polypropylene PP-43 quenched from 120°C to 20°C and preserved time t¢ (days) before loading. Circles show experimental data obtained by Struik (1987a). Curve 1" te = 0.25; Curve 2: te = 1.0; Curve 3: te = 3.0; Curve 4: te = 10.0; Curve 5: te = 30.0. Figure
63
2.3. Creep and Relaxation Kernels
log C
0 0 0 0
0 0 0 0
0 0
-3
I -1
I
I
I
I
0 0 0 0
I
I log t
0 0 0
I
t 2
Figure 2.3.2: The creep measure C(t + re, re) (GPa -1) versus time t (min) for torsion of polypropylene specimens PP-62 quenched from 120 to -20°C and preserved time te ( m i n ) before loading. Circles show experimental data obtained by Struik (1987a). Curve 1" te = 21; Curve 2: te = 45; Curve 3: te -- 90; C u r v e 4: te - 180; C u r v e 5: te = 360.
amorphous and semicrystalline polymers (see, e.g., Figures 2.3.1, 2.3.2, and 2.3.3, which demonstrate affinity of the creep measures). From the physical standpoint, Eq. (2.3.34) may be treated as some version of the separability principle, which states that processes of creep and aging are independent of each other. Evidently, the Arutyunyan formula (2.3.29) is a particular case of Eq. (2.3.34). According to Eq. (2.3.34), the mechanical behavior of an aging viscoelastic medium is determined by three material functions: the "non-aging" creep measure F(t), the aging function ~b(r), and the current Young's modulus E(r). The function F(t) can be presented using one of the expressions discussed earlier. For example,
Chapter 2. Constitutive Models in Linear Viscoelasticity
64
log C
0 o
-2
I
I
I
-1
I log t
2
F i g u r e 2.3.3: The creep measure C(t + te, te) (GPa -1) versus time t (min) for tension of an epoxy adhesive C quenched from 87 to 42°C and preserved time te (min) before loading. Circles show experimental data obtained by Vleeshouwers et al. (1989). Solid lines show their approximation by the exponential function (2.3.36) with M = 2, /31 = 0.3491 GPa -1,/32 = 0.0599 GPa -1, 3'1 = 0.06 min -1, 3'2 = 1.6 min -1. Curve 1: te = 20, r/ = 1.66; Curve 2: te = 40, ~ = 1.53; Curve 3: te = 80, r/ = 1.15; Curve 4: te = 160, r/ = 1.00; Curve 5: te = 320, r / = 0.67.
this function may be approximated by the truncated Prony series (2.3.4) M
F(t) = ~
~3m[1 - exp(-3'mt)],
(2.3.36)
m=l
where ~3m and 3'm are adjustable parameters (see Figure 2.3.3). For a number of polymeric materials, the functions oh(re) and E(te) depend linearly on the logarithm of elapsed time re:
ck(te) = Cl log te + C2,
E(te) = c3 log te + Ca,
(2.3.37)
2.3. Creep and Relaxation Kernels
65 2.0
0.3
E
0.1
I -1
I
I
I
I
I
I log
te
I
1.5
I
2
Figure 2.3.4: The aging function ~b (GPa-') and the current Young's modulus E (GPa) versus elapsed time te (days) for tensile creep in polypropylene PP-43 quenched from 120 to 20°C. Circles show experimental data obtained by Struik (1987a). Solid lines show their approximation by the linear functions (2.3.37) with c, = -0.0395, ca = 0.1799, c3 = 0.1737, and c4 = 1.7316. where Cl to c4 are adjustable parameters. Expressions (2.3.37) provide fair fit of experimental data (see Figures 2.3.4, 2.3.5, and 2.3.6). The model (2.3.36) and (2.3.37) with parameters 13m, Tin, and Cn enables us to determine the creep function Y(t, ~') for an aging viscoelastic medium using data obtained in the standard creep tests. To validate the model, we consider data obtained in the standard relaxation test for the same material (an epoxy adhesive) and compare them with results of numerical simulation. This comparison allows us to check the most important hypothesis of the model regarding multiplicative presentation of the creep measure (2.3.34). To determine the material response in relaxation tests, we solve numerically the Volterra Eq. (2.2.39) for the relaxation measure Q(t, T), and substitute the functions E(I") and Q(t, T) into the constitutive Eq. (2.2.3). Figure 2.3.7 demonstrates good agreement between experimental data and their prediction.
Chapter 2. Constitutive Models in Linear Viscoelasticity
66
1.6
0.2
4,
©
{
{
{
{
{
{
1
{ log te
{
{
] 1.4 3
Figure 2.3.5: The aging function ~b (GPa -1) and the current Young's modulus E (GPa) versus elapsed time te (rain) for torsion of polypropylene specimens PP-62 quenched from 120 to -20°C. Circles show experimental data obtained by Struik (1987a). Solid lines show their approximation by the linear functions (2.3.37) with Cl = -0.0519, c2 = 0.1803, c3 = 0.0723, and ca = 1.3461. 2.3.3
P r o p e r t i e s of C r e e p a n d R e l a x a t i o n M e a s u r e s
We begin with restrictions imposed on creep and relaxation measures of aging viscoelastic media. In this study, we confine ourselves to regular creep and relaxation measures. Experimental data show that the following inequalities are fulfilled for any 0 --- ~" < t < ~ [see, e.g., Drozdov (1996a)],
Y(t, ~') > 0,
(2.3.38)
lim Y(t, T) = Y~(I") < ~,
(2.3.39)
OY - - ( t , ~') >- O, Ot
(2.3.40)
t---,~
67
2.3. Creep and Relaxation Kernels
0.465
0.445 0
log te
3
Figure 2.3.6: The aging function q~ (GPa- 1 ) and the current Young's modulus E (GPa) versus elapsed time te (rain) for tension of an epoxy adhesive C quenched from 87 to 42°C. Circles show experimental data obtained by Vleeshouwers et al. (1989). Solid lines show their approximation by the linear functions (2.3.37) with Cl = -0.8338, c2 = 2.7888, c3 = 0.0085, and ca = 0.4405.
lim ~oY (t, ~') = 0, tgt
(2.3.41)
~Y -(t,~. ~') < 0.
(2.3.42)
t---,~
To explain the mechanical meaning of these conditions, we consider a piecewise constant loading program or(t) = [ 0 , Lo0,
0--< t < ~', ~-<-- t ~ oo,
(2.3.43)
68
Chapter 2. Constitutive Models in Linear Viscoelasticity
1.0 -0
0 0
0
0 0
_%
0
•
© •
0.8
I
O
{
I
I
I
I
I
I
I
0
t z Figure 2.3.7: The dimensionless ratio E = o-/o-(0) versus time t (min) for tensile relaxation of an epoxy adhesive C. Filled circles show experimental data obtained by Vleeshouwers et al. (1989); Unfilled circles show results of numerical simulation.
where o0 and ~" are positive constants. Substitution of expression (2.3.43) into the constitutive Eq. (2.2.19) yields e(t) = [0,
LY(t,
0 -< t < ~', T)~ro,
~" <-- t < ~.
Inequality (2.3.38) means that any tensile stress causes elongation of a viscoelastic specimen. Equation (2.3.39) implies that deformation of a specimen is bounded. According to condition (2.3.40), elongation of a viscoelastic specimen increases in time under tensile loads. The rate of strain is bounded, and it vanishes when time approaches infinity [see Eq. (2.3.41)]. Finally, inequality (2.3.42) implies that the earlier a specimen was loaded, the greater its strain.
69
2.3. Creep and Relaxation Kernels
The following restrictions are imposed on relaxation measures of aging viscoelastic media for any 0 -< r -< t < w: (2.3.44)
0 < X(t, r) <- ~, OX ( t, ~) < OX ~, - -~-(r) 0--~
= 0,
(2.3.45)
OX OX ~ ( t , r) -> -z--(~, r) = 0, OrOr
(2.3.46)
oZx (t, r) <
oZx
(~, r) = 0.
(2.3.47)
The mechanical meaning of restrictions (2.3.44) to (2.3.47) may be easily explained in the framework of the model of adaptive links. According to that model, the function X(t, r) is proportional to the number of adaptive links arising before instant r and existing at instant t. Its derivative OX ~(t, Or
r)
determines the number of links arising in the interval [r, r + dr] and existing at instant t. According to Eqs. (2.3.44) and (2.3.46), for any instants 0 <- r < t < ~, the number of adaptive links arising before instant r (or at instant r) and existing at the current instant t is positive. Inequalities (2.3.45) and (2.3.47) mean that the number of adaptive links arisen before instant r (or at instant r) decreases in t owing to the breakage process. To explain the mechanical meaning of Eqs. (2.3.44) to (2.3.47), we consider the deformation program e(t) = ~0,
LE0,
0 --- t < r, T - <-
t ~
(2.3.48)
c~,
where e0 and r are positive constants. Substitution of Eq. (2.3.48) into the constitutive Eq. (2.2.3) implies the following formula for the stress or(t): or(t)=
0, X(t, r)eo,
0 < - t < r, r <- t < ~,
It follows from Eq. (2.3.44) that elongation of a viscoelastic specimen occurs under the action of tensile load. Equation (2.3.45) describes the relaxation phenomenon: the stress decreases in time at a fixed strain. The rate of relaxation vanishes when time approaches infinity. According to Eq. (2.3.46), the later a viscoelastic specimen is loaded, the larger the stress. To explain the meaning of inequality (2.3.47), we consider the deformation program e(t) = eor(t - r),
(2.3.49)
70
Chapter 2. Constitutive Models in Linear Viscoelasticity
where 8(t) is the Dirac delta function. Equations (2.2.3) and (2.3.49) yield o-(t) = (0,
0 --- t < ~',
~)eo, ~, , k - a xcg.r tt
--Td° &
= {0, o2x
(t)
-
~ < t < ~,
0--< t < r, (t, ~)eo,
~ < t < ~.
It follows from Eq. (2.3.46) that the stress o- is negative. If we present the Dirac function as a limit of a sequence of rectangular impulses with growing intensities and decreasing durations, the negativity of the stress may be treated as a consequence of the principle of fading memory: the effect of the falling branch of any impulse is stronger than the effect of the lifting branch. Inequality (2.3.46) describes the relaxation phenomenon: after loading at instant ~-, the absolute value of stress Icr(t)l decreases monotonically in time and tends to zero. It follows from Eq. (2.3.47) that the rate of relaxation do" decreases monotonically in time and tends to zero as t ~ oo. For non-aging viscoelastic media, conditions (2.3.44) to (2.3.47) are presented as follows: -
1 < Qo(oO) < Q o ( t ) <- Q0(0) = 0,
dao (t) < dQo (~) = O, d--i-
-
--d-i-
d2Q° (t) > d2Q° dt 2 _ dt 2 (oo) = 0.
(2.3.50)
Hrusa and Nohel (1985) and MacCamy (1977) introduced the function Ho(t) = Qo(t) - Qo(~),
and wrote conditions (2.3.50) in the form Ho(t) >- O,
dHo ~(t) dt
<- O,
d2Ho ~(t) dt
>- O.
(2.3.51)
The first inequality in Eq. (2.3.51) implies the non-negativity of the function Ho(t), the second means its monotonicity, and the third yields the convexity of Ho(t). Following Pipkin (1972), Beris and Edwards (1993) imposed an infinite sequence of constitutive inequalities ( - 1 ) ndnH° (t) --- 0 dt n
(n = 0, 1 ...).
(2.3.52)
To explain the mechanical meaning of Eq. (2.3.52), Beris and Edwards (1993) introduced loading programs in the form of the so-called nth derivatives of the Dirac
2.4. Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity
71
function. These functions are treated as idealizations of sequences of n + 1 strain steps with alternating signs, when the length of any step tends to zero. According to Bernstein's theorem, Eqs. (2.3.52) imply that there is a nonnegative function x ( T ) such that Ho(t) =
/0
x(T) exp - ~
dT.
(2.3.53)
Differentiation of expression (2.3.53) yields Eq. (2.3.6) with the nonnegative relaxation spectrum x(T). Positivity of relaxation measures plays the key role in the study of stability of viscoelastic media. An integrable function a(t) is of positive type, if for any complex function rl(t) the dissipative inequality holds, ~
/0
rl(s) ds
/0s
a(s - ~')~(r)dr >- O,
(2.3.54)
where ~ stands for the real part of a complex number p = a + ~o9, ~ = ~ - 1 , and the superscript bar denotes complex conjugate. A function a(t) is of strong positive type, provided there is a positive constant 3 such that function a(t) - ~3exp(-t) is of positive type. Gripenberg et al. (1990) demonstrated that inequalities (2.3.51) guarantee that the function Ho(t) is of strong positive type, provided that the function Ho(t) is triple continuous differentiable, and its second derivative is positive on a set of positive measure. As a consequence, there is a positive 8 such that for any o9
9t/
0(.o) > -- 1 +o92'
(2.3.55)
where the tilde stands for the Laplace transform. Since 9~Ho(~o9) =
Ho(t) cos tot dt,
inequality (2.3.55) implies the strong negativity of the Fourier cosine transform of the function Ho(t). The latter condition was employed by Fabrizio and Morro (1992) to prove the existence and uniqueness of solutions to boundary value problems in linear viscoelasticity.
2.4 Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity This section is concerned with thermodynamic potentials of aging viscoelastic media. We formulate two variational principles for quasi-static loading and discuss their connections with the second law of thermodynamics. Finally, some restrictions are
72
Chapter 2. Constitutive Models in Linear Viscoelasticity
derived that are imposed on the relaxation function by the second law of thermodynamics.
2.4.1 Thermodynamic Potentials of Aging Viscoelastic Media Consider a viscoelastic medium which occupies a connected domain 12 with a smooth boundary F in the natural configuration at temperature 19 = (90. Points of f~ refer to Lagrangian coordinates ~ = {~i}. At the initial instant t = 0, body forces/3 and surface forces b are applied to the medium. The surface traction is prescribed on a part F (~) of the boundary F. The other part F (u) of the boundary is clamped. A thermodynamic theory for materials with fading memory was derived by Coleman (1964). This theory assumes that • Free energy, entropy, and heat flux depend on the history of strains, on the history of temperature, and on the current value of the temperature gradient. • The second law of thermodynamics is fulfilled for any sufficiently smooth thermodynamic process. As a consequence of these hypotheses, it is shown that the stress tensor and the entropy are expressed in terms of the free energy, which, in turn, is independent of the temperature gradient. Wang and Bowen (1966) introduced a concept of quasi-elastic materials and proved that the Coleman conclusions follow from some general hypotheses. Gurtin (1968) suggested a thermodynamic theory of materials with fading memory based on some "chain-rule" property of free energy. A concept of dissipative processes in media with fading memory was developed by Coleman and Mizel (1967, 1968), Coleman and Owen (1970), and Day (1969). For a detailed exposition of this theory, see, e.g., Day (1972). Gurtin and Hrusa (1988) derived some conditions that ensure the existence of a free energy for nonlinear viscoelastic media with infinitesimal strains. We consider quasi-static loading of an aging, linear, viscoelastic medium with small strains. The material behavior obeys the constitutive equations (2.2.72) of a homogeneous, isotropic viscoelastic medium with purely elastic dilatation. Neglecting thermal expansion, we present the specific Helmholtz free energy • (per unit mass) in the form C
= ~0 + 1 W - n0(® - (90) - ~-~0(® - 190)2. /90
(2.4.1)
Here W is strain energy density (per unit volume), ~0, H0, and c are the specific free energy, the specific entropy, and the specific heat capacity (per unit mass) in the initial configuration at the temperature 190, P0 is mass density in the initial configuration, 19 is the absolute temperature. Equation (2.4.1) may be treated as the Taylor expansion of the function • in the vicinity of the initial configuration and the initial temperature, where only the second-order terms are taken into account.
2.4. Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity
73
By analogy with Eq. (2.2.44), we write the following formula for the strain energy density W:
W = 2KeZ(t) + X(t, 0)~(t) • ~(t) +
t OX -0-~T(t, ~-)[O(t)- O(~-)]" [ 0 ( t ) - 0(~-)] d~-,
fo
(2.4.2)
where e and 0 are the first invariant and the deviatoric part of the strain tensor 5, K is a constant bulk modulus, and X(t, z) is a shear relaxation function. The specific entropy H equals 0q~ H = --00"
(2.4.3)
Substitution of (2.4.1) into Eq. (2.4.3) implies that C
H = H0 + ~00(O - O 0 ) .
(2.4.4)
It follows from Eq. (2.4.4) that the specific entropy H is independent of the strain tensor ~ (since thermal expansion is neglected). The specific internal energy ~ is determined by the formula = •
+ HO.
(2.4.5)
Substitution of expressions (2.4.1) and (2.4.4) into Eq. (2.4.5) yields
1
(I) = (I)o + ~ W +
c
( 0 2 -Oo2),
(2.4.6)
po
where ~o = q~o + HoOo. Formulas (2.4.1) to (2.4.6) determine the main thermodynamic potentials of an aging viscoelastic medium.
2.4.2
Variational Principles in Viscoelasticity
As common practice, two types of deformation processes are considered: isothermal and adiabatic [see Arutyunyan and Drozdov (1992)]. We begin with isothermal loading, when the temperature ® remains constant and equal to ®0.
The Principle of Minimum Free Energy
We fix an instant t -> 0 and a deformation
history up to this instant, {o(r, ~), 0 <- r <
t},
(2.4.7)
Chapter2. ConstitutiveModelsin LinearViscoelasticity
74
where fi(t, ~) is the displacement vector from the initial to actual configuration. The set of admissible displacement fields consists of continuously differentiable vector functions fi(t, ~) satisfying the boundary condition Ir(u~= 0.
(2.4.8)
Let fi*(t, ~) be an admissible displacement field, and ~*(t, ~) the corresponding infinitesimal strain field. Here and in the following, asterisks denote admissible thermodynamic quantity. The observed quantity (which occurs in the deformation process) is denoted by the same symbol without an asterisk. It follows from Eq. (2.4.1) that for isothermal processes the free energy of the medium equals
t(t) = jf po~(t) dVo = p0~I,01~l+ w*(t).
(2.4.9)
Here f I~1 =
is volume of the domain ~ ,
~ dVo
dVo is the volume element, Wt(t)
and
= fo W(t)dVo
(2.4.10)
is the potential energy of deformations. The work of external forces on the displacement from the actual configuration at instant t - 0 to an admissible actual configuration at instant t equals
At(t) =/~ po[~(t)" [fi*(t) -
fi(t - 0)] dVo +
fr~, [fit)" [fi*(t)
- fi(t - 0)] dSo, (2.4.11)
where dSo is the surface element. For simplicity, the argument ~ is omitted. The increment of the total free energy for transition from the actual configuration at instant t - 0 to an admissible actual configuration at instant t is calculated as T(t) = [ ~ *(t) - • t (t - 0)] - At (t).
(2.4.12)
The principle of minimum free energy states that given instant t -> 0 and deformation history (2.4.7), the real displacement field fi(t) minimizes the functional T(t) on the set of admissible displacement fields. Our purpose now is to demonstrate that the principle of minimum free energy implies the equilibrium equation and the boundary condition in stresses provided the constitutive relations have the form [see Eq. (2.2.72)],
2.4. Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity
?fit) = 2
tr(t) = 3Ke(t),
(t, t)O(t) -
/0
]
-~z(t, ~')0(~') d~" ,
75
(2.4.13)
where tr and ~ are the spherical and deviatoric parts of the stress tensor 6-. To prove this assertion, we fix an admissible increment aft(t) of the displacement vector fi(t) and the corresponding increment 6~(t) = 1 [~r0afi(t ) + fTOafiT(t)] of the infinitesimal strain tensor ~(t). Here ~70 is the gradient operator in the initial configuration, and T stands for transpose. It follows from Eqs. (2.4.2) and (2.4.9) to (2.4.12) that the increment of the functional T(t) is calculated as
6"f'(t)=/n{Ke(t)ae(t)+2IX(t,O)~(t) +
jo
-~z(t, ~-)(O(t) - ~(~')) dT
]}
• 6O(t) dVo
where 3e and 6~ are the spherical and deviatoric parts of the tensor 6~, respectively. Substitution of expressions (2.4.13) into Eq. (2.4.14) implies that
~'T(t) = / a [~(t) . 6~(t) - po[~(t). 6~(t)] dVo - fv~,~ [fft) . ~fi(t)dSo. (2.4.15) Applying Stokes' formula to Eq. (2.4.15), we find that aT(t) = - f n [~7°" 6(t) + p0/)(t)] • a~(t)dVo +/r~,~ [ h - & ( t ) - b(t)] • 6~(t)dSo,
(2.4.16)
where h is the unit outward normal vector to the boundary F. The necessary condition of minimum for the functional T(t) reads 6 T(t) - 0.
(2.4.17)
Since the increment ~ ( t ) of the displacement field ~(t) is arbitrary, Eqs. (2.4.16) and (2.4.17) imply the equilibrium equation in
fTo " 6-(t) + po[~(t) = 0
(2.4.18)
and the boundary condition in stresses on the surface F <'~> h. 6-(t) = b(t).
(2.4.19)
Chapter 2. Constitutive Models in Linear Viscoelasticity
76
The principle of minimum free energy is widely used for elastic media. However, its application to viscoelastic media has been questionable for a long time. To explain why this principle remains true for quasi-static problems in viscoelasticity, we refer to the concept of adaptive links (see Section 2.2). We introduce two characteristic times. The first, Te, is the characteristic time for elastic deformations, whereas the other, Tv, is the characteristic time for stress relaxation. Since the relaxation process is modeled as breakage and creation of adaptive links, Tv coincides with the characteristic time for the reformation process. At quasi-static loading, the time Te is essentially less than T~ (which is assumed to have the same order of magnitude as the characteristic time for changes in external loads). As a consequence, at any instant t (in the rapid scale with the characteristic time Te), a viscoelastic medium may be treated as a system with a fixed number of adaptive links, i.e., as a purely elastic continuum. Evidently, the principle of minimum free energy can be applied to this elastic medium.
Gibbs' Principle We now consider adiabatic loading, when the heat flux to a medium from the environment vanishes. Introduce three characteristic times" the characteristic time Te for elastic deformations, the characteristic time T~ for stress relaxation, and the characteristic time To for approaching thermodynamic equilibrium. The quantity To is the time necessary to reach a uniform temperature in the medium under consideration. We suppose that To ~ T~. This assumption is rather strong: it means that so weak changes in the temperature 19 are permitted that the time, To, necessary to reach a uniform temperature over the entire domain fl is less than the characteristic time of relaxation, Tv. The set of admissible temperature fields is determined by the inequality ®*(t) -> 0.
(2.4.20)
We fix an arbitrary instant t -> 0 and a thermodynamic history up to the instant t (2.4.21)
{(0(~'), fi(~', so)), 0 -< ~" < t}.
The temperature 19 in Eq. (2.4.21) is independent of Lagrangian coordinates ~, since the thermodynamic history is treated as a sequence of thermodynamically equilibrium states that replace each other. For an admissible thermodynamic state (®*(t), fi*(t, ~)) at instant t, the internal energy ~* and the entropy H t of a viscoelastic medium are determined by the formulas [see Eqs. (2.4.4) and (2.4.6)]
*(t) = pol~l
H*(t)
= polOI
{ ~o + 2~oc [(o*(0) 2 -
o~]
{/40+ ooc [°*(t)- °0]}
} + w*(t),
(2.4.22)
(2.4.23)
77
2.4. Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity
According to the first law of thermodynamics, (I) t (t) - • *(t - 0) = At (t).
(2.4.24)
Gibbs' principle states that given instant t >-- 0 and thermodynamic history (2.4.21), the real displacement field fi(t) and the real temperature ®(t) maximize the entropy H i ( t ) on the set of admissible thermodynamic states that satisfy the balance law (2.4.24). To establish a connection between the principle of minimum free energy and Gibbs' principle, we substitute expression (2.4.22) into Eq. (2.4.24) and find the temperature 19*(t)
®*(t) = ®(t - O)
20o [At (t) + W t (t - 0) - W t (t)] 1 + poclf~102( t _ O)
1/2
. (2.4.25)
It follows from Eqs. (2.4.9) and (2.4.12) that 2r'(t) = W t (t) - W t (t - 0) - At (t). This equality together with Eq. (2.4.25) implies that 2®o "/"(t) ®*(t) = ®(t - 0) 1 - 0oclf~l®2( t _ 0)
.
(2.4.26)
Substitution of expression (2.4.26) into Eq. (2.4.23) yields
H*(t)
= p01f~l
Ho + c
®(t-0)
(9o
1-
poclf~lO2(t- O)
-1
. (2.4.27)
It follows from Eq. (2.4.27) that the displacement field ~ (t) that maximizes the entropy H i ( t ) minimizes also the free energy 9"(t), and vice versa. Therefore, the principle of minimum free energy for isothermal loading and Gibbs' principle for adiabatic loading imply the same displacement field fi(t). As a consequence, we find that Gibbs' principle yields the equilibrium Eq. (2.4.18) and the boundary condition (2.4.19).
2.4.3
Gibbs' Principle and the Second Law of Thermodynamics
To derive connections between Gibbs' principle and the second law of thermodynamics for adiabatic processes, we assume the characteristic time for changes in external loading Tv to exceed essentially the time Te necessary to establish thermodynamic equilibrium in a viscoelastic medium. We divide the interval of loading [0, T] by points tn = nA (n = O, 1 . . . . . N), where A = T / N , and A is a time which is essentially larger than Te and is significantly less than Tv. The body force/) and the surface traction b are approximated by stepwise loads
Ba (t, ~) = B(tn, ~),
bA (t, ~) = [ff tn, ~),
tn <-- t < tn+ 1.
(2.4.28)
Chapter 2. Constitutive Models in Linear Viscoelasticity
78
At instant tn -- O, the medium is in thermodynamic equilibrium under the action of loads [3(tn-1) and b(t~_ 1). The equilibrium state is characterized by the displacement field Ft(tn - - 0 ) and the temperature l~(tn - - 0 ) . At instant tn, new loads B(tn) and b(tn) force the viscoelastic medium to leave its "old" equilibrium state and to move in the interval (tn, tn+l) t o a "new" equilibrium state, which is characterized by a displacement field fi(t~+1) and a temperature O(tn+ 1). Neglecting the inertia forces, we write the first law of thermodynamics in the interval (tn, tn+ 1)
d---7-(t) =
/~
Off
poB(tn) " -~(tl dUo +
fr
Off
(~)
[~(tnl " --~(t) dSo.
(2.4.29)
Integration of Eq. (2.4.29) implies that
dp~(t) - dp*(tn) = ff~ po[3(tn) " [fi(t) -- fi(tn)] dVo + L ~ , [~(tn)" [fi(t)- U(tn)] dSo.
(2.4.30)
The second law of thermodynamics reads
dH* dt
~(t)
-- O.
(2.4.31)
It follows from Eq. (2.4.31) that the entropy H t (t) does not decrease in time in the interval (tn, tn+l). Therefore, at instant tn+l the function Hi(t)reaches its maximal value in the interval (tn, tn+ 1), i.e., its maximal value on a set of displacement fields fi(t, ~) and temperatures O(t, ~), which occur during the transition process from one thermodynamic equilibrium state to another and which satisfy the balance law (2.4.30). Gibbs' principle states that at instant tn+ 1, the function H t (t) reaches its maximal value on the set of admissible displacement fields ~(tn+ 1, ~) and temperatures O(tn+ 1) that obey the balance law (2.4.30) at t = t~+ 1. Gibbs' principle does not imply the second law of thermodynamics and vice versa. On the one hand, the variational principle implies that the functional Ht(t~+l) reaches its maximal value on the set of all admissible displacements. This set may be wider than the set of displacements that take place in the transition process. On the other hand, Gibbs' principle allows only homogeneous temperature fields O(tn+ 1) to be considered, whereas spatially inhomogeneous temperature fields ®(t, ~) occur in the transition process. An analogy may be noted between these assertions and the ergodic property of stochastic processes. A stationary stochastic process is called ergodic provided its mean value in time coincides with its mathematical expectation (the mean value on a probability space). The second law of thermodynamics states that the entropy reaches its maximum in time in the interval (tn, tn+ 1) at the equilibrium state at instant tn+ 1. According to the Gibbs principle, this functional reaches its maximum on the set
2.4. Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity
79
of all admissible thermodynamic states at instant tn+ 1- If these assertions imply the same thermodynamically equilibrium state at instant tn+l, then the nonequilibrium transition processes from one equilibrium state to another has some ergodic property.
2.4.4 Thermodynamic Inequalities in Linear Viscoelasticity Since the potential energy of deformation W is nonnegative for an arbitrary admissible strain field ~(t), Eq. (2.4.2) implies that for any 0 - r < t,
X(t, 0) - 0,
(2.4.32)
OX ~ ( t , r) - 0. 0r
(2.4.33)
Substituting expressions (2.4.32) and (2.4.33) into the formula
X(t, r) = X(t, 0) + fo ~ ~OX (t T1)d'rl, ,
we find that for any 0 - r < t,
X(t, r) >-- O.
(2.4.34)
To derive other constitutive inequalities, we write the first law of thermodynamics in the differential form
d---~(t) =
00[3(0" --~(t) dVo +
(or)
b(t) . --~(t) dSo.
(2.4.35)
Substitution of expression (2.4.22) into Eq. (2.4.35) yields
Polf~.c ®(t)d® ®---o d--t-( t ) =
f ooB(t>.°o -~(t)dVo
+
O~ dW t c~,[fft) " -~(t)dSo - --~-(t).
Combining this equality with Eq. (2.4.23), we obtain
dHt Off Oft ®(t)---~-(t) = fn po[3(t).-~(t)dVo + fr (~) [fft).-~(t)dSo-
dW* (2.4.36) --dT-(t).
It follows from (2.4.2), (2.4.10), and (2.4.13) that
dW*
= / n &(t) " --;-7(t)dVo atO~ + Q(t),
(2.4.37)
where
Q(t) =
-~(t, 0)O(t) • O(t) +
foo' 0-~r(t, °2x r ) [ ~ ( t ) -
~(r)]" [ ~ ( t ) - ~(r)]
}
dr dVo.
(2.4.38)
80
Chapter 2. Constitutive Models in Linear Viscoelasticity
Substitution of expression (2.4.37) into Eq. (2.4.36) with the use of Stokes' formula yields
dH t ®(t) --~- (t) = - Q(t).
(2.4.39)
Equation (2.4.39), together with the second law of thermodynamics (2.4.31), implies that the functional Q is nonpositive for an arbitrary admissible tensor function O(t). It follows from this condition and Eq. (2.4.38) that for any 0 -< ~" < t,
OX O--7(t,0) - 0,
(2.4.40)
oZx ~(t, OtOT
(2.4.41)
~') -< 0.
Substitution of expressions (2.4.40) and (2.4.41) into the formula
OX (t, T) = OX
O---t-
f~
oZx
--~-(t, 0) + J0 OtOT1 (t, T1) d~'l
implies that for any 0 --< r < t,
OX ~ ( t , r) -- 0. Ot
(2.4.42)
Equations (2.4.33), (2.4.34), (2.4.41), and (2.4.42) provide thermodynamic restrictions on the relaxation function X(t, ~') in the constitutive model (2.4.13). These conditions coincide with inequalities (2.3.44) to (2.3.47) developed with reference to experimental data for aging viscoelastic media. Similar inequalities for non-aging viscoelastic media were derived by Day (1971).
2.5
A Model of Adaptive Links for Aging Viscoelastic Media
In this section a version of the model of adaptive links is derived for the mechanical behavior of linear viscoelastic materials subjected to physical aging. Two types of links are distinguished: links arising at the instant of quenching (type I), and links emerging in the quenched material at a constant temperature (type II). The mechanical reponse in an aging medium is determined by three material functions which characterize (i) annihilation of links of type I, (ii) the breakage of links of type II, and (iii) reformation of links of type II. Integral equations are derived for these functions, and their solutions are found by using data in the standard relaxation tests. To verify the model, we calculate the material response in creep tests and compare results of numerical simulation with experimental data for an epoxy adhesive. Fair agreement is demonstrated between observations and their prediction.
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
81
We analyze numerically the behavior of a viscoelastic medium under timevarying loads: elongation of a specimen with a constant rate of strain, its recovery after creep tests, and steady shear oscillations of a layer. In the latter case, numerical results are compared with experimental data for polypropylene. The exposition follows Drozdov (1996b).
2.5.1
A Model of Adaptive Links
The mechanical behavior of physically aged viscoelastic materials was studied experimentally in a number of works [see, e.g., Brennan and Feller (1995), McKenna (1989), Plazek et al. (1984), Struik (1978, 1987a, b), Waldron et al. (1995), etc.] Three main approaches may be distinguished to constructing constitutive models for aged viscoelastic media. According to the first, processes of creep and relaxation are described by simple relationships that express either the strain as a function of time for a given stress, or the stress as a function of time for a given strain. As common practice, the Kohlrausch-Williams-Watts formula is employed both for the creep compliance and for the relaxation modulus [see, e.g., Vleeshouwers et al. (1989)]. Parameters in these relationships are found by fitting experimental data. The main disadvantage of this method is that it makes it impossible to predict the material behavior under time-dependent loads. According to the other approach, stresses and strains obey the standard Volterra constitutive equations in convolution, where the real time t is replaced by the so-called internal (reduced, or pseudo) time
~-(t) =
t ds a(s) '
fo
where a(t) is a material function to be found by fitting experimental data. This method goes back to the time-temperature superposition principle, where a is assumed to be a function of the current temperature. To describe the material aging, the shift function a is assumed to depend on the free (frozen-in) volume which, in turn, is a function of the current temperature and stress intensity [see, e.g., Knauss and Emri (1981) and Losi and Knauss (1992) for details]. In cooling a polymer from a temperature 190 above the rubber-glass transition temperature ®g to a temperature O1 below ®g, its free volume (or enthalphy) deviates from the equilibrium value at 190 toward new equilibrium value. This evolution induces changes in the viscoelastic response, which are treated as physical aging [see, e.g., McKenna (1989) and Struik (1978)]. For uniaxial deformation of a specimen quenched from 190 to O1 and annealed for a time te before loading, this approach leads to the constitutive equation t
o(t) =
fO
E(~'te (t) -- ~'te(S)) de(s),
(2.5.1)
Chapter 2. Constitutive Models in Linear Viscoelasticity
82
where t is time after loading, o- is the stress, e is the strain, and E(t) is the relaxation function. For a relaxation test with a constant e, Eq. (2.5.1) implies that
tr(t) = E(~'te(t))e. At fixed temperatures 190 and only, and we can write
(2.5.2)
O1, the parameter a depends on the elapsed time te t
-
%
-
--.
a(te)
(2.5.3)
It follows from Eqs. (2.5.2) and (2.5.3) that • Young's modulus E = 7?(0) is independent of the material age te. • The relaxation curves or versus log(time) corresponding to different times te may be obtained from each other by a shift (as a rigid body) along the time axis. Numerous experimental studies of amorphous polymers confirm the time-aging time superposition principle [see, e.g., Struik (1978) and Waldron et al. (1995)]. On the other hand, some discrepancies should be mentioned between theoretical predictions based on the superposition principle and observations for a number of crystalline and semicrystalline polymers. For example, Chai and McCrum (1984) referred to poor prediction of experimental data based on the superposition principle in relaxation tests with poly(methyl methacrylate) and polypropylene and suggested specific transformations of compliance curves to obtain better agreement with observations. Struik (1987a, b) proposed to add some "vertical" shift of creep compliance curves obtained at various times te after quenching to the "horizontal" shift in order to derive a master curve with an appropriate level of accuracy. Struik's model is based on the hypothesis that a part of the amorphous phase of a semicrystalline polymer is glassy above the temperature O g and a series of transition temperatures should be introduced. The theories of physical aging in crystalline polymers derived by Chai and McCrum (1984) and Struik (1987a, b) permit master curves for creep experiments to be constructed correctly, but they are not directed to engineering calculations, where explicit stress-strain relations (accounting for the material aging) are necessary for an arbitrary loading. According to the third approach, stresses and strains in aged viscoelastic media obey a linear Volterra equation with coefficients (elastic moduli) dependent on time and with kernels of integral operators (creep and relaxation kernels) that depend on two temporal variables [see, e.g., Arutyunyan et al. (1987)]. For uniaxial deformation, the stress-strain relation is written as follows [see Eq. (2.2.40)]:
or(t) = X(t + te, te)e(t) + fot ~OX (t
+ te, ~" + te)[e(t) -- e(l")] dr,
(2.5.4)
where X(t, T) is a relaxation function and te is a material age at the instant of loading.
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
83
This approach seems more appropriate for applications, but the following three drawbacks should be mentioned: 1. Equation (2.5.4) provides a purely phenomenological model that does not account for mechanisms of deformation in viscoelastic media. For example, model (2.5.4) does not distinguish physical (reversible) and chemical (irreversible) aging. 2. Serious difficulties are encountered in experimental determining the function of two variables X(t, T), which is necessary even for uniaxial loading. 3. Equation (2.5.4) is too complicated for the analytical study of the viscoelastic behavior. Because the Laplace transform method cannot be applied to nonconvolutional integral equations, explicit expressions are required for both the creep and relaxation kernels. Our objective is to construct a model for the viscoelastic behavior of aged polymeric materials which • Ensures fair prediction of experimental data for semicrystalline and crystalline polymers. • Requires minimal experimental data for determining adjustable parameters. • Can be used in engineering calculations for arbitrary loading programs. To derive such a model, we employ the concept of adaptive links between polymeric chains (see Section 2.2). According to this concept, a viscoelastic medium is treated as a network of parallel springs (links) with a constant rigidity c. The links can arise and break. The function X , ( t + te, "r + te) = 1 X ( t + re, "r + te) c
(2.5.5)
equals the number of links arising before instant ~-and existing at the current instant t (time is measured from the instant of loading). In particular, X,(t, t) is the number of links existing at instant t, X,(t, 0) is the number of links arising at the initial instant and existing at instant t, and 0X, ~(t, Or
r) dr
is the number of links arising within the interval [~', ~" + dr] and existing at instant t. These amounts are connected by the formula
X,(t, t) = X,(t, O) +
~0t -~-T ~X, (t, ~') d~'.
(2.$.6)
The process of physical aging is described by means of adaptive links of two types. Links of type I arise during the process of quenching. Owing to the rapid cooling, the state of these links is not thermodynamically stable. Their mechanical properties differ from the properties of links of type II which arise in isothermal conditions at a fixed temperature 191. After quenching, new links of type I do not arise, and the existing links proceed to break in time. The aging is treated as a successive
84
Chapter 2. Constitutive Models in Linear Viscoelasticity
breakage of links of type I and their replacement by links of type II. Deviation of the free volume to a new equilibrium state during physical aging is treated as the breakage of "nonequilibrium" links arising under rapid cooling. The aging process is characterized by three functions of one variable: two rates of breakage for "old" links of both types, and the rate of creation for "new" connections. These functions are found by fitting experimental data in the standard relaxation tests. Unlike the model developed in Section 2.2, all links are assumed to be involved in the process of replacement. In this case, Eqs. (2.2.45) are written as X , ( t , 0) = X,(0, 0)[ 1 - g(t, 0)], OX,
~(t, 0T
~') = ~(~')[1 - g(t - ~, ~')],
(2.5.7)
where X,(0, 0) is the number of links arising at quenching (links of type I), 0X, • (~') = --yz(t, ~')
u{
(2.5.8)
t'-T
is the rate of emergence for new links: ~(~-) d r links of type II arise within the interval [~', ~" + d~'], and g(t - T, r) is the relative number of links existed at instant ~- and broken to the instant t. Substituting expressions (2.5.7) into Eq. (2.5.6) and using Eqs. (2.2.5) and (2.5.5), we obtain the integral equation E ( t ) = E(0)[ 1 - g(t, 0)] + c f0 t q~('r)[1- g(t - "r, "r)] d-r,
(2.5.9)
where E ( t ) is the current Young's modulus. We assume that the breakage process for links arising after quenching is uniform in time g(t - T, r) = N ( t - r),
(2.5.10)
where the function N depends on the difference t - ~"only. Introducing the notation E,(t) -
E(t) E(O)'
dp,(t) -
cdP(t) E(O) '
no(t) = g(t, 0),
(2.5.11)
~,(~-)[1 - N ( t - ~')] dr.
(2.5.12)
and using Eq. (2.5.10), we rewrite Eq. (2.5.9) as E , ( t ) = 1 - no(t) +
J0"t
Equation (2.5.12) may be treated as the Liouville equation for the number no of links of type I between polymeric molecules. Consider a series of relaxation tests with a fixed strain e and various elapsed times te after quenching. It follows from Eq. (2.5.4) that O'te (t) = X ( t + te, te)~,
(2.5.13)
85
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
where trte (t) is the stress in a specimen at instant t after loading. By analogy with Eq. (2.5.6), we write te OX, -~-r (t + te, ~')d~'.
fO
X , ( t + te, te) = X , ( t + te, O) +
(2.5.14)
Substitution of expressions (2.5.7), (2.5.10), and (2.5.11) into Eq. (2.5.14) yields X , ( t + re, re) = X,(0, 0)[1 - no(t + re)] +
~(r)[1 - N ( t + te -- ~')] d~'.
Combining this equality with Eq. (2.5.5) and using Eqs. (2.2.5) and (2.5.11), we arrive at the formula X ( t + te, te) = E(O)
(
1 - no(t + te) +
~te
~,('r)[1
- N ( t + te -- "r)] d'r.
}
(2.5.15) It follows from Eqs. (2.5.13) and (2.5.15) that trte(t) - 1 - no(t + re) + fOte ~,(~')[1 - N ( t + te -- 1")]d~'. E(O)e
(2.5.16)
Equation (2.5.12) implies that 1 -
n o ( t + te) = E , ( t
~0"t+ te
+ te) -
~,(~')[1 - N ( t
+ te --
r)] dr.
Substitution of this expression into Eq. (2.5.16) results in t + te O'te(t)
--
~,(~')[1 - N ( t + te - ~')] d~'.
E , ( t + te) -
Orte(O)
(2.5.17)
d te
Finally, introducing the new variable ~ = ~- - te and omitting the tilde, we find from Eq. (2.5.17) that O'te(t) __ E , ( t + re) -cr,~ (O)
~,(-r + te)[1 -- N ( t - "r)] dr.
(2.5.18)
For a given set of relaxation curves ~i'te(t) = 1 -- Crte(t) Crte(O) '
(2.5.19)
depending on two variables te and t, Eq. (2.5.18) is a functional-integral equation for determining two functions of one variable ~ , ( t ) and N(t). When these functions are found, no(t) can be obtained by solving Eq. (2.5.12). The function no(t) is not significant for the model in that only the functions ~ , ( t ) and N ( t ) determine the material response [see Eqs. (2.5.4), (2.5.7), and (2.5.10)]. The function no(t) characterizes the
86
Chapter 2. Constitutive Models in Linear Viscoelasticity
interval where the material aging is essential: when no(t) vanishes, the aging may be neglected. Equation (2.5.18) is rather complicated to be solved either analytically or numerically. However, one case can be mentioned, in which its solution may be found relatively simply. Namely, we assume that (i) The characteristic time for creation of new links (the time when the function • (t) changes significantly) essentially exceeds the characteristic time for their breakage (the time necessary for the function N(t) to be changed dractically). (ii) The time characterizing changes in Young's modulus E(t) essentially exceeds the characteristic time for breakage of new links. Condition (i) is essential for our analysis, whereas condition (ii) is introduced only for convenience of calculations. According to these assumptions, within the standard relaxation tests (5-15 min), the functions E(t + te) and dp(t + te) remain practically constant. As a result, we obtain from Eq. (2.5.18) E,(te) - dO,(te)
fO t [1 - N(~')] d~"
-
O'te(t)
(2.5.20)
O'te (O ) "
Differentiation of Eq. (2.5.20) implies that ~,(te)[1 - N(t)] = - (rte(t)
O',e(0) '
(2.5.21)
where the superimposed dot denotes differentiation with respect to time. Setting t = 0 in Eq. (2.5.21), we find that d~,(te) = -- O'te(0)
(2.5.22)
O'te (O ) "
Substitution of expression (2.5.22) into Eq. (2.5.21) yields N(t) = 1 -
(rte(t)
(2.5.23)
O'te (O ) "
Equations (2.5.22) and (2.5.23) together with Eqs. (2.5.5), (2.5.7), (2.5.8), (2.5.10), and (2.5.11) enable us to determine the function X(t, r) using experimental data in the standard relaxation tests. Our purpose now is to calculate parameters of the function X(t, T) for an epoxy adhesive. Experimental data for three epoxy adhesives (A, B, and C), as well as a detailed description of their chemical and physical properties are presented in Vleeshouwers et al. (1989). Young's modulus E is plotted versus the material age te in Figure 2.5.1. According to Figure 2.5.1, E changes so weakly that we may assume it to be constant and equal approximately 0.45 GPa. The relaxation curves ~te (t) are plotted in Figure 2.5.2 for various te values. With an appropriate level of accuracy, these curves may be approximated by the truncated
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
87
2.67
logE
o 2.65
I
I
I
I
log te
0.5
I
I
2.5
2.5.1: Young's modulus E (GPa) versus the material age te (min) for epoxy adhesive C quenched from 87 to 42°C. Circles show experimental data obtained by Vleeshouwers et al. (1989). The solid line shows their approximation by the linear function log E = al log te + a2 with al = 0.0082 and a2 = 2.6441. Figure
Prony series M
~te(t) = ['(te) ~
Xm[I - exp(-~/mt)],
(2.5.24)
m=l
where Xm and ~/m are adjustable parameters and M = 3. The value M = 3 corresponds to three different kinds of links mentioned by He and Song (1993). This number is chosen as the minimal number that provides fair agreement between Eq. (2.5.24) and experimental data, on the one hand, and ensures positivity of the coefficients Xm, on the other hand. The function F(te) is defined up to an arbitrary multiplier, which may be found by using various approaches. For example, we may assume additionally that F(~) = 1, which implies that the sum in Eq. (2.5.24) determines the relaxation curve for an old
88
Chapter 2. Constitutive Models in Linear Viscoelasticity 1
2
3
4
0.15
0
l
(~ 0
I
I
I
I
I
I
I t
I
I
Figure 2.5.2: The dimensionless stress Et, versus time t (min) for an e p o x y adhesive C q u e n c h e d from 87 to 42°C and p r e s e r v e d t¢ (min) before loading. Circles s h o w e x p e r i m e n t a l data obtained by V l e e s h o u w e r s et al. (1989). Solid lines s h o w their a p p r o x i m a t i o n by the function (2.5.24) with X1 = 0.1999, )(2 = 0.0429, X3 = 0.0145, a n d 71 = 0.04 (min-1), 3/2 = 1.20 (min-1), 3/3 = 22.00 (min-1). C u r v e 1: t¢ = 5, F = 2.781; C u r v e 2: t¢ = 10, F = 2.077; C u r v e 3: t¢ = 20, F = 1.696; C u r v e 4: t¢ = 40, F = 1.527; C u r v e 5: t¢ = 80, F = 1.115; C u r v e 6: t¢ = 160, F = 1.0. material (as te --* oo). According to another approach, the F(te) value m a y be fixed for a given te. Since this choice does not affect the material function N(t), we set
F(te)]te =
160
(rain) = 1
to simplify calculations. Substitution of expression (2.5.24) into Eqs. (2.5.22) and (2.5.23) implies that M
~ , ( t e ) = F(t~) Z m=l
XmTm,
N(t) = 1 - ~']~mL1Xm'}tm e x p ( - T m t ) . ~-]~mM=l ,¥ m'}tm
(2.5.25)
89
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
0.1
log~,
I
-0.4
I
0.5
I
I
I
I log te
2.5
Figure 2.5.3: The function cI), versus the material age te (min) for an epoxy adhesive C quenched from 87 to 42 °C. Circles show experimental data obtained by Vleeshouwers et al. (1989). The solid line shows their approximation by the linear function log ~,(te) = al log te + a2 w i t h a l = -0.2942 and a 2 : 0.2761. Figure 2.5.3 demonstrates that in the interval of observations, the function dP.(te) decreases monotonically. This function should tend (as te ~ ~ ) to a constant, which determines the rate of reformation in an old material.
2.5.2
Validation
of the Model
To verify the model, we demonstrate that it not only fits experimental data in relaxation tests by using adjustable parameters, but also predicts the material response in other experiments. For this purpose, we choose creep tests for the same epoxy adhesive and show that the model ensures fair agreement between results of numerical simulation and observations. At first sight, this choice of experiments seems oversimplified, since a unique relationship exists between creep and relaxation measures, which allows creep func-
90
Chapter 2. Constitutive Models in Linear Viscoelasticity
tions to be calculated for given relaxation functions. This means that for an arbitrary function X(t, T) which fits experimental data in relaxation tests, the corresponding creep function Y(t, r) is found uniquely and no experimental confirmation is needed. On the other hand, our objective is to avoid the necessity to determine the function of two variables X(t, T) experimentally. For this purpose, the function X(t, T) is replaced by three functions of one argument only [see Eqs. (2.5.7) and (2.5.10)], and, afterward, additional assumptions regarding these three functions are introduced [see Eqs. (2.5.11) and (2.5.25)]. Therefore, the aim is not to check the validity of Boltzmann's principle for epoxy adhesives [it has already been done by Vleeshouwer et al. (1989)], but to validate the preceding hypotheses regarding the functions E(t), N(t), and ~(t). Resolving Eq. (2.5.4) with respect to the strain e, we obtain [see Eq. (2.2.19)], e(t) = Y(t + te, t + te)Or(t)-
t OY -~r(t + te, ~" + te)O'(~')d'r,
fo
(2.5.26)
where Y(t, T) is the creep function. For a creep test with a constant stress o-, Eq. (2.5.26) implies that
(2.5.27)
ete (t) = Y (t + te, te)Or.
To predict results of creep tests, the function Y(t, T) should be calculated. For this purpose, we employ Eq. (2.2.33), which is presented as follows: OY 1 aX E(t + te)--Z--(t + te,S-Jr- te)-k- ~ ~ ( t dS E(s + te) as
-k- te, S + te)
aX + te, "r + te)--~s('r aY + te, S + te)d'r. = fs t -~r(t
(2.5.28)
We integrate Eq. (2.5.28) from T to t and change the order of integration. As a result, we find that ~ m1+ (te)t aX E(t + te)[g(t + te, t + te) - g(t + te, T + te)] + fT t E(s as
+ te, S + te)ds
x = f t a--~r(t + te, "r + te)dr f ~ a Y--~s('r + te, S + te)ds. We calculate the integral in the right-hand side of this equality with the use of the formula [see Eq. (2.2.21)], Y(t + te, t + re) =
E(t + te)'
and obtain E(t + te)Y(t + te, T + t e ) -
(t + te, T + te)Y(T fTt aX aT
+ te, T + te)dT = 1.
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
91
Finally, replacing T by s, we arrive at the linear Volterra integral equation for the function Y(t + te, T + te)
js t --~(t 0X
E(t + te)Y(t + te,S + te) --
+ te, r + te)Y(r + te,S + te)d~" = 1. (2.5.29)
Substitution of expressions (2.5.5), (2.5.7), (2.5.10), and (2.5.11) into Eq. (2.5.29) yields E,(t + te)Y,(t + te,S + te) --
fs t ~,(~" +
te)[1 -- N(t - ~')]Y,(~" + te,S + te)dT = 1, (2.5.30)
where Y,(t, ~) = E(O)Y(t, T).
It follows from Eqs. (2.5.27) and (2.5.30) that the dimensionless creep compliance
E(O)~-te(t) = or
J, te(t) =
Y,(t + te, te)
satisfies the linear Volterra equation E,(t + te)J, t e ( t ) -
f
t ~,(~" + te)[1 - - N ( t -
r)]J, te(T) dr = 1.
(2.5.31)
Since duration of the standard creep test is essentially less than the characteristic time of aging, Eq. (2.5.31) may be simplified. Replacing E,(t + te) by E , ( t e ) and ~ , ( t + te) by dP,(te), we obtain the integral equation in convolution E,(te)J, te(t) - dP,(te)
~00t [1 -
N(t - T)]J, te(T) d T -- 1.
(2.5.32)
Equation (2.5.32) can be solved explicitly provided the function N(t) equals a sum of exponential functions (2.5.25). We do not dwell on this issue and confine ourselves to the numerical analysis of Eq. (2.5.32). It follows from Eqs. (2.5.25) and (2.5.32) that
M E,(te)J, te(t) - r(te) Z X m ' Y m ~
t exp[--'ym(t -- ~)]J, te(7) dT = 1.
m=l Introducing the new variables Zm(t) -
f0 t e x p [ - T m ( t - l")]J, re(l") dl",
92
Chapter 2. Constitutive Models in Linear Viscoelasticity
we obtain J, te(t) -
1 E,(te)
1 + F(te)
Xm'YmZm(t) , m=l
where the auxiliary functions Zm(t) obey the ordinary differential equations dZm = -~/mZm + J, te, dt
Zm(O ) --O.
The compliance c u r v e s Jte (t) are plotted in Figure 2.5.4, together with experimental data obtained in creep tests. Comparison of numerical results with observations
0.39
© -log J
_
o
Q
-
©
-
2'
o
© _
° 0
J
© _
(
o
~o
110000
0
©
0.33
I 0
I
I
I log t
I
I 2
Figure 2.5.4: The creep compliance Y (GPa- 1) versus time t (sec) for an epoxy adhesive C quenched from 87 to 42°C and preserved t¢ (min) before loading. Circles show experimental data obtained by Vleeshouwers et al. (1989). Solid lines show results of numerical simulation. Curve 1" t¢ = 40; Curve 2: t¢ = 160.
93
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
shows that the model provides adequate prediction of experimental data without any additional fitting of adjustable parameters.
2.5.3
Prediction of Stress-Strain Curves for Time-Varying Loads
Using the model of adaptive links, stress-strain curves can be calculated for more complicated programs of loading than employed in the creep and relaxation tests. We confine ourselves to three cases: loading with a constant rate of strain, recovery at unloading, and steady periodic oscillations. For loading with a constant rate of strain d0, we can write e(t) = dot.
(2.5.33)
Substituting expression (2.5.33) into the constitutive Eq. (2.5.4) and using Eqs. (2.5.5), (2.5.7), (2.5.8), (2.5.10), and (2.5.11), we find that
O'te(t)
= E(0)d0
/ f0t t -
~.(1" + te)[1 - N ( t - ~')]~"d r
/
.
(2.5.34)
The stress-strain dependencies calculated with the use of Eqs. (2.5.33) and (2.5.34) are plotted in Figure 2.5.5. This figure demonstrates the behavior typical of nonaged viscoelastic media: when the time of loading increases (which is equivalent to a decrease in the rate of strain), the stress o- decreases due to the stress relaxation. To analyze the material recovery, we consider uniaxial deformation of a specimen of an age te under the action of the longitudinal stress o-(t) =
Oro, O<_t <-- T, 0, t > T,
(2.5.35)
where T is the time of loading. Substitution of expression (2.5.35) into Eq. (2.5.26) implies that ete(t) _ f Y(t + te, te), oro [ Y(t -+- te, t e ) -
Y(t + re, T + te),
0 <-- t <-- T, t > T.
(2.5.36)
According to Eq. (2.5.36), the function Y (t + te, te) determines the creep compliance. This function increases monotonically and tends to some limiting value Y°(te) as t ---, ~. It follows from Eq. (2.5.36) that the function Y (t + te, T + te) determines the material recovery when T exceeds the time necessary for the function Y (t + te, te) to reach its limiting value. The dimensionless creep compliance J*te (t) = E(O)Y (t + te, te) and creep recovery J,t, (t) = E(O)Y(t + te, T + te) calculated for the epoxy adhesive C at T = 10 (min) are plotted in Figure 2.5.6. The numerical results show that • The creep compliance (recovery) for a "young" material exceeds the creep compliance (recovery) for an "old" medium. • The creep compliance and the creep recovery curves increase monotonically in time.
Chapter 2. Constitutive Models in Linear Viscoelasticity
94
4.5
O"
I
I
I
I
I
I
I
0 E 0.01 Figure 2.5.5: The stress cr (MPa) versus the strain e for an epoxy adhesive C quenched from 87 to 42* C, preserved 20 min, and loaded with a constant strain rate 50 (min-1). Curve 1: 5o = 0.1; Curve 2: 5o = 0.01; Curve 3: 5o = 0.001.
• For a fixed age te, the creep compliance curves are located above the creep recovery curves. • With the growth of the aging time, the difference between the creep curves and the recovery curves vanishes. To study oscillations of a specimen of an age te, we consider the deformation program
e(t) = e0 sin(tot), where e0 is the amplitude and to is the frequency of longitudinal vibration. As common practice in the analysis of steady oscillations, we introduce the complex deformation
2.5. A Model of Adaptive Links for Aging Viscoelastic Media
95
1.3
1
J.te 2
3 4
1.0
I
I
I
I
I
I
I
I
I
0 t 10 Figure 2.5.6: The dimensionless creep compliance and creep recovery J*te versus time t (min) for an epoxy adhesive C quenched from 87 to 42°C and preserved te (rnin) before loading. Curves 1 and 3: creep; Curves 2 and 4: recovery. Curves 1 and 2: te = 5; Curves 3 and 4: te = 40.
program E(t) = Eo exp(~tot)
(2.5.37)
and replace the lower limit of integration in Eq. (2.5.4) by - ~ . Assuming additionally that the functions E(t) and cI)(t) change slowly within the interval of observations, we set E(t + te) ~ E(te),
dP(t + te) ~ dP(te).
Combining these equalities with Eqs. (2.5.4), (2.5.5), (2.5.7), (2.5.8), (2.5.10), and (2.5.11), we find that
{
or(t) = Eo E(te) exp(~oot) - d#(te)
i
}
[1 - N(t - 1")] exp(w91")d~" .
96
Chapter 2. Constitutive Models in Linear Viscoelasticity
Finally, we introduce the new variable s = t - ~"and obtain
o'(t) =
{
E(te) - ~(te)
/0
[1 - N ( s ) ] e x p ( - ~ m s ) d s
}
e0exp(~mt)
= [E[e(¢.o ) + ~E"te(m)]E(t),
where
E'te(m) = E(te) -
dP(te)
f~ [1 -
N(s)]cos(ms)ds,
(2.5.38)
EreII (m) = ~(te) ~0 ~ [ 1 - N(s)] sin(ms) ds.
For steady shear oscillations, Young's dynamic moduli E te/ and E"te should be replaced by the shear dynamic moduli G lte and G[~, respectively. Using formula (2.2.70) G ~__
E 2(1 + v)'
where v is Poisson's ratio, and assuming additionally that the material under consideration is incompressible, v = 0.5, we find from Eq. (2.5.38) that
te((-O) = -~
E(te) - ~(te)
[1 - N(s)]cos(ws)ds
1 fo ~ [ 1 - N(s)] sin(ms) ds. Gteii (60) = ~(I)(te)
, (2.5.39)
It is of interest to compare dependencies (2.5.39) with experimental data for semicrystalline polymers. For this purpose, we calculate the loss modulus G"te for polypropylene PP-43, using data for the storage modulus G tet and the loss tangent II I obtained by Struik (1987b). Material properties and details of tan 6 Gte/Gte experimental procedure are discussed in Struik (1978). The dynamic modulus G"te in shear oscillations with frequency 1 Hz is plotted versus the aging time te in Figure 2.5.7. According to this figure, the G"te value is practically independent of the aging time te. This assertion together with the first Eq. (2.5.39) implies that the storage shear modulus G Ire and the static shear modulus Gte =½Ere should differ by a constant, and their graphs (as functions of the aging time re) should be parallel. These graphs are plotted in Figure 2.5.8. The dependence G / against te is adopted from Struik (1987b), where experimental data are obtained for dynamic tests with frequency 1 Hz. The dependence E versus te is adopted from Struik (1987a), where experimental data are obtained for longitudinal tensile creep. The similarity of slopes of these graphs confirms that the model correctly predicts the viscoelastic behavior of polymers under static and dynamic loading. =
Bibliography
97
0.05
G"
_0 . . . .
I
0
I
"'
I
. . .0 . . . .
I
I
O
I
. . . .
I I log te
©
I
4 The loss shear modulus G" (GPa) versus the material age te (hr) for polypropylene PP-43 quenched from 120 to 20°C. Circles show experimental data obtained by Struik (198719). The solid line shows their approximation by the linear function G" = 0.00016 log(te) + 0.0308. -1
Figure
2.5.7:
Bibliography [ 1] Achenbach, J. D. and Chao, C. C. (1962). A three-parameter viscoelastic model particularly suited for dynamic problems. J. Mech. Phys. Solids 10, 245-252. [2] Adeyeri, J. B., Krizek, R. J., and Achenbach, J.D. (1970). Multiple integral description of the nonlinear viscoelastic behavior of a clay soil. Trans. Soc. Rheol. 14, 375-392. [3] Aklonis, J. J., MacKnight, W. J., and Shen, M. (1972). Introduction to Polymer Viscoelasticity. Wiley, New York.
98
Chapter 2. Constitutive Models in Linear Viscoelasticity
GI
1 ~G
I
I
I
I
I
I
I
I
I
log te 4 Figure 2.5.8: The static shear modulus G (GPa) (unfilled circles) and the storage shear modulus G ~ (GPa) (filled circles) versus the material age te (hr) for polypropylene PP-43 quenched from 120 to 20°C. Unfilled circles show experimental data obtained by Struik (1987a); Filled circles show experimental data obtained by Struik (1987b). Solid lines show their approximation by the linear functions G = 0.054 log(re) + 0.542 and G ~ = 0.048 log(te) + 0.625. -1
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Chapter 2. Constitutive Models in Linear Viscoelasticity
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Chapter 2. Constitutive Models in Linear Viscoelasticity
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Chapter 2. ConstitutiveModels in Linear Viscoelasticity
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Chapter 3
Nonlinear Constitutive Models with Small Strains This chapter deals with nonlinear constitutive models in viscoelasticity at small strains. We discuss differential constitutive equations in Section 3.1 and integral relations in Section 3.2. Based on the concept of adaptive links, we derive a new constitutive model for crosslinked polymeric materials in Section 3.3 and for noncrosslinked polymers in Section 3.4. Various approaches to the design of nonlinear constitutive models for viscoelastic media under isothermal loading are discussed in Christensen (1982), Drozdov (1996), Findley et al. (1976), Lockett (1972), and Ward (1971), to mention a few. A survey of nonlinear phenomena in viscoelasticity is provided by Malkin (1995).
3.1
Nonlinear Differential M o d e l s
This section is concerned with nonlinear differential models for viscoelastic media at small strains. As common practice, differential constitutive equations reflect rheological models consisting of elastic elements (springs) and viscoelastic elements (dashpots) connected in series and in parallel. To derive a nonlinear constitutive model, it suffices to replace a linear spring by a nonlinear elastic element, and a linear dashpot by a nonlinear viscoelastic element in an appropriate rheological model. An elastic element establishes a one-to-one connection between the stress o- and the strain ~. A linear spring obeys Hooke's law o- = EE,
(3.1.1)
where E is Young's modulus. A nonlinear spring satisfies the equality o- = ~(e), 107
(3.1.2)
108
Chapter 3. Nonlinear Constitutive Models with Small Strains
where ~(e) is a sufficiently smooth function. A power-law spring with the constitutive equation • (e) = Ae ~
(E > 0),
(3.1.3)
where A and a are adjustable parameters, provides an example of the nonlinear response. A dashpot establishes a one-to-one connection between the stress o" and the rate of strain 5, where the superscript dot denotes differentiation with respect to time. A linear dashpot obeys Newton's law o - = r/S,
(3.1.4)
where r/is the Newtonian viscosity. A nonlinear dashpot satisfies the constitutive equation cr = ,Iff~),
(3.1.5)
where ~(d) is a sufficiently smooth function. Examples of the nonlinear function • (d) are the power-law dashpot with the constitutive equation o- = Bd t3,
(3.1.6)
the Eyring dashpot with the constitutive equation o- = L sinh- 1 C' ~
(3.1.7)
and the Briant dashpot with the constitutive equation o- = D(d + d)--------S'
(3.1.8)
where B, C, D, L, d,/3, and 3' are material parameters and we assume that e > 0. Several expressions for the function o- = ~ ( e ) and its inverse ~ = • (o-) are presented by Halldin and Lo (1985) and Papo (1988). The Eyring equation (3.1.7) is presented in the form [see Eyring (1936)] = Csinh
~
,
(3.1.9)
where V is the Eyring volume, k is Boltzmann's constant, and 19 is the absolute temperature. Since no rational procedures exist for design of rheological models, it is rather difficult to establish reasons why one or another model is chosen to fit experimental data except the taste of the researcher. Thus, we confine ourselves to a classification of differential models according to the number of basic elements (springs and dashpots). We begin with the simplest models, which consist of two viscous elements connected in parallel. These models describe the viscoelastoplastic behavior in the
109
3.1. Nonlinear Differential Models
vicinity of the yield point [see, e.g., Papo (1988), where results of numerical simulation are compared with experimental data for gypsum plaster paste]. Rheological models without elastic elements reflect the response in viscoelastoplastic media, where viscoplastic stresses exceed essentially elastic stresses, and the latter may be neglected. A linear dashpot adequately describes the steady viscoplastic flow, but fails to predict the material response near the yield point, since it implies that the yield stress is proportional to the rate of strain, which contradicts experimental data [see Haward and Thackray (1968)]. To ensure an adequate description of the viscoelastic behavior for a wide range of strains, a linear dashpot is connected in parallel with a nonlinear viscoelastic element, which predicts the material behavior in the vicinity of the yield point. A combination of a linear dashpot (3.1.4) and a power-law dashpot (3.1.6) provides the Sisko model (3.1.10)
or = 'r/e + B 1 5 / 3 ,
where B1, ~3, and r/are adjustable parameters. Combining a linear dashpot with the Eyring dashpot, we arrive at the PowellEyring model or
= r/e + L s i n h - 1 -d-
(3.1.11)
C'
where C, L, and ~ are adjustable parameters. A linear dashpot together with the Briant dashpot provide the Carreau model 4
or = rl i~ + D ( i ~ '~ a+ )"-------~'
(3.1.12)
where D, d, 3', and a~ are adjustable parameters. A particular case of the Carreau model with 3' = 1 is the Williamson model o- = r/e + D
e+d
.
(3.1.13)
A specific nonlinearity in the viscous element is suggested by the ShangrawGrim-Mattocks model o- = r/e + B[ 1 - e x p ( - a ~)],
(3.1.14)
where B, a, and r/are adjustable parameters. A nonlinear Kelvin-Voigt element consists of a nonlinear elastic element connected in parallel with a nonlinear viscous element. The strains in the spring and dashpot coincide, whereas the total stress or equals the sum of the elastic stress ore and the viscous stress o'v, or -- ore "+- O'v.
(3.1.15)
110
Chapter 3. Nonlinear Constitutive Models with Small Strains
Substitution of expressions (3.1.2) and (3.1.5) into Eq. (3.1.15) implies the constitutive equation or = ~(e) + ~
~-
.
(3.1.16)
A nonlinear Maxwell element consists of a nonlinear spring and a nonlinear dashpot connected in series. The stresses or in the elastic and viscous elements coincide, and the total strain e equals the sum of the elastic strain Ee and the viscous strain Ev, = Ee + E~.
(3.1.17)
Assuming the nonlinear dependencies (3.1.2) and (3.1.5) to be valid, we write Ee __
(~)(O"),
dev _ ~(cr),
(3.1.18)
dt
where ~(o") and ~(o") are functions inverse to ~(e) and ~(d), respectively. We differentiate Eq. (3.1.17) with respect to time, use Eq. (3.1.18), and find that de _ x~(o") + E(o")do"
dt
d--7'
(3.1.19)
where
d~ _=(~r) = -d--ff(~r). For a linear elastic element (3.1.1) and the power-law viscous element (3.1.6), Eq. (3.1.19) implies that de
ldo" +(o")1//3
dt
E dt
-B
"
(3.1.20)
The model (3.1.20) provides fair fitting of experimental data for polycarbonate [see Halldin and Lo (1985)]. A generalization of Eq. (3.1.19) was suggested by Mihailescu-Suliciu and Suliciu (1979) and Gurtin et al. (1980) dode dt - Zo(o", e) + ZI(O" , E) d---t'
(3.1.21)
where Z0 and Z 1 a r e material functions. Existence, uniqueness, and stability of solutions to dynamic problems for viscoelastoplastic media with constitutive equation (3.1.21) were studied by Faciu (1991), Faciu and Mihailescu-Suliciu (1987, 1991), Podio-Guidugli and Suliciu (1984), and Suliciu (1984). Another extension of the Maxwell constitutive model (3.1.19) is proposed in the framework of the so-called "unified" approach, developed by Krempl and coauthors [see, e.g., Krempl (1987), Bordonaro and Krempl (1992), Krempl and Kallanpur
111
3.1. Nonlinear Differential Models
(1985), Krempl and Ruggles (1990), Nishiguchi et al. (1990a,b)]. According to that approach, viscous and plastic deformations are not distinguished, and the total strain E equals the sum of the elastic strain Ee and inelastic strain Ei E --
Ee +
(3.1.22)
E i.
The elastic strain is connected with the stress o- by the linear equation (3.1.1). The rate-of-strain for inelastic deformation is assumed to depend on the overstress
dEi dt
N
- ~(o- - o'0),
(3.1.23)
where o0 is some "equilibrium" stress, which is found from the stress-strain curve at extremely low rates of loading. Constitutive equations similar to Eqs. (3.1.22) and (3.1.23) are proposed by Ek et al. (1986) in the framework of the theory of stress-aided thermal activation, which goes back to the Eyring concept of kinetic rates for inelastic processes in solids [see Krauz and Eyring (1975)]. The concept of thermally activated processes is based on the following two hypotheses: 1. Flow processes in solids are characterized by some activation volume V similar to the Eyring volume in Eq. (3.1.9). As common practice, two phenomenological equations are employed for the activation volume [see Johnson and Gilman (1959)], V = K(tr - t r o ) K,
(3.1.24)
g = L exp ~ - (o" m O'0) •
(3.1.25)
Here K, L, K are adjustable material parameters, ® is the absolute temperature, k is Boltzmann's constant, b is the Burgers vector, and S is the activation area. 2. The rate of inelastic deformation is proportional to the activation volume
dEi dt
- aV,
(3.1.26)
where a is a material constant. We differentiate Eq. (3.1.22) with respect to time, replace the derivative of the elastic strain ee with the use of Eq. (3.1.1), and substitute expressions (3.1.24) to (3.1.26) for the rate of inelastic strains. As a result, we obtain either the Hooke-Norton equation
dE
1 do
dt
E dt
+ Ko(tr - o-o)K,
(3.1.27)
or the Hooke-Eyring equation 1 dtr
de m
dt
E dt
o-
+ L0 exp
-- o'0) M
'
(3.1.28)
112
Chapter 3. Nonlinear Constitutive Models with Small Strains
where Ko = c~K,
Lo = aL,
M-
kO bS
Using experimental data for polyethylene, Ek et al. (1986, 1987) demonstrated that Eq. (3.1.27) is acceptable at low stresses and Eq. (3.1.28) provides adequate prediction of observations at high stresses. To fit relaxation curves, the internal stress o'0 was assumed to be an adjustable function of the strain e. Several physical models reflect the concept of thermally activated processes in solids. For example, Johnson and Gilman (1959) treated V as the dislocation velocity, while Amoedo and Lee (1992) treated V as the activation energy. Unlike Eq. (3.1.26), Johnson and Gilman (1959) assumed that the stress o- is proportional to V, cr = -/31V, where the coefficient /31 depends on Young's modulus and dislocation density. De Batist and Callens (1974), Kubat and Rigdahl (1976), and Kubat et al. (1992) suggested that the relaxation rate is proportional to V, dcr dt
--
-/32V
'
where 132 is an adjustable parameter. According to Amoedo and Lee (1992), the effective stress is proportional to the activation energy cr - or0 = 133V. To describe the viscoelastic response in media with several relaxation times, the Maxwell elements are connected in parallel. Generalized nonlinear Maxwell models are considered by Keren et al. (1984), La Mantia (1977), La Mantia and Titomanlio (1979), La Mantia et al. (1981), and Partom and Schanin (1983). In those works, N Maxwell elements with linear functions f~n(E)
'qJ'n(E)
- - En~. ,
= r/,,~
(3.1.29)
are assumed to be connected in series. The latter means that the strains in the Maxwell elements coincide, while the total stress or equals the sum of stresses O"n in the elements N Or = Z
(3.1.30)
O'n.
n=l
It follows from Eqs. (2.1.3) and (3.1.29) that the stress equation Orn +
r/n do',,
E. dt
dE - "O,, •
-d-i
O"n
obeys the differential
(3.1.31)
113
3.1. Nonlinear Differential Models
We introduce the characteristic time of relaxation G m On En and write Eq. (3.1.31) as follows" de orn + Tn dcrn dt - EnTn m dt.
(3.1.32)
La Mantia (1977) suggested that the characteristic times of relaxation Tn are expressed in terms of the free volume fraction fn for the nth spring with the use of the Doolittle equation Tn. Tno
exp ( 1 ~
1) fno
(3,1,33)
'
where the index 0 indicates quantities that refer to the initial (stress-free) configuration, The increment of the free volume fraction fn is proportional to the mechanical energy Wn
Wn
fn = fn0 + 6~--~-,
(3.1.34)
where 6 is a material parameter. It is assumed that the mechanical energy Wn coincides with the stress o- for uniaxial loading, and Wn is proportional to the first invariant of the stress tensor for three-dimensional loading, (3,1,35)
W n --- t . L n p ,
where p is pressure and/xn is an adjustable parameter, La Mantia and Titomanlio (1979) generalized Eq, (3,1,34) and supposed that the free volume fraction fn is governed by the ordinary differential equation dt
rn
-~,
- (In -- fnO)
1
,
(3.1.36)
which implies the steady solution (3.1.34). The model (3.1.30), (3.1.32), (3.1.33), and (3.1.36) predicts correctly the viscoelastic response in polyisobutylene [see La Mantia et al. (1981)]. In the La Mantia constitutive model, the material nonlinearity arises both for uniaxial and three-dimensional loading. Partom and Schanin (1983) proposed a model in which the nonlinearity is essential only for three-dimensional deformations. It is assumed that (i) the volume deformation is linearly elastic; (ii) the deviatoric part ~ of the strain tensor ~ equals the sum of the deviatoric parts of the elastic strain tensor ee and the viscous strain tensor ~ ~ = ~e + ~'~;
(3.1.37)
(iii) for any Maxwell element, the deviatoric parts ~n of the stress tensors d'n coincide for the elastic and viscous elements; (iv) the deviatoric part of the elastic strain tensor
114
Chapter 3. Nonlinear Constitutive Models with Small Strains
ee is proportional to the deviatoric part of the stress tensor Sn
6'e --
~n 2Gn '
(3.1.38)
where Gn is the shear modulus, and (v) the principal axes of the tensors ~,, and dO~/dt coincide dG
dt
- A~n,
(3.1.39)
where An is a material function. Multiplying Eq. (3.1.39) by itself, we obtain H2 =
2 2 A n ~, n ,
(3.1.40)
where ~n = (2Sn :Sn) 1/2
is the stress intensity and
Hv = (2 dev " dev dt ) is the viscous rate-of-strain intensity, which is assumed to be a given function of the stress intensity ~n H v = Fn(~n).
(3.1.41)
Combining Eqs. (3.1.40) and (3.1.41), we find that Fn(~n)
A,, - - - .
(3.1.42)
To account for the effect of hydrostatic pressure p on the viscoelastic shear response, Partom and Schanin (1983) replaced the stress intensity E~ in Eq. (3.1.42) by the "reduced stress" intensity ~ 0 _. ~ n E tZnP,
where/*n is a material parameter. As a result, they arrived at the nonlinear constitutive equation
dOv dt
E
1 Fn(~n £n
- Id,np)Sn.
(3.1.43)
The model (3.1.30), (3.1.37), (3.1.38), and (3.1.43) correctly describes the viscoelastic response in poly(vinyl chloride) under uniaxial and biaxial loading. A similar model for the effect of hydrostatic pressure on the viscoelastoplastic behavior of metals was proposed by Rubin (1987).
3.1. NonlinearDifferentialModels
115
Nonlinear analogs of the standard viscoelastic solid were proposed in 1940s by Eyring and Haxley [see the bibliography in Krauz and Eyring (1975)]. Haward and Thackray (1968) analyzed the simplest model, in which a linear elastic element with Young's modulus E1 was connected in series with the Kelvin-Voigt element, consisting of a linear elastic spring with Young's modulus E2 and an Eyring dashpot with parameters C and L. The total stress o" coincides with the stress O" 1 in the elastic element and the stress Or2 in the Kelvin-Voigt element. The total strain e equals the sum of the strain e~ in the elastic element and the strain E2 in the Kelvin-Voigt element E =
The strain
e1
E1
is connected with the stress
(3.1.44)
-+- E 2.
Or by Hooke's law
O" 1 =
or E 1 --
(3.1.45)
. E1
The strain e2 is expressed in terms of the stress 0"2 = tr by the constitutive equation
cr=E2e2+gsinh-l(lde2) ~--~
.
(3.1.46)
Excluding the variables el and e2 from Eqs. (3.1.44) to (3.1.46), we find that o" = E 2
( e -- ~11) +
L sinh- 1
[l(de dt
_
1 do-)]
E1 dt
"
After simple algebra, we arrive at the nonlinear differential equation
de dt
1 d~r - C sinh II ( ~EI c+ E2 r dt E1
E1
- Eze )1 ,
(3.1.47)
which demonstrates fair agreement with experimental data in tests with constant rate of extension for cellulose derivatives, polycarbonates and poly(vinyl chloride) [see Haward and Thackray (1968)]. Amoedo and Lee (1992) analyzed constitutive relations similar to Eqs. (3.1.44) to (3.1.46). Unlike the Haward-Thackray model, they presented the stress 0"2 in the Kelvin-Voigt element as a sum of the elastic stress O'~ = E 2 E 2
(3.1.48)
and the viscous stress (overstress) o-~~. The rate of strain in the viscous element expressed in terms of the viscous stress O'~ l =
or 2 - - 0"~
E2
is
(3.1.49)
and some internal parameter 0 with the use of an analog of the second equality in Eq. (3.1.18)
de2 _ dt
~(o-~', 0),
(3.1.50)
116
Chapter 3. Nonlinear Constitutive Models with Small Strains
where ~ is a material function. The internal variable 0 obeys a nonlinear ordinary differential equation, the right side of which depends on the viscous stress 0-~
dO - F(O, 0"~'). dt
(3.1.51)
The model (3.1.48) to (3.1.51) correctly predicts experimental data for polycarbonate and polypropylene. Ng and Williams (1986) proposed a similar model, where a nonlinear Maxwell element, consisting of a linear spring with Young's modulus E1 and an Eyring dashpot with parameters C and L, was connected in parallel with an elastic spring with Young's modulus E2. The strains in the Maxwell element and in the spring coincide, e1 =
e2 =
e,
while the total stress or equals the sum of the stress o1 in the Maxwell element and the stress o2 in the spring or - -
(3.1.52)
0-1 4- 0 " 2 .
The constitutive equation for the Maxwell element has the form
1 d0-1
0-1
de
m ~ 4- C sinh ~ = m . E1 dt L dt
(3.1.53)
The response in the elastic element obeys Hooke's law o'2 = E2e.
(3.1.54)
Excluding the variables 0-1 and 0"2 from Eqs. (3.1.52) to (3.1.54), we find the constitutive equation
E1 + E2 de 1 do 0- - E2e + C sinh ~ = . E1 dt L E1 dt
(3.1.55)
Linearization of Eq. (3.1.55) implies the constitutive equation of the standard viscoelastic solid
do + E1 de E1E2 too" - (El + 4- ~ ~ , dt rI E2)--~ ~l
(3.1.56)
where 77-
L C"
Equation (3.1.55) correctly predicts experimental data in tests with constant rate of extension for a number of aromatic polyesters [see Ng and Williams (1986)]. The preceding list of nonlinear differential constitutive models is far from being exhaustive. We deal with the simplest constitutive equations, and do not discuss
3.2. Nonlinear Integral Models
117
rheological models that contain more than three elements [see, for example, So and Chen (1991), where a four-elements Burgers-type model with an Eyring dashpot is studied, and Morimoto et al. (1984), where a five-elements rheological model is applied to describe the viscoelastic response in polyurethane foams]. We confine ourselves to the viscoelastic behavior with small strains, and do not analyze models in which the geometrical nonlinearity becomes significant [see, for example, Haward and Thackray (1968), Ng and Williams (1986), and So and Chen (1991)]. A brief survey of differential constitutive relations in finite viscoelasticity is provided in Chapter 4.
3.2
Nonlinear Integral Models
The objective of this section is to discuss integral constitutive equations for viscoelastic media with small strains. We begin with uniaxial loading, in which the only stress ois connected with the only strain e, and analyze single-integral and multiple-integral constitutive relations. Afterward, the models are generalized to three-dimensional loading, in which the constitutive equations express the stress tensor 6- in terms of the strain tensor 5. Based on phenomenological approach, we concentrate on convenience of constitutive models for engineering calculations and on agreement between experimental data and their predictions.
3.2.1
Uniaxial Loading
We begin with single-integral constitutive models, which may be treated as generalizations of the Boltzmann superposition principle to nonlinear media. The study is concentrated on the stress-strain relations. Construction of strain energy densities for nonlinear viscoelastic materials is beyond the scope of our analysis [see, for example, Gurtin et al. (1980) and Gurtin and Hrusa (1988) for a discussion of this question].
Single-Integral Constitutive Equations According to the Boltzmann superposition principle, the stress o- in a linear viscoelastic medium is connected with the strain e by the formula r(t) =
~00t x ( t ,
de
(3.2.1)
where X(t, ~') is a function of two variables, which is assumed to be twice continuously differentiable. Assuming a specimen to be in the natural (stress-free) state before loading, E(O) = O,
o-(O) = O,
(3.2.2)
118
Chapter 3. Nonlinear Constitutive Models with Small Strains
and integrating Eq. (3.2.1) by parts, we obtain ~r(t) = x ( t , ~)~(r)I "=' r=0 -
f0 t -~y °~X(t,
T)~(T)dr
= X(t, t)E(t) - foot -~r(t, OX r)~(r) dr.
(3.2.3)
X(t, r) = E ( r ) + Q(t, r),
(3.2.4)
Setting
where E(t) is the current Young's modulus and Q(t, 7) is the relaxation measure, which satisfies the condition Q(t, t) = O,
we present Eq. (3.2.3) in the form o"(t) = E ( t ) e ( t ) -
f0 t ~~[ E ( r )
+ Q(t, r)]E(r) dr.
(3.2.5)
Equation (3.2.5) describes the mechanical response in an aging viscoelastic medium. For nonaging materials, (3.2.6)
X(t, 7) = Xo(t - r),
which implies that E(t) = E,
(3.2.7)
Q(t, r) = EQo(t - r).
Combining Eqs. (3.2.5) and (3.2.7), we find that o"(t) = E
e(t) -
J0'
R(t - r ) e ( r ) d r
]
(3.2.8)
,
where dQo R(t) = - ~ ( t ) dt
is the relaxation kernel. An inverse relationship reads E(t) = -~ o'(t) +
K(t - r)o'(r) d
,
(3.2.9)
where K ( t ) is the creep kernel. Equation (3.2.9) follows from the Boltzmann superposition principle do" (T ) dr, e(t) = f0 t Y(t, r)--~r
(3.2.10)
3.2. Nonlinear Integral Models
119
provided that K(t) =
Y ( t , ~) = Yo(t - T),
EdYo
~(t).
(3.2.11)
Equations (3.2.8) and (3.2.9) can be presented in the operator form 1
or = E ( I - R)E,
E = -z(l + K)or,
(3.2.12)
where I is the unit operator and K, R are the creep and relaxation operators with the kernels K ( t ) and R(t), respectively. For any function f ( t ) integrable in [0, oo), we write If = f(t),
Kf
=
fot
K ( t - T)f(~') d~',
Rf
=
fot
R ( t - ~')f(~') d~'.
The operators K and R are connected by the equality (I + K) = ( I - R) -1. One of the first nonlinear constitutive equation was suggested by Guth et al. (1946) for the viscoelastic behavior of rubber. According to Guth et al. (1946), to derive a constitutive equations for a nonlinear medium, the strain e in the linear constitutive equation (3.2.8) should be replaced by some nonlinear function qffe). As a result, we obtain or(t) = E
I
q~(e(t)) -
/0'
R ( t - ~')q~(e(l"))d r
1
(3.2.13)
.
Resolving Eq. (3.2.13) with respect to qffe), we obtain
'E
qff e(t)) = -E or(t) +
/ot
g ( t - r)or(r) d r
1
(3.2.14)
.
Constitutive equations (3.2.13) and (3.2.14) were derived also by Rabotnov (1948), by expanding an arbitrary nonlinear functional into a series in multiple integrals. Rabotnov's approach will be discussed later in detail. Equation (3.2.13) expresses the stress o" in terms of a nonlinear function q~ of the strain e. Talybly (1983) suggested a constitutive equation, where the strain e(t) is expressed in terms of a nonlinear function q~ of the stress o-,
IE
e(t) = E7 ~(or(t)) +
/0
K ( t - 'r)qt(o'('r))d
.
(3.2.15)
Equation (3.2.15) may be treated as a nonlinear version of the constitutive equation (3.2.9). It is of interest to compare Eq. (3.2.15) with the formula for strain in a
120
Chapter 3. Nonlinear Constitutive Models with Small Strains
nonlinear elastic material [see e.g. Truesdell (1975)], aW e = ~(o-), ~o"
(3.2.16)
where W(o-) is the specific potential energy (per unit volume). Setting aW
@(~) - E-~(cr), we present Eq. (3.2.15) as OW e = OW 0o" (o(t)) + f0 t K ( t _ r)-0--~(o'(~'))d~'.
(3.2.17)
A nonlinear analog of the constitutive equation (3.2.10) for nonaging media was proposed by Findley et al. (1976) based on a modified superposition principle [see also Leaderman ( 1943) ],
f0t
e(t) =
(3.2.18)
Yo(t - r ) d d / ( c r ( r ) ) d r . dr
To generalize Eqs. (3.2.13) and (3.2.14), we assume that the current nonlinear response and the influence of the deformation history differ from each other. This implies the constitutive equations
[
o(t) = E qh(e(t)) e(t) = ~
f0t
qq(o'(t)) +
]
R ( t - "r)q~2(e('r))d'r ,
K ( t - r)q~2(o'(~')) d
(3.2.19) ,
(3.2.20)
which are characterized by four nonlinear functions q~l(e), q~2(e), I~1(O'), and qJ2(o'). To validate Eq. (3.2.20), Talybly (1983) determined the functions qq(o'), q~2(o'), and the kernel K ( t ) for high-density polyethylene in tension. Moskvitin (1972) suggested accounting for two types of nonlinearity typical of the constitutive equations (3.2.13) and (3.2.15) by introducing two nonlinear functions qffo') and q~(e),
[
qJ(o'(t)) = E qff e(t)) -
/ot
R ( t - l")qffe(r)) dr
1.
(3.2.21)
Bugakov (1989) demonstrated that Eq. (3.2.21) ensured an acceptable accuracy in predicting experimental data for a number of polymeric materials. Suvorova (1977), Makhmutov et al. (1983), and Viktorova (1983) proposed to replace one integral operator in Eq. (3.2.14) by a sum of several Volterra operators with different kernels to describe nonlinear processes in viscoelastic and viscoelastoplastic media.
121
3.2. N o n l i n e a r Integral M o d e l s
Active loading of a viscoelastoplastic material in the interval [0, T] obeys the formula
1 [o(t) + f0t
qffe(t)) = ~
K l (t - ~')o(~') d~" +
f0t
K z ( t - ~')o'(~') d r
1
,
(3.2.22)
where K1 (t) and Kz(t) are creep kernels. The first kernel, K1 (t), characterizes viscous effects, whereas the other kernel, Kz(t), characterizes plastic effects and/or damage cumulation. The difference between Eqs. (3.2.14) and (3.2.22) becomes evident under unloading, when the intensity of loads decreases in the interval [T, ~). Unlike the constitutive relation (3.2.14), which remains without changes, Eq. (3.2.22) takes the form
1 [o'(t) + ~0t
~(~(t)) = ~
K l (t - ~')o(~') d~" +
fr
1
K 2 ( t - ~')o'(~') d~" .
(3.2.23)
Makhmutov et al. (1983) demonstrated fair agreement between the model (3.2.22), (3.2.23) and experimental data for organoplastics and carbon-fiber and organic-fiber composites loaded and unloaded with constant rates of strains. Experimental data for a number of polymers show that stresses affect mainly creep and relaxation kernels, whereas the instantaneous response remains linear. To account for this phenomenon, Ilyushin and Ogibalov (1966) proposed a model in which the creep and relaxation kernels depended on two variables: the time t - ~-and either the stress intensity ~(~') or the strain intensity F(~-). In this case, Eqs. (3.2.8) and (3.2.9) read o'(t) = E
e(t) -
e ( t ) = -~
o(t) +
R ( t - ~', F(r))e(r)
K ( t - r,
dr ,
~(r))o'(r) dr .
(3.2.24)
Assuming the effect of stresses and strains on the kernels to be rather weak, they expanded these kernels into the Taylor series with respect to the second argument and neglected terms of the second order of smallness. As a result, they arrived at the following constitutive equations: o(t)
= E
e ( t ) = -~
e(t) -
o(t) +
R l (t - ~ ) e ( ~ ) d r -
Kl(t - ~)o(r)dr
Rz(t
+
- ~-)F(r)E(~') dr ,
Kz(t -
-r)~(r)o'(r)dr ,
(3.2.25) which were called the main cubic theories of creep and relaxation [see also Ilyushin and Pobedrya (1970)]. The model (3.2.25) was verified in a number of studies [see,
122
Chapter 3. Nonlinear Constitutive Models with Small Strains
e.g., Malmeister (1982, 1985), Malmeister and Yanson (1979, 1981, 1983), Urzhumtsev (1982), Urzhumtsev and Maksimov (1975), to mention a few]. To account for the effect of stresses and strains on creep and relaxation kernels, Leaderman (1943) proposed to change the time scale in Eq. (3.2.1). Based on the concept of reduced or pseudo-time {~ = {~(t), previously developed to describe the effect of temperature on the viscoelastic behavior, he derived a nonlinear analog of the linear constitutive equation (3.2.1) for nonaging media or(t) =
Xo(~(t) - ~(~'))
(~') d~'.
(3.2.26)
The pseudo-time ~(t) is connected with the real time t by the formula
~(t)
=
~0"ta(e(r))' d~-
(3.2.27)
where a = a(e) is a shift factor. For relaxation tests with
~(t) = {0, t
t-->O,
Eqs. (3.2.26) and (3.2.27) imply that (3.2.28) According to Eq. (3.2.28), the curves o-,/e versus log(time) plotted at various strains e can be obtained from each other by horizontal shift along the time axis. The constitutive equation (3.2.26) was checked experimentally by Dean et al. (1995) for poly(vinyl chloride), by Losi and Knauss (1992a, b) and Knauss and Emri (1987) for poly(vinyl acetate), by Shay and Carruthers (1986) for several amorphous polymers, and by Yanson et al. (1983) for an epoxy resin. Wineman and Waldron (1993) employed Eq. (3.2.26) to describe qualitatively the viscoelastoplastic behavior of polymeric materials. McKenna and Zapas (1979) proposed to treat shift of relaxation spectra caused by applied loads as an apparent rejuvenation of viscoelastic materials [see also Waldron et al. (1995) for a discussion of this issue]. An explicit expression for the shift function a(e) may be derived with the use of the concept of free (freezing-in) volume, which was proposed by Doolittle in 1950s for polymeric liquids. According to the free-volume theory [see, for example, Knauss and Emri (1987) and Losi and Knauss (1992a, b)], the shift factor a in Eq. (3.2.27) is expressed in terms of the free-volume fraction f with the use of the formula [see
123
3.2. Nonlinear Integral Models
Doolittle (1951)] B l°ga=lnl0
(1 1) ~-~ •
(3.2.29)
Here B is a material constant and f0 is the free volume fraction in the reference (stressfree) state. The function f ( t ) at the current instant t equals the sum of the reference value f0 and its increments in the interval [0, t). Any increment of f is proportional to increments of temperature, pressure, and humidity (solvent concentration) with prescribed coefficients. At isothermal loading with a fixed moisture content, f = f0 + c~,
(3.2.30)
where a is a material parameter. Equations (3.2.26), (3.2.27), (3.2.29), and (3.2.30) permit the strain e in a nonlinear viscoelastic medium to be determined for a given program of loading tr(t). Experimental data for poly(vinyl acetate) [see Losi and Knauss (1992a)] and polyethylene [see Chengalva et al. (1995)] demonstrate good agreement with theoretical predictions. By analogy with Eq. (3.2.30), the free volume fraction f can be assumed to depend linearly on the moisture content w f = f0 +/3w,
(3.2.31)
where/3 is a material parameter. Equations (3.2.27), (3.2.29), and (3.2.31) provide a version of the time-moisture superposition principle similar to the time-temperature superposition principle in thermoviscoelasticity. However, experimental data obtained for a polyester resin in creep tests demonstrate poor agreement with predictions based on this principle [see Aniskevich et al. (1992)]. The latter means that more sophisticated relations should be suggested between the free volume fraction and characteristics of a viscoelastic medium. For example, Knauss and Emri (1987) proposed the Volterra equation for the free volume fraction f (t) = fo +
f0t M(t
- ~')o('r) d~"
(3.2.32)
with a prescribed positive kernel M(t). Yanson et al. (1983), analyzing loading programs with constant rates of strains for organic plastics, found that f should depend on the strain rate dE(t)/dt. La Mantia et al. (1981) proposed a nonlinear ordinary differential equation for the free volume fraction f. The fight-hand side of that equation depends on the stress intensity ~ at the current instant t. Experimental data obtained in elongational and shear tests for polyisobutylene melts demonstrate fair agreement with the model predictions. The preceding approaches have merely phenomenological character, since physical concepts are absent for the dependence of the free volume fraction f on stresses (strains). Schapery (1964, 1966, 1969) suggested a constitutive model, which generalized Eqs. (3.2.19) and (3.2.20) on the one hand and Eqs. (3.2.26) on the other. The
124
Chapter 3. N o n l i n e a r Constitutive M o d e l s with Small Strains
Schapery equations read
j0"tX o ( ~ ( t )
o'(t) = q~0(¢(t)) + q~l(E(t)) e(t)
- sO(r)) q~2(E(r))dr,
= ~0(o'(t)) + ~l(O'(t)) ~0"t Yo(~l(t)
-
aTd r/(r))--;-~2(o-(r))dr,
(3.2.33)
where ~(t) =
fo
t
dr a,(E(r)) '
r/(t) = fot . dr a¢ (o-(r))
(3.2.34)
The model (3.2.33) provides rather general stress-strain relations for a nonlinear viscoelastic medium, which are compatible with basic principles of thermodynamics. This is a reason why the Schapery model is widely used for fitting experimental data. Model (3.2.33) was employed by Rand et al. (1996) for polyethylene films, by Schapery (1966) for a polybutadien acrylic acid propellant, by Smart and Williams (1972) for polypropylene and poly(vinyl cloride), by Peretz and Weitsman (1982) for an adhesive material, by Wing e t al. (1995) for polycarbonate and polycarbonate foams, by Wortmann and Schulz (1994a, b) for Nomex, Kevlar, and polypropylene fibers, etc. A shortcoming of the Schapery model is that it requires a large number of material functions to be found in experiments. The latter implies that a wide experimental program should be carded out to determine adjustable parameters, since introduced a priori hypotheses may lead to poor agreement with observations [see a discussion of this question in Smart and Williams (1972)].
ConstitutiveEquations
Multiple-Integral The general nonlinear stress-strain relation in the nonlinear viscoelasticity with small strains is written as o-(t) = G(~(r))
(0 -< r --- t),
(3.2.35)
where G is a nonlinear functional, which satisfies axioms of the constitutive theory [see, for example, Drozdov (1996)]. Expression (3.2.35) was introduced by Volterra in 1930s. Green and Rivlin (1957), Coleman and Noll (1960), Pipkin (1964), and Pobedrya (1967) proposed approximating the functional G by polynomials o0
G(E) = ~
Gm(e),
(3.2.36)
m=l
where
f0t f0t G m ( t , T1, " "
Gm(E) . . . .
, Tm)dE(T1)" " " de(rm).
(3.2.37)
125
3.2. Nonlinear Integral Models
Substitution of expression (3.2.36) into Eq. (3.2.35) implies the constitutive equation or(t) = Z
Gm(e(r))
(3.2.38)
(0 <- ~" <-- t).
m=l
Resolving Eq. (3.2.38) with respect to the strain e, we find that oo
e(t) = Z
9-/'m(or(l"))
(0 --< 1" <-- t),
(3.2.39)
m=l
where
fot foo'
9-(m(or) . . . .
Hm(t,
"rl . . . . .
"rm) do'(7"l)
• • •
dor(~'m).
(3.2.40)
Equations (3.2.37), (3.2.38), (3.2.39), and (3.2.40) require infinite series of functions Gm(t, ~'1. . . . . "rm),
Hm(t, ~'1. . . . . "rm)
to be determined in experiments. To employ these equations in applied problems, additional simplifications should be introduced. For nonaging media with Hm(t, "rl . . . . . ~'m) = Hm(t - "rl . . . . . t - "rm),
Rabotnov (1948) assumed the functions Hm(t - ' r l . . . . . products of some function of one variable by itself
t - ~'m) to be factorized as
m
Hm(t - rl . . . . . t
--
'rm)
=
Olm
H
H ( t - rk),
(3.2.41)
k=l
where Ogm are material parameters. Substituting expression (3.2.41) into Eqs. (3.2.39) and (3.2.40), we obtain e(t) = Z
H ( t - r)
Olm
m=l
We set oo
dp(t) = Z
O~mtm
m=l
and denote by q~(t) the inverse function, ~(th(t)) - t.
(r) d r
.
(3.2.42)
126
Chapter 3. Nonlinear Constitutive Models with Small Strains
It follows from Eq. (3.2.42) that t
(¢(e(t)) =
fo
do" H ( t - r)--dT(r) dr.
(3.2.43)
We integrate Eq. (3.2.43) by parts, use Eq. (3.2.2), introduce the notation E-
1 H(O)'
1 dH K(t) = -H(O---) d--t-(t)
and arrive at the constitutive model (3.2.14). Similar transformations were suggested by Findley and Khosla (1955) and Findley and Lai (1967). Sums of series (3.2.38) and (3.2.39) were calculated explicitly for a nonaging medium by Pobedrya (1981) under the assumption that the mth terms G,n and 94~ are proportional to the ruth power of a small parameter K. As a result, the constitutive equations (3.2.38) and (3.2.39) were presented in the form or(t) =
e(t) =
f0 t Xo(t
E(r)
- r)d
fot Yo(t -
1 - ~/fo ~70(t - "rl) dE('rl) r)d
1 - ~/fO ~'0(t - rl) dor(rl)'
(3.2.44)
where XT0(t)and ~'0(t) are auxiliary functions of creep and relaxation. To validate Eqs. (3.2.44), Sharafutdinov (1987) used experimental data for polycarbonate in creep tests. As common practice, only one to three terms in the series (3.2.38) and (3.2.39) are taken into account. Green and Rivlin (1957), Ward and Onat (1963), Findley and Onaran (1968), and Lai and Findley (1968a, b) proposed the so-called three-integral constitutive equations e(t) =
fOtHi(t, "rl) do'('rl) + ~o'tfo0tH2(t, 'rl, 'r2) dor('rl) dor('r2) + fotfotfot H3(t, T1, T2, '/'3)dor('rl) dor('r2) dor('r3).
(3.2.45)
The model (3.2.45) demonstrates fair prediction of experimental data in uniaxial creep and relaxation tests, in uniaxial tests with constant rate of loading, and in biaxial relaxation tests for polycarbonate [see Lai and Findley (1969) and Haio and Findley (1973)], polypropylene [see Hadley and Ward (1965) and Ward and Wolfe (1966)], poly(vinyl chloride) [see Findley and Onaran (1968)], polyurethane [see Nolte and Findley (1971, 1974)], asphalt concrete [see Vakili (1983)], and clay soils [see Adeyeri et al. (1970) and Krizek et al. (1971)].
3.2.2
Three-Dimensional Loading
As common practice, constitutive equations are proposed first for uniaxial loading and afterward, extended to three-dimensional loading. Three basic approaches may
127
3.2. Nonlinear Integral Models
be distinguished to designing constitutive relations for a viscoelastic medium under arbitrary loads. According to the first, to obtain constitutive equations for three-dimensional loading, it suffices to replace the stress o" by the stress tensor 6-, and the strain E by the strain tensor ~ in the constitutive equations for uniaxial deformation. For example, to generalize Eq. (3.2.17), Malmeister (1982) and Malmeister and Yanson (1983) suggested the following constitutive equation: 0W ~(t) = T - d ( ~ ( t ) ) +
f0 t x ( t
~-) .... (6-(~')) d~'.
-
(3.2.46)
dO"
Equation (3.2.46) with the specific potential energy W(6") = c112(6") + 6"212(0") + c0[c112(0 ") +
6'212(0-)] 2,
where ck are material parameters, and I~ stand for the principal invariants, demonstrates fair prediction of experimental data for polycarbonates and epoxy resins in a wide range of temperatures. According to the second approach, we distinguish the volume deformation E and the shear deformation ~, E = I1(~),
1 ^ ~EI,
~' = ~ -
where i is the unit tensor, and suppose that constitutive equations for the amounts e and ~ have the same form as appropriate constitutive equations for uniaxial loading. This procedure is applicable when the stress-strain relations for uniaxial loading are "quasi-linear" and the nonlinearity is introduced in terms of some internal time. An additional simplification arises when the volume deformation is treated as linearly elastic. For example, three-dimensional analogs of the constitutive equations (3.2.24) read [see Urzhumtsev (1982)], O"(t) = 3Koe(t),
e(t) = fK-oO"(t),
[
}(t) = 2G ~'(t) -
~(t) =
/0t
]
R(t - "r, F(-r))g~(-r) d'r ,
}(t) +
K(t - ~', E(~'))~(r) d r
, (3.2.47)
where K0 is the bulk modulus, G is the shear modulus, o- is the first invariant, ~ is the deviatoric part of the stress tensor,
r =
5~.~
,
~; = ( 2 ~ . ~ ) ' / 2
are intensities of strains and stresses, and K and R are the shear creep and relaxation kernels.
128
Chapter 3. Nonlinear Constitutive Models with Small Strains
The third approach based on a hypothesis regarding the "tensor-linear" character of the stress-strain relations. According to this assumption, the stress tensor (or its deviatoric part) at the current instant t depends linearly on the strain tensor (or its deviatoric part) at previous instants, whereas coefficients in this dependence may be arbitrary (nonlinear) functions of the principal invariants of the strain tensor. To extend constitutive equations for uniaxial loading to three-dimensional case, it suffices 1. To present any nonlinear function q~(E)in scalar constitutive equations as a product of linear and nonlinear functions q~(e) = ~(e)e.
(3.2.48)
2. To replace the strain E in the linear function by the strain tensor (or its deviatoric part), and to replace the strain in the nonlinear function by some principal invariant of the strain tensor. 3. To repeat the same proceduce with any nonlinear function of the stress o'. 4. To substitute refined expressions into the constitutive equations. As an example, we consider the constitutive equation (3.2.13) for an incompressible viscoelastic bar. Points of the bar refer to Cartesian coordinates {Xl, x2, x3}. For uniaxial extension in the Xl direction, the incompressibility condition implies that the strain tensor ~ has three nonzero components Ell = E and ~22 = E33 - - - - ~ E1 , and the strain intensity equals F = e. Substitution of these expressions into Eq. (3.2.48) yields q~(E) = ~(F)e,
(3.2.49)
where
q,(F)- ,p(F) F Finally, combining Eqs. (3.2.13), (3.2.14), and (3.2.49) and using the incompressibility condition, we obtain the constitutive equations
[
/0t 1[ /0t
~(t) = E ~(F(t))0(t) -
• (F(t))0(t) = b7 ~(t) +
R(t -
]
~')~(F(~'))O(-r) d~" ,
K(t - T)~(T)dr
J
.
(3.2.50)
Equations (3.2.50) demonstrate fair prediction of experimental data for polymers [see, e.g., Bugakov and Chepovetskii (1984) and Drescher and Michalski (1971)].
3.2. Nonlinear Integral Models
129
Applying this method to constitutive equation (3.2.25), and assuming the volume response to be linearly elastic, we find that 1
e(t) = yGo~(t),
'[ ~(t) + /0tKl(t - r)~(r) dr + /0 K2(t - r)~(r)~(r) dr
~(t) = - ~
. (3.2.51)
Equations (3.2.51) were verified by Malmeister (1985), Malmeister and Yanson (1979, 1981), and Urzhumtsev (1982) using experimental data for polycarbonate. These equations were extended to aging viscoelastic media by Kregers and Yanson (1985). For uniaxial loading, we cannot distinguish the effect of pressure and the effect of shear stresses on the nonlinearity of response. For three-dimensional loading, Goldman (1984) and Goldman et al. (1982) demonstrated that the first invariant o- of the stress tensor (hydrostatic pressure) and the stress intensity £ affected the viscoelastic behavior by different ways, and suggested the following constitutive equation:
~(t) = f(tr(t), X(t))~(t) + ~0 t K(t - r)f(tr(r), X(r))~(r) dr,
(3.2.52)
where f(o-, E) is a given function. Equation (3.2.52) with a function f exponential in o- and polynomial in ~, correctly predicts experimental data in shear tests for polyethylene, polyoxymethylene, poly(methyl methacrylate), and polypropylene. There is no regular procedure to extend the multiple-integral constitutive equations to three-dimensional loading. As an example of a tensor multiple-integral model in nonlinear viscoelasticity, we present equations proposed by Lai and Findley (1969) and Nolte and Findley (1974)
~(t) =
/o'E
1
Yl(t - ~')I1 --~-(~') i + Y2(t- I")---d7(~') d~"
+ fotjo't I I Y3(t - T l , t - 72)11 (d~" ---~-(T1)) I1 (d~" --~-(T2)) + Y4(t - T l , t - r2)I1 -~(T1) " --~(1"2) Jr- r 5 ( t -
(d6")d~"
Tl,t - r2)I1 --~-(T1) --~(T2) d#
d#
}
X Y6(t - r l , t - T2)-~-(T1) " --~-(T2) drldT2
Chapter 3. Nonlinear Constitutive Models with Small Strains
130
+
foot~otfotI Y7(t-T l , t -
T2, t -
T3)I1
Ido. do. " --~-(T3) do. I --~-(T1) " ---~(T2)
+ Y8(t - r l , t -
r2, t -
r3)I1 --~-(rl)
+ Y9(t-
r2, t -
r3)I1 --~-(rl) I1 --~-('r2) --~-(r3)
rl,t-
I1
--~-(r2)" --~-(r3)
+ Ylo(t - r l , t -
d& (d& d& ) r2, t - r3)---dT-(r1)I1 -d-7(r2) • --~-(r3)
+ Yll(t - r l , t -
r2, t -
(d&)d& r3)I1 --~-(rl)
d& --dT-(r2) • --~-(r3)
do. dO" dO" ) + Y12(t - rl,t - r2, t - r3)--d-7(rl) • --dT-(r2) • --dT-(r3) ~ dr1 dr2 dr3, where Yl(t) to Y12(tl,t2, t3) are creep functions to be found by fitting experimental data.
3.3
A Model for Crosslinked Polymers
In this section a new constitute model is derived for the nonlinear viscoelastic behavior of polymers. Using data for polypropylene and polyurethane in relaxation tests, material parameters are found. To verify the model, loading-unloading tests with constant rates of strains are employed, which demonstrate fair agreement between experimental data and their prediction [see Drozdov (1997)]. Two Alfrey-type theorems are proved that permit a solution to a viscoelastic problem to be found explicitly in terms of solutions to appropriate elastic problems. Some restrictions on the model nonlinearity are determined that imply the correspondence principles. Constitutive models for nonlinear viscoelastic media under isothermal loading have been the focus of attention in the past three decades owing to their numerous applications in polymer engineering. Despite a number of publications on this subject, it is rather difficult to mention a model which satisfies the following conditions: (i) the model is comparatively simple to be employed for the analysis of stresses and displacements in solids with an arbitrary geometry, (ii) its adjustable parameters can be determined by using the standard tests, and (iii) the model reflects adequately the material behavior under time-varying (in general, nonmonotonic) loads. These conditions impose rather strong restrictions on constitutive models. For example, in multiple-integral constitutive models, several functions of several variables should be found experimentally. This requires such a number of experimental data that are not available in the standard tests [see, e.g., Ward and Wolfe (1966)]. The number of variables even increases if we deal with aging viscoelastic materi-
3.3. A Model for Crosslinked Polymers
131
als. Thus, conditions (i) and (ii) confine us to single-integral constitutive models. Several single-integral models demonstrate excellent fit of data used for determining adjustable parameters, but fail to predict results in other experiments [see, e.g., a discussion of this question in Smart and Williams (1972) and Fiegl et al. (1993)]. This implies that condition (iii) becomes extremely important for validating constitutive models.
3.3.1
A Model
of Adaptive
Links
We begin with uniaxial extension of a specimen made of an aging linear viscoelastic material. According to Boltzmann's superposition principle, the stress o- is expressed in terms of the strain E by the formula (2.2.42)
or(t) = X(t, 0)e(t) +
f0 t -~-r ~X (t, l")[e(t)
- e(~')] dl".
(3.3.1)
To provide a mechanical interpretation of Eq. (3.3.1), we consider a network of parallel springs (links between chain molecules) with rigidity c. The links can arise and break. The reformation process is determined by the function X,(t, ~') -
X(t, "r)
C
.
(3.3.2)
At instant ~-, the system contains X,(~-, ~-) links, within the interval D', ~" + d~'], 0X, - - ( t , "1")It=Td~0T new links merge with the system. All the links arise being stress-free, which means that the strain at instant t in springs arising at instant ~"equals
E<>(t, 1") = e(t) - e(l"). The quantity
X,(t, O) equals the number of initial springs existing at instant t, while 0X, - - ( t , r) d r 0T determines the number of springs arising within the interval [~', ~" + d~-] and existing at instant t. The stress or(t) equals the sum of stresses in individual links
or(t) = oro(t) +
/o'
dor(t, "r),
(3.3.3)
where oro(t) is the stress in the initial springs, and dor(t, ~) is the stress in the springs arising in the interval [1", -r + dr]. We assume that the response in any link is governed
132
Chapter 3. Nonlinear Constitutive Models with Small Strains
by Hooke's law o" = ce,
(3.3.4)
and find that ~ro(t) = cX,(t, 0)e(t) = X(t, 0)e(t),
aX, (t, T)e~(t, ~') d r = aX &r(t, ~') = c---~-r a---~(t, -r)[e(t) - e(l")] d-r.
(3.3.5)
Substitution of expressions (3.3.5) into Eq. (3.3.3) implies the constitutive equation (3.3.1). The relaxation function X(t, T) is presented as X(t, ~') = E(~-) + Q(t, ~'),
(3.3.6)
where E(t) = X(t, t)
is the current Young's modulus and Q(t, T) = x ( t , r) - x(~', T)
is the relaxation measure. It follows from Eqs. (3.3.2) and (3.3.6) that C
--
x(0, 0) x,(o, o)
E(0) x,(o, o )
Substitution of this expression into Eq. (3.3.4) implies that o" =
E(0)e x,(0, o )
(3.3.7)
To extend the constitutive equation (3.3.1) to nonlinear materials, we replace Hooke's law (3.3.7) by the nonlinear dependence (3.1.2) o" =
~(~) x,(0, 0)'
(3.3.8)
where q~(e) is a function, which is assumed to be odd and piecewise continuously differentiable. As common practice, the power law (3.1.3) is employed
q~(e) = Alel ~ sign e,
(3.3.9)
where A and a are material parameters. According to Eq. (3.3.8), the response of nonlinear elastic links is determined by the equalities similar to Eq. (3.3.5) X,(t, O) oro(t ) = ~ q ~ ( e ( t ) ) X,(O, O)
-
X(t, O) q~(e(t)) E(0)
133
3.3. A Model for Crosslinked Polymers
dot(t, r) =
1
OX, --(t,
r)qff e~ (t, r)) d r
x,(o, O) ,gr 10X E(O) Or
(3.3.10)
(t, r)qff e(t) - e(r)) dr.
Combining Eqs. (3.3.3) and (3.3.10), we arrive at the constitutive relation o-(t) = E - ~
(t, 0)qffe(t)) +
-~r (t, r)q~(e(t) - e(r)) d r
(3.3.11)
,
which will be the subject of our analysis. For a nonaging medium with X(t, r) = Xo(t - r),
E(t) = E,
Q(t, r) = E Q o ( t -
r),
Eq. (3.3.11) read o-(t) = [1 + Qo(t)]qffe(t)) -
f0t Qo(t -
r)qffe(t) - e ( r ) ) d r ,
(3.3.12)
where the superposed dot denotes differentiation with respect to time. A linear elastic spring with the constitutive equation (3.3.7) is characterized by the potential energy
f~ c~d~ = X,(O, #(e)0)' where ff'(e) = ~E(0) E2.
(3.3.13)
By analogy with Eq. (3.3.13), potential energy of a nonlinear elastic spring (3.3.8) is calculated as
x,(0, 0)' where W(e) =
/0
q~(~) d~.
(3.3.14)
Since the function q~(e) is odd, the function ff'(e) is even, and we set
• <E> = #
Chapter 3. Nonlinear Constitutive Models with Small Strains
134
Eqs. (3.3.13) and (3.3.14) implies that for a linear viscoelastic medium,
E(O)0) {X,(t, O)e2(t) + fot --~-r aX, (t, r)[e(t) - e(r)] 2 dr } W(t) = 2X,(0, _
1 X(t, O)e2(t) + 2
~
(t, ~')[~(t) - e(~')]2 d~"
(3.3.15)
and for a nonlinear viscoelastic medium, w(t)
=
1 O) [X*(t, O)ff'(e(tl) + X,(O,
fo tOX* - ~ r (t, ~')ff'(e(t) - e(r))d ~'1
1 IX(t, 0)W(le(t)l) + fotOX -0--r-T(t,T)W(le<>(t,T)I)d 7-] .
E(0)
(3.3.16)
According to Eqs. (3.3.15) and (3.3.16), W(t) depends on the strain e(t) at the current instant t and on the entire strain history before that instant. Equation (3.3.15) was suggested as the Lyapunov functional for an aging linear viscoelastic solid by Dafermos (1970), whereas expression (3.3.16) is new. It follows from Eqs. (3.3.11) and (3.3.16), that the formula OW(t)
tr(t)-
Oe(t) '
(3.3.17)
previously derived for elastic media [see, for example, Truesdell (1975)], remains valid for an aging viscoelastic medium as well. According to the concept of adaptive links, the energy dissipated per unit volume and per unit time equals the energy of links (located in the unit volume) that break within the unit interval of time. Since the number of initial links lost during the interval [t, t + dr] equals
OX, ~,(t, O) dt, Ot and the number of links arising within the interval D', 1- + dT] and broken during the interval [t, t + dr] equals
O2X,
~(t,
~')dt dT,
the rate of energy dissipation D for a nonlinear viscoelastic medium is calculated as:
1
[OX,
D(t) = -X,(O, 0----~[-ffi-(t' O)ff'(le(t){) + 1
-E(0) --~-(t,0)~(le(t)l)
+
fot 02X, o~t&- (t, r)ff'(l~<>(t, 1")1)d~"1
t 02X t
fo
a-~T(, r)ff'({ e<>(t, ~')l) d
m-].
(3.3.18)
135
3.3. A Model for Crosslinked Polymers
In particular, for a linear viscoelastic medium, expression (3.3.18) is transformed into the formula [c.f. Eq. (2.4.38)]
D(t) = - ~ 1 {OX -~- ( t , O)E2(t) +
f0 t -~-~r(t, o32X ~')[e(t) -
}
e('i')] 2 d~- ,
(3.3.19)
which was discussed by Arutyunyan and Drozdov (1992). Equation (3.3.11) differs from the integral models exposed in Section 3.2 by the argument of the nonlinear function in the integrand: those models require the nonlinear function q~ to depend on e(~') only, whereas in our model q~ depends on e(t) - e(z). For the linear constitutive equation (3.3.1), this difference is not significant, but for the nonlinear equation (3.3.11) it becomes essential. To our knowledge, model (3.3.11) is the first constitutive model that employs the difference e(t) - e(~') in order to account for the effect of the strain history on stresses in physically nonlinear media with small strains. A similar variable, the so-called differential strain history, has been used in finite viscoelasticity [see, e.g., Bernstein (1966), Coleman and Noll (1961), and Green and Rivlin (1957)]. In those works (and many others), the Weierstrass theorem was used to derive constitutive relations in the form of truncated Taylor series in multiple integrals. Despite their generality, those models cannot be reduced to the model (3.3.11), since our model contains a nonlinear function (3.3.9) with a weak singularity at zero. In the general case, two different ways exist to account for the nonlinearity in constitutive relations. According to the first, a nonlinear behavior of adaptive links is introduced, whereas the reformation and breakage processes are assumed to be independent of stresses; see Eq. (3.3.11). Such a hypothesis reflects separability of elastic and viscous effects. According to the other approach, the process of replacing new links is assumed to depend on the stress intensity, whereas the links can be linear or nonlinear. This implies a model, where the function X depends on variables t and ~-, as well as on the stress (or strain) intensity at some intermediate instant. Phenomenological models with finite strains, where the effect of stresses (strains) on the function X is taken into account, are suggested by Kaye and Kennett (1974), Wagner (1976), Wagner et al. (1979), and Winter (1978). Another way to describe the effect of stresses on the function X goes back to Eyring (1936), who proposed to treat inelastic deformation of a solid as a kinetic process. The processes of formation and breakage of links between polymeric molecules are characterized by some activation energies and initiation times, which determine the relaxation spectrum of a viscoelastic material. This concept will be studied in detail in Section 3.3.4 [see also Drozdov and Kalamkarov (1995b)]. We confine ourselves to constitutive models that obey the separability principle, because these models require the minimal number of adjustable parameters compared to models in which the relaxation measures depend on stresses.
136
3.3.2
Chapter 3. Nonlinear Constitutive Models with Small Strains
Determination of Adjustable Parameters
Let us consider the standard relaxation test
e(t) = {0, t
t -> O.
(3.3.20)
Substitution of expression (3.3.20) into the constitutive equation for a nonaging viscoelastic medium (3.3.12) yields o-,(t) = [1 + Qo(t)]qffe).
(3.3.21)
Bearing in mind that Q0(0) = 0, we find from Eq. (3.3.21) that Q0(t) = 1 -
o-E(t) ~r,(0)'
(3.3.22)
where Qo(t) = -Qo(t).
We consider experimental data obtained by Smart and Williams (1972) for polypropylene specimens and by Partom and Schanin (1983) for polyurethane specimens. First, we calculate the ratios rE1,E2(t) of stresses corresponding to different strains E1 and E2. According to Eq. (3.3.21), these ratios should be independent of time t, provided the relaxation measure Qo(t) is independent of strains. The dependencies plotted in Figure 3.3.1 confirm that the process of replacing adaptive links is independent of strains. This is a significant advantage of the model (3.3.12), because other nonlinear constitutive models imply that the strain intensity significantly affects the reformation and breakage processes. Since the relaxation measure Qo(t) is independent of strains, it may be determined with the use of Eq. (3.3.22) for an arbitrary strain e. For convenience, we choose = 0.01 for polypropylene specimens and e = 0.02 for polyurethane specimens (the lowest level of strains in appropriate tests). For these strains, we set o0.01(0) = 12.4 MPa for polypropylene and o'0.02(0) = 4.24 MPa for polyurethane. These data are consistent with Young's moduli obtained in creep tests for a polypropylene monofilament [see Ward and Wolfe (1966)], and for solid polyurethane [see Partom and Schanin (1983)]. We present the relaxation measure Qo(t) in the form of a truncated Prony series (2.3.3) M
Qo(t) = - Z
Xm[1 -- exp(--~/mt)],
(3.3.23)
m=l
where M, Xm, and ~/m are adjustable parameters. Since the number of experimental data in relaxation tests is rather small (about 10), we confine ourselves to M = 2 in
3.3. A Model for Crosslinked Polymers
137
1.0
~o
o
()
o
()
0.5
I
I
I
I
I
I
I
I
I
100 1.0 ~o o
0
0
0
0
.00
0.5
I 0
i
i
i
i
i
i
i t
i
i 15
3.3.1: The ratios of stresses r versus time t (min) in relaxation tests at various strains E. Circles and asterisks correspond to experimental data for a polypropylene specimen (A) and for a polyurethane specimen (B). Polypropylene m unfilled circles: ro.ol,0.02; filled circles: ro.ol,0.03; asterisks: ro.ol,0.04. Polyurethane m unfilled circles: ro.o2,0.03; filled circles: r0.o2,0.04.
Figure
Eq. (3.3.23). Experimental data and their approximation by the exponential function (3.3.23) are plotted in Figure 3.3.2. To find the function q~(e), we use the equality ~r,(0) = ¢(e),
which follows from Eq. (3.3.21). The corresponding dependencies are plotted in Figure 3.3.3. The relaxation curves for polypropylene and polyurethane are plotted in Figures 3.3.4 and 3.3.5. These figures demonstrate fair agreement between experimental data and their prediction by the model (3.3.12).
138
Chapter 3. Nonlinear Constitutive Models with Small Strains
0.5
Q0
I
I
I
I
I
I
0
I
I
I
t
100
0.5
Q,
r
I
I
I
I
I
I
0
I
t
I
I
15
Figure 3.3.2: The dimensionless relaxation measure Q0 versus time t (min). Circles show experimental data for a polypropylene specimen (A) and for a polyurethane specimen (B). Solid lines show their approximation by the exponential function (3.3.23) with X1 = 0.210, X2 = 0.210, ~/1 = 0.06 (min- 1), ~/2 = 1.90 (min- 1) for polypropylene and X1 = 0.258, X2 = 0.312, 3/1 = 0.03 (rain-l), ~/2 = 1.70 (min-1) for polyurethane.
To verify the model in experiments, where both stresses and strains depend on time, we consider loading and unloading with a piecewise constant rate of strain,
~(t)={ ~0, 0-
To <- t <---2To,
where 50 and To are constants. Experimental data and results of numerical simulation with the use of Eq. (3.3.12) are plotted in Figure 3.3.6. According to this figure, the model ensures fair prediction of experimental data under time-dependent loading without any additional fit of adjustable parameters.
3.3. A Model for Crosslinked Polymers
139
25
f
I
0
I
I
I
I
I
I
e
I
I
0.04
3.3.3: The function q~(~) (MPa). Circles show experimental data. Solid lines show their approximation by the power-law function (3.3.9). Curve 1: a polypropylene specimen with A = 157.48 MPa and c~ = 0.5519. Curve 2: a polyurethane specimen with A = 16.88 MPa and c~ = 0.3525. Figure
Two sources of small deviations between experimental data and numerical results may be mentioned. The first may explain discrepancies between experimental data and their prediction in the region of small strains (less than 0.5 %). As is well known, majority of polymeric materials demonstrate the linear response at very small strains. Since the power function (3.3.9) is not reduced to the linear dependence in the vicinity of the point e = 0, the use of this function can cause a bias to larger stresses which is demonstrated in Figure 3.3.6. To avoid these discrepancies, a more sophisticated function q~(E) should be chosen with a finite derivative with respect to ~ (Young's modulus) at E = 0. The discrepancies may be also caused by an insufficient number of terms (M = 2) in the truncated Prony series (3.3.23). To make the prediction more accurate, this
Chapter 3. Nonlinear Constitutive Models with Small Strains
140 25
i
o Oeee 0 eOo°eeeee
0
©
eeoeeeeeeoee 0
ee°e°eeeeeeeoeeeeeeeeel
~°°eeeeeoeeeeoeoeeeeeoeeeeeoeoeeeeeooeeeeeeeeoeeee,
o
(
eeeeo'leeeeeeeeol,,Oeeeee6~leoeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeel~
--
I
I
I
0
I
I
I
I
I
I
t
100
Figure 3.3.4: The stress o" (MPa) versus time t (min) in relaxation test. Circles show experimental data for polypropylene specimens. Dotted lines show prediction of the model. Curve 1" E = 0.01. Curve 2: E = 0..03.
number should be increased to determine the material viscosities Xm and the relaxation times Tm = 7m I more precisely. However, this refinement significantly increases the number of adjustable parameters, and we do not dwell on it.
3.3.3
Constitutive
Equations
for Three-Dimensional
Loading
To extend the model (3.3.11) to three-dimensional loading, we use an approach suggested by Drozdov (1993). For simplicity we confine ourselves to incompressible media. Consider uniaxial extension of a viscoelastic rod as a three-dimensional deformation. Points of the rod refer to Cartesian coordinates {xi}, where the axis xl coincides with the longitudinal axis of the rod; the axes x2 and x3 are located in the cross-sectional plane. For an incompressible medium, the strain tensor ~(t) has the
141
3.3. A Model for Crosslinked Polymers
) I
I0.
-
o6
"0
"9
"..~..
".©
°oo 0 °°oooo °
--
0
.....~ ......
"6".....................
"........"6"......... "6"...................... °°°°•OooooooooOOOoo
....a ...........~ .................. 0 °°OOOooooooo
0
0 ooooooooo•
0
O
°°°°°°°oooooooooooooooooooo
°°°°°°°oooooooooooooooooooooooooq
I
I
I
I
I
I
0
I
I
t
I
15
F i g u r e 3.3.5: T h e s t r e s s o- ( M P a ) v e r s u s t i m e t (rain) in r e l a x a t i o n tests. C i r c l e s s h o w e x p e r i m e n t a l d a t a for p o l y u r e t h a n e s p e c i m e n s . D o t t e d lines s h o w p r e d i c t i o n of t h e m o d e l . C u r v e 1: e = 0.02. C u r v e 2: E = 0.03. C u r v e 3: E = 0.04.
nonzero components Eli (t) = e(t),
E22(t) = E33(t) -- --
1E(t),
the deviatoric part O(t) of the strain tensor coincides with ~(t), and the strain intensity 2 ) 1/2 ~ , ° ( t , 1-)" O<>(t, ~)
(3.3.24)
F<>(t, T) = le°(t, ~)l = le(t) - e(l-)l.
(3.3.25)
F<>(t, ~-) = is calculated as
Chapter 3. Nonlinear Constitutive Models with Small Strains
142
20
m
0
0
E
0.02
Figure 3.3,6: The stress u (MPa) versus strain E in loading and unloading tests with $0 = 0.0033 min-' and To = 6 min. Circles show experimental data. Dotted lines show prediction of the model.
Substituting expression (3.3.24) into Eq. (3.3.16j, we find the strain energy density (per unit volume)
By analogy with Eq. (3.3.19), the rate of energy dissipation (per unit volume) is calculated as
In an incompressible viscoelastic medium, the stress tensor &(tjequals B(tj = -p(tji
+ j.(tj,
(3.3.27j
143
3.3. A Model for Crosslinked Polymers
where p is pressure, I is the unit tensor, and the deviatoric part ~(t) of the stress tensor is determined by the formula similar to Eq. (3.3.17)
ow(t) ~(t)
-
.
(3.3.28)
at'U)
According to Lurie (1990), we write 0F<>(t, T) = ~ 2 ~<>(t, ~-) = ~ 2 [~(t) - ~(~')]. 0~(t) 3F~(t, ~') 3F~(t, ~')
(3.3.29)
It follows from Eqs. (3.3.26), (3.3.28), and (3.3.29) that 2 ~(t) = 3X,(0, 0)
+
X,(t, O)
W'(F(t)) O(t) F(t)
f0' -~T ax, (t,-r) W'(F*(t, 1"))_ F¢(t ' ~') [~,(t)
~(~-)]
dT},
(3.3.30)
J
where the prime denotes differentiation. Finally, substituting expression (3.3.30) into Eq. (3.3.27) and bearing in mind Eq. (3.3.14), we obtain the constitutive equation of an aging, incompressible, nonlinear, viscoelastic medium d-(t) = - p(t)] + 3X,(O, 0---------~X,(t, O)~(r(t))~(t)
+
/o
}
--~-r(t, ~')~(F <>(t, ~'))[O(t) - ~(~-)1dr ,
(3.3.31)
where q'(F) -
W'(F) ~(r) F F "
(3.3.32)
In particular, for the nonlinear function (3.3.9), Eq. (3.3.32) implies that • (F) = AF a-1.
(3.3.33)
3.3.4 Correspondence Principles in Nonlinear Viscoelasticity Our objective now is to formulate and to prove two correspondence principles [the Alfrey-type theorems; see Alfrey (1944)] for viscoelastic media governed by the constitutive relation (3.3.31) and to analyze restrictions on the nonlinearity (3.3.32) imposed by these principles.
Theorem 3.3.1 Let continuously differentiable functions q~ and ~ exist such that for any/3 and F
q'(13F) = ~(/3)_=(F).
(3.3.34)
144
Chapter 3. Nonlinear Constitutive Models with Small Strains
Suppose that inertia forces can be neglected, a body force/3 is given, a surface traction b is prescribed on the entire surface of a solid, and the vectors/) and b are independent of time. Then the stress tensor in an aging nonlinear viscoelastic medium with constitutive equation (3.3.31) coincides with the stress tensor in a nonlinear elastic medium with the constitutive law 2 #(t) = -p(t)~l + -~alr(F(t))E(t).
(3.3.35)
The displacement vector fio(t) in the viscoelastic solid is proportional to the displacement vector fie in the elastic medium
fv(t) = K(t)fe,
(3.3.36)
where the function K(t) satisfies the nonlinear integral equation
°~X* ~')q,(l~(t) X,(t, O)¢(IK(t)l)K(t) + f0 t ---~-z(t,
- K(T)I)[K(t) -- K(T)] d r = X , ( 0 , 0).
(3.3.37) T h e o r e m 3.3.2 Let Eq. (3.3.34) hold. Suppose that inertia forces are neglected, body forces vanish, displacements (J are given on the entire boundary of a body, and the vector (1 is independent of time. Then the displacement vector fv in the viscoelastic medium (3.3.31) coincides with the displacement vector fie in the elastic solid (3.3.35), whereas the stress tensor o-v(t) in the viscoelastic solid is proportional to the stress tensor O'e in the elastic medium
x,(t, 0)
o'v(t) = ~ O ' e .
x , ( o , o)
(3.3.38)
To prove these assertions, it suffices to substitute expressions (3.3.36) and (3.3.38) into the governing equations and boundary conditions for a viscoelastic medium and check that these equalities turn into identities provided the elastic displacements and stresses satisfy the same governing equations. We now concentrate on functions that satisfy condition (3.3.34). The most surprising result is that despite the presence of two arbitrary functions in the right-hand side of this relationship, the power-law function (3.3.33) provides the only solution of Eq. (3.3.34). T h e o r e m 3.3.3 Any continuously differentiable function air(F) satisfying Eq. (3.3.34) has the form xtr(F) = DF 8, where D and 6 are arbitrary constants.
(3.3.39)
145
3.4. A Model for Non-Crosslinked Polymers To prove Theorem 3.3.3, we first set 13 = 1 in Eq. (3.3.34) and find that
~(r)
,I,(F)
/--¢
~(1)
Substitution of this expression into Eq. (3.3.34) implies that
q'(/3),i,(r).
• (/3F) = qJ(1)
(3.3.40)
We set F = 1 in Eq. (3.3.40) to obtain
'I'(/3) g~(13) = ~xIt(1) g~(1). This equality together with Eq. (3.3.40) results in
q'(/3) ~I,(F). q'(/3 F) - ~I'(1)
(3.3.41)
We differentiate Eq. (3.3.41) with respect to/3, set/3 = 1, and find that ~ ' ( F ) F = 6~(F),
(3.3.42)
where
~'(1) ,I~(1) Integrating Eq. (3.3.42), we arrive at the formula (3.3.39), which completes the proof. Theorem 3.3.3 can serve as an additional source of justification for the model, since it shows that the dependence (3.3.33) provides the only nonlinearity that implies the correspondence principles.
3.4
A Model for Non-Crosslinked Polymers
A nonlinear model of adaptive links, which has been developed in the previous section, is based on the following assumptions" • The processes of reformation and breakage for adaptive links are independent of stresses and strains (the separability principle). • Nonlinearity in the material response is a consequence of the nonlinearity in the response of each elastic link. • For any class of adaptive links, nonreplacing links have a positive concentration, which implies that the creep curves tend to some horizontal asymptotes as time tends to infinity. That model correctly describes the nonlinear behavior of crosslinked polymeric materials.
Chapter 3. Nonlinear Constitutive Models with Small Strains
146
In this section, we derive a constitutive model, which extends the concept of adaptive links to non-crosslinked polymers. Our purpose is to describe the entire spectrum of the viscoelastic responses from the elastic behavior at the initial instant to steady creep flow after some transition period. The model is based on the following hypotheses: • Adaptive links existing at the initial instant are divided into two types. Links of type I break under loading, whereas links of type II replace each other. All adaptive links of type II are involved into the process of replacement, and concentration of nonreplacing links equals zero. • Breakage and reformation of adaptive links are treated as kinetic processes, and kinetic equations are introduced for the numbers of links of type I and type II. • Nonlinearity arises because the rates of kinetic processes depend on the stress intensity. This approach may be treated as an attempt to combine the concept of adaptive links [see, e.g., Green and Tobolsky (1946)] with the concept of internal kinetics of inelastic processes in solids [see, e.g., Eyring (1936) and Krauz and Eyring (1975)]. As a result, we arrive at a model without the separability property, where the nonlinearity of the material response is reflected in the effect of stresses on the rates of reformation and breakage of adaptive links. To verify the model, we compare numerical predictions with experimental data for polypropylene fibers. The results demonstrate fair agreement between observations and results of numerical simulation [see Drozdov and Kalamkarov (1995b, 1996)].
3.4.1
A Model of Adaptive Links
A model of adaptive links for an aging linear viscoelastic medium has been discussed in Chapter 2. That model is characterized by a function X(t, T), which equals the number of links arising before instant T and existing at instant t. In Chapter 2, a multiplicative presentation has been derived for the function X(t, T), assuming that concentration X of nonreplacing links is positive. This hypothesis fails for noncrosslinked media, where the parameter X equals zero. Since the formula for the function X(t, z) does not permit the limit as X ~ 0 to be calculated directly, we repeat calculations for the case X = 0. For uniaxial extension of a viscoelastic specimen the constitutive equation reads
tr(t) = X(t, O)e(t) +
{
t OX -~T(t, ~')[e(t) - e(~')] dr
fo
= c X,(t, O)e(t) +
/o
}
-~T (t, ~')[e(t)- e(r)] d~" .
(3.4.1)
147
3.4. A Model for Non-Crosslinked Polymers
Here or is the stress, e is the strain, c is rigidity of a link, and X ( t , 7")
X , ( t , 7.) -
is the number of links that have arisen before instant 7. and exist at instant t. The amount X , ( t , 0) is the number of initial links that exist at instant t, and the amount 0X, ~(t, 07.
7.) dT.
is the number of links that arose within the interval [7., 7. + d7.] and exist at instant t. These quantities are presented in the form X , ( t , 0) = X,(0, 0)[ 1 - g(t, 0)],
0X, OX, (t, 7.) = --~--T(7., 7.)[1 - g(t - 7., 7.)] = ~(7.)[1 - g(t - 7., 7.)]. 07.
(3.4.2)
Here g(t - 7., 7.) is the relative number of links that arose at instant 7. and have broken to the instant t, and OX, d~ ( 7.) = -7-- ( t , 7.) tiT.
t=,r
is the rate of reformation of adaptive links. Substitution of expressions (3.4.2) into the formula X , ( t , t) = X , ( t , O) +
t OX, (t ~ , ~') dT.
fo
implies the balance law for the number of adaptive links X , ( t , t) = X,(0, 0)[ 1 - g(t, 0)] +
f0 t ~(7.)[ 1 -
g(t - 7., 7.)] d7..
(3.4.3)
For an aging viscoelastic material, the function X ( t , 7.) is presented in the form (3.4.4)
X ( t , 7.) = E(7.) + Q(t, 7.),
where E(7.) = X(7., 7.) is the current Young's modulus and Q(t, T) = X ( t , 7.) - X(7., 7.) is the relaxation measure, which satisfies the condition Q(t, t) = O.
The mechanical properties of nonaging media are independent of time, which implies that E(t) = E,
Q(t, r) = E Q o ( t -
r).
(3.4.5)
148
Chapter 3. Nonlinear Constitutive Models with Small Strains
Combining Eqs. (3.4.2), (3.4.4), and (3.4.5), we find that for nonaging viscoelastic media X , ( t , t) = N,
d~(t) = alp,
g(t - ~', ~') = go(t - ~'),
(3.4.6)
where • is a material parameter and N-
E C
is the total number of adaptive links. Substitution of expressions (3.4.6) into Eq. (3.4.3) yields go(t) = ~ , f0 t [1 - go(t - ~')] d~',
(3.4.7)
where
I), D
c~ • E
The integral equation (3.4.7) is equivalent to the differential equation dgo (t) = ~ , [ 1 - g0(t)] dt
(3.4.8)
with the initial condition g0(0) = 0. The solution of Eq. (3.4.8) reads go(t) = 1 - e x p ( - ~ , t ) .
(3.4.9)
It follows from Eqs. (3.4.2), (3.4.6), and (3.4.9) that X(t, O) = E e x p ( - ~ , t ) ,
OX -~r (t, ~') = E ~ , e x p [ - ~ , ( t - ~')]. (3.4.10)
Substitution of expressions (3.4.10) into the constitutive equation (3.4.1) implies that o'(t) _ e x p ( - ~ , t ) E ( t ) + ~ , E = e(t) - ~ ,
e x p [ - ~ , ( t - ~')][e(t) - e(-r)] d~"
e x p [ - ~ , ( t - ~-)le(~-)d~'.
(3.4.11)
We differentiate Eq. (3.4.11) with respect to time, and exclude the integral term from the obtained equality with the use of Eq. (3.4.11). As a result, we obtain constitutive equation of the Maxwell solid do" Ede(t)at = -dT (t) + ~ , t r ( t ) .
(3.4.12)
Equation (3.4.12) implies the linear growth of strains in time in creep tests with a constant stress tr. Such a behavior is observed in experiments on non-crosslinked
3.4. A Model for Non-Crosslinked Polymers
149
polymers after some transition period (see Figures 3.4.1 and 3.4.8). On the other hand, within the transition period, results of numerical simulation with the use of Eq. (3.4.12) are far away from experimental creep curves. To explain this phenomenon, we assume that not all the links of the network are included in the process of replacement within the transition period. This hypothesis is in fair agreement with the Eyring concept of internal kinetics, which states that some initiation time is necessary to activate kinetic processes. Our purpose now is extend Eq. (3.4.12) by introducing kinetic equations to describe breakage and reformation of adaptive links.
3.4.2
A Generalized Model of Adaptive Links
We assume that two types of initial links are distinguished. Links of type I break under the action of external forces, and they are not included into the process of replacement. We denote by nl(t)N the number of initial links that have lost to instant t, where N is the total number of links at the initial instant. The limiting number of broken links as time tends to infinity is denoted as n looN. Links of type II replace each other according to the preceding scenario. Denote by n2(t)N the number of initial links that have been included into the process of replacement to instant t. The limiting number of these links as time approaches infinity and steady flow of a non-crosslinked polymer is observed is denoted as n 2 ~N.
At the initial instant, the system consists of N links that should "decide" whether they belong to type I or type II. Before such a decision is made, these links demonstrate the linear elastic response and are characterized by the rigidity c = E / N . At instant t >- 0, the number of links that should make their decision equals [1 - n l ( t ) - n2(t)]N. The concentrations nl (t) and n2(t) satisfy the conditions n l ( 0 - 0) = n 2 ( 0 - 0 ) - - 0 ,
nl(oO) -b n 2 ( ~ ) = 1.
(3.4.13)
The first equality in Eq. (3.4.13) means that no links make their choice before a viscoelastic specimen is loaded. The other equality in Eq. (3.4.13) means that all the links should choose their type in the infinite interval of time. We present the stress o- as a sum of three terms tr(t) = o-(°)(t) + o-(1)(t) + 0"(2)(0,
(3.4.14)
where cr (°) is the total stress in links that have not yet chosen their type, 0 "(1) is the total stress in links of type I, and or(2) is the total stress in links of type II. Before making their decision, links demonstrate the linear elastic behavior
~r(°)(t) = cN[1 - n l ( t ) - n2(t)]E(t) = E[1 - n l ( t ) - nz(t)]~(t).
(3.4.15)
Links of type I have broken, and they produce no response o'(1)(t) = 0.
(3.4.16)
150
C h a p t e r 3. N o n l i n e a r C o n s t i t u t i v e M o d e l s w i t h S m a l l S t r a i n s
To calculate the total response in links of type II, we consider links that choose to belong to type II within the interval D', ~" + dr]. The number of these links is dn2 -ffi- ( r ) d r ,
and the stress per link can be found by using formula similar to Eq. (3.4.1) T, 0)e(t) +
X(t -
-~-s(t - l " , s - ~')[e(t) - e(s)] d s
.
The stress at instant t in links that joined type II within the interval [~', • + d~'] equals
{
~', O)~(t) +
X(t -
[x
=
(t -
ftox
-~-s(t - r , s -
r, t -
~')[~(t) - ~(s)] d s
f'ax(,as
~')e(t) -
-
r,s-
}dn2 ]dn2
~')e(s)ds
---d-i-(r) d r
--d-~-(T)dr.
Summing up stresses in links that have chosen to be of type II before instant t, we find that cr(2)(t) =
(t -
~', t -
~')e(t) -
=
X(t -
z, t -
r)--d-i-(r) d r
-
--~-(r) d r
=
X(t -
-
r, t -
e(s)ds
--~s (t -
~', s -
r)e(s) ds
--~(r)
dr
e(t)
-~-s(t - r, s - r)e(s) d s
r)--d-[-(~') d r
-~s(t -
r,s-
e(t)
(3.4.17)
r)--d-~(r)dr.
According to Eqs. (3.4.6) and (3.4.10), for a nonaging viscoelastic medium X(t -
r, t -
r) = E,
OX --(t Os
-
r,s-
z) = Erb, e x p [ - ~ , ( t
- s)].
Substitution of these expressions into Eq. (3.4.17) implies that (r(Z)(t) = E
{
nz(t)e(t)
- ~,
f0t
}
exp[-@,(t - s)]nz(s)E(s)ds
.
(3.4.18)
Finally, substituting expressions (3.4.15), (3.4.16), and (3.4.18) into Eq. (3.4.14), we obtain or(t) = E
{
[1 - n l ( t ) ] e ( t ) - dO,
f0t
exp[-~,(t - s)]n2(s)e(s)ds
}
.
(3.4.19)
3.4. A Model for Non-Crosslinked Polymers
151
The constitutive model (3.4.19) is determined by two adjustable parameters, E and • ., and by two adjustable functions, nl(t) and n2(t), which are found by fitting experimental data. Since nl(t) and n2(t) characterize the kinetics of formation and breakage for adaptive links between chain molecules, it is natural to assume that these functions satisfy the standard equations in the chemical kinetics:
dnl dt (t) = al[nl ~ - nl(t)] ~1, dn2 (t) = a2[n2 ~ - n2(t)] ~2 dt
(3.4.20)
where al, a2, al, and a2 are parameters to be determined. According to Eqs. (3.4.19) and (3.4.20), only six adjustable parameters should be found in the standard creep and relaxation tests to characterize the model. Experimental data for polypropylene fibers show that the characteristic rate for joining links of type II essentially exceeds the characteristic rate for joining links of type I. In this case, we can set n2(t) = n2 oo
(t > 0),
(3.4.21)
which reduces the number of adjustable parameters and permits the constitutive equation (3.4.19) to be written as or(t) = E
{ [1 -
nl(t)]a(t) - ~ , ( 1 - nl o~)
f0t e x p [ - ~ , ( t
- s)]a(s)
ds } . (3.4.22)
We differentiate Eq. (3.4.22) with respect to time and exclude the integral term from the obtained equality and Eq. (3.4.22). As a result, we arrive at the differential equation
nl] }
dor (t) + cI),o-(t) = E { [1 - nl(t)]-:-(t) ---at--de + dP,(nl~ -- nl(t))-- -~-(t) e(t) . dt (3.4.23) For the standard creep test with a constant stress o-, Eq. (3.4.23) is simplified as [1 -
dE dnl ] ~, nl (t)]-zT(t) + ~ , ( n l o~ -- nl(t)) -- --~-(t) a(t) = ~or. E dt
At large times t, when nl(t) ~ nl ~ and
EO0
dnl(t)/dt ~ 0, Eq. (3.4.24) implies that
E(1 - nl ~)'
where
dE is the limiting rate of creep.
(3.4.24)
(3.4.25)
7
0.1
E
6 5
4 3 2 1 v
0
-
I
0
I
I
120
I
I
I
I
240
I
I
I
I
360
I
I
t(min)
I
480
Figure 3.4.1: Creep curves for a polypropylene specimen. Circles show experimental data obtained by Barenblatt et al. (1974). Solid lines show prediction of the model. Curve 1:u = 7.46 MPa. Curve 2: u = 11.28 MPa. Curve 3: D = 13.54 m a . Curve 4: u = 15.01 MPa. Curve 5: u = 16.82 MPa. Curve 6: u = 18.05 MPa. Curve 7 u = 20.31 MPa.
153
3.4. A Model for Non-Crosslinked Polymers 3.4.3
Validation
of the Model
We begin with experimental data obtained by Barenblatt et al. (1974) in creep tests for polypropylene films within a relatively large time interval, 8 hr, while the transition period is estimated as about 2 hr (see Figure 3.4.1). According to Eq. (3.4.22), the strain E at the initial instant t = 0 is calculated as O"
E(0) = E[1 - nl 0(o')]'
(3.4.26)
where nl 0(o') equals the concentration nl of links of type I at the initial instant t = 0 for a specimen loaded by the stress o-. The parameter (3.4.27)
E~ = E[1 - nl 0(or)]
is plotted versus the stress o- in Figure 3.4.2. Experimental data are approximated by
m
E
o
I
0
I
I
I
I
I
©
I o-
I
I
25
Figure 3.4.2: Young's modulus E¢ (GPa) versus the stress cr (MPa) for polypropylene specimens. Unfilled circles: experimental data obtained by Barenblatt et al. (1974). Filled circles: experimental data obtained by Ward and Wolfe (1966). Solid lines I and 2: their approximations by the linear functions (3.4.28) and (3.4.37).
154
Chapter 3. Nonlinear Constitutive Models with Small Strains
the linear function E~ = 2 . 2 7 - 7.34.10-2or,
(3.4.28)
where E~ is measured in gigapascals, and o- is measures in megapascals. We assume that Young's modulus E is independent of stresses, and that the initial concentration n l 0 of links of type I vanishes when the stress o" equals zero. It follows from Eqs. (3.4.27) and (3.4.28) that E = 2.27 GPa, and the function nl 0 increases linearly in onl 0(or)-- 3.233" 10-2o ",
(3.4.29)
where o- is measured in megapascals. Experimental data for the initial concentration of links of type I together with their approximation by Eq. (3.4.29) are plotted in Figure 3.4.3.
nl 0
I
0
I
I
I
I
¢r
I
I
25
3.4.3: The initial concentration nl o of adaptive links of type I versus the stress cr (MPa). Unfilled circles: experimental data obtained by Barenblatt et al. (1974). Filled circles: experimental data obtained by Ward and Wolfe (1966). Solid lines I and 2 show their approximation by the linear functions (3.4.29) and (3.4.38). Figure
155
3.4. A Model for Non-Crosslinked Polymers
The value E = 2.27 GPa is in good agreement with data for rigidity ofpolypropylene monofilaments provided by other sources [see, e.g., Bataille et al. (1987) and Hartman et al. (1987)]. The limiting rate of creep eo~ is calculated using experimental data obtained for t > 2 hr after loading. The value 5oo(o') is plotted versus the stress o- in Figure 3.4.4. We approximate experimental data by the hyperbolic function O"
e~(or) = A sinh -K
(3.4.30)
4~(o-) = Bcr/3,
(3.4.31)
and by the power-law function
tl
-3
log ~
-6
I
0
q
I/
1
1
1
I
o-
25
3.4.4: The rate of limiting creep flow e~ (min -1) versus the stress ~r (MPa). Circles show experimental data obtained by Barenblatt et al. (1974). Solid lines show their approximation by the hyperbolic function (3.4.30), curve 1, and by the power function (3.4.31), curve 2.
Figure
156
Chapter 3. Nonlinear Constitutive Models with Small Strains
and find the following values of adjustable parameters: A = 98.10- 10 -9 B = 8 . 9 9 . 1 0 -9
min-1,
K = 2.57
min -1,
MPa,
/3 = 2.95.
It follows from Eq. (3.4.25) that 1 - nl oo(o')
~,(~r)
= Too(o-),
(3.4.32)
where Or
Too(o-) =
(3.4.33)
E4oo(o-)
is the characteristic time of the limiting creep flow. The parameter Too is plotted versus the stress o" in Figure 3.4.5. Experimental data are approximated by the linear 1500
T~
2\
I
0
I
I
I
N
I~
I
I
I
o-
I
~
I
25
Figure 3.4.5: The characteristic time of the limiting creep flow T~o (min) versus the stress tr (MPa). Unfilled circles: experimental data obtained by Barenblatt et al. (1974). Filled circles: experimental data obtained by Ward and Wolfe (1966). Solid lines I and 2: their approximation by the linear functions (3.4.34) and (3.4.39).
3.4. A Model for Non-Crosslinked Polymers
157
function T~(~r) = (1.08 - 0.50or) • 103,
(3.4.34)
where Too is measured in minutes, and o- is measured in megapascals. For a given function Too(o-), Eq. (3.4.32) does not allows us to determine two adjustable functions: the rate of reformation for adaptive links cI), (or), and the limiting concentration nl oo(o') of links of type I. These functions are found by fitting experimental data obtained in creep tests. We confine ourselves to the kinetic equations of the first order, and set al = 1. For a given stress or, we seek parameters al = al (o-) and nl oo = nl oo(o-), which ensure the best agreement between experimental data and their numerical prediction. Given nl oo and Too values, the rate of reformation cI),(o-) is determined from Eq. (3.4.32). The limiting concentration n l ooof links of type I is plotted versus the stress o- in Figure 3.4.6. Experimental data are approximated by the linear function
1
nloo
0
I
0
i
I
I
I
I
I
o-
I
I
25
3.4.6: The limiting concentration nl ~ of adaptive links of type I versus the stress cr (MPa). Unfilled circles: experimental data obtained by Barenblatt et al. (1974). Filled circles: experimental data obtained by Ward and Wolfe (1966). Solid lines I and 2: their approximation by the linear functions (3.4.35) and (3.4.40). Figure
158
Chapter 3. Nonlinear Constitutive Models with Small Strains
0.1
al
•
I
I
~0
I
[I
I
I
0
I ~
o-
I
I
25
Figure 3.4.7: The rate of breakage al (min -1) for adaptive links of type I versus the stress or (MPa). Unfilled circles: experimental data obtained by Barenblatt et al. (1974). Filled circles: experimental data obtained by Ward and Wolfe (1966). Solid lines: their approximation by the linear functions (3.4.36) and (3.4.41). nl o~(o-) = 0.412 + 0.021o-,
(3.4.35)
where o- is measured in megapascals. The rate of breakage al is plotted versus the stress o- in Figure 3.4.7. Experimental data are approximated by the linear function al = (-5.09 + 0.56o-). 10 -2,
(3.4.36)
where a l is measured in minutes -1, and or is measured in megapascals. Formula (3.4.36) is true only for sufficiently large stresses, when o- exceeds 13 MPa and the fight-hand side of Eq. (3.4.36) is positive. The strains plotted in Figure 3.4.1 increase monotonically in time for the entire process of loading. This distinguishes the proposed model from other approaches, which lead to nonmonotonic strain-time dependencies [see, e.g., Taub and Spaepen (1981) for a discussion of this question].
3.4. A Model for Non-Crosslinked Polymers
159
The preceding procedure is repeated for experimental data obtained by Ward and Wolfe (1966) and plotted in Figure 3.4.8. The parameter E,~ is plotted versus the stress o- in Figure 3.4.2. Experimental data are approximated by the linear function E~ = 0.97 - 2.33 • 10 -2o-,
(3.4.37)
where E~ is measured in gigapascals, and cr is measured in megapascals. Combining Eq. (3.4.27) with Eq. (3.4.37), we find that Young's modulus E equals 0.97 GPa, and the initial number of adaptive links of type I is calculated as nl 0(o')= 2 . 4 0 . 1 0 - 2 o -,
(3.4.38)
which is rather close to formula (3.4.29). Experimental data for the initial concentration of adaptive links of type I and their prediction with the use of Eq. (3.4.38) are presented in Figure 3.4.3. The characteristic time of creep flow T~ is plotted versus the stress o- in Figure 3.4.5. Experimental data are approximated by the linear dependence T~(o-) = (1.40 - 0.13o-)- 103,
(3.4.39)
where T~ is measured in minutes, and o- is measured in megapascals. The limiting concentration of links of type I is plotted in Figure 3.4.6. Experimental data are approximated by the linear dependence nl ~(o-) = 0.295 + 0.029o-,
(3.4.40)
which is rather close to Eq. (3.4.35). The kinetic coefficient a l, which characterizes the rate of breakage for adaptive links of type I, is plotted versus the applied stress o" in Figure 3.4.7. Experimental data are approximated by the linear function al(o) = (116.0 - 6.2o-). 10 -3,
(3.4.41)
where a l is measured in minutes -1, and o- is measured in megapascals. Creep curves for two types of polypropylene and their approximation by the model are depicted in Figures 3.4.1 and 3.4.8. These figures demonstrate fair agreement between experimental data and their numerical predictions. Experimental data obtained by Barenblatt et al. (1974) and Ward and Wolfe (1966) describe the mechanical response in polypropylene with high (Figure 3.4.1) and low (Figure 3.4.8) molecular weight. The standard characteristics of these polymers, e.g., Young's modulus and the characteristic rate of limiting creep d~, differ significantly. However, the kinetic characteristics of these specimens, n l 0, n l ~, and al, are relatively close to each other (see Figures 3.4.3, 3.4.6, and 3.4.7), which may serve as an indirect confirmation of the model. The plot presented in Figure 3.4.7 demonstrates that the rate of breakage for adaptive links of type I depends nonmonotonically on the stress o-. For relatively small stresses (which exceed the yield stress), the kinetic parameter al decreases in o-, reaches its minimum and afterwards increases in the region of large stresses.
0.05
I
4
3 2
1
0 0
120
240
360
t
480
Figure 3.4.8: Creep curves for a polypropylene specimen. Circles show experimental data obtained by Ward and Wolfe (1966). Solid lines show prediction of the model. Curve 1:(T = 2.0 MPa. Curve 2: (T = 4.0 MPa. Curve 3: u = 6.0 MPa. Curve 4:u = 8.0 MPa.
Bibliography
161
It is of special interest that at small stresses, experimental data obtained by two independent sources for two different kinds of polypropylene provide practically the same approximating curve. To our knowledge, the nonmonotonic dependence of the rate of breakage for adaptive links on stresses has not yet been studied. As a possible mechanism of this phenomenon, we suggest stress-induced crystallization of polypropylene under large stresses, which leads to structural changes in material. However, available experimental data are not sufficient to confirm this hypothesis.
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166
Chapter 3. Nonlinear Constitutive Models with Small Strains
[68] La Mantia, E P., Titomanlio, G., and Acierno, D. (1981). The non-isothermal rheological behaviour of molten polymers: shear and elongational stress growth of polyisobutylene under heating. Rheol. Acta 20, 458-462. [69] Leaderman, H. (1943). Elastic and Creep Properties of Filamentous Materials. Textile Foundation, Washington, D.C. [70] Lockett, E J. (1972). Nonlinear Viscoelastic Solids. Academic Press, London. [71] Losi, G. U. and Knauss, W. G. (1992a). Thermal stresses in nonlinearly viscoelastic solids. Trans. ASME J. Appl. Mech. 59, $43-$49. [72] Losi, G. U. and Knauss, W. G. (1992b). Free volume theory and nonlinear thermoviscoelasticity. Polym. Eng. Sci. 32, 542-557. [73] Lurie, A. I. (1990). Non-linear Theory of Elasticity. North-Holland, Amsterdam. [74] Makhmutov, I. M., Sorina, T. G., Suvorova, Y. V., and Surgucheva, A. I. (1983). Failure of composites with account for the effects of temperature and moisture. Mech. Composite Mater. 19, 175-180. [75] Malkin, A. Y. (1995). Non-linearity in rheology - - an essey of classification. Rheol. Acta 34, 27-39. [76] Malmeister, A. A. (1982). Predicting the nonlinear thermoviscoelasticity of the "Diflon" polycarbonate in a complex stressed state during stress relaxation. Mech. Composite Mater. 18, 499-502. [77] Malmeister, A. A. (1985). Predicting the thermoviscoelastic strength of polymer materials in a complex stressed state. Mech. Composite Mater. 21,768774. [78] Malmeister, A. A. and Yanson, Y. O. (1979). Nonisothermal deformation of a physically nonlinear material (polycarbonate) in a complex stressed state. 1. Basic experiments. Mech. Composite Mater 15,659-663. [79] Malmeister, A. A. and Yanson, Y. O. (1981). Predicting the deformability of physically nonlinear materials in a complex stressed state. Mech. Composite Mater 17, 226-231. [80] Malmeister, A. A. and Janson, Y. O. (1983). Predicting the relaxation properties of EDT- 10 epoxy binder in the complex stressed state. Mech. Composite Mater 19, 663-667. [81] McKenna, G. B. and Zapas, L. J. (1979). Nonlinear viscoelastic behavior of poly(methyl methacrylate) in torsion. J. Rheol. 23, 151-166.
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[82] Mihailescu-Suliciu, M. and Suliciu, I. (1979). Energy for hypoelastic constitutive equations. Arch. Rational Mech. Anal. 71,327-344. [83] Morimoto, K., Suzuki, T., and Yosomiya, R. (1984). Stress relaxation of glassfiber-reinforced rigid polyurethane foam. Polym. Eng. Sci. 24, 1000-1005. [84] Moskvitin, V. V. (1972). Strength of Viscoelastic Materials. Nauka, Moscow [in Russian]. [85] Ng, T. H. and Williams, H. L. (1986). Stress-strain properties of linear aromatic polyesters in the nonlinear viscoelastic range. J. Appl. Polym. Sci. 32, 48834896. [86] Nishiguchi, I., Sham, T.-L., and Krempl, E. (1990a). A finite deformation theory of viscoplasticity based on overstress: Part 1 - Constitutive equations. Trans. ASME J. Appl. Mech. 57, 548-552. [87] Nishiguchi, I., Sham, T.-L., and Krempl, E. (1990b). A finite deformation theory of viscoplasticity based on overstress: Part 2 - Finite element implementation and numerical experiments. Trans. ASME J. Appl. Mech. 57, 553-561. [88] Nolte, K. G. and Findley, W. N. (1971). Multiple step, nonlinear creep of polyurethane predicted from constant stress creep by three integral representation. Trans. Soc. Rheol. 15, 111-133. [89] Nolte, K. G. and Findley, W. N. (1974). Approximation of irregular loading by intervals of constant stress rate to predict creep and relaxation of polyurethane by three integral representation. Trans. Soc. Rheol. 18, 123-143. [90] Papo, A. (1988). Rheological models for gypsum plaster pastes. Rheol. Acta 27, 320-325. [91 ] Partom, Y. and Schanin, I. (1983). Modeling nonlinear viscoelastic response. Polym. Eng. Sci. 23,849-859. [92] Peretz, D. and Weitsman, Y. (1982). Nonlinear viscoelastic characterization of FM-73 adhesive. J. Rheol. 26, 245-261. [93] Pipkin, A. C. (1964). Small finite deformations of viscoelastic solids. Rev. Modern Phys. 36, 1034-1041. [94] Pobedrya, B.E. (1967). The stress-strain relation in nonlinear viscous elasticity. Soviet Phys. Doklady 12, 287-288. [95] Pobedrya, B. E. (1981). Numerical Methods in Elasticity and Plasticity. Moscow University Press, Moscow [in Russian]. [96] Podiu-Guidugli, P. and Suliciu, I. (1984). On rate-type viscoelasticity and the second law of thermodynamics. Int. J. Non-Linear Mech. 19, 545-564.
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[97] Rabotnov, Y. N. (1948). Some problems in the creep theory. Vesta. Mosc. Univ. 10, 81-91 [in Russian]. [98] Rand, J. L., Henderson, J. K., and Grant, D. A. (1996). Nonlinear behavior of linear low-density polyethylene. Polym. Eng. Sci. 36, 1058-1064. [99] Rubin, M. B. (1987). An elastic-viscoplastic model for metals subjected to high compression. Trans. ASME J. Appl. Mech. 54, 532-538. [ 100] Schapery, R. A. (1964). Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. J. Appl. Phys. 35, 1451-1465. [ 101 ] Schapery, R. A. (1966). An engineering theory of nonlinear viscoelasticity with applications. Int. J. Solids Structures 2, 407-425. [ 102] Schapery, R. A. (1969). On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 9, 295-310. [ 103] Sharafutdinov, G. Z. (1987). Constitutive relations in viscoelasticity and viscoplasticity. Mech. Solids 22(3), 120-128. [ 104] Shay, R. M. and Caruthers, J. M. (1986). A new nonlinear viscoelastic constitutive equation for predicting yield in amorphous solid polymers. J. Rheol. 30, 781-827. [105] Smart, J. and Williams, J. G. (1972). A comparison of single-integral nonlinear viscoelasticity theories. J. Mech. Phys. Solids 20, 313-324. [ 106] So, H. and Chen, U. D. (1991). A nonlinear mechanical model for solid-filled rubbers. Polym. Eng. Sci. 31,410-4 16. [ 107] Suliciu, I. (1984). Some energetic properties of smooth solutions in rate-type viscoelasticity. Int. J. Non-Linear Mech. 19, 525-544. [ 108] Suvorova, Y. V. (1977) Nonlinear effects in deforming hereditary media. Mech. Polym. 13(6), 976-980 [in Russian]. [109] Talybly, L. K. (1983). Nonlinear theory of thermal stresses in viscoelastic bodies. Mech. Composite Mater. 19, 419-425. [110] Taub, A. I. and Spaepen, E (1981). Ideal elastic, anelastic and viscoelastic deformation of a metal glass. J. Mater. Sci. 16, 3087-3092.
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170
Chapter 3. Nonlinear Constitutive Models with Small Strains
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Chapter 4
Nonlinear Constitutive Models with Finite Strains This chapter deals with constitutive relations for viscoelastic media with finite strains. In Section 4.1, a brief survey is presented of differential models in finite viscoelasticity. In Section 4.2, a fractional derivative of an objective tensor is introduced, and fractional analogs are constructed for differential models with finite strains. It is demonstrated that fractional differential models provide fair agreement between numerical prediction and experimental data for viscoelastic solids and fluids. Section 4.3 is concerned with integral models for nonlinear viscoelastic media with large deformations. In Section 4.4, a model of adaptive links is proposed, and constitutive equations are derived based on the Lagrange variational principle. Optimal choice of a strain energy density for adaptive links is discussed in Section 4.5.
4.1
Differential Constitutive M o d e l s
This section is concerned with differential models in finite viscoelasticity. Two basic methods may be distinguished for constructing differential constitutive relations for viscoelastic media with finite strains. According to the first method, to derive a constitutive equation in finite viscoelasticity, scalar stresses and strains in an appropriate constitutive equation at small strains should be replaced by "finite" tensors of stresses and strains. Since a number of strain tensors and corotational derivatives exist, different versions of the same "infinitesimal" constitutive equation arise. Every version corresponds to a particular strain tensor and a particular corotational derivative. As common practice, the choice of a strain tensor and a corotational derivative is between equally acceptable alternatives, and it is a matter of taste and convenience of the researcher. 171
172
Chapter 4. Nonlinear Constitutive Models with Finite Strains
According to the other approach, a rheological model consisting of springs and dashpots is employed to design constitutive equations in finite viscoelasticity. Linear springs and dashpots in this model are replaced by nonlinear elastic and viscous elements, whereas the rules of summation (for stresses in elements connected in parallel and for strains in elements connected in series) remain unchanged. As is well known, any rheological model with infinitesimal strains is equivalent to a differential model. An analog of this model in finite viscoelasticity is described by some differential equations as well, but the total number of these equations essentially exceeds the number of constitutive relations in the linear theory, since rheological models, which are equivalent to each other at infinitesimal strains, differ significantly at finite strains. To the best of our knowledge, no rational classification exists for differential constitutive models in finite viscoelasticity. Thus, we confine ourselves to several examples.
4.1.1
The Rivlin-Ericksen Model
An arbitrary differential model [of the order (m, n)] in linear viscoelasticity is described by the equation do" a~tr + a~ dt + " "
dm or - b~e + b~ de + amt dt m ~ +...
'dnE + bn dt m'
(4.1 1) .
where o- is the stress, e is the strain, and a~ to b~ are adjustable parameters that satisfy the conditions a m 4: 0, bn~ 4: 0. In particular, an arbitrary model of the order (0, n) reads de. dne tr = boe + bl d t + "'" + bn dt n , I where bk = b kI / a o. For viscoelastic fluids, the first term in the fight-hand side of this equality vanishes, and we obtain
de dne or = bl dt + "'" + bn dt---;"
(4.1.2)
To derive a Rivlin-Ericksen model, we replace the stress tr in Eq. (4.1.2) by the Cauchy stress tensor 6", and the kth derivative of the strain e by the kth RivlinEricksen tensor Ak [see Eq. (1.1.122)]. As a result, we arrive at the constitutive relation O" = ~
bkAk.
(4.1.3)
k=l
The model with n = 1 corresponds to the Newtonian fluid = */,41 = 2r/b, where r/is the Newtonian viscosity.
(4.1.4)
4.1. Differential Constitutive Models
173
More complicated models are derived when we permit the stress tensor to be a polynomial in the tensors Ak n
L
O"-- Z ~ bkl~i"
(4.1.5)
k=l l=l
In particular, the second order model (with respect to I~r ~1) obeys the equation (4.1.6)
O" = ~A1 -at- b12 ~2 + b21,~2,
where ~/, b12, and b21 are adjustable parameters. The model (4.1.6) was employed by Astarita and Marrucci (1974), Ballal and Rivlin (1979), Coleman et al. (1966), and Rivlin and Ericksen (1955) to study laminar flows of viscoelastic fluids. Replacing the kth derivative of the strain e in the constitutive Eq. (4.1.2) by the White-Metzner tensors/)k [see Eq. (1.1.123)] we obtain the White-Metzner model
O"= ~ bkBk,
(4.1.7)
k=l
which was used by Astarita and Marrucci (1974), Huilgol (1979), and White and Metzner (1963).
4.1.2
The Kelvin-Voigt Model
The Kelvin-Voigt model consists of a spring and a dashpot connected in parallel, which implies that the Cauchy stress tensor 8 equals the sum 6- = 0"e + 0"v,
(4.1.8)
where the tensor O"e determines the response in an elastic element, and the tensor 6"v characterizes stresses in a viscous element. Model (4.1.8) treats a viscoelastic medium as a mixture of two continua: elastic and viscous, that cannot slide with respect to each other. These continua have the same strains, whereas the resulting stress equals the sum of stresses in the continua. Assuming the elastic medium to be homogeneous, isotropic, and to possess a strain energy density W(I1,I2,I3), we write [see, e.g., Lurie (1990)], 2
(4.1.9)
O"e -- ~ 3 3 (t//0? + ~1 k + I/t2F2),
where I is the unit tensor, P is the Finger tensor for transition from the initial to actual configuration, Ik is the kth principal invariant of the Finger tensor, and
~0
--
OW ~ 13 013'
qtl
--
OW 011
OW 012
at- I 1 ~ ,
OW 012
~2 = - - ~ .
(4.1.10)
174
Chapter 4. Nonlinear Constitutive Models with Finite Strains
To describe the material viscosity, the Newton law (4.1.4) is employed 6-~ = 2r/D.
(4.1.11)
Substitution of expressions (4.1.9) to (4.1.11) into Eq. (4.1.8) implies the constitutive relation 2 #(t) = - p 7 + - ~ 3 (~07 + ~,~P + ~2P 2) + 2rib.
(4.1.12)
Equation (4.1.12) was used by Engler (1989) and Renardy et al. (1987) to study the existence and uniqueness of solutions to boundary value problems in finite viscoelasticity.
4.1.3
The Maxwell Model
The Maxwell model consists of an elastic and a viscous element connected in series. Unlike the Kelvin-Voigt model, two approaches are distinguished in the design of the Maxwell models with finite strains. The first is based on a rheological model consisting of two elements connected in series, which implies that a specific intermediate configuration is introduced. We do not dwell on this class of models, referring to Leonov (1976) and Leonov et al. (1976), where it is discussed in detail. That version of the Maxwell model adequately predicts experimental data for a number of polymeric fluids. However, it contains such a number of adjustable functions that it is too difficult to expect that this model can be employed in applications. Nishiguchi et al. (1990a, b) proposed another version of the Maxwell model, where the total rate-of-strain tensor b equals the sum of the elastic rate-of-strain tensor/~)e and the viscous (or plastic) rate-of-strain tensor by b=be+b~.
The elastic tensor/~)e is expressed in terms of some corotational derivative of the Cauchy stress with the use of the constitutive relations for a hypoelastic solid [see, e.g., Truesdell (1975)]. The viscous rate-of-strain tensor Dv is assumed to be a nonlinear tensor-valued function of the so-called overstress tensor P, which equals the difference between the Cauchy stress tensor 6- and an "equilibrium" Cauchy stress (which describes the response of a viscoelastoplastic medium at very low rates of loading). The constitutive model based on the overstress concept was derived to describe the mechanical response in metals [see, e.g., Krempl (1987)]. However, it predicts the response in some polymeric materials as well [see, e.g., Bordonaro and Krempl (1992) and the bibliography therein]. The other approach to constructing Maxwell models with finite strains is based on replacing the stress tr and the strain • in the constitutive Eq. of the Maxwell solid
175
4.1. Differential Constitutive Models
with small strains rl do" E dt
+ o- =
de rl-dt
(4.1 13)
by appropriate finite stress and strain tensors. Here E is Young's modulus, and rl is the Newtonian viscosity. Replacing the material derivative of the stress o- by the Jaumann derivative of the Cauchy stress tensor 6. [see Eq. (1.1.113)] and the material derivative of the strain e by the rate-of-strain tensor D, we obtain the constitutive model 1
~<> + - ~ T
= 2/xb,
(4.1.14)
where T-
rI E,
I.z-
E 2.
Equation (4.1.4) was used by Johnson and Segalman (1977) and Pearson and Middleman (1978)to describe flows of viscoelastic fluids. Using the Oldroyd corotational derivatives (1.1.119) and (1.1.120) of the Cauchy stress tensor 6., we arrive at the constitutive models Tr. v + 6. = 2 ~ T b ,
(4.1.15)
T6 -A + 6- = 2/xTD.
(4.1.16)
Model (4.1.15) with coefficients T and/x depending on the second invariant of the rate-of-strain tensor b was employed by White and Metzner (1963) and Pearson and Middleman (1978). Experimental verification of the constitutive Eq. (4.1.16) was carried out by Pearson and Middleman (1978) and Janssen and Janssen-van Rosmalen (1978). To generalize the constitutive Eq. (4.1.16), the Maxwell element is replaced by a set of parallel Maxwell elements (the Maxwell-Weichert model) N
6. = Z
6. n ,
Tn 6.nA + 6. n = 21.LnTn D,
(4.1.17)
n=l
where N in the number of the Maxwell elements, Tn and ~tJ,n are their relaxation times and elastic moduli, respectively. Model (4.1.17) with coefficients 'l~n and ].l,n depending on the principal invariants of the stress tensor 6. was proposed and verified by La Mantia (1977) and Giacomin and Jeyaseelan (1995). Astarita and Marrucci (1974) generalized the constitutive Eq. (4.1.16) by replacing the upper convected derivative 6./\ of the stress tensor 6. by the general corotational derivative 6.[] [see Eq. (1.1.121)]. Equations (4.1.14) to (4.1.16) present various versions of the Maxwell model for compressible viscoelastic media. For incompressible materials, we should replace in
176
Chapter 4. Nonlinear Constitutive Models with Finite Strains
these relationships the Cauchy stress tensor # by its deviatoric part ~ [see a discussion of this procedure by Astarita and Marrucci (1974)].
4.1.4
The Standard Viscoelastic Solid
The standard viscoelastic solid is treated as a system consisting of two springs and a dashpot. Two rheological versions of this model are distinguished (see Figure 4.1.1). Version A was extended to large deformations by Haward and Thackray (1968). In that work, the linear dashpot was replaced by the Eyring viscous element, and the linear spring with Young's modulus Ee was replaced by a nonlinear elastic element with the constitutive Eq. (4.1.9), where the strain energy density W was taken in the Langevin form. The Haward-Thackray constitutive model was employed to describe the effect of strain rate of the yield stress in cellulose derivatives and poly(vinyl chloride). Version B was extended to finite strains by Buckley and Jones (1995). In that work, the linear spring with Young's modulus E1 was replaced by a nonlinear elastic element with the constitutive Eq. (4.1.9). That element describes the response caused by changes in molecular conformations. The Maxwell element connected in parallel with the spring was replaced by a nonlinear Maxwell element of Leonov's type. The nonlinear Maxwell element is characterized by some intermediate configuration, where we can arrive at after viscous deformation of the initial configuration. The viscous deformation is assumed to be isochoric, and its rate-of-strain tensor Dv obeys Newton's law (4.1.11) with an Eyring-type dependence of the material viscosity r/ on the stress intensity. The nonlinear Maxwell element describes the response in a viscoelastic medium caused by deformation of entanglements and crosslinks between chain molecules. Numerical simulation demonstrates that the Buckley-Jones model provides qualitative agreement with experimental data for polymeric materials.
J
E1
j
E1 ~
iJj ~,
Figure 4.1.1: Two versions of the standard viscoelastic solid.
4.2. Fractional Differential Models
177
At small strains, both rheological models A and B are described by the linear constitutive equation
do de. a~o" + a{-d7 = b~e + b~ dt'
(4.1 18)
where a~, a{, b~, and b~ are expressed in terms of the moduli El, E2, and r/. To extend the model (4.1.18) to finite strains, we should replace the infinitesimal strain • by some finite strain tensor, the material derivative of the infinitesimal strain tensor by the rate-of-strain tensor b, the stress ~r by the Cauchy stress tensor, and its material derivative by an appropriate corotational derivative. This procedure is not unique, since various strain tensors and corotational derivatives can be used. For example, Hausler and Sayir (1995) proposed to employ the Finger strain tensor EF [see Eq. (1.1.50)], and the upper convected derivative of the stress tensor &A [see Eq. (1.1.119)]. Assuming the viscoelastic material to be incompressible, they arrived at the constitutive equation
aD?~+ a~?~A = bDF,F + b~D,
(4.1.19)
which adequately describes the response in butyl rubber. To generalize "linear" constitutive relations, higher order terms compared to the strain tensor and the rate-of-strain tensor may be added to the fight-hand side of Eq. (4.1.19). For example, the generalized Hausler-Sayir model takes into account terms of the second order with respect to the tensors/~F and D. Assuming the fight side of the constitutive equation (4.1.19) to be an isotropic function of these tensors, we obtain
aD~ + a~?~A = bDF,F + b([)
+
Cll b2 -+-c12(O-/~F + EF" D) + c22/~2,
(4.1.20)
where a~, b~, and Ckl are adjustable functions of the principal invariants of the Finger tensor. Model (4.1.20) correctly predicts the viscoelastic behavior of carbon black reinforced rubber [see Hausler and Sayir (1995)].
4.2
Fractional Differential Models
In this section a new class of constitutive models is derived for viscoelastic media with finite strains. The models employ the so-called fractional derivatives of tensor functions. First, we introduce fractional derivatives for an objective tensor, which satisfy some natural assumptions. Afterward, fractional differential analogs are constructed for the Kelvin-Voigt, Maxwell, and Maxwell-Weichert rheological elements. The models are verified by comparison with experimental data for viscoelastic solids and fluids. We consider uniaxial extension of a bar and radial oscillations of a thick-walled spherical shell made of an incompressible fractional Kelvin-Voigt material. Explicit solutions to these problems are derived and compared with experimental data for
178
Chapter 4. Nonlinear Constitutive Models with Finite Strains
styrene butadiene rubber and synthetic rubber. It is shown that the fractional KelvinVoigt model provides fair prediction of experimental data. For uniaxial extension of a bar and simple shear of a layer made of a compressible fractional Maxwell material, we develop explicit solutions and compare them with experimental data for polyisobutylene. It is shown that the fractional Maxwell model ensures fair agreement between experimental data and results of numerical simulation. This model allows the number of adjustable parameters to be reduced significantly compared to other models, which provide the same level of accuracy in predicting experimental data. The exposition follows Drozdov (1997).
4.2.1
Fractional Differential Operators with Finite Strains
Fractional differential models for viscoelastic media have attracted essential attention in the past decade. These models are widely spread in engineering because of their simplicity and adequate prediction of experimental data in dynamic tests. However, application of fractional models is confined to small strains, since the standard definition of the fractional derivative as a Volterra integral operator with the Abel kernel implies that the fractional derivative of an objective tensor is nonobjective. Our aim is to introduce a new operator of fractional differentiation, which (i) coincides with the standard fractional derivative at infinitesimal strains, (ii) maps an objective tensor function into an objective tensor function. Based on this operator, we propose several fractional differential models in finite viscoelasticity and compare results of numerical simulation with experimental data. Fractional derivatives with infinitesimal strains have been discussed in detail in Chapter 2. We recall that for a function f ( t ) , continuously differentiable in [0, oo) and equal zero at t - 0, the fractional derivative of the order a ~ (0, 1) is defined as f{~}(t) =
/0 tJ _ ~ ( t
- r)
(r) dr,
(4.2.1)
where
t o~ J~(t) =
F(1 +a)'
(4.2.2)
is the Abel kernel, and F(z) =
t z- 1 e x p ( - t) dt.
f0 X) is the Euler gamma function of a complex variable z. To extend formula (4.2.1) to an arbitrary objective tensor-valued function f'(t), the following formula is proposed:
t
~}(t) =
fo
J - ~ ( t - r)[~'~?(t)] r " (/D(r)" (7~?(t)d$.
(4.2.3)
4.2. Fractional Differential Models
179
Here f'[](t) is some corotational derivative of f'(t), V~?(t) is the relative deformation gradient for transition from the actual configuration at instant ~- to the actual configuration at instant t, and T stands for transpose. First, let us check that condition (i) is fulfilled, i.e., that Eq. (4.2.3) is reduced to Eq. (4.2.1) at infinitesimal strains. Indeed, according to Eq. (1.1.59), the deformation gradient ~'~?(t) coincides with the unit tensor I at infinitesimal strains. Since the corotational derivative f'[] is reduced to the material derivative
Ot' [see Eqs. (1.1.117) and (1.1.121)], our definition of the fractional derivative (4.2.3) coincides with the standard definition (4.2.1) at infinitesimal strains. To check condition (ii), we consider two motions of a medium, which differ from each other by a rigid motion. Tangent vectors gi and ~,[ and dual vectors ~i and ~it for these motions are connected by formulas (1.1.89) and (1.1.90)
a~[(t) = 0 T(t). g,i(t),
git(t) = gi(t)" O(t),
(4.2.4)
where 0 = O(t) is an orthogonal tensor function of time. Substituting expressions (4.2.4) into Eq. (1.1.59), we find that
fT~?'(t) = Or(T) • fT~?(t). O(t).
(4.2.5)
Any corotational derivative of an objective tensor f' is indifferent with respect to rigid motion,
f/,D '(t) = Or(t) • f'n(t). O(t).
(4.2.6)
Substitution of expressions (4.2.5) and (4.2.6) into Eq. (4.2.3) yields
~ } '(t) = Or(t) • fr{~}(t)" O(t),
(4.2.7)
which means that the operator (4.2.3) maps an objective tensor function into an objective tensor function, and formula (4.2.3) determines a fractional derivative for an objective tensor, which satisfies conditions (i) and (ii). The definition (4.2.3) is nonunique. This may be explained by the following: 1. The nonuniqueness in the choice of a corotational derivative for an objective tensor. 2. The nonuniqueness in the structure of Eq. (4.2.3). For any objective tensor function CJ(t), which reduces to the unit tensor at infinitesimal strains, the expression ~}(t) =
J_~(t-
r)[V,f(t)] r - ~(~-). 9n(~-) • fJ(~-). V,f(t)d~"
(4.2.8)
provides another formula for a fractional derivative, which satisfies conditions (i) and (ii).
180
Chapter 4. Nonlinear Constitutive Models with Finite Strains
According to Eq. (4.2.8), to determine a fractional derivative of an objective tensor f', we should fix its corotational derivative ¢¢D and an objective function U. In particular, for the Cauchy stress tensor &(t) we employ the upper corotational derivative 6"A [see Eq. (1.1.119)] and the unit tensor tJ. As a result, we obtain
1 f0'(t -
dl~I(t) = F(1 - ~)
r)-~[V,f(t)] r . d'zx(r) • V , f ( t ) d r .
(4.2.9)
By analogy with Eq. (4.2.9), the fractional rate-of-strain tensor b {~t is defined as follows" bl~I(t) = r(1 - ~)
'
4.2.2
f0'(t -
r)-~[¢,f(t)] r . b ( r ) . ¢ , f ( t ) d r .
(4.2.10)
Fractional Differential Models
In Section 4.1, we discussed differential constitutive models in finite viscoelasticity, which are treated as combinations of the simplest rheological elements: springs and dashpots. At small strains and uniaxial loading, the constitutive equation of a linear elastic element reads or e = EE,
where O"e is the stress, e is the strain, and E is Young's modulus. At finite strains, the spring is treated as a homogeneous, isotropic, hyperelastic medium with the constitutive equation 2
O"e = - ~ 3 (I/t0I -+- ~1 p + I/t2i~-'2),
(4.2.11)
where 6"e is the Cauchy stress tensor, P = F(t) is the Finger tensor for transition from the initial to the actual configuration, Ik is the kth principal invariant of F, and functions qti are expressed in terms of a strain energy density W by the formulas OW ~0 = 13(F)-~-3,
^ 0W OW + I I ( F ) ~ I/tl = Oil 012'
0W i//2 = - ~ . 012
(4.2.12)
At small strains and uniaxial loading, the constitutive equation for a linear viscoelastic element reads 0% = r/d,
(4.2.13)
where o'v is the stress, d is the rate of strain, r/is the Newtonian viscosity, and the superscript dot denotes differentiation with respect to time. As a natural generalization of the Newtonian dashpot (4.2.13), where the stress o'v is proportional to the first derivative of the strain e, we consider the fractional viscoelastic element with the constitutive equation [see, e.g., Bagley and Torvik (1983), Glockle and Nonnenmacher
4.2. Fractional Differential Models
181
(1994), and Koeller (1984)] o.v = r/e {~}.
(4.2.14)
The fractional dashpot (4.2.14) is characterized by two parameters: the order a E (0, 1) of the derivative and the material viscosity r/. To extend the constitutive equation (4.2.14) to finite strains, we replace the fractional derivative e{~} by the fractional rate-of-strain tensor b {~} [see Eq. (4.2.10)] and write 6-~ = 2r//) {~}.
(4.2.15)
As common practice, the viscosity coefficient 2r/is used for three-dimensional loading instead of the coefficient rt for uniaxial loading [cf. Eqs. (4.2.14) and (4.2.15)]. Our purpose now is to introduce analogs of the Kelvin-Voigt and Maxwell elements.
The Kelvin-Voigt Model The Kelvin-Voigt model consists of an elastic element (spring) and a viscous element (dashpot) connected in parallel. The Cauchy stress tensor 6- equals the sum (4.2.16)
6- = 6-e + 6-v,
where the tensor 6-e determines the response in the elastic element and the tensor 6-v determines the response in the viscous element. We confine ourselves to incompressible media with
I3°(t, ~') = 1,
(4.2.17)
where l~(t, ~-) is the kth principal invariant of the Finger tensor F<>(t, ~-) for transition from the actual configuration at instant T to the actual configuration at instant t. Substitution of expressions (4.2.11), (4.2.15), and (4.2.17) into Eq. (4.2.16) implies the constitutive equation 6-(t) = - p I + 2(qqF + ~2p2) + 2r/D {~}
(4.2.18)
of a fractional Kelvin-Voigt model with finite strains.
The Maxwell Model The Maxwell model consists of an elastic element (spring) and a viscous element (dashpot) connected in series. To construct a Maxwell model with finite strains, we replace the stress or and the strain e in the Maxwell constitutive equation at small strains (r/stands for the relaxation time)
do-
de
~)-d~- + o- = / x n ~ -
(4.2.19)
by appropriate finite stress and strain tensors. Using the Oldroyd corotational derivatives (1.1.119) and (1.1.120) of the Cauchy stress tensor 6- and the rate-of-strain
182
Chapter 4. Nonlinear Constitutive Models with Finite Strains
tensor/3, we arrive at the constitutive models r/& v + 8 = 2/xr/D,
(4.2.20)
r/8 zx + 6- = 2/xr/D.
(4.2.21)
A natural generalization of Eqs. (4.2.20) and (4.2.21) is the fractional Maxwell model r/& {~} + & = 2/xr/D,
(4.2.22)
which is determined by three adjustable material parameters c~, r/, and/,. To extend the constitutive Eq. (4.2.22), the Maxwell element is replaced by a set of parallel Maxwell elements (the Maxwell-Weichert model) N & = Zr., n=l
tinct^A n + 6"n = 21,*nrlnD,
(4.2.23)
where N is the number of the Maxwell elements and ~n,/*, are adjustable parameters. By analogy with Eq. (4.2.23), the fractional Maxwell-Weichert model is governed by the constitutive equations N O" = Z O'n' n=l
'OnO'{n%} -Jr-O"n -- 2t.l,n'Onb.
(4.2.24)
Our purpose now is to analyze several problems of practical interest using the constitutive Eqs. (4.2.18), (4.2.22), and (4.2.24) and to verify the fractional differential models by comparison results of numerical simulation with experimental data.
4.2.3
Uniaxial Extension of an Incompressible Bar
Let us consider a bar with length 10 and cross-sectional area So made of a fractional Kelvin-Voigt material (4.2.18). At instant t = 0, tensile loads P are applied to the bar ends. Under their action, uniaxial deformation occurs in the bar x 1 = k ( t ) X 1,
x 2 = k o ( t ) X 2,
x 3 = k o ( t ) X 3,
(4.2.25)
where x i and X i are Cartesian coordinates in the initial and actual configurations with unit vectors ~i; k(t) and ko(t) are functions to be found. The radius vectors of a point {X i} in the initial and actual configurations are r0 "- X16'I q- X26'2 "+" X36'3,
r(t) = k(t)Xl~l + ko(t)(X2~2 + X36'3).
(4.2.26)
Differentiation of Eqs. (4.2.26) implies that V0r(t) -- k(t)¢'le'l 4- k0(t)(¢'2e'2 4- 6'36'3).
(4.2.27)
Substitution of Eq. (4.2.27) into Eq. (1.1.60) yields the relative deformation gradient for transition from the actual configuration at instant r to the actual configuration at
4.2. Fractional Differential Models
183
instant t
-
k(t)
~0(t)
to(r)
~o(T)
V~?(t) = ,--7--7,elel +
(4.2.28)
(~'26'2 -k-6'36'3).
It follows from Eqs. (1.1.61) and (4.2.28) that the relative Finger tensor equals
k(t) ]
T) -" k--~J
P~(t,
ko(t)12(6'26'2 + 6,36,3).
2
~1~1 -k- ko(g )
(4.2.29)
According to Eq. (4.2.29), the principal invariants of the tensor F<> are calculated as 11<>(t,
"r)
=
I2~ (t, r) =
I3~ (t, T) =
k-~jk(t)]2 + 2 [ 2ko('r) k°(t) ] k°(t) 12 { 2 [ k(t)
ko(~')
L-k-(~
+
l~0(r)J
'
k(/)] 2 [ k0(t) ] 4
(4.2.30)
Equations (4.2.30) together with the incompressibility condition (4.2.17) imply that
ko(t) = k-1/2(t).
(4.2.31)
Substitution of expression (4.2.31) into Eqs. (4.2.28) to (4.2.30) yields
k(t) Ik("t') ] 1/2 ~rr?(t) = k - ~ elel + k(t) J (6'2~'2 + ~'3~'3),
P*
=/;~[k(t> ] 2
k(r) 6'16'1 -k- -7--7~.,(6'26'2 -+-6'36'3),
Kit)
[k(t) ] 2
I i<>( t , T) = [ ~-~
k(~')
+ 2 k (t--S'
12<>(t,r) = z ~
+ [ k(t) J . (4.2.32)
Differentiating the second equality in Eq. (4.2.26) with respect to time and using Eq. (4.2.31), we find that
= k(t) IXle 1
--
1 _ 3/2(t)(X2~. 2 -~k
+
X36'3)
(4.2.33)
We calculate the covariant derivative of the velocity vector (4.2.33) in the actual configuration and obtain V V = k-~
~1~1 - 2 (~2~2 -I- e3e3) .
(4.2.34)
184
Chapter 4. Nonlinear Constitutive Models with Finite Strains
Substitution of expression (4.2.34) into Eq. (1.1.99) implies that
o(t) = ~
~
(4.2.35)
- ~(~2~2 + ~3~3) •
Finally, Eq. (4.2.35) together with Eqs. (4.2.10) and (4.2.28) yields
b{,,}(t) =
1
r(1-~)
/ot( t - r ) -~ ( [k(t) Lk-~
2
k(g)
e l e l - 2k(t) (e2g' 2 + 6'3g'3)
~
dr.
(4.2.36) To calculate the Cauchy stress tensor &, we substitute expressions (4.2.32) and (4.2.36) into the constitutive Eq. (4.2.18) and obtain (4.2.37)
O" = orlele 1 + or2(g'2g' 2 + g'3g'3).
Here
[ 21
Or1(t) = --p(t) + 2k2(t) W1 + k-~ W2
2r/k2(t) f t (t - r)- a k(r) + F(1 - a) k3(,r) dr, -- p(t) + ~-~
or2(t ) =
W1 +
- F(1 - a ) k ( t )
k2(t) +
W2
(4.2.38)
(t - r ) - ~ k ( r ) d r ,
where W1-
OW
011
,
W2 -
OW
012
•
The boundary condition on the stress-free lateral surface reads o'2 = 0 . Combining this equality with Eq. (4.2.38), we find that
E 1][
O"1 = 2 W 1 nt- k--~W2
k2(t)-
n
+ F(1 - a) f0 t(t
k(~) -
-
Fk(t~ q 2
k(~)
r) -~ { k - ~ + 2 /[k-~]/ } k - ~ dr.
(4.2.39)
According to Eq. (4.2.39), the longitudinal stress o'1 is characterized by the strain energy density W, which may be found in static tests with finite strains when the
185
4.2. Fractional Differential Models
material viscosity is neglected, and by the parameters r/and a, which are determined in tests with time-varying loads either with small or with finite deformations. As an example, we consider uniaxial extension of styrene butadiene rubber [see Bloch et al. (1978)]. We assume that strain energy density W has the Mooney-Rivlin form W
--
Cl0(/1
-
(4.2.40)
3) + c01(I2 - 3).
Adjustable parameters C10 and C01 are found by fitting experimental data for slow loading with the extension rate k - 0.001134 min -1. The longitudinal stress (7"1 is plotted versus the extension ratio k in Figure 4.2.1. This figure demonstrates fair agreement between experimental data and their prediction by Eqs. (4.2.39) and (4.2.40).
0.6
(7" 1
ooooooooOOOo°°°°°°°°°°°°°~ ~...~'° o(~Oo°°
oo~o o°°°
oO~°°° ooOo(~°
m
° 1
~o
I
ooO°6° ooO°~ oO~
n
I
n
i
n
i
k
i
I
2
Figure 4.2.1: The longitudinal stress ~rl (MPa) versus the extension ratio k for plasticized styrene butadiene rubber. Circles show experimental data obtained by Bloch et al. (1978) in uniaxial tests with the extension rate k = 0.001134 min-1; Dotted line shows prediction of the Mooney-Rivlin model with c10 = -0.035 and c01 = 0.178.
186
Chapter 4. Nonlinear Constitutive Models with Finite Strains
06 oooooooO~
.~.......'" 0
_
.~.'" -
~
o•
o oo°
...~.'"
a,.....
©..." ooO°
° ~°
-
o e°
....~..'.o..-'"
~
........
.....
...."
...~ .... ooOo°°°
~" ~" .~." ....... ~...'" ° oo°
ooOoo o o O ~ go
ooOgo
oooO°°~
ooOo°
°oO°° ooOo°
..'.6.~"
..:.'~."
• gO ° o
o o o~l~°
.~ oo • 3:" oOo
m
oo
8o
t" 1
I
I
I
I
I
I
I k
I
I 2
Figure 4.2.2: The longitudinal stress or1 (MPa) versus the extension ratio k for plasticized styrene butadiene rubber. Circles show experimental data obtained by Bloch et al. (1978) in tmiaxial tests with a constant rate of extension k min-1; Dotted lines show prediction of the fractional Kelvin-Voigt model consisting of the Mooney-Rivlin spring with cl0 = -0.035 and c01 = 0.178 and the fractional dashpot with parameters c~ and r/MPa-hour ~. Curve 1: k = 0.02268, a = 0.65, ~7 = 0.010; Curve 2: k = 0.4536, a = 0.52, r / = 0.006; Curve 3: k = 4.536, a = 0.46, ~7 = 0.004.
The parameters a and r/are determined by fitting experimental data for rapid loading, when the viscous effects should be taken into account. The corresponding results are presented in Figure 4.2.2. This figure shows that a and ~7values, found with the use of the least-squares method, ensure excellent agreement between experimental data and their prediction by the Kelvin-Voigt model with the Mooney-Rivlin spring and the fractional dashpot. The obtained results demonstrate that both parameters c~ and ~7 change with an increase in the extension rate. It is of interest to fix the value of a , found by fitting data in a test with a given extension rate, and to study the effect of the extension rate in other experiments on the material viscosity r/. As an example, we choose an a value
187
4.2. Fractional Differential Models
06 c OoooOO o ° ° C)
O" 1
ooo°°°°°°
C) o o ° ° ° ° ° oOO o° C) o ° ° ° oo o°
ooo o~U~ooe°
• °°°° oee~ P
.(~.'"'" ..'~ .... ... ..-~...." . .... ~ .... ...~.'"" .~...."" ...~ ....
..~" ..~." .'"" ..~3"" ee°~
o o°
.." ..'6 o O ° ( ~ e o° e°
eo
.~ ........ ~ .......
~'"'"
coo ° °
.o"
oeO o ° ° oo
•" . " 0 .,~" o° ° • oo ° ° • ° • ~o ° • e ° C~ •
..-..oo ° • e• • •e
I 1
I
I
I
I
I
I
k
I
I
2
4.2.3: The longitudinal stress or1 (MPa) versus the extension ratio k for plasticized styrene butadiene rubber. Circles show experimental data obtained by Bloch et al. (1978) in uniaxial tests with a constant rate of extension k min-1; Dotted lines show prediction to the fractional Kelvin-Voigt model consisting of the Mooney-Rivlin spring with c10 = -0.035 and c01 = 0.178 together with the fractional dashpot of the order a = 0.65 with a material viscosity ~ MPa.hour ~ depending on the rate of loading. Curve 1: k = 0.02268, ~/= 0.010; Curve 2: k = 0.4536, ~/= 0.004; Curve 3: k = 4.536, rl = 0.0014.
Figure
in the test with k = 0.02268 min-1 and fit experimental data for other extension rates by the only material parameter rl. The corresponding data are plotted in Figure 4.2.3. Comparing Figures 4.2.2 and 4.2.3, we draw the following conclusions: 1. Fixing an c~ value found in one test, we reduce accuracy of fitting in other tests nonsignificantly. This means that the order c~ of the fractional derivative may be treated as a parameter independent of the rate of loading; 2. For a fixed a value, an appropriate material viscosity rl decreases with an increase in the rate of extension (by an order of magnitude, when the rate increases by two orders). Some dependence of the material viscosity on the rate of strain
188
Chapter 4. Nonlinear Constitutive Models with Finite Strains
is a characteristic feature of non-Newtonian fluids. Therefore, styrene butadiene rubber demonstrates the non-Newtonian behavior under sufficiently rapid loading. 3. The effect of the rate of loading grows with an increase in the extension ratio k. The Kelvin-Voigt model with the standard rate-of-strain tensor b leads to the opposite result (the influence of the rate of loading decreases with an increase in longitudinal deformation). Therefore, only the fractional Kelvin-Voigt model provides an adequate description of experimental data. As another example, we consider uniaxial extension of a specimen made of a synthetic rubber [see Derman et al. (1978)]. The corresponding results are plotted in Figure 4.2.4. Curve 1 is obtained for slow loading with the extension rate k = 0.002 min -1 when the material viscosity may be neglected. Confining ourselves to the Mooney-Rivlin medium (4.2.40), we find material parameters c10 and c01 by fitting experimental data and demonstrate fair agreement between numerical and experimental results (curve 1). Afterward, we consider rapid loading with the extension rate k = 0.2 min -1 and find parameters c~ and r/by fitting experimental data for curve 2. Figure 4.2.4 demonstrates good correspondence between experimental data and their prediction both for slow and rapid regimes of loading.
4.2.4
Radial Deformation of a Spherical Shell
Let us consider a hollow sphere with inner radius R1 and outer radius R2 made of an incompressible fractional Kevlin-Voigt material. At the initial instant t = 0, internal pressure P0 = po(t) is applied to the sphere. External surface is traction-free; body forces are absent. The pressure P0 changes in time so slowly that the inertia forces may be neglected. Our objective is to establish a connection between the radial displacement uo(t) on the internal surface and the pressure po(t), as well as to find stress distribution in the sphere. Under the action of internal pressure, spherically symmetrical deformation occurs in the shell r = f ( t , R),
0 = (9,
q) = ~,
(4.2.41)
where {R, (9, ~ } and {r, 0, ~b} are spherical coordinates in the initial and actual configurations with unit vectors ~R, ~O, ~, and G, ~0, ~6, respectively, and f ( t , R ) is a function to be found. The radius vectors in the initial and actual configurations are ?o = RF.R,
? = f(t,R)G.
(4.2.42)
Differentiation of Eqs. (4.2.42) implies that Vor = h(t)~RG + f(t)(eoeo + e~e4~),
where
h(t) = -a~f (t),
(4.2.43)
4.2. Fractional Differential Models
189
ooO"~ •
•
•
oo~
~
or1
•
•
•
o o°°
•
•
• °o •
o~o
e° o
•
•
oo °
~}o o° •
•
••
.~
•
oo °
° o°
°°
•
•
•
° o°°
°° o° •
•
o(~ °°
..'~"
•
° ~°
° o(~
°•
•
ee
•
° o°
o°
(~o o° • •
~oo °° •
oo °
•o°
o°
•
•
ee
°•
•
•
o°
•
•
•
•
ee
•
•
• o'
°~°
•
•
.'0
.©"
o•
i
I
I
I
n
I
I
I
k
1
I
1.5
Figure 4.2.4: The longitudinal stress O" 1 (MPa) versus the extension ratio k for synthetic rubber. Circles show experimental data obtained by Derman et al. (1978) in uniaxial tests with a constant rate of extension k min-1; Dotted lines show prediction of the fractional Kelvin-Voigt model, which consists of the Mooney-Rivlin spring with c10 = 0.551 and c01 = 0.089 and the fractional dashpot with c¢ = 0.84 and r/ = 0.32 MPa.min ~. Curve 1: determining the parameters c10 and c01 at k = 0.002; Curve 2: determining the parameters c~ and ~/at k = 0.2.
and the argument R is omitted for simplicity. Substitution of expression (4.2.43) into Eqs. (1.1.60) and (1.1.61) yields
h(t) f(t) fTT?(t) = ~('~e.re.r "Jr-- ~ ( e o e o
F°(t, T) =
h(T)J erer + [ - ~ J
+ edpe.th), (e'oe'o "~- e'~be'qb)"
(4.2.44)
190
Chapter 4. Nonlinear Constitutive Models with Finite Strains
It follows from Eqs. (4.2.44) that I?(t, r) = \ ~ ( - ~ j
+2
~
,
h (I(,> 54 \T~/
i¢(t, ~> = \ ~ /
Equations (4.2.45) and the incompressibility condition (4.2.17) imply that f2 -~ Of = R2
.
Integration of this equation results in f ( t , R ) = [R3 + C(t)] 1/3,
h(t,R) = RZ[R3 + C(t)] -2/3,
(4.2.46)
where C = C(t) is a function to be found. Substitution of expressions (4.2.46) into Eq. (4.2.44) yields /~¢(t, ~') =
~-5 + C-~ ]
~r~.r +
R3 + C(~')
(~.o~.o + ~.4~.4~).
(4.2.47)
Differentiating the second equality in Eq. (4.2.42) with respect to time, we find that 0 = O / ( t ) ~ r.
ot
Calculation of the covariant derivative of this vector implies that fT O -
1 Oh
h(t) Ot
(t)GG.
(4.2.48)
Combining Eq. (4.2.48)with Eq. (1.1.99)and using Eq. (4.2.46), we obtain 1 Oh(t)GG = 2 C'(t) f) - h(t) Ot --3 R 3 + C(t)
e.rOr.
(4.2.49)
Substitution of expressions (4.2.44) and (4.2.49) into Eq. (4.2.10) yields D{a}(t) =
-
2[R3 + C(t)] -4/3 fot 3F(1 - or) (t - s)-a[R 3 + C(s)]l/3c(s)ds erer.
(4.2.50)
Equations (4.2.47) and (4.2.50) together with the constitutive Eq. (4.2.18) imply that O" ~ O're.re. r -]- oroe.oe. 0 Jr-
tr4,e6e4~,
(4.2.51)
4.2. Fractional Differential Models where
Or r
[
=
--
p(t) + 2 Wl(t, O) + 2
191
(g3+c(t))2/3 R3
t
4rl
[R 3 + C(t)] -4/3
3F(1 - c~)
1(
W2(t, O)
R 3 + C(t)
)
4/3
fo
(t -- s ) - a [ R 3 + C ( s ) ] l / 3 C ( s ) d s ,
~ro = or6 = -p(t) + 2 { Wl (t, O) 2/3 +
R 3 + C(t)
(4.2.52) We integrate the equilibrium equation
030"r -~-2--(O"r -- O'0) ~--0 Or
(4.2.53)
r
from rl = f(t, R1) to r2 = f(t, R2) and use the boundary conditions
O'rIR=R= = O.
OrrlR=R1 = --Po, As a result, we obtain
po(t) = 2 I1 r2 ~tro-r d rO ' r
= 2 fR R2(tro
1
R2dR --
O'r)
R 3 + C(t)
Substitution of expressions (4.2.52) into this equality implies that
po(t) = 4
1
W1 (t, O) + W2(t, O)
R3
[( R 3 + C(t) ) 2/3 ( R 3 + C(t) )-4/3] X
R3
8T~ + 3F(1 - a )
-
fot
R3
R2dR R 3 + C(t)
fRR2 [R3 + C(~')] 1/3R2dR • (4.2.54) [R 3 + C(t)]7/3
(t - 1")-'~(7(1")d~"
1
Given internal pressure po(t) and strain energy density W(I1, •2), Eq. (4.2.54) is a nonlinear integro-differential equation for the function C(t). After determining this function, the stress distribution in the shell can be found from Eqs. (4.2.52). We confine ourselves to the Mooney-Rivlin media with strain energy density (4.2.40). By introducing the new variables R 3 --- R~x and C = R~A, we present Eq.
Chapter 4. Nonlinear Constitutive Models with Finite Strains
192 (4.2.54) as follows:
flblCl0+C01 (x+A(t))2/31
2rl
+3F(1 - ~)
fot
(t - T)-~f4(T)dT
(
_
x
x
x + A(t)
/
flb (X + A(~')) 1/3 dx 3 (X + A(t)) 7/3 - 4P0(t),
dx x + A(t) (4.2.55)
where b = (R2/R1) 3. At small strains, when A(t) ~ 1, Eq. (4.2.55) is reduced to the linear integro-differential equation with the Abel kernel
2rt c~) fOt(t -
I~A(t) + 3F(1
-
r)-~A(~')d~ " =
3bpo(t) 4(b
-
1)'
(4.2.56)
where ~ = 2(Cl0 + c01) is the Lame parameter (shear modulus). Let us analyze small steady oscillations of the shell (sufficiently slow to neglect the inertia forces) under the internal pressure
po(t) = P sin wt,
(4.2.57)
where P is the amplitude, and o) is the frequency of oscillations. Since we are interested in steady vibration, we replace the lower limit of integration 0 by - ~ [see, e.g., Burton (1983)]. As a result, we obtain
2 rt fj 3bpo(t) txA(t) + 3F(1 - c~) co (t - r)-'~.,~(r)d~ - = 4 ( b - 1 ) .
(4.2.58)
We replace expression (4.2.57) by the formula
po(t) = P exp(tcot), where ~ = x / ~ ,
(4.2.59)
and seek a periodic solution of Eq. (4.2.58) in the form
A(t) = A, exp(t~ot),
(4.2.60)
where A. is an unknown parameter. We substitute expressions (4.2.59) and (4.2.60) into Eq. (4.2.58) and transform the integral term as
S
(t - ~')-~A('r) d'r = A, to9 = A,~o
i
(t - "r)-~ exp(wJ~') d~" ~-~ e x p [ ~ o ( t - ~)] d~
= A,~co exp(~ot)
~-~ e x p ( - w ~ ) d~
193
4.2. Fractional Differential Models
= A,(~to) ~ exp(~tot)
f0 ~
~1 ~ e x p ( - ~ l ) d~l
= A,(~to)~l-'(1 - c~) exp(~tot).
(4.2.61)
(We introduced the new variables ~ = t - • and ~1 = Sto~.) As a result, we find that
A:~
4(b-
E 2 ] E(
-1
1) ~ + 3~/(rto)~
4(b - 1)
/x + ~ r/to ~ cos T
+ ~ ~r/to~ sin T
o]1
.
(4.2.62)
Substituting Eq. (4.2.62) into Eq. (4.2.60) and calculating the imaginary part of the obtained expression [which corresponds to the load (4.2.57)], we arrive at the periodic solution 3be A(t) = 4(b - 1)
×
E//x + ~2rlto'~ cos -"/]'~ og/2+ /2~ rtto'~ sin T,'/tog/ 2]-1
/z + ~ rlto ~ cos T
sin(tot) - ~ rlto ~ sin --~ cos(tot) . (4.2.63)
Let us consider the periodic function po(t) in the form po(t) = P[1 + sin(tot)],
(4.2.64)
which corresponds to oscillations of the internal pressure from 0 to 2P. Since any steady solution corresponding to a constant internal pressure is independent of time, and the governing Eq. (4.2.58) is linear, oscillations (4.2.64) of the internal pressure cause the following periodic solution: 3bP 1 A(t) = 4 ( b - 1--------~ ~ +
2 7rc~ tx + ~rlto ~ cos T
/x + ~ r/to ~ cos
+
r/to ~ sin ~-~
sin(tot) - ~ r/to ~ sin - ~ cos(tot)
/
-1 .(4.2.65)
It is of interest to compare the obtained solution (4.2.65) (with parameters calculated in the previous section for uniaxial extension) with experimental data for a thick-walled spherical shell (R2 = 2R1) made of a synthetic rubber. The internal pressure po(t) is plotted versus the radial displacement of the internal surface uo(t) - f ( t , R 1 ) -
R1 = RI[(1 + A(t)) 1/3 - 1]
in Figure 4.2.5. This figure demonstrates fair agreement between observations and their prediction with the use of the fractional Kelvin-Voigt model. The results of
194
Chapter 4. Nonlinear Constitutive Models with Finite Strains
0
0.4
0 0
Po
o H °°." -
~'$"
o
_
0 o
d;
Ho
....
OZ. [~'"
oO,O
o..
©
d~l 0
I
I
I
I
I
I u0
I
I
3
Figure 4.2.5: Internal pressure P0 (MPa) versus the radial displacement u0 m m of the inner surface of a thick-walled spherical shell. Circles show experimental data obtained by Derman et al. (1978) for a shell made of a synthetic rubber loaded by the internal pressure p0 = P[1 + sin(rot)] with P = 0.2 MPa and ro = 0.151 min-1; Dotted lines show prediction of the fractional Kelvin-Voigt model, parameters of which were determined in experiments on uniaxial extension.
numerical simulation describe adequately slope of the graph and dimensions of the ellipse-type curve. Small deviations (about 10%) between experimental data and numerical results may be explained by (a) the neglect of inertia forces in our model, (b) linearization of the nonlinear Eq. (4.2.55), (c) errors of measurements, and (d) small number of experimental data in uniaxial tests. Our study of the fractional Kelvin-Voigt model draws the following conclusions: 1. The model describes adequately experimental data both for uniaxial extension of a bar and for radial oscillations of a thick-walled sphere. 2. The fractional Kelvin-Voigt model is rather simple for the analytical study and numerical simulation.
4.2. Fractional Differential Models
195
3. Material parameters found by fitting experimental data in one test may be employed for the prediction of results in other tests. 4.2.5
Uniaxial Extension of a Compressible
Bar
We consider uniaxial extension a bar with length l0 and cross-sectional area So made of a compressible fractional Maxwell material. At instant t = 0, tensile loads P = P(t) are applied to the bar ends. The function P(t) changes in time so slowly that the inertia forces may be neglected. We assume that uniaxial deformation (4.2.25) occurs in the bar, where k(t) and ko(t) are functions to be found. The radius vectors of a point with Cartesian coordinates {X i} in the initial and actual configurations are calculated by formulas (4.2.26). It follows from these equalities that
k(t)
ko(t) ,_ _
~'~ = k - - ~ e l e l + k - - ~ t e 2 e 2
+ e3e3),
(4.2.66)
6'i are unit vectors of the Cartesian coordinate frame. The Cauchy stress tensor 6" is assumed to have the form
where
6. -- o-lele I q- o-2(g,2g, 2 q- g,3g,3),
(4.2.67)
where O"i : cri(t) (i = 1, 2) are unknown functions. Tensor (4.2.67) satisfies identically the equilibrium equations in the absence of body forces. Combining expressions (4.2.28), (4.2.66), and (4.2.67) with Eq. (1.1.119), we calculate the upper Oldroyd derivative of the Cauchy stress tensor 6"n =
0"1 -
~lel +
0"2 - 2~o-2
(~2~2 -+- g'3g'3) •
(4.2.68)
Substitution of expressions (4.2.28) and (4.2.66) to (4.2.68) into the constitutive Eq. (4.2.22) implies the following governing equations: r/ k (t)) [ 2 0"1(T)_ 2 _k(r)" F(1 - a ) ~oot (t - r) -~ (\k--~ _ t r l ( r ) 1 dr k(r)
k(t) + ~rl(t) = 2r11*k(t---7' F ( 1 r/ -a)
(4.2.69)
f0t ( t - r ) -~ ( ko(t) ) 2
ko(r)
k°(r) °'2(r)] dr 0"2(r) - 2 ko(r)
&(t)
+o-2(t) = 2 r l ~ ~ .
(4.2.70)
ko(t)
Boundary conditions at the ends are written in the integral form
f s O'l dx 2 dx 3 = P.
(4.2.71)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
196
Since the lateral surface is stress-free, we can set 0"2 = 0 .
It follows from this equality and Eq. (4.2.70) that for any t >- 0,
ko(t) = 1. In this case,
X 2 = X 2, x 3 = X 3,
(4.2.72)
and the boundary condition (4.2.71) reads (4.2.73)
Orl = p ,
where p = P/So. We consider two types of loading corresponding to creep and relaxation tests. For creep tests, the force P is assumed to be constant. In this case, Eqs. (4.2.69) and (4.2.73) imply that
k(t) /x ~
pk2(t)
ft
k("r)
p
+ F(1 -- a) J0 (t - r) -~ k3('r) d r - 2,/ .
(4.2.74)
At infinitesimal strains, when the ratio p/Ix is small, Eq. (4.2.74) is reduced to the linear differential equation
k(t) k(t)
p 2tit,
(4.2.75)
The solution of Eq. (4.2.75) with the initial condition k(0) = 1 describes the exponential creep typical of the standard (nonfractional) Maxwell model
k(t) = exp
2-~
"
(4.2.76)
It follows from Eqs. (4.2.74) to (4.2.76) that in creep tests, the effect of the fractional derivative in the Maxwell model has the second order of magnitude compared to the ratio p/Ix. Let us now analyze relaxation tests in which the extension ratio k is fixed. In this case, Eq. (4.2.69) can be presented as Orl(t) +
F(1 rt- c~) f0t (t
-- , r ) - a O - l ( , r ) d , r
= O.
(4.2.77)
To find an explicit solution of Eq. (4.2.77) with the initial condition o-(0) = p, we use the Laplace transform method. The Laplace transform of the second term in Eq. (4.2.77) is found as follows:
fo
exp(-zt) dt
f0'
(t
- - "i') -t~ O" 1 ( T )
dT
4.2. Fractional Differential Models
f0 /o
=
=
197
(t - ~')-~ exp(-zt) dt
~1(~') d~"
O"1(1")exp(--Zr) dl"
/o
~-~ exp(-z~) d~.
(4.2.78)
To calculate the first integral in the fight-hand side, we use the standard formula for the Laplace transform of the derivative f0 ~ o1(~') exp(-zl") dl"
Z6"l(Z)
crl(0),
(4.2.79)
where 6"1(z) is the Laplace transform of O-l(t). The second integral is calculated similar to Eq. (4.2.61) f0 ~ ~-'~ exp(-z~)d~ = r(1 - a) Z1-a
(4.2.80)
"
Substitution of expressions (4.2.79) and (4.2.80) into Eq. (4.2.78) yields
/0
exp(-zt)dt
/0 t( t -
~')-~d-l(r) d~" = F(1 - a)[Z0l(Z)- p]z ~-1.
It follows from this equality and Eq. (4.2.77) that 6"l(z)=p
(
~
1 ) (
1 ) z~ + r / _ 1 .
(4.2.81)
The inverse transform for z -(1-~) is t-o~
F(1 - a ) ' whereas the inverse transform for (z~ + a)-1 equals sin( 7rc~) f ~ x ~ exp( - xt) dx 7r JO X2a + 2ax~ cos(Tret) +
a 2"
Since the inverse transform for the product equals convolution of the inverse transforms for multipliers, we find from Eq. (4.2.81) that
sin(1ra) f o t f o ~ X ~ e x p ( - x ( t - s ) ) d x trl(t) = p 7r-F((1~ a) s -~ ds x 2ct + 2r/-lx a cos('rra)
+ 11-2.
(4.2.82)
It follows from Eq. (4.2.82) that for the fractional Maxwell model, relaxation curves corresponding to different p values (i.e., to different extension ratios) are similar to one another. Numerous experimental studies for nonaging polymeric materials confirm this assertion [see, e.g., Glucklich and Landel (1977) and Titomanlio et al. (1980)].
Chapter 4. Nonlinear Constitutive Models with Finite Strains
198
0.8
m
o n
O0 O O00 O 00 O °o o
°°Oooo ° °°°OOooeo 0
°°°°°°°°°OOoooooooeoooooooooooooooooooooooo
©
I
I
I
I
I
©
I
I t
I
I 1
Figure 4.2.6: The dimensionless longitudinal stress E = o'1/o'1(0) versus time t hrs for polyisobutylene. Circles show experimental data obtained by Titomanlio et al. (1980) in uniaxial relaxation tests with the extension ratio k = 1.75; Dotted line shows prediction of the fractional Maxwell model with a = 0.4 and rl = 20.0 hours". To compare the obtained result with experimental data, we plot the dimensionless longitudinal stress E = o'1/p versus time t in Figure 4.2.6. This figure demonstrates fair agreement between experimental data for polyisobutylene at large strains and their prediction by the fractional Maxwell model.
4.2.6
Simple
Shear
of a Compressible
Medium
We consider simple shear of a fractional Maxwell medium that occupies the infinite layer {-cx~ < X 1 < 0%
0 <- X 2 --- H,
- o9 < X 3 < o9}.
Here H is thickness of the layer, and {X i} are Cartesian coordinates in the initial configuration. At the instant t = 0, external loads are applied to the medium and
4.2. Fractional Differential Models
199
cause its shear X3 = X 3,
X2 = X 2,
X 1 "- X 1 + K ( t ) X 2,
(4.2.83)
where K is the coefficient of shear, and {X i} are Cartesian coordinates in the actual configuration. The radius vectors of a point with Cartesian coordinates {X i} in the initial and actual configurations equal ? ~- (X 1 q- KX2)~I q- X2~2 -1- X3~3"
Fo = X l e l q- X26'2 q- X3e3,
(4.2.84)
Differentiation of Eqs. (4.2.84) implies that ~707(t ) = I -k- K(t)g'2g'l.
Combining this equality with Eq. (1.1.60), we obtain ~r~?(t) = I + [ K ( t ) - K(T)]6'26'1.
(4.2.85)
Differentiation of the second equality in Eq. (4.2.84) with respect to time yields = K(t)X2~I .
Calculating the covariant derivative of the velocity vector, we find that ~r0 = K(t)g'2el,
~0T
= K(t)¢'le2,
1
/~) = ~K(t)(¢'l~'2 -k- ~'2¢'1).
(4.2.86)
We assume that the Cauchy stress tensor ~- = O-11~1~,1 -Jr- or22e2e 2 q- o-33e3e 3 -Jr- O'12(6'16' 2 + 6'26'1) + O'13(6'16' 3 + 6'36'1) + Or23(6'26' 3 -k- 6'36'2)
(4.2.87)
depends on time t only. Substitution of expressions (4.2.86) and (4.2.87) into Eq. (1.1.119) implies the following formula for the upper convected derivative of the stress tensor: (~A ~_ ((i'll -- 2 k o r l 2 ) e l e l
+ ~22e2e2 -1- &33e3e3
q- (0"12 -- Kor22)(ele2 + ~'2el) q- (0"13 -- Ko'23)(ele3 q- e3el) -+- 0"23(~,2~,3 -+- ~,3~,2).
(4.2.88)
It follows from Eqs. (4.2.9), (4.2.85), and (4.2.88) that d'{~I(t) = F(1 1- ~)
]0.t(t -
~')-~{[d'11(r) - 2~:(~')o12(~')
+ 2(K(t) - K(T))(O'12('r ) -- /~(T)O'22(T)) + (K(t) - K(T))20"22(T)]e16'l -Jr- [O'12(T ) -- K(T)O'zz(T ) -+- ( K ( t ) - K(T))O'zz(T)](ele. 2 -~ 6'2el)
200
Chapter 4. Nonlinear Constitutive Models with Finite Strains
-]- [O'13(T ) -- /<(T)O'23(T ) -]- (K(t) - K(T))0"23(T)](~'l~' 3 -q- ~'3~'1) + 0"22('r)~'2~'2 + 0"33('1")~'3~'3 + 0"23('t')(~'2~'3 +
~3~'2)}dr.
(4.2.89)
Substitution of expressions (4.2.86), (4.2.87), and (4.2.89) into the constitutive Eq. (4.2.22) implies the following system of integro-differential equations: F(1 '1-7 a ) f 0 t (t -- T)-o~[0"ll(T) -- 2k('r)o'12('r) + 2(K(t) -- K(~')) × (O12(~') -- K(~')o'22(~')) + (K(t) -- K(~'))2&22(~-)] dl " + o'11(t) = 0,
(4.2.90)
F(1 - c~)
(t - ~')-~0"22(~')d~" + o22(t) = 0,
(4.2.91)
F(1 - c~)
(t -- 7") - a 0"33 ('r) d'r + 0"33(t) = 0,
(4.2.92)
F(1 'I-"1 c~) f 0 t (t - "r)-~[0"12(1 ") -/c(~')o'22(~') + (K(t) - K(~'))0"22(~')] d r + oq2(t) = IxrIK(t), F(1 - ~)
f0
(4.2.93)
(t - ~')-~[d'13(~') - K(~')o'23(~') + (K(t) - K(~'))#23(~')] d'r
+ o'13(t) = 0, F(1 - ~)
f0
(4.2.94) (t - "r)-~d'23('r) d'r + o'23(t) = 0.
(4.2.95)
It follows from Eqs. (4.2.91), (4.2.92), (4.2.94), and (4.2.95) that o'13(t) = o'22(t) = o'23(t) = o'33(t) = 0.
(4.2.96)
Substituting expressions (4.2.96) into Eqs. (4.2.90) and (4.2.93), we obtain the linear integro-differential equation for the shear stress F(1 - c~)
(t - ~')-~#~(~')d~" + Ol~(t) = t.~K(t)
(4.2.97)
and the linear integro-differential equation for the normal stress F(1 - ~)
f0
(t - "r)-a[d'll('r) - 2K('r)o'~2('r)
+ 2 ( ~ ( t ) - ~(~'))d-12(~-)] d~- + O-ll(t) = 0.
(4.2.98)
Given coefficient of shear ~(t), Eq. (4.2.97) may be solved independently of Eq. (4.2.98). After finding the function o-12(t) from Eq. (4.2.97), Eq. (4.2.98) is solved by using the same procedure as for Eq. (4.2.97). We confine ourselves to the analysis of Eq. (4.2.97).
4.2. Fractional Differential Models
201
Periodic oscillations K(t) = K0 sin(~ot)
(4.2.99)
are one of the most interesting types of shear motions. Here K0 is the amplitude and ~o is the frequency of vibration. Substitution of expression (4.2.99) into Eq. (4.2.97) yields F(1 - o~)
(t - ~')-~d'12('r) d'r + o'12(0 = IXrlK0~ocos(rot).
(4.2.100)
We concentrate on steady oscillations of the medium, in which the lower limit of integration 0 may be replaced by - ~ . Replacing cos(cot) is the right-hand side of Eq. (4.2.100) by exp(w)t), we obtain F(1 - a)
~ ( t - "r)-~d-12(l") d'r + o.12(t) = IX.~K0~oexp(~mt).
(4.2.101)
We seek a periodic solution of Eq. (4.2.101) in the form o.12(t)
--
o.,
exp(w~t).
(4.2.102)
Substituting expression (4.2.102) into Eq. (4.2.101) and using Eq. (4.2.61), we find that O"~g
--
o
1 + ~(~w) '~
Since 'WOg
~=cos
"/'gO~
~ +~sin
2'
the latter equality can be written as
[/
o., = ~Koo9
1 + ~ o ~ cos --~
+
7ro~
(
rloo~ sin --~ "/'i'og
× E(~ +,~o cos v ) - s i n
V]"
,4.~.1o~,
Finally, substituting Eq. (4.2.103) into Eq. (4.2.102) and calculating the real part of the obtained equality, we arrive at the formula ola(t) =/XrlKoto ×
[(
1 + rico~ cos --~
1 + rico~ cos --~
+
(
oo)1-1
rltO~ sin --~
cos(o)t) + rico~ sin --~ sin(o~t) . (4.2.104)
To verify Eq. (4.2.104), we use experimental data for shear oscillations of a polyisobutylene layer. The shear stress o'12 is plotted versus the rate of shear k in Figure 4.2.7. This figure demonstrates fair agreement between experimental data and their prediction by the fractional Maxwell model. It is worth noting that fitting of
Chapter 4. Nonlinear Constitutive Models with Finite Strains
202
80 o
o
o
O'12
/
/
o.."
"o
/
/ o
-
/ 0
..."
o
0
I
-80 -20
k
20
4.2.7: The shear stress O'12 (kPa) versus the shear rate k (sec- 1) for polyisobutylene Vintanex LM-MS. Circles show experimental data obtained by Giacomin and Jeyaseelan (1995) in shear test with K0 = 16.38 and 00 = 0.88 sec-1; Dotted line shows prediction of the fractional Maxwell model with/z = 7.8 kPa, a = 0.4, and 71 = 2.0 Figure
sec
a .
experimental data was carried out by using two parameters/x and r/only. The order a of the fractional derivative was taken without changes from experimental data in relaxation tests for a polyisobutylene bar carried out by Titomanlio et al. (1980) [see Figure 4.2.6]. Discrepancies between observations and their predictions have the same order of magnitude in our model and in the model suggested by Giacomin and Jeyaseelan (1995). The difference between these models is in the number of adjustable parameters. For the fractional Maxwell model we find two parameters, whereas the Giacomin-Jeyaseelan model employs 11 material constants. This means that the use of fractional derivatives permits the number of adjustable parameters to be reduced significantly.
4.3. Integral Constitutive Models
203
Governing equations for fractional differential models are not very complicated compared to equations based on "conventional" constitutive models in finite viscoelasticity. For example, the fractional Kelvin-Voigt model implies governing equations similar to the equations developed by using the K-BKZ constitutive model, whereas the fractional Maxwell model leads to the governing equations similar to the equations developed by using the standard Maxwell model with finite strains.
4.3
Integral Constitutive Models
This section provides a brief survey of integral constitutive models in finite viscoelasticity. Since a number of viscoelastic materials permitting large deformations are practically incompressible, we confine ourselves to isotropic viscoelastic media satisfying the incompressibility condition.
4.3.1
Linear Constitutive Equations
According to Boltzmann's superposition principle, the stress-strain relation for an aging, linear, isotropic, incompressible, viscoelastic medium with small strains can be presented as #(t) = -p(t)7l +
f0t X(t, ~') d 5(~').
(4.3.1)
Here 6" is the stress tensor, 5 is the strain tensor, I is the unit tensor, and X(t, T) is a scalar function of two variables. We assume that the strain history 5(t) is sufficiently smooth, and 5(0) = 0. Integrating Eq. (4.3.1) by parts, we obtain &(t) = - p ( t ) ] + X(t, t)5(t) -
t OX --~T (t, r)5(r) dr
fo
= - p ( t ) I + X(t, 0)[5(t) - 5(0)] +
fO t -~T OX(t, ~')[5(t)
- 5(1")] d~-. (4.3.2)
Since 5 ( t ) - 5(0) = 2
f0t 6(~-)[5(t)-
5(~')] d~',
where 3(t) is the Dirac delta function, Eq. (4.3.2) reads d(t) = - p ( t ) ] + 2
H(t, ~')~o(t, ~') d'r,
(4.3.3)
where 10X H(t, ~') = X(t, 0)6(1") + ~ -7-(t, ~')
8T
(4.3.4)
204
Chapter 4. Nonlinear Constitutive Models with Finite Strains
and ~<>(t, ~-) = ~(t) - ~(~-)
(4.3.5)
is the infinitesimal strain tensor for transition from the actual configuration at instant ~-to the actual configuration at instant t. To construct a linear constitutive equation with finite strains, the infinitesimal strain tensor ~<>in Eq. (4.3.3) should be replaced by an appropriate finite strain tensor. Lodge (1964) proposed replacing the tensor ~<> by the strain tensor, [cf. Eq. (1.1.50)] 1
E~F(t,'r) = -~ [I31(p~(t,r))P~(t,'r) - 7],
(4.3.6)
where F¢(t, ~') is the relative Finger tensor for transition from the actual configuration at instant r to the actual configuration at instant t, and Ik stands for the kth principal invariant. Substituting expression (4.3.6) into Eq. (4.3.3) and using the incompressibility condition I3(/2"~(t, ~')) = 1,
(4.3.7)
#(t) = - p ( t ) ] + f0 t H(t, ~')[P<>(t, r) - ]] dl".
(4.3.8)
we arrive at the Lodge equation
Since pressure p is an undetermined scalar parameter, Eq. (4.3.8) can also be written
as #(t) = - p ( t ) ] +
/o'
H(t, ~')P<>(t, ~') d~'.
(4.3.9)
Substitution of expression (1.1.61) into Eq. (4.3.9) yields
~(t) = - p ( t ) ] + ¢0fr(t) •
/o t H(t, ~')~,-1(~')d~-. ~7of(t ).
(4.3.10)
It follows from Eqs. (1.1.44), (1.1.51), and (4.3.7) that
g-l(T) = i~'0(T) -- l -
2/~Fo('r),
(4.3.11)
where Fo is the Piola tensor and/~Fo is the Piola strain tensor. We substitute expressions (4.3.4) and (4.3.11) into Eq. (4.3.10) and obtain the constitutive equation [see Christensen (1980)]
[1
#(t) = - p ( t ) ] + ¢0~r(t) • -~X(t,t)?l- 2
/otH(t, r)~Fo(r)dr ] • ¢0f(t).
(4.3.12)
Coleman and Noll (1961) suggested to replace the infinitesimal strain tensor ~<>(t, ~') in Eq. (4.3.3) by the relative Almansi strain tensor ,2t<>(t, r). Substitution of
4.3. Integral Constitutive Models
205
expression (1.1.63) into Eq. (4.3.3) results in d'(t) = - p ( t ) I +
/0
H(t, -r)[] - ( f ~ ( t , -r)) -1] dr.
(4.3.13)
Since pressure p is an undetermined parameter, Eq. (4.3.13) is equivalent to the constitutive equation d'(t) = - p ( t ) I -
H(t,-r)[Po(t, ~.)]-1 dr.
(4.3.14)
To generalize models (4.3.9) and (4.3.14), we account for integral terms with both the relative Finger tensor F<>(t, ~') and its inverse (F¢(t, ~.))-1. The Ward-Jenkis constitutive equation reads [see Spriggs et al. (1966)] d(t) = - p ( t ) i +
j0
[Hi(t, ~')fo(t, ~') - H2(t, ~')(F~(t, 'r))-l] d'r,
(4.3.15)
where Hi(t, ~') and H2(t, ~') are adjustable functions. Tanner (1968) suggested a particular case of model (4.3.15) with one relaxation kernel d'(t) = - p ( t ) ] +
H(t,-r)[(1 - a)P~(t, "r) - a(PO(t, "r))-1] dr, (4.3.16)
where a is an adjustable parameter.
4.3.2
Constitutive Equations in the Form of Taylor Series
A nonlinear stress-strain relation in finite viscoelasticity can be presented in the form 6"(t) = G(F<>(t, r)),
0 --- r -< t < m,
(4.3.17)
where G is an arbitrary tensor-valued functional of the relative Finger tensor F<>, which satisfies axioms of the mechanics of continua [see, e.g., Drozdov (1996)]. According to the Weierstrass theorem [see Green and Rivlin (1957) and Coleman and Noll (1960)], any sufficiently smooth functional G may be approximated by polynomials
oo (4.3.18)
m=l Here
m--j0 "t• "" f0 t Hm(t, T 1 , " " " ,
rm)A~(t, T1)""" A¢(t, r m ) d r l " "
drm, (4.3.19)
where ,4<>(t, r) is the relative Almansi strain tensor, and Hm(t, T1 . . . . . Tin) is a relaxation function. Substitution of expression (4.3.19) into Eq. (4.3.18) implies a
206
Chapter 4. Nonlinear Constitutive Models with Finite Strains
constitutive model with an infinite series of multiple integrals. As common practice, only several terms in this series are taken into account. Coleman and Noll (1961) and Pipkin and Rivlin (1961) proposed a linear finite viscoelasticity model with linear terms only. To derive more sophisticated relations, nonlinear terms in Eq. (4.3.18) should be taken into account. By including the second order terms, we arrive at the second-order theory of viscoelasticity [see Coleman and Noll (1961)], #(t) = -p(t)?. - 2
+
fOt H(t,
7")A¢(t, 7")dT"
/o'/ot[
Hi(t, 7.1,7"2)A~(t, 7"1)" A~(t, 7"2)
+H2(t,
7"1,
7"2)II(A~(t, 7"1))A~(t, 7"2)] d7.1d7.2
(4.3.20)
with three adjustable functions H(t, 7"),Hi (t, 7"1,7"2),and H2(t, 7"1,7"2). Onogi et al. (1970) suggested the third-order theory of viscoelasticity d'(t) = - p ( t ) I +
fOt Hi(t,
7")AO(t, 7")d7"
+
lotlo t Hz(t, 7.1,7.2) A~(t, 7.1)" A~(t, 7"2)+ A¢(t, 7.2)" A~(t, 7.1) dT.ad7.2
+
fot~o'tfot H3(t,T1, 7.2,'/'3) IA~(t, 7.1)" A~(t, 7"2)",~(t, '/'3)
+ A~(t, 7.1)" A~(t, 7.3)"/]~(t, 7.2) + A~(t, 7.2)" A~(t, 7"1)" A~(t, 7"3) + A~(t, 7.2)" A¢(t, 7.3)"/k~(t, 7.1) + A~(t, 7.3)" A¢(t, 7.1)" A~(t, 7.1) +A~(t, 7.3)" A~(t, 7.2)" A¢(t, 7.1)] d7.1d7.2d7.3.
(4.3.21)
Bemstein (1966) proposed a similar expression, where polynomials in the relative Almansi strain tensor were replaced by polynomials in the relative Cauchy strain tensor t~~(t, 7.). The models (4.3.20) and (4.3.21) are mainly of theoretical interest, in that a huge number of experimental data is necessary to determine functions Hk of several variables.
4.3.3 BKZ-Type Constitutive Equations To derive constitutive equations in linear viscoelasticity with small strains, material parameters in constitutive equations for appropriate elastic media should be replaced by integral operators. Bernstein et al. (1963) suggested applying this procedure to hyperelastic materials to derive constitutive models in finite viscoelasticity. For an isotropic, homogeneous, and incompressible hyperelastic medium, the Cauchy stress tensor # is determined by the Finger formula [see, e.g., Lurie (1990)]
4.3. Integral Constitutive Models
207
~ = -pI + 2 ( 0Wp-011 0WP-1)012 ,
(4.3.22)
where strain energy density W(I1,12) depends on the principal invariants of the Finger tensor F. By analogy with Eq. (4.3.22), the BKZ-type constitutive model presumes that for an isotropic, homogeneous, incompressible viscoelastic medium there is a function U of instants t and ~-and the principal invariants Iff(t, ~-) of the relative Finger tensor F<>(t, ~-) such that the Cauchy stress tensor obeys the constitutive equation
a(t) = -p(t)I + 2 ft
OU -~l (t, ~-,I1~ (t, ~), l? (t, ,r))F ~ (t, ~)
-O---U-U(t,~',Ii<>(t,r),I2¢(t, ~'))(/~(t, 7)) -1 dr. ai2
(4.3.23)
For experimental validation of Eq. (4.3.23), see, e.g., Bernstein et al. (1963) and Zapas and Craft (1965). Two versions of the B KZ-type models are distinguished. The first is based on the separability principle, which states that the function U(t, 7",11,12) characterizing the viscoelastic response may be factorized as a product of a function H(t, ~') (which describes viscous properties of a medium) and a function W(I1,12) (which determines the instantaneous elastic response)
U(t, "r, Ii,I2) = 2H(t, "r)W(Ii, I2).
(4.3.24)
According to the other approach, no separability hypotheses are introduced, and the function U is presented either in the form
U = 2H(t, ~',I1,I2)W(I1,I2),
(4.3.25)
U = 2H(t, ~',I2(D),I3(D))w(II,I2).
(4.3.26)
or in the form
Here Ik(b) is the kth principal invariant of the rate-of-strain tensor D. As is well known, 11(D) = 0 for an incompressible medium. Let us assume that the separability principle holds. Substituting expressions (4.3.4) and (4.3.24) into the constitutive Eq. (4.3.23), we find that
[OWi( 1(t ),I2(t))F(t)- -~2(Ii(t),Iz(t))p-l(t) OW ] #(t) = -p(t)I + 2 { X(t,O) [-~-1
+
0X f0 t ~(t,
~)
0W
(I1"(t, ~'),
(t, ,/'))P~ (t, r)
_ 0W012(I~(t, ~'),I2<>(t,~-))(P¢(t, ~.))-1 d r } ,
(4.3.27)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
208
where F(t) is the Finger tensor for transition from the initial to the actual configuration at instant t, and Ik(t) is the kth principal invariant of the tensor F(t). We now concentrate on the constitutive assumption (4.3.25). For a neo-Hookean viscoelastic medium with the strain energy density 1
W(11,I2) = ~(I1 - 3),
(4.3.28)
we substitute Eqs. (4.3.25) and (4.3.28) into the constitutive Eq. (4.3.23) and arrive at the generalized Lodge model
d'(t) = -p(t)]I +
/t
H(t, "r,I~(t, ~'),I~(t, ~'))P¢(t,-r) d~'.
(4.3.29)
For viscoelastic media obeying the separability principle, Eq. (4.3.29) is transformed into the Lodge model (4.3.9). Assuming additionally in Eq. (4.3.29) that
H(t, 1",I1,12) = Ho(t, ~')~(I1,/2),
(4.3.30)
where H0(t, ~') is a memory function, and ~(I1,/2) is a damping function, we obtain the Wagner model
d'(t) = -p(t)] +
/0'
Ito(t, "r)~(I "' 1¢ (t, 'r), I~ (t, "r))F¢ (t, 'I")d'r.
(4.3.31)
The model (4.3.31) demonstrates fair prediction of experimental data for polyethylene melts [see Wagner (1976) and Laun (1978)]. As common practice, the memory function IIo(t, ~') has the form of a truncated Prony series. There is no rational procedure to choose the function ~(I1,12). Kaye and Kennett (1974) proposed the expression
~(Ii,I2) = exp ( - a V / l l
- 3) ,
(4.3.32)
where a is an adjustable parameter. Wagner (1976) suggested replacing the first invariant in Eq. (4.3.32) by the second invariant
~(Ii,I2) = exp ( - a v / 1 2 - 3) .
(4.3.33)
A similar expression was also proposed by Winter (1978),
~(11,/2)
=
exp
-a
.
(4.3.34)
Wagner et al. (1979) suggested employing a linear combination of expressions (4.3.32) and (4.3.33) ~(I1,I2) = c l e x p ( - a l V / l
- 3) + c 2 e x p ( - a 2 v / I
- 3),
(4.3.35)
4.3. Integral Constitutive Models
209
where I
= od1 + (1 -
cz)I2,
and a, al, a2, C1, C2 are adjustable parameters. For a Mooney-Rivlin viscoelastic medium with the strain energy density
W(I1,I2) =
(4.3.36)
Cl(I1 - 3) + c2(12 - 3),
Emery and White (1969) substituted Eqs. (4.3.25) and (4.3.36) into Eq. (4.3.23) to derive the so-called additive functional constitutive law of the first kind
&(t) = --p(t)I + fOt H(t,T,I?(t,T),I2<>(t,T)) [ClPO(/, T)- c2(ti'~(t,T)) -1] aT. (4.3.37) Equation (4.3.37) correctly predicts experimental data for a silicon polymer. To account for the effect of the strain history on the relaxation function, Waldron and Wineman (1996) and Wineman and Waldron (1995) applied the free volume concept, where the shift factor for an internal time depended on the first and second invariants of the Finger tensor. Constitutive models obeying equality (4.3.26) have been discussed in detail by Astarita and Marrucci (1974). We present only two models extending linear models in finite viscoelasticity. Middleman (1969) employed the Bogue model (a generalization of the Lodge model)
#(t) = - p ( t ) ] +
f0 t H(t,
r, I2(D))P <>(t, 1")d~',
(4.3.38)
and Goldstein (1974) and Macdonald (1976) applied the Carreau model (a generalization of the Tanner model)
gr(t) = - p ( t ) I +
f0 t H(t,
r, I2(b(T)))[(1- a)F°(t, T ) -
a(F°(t, T)) -1]
dT
(4.3.39) to analyze the response in polymeric melts. The mechanical behavior of the BKZ-type models is determined by one function U(t, T, I1,12). Wagner (1977) proposed generalizing Eq. (4.3.23) and considering the integral constitutive model
d'(t) = -p(t)]l + 2 j~ot[ Ul(t, "r,I~(t, ~'),I~(t, ~'))FO(t, "r) -U2(t, T, Ii¢(t, T),I~(t, ~'))(P<>(t, 1"))-1 ] d~',
(4.3.40)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
210
which is characterized by two adjustable functions U1 and U2. By setting
OU
U~ -
~I~
,
U2 -
,gU ~I2
,
(4.3.41)
we reduce Eq. (4.3.40) to the standard B KZ constitutive Eq. (4.3.23).
4.3.4
Semilinear Constitutive Equations
To extend constitutive equation (4.3.1) to finite strains, Chang et al. (1976) suggested (i) replacing the infinitesimal strain tensor ~ by an appropriate strain tensor at finite strains, and (ii) making necessary transformations of the obtained equation, confining ourselves to affine deformations ?(t) = f(t)~o" A.
(4.3.42)
Here ?0 is the radius vector in the initial configuration, ~(t) is the radius vector in the actual configuration at instant t, f(t) is a smooth scalar function, and A is a constant tensor. We differentiate Eq. (4.3.42) and, after simple algebra, obtain
(TrY(t) = -f ~(t) ?I,
~'0?(t) = f(t)~k,
(4.3.43)
where V0?(t) is the deformation gradient for transition from the initial to actual configuration, and V~?(t) is the deformation gradient for transition from the actual configuration at instant r to the actual configuration at instant t. Combining Eqs. (1.1.42) and (1.1.61) with (4.3.43), we find that F(t) = f2(t)Ar • A,
~ ( t , r) = F¢(t, r) =
~-~
].
(4.3.44)
We replace the tensor ~(r) in Eq. (4.3.1) by the Finger strain tensor Ev(r), use the incompressibility condition (4.3.7), and arrive at the formula
d(t) = -p(t)] +
/0
X(t, r)
(r) dr.
Substitution of expressions (4.3.44) into Eq. (4.3.45) implies that 8 ( 0 = -p(t)Y +
t
fo
= - p(t)] +
+ ~T
-~
x(t, r)
Of 2('r)/~ r . /~ dr
ar
X(t, r)
f2(t)
f2(t)-~T
Ar'Adr
(4.3.45)
211
4.3. Integral Constitutive Models
= - p(t)~, + -~ = - p(t)] +
X(t, r) F(t). --~r~r, t) + --~-r(r, t)" F(t) dr X(t, "r) F(t)"
Or
---~--('r, t)" F(t) dr,
(4.3.46) where C<>(t, r) is the relative Cauchy strain tensor. Equation (4.3.46) is called the semilinear constitutive model with finite strains. Two approaches are employed to generalize Eq. (4.3.46). Chang et al. (1976) and Bloch et al. (1978) proposed replacing the relative Cauchy strain tensor (7<>(t, r) by the relative Eulerian m-tensor ~(m)~(t, r) [see Eq. (1.1.68)]. As a result, we obtain the constitutive relation &(t) = - p ( t ) I +
fot
[^
X(t, 7") F m / z ( t ) "
03"~"(m)<~
m'°
07"
(~'' t) +
"E OT
]
('r, t) " Fm/z(t)
dr,
(4.3.47) which is called the linear viscoelastic model with moderately large deformations. According to the other approach, second-order terms (compared with the Finger tensor) are introduced into the constitutive relation (4.3.46). We do not discuss this issue in detail, and confine ourselves to two examples. McGuirt and Lianis (1970) introduced the constitutive equation OW 8W #(t) = - p ( t ) ? + 2 -27--.(I1 (t), I2(t)) + I1 (t)--z-7-.(I1 (t), I2(t)) F(t) 011 012 - aW(II(t),I2(t))k2(t )
~I2
2
t
+ k=0 fo Hk
Pk('r) . -~r (t, r) +
Hkl(t, r)/~/('r)I1
(t, r) . Pk(r) dr
Fl('r). - ~ r (t, r)
(4.3.48)
dr.
k,l =O
DeHoff et al. (1966) developed the constitutive model • F0W I t 0W ] #(t) = -p(t)~l + 2 [-~-1( 1(),I2(t)) + 11(t)~2(I1(t),I2(t)) F(t) O--W-W(ll(t),I2(t))F2(t) +
aI2
Hk(t, T)Fk(T)
k=O
• --~-r (t, ~)
212
Chapter 4. Nonlinear Constitutive Models with Finite Strains
+ Or
+ ~ Fk(t) fO I1
(t, 7)"
k=0
Hkl(t, r)/~'k(r)
dr.
(4.3.49)
/=0
Here Hk(t, T), Hkl(t, 7) are kernels of the integral operators and W(II,/2) is a strain energy density. Adjustable parameters and functions in Eqs. (4.3.48) and (4.3.49) are found by using experimental data for styrene-butadiene rubber, ethylene propylene rubber, and polyurethane. It has been shown that Eqs. (4.3.48) and (4.3.49) predict adequately the mechanical response in polymeric materials at deformations up to 200%.
4.4
A Model of Adaptive Links
Several integral models considered in the previous section do not presume the existence of strain energy density. This is a shortcoming of those models, since potential energy of deformations is helpful for the analysis of • stability of viscoelastic media, • phase transitions in solids, • crack propagation. In this section, we develop a constitutive model for an aging viscoelastic medium based on the concept of adaptive links. To derive constitutive relations, the principle of minimum free energy for isothermal processes is employed. As an example, extension of a neo-Hookean viscoelastic bar is studied, and the effect of material parameters on the extension ratio is analyzed numerically. The exposition follows Drozdov (1992, 1993).
4.4.1
A Model of Adaptive Links
A concept of adaptive links was introduced in Chapter 2 to describe the mechanical behavior (linear and nonlinear) of viscoelastic materials with small strains. In this section, we extend this approach to viscoelastic media with finite strains. A nonlinear viscoelastic material is treated as a network of parallel, nonlinear, elastic links (springs). The links are made of a hyperelastic material with a strain energy density W. The natural configuration of a link existing at the initial instant t = 0 coincides with the initial configuration of the system. For an isotropic medium, strain energy density (per unit volume in the initial configuration) of initial links equals W(II (t), I2 (t), I3 (t)),
(4.4.1)
4.4. A Model of Adaptive Links
213
where Ik(t) is the kth principal invariant of the Finger tensor F(t) for transition from the initial to the actual configuration at instant t. The natural configuration of a link arising at instant ~- coincides with the actual configuration of a viscoelastic medium at that instant. By analogy with Eq. (4.4.1), strain energy density of emerging links (per unit volume in the initial configuration) equals
W(I1<>(t, ~'), 12~ (t, ~'), 13~ (t, ~')),
(4.4.2)
where Ik~(t, T) is the kth principal invariant of the relative Finger tensor F<>(t, ~-) for transition from the actual configuration at instant ~" to the actual configuration at instant t. Formation and breakage of adaptive links are governed by the function X.(t, r), which equals the number of links arisen before the instant T and existing at instant t. Potential energy of a network of parallel links equals the sum of energies for individual links
°~X*(t, ~')W(Ik¢(t, ~'))dr. W(t) = X.(t, O)W(Ik(t)) + f0 t --~z
(4.4.3)
To extend the constitutive model (4.4.3), we assume that M different types of links exist. The strain energy density for links of the mth type is denoted as Wm, the processes of breakage and reformation for links of the mth type are governed by the functions Xm.(t, r). Similarly to Eq. (4.4.3), strain energy density of an isotropic, aging, hyperviscoelastic medium equals
M[Xm,(t, O)Wm(Ik(t))h-fot°3xm* (t, ~')Wm(Ik<>(t, ~'))d~"1 . ~gT
W(t) = ~
(4.4.4)
m=l
Employing formula (4.4.4), one can derive constitutive equations for viscoelastic media with finite strains. Assuming the reformation process to be strain-independent, we arrive at the so-called "operator-linear" model in finite viscoelasticity (the separability principle). In the general case, both the strain energy densities Wm and the functions Xm depend on the Finger tensor F.
4.4.2 The Lagrange Variational Principle Consider a viscoelastic medium that is in its natural state and occupies a bounded, connected domain lI0 with a smooth boundary F0. Points of 120 refer to Lagrangian coordinates ~ = {~/}. At the initial instant t = 0, dead body forces/) and dead surface forces b are applied to the medium. The surface traction is prescribed on a part F0(~) of the boundary. The other part F0(u) = F0 \ F0(~) of the boundary is clamped Ir~u,= 0.
(4.4.5)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
214
For dead loads/) and b, there are vector functions/)0(t, ~) and b0(t, ~) such that for any t -----0 and for any ~ E ~0
p(t)[~(t) dV(t) = po[3o(t)dVo,
b(t) dS(t) = bo(t) dSo.
Here P0 and p are mass densities, dVo and dV are volume elements, and are surface elements in the initial and actual configurations, respectively. Potential energy of deformation ff'(t) is determined by the formula
(4.4.6)
dSo and dS
(V(t) = / ~ W(t) dVo.
(4.4.7)
Combining Eqs. (4.4.4) and (4.4.7), we find that
~V(t) = ZM £ m=l
o
IXm.(t, O)Wm(Ik(t)) + fO0t oxm*(t, r)Wm(I~(t, r))d T]
dVo.
OT
(4.4.8) We fix an instant t > 0 and some deformation history before this instant. Let fi(t, ~) be an arbitrary displacement field at instant t, and R(t - 0, ~) the limit of the displacement field ~(s, ~) as s ~ t. The corresponding Finger tensors are F(t) and F(t - 0), and their principal invafiants are denoted as Ik(t) and Ik(t - 0), respectively. The work A t (t) of external forces on the displacement from a configuration characterized by the vector R(t - 0, ~) to a configuration determined by the vector R(t, ~) equals
/.
/.
At(t) = [
polio(t)" [fi(t) - fi(t - 0)] dVo + I
b0(t) • [fi(t) - fi(t - 0)]
dr 0~
J~0
dSo. (4.4.9)
At isothermal processes, free energy of a viscoelastic medium differs from its mechanical energy if" by a constant. Neglecting this constant, we find the increment of the free energy W t for transition from the actual configuration characterized by the displacement vector fi(t - 0, ~) to the actual configuration characterized by the displacement vector fi(t, ~)
M W*(t) =
Z/f~ m=l
[Xm,(t,O)Wm(Ik(t)) + o
fOt OXm* (t, r)Wm(Ik~(t, r))dr ] aVo Or
M 0,, m=l
o
+ ~0t c]Xm,(t, r)Wm(I~(t - O, r))d T] dVo. Or
(4.4.10)
4.4. A Model of Adaptive Links
215
The increment of the total free energy is calculated as T(t) = Wt (t) - At (t).
(4.4.11)
Substitution of expressions (4.4.9) and (4.4.10) into Eq. (4.4.11) implies that M T(t)=
(t, "r)Wm(I~(t, r ) ) d r 1 dVo Z ff~ [xm*(t'O)Wm(Ik(t))+ ~0"t OXm* Or M
m=l
0
m--1
+
0
~0"t ---~-r(t, OXm* ~')Wm(I~(t /,
- O, "r))dr ] dVo /,
- / poBo(t), t o ( t ) - ~ ( t - o)1 dVo - I bo(t).{~,(O- ~,(t- o)1 dSo . Jr(oor) J~ 0 (4.4.12) The Lagrange principle states that for any instant t - 0 and for any prescribed deformation history before this instant, the real displacement field fi(t, ~) minimizes the total free energy T(t) on the set of continuously differentiable displacement fields that obey the boundary condition (4.4.5). Variational principles in finite viscoelasticity are analyzed in a number of studies. The preceding formulation has been discussed in detail by Arutyunyan et al. (1987) [see also Arutyunyan and Drozdov (1992), Drozdov (1992, 1993), and Drozdov and Gertsbakh (1993)]. To derive constitutive equations for an aging viscoelastic medium, we calculate the increment BT(t) of the functional T(t) caused by an admissible perturbation 6fi(t) of the displacement field ~(t). This value is calculated up to the second-order terms compared to 3~. It follows from Eqs. (4.4.6) and (4.4.12) that
0
0
0
(t)
~r)
(~')(t)
(4.4.13) The increment 6W(t) is found in Drozdov (1993). Omitting cumbersome calculations, we present the final result
Chapter 4. Nonlinear Constitutive Models with Finite Strains
216 M
aw(t) = ZXm,(t, O){2{~m(t) " 8~(t) m=l
OWm + 2 -~-2(Ik(t)) + ll(t)~3(Ik(t)) ) [(/~(t)" 8~(t)) 2 -(F(t)" 8~(t))" (/>(t) • 8~(t))] _4aWm 013 (Ik(t)) [(F(t)" 8~(t))(p2(t)" 8~(t)) --(/7(t)" a~(t))" (F2(t)"
a~(t))]
+ I1(f7 a~ T(t)" Ore(t)" V7a~(t)) + 2 [(P(t) " 6e(t)) (~---i~m+ II(t)~2 ) - (Pz(t).8~(t))-~2 + 13(t)Ii(8~(t))-~3 +Z m=l
ar
Wm(Ik(t))}
(t, r) 2{~mO(t,7")" 8~(t)
(Ik<>(t,r))) [(PO(t, r)" 8~(t)) 2 v*3 + 2 ( ~OWm(ikO(t' r)) + I10(', r)OW_--~rm -(P<>(t, r). 8~(t))" (P<>(t,r) • a~(t))]
- 40Wm (I~(t, r)) [(P~(t, r)" 3~(t))((Po(t, T)) 2 " aE(t)) aI3
--(PO(t, r)" 8~(t))" ((PO(t, "r))2 • 8~(t))] + I1((7 8fiT(t)" ~)0m (t, "r)" (78fi(t))
+ (k~(t, ~) ae(O) ~ + I~(t, ~)~ -((Po(t, r)) 2" 8~(t))-~2 + I3<>(t,r)Ii(8~(t))
Wm(I~(t,~))}d~. (4.4.14)
4.4. A Model of Adaptive Links
217
Here 1
at(t) = ~[Vfi(t) + Vfir(t)],
o3Wm o~Wm 6m(t) = --5-ff (Ik(t)) + Ii(t)--g-~2 (Ik(O) k(t)
- OWm 012 (Ik(t))p2(t) + h(t)OWn (Ik(t))], 013
~m<>(t,T)= /[~lm (/~(t'o'l
W))+ I~(t, T)--~-2('~(t,°3WmT))lk ~ (t, T) _. o~Wm,..~
_ OWmoi(I~(t, 2 ~'))(P<>(t, ~.))2 + l~(t, ~-)--~-3tlk (t, ~'))7,
(4.4.15)
where ] is the unit tensor, T denotes transpose, and ~' stands for the Hamilton operator in the nonperturbed actual configuration. We introduce the notation
8 ( 0 - 2 ~/v/~3(t) [Xm,(t,O)~m(t)+fo m=l
tOXm* --~-~--z(t, ~')~m~(t, z)d~"1 .
(4.4.16)
It follows from Eqs. (4.4.15) and (4.4.16) that
#(t) _
2 ~--~{Xm,(t, O) [ ( -ffi~l aWm (I~(t)) + Ii(t)--~Z aWm (Ik(t))) F(t) lx/~3(t) m=l -- °~Wm(Ik(t))p2(t) + 13(0
012
×
(Ik(t))] +
(t ~') 07" '
I( OWn (IkO(t' T)) -[- I?(t, T)°3Wm(Ik<)(t,T))) k~(t, T)
] /
_ OWnoI2(I~(t, ~'))(P<>(t, ~.))2 + I~(t, ~-)-~3tlk (t, ~'))] d~" . (4.4.17) We substitute expressions (4.4.14) to (4.4.16) into Eq. (4.4.13) and transform the integrals over 1)0 with the use of the mass balance law. As a result, we obtain
aT(t) = L(afc(t)) + N(afi(t)) + . - . ,
(4.4.18)
where
L(a~(t)) = f~ ~(t) " 6~(t) dV(t) - ~ (t)
-
j (~)(t) ;
b ( t ) . 6fi(t) dS(t),
(t)
p(t)B(t) " 6~(t) dV(t) (4.4.19)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
218
M/~ F
N(Sfi(t)) = 2 Z m=l
o
,(t, O)~m(6~(t), t) +
jO"t°~Xm* 037"
]
(t, 7")~,m~(6~(t),t, 7")dT" dVo
+ 21/~(t) I1 (V t~ r (t)- ~-(t) • ~' t~fi(t)) dV(t),
(4.4.20)
and the dots stand for terms of the higher order of magnitude. The function ~m~(6~, t, 7") is determined as t~,(~~m ,~" ,
T) =
L-~-2r ~Wm(¢ (t, r)) + I? (t, T)-~3~Wm (¢ (t, r))]
X (P~(t, r)" t ~ ) 2 -- (/~"O(t, 7.)" 8~)" (F¢(t, r)" 8~) 1
- 20Wm (I~(t, 7.)) [(P~(t, 7.)" 6~)((Po(t,
a13
T)) 2"
t~E)
--(/~'~(t, T)" ~ ) " ((~P'~(t, T))2 " ~E)] + (F~(t, r)" 3e)
~
+ Ii<>(t,r)-~2
0 2 +13~(t, 7.)I1(3~)~3 Wm(I~(t, 7")).
012 (4.4.21)
To find an expression for the function ~--~m(6E, t), it suffices to replace the relative Finger tensor F<>(t, 7.) in Eq. (4.4.21) by the Finger tensor F(t). The Euler-Lagrange necessary condition of minimum for the functional T(t) reads
L(6~(t)) = 0.
(4.4.22)
We substitute expression (4.4.19) into Eq. (4.4.22), transform the first term with the use of the Stokes formula, and obtain
fr
0r)(t)
[h(t). & ( t ) - b(t)]. 8fi(t)dS(t)
- f~ [fT. &(t) + O(t)B(t)]" 6~t(t)dV(t) = O, (t)
(4.4.23)
where h(t) is the unit outward normal vector to F(t). Since 3fi(t) is an arbitrary displacement field, Eq. (4.4.23) implies the equilibrium Eq. in l~(t)
fT. &(t) + p(t)[~(t) = 0
(4.4.24)
219
4.4. A Model of Adaptive Links
and the boundary condition in stresses on the surface F(~)(t) h(t)" ~(t) = b(t).
(4.4.25)
Equations (4.4.24) and (4.4.25) imply that the tensor 6"(t) determined by the formula (4.4.16) is the Cauchy stress tensor.
4.4.3
T h e r m o d y n a m i c Stability of a Viscoelastic M e d i u m
A quasi-static motion fi(t, ~) of a viscoelastic medium is thermodynamically stable if for any instant t >- 0, the displacement field fi(t, ~) minimizes (locally) the total free energy T(t) on the set of admissible displacement fields. It follows from Eq. (4.4.18) and the Legendre-Hadamard condition that the motion ~(t) is thermodinamically stable provided for any t -> 0 and any admissible perturbation of the displacement field 3~(t), N(3fi(t)) > O.
(4.4.26)
Combining Eqs. (4.4.20) and (4.4.26), we find that
M
O(t~Ft(t),t)< 4 Z f~ [Xm,(t,O)~m(t~(t),t) m=l
0
+ ~0"t cgXm*(t T)_=m~(a~(t),t, T) d~1 dr0,
07" '
(4.4.27)
where f D(O, t) = - ] 11(~'oT . 0"(t) • ~70) dV(t). Ja (t)
(4.4.28)
We suppose that for any integer m, any instant t, and any admissible displacement field ~,
~.~m(~(V),t) > 0,
(4.4.29)
where _
~(o) = ~(V 0 + ¢ o~) is the infinitesimal strain tensor (in the basis of the actual nonperturbed configuration) corresponding to the displacement vector ~. Introduce the notation
Am(t) = inf f ~ ~'~m(~(V)' t) d V 0 ID(0, t)l
Chapter 4. Nonlinear Constitutive Models with Finite Strains
220
IIm(t, 7")
=
inf fno ~(~(~)' t, r)dVo
fl~o ~m(E(~)), t) dVo
(4.4.30) '
where the minimum is calculated on the set of admissible displacement fields ft. According to Eqs. (4.4.30), the fight-hand side of Eq. (4.4.27) is estimated as
Mf~[Xm,(t, O)~m(t~(t), t) + ~otaxm* 0'I" (t, ~')~m°(3~(t), t, r) dr ] dVo
4~
m=l
0
M IXm,(t, O) + fOtaNn* aT (t, "r)IIm(t, "r)dr ] L
~m(~(V),t) dVo
> 4Z --
m-1
0
M [Xm,(t, 0) + lot aNn*(t, ~')IIm(t, r) d T1Am(t).
--> 4[D(O t)[ Z '
(4.4.31)
07"
m=l
It follows from Eqs. (4.4.27) and (4.4.31) that the motion fi(t) is thermodynamically stable provided that for any t -> 0,
EXm
4Z
,(t, 0) +
/0toxm*
(4.4.32)
0~. (t, ~')IIm(t, ~') dr Am(t) > 1.
m=l
Let us consider two particular cases. For an elastic medium with M = 1,
Xl,(t, r) = 1, and W1 = W, the condition of thermodynamic stability (4.4.32) reads sup sup
] ft~(t)11(¢ oT. &(t)" (7 ~) dV(t)]
,>_o ~
fno =-(~(~'), t) alVa
< 1,
(4.4.33)
where
=_(?:,t) = 4
-~-~2(Ik(t))+ Ii(t)
(Ik(t)) [(/2"(t) " ~)2 _ (/~(t)" ~)" (/2"(t)" ~)]
-- 2 aw(ik(t))[(p(t). ~)(p2(t) • ~) -- (F(t). ~)" (F2(t). ~)] a13
O
3
2
+ [~" (F(t)--~I +(Ii(t)-~(t)-F2(t))-~2 + I 3 ( t ) ' ~ / 3 ) ] W(Ik(t)) 1. (4.4.34) For a viscoelastic medium with infinitesimal strains for transition from the initial to the actual configuration, we set P<>(t, r) = I,
Ii<>(t,r) = 3,
I~(t, r) = 3,
I3~(t, r) = 1.
(4.4.35)
4.4. A Model of Adaptive Links
221
Substitution of expressions (4.4.35) into Eq. (4.4.21) implies that
--m(e,t) = ~ ( 5 ,
t, "r) = [12(5) -- 11(52)1
+ I2(5)
c9
012 +
013
WOm
t9 nt- 0 ) 2
OI---~+ 2-~2
013
W°m' (4.4.36)
where OWOm _ OWm
Olk
OIk (3,3,1).
We introduce the dimensionless Lame parameters X° and C ° by the formulas [see Lurie (1990)], A° + 2C ° = 4
0I---1
-~2
013
W°m'
C° = - 2
012
013 W°m" (4.4.37)
Substitution of expressions (4.4.37) into Eq. (4.4.36) yields
1 0 ~m(5, t) = ~m(5, t,'r)= ~1 )t°I2(5) + ~ Cmll (5 2 ) .
(4.4.38)
We combine Eqs. (4.4.30), (4.4.32), and (4.4.38) to obtain the following condition of thermodynamic stability:
[ faoll(fTofJ r . &(t). ~700) dVo[ < 1, sup sup t_>o o fao [h°(t)IE(5(v)) + 2C°(t)I1 (~2(fi))] dVo
(4.4.39)
where M X°(t) = Z XmXm,(t, o t),
m=l
M C0(t) = Z COXm*(t' t).
m=l
For an elastic medium with infinitesimal strains, we set M = 1, A = A°, C = C °, and arrive at the well-known stability condition sup sup t_>0
0
I faoll(fTof) T" &(t)" ¢o0) dVol fao[M2(5(~)) + 2CI1(52(©))] dVo < 1,
(4.4.40)
where X and C are the Lame parameters [see, e.g., Drozdov and Kolmanovskii (1994)].
4.4.4
Constitutive Equations for Incompressible Media
For incompressible viscoelastic media, the Lagrange principle states that the displacement field fi(t, ~) minimizes the functional T(t) on a subset Y of the set of admissible
Chapter 4. Nonlinear Constitutive Models with Finite Strains
222
displacement fields, elements of which satisfy the incompressibility conditions I3(t) = 0,
I3~(t, ~') = 0.
(4.4.41)
It follows from Eq. (4.4.41) that the strain energy densities Wm depend on the first two principal invariants, Wm = Wm(II,I2).
Repeating the preceding transformations, it can be shown that the Lagrange principle implies the constitutive equation [see, e.g., Drozdov (1992,1993)]
M [xm.(t, O)Om(t) + J~ toxm* (t, ~')OCm(t, ~')d r] .
~(t) = -p(t)i + 2 Z
m=l
(4.4.42)
03T
Here p(t) is pressure [a Lagrange coefficient for the restriction (4.4.41)], and Ore(t) =
[0m
1
- ~ ( I ~ (t), h(t)) +/1 (t)--~-2 (/1 (t), h(t)) P(t)
_ OW~ (11(t), h(t))P2(t), 012
o/2tgWm
6era(t, T ) : [ ~(Ii<>(t, T),I?(t, T)) + II<>(t,T)--Z7" (ii~(t ' ~.),i2<>(t' r))] F<>(t, ~-) ~1
_ OWm (I~(t, T),I~(t, r))(P<>(t, r)) 2.
(4.4.43)
~12
Let only one type of adaptive links exist, and the mechanical behavior of links obey the constitutive equation of a neo-Hookean elastic medium with the strain energy density Wl -- T/./,1 (I1 --
3),
(4.4.44)
where/.1,1 is the rigidity per link. We set M = l, substitute expression (4.4.44) into Eq. (4.4.42), and use Eqs. (4.4.43). After simple algebra, we obtain
d'(t) = -p(t)]l + Aq
1,(t, 0)F(t) +
---~--T(t, ~')P¢(t, 1")dr .
(4.4.45)
Introducing the notation
I-~ = tXlXl,(O, 0),
X l , ( t , 'r)
X(t, ~') = XI,(0, 0)'
(4.4.46)
we present the constitutive equation of a neo-Hookean viscoelastic medium in the form
d'(t) = -p(t)] + tx
(t, 0)F(t) +
-0--~T(t, ~')P¢(t, ~')d~" .
(4.4.47)
223
4.4. A Model of Adaptive Links
For nonaging media, the constitutive relation (4.4.47) is simplified. We set (4.4.48)
X(t, r) = 1 + Qo(t - r),
and obtain ~(t) = - p ( t ) ? + I~
(
[1 + Qo(t)]P(t) -
/0
Qo(t - r)F <>(t, r) d r
}
,
(4.4.49)
where the superimposed dot denotes differentiation with respect to time.
4.4.5
E x t e n s i o n of a Viscoelastic Bar
To verify the constitutive Eq. (4.4.49), we consider tension of a rectilinear bar made of an incompressible neo-Hookean viscoelastic material [see Drozdov (1994)]. The bar is in its natural state and occupies a domain ~~0 -- {0 ~ X 1 ~ L0, (X2, X 3) E 000} , where X i are Cartesian coordinates in the initial configuration with unit vectors ~i, L0 is the bar length, and coo is the bar cross section. At the initial instant t = 0, tensile loads P(t) are applied to the ends of the bar. The lateral surface is stress-free; body forces are absent. In the actual configuration at instant t -> 0, the bar occupies a domain 12(t) = {0 -< S 1 -< L(t),
(X2,X 3) ~ col(t)}.
Deformation of the bar is determined by the formulas X 1 = k ( t ) X 1,
x 2 = k o ( t ) X 2,
x 3 = k o ( t ) X 3,
(4.4.50)
where x i are Cartesian coordinates in the actual configuration, and k(t), ko(t) are functions to be found. It follows from Eq. (4.4.50) that the radius vectors in the initial and actual configurations equal r0 -- X16'l -t--X26,2 + X36,3,
r(t) = k ( t ) X l ~ l + ko(t)X2~2 + ko(t)X3e3 .
(4.4.51)
Differentiation of Eqs. (4.4.51) implies that V0r(t) -- k(t)6'l~'l + ko(t)(~'2~'2 + ~'3~'3), -
k(t)
ko(t)
V~?(t) = k--~elel + k0(T ) (~'2~'2 + 6'3~'3).
(4.4.52)
According to Eqs. (4.4.52), the Finger tensor F(t) and the relative Finger tensor P~(t, r) are calculated as F(t) = k2(t)g'lg'l + k2(t)(~'2~'2 + 6'36'3),
2
P<>(t, r) = \ k - ~
6'16'1 +
ko(r)
(6'26'2-]" 6'36'3)"
(4.4.53)
Chapter 4. NonlinearConstitutive Models with Finite Strains
224
Combining Eqs. (4.4.53) with the incompressibility condition (4.4.41), we obtain
ko(t) = k-1/2(t).
(4.4.54)
This equality together with Eqs. (4.4.53) yields 1
F(t) "- k2(t)~,l¢,l -{- k--~(~,2~,2 q- ~,3~,3),
(k(t) ) 2
k(~') ~'16'1 + k--~ (~'2~'2 q- 6'3~'3)"
F ° ( t ' ~')= k,k--~
(4.4.55)
Substituting expressions (4.4.55) into the constitutive Eq. (4.4.49), we find the Cauchy stress tensor
O" "-" o-lele 1 + 0-26,26,2 + o'3e3e3,
(4.4.56)
where
trl(t) = -p(t) + ~kZ(t) { [1 + trz(t) = cr3(t) =
--p(t) + ~-~
Qo(t)] - f0 t Qo(t - ~-)k-2(~-) d~-} , [1 + Qo(t)] -
/o
Qo(t - l")k(T)dT
}
.
(4.4.57) The boundary conditions on the lateral surface of the bar read
(4.4.58)
or2(t) = tr3(t) = 0.
It follows from Eqs. (4.4.57) and (4.4.58) that the only nonzero component of the stress tensor equals
Orl(t ) = ~
{ [1 + Qo(t)][kZ(t)- k-l(t)] - fOtQo(t -
k2(t) 1-) I kZ(,r)
k(r) k(t)] d~-}. (4.4.59)
Boundary conditions on the edges of the bar are written in the integral form
f~
O" 1
l(t)
(t) dx2 dx3 = P(t).
(4.4.60)
Substitution of Eq. (4.4.59) into Eq. (4.4.60) with the use of Eqs. (4.4.50) and (4.4.54) yields [1 + Qo(t)]
[k(t) - k-~(t) llft -
Qo(t- r) I ~k ( t ) (-k (\T~) ) 2 1 d T ~-~ = P,(t), (4.4.61)
4.4. A Model of Adaptive Links
225
where P
p,-
~s0 is the dimensionless tensile force, and So is the cross-sectional area in the initial configuration. Given tensile force P(t), Eq. (4.4.61) is a nonlinear Volterra integral equation for the extension ratio k(t). We introduce a new function K(t) = k - l ( t ) and obtain the cubic equation K3 + bl(t)K - b2(t) = 0,
(4.4.62)
where
E
/o
bl(t) = P,(t) 1 + Qo(t) b2(t) =
[
1+
Qo(t) -
Qo(t -
/o'
The functions 1 +
Qo(t -
]1
l")K-l(~-)dl -
~')K2(~-)d~ -
Qo(t) and
1+
,
Qo(t) -
]'
Jo
Qo(t-
~-)K-l(~)d~ -
- Q o ( t - ~') are positive, which implies that for any
t-0, bl(t) -> 0,
b2(t) > 0.
It follows from these inequalities that for a positive tensile load P(t), the algebraic Eq. (4.4.62) has the only real positive solution
I~/b~(t) K(t)=
V
27
bZ(t)b2(t) +
4
+
l~/b~(t)bZ(t)b2(t)
2
-
V
27
+
4
(4.4.63) 2
"
To validate the constitutive model (4.4.49), we compare results of numerical simulation with experimental data for styrene butadiene rubber under the piecewise constant loading
k(t)=
kl, 0 - t < T1, k2, T1 -
(4.4.64)
where T1 = 30 sec, T2 = 80 sec, kl = 1.393, and k2 = 1.819 [see Stafford (1969)]. It follows from Eq. (4.4.59) that O'l(l )
-
-
/x[1 + Q0(t)]
.
k2 - ~
,
0 --- t -< T1,
226
Chapter 4. Nonlinear Constitutive Models with Finite Strains
-
- ~
[Q0(t) - Qo(t - T1)]
/
,
T1 < t -< T2.
(4.4.65)
The shear modulus/x is found from the first equality in Eq. (4.4.65) ( 1 )
-1
,
which implies that/, = 0.605 MPa. The relaxation measure Qo(t) is presented as a sum of two exponential functions [see Eq. (2.3.3)], 2
Qo(t) = - Z
Xm[1 - e x p ( -
(4.4.66)
Tmt)].
m=l
The adjustable parameters Xm and ~/m are found by fitting experimental data, X1 = 0.266,
X2 = 0.079,
3'1 = 0.037 s -1,
3t2 = 0.701 s -1.
The longitudinal stress o'1 is plotted versus time t in Figure 4.4.1, which demonstrates good agreement between experimental data and their prediction. The extension ratio k is plotted versus time t in Figure 4.4.2 for creep test, in which the tensile force P remains constant. The curves presented in this figure are typical of creep in polymeric materials with finite strains [see, e.g., experimental data in Zapas and Craft (1965) and Zdunek (1992)]. The extension ratio k is plotted versus time t in Figures 4.4.3 and 4.4.4 for time-periodic tensile load P , ( t ) = {P0,
0,
2 k T < t < (2k + 1)T,
(2k + 1)T < t < 2(k + 1)T.
(4.4.67)
Results of numerical simulation demonstrate that a steady periodic regime of deformation arises in a viscoelastic specimen. This regime corresponds to periodic excitations, and it is practically independent of the history of loading. Axial elongation of a specimen is determined by the tensile force P0 at the current cycle of loading. About three cycles of loading are necessary to establish the steady regime of extension for the maximal dimensionless tensile force P0 up to unity.
4.5
A C o n s t i t u t i v e M o d e l in Finite V i s c o e l a s t i c i t y
In Section 4.4, a constitutive model was derived for viscoelastic media with finite strains. According to it, a polymeric material is treated as a network with a variable number of elastic links that arise and break owing to micro-Brownian motion. Assuming the processes of breakage and reformation of adaptive links to be stressindependent, we arrive at operator linear constitutive relations in finite viscoelasticity. That model implies that two material functions (the relaxation measure and the strain energy density) determine the viscoelastic response.
227
4.5. A Constitutive Model in Finite Viscoelasticity
1.5
Or 1
0.5
I 0
i
I
i
I
I
I
i t
i
i 80
Figure 4.4.1: The longitudinal stress o-1 MPa versus time t s for uniaxial extension of styrene butadiene rubber. Circles show experimental data presented by Stafford (1969). Solid line shows prediction of the model. As common practice, strain energy density is assumed to be prescribed, whereas its adjustable parameters are found by fitting experimental data. We do not suppose any expression for the strain energy density a priori, and accept the Rivlin hypothesis that potential energy of deformations can be presented as a sum of two adjustable functions of one variable. This approach allows us to predict correctly data obtained in several tests (uniaxial extension, biaxial extension, and torsion) for styrene butadiene rubber and butyl rubber [see Drozdov (1995)].
4.5.1
A Model of Adaptive Links
The operator linear theory of viscoelasticity is based on the concept of adaptive links, where M different types of links exist that arise and break during the loading process. Any link is treated as purely elastic with a strain energy density Wm (m = 1 . . . . , M). The natural configuration of a link that arises at instant ~"coincides with the actual
228
Chapter 4. Nonlinear Constitutive Models with Finite Strains
1.7
1.0
I
I
I
I
I
I
0
I
I
I
t
20
Figure 4.4.2: The extension ratio k versus time t s for a neo-Hookean viscoelastic bar loaded by the dimensionless tensile force P,. Curve 1: P, = 0.1; Curve 2: P, = 0.5; Curve 3: P, = 1.0.
configuration of the medium at that instant. This hypothesis implies that the functions Wm depend on both the initial ~- and current t instants, Wm = Wm(t, T). Breakage and reformation of adaptive links are characterized by the functions Xm,(t, ~). The function Xm,(t, ~) equals the number of links (of mth type) that arise before instant ~and exist at instant t. The strain energy density W(t) of a viscoelastic medium with finite strains is presented in the form
MIXm,(t, O)Wm(t, O) + fOt°~xm*(t, T)Wm(t, T) d T1
W(t) = ~
m=l
O~'/"
(4.5.1) "
We confine ourselves to isotropic and homogeneous media, and suppose that the functions Wm depend on the principal invariants of some relative deformation tensor. Referring to the classification suggested by Zdunek (1992), we distinguish three different cases:
229
4.5. A Constitutive Model in Finite Viscoelasticity
/, f
/" f r
,,... I
0
M, I
I
I
I
M,
M,
M, I
I
t
I
I
200
Figure 4.4.3: The extension ratio k versus time t s for a neo-Hookean viscoelastic bar loaded by the time-periodic tensile force (4.4.67) with P0 = 1.0 and T = 20 s.
1. The invariants I~ (k = 1, 2, 3) of the relative Finger tensor P<> are used as arguments of the functions Wm. This approach was developed by Bernstein et al. (1963), Coleman and Noll (1961), Christensen (1980), Green and Rivlin (1957), Pipkin (1964), and Tanner (1968). 2. The invariants of generalized tensors of strains are used as arguments of the functions Wm. This concept was employed by Bloch et al. (1978), Chang et al. (1976), and Morman (1988). 3. The eigenvalues A~ of the Finger strain tensor P<> are used as arguments of the functions Wm. This method goes back to Valanis and Landel (1967), see also Ogden (1972). It was employed to describe the response in rubberlike materials by Glucklich and Landel (1977), and in polymeric melts by Leblans (1987) and Feigl et al. (1993).
230
Chapter 4. Nonlinear Constitutive Models with Finite Strains
1.5
f
f
f
f L
M. 1.0
I
0
I
I
M. I
I
M. I
I
t
I
I
200
Figure 4.4.4: The extension ratio k versus time t s for a neo-Hookean viscoelastic bar loaded by the time-periodic tensile force (4.4.67) with P0 = 0.5 and T = 20 s.
The first approach seems quite natural from the mathematical standpoint. Nevertheless, it is not widespread in applications, which may be explained by the following reasons: 1. A limited number of expressions for strain energy densities Wm are employed to fit experimental data [see, e.g., their list in Drozdov (1996)]. In some cases, all those expressions are not sufficient to fit experimental data. 2. A limited number of experimental data and specific character of experiments (preferably, uniaxial extension) are not sufficient to choose which constitutive model provides the best agreement with observations [see a discussion of this issue by Alexander (1968)]. 3. Significant discrepancies between experimental data and their theoretical prediction arise when a model developed for one loading program is applied to predict results of another program.
231
4.5. A Constitutive Model in Finite Viscoelasticity
These assertions are shortcomings of the viscoelasticity theory based on the use of the principal invariants of the Finger tensor. We try to argue them in the following by using experimental data obtained by Glucklich and Landel (1977) for styrene butadiene rubber. These shortcomings are typical (to certain extent) of other approaches as well. For example, the method based on the use of generalized tensors of strains is reduced in applications to the expansion of the strain energy density W(t) into a series in multiple integrals (where, as common practice, the nonlinear terms are neglected). As a result, a set of strain energy functions is reduced to several modifications of the Mooney-Rivlin constitutive model, which cannot exhaust the entire spectrum of material responses. As another example, we refer to Feigl et al. (1993) in which significant descrepancies are demonstrated between experimental data for low-density polyethylene and their prediction based on the Valanis-Landel model for shift and elongation tests. One of the reasons for the failure of techniques using the relative Finger tensor is their restriction to a limited set of"classical" strain energy densities. On the contrary, the Valanis-Landel approach postulates a priori that some material functions should be determined by fitting experimental data. We propose to combine these methods and to construct a new model which (i) employs the principal invariants of the relative Finger tensor, and (ii) allows some material functions to be found from observations. Apparently, the latter condition becomes trivial if the functions Wm(I?,12° , 13° ) of three variables are assumed to be measured. However, this procedure requires an amount of experimental data, which is not available. To simplify the problem, we confine ourselves to incompressible media with
13~(t, "r)
= i,
(4.5.2)
which implies that the strain energy densities W m depend on the first two principal invariants
Wm-- Wm(I?,I2<>). We assume additionally that these functions can be presented in the form
Wm(l?,I2 ~)
=
~ W(ml)(I?) q-- ~jl/(2) ,,m t'l kJt2 ),
(4.5.3)
where the functions of one variable W m(k)(l~ ~ , ' k ) are found by fitting experimental data. An assumption similar to Eq. (4.5.3) was introduced by Rivlin (1956) for purely elastic media [see also Stafford (1969)].
4.5.2
Uniaxial Extension of a Viscoelastic Bar
We begin with uniaxial extension of a viscoelastic bar at finite strains. The bar is in its natural (stress-free) configuration. At the instant t = 0, tensile forces P are applied to the ends. Elongation of the bar is described by the equations x 1 = k(t)X 1,
x 2 = ko(t)X 2,
x 3 = ko(t)X 3,
(4.5.4)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
232
where X i and X i a r e Cartesian coordinates in the actual and reference configurations, and k(t) and ko(t) are functions of time to be found. It follows from Eq. (4.5.4) that the relative Finger tensor/~<>(t, ~') equals
P~(t,
\k(T) J
~,1~1 q- ( k ° ( t ) ) 2 ko(?) (g'2g'2 nt- g'3g'3)'
(4.5.5)
where 6'i are unit vectors of the Cartesian coordinates in the initial configuration. Calculation of the principal invariants of the tensor F<> with the use of Eq. (4.5.5) yields
(k(t)) 2
Ii<>(t,T) = \ ~
( ko(t) ) 2 + 2 k, ko(?) '
I 2 ° ( t " r ) = 2 ( k (2k('r)kO('r) t)k°(t))
+ (k°(t)) 4
ko(r)
'
(k(t))2 (k°(t)) 4 I30(t' "r) =
k-~
(4.5.6)
k0('r)
Equations (4.5.6) together with the incompressibility condition (4.5.2) imply that
ko(t) - k-1/2(t).
(4.5.7)
It follows from Eqs. (4.5.5) to (4.5.7) that
P<>(t,
~> =
(~}:' \k(r)
k(r) ~'le'l + ---A-2-(6'26'2 at- ~,3~,3), tort) k(?) + 2 k(t---)'
i2o(t, r) = 2 k(t)
k-~ +
(k0-)) 2 ~ . (4.5.8)
For a viscoelastic medium with the strain energy density (4.5.1) the constitutive equation (4.4.42) reads
6-(t) = --p(t)I + 2 Z
m=l
,(t, O)~)m(t) +
/0
¢9T
]
(t, "r)~)~m(t, "r)d'r .
(4.5.9)
Here 6- is the Cauchy stress tensor, p is pressure, and
Om(t) = Win,1(t)F(t) + Wm,2(t)[I1 (t)F(t) - F2(t)], ~m~
Wm~,
(4.5.10)
4.5. A Constitutive Model in Finite Viscoelasticity
233
where
°~Wm t Wm,k(t)= W ( I 1(),/2(t)),
Wm,k(t,O
1") = 8_-~Wrm(Ii<>(t,1"),I2<>(t, 1")).
(4.5.11)
u1 k
Substitution of expressions (4.5.8) into Eqs. (4.5.9) and (4.5.10) yields (4.5.12)
6-(t) = O'l(t)~l~l + o'2(t)g'2g'2 -+- O'3(t)g'3g'3, where
M
crl (t) = - p ( t ) + 2 Z
{Xm,(t, 0)[Wm,1(t)k(t) + 2Wm,2(t)]k(t)
m=l
+ jot °~Xm*81" (t, ~) o'2(t) = or3(t)= --p(t)
Wm~l(t, ~)~-~ +
[
~(t)
(
+ 2 Z Xm*(t' O) Wm,1(t) -+- k2(t) -1- ~ m=l
1)
]1
Wm,2(t) k(t)
_3r_fotOXm,(t,1-)[Wm~,l(t,1").3r_(Ik(t)~2 + k(1")) ~Wm,2(/,1")] k(1-)/61" ~(~))
a~
~(t)
~
"
(4.5.13) The boundary conditions on the lateral surface read
Or2 = tT3 = O. It follows from these equalities and Eq. (4.5.13) that the Cauchy stress tensor 6" has the only nonzero component
o'1 (t) = 2 Z
Xm,(t, O) Wm,1(t) +
Wm,2(t)
k2(t) -
-]
_Jr_fOt OXm*(t, 1")[Wm'l(t' ~ 1-) + kk(1")W~m,2(t, 1")] [(k(t)~2 k(T) 01" -~ k(-~J - k(t) (4.5.14) We confine ourselves to viscoelastic media with one type of adaptive links, M = 1. For nonaging materials, when the function Xl,(t, 1") depends on the difference t - 1"only, we write Xl,(t, 1") =
XI,(0, 0)[1
+ Qo(t - 1")],
(4.5.15)
where XI,(0 , 0) is the initial number of adaptive links and Qo(t) is a relaxation measure.
Chapter 4. Nonlinear Constitutive Models with Finite Strains
234
We choose three well-known expressions for the strain energy density W1 (per unit link): • The neo-Hookean material
C1 Wl ~- 2Xl,(0, 0) (I1 -- 3).
(4.5.16)
• The Mooney-Rivlin material 1
Wl - 2Xl,(0, 0) [Cl(I1 - 3) + 2C2(I2 - 3)].
(4.5.17)
• The Knowles material C1 W1
--"
2bXl,(O, O)
{[
1 + b(I 1 - 3 )
n
I n- 1 } .
(4.5.18)
Parameters C1, C2, b and n are found by fitting experimental data. Substitution of expressions (4.5.16) to (4.5.18) into Eq. (4.5.14) implies that in relaxation tests with
k(t) =
1,
t-<0,
A,
t > O,
the longitudinal stress (7"1(t) is calculated as follows: • For the neo-Hookean material
o-l(t) = Cl[l + Qo(t)] ( A 2 -
l)
.
(4.5.19)
2C2)(1) C1 + - ~ A2 _
(4.5.20)
• For the Mooney-Rivlin material O'l (t) = [1 + Q0(t)]
(
• For the Knowles material
( O'l(t)=Cl[l+Q0(t)]
1 + .b n
)]n-i ( 1 ) A2+ . 2. A
3 .
A2
.
(4.5.21)
Experimental data for styrene butadiene rubber together with their theoretical predictions with the use of Eqs. (4.5.19) to (4.5.21) are plotted in Figure 4.5.1. The following conclusions may be drawn: 1. The neo-Hookean model does not describe adequately the experimental data. For example, observations demonstrate a convex dependence of the longitudinal stress on the extension ratio, whereas the neo-Hookean model implies a concave dependence.
4.5. A Constitutive Model in Finite Viscoelasticity
235 1
•""
1.2
2
.G:"""
oOOllO°
--
.el ~'' O"1
-
oo:o ° •
..:::::."...." o...:.'."
-
9::"
eee • e ee
0:::" ° 885
o• o• o• o• o•
•
Oo oo
--
ee
::b
"
•
° •
•
e
•
•
•
o•
ooo ° ° •
e
o~ °
0o ° •
ee
I!..
•
•
e•
oo~o•
--
1
."
••
° ~O°
__
-
.'"
ooo °
_
"::i""" I
I
I
I
I
I
I
)t
I
I
3
Figure 4.5.1: The principal stress o-, (MPa) versus the extension ratio ~ for styrene butadiene rubber. Circles show experimental data obtained by Glucklich and Landel (1977) after ~- = 10 rain of loading. Dotted lines show their approximation. Curve 1: the neo-Hookean model (4.5.19) with C1[1 + Q0(~')] = 0.180 MPa; Curve 2: the Mooney-Rivlin model (4.5.20) with C1 = 0 MPa and C2[1 + Q0(l")] = 0.219 MPa; Curve 3: the Knowles model (4.5.21) with C1[1 + Q0(~)] = 0.322 MPa, b -- 0.06, and n = 0.21.
2. Both the Mooney-Rivlin and the Knowles models show fair agreement between experimental data and their prediction. Despite the difference between these models (the former model depends on the second principal invariant 12, whereas the latter depends on the first principal invariant 11), the material responses predicted by these models are extremely close to each other. The latter means that uniaxial extension is not sufficient to choose an adequate model, and more complicated tests are necessary.
Chapter4. NonlinearConstitutiveModels with Finite Strains
236 4.5.3
Biaxial Extension of a Viscoelastic Sheet
Let us consider biaxial extension of a viscoelastic plate with length/1, width/2, and thickness h. The plate is in its natural state and occupies the domain h2
0 ~ X 1 ~ 11, 0 ~ X 2 ~ 12,
X3
h2 } "
At the instant t = 0, tensile distributed forces are applied to edges of the plate and cause its deformation
xl "- kl(t) X1, where
x2 = k2(t)X 2,
x 3 = k3(t)X 3,
(4.5.22)
ki(t) are functions to be found. Equations (4.5.22) imply that FO(t,T) = (kl(t)) 2 kl(r)
(k2(t)) 2 (k3(t)) 2 g'1g'1 + \ k2(r) g'2e2+ \k3(r) g'3e3'
Ii<>(t,T) = (kl(t)) 2 km(r)
(k2(t)) 2 +
kz(r)
(k3(t)) 2 k3(r)
+
'
(kl(t)k2(t))2+ [(kl(t))2+ (k2(t))21 I~(t, r) = kl(r)k2(r) k, kl(r) \k2('r) I3<>(t,r)= (kl(t)k2(t)k3(t) k,~-1( ~ )
)2
(k3(t)) 2 k3('r) '
"
(4.5.23)
It follows from Eqs. (4.5.23) and the incompressibility condition (4.5.2) that 1
k3(t) =
kl(t)k2(t~)"
(4.5.24)
Combining expressions (4.5.23) and (4.5.24), we find that
el(T)
I?(t'~')=(kl(t)) 2kl(g)
elel + ~k2(g)
e3e3'
+ (k2(t)) 2 + (kl('r)k2(T)) 2 kl(t)k2(t) ' k2(r)
I2<>(t,T ) - - ( k l ( T ) ) 2 kl(t)
e2e2 + ~, k l l ~
(k2(g)) 2 +
k2(t)
(kl(t)k2(t)) 2 + \kl(~)
"
(4.5.25)
Substitution of Eq. (4.5.25) into the constitutive equations (4.5.9) and (4.5.10) implies formula (4.5.12) with
M
oq(t) = -p(t) + 2 _ Xm.(t, O) Wm,l(t)k2(t) + Wm,2(t) k2(t)k2(t) +
4.5. A Constitutive Model in Finite Viscoelasticity +
lot OXm*(t, r) [Wm, ~ 1(t, r) (kl(t)) 2 Or
kl(r)
((kl(t)k2(t)) 2 (k2('r))2)
0 + Wm,a(t, r) o'2(t) =
(
')
Xm,(t, O) Wm,l(t)k2(t) + Wm,g(t) k2(t)k2(t) + k - ~
Or
k2(~')
--p(t) + 2 Z +
+ k, k2(t)
m=l jo't~Xm*[w ~ 1(t, r) (k2(t) ) 2 (t, r) m,
+ Wm<>2(t,r) o'3(t) =
kl (r)ka(r)
E
-p(t) + 2 Z +
237
kl(r)k2(r) Xm,(t, O) Wm,l(t) k2(t)k2(t)
m=l ~ot°~Xm*[~
Or (t, r) Win,l(t, r)
+Wm<>,2(t,r)
(()2
kl ('r) kl(t)
+
')]
-k Wm,2(t) k-~ q- k-~
(kl (T)k2(T))2 kl(t)kz(t)
()2)
ka(r) k2(t)
dr}.
(4.5.26)
X3 =
It follows from the boundary conditions at the surfaces +_h/2 that o'3 = O. Excluding pressure p from this equality and Eqs. (4.5.26), we obtain
M o-l(t) = 2Z{Xm,(t,O)[Wm,l(t) + Wm,2(t)k2(t)] k2(t)m=l
1]
k2(t)k2(t)
E [(k,(t)):_kl.(T)k2(T)2]
-'1-fot OXm*(t, r) Or
kl(r)
o-2(t) = 2 Z
kl(t'k2(t')
dr},
{Xm,(t, O)[Wm,l(t) + Wm,z(t)kZ(t)]
m=l
+
Wm~l(t, r) + Wm,2(t,r) k2(r)
(t) -
Or (t, r) Wm,l(t, r) + Wm,2(t, 7") kl ('r)
1]
k~(t)k~(t)
238
Chapter4. NonlinearConstitutiveModels with Finite Strains kl (t)k2(t) )
"
(4.5.27)
Given extension ratios kl (t) and k2(t), Eqs. (4.5.27) determine nonzero components of the Cauchy stress tensor 6"(0. Let us consider relaxation tests with ki(t) =
1,
t_<0,
hi,
t > O.
It follows from Eqs. (4.5.15) and (4.5.27) that for nonaging viscoelastic media with M=I, trl (t) = 2X1,(0, 0)[ 1 + Qo(t)] trz(t) = 2X1,(0, 0)[1 + Qo(t)]
OWl
011
"+ OWlx
011
+ OWl A 012
/~12-
012
2 2
'
A mA 2
21)( A2
-
1)
2 2 A 1A 2
"
(4.5.28)
To fit experimental data for biaxial extension, it is quite natural to begin with strain energy densities (4.5.16) to (4.5.18) with the material parameters found for uniaxial extension. The corresponding results are presented in Figures 4.5.2 and 4.5.3 for the Mooney-Rivlin and Knowles materials. The figures demonstrate significant discrepancies between experimental data and their theoretical predictions. The same is also true for the neo-Hookean model (4.5.16), but an appropriate figure is omitted. The following conclusions may be drawn: 1. Simple uniaxial experiments are not sufficient to validate constitutive models in nonlinear viscoelasticity. 2. The standard constitutive equations do not adequately describe both uniaxial and biaxial extension, and more sophisticated models should be developed. We assume that the function of two variables W1(I1, I2) can be reduced to several functions of one variable, which are found by fitting experimental data. Four different versions of this reduction are proposed: (i) W1 depends on the first principal invariant I1 only. (ii) W1 depends on the second principal invariant I2 only. (iii) W1 depends on the ratio similarity condition).
I1/If
only, where v is a material constant (a self-
(iv) W1 is presented as a sum (4.5.3) of functions h,~k) that depend on the kth principal invariant Ik only. To verify hypotheses (i) to (iv), we find derivatives of the strain energy density from Eqs. (4.5.28)
239
4.5. A Constitutive Model in Finite Viscoelasticity
1.2
©
O"i
Oo. •
•
•
• •
•
0
•
oo
(~o °
oo °
•
•
•
o•
• •
o•
o°
•
• o°
•
•
• O
• •
oo
•
•
o°
gO
o°
•
o°
•
o•
•
•
•
•
•°
°q °
oo ° o•
• ° O
• •
•
8~°"
w,-
o•
o°
• •
oo ° oo
o• o•
o•
I
I
I
I
I
I
I
I
1
I
I
I
A1
2.2
Figure 4.5.2: The principal stresses 0-1 and 0"2 (MPa) versus the extension ratio /~1 for styrene butadiene rubber. Circles show experimental data for/~2 ~'~ 1 obtained by Glucklich and Landel (1977) after r = 10 min of loading. Unfilled circles: 0"1; filled circles: 0-2. Dotted lines show prediction of experimental data by the Mooney-Rivlin model (4.5.17). Curve 1: 0-1. Curve 2: 0-2.
OWl 0/1 OWl _ 0/2
2X1,(0, 0)[1 + Qo(t)l(A 2 - A2) [ AI-~A22Z 1 ~ 2~.:~
[
2Xl,(O, 0)[1 + Qo(/)]O t2 - A 2 )
Or2(t)
/~12/~24_ 1
2 4 A1A 2 -- 1 Orl(t)
]
~14/~22_ 1 "
(4.5.29)
First, we accept hypothesis (i) and assume that the function W1 depends on 11 only. Equations (4.5.29) imply that Or1 0"2
_
4 2
~.1~.2
--
1
2 4 -- 1 /~'1/~'2
(4.5.30)
240
Chapter 4. Nonlinear Constitutive Models with Finite Strains 1.2
o- i O
_
0
-
ooo°°°°°°°°°°°°°°°°i
0
ooo°,°°°°°°
_
ooooo°
_
° oo°°
0
goDgo 0
•~o °
~ -
o• ° oo°O
•
• •
•
•
o°°°°
gig °°°
~
o
~
o
~
~
~
~
o
~
~
~
~
o
~
...:::i........................ °°° I
i
I
I
I
I
I
I
I
I
A1
I 2.2
Figure 4.5.3: The principal stresses 0-1 and 0" 2 (MPa) versus the extension ratio •1 for styrene butadiene rubber. Circles show experimental data for ~ 2 -" 1 obtained by Glucklich and Landel (1977) after T = l0 min of loading. Unfilled circles: 0"1; filled circles: 0"2. Dotted lines show prediction of experimental data by the Knowles model (4.5.18). Curve 1: 0"1. Curve 2: 0"2.
Second, we consider hypothesis (ii) and assume that the function W1 depends on 12 only. It follows from Eqs. (4.5.29) that ~.2(~.?~.22- 1) o2 - A12(A12A4 - 1)"
O" 1
(4.5.31)
Third, we accept condition (iii) regarding self-similarity of the strain energy density. According to Eqs. (4.5.29), 4 4 O'1(/~2/~4 -- 1 ) - o'2(AaA 2 - 1) v = 1 + A2A4 + A4A2 o.lA2(A~)t4 _ 1 ) - o'2A~(A4A2 - 1)"
(4.5.32)
Experimental data obtained by Glucklich and Landel (1977) demonstrate that Eqs. (4.5.30) to (4.5.32) are far from being fulfilled. This is a reason why the neo-Hookean
4.5. A Constitutive Model in Finite Viscoelasticity
241
model (4.5.16) and the Knowles model (4.5.18) based on the hypothesis (i), as well as the Mooney-Rivlin model (4.5.17) satisfying assumption (ii) fail to predict biaxial extension. Finally, we consider assumption (iv). According to this hypothesis, Eqs. (4.5.29) can be treated as two independent equations for determining the functions W~11)(I1) and w. 1(2) (/2). The corresponding dependences are plotted in Figures 4.5.4 and 4.5.5. Figure 4.5.4 shows that the function
aWl
0)[1 + Q 0 ( r ) ] ~
(I) 1 = X I , ( 0 ,
tgI 1
depends weakly on the second invariant 12. Deviations from the approximating curve, which correspond to both errors in measurements and the effect of argument 12, are 0.2 _D m
© g. °o ° °oo? •
°°Oo ° "°°°Ooo 0
•
°°°OOooooo •
,
..... .~, ............ ,~ ................. °°°OOOoooooooooooooo
o
I
3
I
I
I
I
I
n
ll
I
i
6
Figure 4.5.4: The function ~1 versus the first invariant I1 of the Finger tensor. Circles and asterisks show experimental data for styrene butadiene rubber obtained by Glucklich and Landel (1977) after r = 10 min of loading for different programs of extension. Dotted line shows approximation of experimental data by the power-law function (4.5.33) with A~ = 0.0634 and cq = 0.389.
242
Chapter 4. Nonlinear Constitutive Models with Finite Strains
02
~2
o
°°°°°°OOoooooo° °°~OOoooo • °°°°°°°°°Oo° o
°°°°°°°°~y°°o°
°°°°°°OOooooo
•
I
I
I
, ........... °Oo° 0o Oo
•
I
I
0 °°°°°°OOoooooq
I
I
3
I
/2
I
6
Figure 4.5.5: The function (I) 2 v e r s u s the second invariant/2 of the Finger tensor. Circles and asterisks show experimental data for styrene butadiene rubber obtained by Glucklich and Landel (1977) after T = l0 min of loading for various programs of extension. Dotted line shows approximation of experimental data by the linear function (4.5.34) with B~ = 0.2259 and B~ = 0.0243.
less than 10% (taking into account that 12 changes in tests from 3.04 to 5.81). This allows the dependence of ~1 on 12 to be neglected, which is equivalent to assumption (iv). The function ~1 can be approximated by the power function XI,(O, 0)[1 + Qo('r)] OWl _ all
A~
.
(4.5.33)
(I1 - 3) ~
The least-squares method implies that A~ = 0.0634 and c~1 = 0.389. Formula (4.5.33) can be employed sufficiently far away from the point I1 = 3, which corresponds to the initial configuration. This means that our approximation is valid only for large deformations and cannot be used at small strains (in the latter case the derivative 0W1/~I1 should tend to a finite constant as I1 ---* 3).
4.5. A Constitutive Model in Finite Viscoelasticity
243
Figure 4.5.5 shows that the function
aw~
(I) 2 - - X I , ( 0 , 0 ) [ 1
q-
Q0(l")]--
depends extremely weakly on argument I1, and this dependence can be neglected as well. For example, both errors of measurements and the dependence on the first invariant I1 lead to deviations of ~2 from the approximating curve of about 10%. Up to this level of accuracy, the function ~2 can be approximated by the linear function XI.(0, 0)[1 + Q0(~')]
awl = B~ - B ~ I 2 . ai2
(4.5.34)
The least-squares method implies that B~' = 0.2259 and B~ = 0.0243. It is worth to compare Eqs. (4.5.33) and (4.5.34) with Rivlin's results for purely elastic rubber specimens [see, e.g., Rivlin (1956) and Rivlin and Saunders (1951)]. In that experiment, the function 03W1/0311 was found to be practically constant, whereas the function 03W 1//0312 decreased according to the power law. As is well known [see, e.g., Lurie (1990)], negativity of the second derivative of strain energy density may lead to the loss of ellipticity of the governing equations, and, in turn, to the material instability. Explicit sufficient conditions for ellipticity and strong ellipticity of nolinear partial differential equations were derived by Zee and Sternberg (1983). Numerical analysis shows that expressions (4.5.33) and (4.5.34) with parameters found by fitting experimental data for styrene butadiene rubber satisfy those conditions both for uniaxial and biaxial extension. The next step in our analysis consists in validation of Eqs. (4.5.33) and (4.5.34) for other types of loading, which have not been used in constructing the model. For this purpose, we employ biaxial extension of a plate with equal extension ratios )kl - /~2 = /~ and uniaxial extension of a bar. The results for biaxial extension are plotted in Figure 4.5.6. They demonstrate fair agreement between experimental data and their prediction with the use of Eqs. (4.5.3), (4.5.33), and (4.5.34). The results for uniaxial extension of a bar are plotted in Figure 4.5.7. This figure demonstrates fair correspondence between experimental data and their prediction, except for a narrow region of relatively small deformations, where 1 -< /~1 < 1.5. TO exclude singularity in Eq. (4.5.33), as well as to diminish discrepancies between experimental data and their prediction at small strains, more accurate approximation of the derivative 03W1/0311 is necessary. For example, instead of Eq. (4.5.33), one can employ the dependence XI.(0, 0)[1 + Q0(~')] 03w1 _ o3/1
A~' A2 + (I1 - 3) '~
(4.5.35)
A~' + A3, Ae + (I1 - 3)
(4.5.36)
or
XI.(0, 0)[1 + Q0(~)] 03W1 o311
--
244
Chapter 4. Nonlinear Constitutive Models with Finite Strains
oeooeoooo°o°eO°°°°~ ~ oo°°°°
0
ooOo°°
(~
ooOo°°
O" •°
•°
(~
0
0•
e e
•°
0
•
"O-
I
I
I
I
I
I
1
I )t
I
I 1.5
Figure 4.5.6: The principal stresses 0-1 = 0 " 2 = 0" (MPa) versus the extension ratios )t 1 = } k 2 "-" )k for styrene butadiene rubber. Circles show experimental data obtained by Glucklich and Landel (1977) after T = l0 min of loading. Dotted line shows their prediction with the use of the model (4.5.3), (4.5.39), and (4.5.40).
where Ai and c~ are adjustible parameters [see Alexander (1968)]. We do not employ dependences (4.5.35) and (4.5.36), since they lead to an increase in the number of material parameters to be found. Confining ourselves to the model (4.5.33) and (4.5.34), we conclude that this model (which characterizes the strain energy density W1 by only four material parameters A~', c~1, B~', and B~) correctly predicts both uniaxial and biaxial deformations of styrene butadiene rubber. The relaxation measure Qo(t) is determined by using experimental data for biaxial extension with equal extension ratios ~1 = ~2 = )t for )t = 1.461 (see Figure 4.5.8). The obtained results are verified for other ~ values (see Figure 4.5.9). It follows from Eqs. (4.5.29), (4.5.33), and (4.5.34) that or(t) =
1 + Qo(t) 1 + Q0(r)
F(•),
(4.5.37)
245
4.5. A Constitutive Model in Finite Viscoelasticity 5 ooO~n
1.2 oooooOOo ° ° ° °
O" 1 ° oo°
oo
oo
oo
oo
oo
oo
oo
o oo ° ° ° ° °
oooooOO°°°°
ooOOO°OO°O°O°°°O°O°
eeeeeeee
eeeeee
eeeoeeOe°
ooOo ° ° ° ~ 0
e~eee
oo
0
o
ooooOO°ioooOO ~ o o°
0
0000000 o ° ° ° ° ° ° ° ° ° 0 ° ° ° ° ° ( oooOOo ° ° ° ° °
_oooooooooooooo°°°°°(
^oooooOO°°°°
ee ee
^ooO oo° oo oo e'ea" o o,o ou ^oooOOou o o° ee e o o°° ooOo°u oO°o o°° e e ^oO°°~o o o ° ° ° o0
o °o
_o 00
~
~-.~eeee
o o ~ ooOv
o° o e e O o o° o o° o o ° e e _o o o o° o o • oo oo o o e e ^o o o o o o e e ^oO~oO Oo e • oO~ooOo o~ 0 o ° Qo eeo 0 _~Ue°oOo°
Is~l 88° ~;~o o
Oeeoo o oo oo
Ue
o
8o
0
I
I
I
I
I
I
1
I A1
I
I 3
Figure 4.5.7: The principal stress or1 (MPa) versus the extension ratio )q for styrene butadiene rubber. Circles show experimental data obtained by Glucklich and Landel (1977) after T = l0 min of loading. Curve 1: approximation of experimental data with the use of the model (4.5.3), (4.5.39), and (4.5.40). Curves 2 to 5: deviations from the predicted values caused by discrepancies in experimental data presented in Figure 4.5.4 (curves 2 and 3) and in Figure 4.5.5 (curves 4 and 5).
where F(~) = 2
A~' (2~2 + / ~ - 4
_
3)~1
+ [B~ - B~ (2~ -2
+/~4)]/~2 } (/~2 _ X-4).
Equation (4.5.37) implies that F 0 t ) coincides with o-(~-). Data provided by Glucklich and Landel (1977) show that this is true with a high level of accuracy. For example, for )t = 1.461, we have F(A) = 0.965 MPa according to the model, and o'(~') = 0.989 MPa in experiments. The difference between these amounts is less than 3%. Taking
246
Chapter 4. Nonlinear Constitutive Models with Finite Strains
0.4
-a0
ooooooOOo°°°°°°°°°°°°°°°°°°q( oooooooooO°~°°°°° 0
© © Oo.O"°'
o• --
oo°
o•
~oo°°°°°°°°°°°°°°°e
ooo°°°°
Qo •
m .b
m
o
n
i
i
I
i
i
0
n
t
n
n
30
Figure 4.5.8: The relaxation measure Q0 versus time t min for styrene butadiene rubber. Circles show experimental data obtained by Glucklich and Landel (1977) for = 1.461. Dotted line shows approximation of experimental data by the power-law function (4.5.39) with X = 0.126 and 3' = 0.297.
this assertion into account, we write Eq. (4.5.37) as follows: or(t)
or(r)
-
1 + Qo(t)
1 + Q0(r)
.
(4.5.38)
Experimental data imply that or(0) = 1.34 MPa. Since Q0(0) = 0, it follows from Eq. (4.5.38) that Qo(t)-
or(t)
or(0)
- 1.
The relaxation measure Q0 is plotted versus time t in Figure 4.5.8. The dependence Qo(t) is approximated by the power function Qo(t) = - x tv,
(4.5.39)
247
4.5. A Constitutive Model in Finite Viscoelasticity 1.2 o
mo o 0 °o 0 °o° O0 0 °°°Ooo "...
O0
0
°°°°°°°°°°°°°°OOoooooOooOooooooOOOooo 0 ( °°°°°°°°oooooOOOoooooo °°°°°°°ooooooooooOOoo
_0 o -
o
o
o
•
o
OOoo
°°°°OOoooo° °°°°°°°°°°°OooeoOOOOOOoooo °°°°°°°*OOoooooOOooooo
000000000000000000000000000 000000000
0.4
I
I
I
0
I
I
I
I
I
t
I
30
4.5.9: The principal stress ~r (MPa) versus time t min for styrene butadiene rubber. Curve 1: ~ = 1.376. Curve 2: ~ = 1.192. Circles show experimental data obtained by Glucklich and Landel (1977). Dotted lines show prediction of the model (4.5.3), (4.5.39), and (4.5.40).
Figure
where the adjustable parameters X and 3/are found by fitting experimental data. The least-squares method implies that X = 0.126 and 3' = 0.297. Figure 4.5.8 demonstrates fair agreement between experimental data and their prediction with the use of Eq. (4.5.39). Substitution of expression (4.5.39) into Eqs. (4.5.33) and (4.5.34) yields
OWl XI,(0, 0) 011
A1 (I1 - 3) a '
OWl XI,(0, 0) 012 = B1 - B212,
(4.5.40)
where A1 = 0.0845, B1 = 0.3011, and B2 = 0.0324. After determining parameters of the model, we compare results of numerical simulation with experimental data obtained by Glucklich and Landel (1977) for biaxial extension with )tl = / ~ , 2 -- /~- (see Figure 4.5.9). This figure demonstrates fair agreement between experimental data and their prediction, especially for relatively large deformations. For example, for ~ = 1.376, the maximal deviation of predicted
Chapter 4. Nonlinear Constitutive Models with Finite Strains
248
values from experimental data equals 9.5%. For relatively small deformations the discrepancy increases up to 19.5% for A = 1.192. Accounting for relatively large errors of measurements, we treat the accuracy of predictions as acceptable.
4.5.4
Torsion of a Viscoelastic Cylinder
Let us consider torsion of a circular cylinder with length I and radius R0. The cylinder is in its natural state. At the instant t = 0, torques M and compressive forces P are applied to the ends of the specimen. Under their action the following deformation occurs: r = R,
0 = 19 + c~Z,
z = Z.
(4.5.41)
Here {R, 19, Z} and {r, 0, z} are cylindrical coordinates in the initial and actual configurations with unit vectors ~R, ~o, ~z and G, ~0, ~z, respectively; c~ = c~(t) is twist per unit length. Torsion of a circular viscoelastic cylinder was considered by Duran and McKenna (1990), McKenna and Kovacs (1984), McKenna and Zapas (1979), Santore and McKenna (1991), Waldron and Wineman (1996), and Wineman and Waldron (1993, 1995). It follows from Eqs. (4.5.41) that
F¢(t, 7") = e.re.r nt- [1 + (c~(t)
-
ot(7"))2R2]e.oe.o nt- e.ze.z
+ [c~(t)- ~(7")]R(P.O~z + eze.o).
(4.5.42)
Calculation of the principal invariants of the relative Finger tensor F<> with the use of Eq. (4.5.42) yields
Ii<>(t, 7") = I2<>(t,7") = 3 + [ a ( t ) - a(7")]2R 2.
(4.5.43)
Substitution of expressions (4.5.42) and (4.5.43) into Eqs. (4.5.9) and (4.5.10) implies that
o-
=
O ' r r e . r e . r "Jr-
orooeoe o + Orzz~.z~z + OrOz(~O~z + ezeo),
(4.5.44)
where M
Orrr(t) -- - p ( t ) + 2 Z{Xm,(t,O)[Wm, l(t) + (2
+
ot2(t)R2)Wm,2(t)]
m=l
+
(t, 07"
7 . ) [ W m < > l ( t , 7") -+'
(2 + (c~(t)- ot(7"))2R2)W~2(t, 7")]dT.
M
OrrO(t ) = 2r Z { X m , ( t , m=l
O)ot(t)[Wm, l(t) + Wm,z(t)]
'
'
4.5. A Constitutive Model in Finite Viscoelasticity
+
249
t °3Xm*(t, l")[c~(t) - ot('r)][Wm<>l(t, I") + Wm2(t, ~')] dl" I ,
fO
0"r
'
'
M
troo(t) = ~rr(t)
-t- 2R 2 Z (Xm,(t, O)ot2(t)Wm,1(t) m=l
+
/o
(t, T)(O~(t) -- Og(T)) 2 Wm<>,(t, l "r) d r
0r
}
,
M
trzz(t) = °rrr(t) - 2R2 Z
{Xm,(t, O)ot2(t)Wm,2(t)
m=l
+
/ot°m"
(t, r)(~(t)
0T
-
~') dr ~(~.))2 W~,2(t, ~
}
.
(4.5.45)
We integrate the equilibrium equation
tgO'rr Or
+
Orrr
-
-
r
0"00
= 0
(4.5.46)
from r to Ro and use Eqs. (4.5.41), (4.5.45), and the boundary condition O'rr {R=Ro = 0. As a result, we obtain
Orrr(t) =
-2 ~
,(t,
O)ot2(t)Wm,1(t)
m=l
+
0T
(t, r)(a(t) - O~(T)) 2 W~m,1(t, r) d
pdp.
(4.5.47)
We now calculate the torque M and the compressive force P according to the formulas"
M = 27r
~oR°trOzR2 dR,
P = -27r
fro"R°trzzRdR.
(4.5.48)
Substitution of expressions (4.5.45) and (4.5.47) into Eqs. (4.5.48) implies that
M(t) = 4~r Z
Xm,(t, O)oz(t)[Wm,1(t) + Wm,2(t)]
m=l
+
01"
(t, ~')[c~(t) - ~(-r)][W~l(t, "r) + W~2(t, ~')]d~" g 3 dR, '
M f0Ro{ X~,(t, 0)c~2(t)[W~,l(t) + 2Wm,2(t)]
P(t) = 2~r Z
m=l +
----~-T (t, T) [if(t) -- if(T)] 2 [Wm~,l(t, T) + 2Wm~,2(t, r)] d r
R 3 dR.
(4.5.49)
Chapter 4. Nonlinear Constitutive Models with Finite Strains
250
Given deformation program ct = c~(t), Eqs. (4.5.49) determine the torque M(t) and the compressive force P(t). For relaxation tests with
a(t) = ~0, t <-- O, L c~, t > 0 , formulas (4.5.49) are simplified
M(t) =
M Xm,(t, O) foROL[ t'Jtlt~Wm (3 +
2R2,3 + 2g2)
m=l
+ OWm o912(3 +
o~2R2, 3 + c~2R2)] R 3 dR,
P(t) = 27ra: ZM Xm,(t, 0) fOR°[OWm[ Oil (3 + a2R2, 3 + a2R 2)
m=l
1
+ 2--~2 (3 + c~2R2, 3 + aZR 2) R 3 dR.
(4.5.50)
Given functions M(t) and P(t), Eqs. (4.5.50) may be treated as integral equations for determining strain energy densities Wm(I1, Ia). This problem is ill-posed, and it is difficult to presume that even for M = 1, a regular solution can be obtained. However, a significant advance can be done for moderately large strains, in which the twist O - ~l is sufficiently small. Expanding the functions Wm into series in I1 - 3 and I2 - 3 and neglecting terms of the second order of magnitude compared to O, we arrive at the formulas O M M(t) = 2J Z Z Xm,(t, O)(Cm,1 + Cm,2),
m=l
P(t) = J
Xm,(t, 0)(Cm,1 +
2Cm,2),
(4.5.51)
m=l where
"it 4
J -- -~Ro,
Cm,k -
c)Wm --~k (3, 3),
k - 1, 2.
For M = 1, Eqs. (4.5.51) coincide with governing equations for torsion of a MooneyRivlin viscoelastic circular cylinder. Setting
C1 CI,1 -" 2Xl,(0, 0)'
C2,1
C2 XI,(0, 0)'
where Ck are parameters in Eq. (4.5.17), we rewrite Eqs. (4.5.51) as
M(t) = 2J-;-[1 + ao(t)l
+G
9
4.5. A Constitutive Model in Finite Viscoelasticity
P(t) = J
251
[] + Q0(t)]
~
+ 2C2
.
(4.5.52)
The relaxation measure Qo(t) is found by using experimental data for torsion at small strains. In the latter case, the torque M(t) is expressed in terms of twist O as
M(t) = GJ ~ [1 + Q0(t)], where G is the shear modulus. It follows from this equality that
ao(t)-
M(t)
M(0)
- 1.
(4.5.53)
Dependence Q0 = Qo(t) for butyl rubber at O = 0.00181 is plotted in Figure 4.5.10. 0.4 ooooooooooooO°°°°°°°°' -
-ao
o
............ ~
o
ooOO~O o°°°°°° ................. ~::8o08: ......................................................... 0 oo o°°e o°°°°°°e
i
° °°
Oooo°°°° 0 o°°
• •
•
Q"
6" 9 G
I
0
I
I
I
I
I
I
t
I
1
20
Figure 4.5.10: The relaxation measure Q0 versus time t s for a butyl rubber specimen in torsion at small strains. Circles: experimental data obtained by Hausler and Sayir (1995) for twist angle O = 0.0081. Dotted lines: the exponential function (4.5.54) with X1 = 0.318 and 71 = 1.610 (curve 1), and the power-law function (4.5.55) with X2 = 0.205 and 72 = 0.212 (curve 2).
Chapter 4. Nonlinear Constitutive Models with Finite Strains
252
Experimental data are approximated by the exponential function
Qo(t) = - X l [ 1 - exp(-3,1t)],
(4.5.54)
and the power-law function
Qo(t) = -)(2 ty2
(4.5.55)
with adjustable parameters Xk and ~/k. Large discrepancies can be seen between experimental data and their prediction by the exponential function (4.5.54). The power-law function (4.5.55) with )(2 = 0.205 and ~/2 = 0.212 provides good agreement with observations. To find the adjustable parameters C1 and C2, we approximate the values of M and P obtained at ~" = 2 s by the linear function M(~') = aMO
(4.5.56)
10
O•
gO°°C~
M
o °
O•
O
°• 0
I
I
•
I
I
I
I
I O
I
I 0.2
Figure 4.5.11: The torque M N.m versus twist angle O. Circles show experimental data obtained by Hausler and Sayir (1995). Dotted line shows their approximation by the linear function (4.5.56) with aM = 57.68 N.m.
253
4.5. A Constitutive Model in Finite Viscoelasticity
150 o
,°C)
o°
°oo°
0
~
......... ~......
°°°°°°°°°°oo°°°°°°°°e n
n
u
n
0
i
i
O
i
i
0.2
4.5.12: The compressive force P N versus twist angle O. Circles show experimental data" obtained by Hausler and Sayir (1995). Dotted line shows their approximation by the quadratic function (4.5.57) with ap = 3857.55 N.
Figure
and by the quadratic function P('r) = a e O 2.
(4.5.57)
Experimental data and their predictions are plotted in Figures 4.5.11 and 4.5.12. These figures demonstrate fair agreement between experimental data for the torque M and the linear dependence (4.5.56) with aM = 57.68 (N.m). Fitting of experimental data for the compressive force P is worse. The least-squares method provides the value ae = 3857.55 N. Results presented by Hausler and Sayir (1995) demonstrate large deviations between data for different specimens. The discrepancies between experimental data and their approximation by the quadratic function (4.5.57) are located in the interval of errors in measurements (at least for O > 0.05). The torque M and the compressive load P are plotted versus time t in Figures 4.5.13 and 4.5.14. These figures demonstrate fair agreement between experi-
Chapter 4. Nonlinear Constitutive Models with Finite Strains
254 15
0
M
".0
_
0
O°°°OOoooo 0
............ °OOooooo,
0
°°°°°OOoooooooooo °°°°°°°OOOooooooooooooo,
0
0
0
°°°°°°°OOOoooooooooooooo,
0
0
0
% --
000o600000000000
o
I
0
"'~ ........ "6"........... "o"............. e~............ ~ .......... ~. ......
I
I
I
I
I
I
t
I
I
20
Figure 4.5.13: The torque M N.m versus time t s. Circles show experimental data obtained by Hausler and Sayir (1995). Dotted lines show prediction of the model (4.5.52). Curve 1: T -- 0.102. Curve 2: T = 0.172.
mental data and their prediction with the use of the model (4.5.52). In particular, excellent fitting is obtained for torque M at a relatively small twist (7 = 0.102). Prediction of compressive load P leads to small discrepancies between theory and experiments, which can be explained by errors in measurements of P at instant r [see Figure 4.5.12]. Deviations of experimental data from their prediction by our model and by the model derived by Hausler and Sayir (1995) are extremely similar. The difference consists in the number of adjustable parameters in these models. The proposed model contains only four parameters: X2, 3'2 for the relaxation measure Qo(t), and C1 and 6'2 for the strain energy density W1(I1,I2), whereas the Hausler-Sayir model contains 10 parameters that are determined by fitting experimental data.
Bibliography
255
150 © P
e~ ~dOo °°°°Ooooo ° 0
°°°°°°°°°°o OOooo
0
mo
o
0
......... "............... °°°°°°OOOooooooooooo 0 o o
............
.......
o o
o
o
o
o
o
o
"e"~°°ooOOOOooooooo° •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••'
I
I
I
0
I
I
I
I t
I
I
20
Figure 4.5.14: The compressive force P N versus time t s. Circles show experimental data obtained by Hausler and Sayir (1995). Dotted lines show prediction of the model (4.5.52). Curve 1: 3, = 0.102. Curve 2: 3, = 0.172.
Bibliography [1] Alexander,
H. (1968).
A constitutive relation for rubber-like materials. Int. J.
Engng. Sci. 6, 549-563. [2] Arutyunyan, N. K. and Drozdov, A. D. (1992). Phase transitions in nonhomogeneous, aging, viscoelastic bodies. Int. J. Solids Structures 29, 783-797. [3] Arutyunyan, N. K., Drozdov, A. D., and Naumov, V. E. (1987). Mechanics of Growing Viscoelastoplastic Solids. Nauka, Moscow [in Russian]. [4] Astarita, G. and Marrucci, G. (1974). Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London.
256
Chapter 4. Nonlinear Constitutive Models with Finite Strains
[5] Bagley, R. L. and Torvik, E J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201-210. [6] Ballal, B. Y. and Rivlin, R. S. (1979). Flow of a viscous liquid between eccentric cylinders. II. Poisseuille flow. Rheol. Acta 18, 311-322. [7] Bernstein, B. (1966). Time-dependent behavior of an incompressible elastic fluid. Some homogeneous deformation histories. Acta Mech. 2, 329-354. [8] Bernstein, B., Kearsley, E. A., and Zapas, L. J. (1963). A study of stress relaxation with finite strains. Trans. Soc. Rheol. 7, 391-410. [9] Bloch, R., Chang, W. V., and Tschoegl, N. W. (1978). The behavior ofrubberlike materials in moderately large deformations. J. Rheol. 22, 1-32. [10] Bordonaro, C. M. and Krempl, E. (1992). The effect of strain rate of the deformation and relaxation behavior of 6/6 nylon at room temperature. Polym. Eng. Sci. 32, 1066-1072. [11] Buckley, C. P. and Jones, D. C. (1995). Glass-rubber constitutive model for amorphous polymers near the glass transition. Polymer 36, 3301-3312. [12] Burton, T. A. (1983). Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press, Orlando. [13] Chang, W. V., Bloch, R., and Tschoegl, N. W. (1976). On the theory of the viscoelastic behavior of soft polymers in moderately large deformations. Rheol. Acta 15, 367-378. [ 14] Christensen, R. M. (1980). A nonlinear theory of viscoelasticity for application to elastomers. Trans. ASME J. Appl. Mech. 47, 762-768. [15] Coleman, B. D., Markowitz, H., and Noll, W. (1966). Viscometric Flows of Non-Newtonian Fluids. Springer-Verlag, New York. [ 16] Coleman, B. D. and Noll, W. (1960) An approximation theorem for functionals with applications to continuum mechanics. Arch. Rational Mech. Anal. 6, 355370. [ 17] Coleman, B. D. and Noll, W. (1961). Foundations of linear viscoelasticity. Rev. Modern Phys. 33, 239-249. [18] DeHoff, E H., Lianis, G., and Goldberg, W. (1966). An experimental program for finite linear viscoelasticity. Trans. Soc. Rheol. 10, 385-397. [ 19] Derman, D., Zaphir, Z., and Bodner, S. R. (1978). Nonlinear anelastic behavior of a synthetic rubber at finite strains. J. Rheol. 22, 239-258. [20] Drozdov, A. (1992). Constitutive model of a viscoelastic material at finite strains. Mech. Research Comm. 19, 535-540.
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Chapter 4. Nonlinear Constitutive Models with Finite Strains
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[52] MacDonald, I. E (1976). On the admissibility of rate-dependent viscoelastic models. Rheol. Acta 15, 223-230. [53] McGuirt, C. W. and Lianis, G. (1970). Constitutive equations for viscoelastic solids under finite uniaxial and biaxial deformations. Trans. Soc. Rheol. 14, 117-134. [54] McKenna, G. B. and Kovacs, A. J. (1984). Physical aging of poly(methyl metacrylate) in the nonlinear range: torque and normal force measurements. Polym. Eng. Sci. 24, 1138-1141. [55] McKenna, G. B. and Zapas, L. J. (1979). Nonlinear viscoelastic behavior of poly(methyl metacrylate) in torsion. J. Rheol. 23, 151-166. [56] Middleman, S. (1969). Transient response of an elastomer to large shearing and stretching deformations. Trans. Soc. Rheol. 13, 123-139. [57] Morman, K. N. (1988). An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure. Rheol. Acta 27, 3-14. [58] Nishiguchi, I., Sham, T.-L., and Krempl, E. (1990a). A finite deformation theory of viscoplasticity based on overstress: Part I-Constitutive Equations. Trans. ASME J Appl. Mech. 57, 548-552. [59] Nishiguchi, I., Sham, T.-L., and Krempl, E. (1990b). A finite deformation theory of viscoplasticity based on overstress: Part ll-Finite element implementation and numerical experiments. Trans. ASME J Appl. Mech. 57, 553-561. [60] Ogden, R. W. (1972). Non-linear Elastic Deformations. Ellis-Horwood, Chichester. [61] Onogi, S., Masuda, T., and Matsumoto, T. (1970). Nonlinear behavior of viscoelastic materials. 1. Disperse systems of polytirene solution and carbon black. Trans. Soc. Rheol. 14, 275-294. [62] Pearson, G. and Middleman, S. (1978). Elongation flow behavior of viscoelastic liquids: modelling bubble dynamics with viscoelastic constitutive relations. Rheol. Acta 17, 500-510. [63] Pipkin, A. C. (1964). Small finite deformations of viscoelastic solids. Rev. Modern Phys. 36, 1034-1041. [64] Pipkin, A. C. and Rivlin, R. S. (1961). Small deformations superposed on large deformations in materials with fading memory. Arch. Rational Mech. Anal. 8, 297-310. [65] Renardy, M., Hrusa, W. J., and Nohel, J. A. (1987). Mathematical Problems in Viscoelasticity. Longman, New York.
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Chapter 4. Nonlinear Constitutive Models with Finite Strains
[66] Rivlin, R. S. (1956). Large elastic deformations. In Rheology. Theory and Applications (E R. Eirich, ed.), Vol. 1. Academic Press, New York. [67] Rivlin, R. S. and Ericksen, J. L. (1955). Stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4, 323-425. [68] Rivlin, R. S. and Saunders, D. W. (1951). Large elastic deformations of isotropic materials. 7. Experiments on the deformation of rubber. Phyl. Trans. Roy. Soc. London 243, 251-288. [69] Santore, M. M. and McKenna, G. B. (1991). Torsional relaxation and volume response during physical aging in epoxy glasses subjected to large torsional deformations. Proc. Mater. Res. Soc. 215, 81-86. [70] Spriggs, T. W., Huppler, J. D., and Bird, R. B. (1966). An experimental appraisal of viscoelastic models. Trans. Soc. Rheol. 10, 191-213. [71] Stafford, R. O. (1969). On mathematical forms for the material functions in nonlinear viscoelasticity. J. Mech. Phys. Solids 17, 339-358. [72] Tanner, R. I. (1968). Comparative studies of some simple viscoelastic theories. Trans. Soc. Rheol. 12, 155-182. [73] Titomanlio, G., Spadaro, G., and La Mantia, E R (1980). Stress relaxation of a polyisobutylene under large strains. Rheol. Acta 19, 477-481. [74] Truesdell, C. (1975). A First Course in Rational Continuum Mechanics. Academic Press, New York. [75] Valanis, K. C. and Landel, R. E (1967). The strain-energy function of a hyperelastic material in terms of the extension ratios. J. Appl. Phys. 38, 2997-3002. [76] Wagner, M. H. (1976). Analysis of time-dependent non-linear stress-growth data for shear and elongation flow of a low-density branched polyethylene melt. Rheol. Acta 15, 136-142. [77] Wagner, M. H. (1977). Prediction of primary normal stress difference from shear viscosity data using a single integral constitutive equation. Rheol. Acta 16, 43-50. [78] Wagner, M. H., Raible, T., and Meissner, J. (1979). Tensile stress overshoot in uniaxial extension of a LDPE melt. Rheol. Acta 18, 427-428. [79] Waldron, W. K. and Wineman, A. (1996). Shear and normal stress effects in finite circular shear of a compressible non-linear viscoelastic solid. Int. J. NonLinear Mech. 31,345-369. [80] White, J. L. and Metzner, A. B. (1963). Development of constitutive equations for polymeric melts and solutions. J. Appl. Polym. Sci. 7, 1867-1889.
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Chapter 5
Constitutive Relations for Thermoviscoelastic Media This chapter is concerned with constitutive models in linear thermoviscoelasticity. Section 5.1 provides a brief survey of constitutive models for isothermal loading and demonstrates the effect of temperature on the mechanical response in viscoelastic media with small strains. In Section 5.2, two versions of the model of adaptive links for isothermal loading are derived, and results of numerical simulation are compared with experimental data. Section 5.3 deals with nonisothermal loading of viscoelastic media. We introduce several hypotheses regarding the influence of temperature on coefficients of differential equations for concentrations of adaptive links, validate these assumptions by comparison with experimental data, and apply a model of the standard thermoviscoelastic medium to calculate residual stresses in a polymeric pressure vessel at cooling.
5.1
Constitutive Models in Thermoviscoelasticity
In this section we provide a brief survey of constitutive equations for the mechanical behavior of viscoelastic media under nonisothermal conditions. The main attention is focused on a model of thermorheologically simple media, on models based on the proportionality hypothesis, and on the McCrum model.
5.1.1 Thermorheologically Simple Media Analysis of the influence of temperature on the mechanical response in viscoelastic media is of essential interest because of the wide use of nonisothermal technological processes in manufacturing viscoelastic structures [see, e.g., Advani (1994)]. However, in spite of the essential importance of modeling thermorheological processes, 262
5.1. Constitutive Models in Thermoviscoelasticity
263
it is difficult to mention a model that adequately predicts the thermoviscoelastic behavior. As is well known, elastic moduli, as well as the relaxation (creep) kernels of viscoelastic materials, depend drastically on the absolute temperature (9. One of the reasons for this phenomenon is an increase in the amplitude of random microBrownian oscillations of molecules with the growth of temperature. Since there is no theory that permits the viscoelastic response at the macrolevel to be predicted based on the description of microlevel events, phenomenological approaches are suggested to account for the effect of temperature. Eringen (1960) proposed the simplest model, which neglected the influence of temperature on elastic moduli and relaxation kernels and accounted for thermal expansion only. That model is convenient for calculating thermal stresses, but its predictions are questionable, since viscoelastic properties of polymers are highly sensitive to the temperature changes. The model of thermorheologically simple materials takes into account changes in relaxation (creep) kernels and neglects the effect of temperature on elastic moduli. That model is based on the time-temperature superposition principle proposed by Leaderman (1943). According to the superposition principle, the effect of temperature on the creep and relaxation curves is accounted for by an appropriate shift of them along the logarithmic time scale. The same postulate (in a slightly different form) was also suggested by Ferry (1950). To discuss the time-temperature superposition principle, we consider uniaxial extension of a nonaging, linear, viscoelastic specimen at some reference temperature 190 and at a temperature (9. The constitutive relation between the stress or and the strain e at temperature 190 reads [see Eq. (2.2.16)]
E /0t0 0 ( t -
or(t) = E e(t) +
1
~')E(I")dr ,
(5.1.1)
where E is a constant Young's modulus, Qo(t) is a relaxation measure that satisfies the condition Q0(0) = 0, and the superposed dot denotes differentiation with respect to time. To transform Eq. (5.1.1), we integrate it by parts and use the initial condition e(0) = 0. As a result, we find that t
or(t) = E
fo
[1 + Qo(t - r)]E(r) dr.
(5.1.2)
The time-temperature superposition principle states that Young's modulus E is independent of temperature, whereas the relaxation measure Qo(t, 19) at temperature 19 is obtained from the relaxation measure Qo(t, 190) at the reference temperature 190 by shift of its argument Qo(t, 19) = Qo(~(t, 19), 19o),
(5.1.3)
~(t, 19) = ~b(®)t.
(5.1.4)
where
264
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
The shift factor oh(®) satisfies the following conditions [see, e.g., Morland and Lee (1960) and Muki and Sternberg (1961)]: th(Oo) = 1,
th(O) > Or
d+
d--~(O) > O.
(5.1.5)
To explain the physical meaning of Eq. (5.1.4), it is convenient to present Qo(t) as a function of "dimensionless" variables Qo(t) = Qo
(t
T1 'T1 . . . . . T1
)
where Tn is the nth characteristic time of stress relaxation. Assumption (5.1.4) means that with changes in temperature ®, all the relaxation times Tn change similarly according to the formula Tn(O) = a(O)Tn(OO),
(5.1.6)
where 1
a ( ® ) - oh(®)"
(5.1.7)
The thermal shift function a(O) equals the ratio of any relaxation time at temperature ® to its value at the reference temperature O0 [see, e.g., Williams et al. (1955)]. It reflects the temperature dependence of a segment friction coefficient or a mobility of configurational rearrangements [see Rouse (1953) and Bouche (1953) for a discussion of this issue]. Materials that satisfy the time-temperature superposition principle are called thermorheologically simple [see Schwarzl and Staverman (1952)]. In thermorheologically simple media, all material functions (mechanical, dielectrical, optical, etc.) depend on the internal or "pseudo" time {~(t, 0). For experimental confirmation of the time-temperature superposition principle, see, e.g., Aklonis et al. (1972), Ferry (1980), Han and Kim (1993), Urzhumtsev (1975, 1982), Ward (1971), and the bibliography therein. Schapery (1964) discussed this principle from the standpoint of thermodynamics of irreversible processes. For a relaxation measure Qo(t) in the form of a truncated Prony series, the integral constitutive equation (5.1.1) is equivalent to a differential constitutive equation with constant coefficients (which depend on Young's moduli En of springs and Newtonian viscosities r/n of dashpots). The time-temperature superposition principle states that the Young moduli preserve their values with changes in temperature whereas the material viscosities are changed according to the formula "On(O) =
a(O)nn(O0).
(5.1.8)
As an example, we consider a standard viscoelastic medium that consists of an elastic element with Young's modulus E1 connected in parallel with a Maxwell element with the Young's modulus E2 and Newtonian viscosity rl (see Figure 2.1.3). The integral
5.1. Constitutive Models in Thermoviscoelasticity
265
constitutive model is determined by Young's modulus E = E1 + E2 and the relaxation measure [see Eq. (2.1.7)] E2 t Q°(t) = - E l + E--------~[ 1 - exp ( - T ) I
'
(5.1.9)
where the characteristic time equals T-E2.
n
(5 1 10) ..
This model is equivalent to the differential equation [see Eq. (2.1.5)] d o + o" de e dt T = E-d-i + EI-~.
(5.1.11)
The time-temperature superposition principle implies that the constitutive equation (5.1.11) remains true for an arbitrary temperature 19 provided the moduli E1 and E2 are temperature-independent and the relaxation time T obeys the law (5.1.6) (5.1.12)
T(®) = a(®)T(®o).
To predict experimental data, Tobolsky (1960) (with reference to a network model) proposed the following formula for Young's modulus E: p(O)O E(®) = ~ E ( ® 0 ) ,
(5.1.13)
p(Oo)O0
where p is mass density. When the coefficient of thermal expansion c~ is constant within the temperature interval under consideration, the specific volume V(O) at an arbitrary temperature 0 is calculated as V(O) = V(Oo)[1 + a ( O -
00)].
This equality together with the mass conservation law
p(O)v(O)
=
p(Oo)V(Oo)
implies that p(O) =
p(Oo)
(5.1.14)
1 + a ( O - O0)"
Combining Eqs. (5.1.13) and (5.1.14), we find that E(®) -
19 E(®0) 0ol +a(O-Oo)
.
(5.1.15)
Equation (5.1.15) is rather far from being satisfied in experiments. For example, experimental data obtained for crosslinked polyethylene by Narkis and Tobolsky (1969) demonstrate sharp decrease of the shear modulus G in temperature 19. By assuming Poisson's ratio v to be constant and independent of temperature (u = 0.3),
266
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
we obtain the plot E versus (9 depicted in Figure 5.1.1. Using this dependence and Eq. (5.1.15), the coefficient of thermal expansion a is calculated. Numerical simulation demonstrates that below the glass transition temperature ®g, the parameter ct should increase by more than a decade with the growth of temperature. An equation similar to Eq. (5.1.13) was suggested by Ferry (1950) and Williams et al. (1955). They assumed that Eq. (5.1.13) was true for the modulus E2
p(O)O E2(O) = ~ E 2 ( O o ) , p(Oo)Oo
l00
103
E
_
0
•
0
O
10-
I
250
I
I
o¢
I
I
I
O
I
®
Io
olo
10-2 450
Figure 5.1.1: Young's modulus E MPa and the coefficient of thermal expansion a 1/K versus temperature OK for crosslinked polyethylene. Unfilled circles show experimental data obtained by Narkis and Tobolsky (1969). Filled circles show prediction with the use of Eq. (5.1.15).
5.1. Constitutive Models in Thermoviscoelasticity
267
and combined this equality with Eqs. (5.1.10) and (5.1.12). As a result, they arrived at the equation a(®) = p(O0)®0 r/(®) p(O)O n(O0)'
(5.1.16)
which permits the thermal shift function a(O) to be evaluated by employing experimental data for the material viscosity r/(O). We now concentrate on explicit expressions for the thermal shift function a(®). Several formulas for the function a(O) can be found in Schwartzl and Staverman (1952) with references to previous investigations. One of the simplest relationships for the function a(®) is based on the assumption regarding proportionality of the rate of any kinetic process to the reciprocate temperature. Assuming r/to be proportional to O - 1 and E2 to be temperature-independent, we obtain the Arrhenius equation
Tn(O) - A ( ~ In a(®) = log Tn(®0) R
1 ) - 190 "
(5.1.17)
Here A is an apparent activation enthalpy and R is Boltzmann's constant. Equation (5.1.17) with a constant A ensures excellent fit of experimental data below the glass transition temperature ®g for amorphous polyolefins [see Lacabanne et al. (1978)], poly (methyl methacrylate) [see McCrum and Morris (1964)], polypropylene [see McCrum et al. (1982)], and several other polymers. For a discussion of the Arrhenius equation, see also McCrum et al. (1967). However, the data for polyisobutylene and polystyrene obtained by Fox and Florry (1948, 1950, 1951) demonstrate that the activation enthalpy A strictly depends on temperature O and is proportional to 19-1 and O-5, respectively. The latter means that Eq. (5.1.17) should be refined to ensure better agreement with experimental observations. Schwartzl and Staverman (1952) suggested two revised relationships lna(®) = A
(1 1) ®2 - ®2 '
(5.1.18)
and ln a ( ® ) = B
t9-®~-
t90-®~
'
(5.1.19)
where A, B, and ®~ are adjustable parameters. They recommended Eq. (5.1.18) for polyisobutylene and Eq. (5.1.19) for rubber and poly (methyl methacrylate). Equation (5.1.19) can be presented in the form lna(®) = -Cl
19 - 190 C2 + 0
-- ( ~ 0 '
(5.1.20)
268
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
where B C1 ---- O 0 _ _ O
~,
C2 - - O 0 - - O oo.
Equation (5.1.20) [called the Williams-Landel-Ferry (WLF) equation] ensures fair fit of experimental data for a number of polymeric materials (polyisobutylene, polystyrene, organic glasses) and several inorganic glasses above the glass transition temperature provided the "separate reference" temperature O0 is chosen appropriately [see Williams et al. (1955)]. Further analysis has shown that Eq. (5.1.20) may be applied to a wide range of viscoelastic and composite materials [see, e.g., Ferry (1980), Ilyushin and Pobedrya (1970), Koltunov (1976), and Urzhumtsev (1975)]. On the other hand, several studies demonstrate that Eq. (5.1.20) fails to predict the material response below the glass transition temperature Og. For example, predictions of the WLF equation differ from experimental data for an epoxy resin by 10 orders [see Arridge (1985)]. Low reliability of the WLF formula was shown for polystyrene and polyisobytylene by Williams et al. (1955) and for an NBR elastomer by McCrum (1984). Formula (5.1.20) is closely related to the free-volume theory of liquids [see, e.g., Doolittle (195 la,b, 1952)], where the constitutive relation In r/(O) = A +
B
(5.1.21)
f(O)
replaces the Andrade equation B
In r/(O) = A + ~-.
(5.1.22)
Here A and B are adjustable parameters, and f is the fractional free volume (the ratio of volume of the space available for a polymeric molecule to move freely to the total volume occupied by the molecule). Combining Eqs. (5.1.10), (5.1.12), and (5.1.21), we obtain the Doolittle formula T(O) _ l n r/(O) _ B I 1 In a(O) - In T(Oo) ~/(Oo) f(O)
1 ] f(Oo) "
(5.1.23)
Let the specific free volume depends linearly on temperature, (5.1.24)
f(O) = f(Oo) + a(O - 0o),
where a is a coefficient of thermal expansion. Then Eq. (5.1.23) together with Eq. (5.1.24) implies Eq. (5.1.20) with B Cl-
f(O0)'
f(®0) c2-
c~
"
Because the WLF equation (5.1.20) correctly predicts the material behavior in the rubber region, whereas the Arrhenius law (5.1.17) describes the material response
269
5.1. Constitutive Models in Thermoviscoelasticity
in the glassy region, it is natural to combine these equations. A version of a hybrid equation for the temperature shift factor was proposed by McCrum (1984) 1
ln a(®) = ~
~- - O0
f(®) - f(®0)
'
(5.1.25)
where B is an adjustable parameter. The same equality was suggested earlier by Dienes (1953) for viscous liquids. The McCrum parameters A, B, and f(®0) are expressed in terms of the Dienes parameters as follows: A = W,
B-
~U R '
f(®0)= a(®0-O,),
where U is an activation energy for structural changes associated with local ordering, W is an activation energy for viscous flow of a disordered liquid, and 19, is a temperature below which short-range order is complete. Among other expressions for the temperature shift function, we should mention the formula [see Bueche (1953)] lna(O)=-ln
[ (a 1-erf
-~-B
,
(5.1.26)
and the phenomenological equation [see Khristova and Aniskevich (1995), Sharko and Yanson (1980), and Yanson et al. (1983)] ln a(®) = - A ( O - Oo) - B(O - (90)2.
(5.1.27)
Here A and B are adjustable parameters, and e r f ( x ) - ~ _1~ f0x e x p -( ~ z2 ) dz. Two serious drawbacks of the theory of thermorheologically simple media may be mentioned: 1. The model of thermorheologically simple media neglects the effect of temperature on elastic moduli, which contradicts experimental data. 2. The model of thermorheologically simple media presumes similarity in the ratios of relaxation times TI(O) T2(O) a(O) = T~(Oo) - T2(Oo) - -
~
°
•
°
- -
Tn(O) Tn(OO)
_ _
°
°
°
(5.1.28)
The hypothesis (5.1.28) was questioned by Lacabanne et al. (1978) and McCrum (1984). Experimental data for poly (methyl methacrylate) obtained by Read (1981) demonstrate that the ratio (5.1.28) depends essentially on n and increases with the growth of Tn. The necessity to verify assumption (5.1.28) for polymeric materials was also discussed by Arridge (1985).
270
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
5.1.2 The Proportionality Hypothesis Since the model of thermorheologically simple media takes into account only the effect of temperature on the characteristic times of relaxation and retardation, an alternative model presumes that the characteristic times are temperature-independent, whereas elastic moduli depend significantly on temperature. Such a model was proposed by Suvorova (1977a, b) and developed by Suvorova et al. (1980), Makhmutov et al. (1983), and Viktorova (1983). It is of interest to discuss the arguments that lead to the proportionality assumption. Suvorova (1977a, b) considered viscoelastic media governed (at some temperature 190) by the nonlinear constitutive equation [cf. Eq. (3.2.14)] Eqff E(t)) = o.(t) + [3 f0 t J _ ~ ( t - T)O.(T) dT.
(5.1.29)
Here E is Young's modulus, q~(e) is a function that characterizes the material nonlinearity, ta
J~(t) =
F(1 + a )
is the Abel kernel, F(Z) -
f0 °°
t z-1 exp(-t) dt
is the Euler gamma function, and a E (0, 1] and/3 > 0 are material parameters. Integration of Eq. (5.1.29) by parts implies that E~o(E(t)) = o-(t) + F(1 /3- a)
= o ' ( t ) - (1
f' (t ~r(T) - ~.)----------Z dr
a)F(1 ~ a) ( t - T)I-ao.(T){ T~-0 T=t
do- d ~-] . - f0 t (t-~)' -a -d-7(T) Bearing in mind that o.(0) = 0 and zF(z) = F(1 + z), we obtain Eqff e(t)) = o.(t) +
/3
F(2 - a)
do" 0~~t(t - T) 1-a -dT(r) dr
do"-(~')d~'" = f 0 t [ 1 + F ( 2 /3 - a-------~( t - ~.)1_~1 -d-t
(5.1.30)
If the time-temperature superposition principle is fulfilled, than it follows from Eqs. (5.1.3), (5.1.4), (5.1.7), and (5.1.30) that the material response at temperature 19
271
5.1. Constitutive Models in Thermoviscoelasticity
is governed by the constitutive equation Eq~(e(t)) =
/0'[1 +
/3al-a(O)(t -- r)l-a I dor --dT(r)dr. F ( 2 - c~)
(5.1.31)
According to Eqs. (5.1.30) and (5.1.31), the temperature 19 affects only the coefficient/3. Based on this (merely formal) correspondence, Suvorova and coauthors proposed employing the same procedure for an arbitrary creep kernel K(t), i.e., replacing the constitutive equation Eqff E(t)) = or(t) + ~
f0t K(t
(5.1.32)
- r)~r(r) d r
at the reference temperature Oo by the constitutive equation
/ 0 tK ( t -
Eq~(e(t)) = or(t) +/3(0)
(5.1.33)
r)~r(r) d r
at an arbitrary temperature O. Here/}(O) is an adjustable function that satisfies the condition/}(®0) =/3. Fair agreement between the model prediction and experimental data is demonstrated for several plastics reinforced by carbon fibers [see Suvorova (1977a) and Suvorova et al. (1980)]. Using the proportionality hypothesis for elastic moduli at an arbitrary temperature O and at the reference temperature 00, constitutive equations may be refined to account for the effect of temperature. For example, an analog (at an arbitrary temperature 19) of the linear constitutive equation (at temperature ®0)
'E
E(t) = -~ or(t) +
J0'
(5.1.34)
K(t - r)or(r) d r
is written as [see Talybly (1983)],
'E
E(t) = ~
/3~(O)~r(t) -+- ~32(O )
/0t
K(t - r)~r(r) d r
1
,
(5.1.35)
where ~31(1~) and/32(0) are adjustable functions. The model (5.1.35) ensures good fit of experimental data for polyethylene in the temperature range from 20 to 100 ° C. The following conclusions may be drawn: 1. There is no rational explanation for the proportionality hypothesis. 2. For non-Abelian kernels, this hypothesis does not take into account changes in the characteristic times of creep and relaxation observed in experimental studies. 3. The proportionality principle should be checked experimentally for viscoelastic materials, the mechanical behavior of which is not described by the power-law creep measure.
272 5.1.3
Chapter 5. Constitutive Relations for Thermoviscoelastic Media The McCrum Model
To describe the effect of temperature on elastic moduli and creep (relaxation) measures, McCrum and Morris (1964) introduced three basic hypotheses. We discuss those assumptions with reference to the linear constitutive equation [see Eq. (2.2.29)]
1[ /0/ lfo'
e(t) = ft. or(t) +
Co(t- r)cr(r) d
[1 + Co(t- r)10"(r)dr,
- E
(5.1.36)
where E is Young's modulus and Co(t) is a creep measure. The relaxation kernel
dC° (t ) K(t) = ---dTis expressed in terms of the retardation spectrum/3(T) by formula (2.3.6). Integration of Eq. (2.3.6) results in
Co(t) =
/3(T) [ 1 - exp ( - T ) ]
dT.
(5.1.37)
Equations (5.1.36) and (5.1.37) are valid at some reference temperature Oo with E = E(®o), C(t) = C(t, ®0), and/3(T) =/3(T, ®0). At an arbitrary temperature 0, these functions should be replaced by a modulus E(O), a creep measure C(t, O), and a retardation spectrum/3(T, 0). McCrum and Morris (1964) suggested that two functions a(O) and c(O) exist such that for any temperature O and for any T, T @0). /3(T, O) = c(O)j3 ( a(O)'
(5.1.38)
Substitution of expression (5.1.38) into Eq. (5.1.37) implies that for any t --> 0 and any ®, f0°°
( Ta---~'
) [1
exp t
= a(®)c(O) ~o'~13(r,®o) [ 1 - e x p ( - a ( O ) r ) l = a(O)c(O)Co
a--~' Oo •
Consider the standard creep test with or(t) = [0' t < 0, t tr0, t > 0,
dr (5.1.39)
5.1. Constitutive Models in Thermoviscoelasticity
273
where Oo is a given stress. It follows from Eq. (5.1.36) that the creep compliance
J(t) -
E(t)
oro
equals
J(t, O) =
1
E(O)
[1 + Co(t, O)].
(5.1.40)
The ratios of the limiting compliances at temperatures O and ®0 are determined as
J(O, 0o) bl(O) = J(O, 0 ) '
J(~, 0o) b2(O)= j ( ~ , O ) "
(5.1.41)
Substitution of expression (5.1.40) into Eq. (5.1.41) with the use of Eq. (5.1.39) yields
bl(O) --
E(®)
E(O) 1 + E(Oo)Co(c% ®0) b 2 ( O ) - E(O0)1 + a(O)c(®)Co(o% 0o)"
E(Oo)'
(5.1.42)
According to McCrum and Morris (1964), the functions bl(O) and b2(®) coincide
bl(O) -- b2(O) = b(O).
(5.1.43)
This assertion together with Eq. (5.1.42) implies that 1
c(®) - a(@)"
(5.1.44)
Combining this equality with Eq. (5.1.39), we arrive at the formula
(t)
Co(t, O) = Co a--~' O° •
(5.1.45)
It follows from Eqs. (5.1.42) and (5.1.43) that E(O) = b(®)E(O0).
(5.1.46)
The constitutive equation (5.1.36) of a nonaging, linear, viscoelastic medium reads
1
e(t) = b(O)E(Oo)
lot [ 1 +
Co
(t-'T Oo) a(O)'
O-(~')d~-.
(5.1.47)
The inverse relation can be written as O'(t) = b(O)E(O0)
fot [1 + Q0 ( ta(®)' -- T OO) l i~(T)dT,
(5.1.48)
Chapter 5. Constitutive Relationsfor ThermoviscoelasticMedia
274
where the relaxation measure Q0(t, lg0) is connected with the creep measure C0(t, ®0) by the integral equation [see Eq. (2.2.37)]
Qo(t, 0o) + Co(t, 0o) +
f0t Qo(t -
"r, O0)d'o('r, ®0) d'r = 0.
(5.1.49)
It follows from Eqs. (5.1.45) and (5.1.49) that the thermal shift factors a(l~) for creep and relaxation coincide
Qo(t, O) = Q0 a - - ~ ' (90 .
(5.1.50)
Experimental data for poly (methyl methacrylate) confirm this conclusion [see McCrum and Morris (1964)]. The McCrum constitutive model (5.1.47) and (5.1.48) contains two material functions a(O) and b((9), which are easily found in standard tests. This model was verified by McCrum (1984), McCrum and Morris (1964), and McCrum et al. (1967). The McCrum model generalizes the model of thermorheologically simple media and the model based on the proportionality hypothesis. On the one hand, this is an important advantage, since the McCrum model accounts for both changes in the relaxation (retardation) times and elastic moduli. On the other hand, this implies a shortcoming of the model, since a certain ambiguity arises when it is extended to nonisothermal processes. To discuss this question, we confine ourselves to a standard viscoelastic solid with the relaxation measure (5.1.9). The McCrum assumptions Eqs. (5.1.46) and (5.1.50) together with Eq. (5.1.9) imply that El(O) + E2(O) = b((O)[El(O0) + E2((90)], E2(O) El(O) + E2(O)
E2(Oo) EI(®o) + E2(O0)'
T(O) = a(®)T(O0). Resolving these equations with respect to E1(6)) and E2(O), we find that El(O) = b(O)El(O0),
E2(O) = b(O)E2(O0).
(5.1.51)
Substitution of expressions (5.1.51) into the differential constitutive equation (5.1.11) yields do- +
dt
de + E1 (O0) e 1 . 1 o- - b(O) IE(O0) m a(®) T(®0) dt a(®) T(¢9o)
(5.1.52)
For a nonisothermal process ® = ®(t), solutions of the differential equation (5.1.52) and the integral equation (5.1.48) can differ from each other. A disadvantage of the McCrum model is that it does not provide any criterion that enables us to choose either the integral or the differential model.
5.2. A Model of Adaptive Links in Thermoviscoelasticity
275
To derive such a criterion, a model of adaptive links may be employed in which the McCrum assumptions are interpreted in terms of adaptive links that replace each other.
5.2
A Model of Adaptive Links in Thermoviscoelasticity
A model of adaptive links for an aging, linear, viscoelastic meduim at isothermal loading has been discussed in detail in Chapter 2. In this section, we analyze the effect of temperature of the rates of breakage and reformation for adaptive links. For this purpose, two versions of the model of adaptive links are introduced and results of numerical simulation are compared with experimental data [see Drozdov (1996, 1997d) and Drozdov and Kalamkarov (1995)].
5.2.1 Governing Equations According to the concept of adaptive links, a viscoelastic medium is treated as a network of linear elastic springs (links) that replace each other. It is assumed that M different kinds of links exist, which are characterized by the functions Xm,(t, "r) and rigidities Cm (m = 1. . . . . M). The function Xm.(t, T) equals the number of links of the mth kind that have arisen before instant ~-and exist at instant t. In particular, Xm.(t, O) is the number of initial links of the mth kind that exist at instant t, and
°3Xm*(t, T)dT OT
is the number of links of the mth kind that arose within the interval [~-, ~- + d~'] and exist at instant t. The initial links are divided into two types. Links of type I are not involved in the process of replacement, and their concentration equals Xm. Links of type II replace each other, and their concentration is 1 - Xm. Breakage of adaptive links is characterized by the function gm(t, T), which equals the relative number of links arisen at instant ~"and broken before instant t. To emphasize the effect of the absolute temperature 19 on reformation of adaptive links, we write Xm, = Xm,(t, T, ~)),
gm = gm( t, T, ~ ) ,
Cm = Cm(l~)),
Xm -- Xm(~)) •
These functions are connected by the formulas similar to Eqs. (2.2.45) Xm,(t , O, ~)) -" Xm,(O, O, O ) { X m ( ~ ) q- [1 - Xm(O)][ 1 - gm(t, O, O)]}, OXm, (t, r) = dPm(r, O ) [ 1 - gm(t, "r, O)],
0T
(5.2.1)
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
276 where
~m(t, O) = °3Xm* tg"r (t, t)
(5.2.2)
is the rate of reformation for adaptive links of the mth kind. In this section, we confine ourselves to isothermal loading and assume that the temperature ® is time-independent. The constitutive equation of an aging, linear viscoelastic medium (2.2.42) reads
~(t) = ~M Cm(®) i Xm.(t, O, ®)e(t) +
m--1
--"
~ [ XOn(O) m
.(t, t, O)e(t) --
fOtt~xm*(t, r, O)[e(t) 03T
-- E(r)] dr
I
~o'tt~Xm*(t,T,O)E(T)dTl C~r
m=l M
t
"- ~ Cm(O)fO Xm*(t'T, O)~(T)dT,
(5.2.3)
m=l where the superposed dot denotes differentiation with respect to time. Our objective is to derive a model that describes the effect of temperature on the viscoelastic behavior of polymers below the glass transition temperature. The initial number of adaptive links is treated as a temperature-independent parameter
Xm,(O, O, ®) = XOm,.
(5.2.4)
Assumption (5.2.4) excludes from our consideration such irreversible physical processes as curing [see, e.g., Buckley and Salem (1987)] and gel formation [see, e.g., De Rosa and Winter (1994)], in which the number of adaptive links increases drastically when the temperature decreases. Hypothesis (5.2.4) is quite acceptable below the glass transition temperature, since in the model of adaptive links only the products
Cm(O)Xm,(O,0, O)
(5.2.5)
have some physical meaning. Evidently, one term in the product (5.2.5) may always be chosen as temperature-independent, but the other bears the entire dependence on itself. For nonaging viscoelastic media, we set
Xm,(t, t, O) = X°m,, gm(t,7, O) = gm,o(t- 7, 0),
(I)m(T, O) = (I)m(O).
Substituting expressions (5.2.1), (5.2.4), and (5.2.6) into the equality Xm,(t, t, 19) = Xm,(t, 0, O) W f0 t °3Xm*(t, r, 19)dr,
~gr
(5.2.6)
5.2. A Model of Adaptive Links in Thermoviscoelasticity
277
we find that X°m, : X0m,{Xm(O) + [1 - Xm((~)][1 -
gm,o(t, O)]}
[1 - gm,o(t - r, O)] dr.
+ ~m(®)
This equality is equivalent to the linear integral equation
~I'm(O)
gm,o(t) = XOm,[1 _ Xm(O)]
f0 ~[1 -- gm,o(r)]dr.
(5.2.7)
Differentiation of Eq. (5.2.7) results in
dgm,o _ ¢~m(O) (1 - gm,o), dt X°,n,[1- Xm(O)]
gm,o(O) = O.
(5.2.8)
The solution of Eq. (5.2.8) reads
gm,o(t) = 1 - exp
[ Ore,O,' ] - X O , ( 1 _ t'm(O))
"
(5.2.9)
Substitution of expressions (5.2.1), (5.2.6), and (5.2.9) into the formula
Xm,(t, "r, O) = Xm,(t, t, O) -
~
t OXm, (t, Os
S, 19) ds
implies that
Xm,(t , T, O )
-- XOm, -
ft
(I)m(O) exp
[
= X°~, -X°~,[1 - 1"re(O)]
dpm(O)(t _ S) ]
- XOm---~i--- Xm(O-))J
1 -exp
ds
- X O , ( 1 _ 1"m(O))
" (5.2.10)
Finally, combining Eqs. (5.2.3) and (5.2.10), we arrive at the constitutive equation
O'(t) -- ~ Cm(~))xOm,1^ m=l
1 - [1 - 1"m(O)]
(5.2.11) Introducing the notation M e(l~)) = Z Cm(O)X0m*' m=l
278
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
~ m ( O ) -- Cm(-'-')X°m,[l{'~ ~" - Xm(O)]
E(O)
(I)m (O)
(5.2.12)
~/m(®) = X0m.[1 _ Xm(O)]' we present the constitutive equation (5.2.11) in the form
or(t) = E(@) fo
t{
1 --
]./~m(O)[1 - exp(-3~m(O)(t - r))]
}
~(r) dr.
(5.2.13)
m=l
Comparison of Eqs. (5.2.13) and (5.1.48) implies that the McCrum model follows from the model of adaptive links provided that 1. For any kind of links, the relative number of nonreplacing links is temperature independent Xm(O) ~- Xm(OO).
(5.2.14)
2. The rates of forming new links (~)m([~) satisfy the time-temperature superposition principle (I)m (O) -
(I)m(O0)
a(O)
(5.2.15)
with the same shift factor a(O) for all kinds of links. 3. Rigidities of links Crn([~)) satisfy the McCrum equation Cm(O) = b(O)Cm(OO),
(5.2.16)
with the same shift factor b(O) for all kinds of links. It is noteworthy that items 1 and 3 contradict each other to a certain extent. Concentrations of nonreplacing links Xm reflect the strength of adaptive links, although the parameters Cm determine their rigidity. According to the McCrum hypotheses, changes in temperature significantly affect rigidities of adaptive links, although their strength remains independent of temperature. The latter assertion seems rather questionable. To check assumptions of the McCrum model, we consider experimental data obtained by La Mantia et al. (1980) for Nylon-6 in a wide range of temperatures in the vicinity of the glass transition temperature ®g. Because the number of experimental data for any relaxation curve is comparatively small (about 10), we confine ourselves to nonaging media (5.2.6) with two different kinds of adaptive links: M = 2.
(5.2.17)
5.2. A Model of Adaptive Links in Thermoviscoelasticity
279
Links of the first kind correspond to strong chemical crosslinks and links of the other kind model relatively weak entanglements. The terms "strong" and "weak" are related to the strength of links modeled as elastic springs. Strong links are characterized by a small rate of relaxation 3"1 and a large relaxation time T1 = 3,11. The reformation process for strong links determines reduction of stresses observed in relaxation tests with the characteristic time of about 10 min. Weak links are characterized by a high rate of relaxation 3,2 >~> 3,1 and a small relaxation time T2 = 3,21. The reformation process for weak links determines the material response in dynamic tests with the frequency from 1 to 100 Hz. With the growth of temperature, intensity of micro-Brownian motion increases. This leads to an increase in the rate of breakage for adaptive links (I) m and to a decrease in their rigidities Cm. Since the concentrations Xm of nonreplacing links are independent of temperature, this assertion together with Eq. (5.2.12) implies that Young's modulus E decreases with the growth of temperature 19. Experimental data confirm this conclusion (see Figure 5.2.1). Substituting expressions (5.2.4), (5.2.14), and (5.2.16) into the second equality in Eq. (5.2.12), we find that the parameters ILm are temperature-independent ILm(O) = ILm(O0).
(5.2.18)
Experimental data show that the parameter ILl is practically independent of temperature, and the parameter IL2 decreases in temperature, however, rather weakly (see Figure 5.2.2). The rates of reformation (I9n for adaptive links are determined by mobility of chains caused by micro-Brownian motion, and they increase with the growth of temperature 19. Combining the third equality in Eq. (5.2.12) with Eqs. (5.2.4) and (5.2.14), we obtain that the rates of relaxation 3'm increase in temperature as well. Experimental data confirm this hypothesis for the rubber state, when 19 > 19g. In the glassy state in the vicinity of ®g, an anomalous behavior is observed for weak links: the rate of relaxation 3'2 decreases rapidly in temperature (see Figure 5.2.3). The data presented in Figure 5.2.3 contradict Eq. (5.2.15) and demonstrate that the rates of reformation ~m for adaptive links of different kinds change independently of one another. The same conclusion was demonstrated by Lacabanne et al. (1978) for polyolefines, and by Read (1981) for poly (methyl methacrylate). Experimental data demonstrate that only three material parameters depend on temperature: the current Young's modulus E and the rates of relaxation 3'1 and 3'2. The model may be simplified additionally by assuming (in good agreement with observations) the the rate of relaxation for strong links ~1 is independent of temperature. For the standard relaxation test with E(t) = {0' e0,
t < 0, t > 0,
(5.2.19)
280
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
4.5
E
-I
I I I I I I
!
0.5
I
I
I
I
I
I
40
I
(9
I
I~ 150
Figure 5.2.1: Young's modulus E GPa versus temperature (9 °C for Nylon-6. Circles show experimental data obtained by La Mantia et al. (1980). The solid line shows their approximation by the function E((9) = E0(a(9 - 1)-1 with E0 = 2.756 GPa and a = 0.0397 1/K.
Equations (5.2.13) and (5.2.19) imply that
tr(t) = E(O)
1-
/-~m[1 - exp(-3'm(®)t)]
e0.
(5.2.20)
m=l
To study steady oscillations, we replace zero as the lower limit of integration in the constitutive equation (5.2.13) by - ~ and consider the loading program E(t) = eo exp(~ot)
(5.2.21)
281
5.2. A Model of Adaptive Links in Thermoviscoelasticity
©
0.6
]-6n
0.0
I
I
I
I
I
I
I
40
I
19
I
I
150
Figure 5.2.2: The dimensionless parameters ].L 1 (filled circles) and ].62 (unfilled circles) versus temperature ® °C for Nylon-6. Circles show experimental data obtained by La Manila et al. (1980). Solid lines show their approximation by the constant ] . L 1 - - 0.0818 and by the linear function ~2(19) = 0.6296 - 0.001319.
with a given frequency of vibration co. As a result, we find from Eqs. (5.2.13) and (5.2.21) that in the standard dynamic test
o'(t) = E(O)
exp(~wt)
Id~m'Ym([~))
--
m= 1
exp[-Tm(®)(t - ~') + wgz] d~" Co. oo
Calculating the integral and introducing the complex modulus
E*(CO, ®) -
or(t) ~(t) '
282
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
. Q
71
600
-:
o
w
0 •
0
0
@
I
•
I
I
I
I
I
I
I
40
@
I
I
150
Figure 5.2.3: The rates of relaxation 7, min-1 (filled circles) and 72 (s-') (unfilled circles) versus temperature (9 ° C for Nylon-6. Circles show experimental data obtained by La Mantia et al. (1980). Solid lines show their approximations by the constant 71 = 0.187 and by the function 72 = C1(9 + (C2/(9) m with m = l l , C1 = 2.8626, C2 = 72.8360.
we find that
E* (co, 19) = E'(co, 19) + ~,E"(co, 0), where
[ e'(oo, e ) = e ( e )
(5.2.22)
2 ]
]./,m7m (1~)
1- ~
V2m(O) + 0o2 '
m=l M
I,J,mTm(O)
E"(w,19) = E(O)wZ 72((9) + 0)2" m=l
(5.2.23)
5.2. A Model of Adaptive Links in Thermoviscoelasticity
283
The adjustable parameters ~m and functions E(O) and ~/m(®) are determined by fitting experimental data in relaxation tests and in dynamic tests at frequency to - 3.5 Hz with the use of Eqs. (5.2.20) and (5.2.23). These parameters are plotted versus temperature 19 in Figures 5.2.1 to 5.2.3. To validate the model, we calculate the material response in dynamic tests with frequency to = 110 Hz (employing the material parameters found in previous experiments), and compare results of numerical simulation with experimental data. The storage modulus E ~ and the loss tangent tan 6 = E " / U are plotted versus temperature 19 in Figures 5.2.4 and 5.2.5. Figure 5.2.4 demonstrates fair agreement between experimental data and their prediction by the model, but Figure 5.2.5 shows small discrepancies between numerical results and measurements.
G/
m
o
-
0
I
50
I
I
I
I
I
I
(9
0
I
0
I
150
Figure 5.2.4: The storage modulus G ~GPa versus temperature ®°C for Nylon-6 at frequency co = 110 Hz. Circles show experimental data obtained by La Manila et al. (1980). The solid line shows prediction of the model.
284
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
0
log tan 8 _
0
0
0
--2
I
u
I
I
I
I
50
I
I {9
I
0
I
150
Figure 5.2.5: The loss factor tan/5 versus temperature (9 °C for Nylon-6 at frequency oJ = 110 Hz. Circles show experimental data obtained by La Mantia et al. (1980). Solid line shows numerical prediction.
Comparing experimental data with results of numerical simulation, we may conclude that the generalized McCrum model (without the assumption regarding similarity of relaxation times) with two different kinds of links (strong crosslinks and entanglements) adequately predicts the mechanical response in a viscoelastic material at temperatures near the rubber-glass transition point.
5.2.2
A Refined
Model
of Adaptive
Links
A question of essential interest for applications is whether the generalized McCrum model is the unique model compatible with experimental data for polymeric materials. To show that the answer is negative, we propose another constitutive model (in the framework of the theory of adaptive links), and demonstrate that the new model
285
5.2. A Model of Adaptive Links in Thermoviscoelasticity
ensures the same level of accuracy in predicting experimental data as the McCrum model. According to the model of adaptive links, there are M different kinds of links. Any kind is characterized by the initial number of links Xm,(0, 0, ~ ) , rigidity of a link Cm(O), concentration of nonreplacing links Xm(®), and the rate of reformation On(O). Instead of the function Xm.(t, t, O), which equals the number of adaptive links of the mth kind, it is convenient to introduce the total number of adaptive links M
X,(t, t, 0 ) = Z
Xm,(t, t, ~))
(5.2.24)
m=l
and concentrations of links of the mth kind
Xm.(t, t, 19) X,(t, t, 19)
Tim
(5.2.25)
For nonaging materials, the parameters X, and X~, depend on temperature (9 only X,
= X,(O),
X m , -- X m , ( O ) .
According to Eq. (5.2.25), the same is true for concentrations ~m -- 'lr]m((~))We confine ourselves to nonaging viscoelastic media and introduce the following hypotheses to be verified by experimental data: (i) Rigidities Cm coincide for adaptive links of different kinds
CM = C,
C1 = C2 . . . . .
(5.2.26)
where the parameter c is independent of temperature. Since the total rigidity of the network of adaptive links (Young's modulus E) depends on temperature, Eq. (5.2.26) implies that the total number of links E(O) X,(O)
C
is a function of temperature 19. (ii) Concentrations of nonreplacing links Xm coincide for different kinds of links X1 =X2 . . . . .
XM=X,
(5.2.27)
and the parameter X decreases in 6). Since links of different kinds have the same rigidity c, it is natural to assume the strength distributions for links of different kinds to coincide as well. The strength of an elastic link is characterized by its ultimate strain: a link breaks when its length exceeds some critical value due to micro-Brownian motion of molecules. Since the growth of temperature leads to an increase in amplitudes of random oscillations, this growth enlarges average elongations of links and reduces the number of links that can bear these deformations without failure.
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
286
(iii) Concentrations T/m of adaptive links of various kinds are independent of temperature. Substitution of expressions (5.2.24) to (5.2.27) into Eq. (5.2.12) implies that 1 - x(o) ]J'm(O) = TIm
f~m,(O)
E(O)
Tm(O) -" rlm[1- X(O)]'
(5.2.28)
where E(O) = cX,(O),
~m,(O) -
(I)m(O) x,(o)
(5.2.29)
According to Eq. (5.2.28), assumption (iii) implies that for any temperatures O1 and 02 and for any m = 1,..., M, the ratios
/-Lm(O1) ]-Lm(O2) remain constant. (iv) The rates of reformation (I)m, increase in temperature ®. To verify assumptions (i)-(iv), we fit experimental data obtained in the standard relaxation test (5.2.19) for Ny 6+4%LiC1 mixture by La Mantia et al. (1980). We confine ourselves to a viscoelastic medium with two kinds of links, M = 2. To calculate Young's modulus, we employ the formula E(®) -
~r(o) E0
.
(5.2.30)
The parameter E is plotted versus temperature 19 in Figure 5.2.6. Experimental data show that Young's modulus E(O) decreases monotonically in 19. The dependence E(O) may be approximated by the linear function E(®) = al - a20
(5.2.31)
both in the rubber and glass regions. The relaxation measure is determined by the formula (2o(0 = 1 -
o'(t) E~o
where
Qo(t) = -Qo(t). It follows from Eq. (5.2.20) that
M Qo(t) = - E
m=l
~m[1 -- exp(--ym(O)t)].
(5.2.32)
287
5.2. A Model of Adaptive Links in Thermoviscoelasticity
E
I
I
I
30
1
I
®
I
100
5.2.6: Young's modulus E GPa versus temperature (9 °C for Nylon-6. Circles show experimental data. Dotted line shows their approximation by Eq. (5.2.31) with al = 6.017, a 2 - - 0.056. Figure
The adjustable parameters/.L m and ~/m ensure the best fit of experimental data for the function Qo(t) with the use of Eq. (5.2.32). Data obtained in relaxation tests and their approximation by Eq. (5.2.32) are plotted in Figure 5.2.7. To calculate the parameters ~m,, we multiply equalities (5.2.28) and obtain
fIkm,(O )
=
E(O)t.~m(O)'ym(O ).
(5.2.33)
The rates of manufacturing new links (I)1, and (I92, are plotted versus temperature 19 in Figure 5.2.8. Experimental data show that the functions ~m,(O) increase in temperature, reach their maxima ~bm, at the glass transition temperature O g, and remain constant in the glassy state. They are approximated by the piecewise continuously
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
288
ao
~3
,e~oeooooooooooooeoooooooooooeoooooooooooooeo°'
oooooooooOOO°
. ~ " ' ~ .... ~ ............... ~'""
oooooooooooooooooo°°oooooooooo~oooooooooooooo°°°°°°°°°° ,@...oG~. . . . . . . . . . . . . . . . . . .
I
I
'@' . . . . . . . . . .
I
I
I
I
0
I
t
I
I
10
Figure 5.2.7: The dimensionless relaxation measure Q0 versus time t rain for Nylon-6 at various temperatures ®. Symbols show experimental data. Dotted lines show their approximation by Eq. (5.2.32) with 711 = 0.298 and r12 = 0.702. Curve 1: ® = 42°C. Curve 2: ® = 59°C. Curve 3: unfilled circles - - (9 = 71°C, filled circles m (9 = 80oc, asterisks m (9 = 87°C, diamonds m (9 = 950C.
differentiable functions
f~m,(O)
=
[~bm, exp[--Km(1 -- O / O g ) ] , t, ~bm,,
® < ®g' (9 > O~.
(5.2.34)
Summing up the first equalities in Eq. (5.2.28) with respect to m and using the condition
~--~ r/m = 1, m=l
5.2. A Model of Adaptive Links in Thermoviscoelasticity
289
2.4
130
~
ooooooooooooooooo
q~2, •
©
."~' ................
/
..".
©
g g
30
19
100
F i g u r e 5.2.8: The rates of reformation (I)l, (filled circles) and q~2, (unfilled circles) versus temperature 19°C for Nylon-6. Circles show experimental data. Dotted lines show their approximation by Eq. (5.2.34) with qbl, = 2.18, K 1 "- 9.329 and 4~, = 89.37, K2 = 1.612.
we obtain M
X(O) = 1 - E(®) Z
]'km(O)"
(5.2.35)
m=l
The concentration of nonreplacing links X is plotted versus temperature 19 in Figure 5.2.9. Experimental data show that the function X(®) decreases in temperature, tends to some limiting value X~ as 19 ---, ®g, and remains constant and equal to X~ above the glass transition temperature. This function is approximated by the piecewise
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
290 linear function
x(O) = {bl X~, -
b20,
00 >< Og, (~g.
(5.2.36)
After determining the functions E(O), x(O), and ]2,m(O), the parameters '0m are calculated as
E(O)/a,m(O) "17m
1 - X(O) "
Figures 5.2.7 to 5.2.9 demonstrate physically correct behavior of the material functions in the framework of the model of adaptive links:
6
..o
"'6" ......
n 30
n
n
i O
............ "?"" ©
n
I 100
Figure 5.2.9: The concentration of nonreplacing links X versus temperature O °C for Nylon-6. Circles show experimental data. Dotted line shows their approximation by Eq. (5.2.36) with bl = 1.241, b2 = 0.0144, and X~ = 0.212.
5.2. A Model of Adaptive Links in Thermoviscoelasticity
291
1. Young's modulus E decreases monotonically in temperature. 2. The concentration of nonreplacing links X decreases in temperature and tends to a limiting value Xo~close to zero. 3. The rates of reformation ~m. increase in temperature and tend to some limiting values that depend weakly on temperature in the glassy state. 4. Relaxation curves obtained at various temperatures above the glass transition temperature ®g practically coincide with one another and determine a unique relaxation curve in the glassy state. To demonstrate that fair agreement between theoretical and experimental results for Nylon-6 is more than a simple coincidence, we repeat the preceding calculations for polyisobutylene with @g = - 7 4 ° C , using Eqs. (5.2.30), (5.2.32), (5.2.33), (5.2.35), and experimental data presented in Aklonis et al. (1972). The Young modulus E is plotted in Figure 5.2.10, the relaxation measure ~)0(t) is depicted in Fig0 0
"
E
I
-90
I
I
I
I
I
I
(9
I
I0
-60
Figure 5.2.10: Young's modulus E GPa versus temperature (9 °C for polyisobutylene. Circles show experimental data. Dotted line shows their approximation by Eq. (5.2.31) with al = - 14.226, a 2 = 0.223.
292
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
I
~ ' ~
.....~
........ ~.a
.................
2....*. ....................................
o(~OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
Oo ooo~OooOOO
%
S
~" "
O"
• •
O0
oooooooooooooooooooo(~eooooooooooooooooooooooooooooooooooooo '
oooooooO ooo°°°°°°°°° o
....'" o
....~ ............................................
.6
oo
•
o
,6
©.""
ooO~o°°°°°°°°
.~..
......
°°
©... o •
)
I
0
I
I
I
I
I
I
t
I
I
100
Figure 5.2.11: The dimensionless relaxation measure Q0 versus time t min for polyisobutylene at various temperatures (9. Symbols show experimental data. Dotted lines show their approximation by Eq. (5.2.32) with ~71 = 0.547 and r/2 = 0.453. Curve 1" t9 = -82.6°C. Curve 2:19 = -79.3°C. Curve 3:19 = -76.7°C. Curve 4: unfilled circles 19 = -74.1°C, filled circles m 19 = _70.6oc, asterisks - - 19 = -66.5°C, diamonds m19 = -62.5oc.
ure 5.2.11, and the parameters CI)m, and X are presented in Figures 5.2.12 and 5.2.13. These figures demonstrate fair fit of experimental data as well. To develop the model of adaptive links, we employ results of quasi-static relaxation tests. Thus, it is of interest to demonstrate its ability to adequately predict experimental data in dynamic tests under the action of periodic loads. Let us consider the deformation program (5.2.21), which is determined by the amplitude c0 and the frequency ~o. Confining ourselves to steady oscillations, we arrive at formulas (5.2.23) for the storage modulus Et(®, ~) and the loss modulus E ' ( ® , ~). We calculate the modulus E ~ and the loss tangent tan 3 = E ' / E ~ for Ny 6+4%LIC1 mixture at various temperatures 19 and various frequencies ~o. The
5.2. A Model of Adaptive Links in Thermoviscoelasticity 0.4
IP .........
293 • ...........
.0 ...........
• ........
5.0
(I)2,
/
/
I
-90
I
I
I
I
I
(9
I
I
-60
Figure 5.2.12: The rates of reformation ~1, (filled circles) and ~2, (unfilled circles) versus temperature 19°C for polyisobutylene. Circles show experimental data. Dotted lines show their approximation by Eq. (5.2.34) with qbl, = 0.39, K1 = 22.834, and q~2, = 3.94, K2 = 5.848.
results of numerical simulation are plotted in Figure 5.2.14. The storage modulus E t decreases significantly (by a decade) in the vicinity of the glass transition temperature. At a fixed temperature 19, E ~increases in ~o; however, this dependence is rather weak. The frequency of oscillations to affects significantly the dependence tan 8 versus 0 . At low frequencies, the loss tangent increases monotonically in temperature, reaches its maximum in the vicinity of the glass transition temperature and decreases with the growth of temperature in the glassy state. With an increase in oJ, the point of maximum is replaced into the region of higher temperatures. At high frequencies, the dependence of the loss tangent on temperature becomes monotonic. A small nonmonotonicity may be seen in the neighborhood of the glass transition temperature only, but the maximal losses occur in the glassy state at higher temperatures.These
294
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
0.3
©
.©
0
I
I
n
I
I
........ ( ~ .......... ~ ) ........... ( ~ ......
n
-90
I
(9
I
I -60
Figure 5.2.13: The concentration of nonreplacing links X versus temperature (9 °C for polyisobutylene. Circles show experimental data. Dotted line shows their approximation by Eq. (5.2.36) with bl = -2.0697, b2 = 0.0288, and X~ = 0.040.
qualitative results are in good agreement with experimental data for other polymeric materials [see, e.g., dependencies for poly (vinyl chloride) in Aklonis et al. (1972) and for poly (chloro-tri-fluoroethylene) and poly (vinyl fluoride) in Ward (1971)].
5.3
Constitutive Models for the Nonisothermal Behavior
This section is concerned with nonisothermal behavior of viscoelastic media in the vicinity of the glass transition temperature. The subject is of essential interest owing to its numerous applications in polymer engineering for predicting residual stresses
295
5.3. Constitutive Models for the Nonisothermal Behavior
101
E/
-
10-1 i 0
"...
I
I
I
3
I0
I
I
0.5
tan 3
0 30
Figure 5.2.14:
®
100
The storage m o d u l u s E ~(GPa) and the loss tangent tan ~ versus temperature t9 °C for a Nylon-6 specimen driven by periodic tensile load. Curve 1: to = 2 Hz, curve 2: to = 10 Hz, curve 3: to = 100 Hz
296
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
built up in polymers and polymeric composites [see, e.g., Advani (1994), Kenny and Opalicki (1996), Kominar (1996), Unger and Hansen (1993a, b)]. Our objective is to derive a constitutive model that (i) Adequately predicts the response in viscoelastic media under nonisothermal loading. (ii) Is relatively simple to be employed for numerical simulation of manufacturing processes. The first model for the effect of temperature on the viscoelastic behavior was proposed by Leaderman (1943), who assumed that elastic moduli are temperature-independent and the relaxation times Tm change similarly to each other
Tm(O) Tm(Oo)
a(O).
(5.3.1)
Here (9o is some reference temperature, and a(O) is a temperature shift factor. According to Eq. (5.3.1), the viscoelastic response at the current temperature 19 coincides with the response at the initial temperature (90, provided it is observed in the pseudo-time t
so(t)- a(O)"
(5.3.2)
The concept of pseudo-time was extended to nonisothermal loadings by Morland and Lee (1960), where the formula t
~(t) =
fo
ds a(O(s))
(5.3.3)
was suggested instead of Eq. (5.3.2). Nonlinear constitutive equations in viscoelasticity based on the concept of pseudo-time (internal time) were derived by Schapery (1964, 1966). Other environmental effects on the viscoelastic response were accounted by Chapman (1974) in the framework of Eq. (5.3.3). With reference to the free volume theory [see Doolittle (1951b)], Losi and Knauss (1992a, b) suggested that the temperature shift factor a is a function of the free volume fraction f, which, in turn, is connected with the thermal history by a linear integral equation. A similar approach was proposed by La Mantia et al. (1981), where an ordinary differential equation were developed for the free volume fraction f. Experimental data show that constitutive models based on the time-temperature superposition principle (5.3.1) ensure an acceptable accuracy within a restricted range of temperatures only. In larger intervals, this assertion leads to significant discrepancies between observations and their prediction. To reduce these discrepancies, the effect of temperature on elastic moduli should be taken into account. The simplest way to account for the thermal effects consists in assuming some dependence of Young's modulus E on temperature, while other material parameters (except for the relaxation times) are treated as temperature-independent [see Aklonis et al. (1972)]. This hypothesis introduces an additional vertical shift of compliance
Wineman (1971)]. Stouffer (1972) called that model the thermal-hereditary theory. Three shortcomings of the thermal-hereditary model may be mentioned: 1. There is no experimental validation of basic assumptions and their conclusions. 2. The model essentially employs the linearity of the stress-strain relations and cannot be extended to nonlinear constitutive equations. 3. The model is indifferent to what standard tests are selected as basic. The latter means that choosing either creep tests or relaxation tests as experiments in which the material parameters are determined, we arrive at two constitutive models that are not equivalent to each other. Our purpose is to derive a new model for the nonisothermal viscoelastic behavior of polymers that accounts for both changes in elastic moduli and relaxation times. We concentrate on nonaging viscoelastic materials, but the exposition begins with aging media, where the response explicitly depends on time. Based on the multiplicative presentation for the function X at an arbitrary temperature 19 (see Chapter 2), we describe formation and breakage of adaptive links by a system of kinetic equations. For isothermal loading, coefficients in these equations are known functions of temperature. Assuming these equations to be fulfilled under nonisothermal conditions as well, we arrive at a new constitutive model for thermoviscoelastic media. A similar approach, but without account for breakage of adaptive links, was proposed by Buckley (1988) and Buckley and Jones (1995). The exposition follows Drozdov (1997a, b, c).
5.3.1
Constitutive Equations for Isothermal Loading
Based on Boltzmann's superposition principle and neglecting thermal expansion of a specimen, we present the constitutive equation of an aging, linear, viscoelastic medium under uniaxial loading at a fixed temperature 19 in the form (5.2.3)
M Cm(O) { Xm,(t, O, O)e(t) + fo, OXm,(t, ~', ®)[e(t) -
tr(t) = ~
e(~')] d~-}, (5.3.4)
m=l
where or is the stress, e is the strain, M is the number of different kinds of adaptive links, Cm(®) is the rigidity of a link, and Xm,(t, ~', 19) is the number of links of the mth kind arisen before instant ~- and existing at instant t.
Chapter 5. Constitutive Relationsfor ThermoviscoelasticMedia
298
The function Xm.(t, ~', 19) is expressed in terms of the rate of reformation of adaptive links dPm(t,19) and the breakage function gm(t, ~', 19) with the use of Eqs. (5.2.1). For a nonaging viscoelastic medium, see Eq. (5.2.6), the function
gm,o(t -- T, O) = gm(t, "r, O)
(5.3.5)
satisfies the ordinary differential equation (5.2.8)
Ogm,o(t, 19) = 3'm(O)[ 1 -- gm,o(t, O)],
gm 0(0, 19) = 0,
0t
(5.3.6)
where the relaxation rate ~/m(®) is determined by Eq. (5.2.12). In the general case of an aging viscoelastic medium, we return to the initial notation (5.3.5) and present Eq. (5.3.6) in the form
O~gm 0----~(t,"r, O) = "ym(O)[1 -- gm(t, r, O)],
gm(r, ~', ®) = 0.
(5.3.7)
The function gm(t, r, 19) characterizes reduction in the number of links (owing to their breakage) in any subsystem containing links of the mth kind. For example, if a subsystem consists of ffq'm('r, O) links at instant ~', then the number of links at instant t becomes [see Eq. (5.2.1)]
ffq'm(t, O) = ffq'm(T, O)[ 1 -- gm(t, T, O)].
(5.3.8)
Differentiating Eq. (5.3.8) with respect to time and using Eq. (5.3.7), we obtain
OSV'm(t, 19) = --..~/'m(T,o)Ogm(t, "r, O) 0t dt = -JV'm('r, O)'ym(O)[1 - gm(t, r, O)].
(5.3.9)
Combining Eqs. (5.3.8) and (5.3.9), we arrive at the formula
1
SV'm(t, ~))
°~'~m (t, 19) Ot
= -'ym(O),
(5.3 10)
which implies that the relative rate of decrease in the number of links in any subsystem of adaptive links depends only on the current temperature O. We recall that the initial links (existing at the instant t = 0) are divided into two types: links of type I are not involved in the process of replacement, whereas links of type II replace each other. Denote by Nm,1(t, ®) and Nm,z(t, O) the numbers of initial links of types I and II, respectively, and by Nm(t, T, ®) the number of links of type II that arose (per unit time) at instant ~- and exist at instant t. The subscript index m means that these quantities are calculated for adaptive links of the mth kind. Since the amount Nm,1 is independent of time, we can write
ONm'l(t,O) = 0, 0t
Nm 1(0, ®) = Xm(O)Xm,(O, O, O) ' '
(5.3.11)
where Xm(®) is concentration of nonreplacing links and Xm,(O, O, O) is the initial number of links of the mth kind.
5.3. Constitutive Models for the Nonisothermal Behavior
299
By analogy with Eq. (5.3.10), the functions Nm,2 and Nm are governed by the differential equations
1 ONm'2(t, 19) = - ' y m ( O ) , Nm,2(t, ~)) ot
ONm
1
--(t, Nm(t, T, ~)) Ot
~-,®) = -Tm(®)
(5.3.12)
with the initial conditions Nm,2(0 , O ) -~
[1
- Xm(O)]Xm,(O, 0, O),
Nm(T , T, O ) "-" (I)m(T, 1~),
(5.3.13)
where dPm(t, O) is the rate of reformation of adaptive links of the mth kind (the number of links arising per unit time). The total number of the initial links of the mth kind equals the sum of the number Nm,1 of links of type I and the number Nm,2 of existing links of type II
Xm,(t, O, 19) = Nm,1 (t, 19) + Nm,2(t, 0).
(5.3.14)
The number of links of the mth kind arisen at instant ~" and existing at the current instant t equals
OXm* (t, T, 19) = Nm(t, ~, 6)).
(5.3.15)
Equation (5.2.3) together with Eqs. (5.3.14) and (5.3.15) implies the constitutive equation of an aging viscoelastic medium (
M
Or(t) -- Z Cm(O)~ [Nm'l(t, k m=l +
= Z
/o
O) +
Nm,2(t, O)]E(t)
Nm(t, ~, ®)[e(t) - e(-r)] dr
Cm(O)Xm,(O, O, 0)
}
[nm,1(t, 0) + nm2(t, O)]e(t)
m=l
+
nm(t, ~, ® ) [ e ( t )
-
~(~')] dl"
}
where
nm,l(t,O) =
Nm,l (t, O) Xm,(0, 0, O)'
Nm,2(t, O) nm,2(t, O) = Xm,(O, O, 0 ) '
,
(5.3.16)
300
Chapter 5. Constitutive Relations for Thermoviscoelastic Media nm(t, % O) nm(t, T, O) = Xm,(O, O, 0)"
(5.3.17)
We substitute expressions (5.3.17) into Eqs. (5.3.11) to (5.3.13), employ Eqs. (5.2.12), and find that the functions nm,1, nm,2, and nm satisfy the equations 1
Onm,1 (t, O) = O,
nm,l (t, 19) 3t 1
nm,2(t, ~)
3nm'2 (t, 19) = -~/m([~)), Ot
1 Onm(t, ~', 19) = -- ~/m(®) nm(t, "r, O) Ot
(5.3.{8)
with the initial conditions nm,~(0, O) = Xm(O),
nm,2(0, 1~) = 1 -- Xm(~),
nm(~, ~', 19) = ~/m(O)[1 -- Xm(®)].
(5.3.19)
Suppose that adaptive links of different kinds have the same rigidity [see Eq. (5.2.26)] Cm([~)) = C([~).
(5.3.20)
Xm,(O, O, O) = TIm(O)X,(O),
(5.3.21)
We set
where M
X,(O) = ~ Xm,(O, O, 6))
(5.3.22)
m=l
is the total number of initial links, and 7~m(~) is concentration of initial links of the mth kind. Combining Eqs. (5.3.20) to (5.3.22) with Eq. (5.3.16) and introducing Young's modulus E(®) = c(®)X,(®),
(5.3.23)
we present the constitutive equation for an aging, linear, viscoelastic medium under isothermal loading in the following form: M
~r(t) = E(®) Z m=l
+
f
~m(O)~ [nm,l(t, O) -+-nm,2(t, ®)]E(t) k.
nm(t, ~', ®)[e(t) - E(~')] d~" .
(5.3.24)
301
5.3. Constitutive Models for the Nonisothermal Behavior
Integration of Eqs. (5.3.18) with the initial conditions (5.3.19) yields rim, 1 (t,
19)
= Xm({~),
nm,2(t, 19) = [1
-
Xm(~)]
exp[-3'm(®)t],
nm(t, 1", ~) = [1 - Xm(lO)]~/m(O)exp[-~/m(t - ~')].
(5.3.25)
Substituting expressions (5.3.25) into Eq. (5.3.24), we find that
or(t) = E(O)
(
e(t)
-
~m(O)[1
--
Xm(~))]Tm(O)
t
exp[-Tm(t - ~')]e(r)d~"
/
.
m=l
(5.3.26) According to Eqs. (5.3.23) and (5.3.26), not all the parameters of the model of adaptive links can be measured in experiments. For example, Young's modulus E(®) is determined by two independent parameters c(®) and X,(0, 0, ®), and the material viscosity ]-Lm(O) = T~m(O)[1 -- X m ( O ) ]
is expressed in terms of two parameters T~m and Xm. To eliminate uncertainties, we suppose that under heating and cooling the total number of links X, remains unchanged X,(®) = X,,
(5.3.27)
whereas the temperature affects rigidity of links c only. The latter means that links become weaker (at heating) or stronger (at cooling). Assumption (5.3.27) enables us to distinguish (i) curing (polymerization) of viscoelastic materials, when new crosslinks arise and the parameter X, increases, and (ii) heating and cooling, in which the total number of links remains fixed. It follows from Eq. (5.3.23) that at an arbitrary temperature 19, c(®) -
E(®) . X,
(5.3.28)
To be consistent, by presuming the total number of links X, to be temperatureindependent, we should assume the numbers of links of each kind to have the same property. This implies that the concentrations T~mshould be independent of temperature T~m(~ ) -- T/m.
(5.3.29)
Hypotheses (5.3.27) and (5.3.29) eliminate ambiguities in the model and permit material parameters to be found in standard tests.
302
5.3.2
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
Constitutive Equations for Nonisothermal Loading
We begin with a thermorheologically simple medium, where Young's modulus E and the intensities of relaxation ~L/,m are independent of temperature, whereas the characteristic times of relaxation Tm change in temperature in accordance with Eq. (5.3.1). In the framework of the model of adaptive links, these assumptions mean that the parameters c and Xm are temperature-independent, whereas the dependence of ~/m on temperature has the form ~/m(O) =
'~/m(O0) a(O) "
(5.3.30)
According to the McCrum model, Young's modulus E depends on temperature, the characteristic times of relaxation Tm satisfy the time-temperature superposition principle (5.3.1), and the intensifies of relaxation ~m remain constant. The latter is equivalent to the assumption that the ratio of relaxed and nonrelaxed compliances does not vary in temperature. In the framework of the model of adaptive links, these assumptions imply that the parameters Xm are temperature-independent. We assume that all the parameters E, Xm, and ~/m depend on temperature, which enables us to account for the effect of temperature on the ratio of relaxed and nonrelaxed compliances [the so-called mapping hypothesis; see Stouffer and Wineman (1971)]. In the model of adaptive links, the effect of temperature on the parameters Xm may be explained as follows: since rigidity of links depends on temperature, it is natural to treat their strength as temperature-dependent as well. Strength of a link is characterized by the ultimate strain: a link breaks when its length exceeds some critical value due to micro-Brownian motion of molecules. The growth of temperature leads to an increase in amplitudes of random oscillations and enlarges "average" elongations of links. As a result, it reduces the number of links that can bear these strains without rupture. Since the latter is characterized by the concentration of nonreplacing links (i.e., the links that do not break at a given temperature), we find that Xm(®) should decrease monotonically in (9. We introduce the following hypotheses: (H 1) The parameters X. and r/m are temperature-independent, whereas the parameters E, Xm, and ~/m are functions of the current temperature O(t). (H2) The time-temperature superposition principle (5.3.30) is valid with some shift factor a(19). (H3) The material functions E(O) and a(®) are found by fitting data in isothermal creep and relaxation tests. (H4) Concentrations of nonreplacing links Xm coincide for different kinds of links Xl = )(2 . . . . .
XM = X.
(5.3.31)
The function X(®) decreases monotonically in O below the glass transition temperature ®g, vanishes above 0 8, and it is continuous at 08 x(O 8) = 0.
(5.3.32)
5.3. Constitutive Models for the Nonisothermal Behavior
303
To introduce other hypotheses regarding X(®), we recall that this function characterizes strength of adaptive links. Links arising at a higher temperature are assumed to have a higher strength, which means that some links of type II created at a temperature O, and destined to break at that temperature become links of type I (i.e., links not involved in the process of replacement) at a temperature 19 < 19,. In the model of adaptive links, strength of a link is characterized by the ultimate elongation, which the link can bear without rupture. We suppose that some links arisen at the temperature 19, are so firm that can bear any deformations caused by micro-Brownian motion at a temperature 19 < O,, whereas links arisen at the temperature 19 break because of thermal motion. To avoid overcomplication of the model (where the parameter X becomes a function of two variables: the temperature 19, at which links have been formed, and the current temperature t9), we assume that (H5) The strength of links has a threshold character: links arising at any temperature 19, _> Og can become nonreplacing below the glass transition temperature ®g, whereas links arising at a temperature ®, < ®g annihilate with the growth of time. According to this hypothesis, the concentration of nonreplacing links X is not a material function, since it depends on the rate of cooling for a viscoelastic specimen manufactured at some temperature 19, above the glass transition temperature Og. It is assumed that (H6) The derivative
~(o)
=
dx -d-~(o)
is a material function; i.e., the function 8(19) is independent of the rate of cooling or heating. Let us consider rapid cooling of a viscoelastic specimen from the glass transition temperature Og to some temperature O. A rapid change in temperature means that the characteristic time of quenching is essentially less than the minimal characteristic time of the stress relaxation Tm, and no links of type II break during cooling. Denote by X ° (t9) concentration of nonreplacing links determined in the standard relaxation tests immediately after quenching (when physical aging does not affect the relaxation curves). Since 8(®) is independent of the loading program, we can write s(o)
dx ° = -d-~(o).
Integration of this equation with the boundary condition (5.3.32) yields X°(®)
=
-
~
g
8(O)dO.
(5.3.33)
The function 3(O) characterizes the rate of transforming links of type II into links of type I under cooling. It follows from Eqs. (5.3.19) and (5.3.31) that the
304
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
concentration nm,1 (0, O) of links of type I and the initial concentration nm,2(O, O) of links of type II satisfy the equations
Onm,1 _ t~(O), ~0
(5.3.34)
Onm,2 _ --t~(O) 80
(5.3.35)
with the initial conditions
nm,l(O, Og) "- O,
nm,2(O , Og) = 1.
(5.3.36)
Integration of Eq. (5.3.35) with the use of Eqs. (5.3.33) and (5.3.36) implies that nm,2(0, O) -- nm,2(0, Og) -au
t~(0) dO = 1 - X ° ( 0 ) .
(5.3.37)
Combining this equality with Eqs. (5.3.34) and (5.3.35), we find that
1 ognm,1 nm,2 a O
~(0) 1 -- X* ( O ) '
6(0)
10nm,2 m
nm,2 a~)
1 - X 0(19)"
(5.3.38)
Replacing the derivative with respect to temperature in Eqs. (5.3.38) by the derivative with respect to time, we arrive at the formulas
1 Onto,l _ nm,2 0t
10nm,2 nm,2 0t
t~(O(t))
dO --(t),
1 -- X ° (O(t)) dt
-
t~({~(t))
dO ~(t).
1 - xO ( ® ( t ) ) dt
(5.3.39)
Equations (5.3.39) determine changes in the concentrations of links of type I and the initial concentrations of links of type II caused by transformation of links of type II into links of type I under rapid cooling. Under slow cooling, two processes occur simultaneously: breakage of links of type II and their transformation into links of type I. We assume that (H7) Under cooling, the processes of breakage of links of type II and their transformation into links of type I are independent of each other. This hypothesis together with Eqs. (5.3.18) and (5.3.39) implies the following equations for the functions nm,l(t) =nm, l(t, 0(')) and nm,2(t) = nm,2(t, O(')):
1 nm,2(t) 1
nm,2(t)
dnm,1(t) = dt
6(O(t))
~dO (t),
1 - X ° (®(t)) dt
dnm,2 (t) = -Tm(O(t)) dt
1 6(O(t)) X° (®(t)) dO d--t-(t).
(5.3.40)
5.3. Constitutive Models for the Nonisothermal Behavior
305
Equation (5.3.18) for the function nm(t, "r) = nm(t, "r, O(')) is valid for nonisothermal processes as well as for isothermal processes
1 Onm (t, ~') = --Tm(O(t)). nm(t, ~') Ot
(5.3.41)
The initial condition (5.3.19) is transformed with the use of Eq. (5.3.31) and Eqs. (5.3.34) to (5.3.37)
nm('r, 7") = Tm(l~('r))[1 - Xm(l~('r))] = "ym(O(T))[
1 - X ° (®(~'))]
= Tm(®(~'))[1 - n m , l(~')].
(5.3.42)
The constitutive equation (5.3.24) together with the ordinary differential equations (5.3.40), the partial differential equation (5.3.41), and the initial conditions (5.3.36) and (5.3.42) determines the response in a nonaging, linear, viscoelastic medium under cooling from the glass transition temperature ®g. If a specimen is cooled from some initial temperature 190 < O g, then the governing equations (5.3.24), (5.3.40), (5.3.41), and the initial condition (5.3.42) remain unchanged, while the conditions (5.3.36) read
nm, 1(0) -- )('(O0),
nm,2(0 ) = 1 -- X(®0),
(5.3.43)
where X(®0) is given. Let us consider heating of a viscoelastic specimen from some temperature ®0 < ¢9g. Since the number of adaptive links of type I decreases under heating, we employ the differential equation (5.3.34) for the function nm,1(t) dO
dnm,1 (t) = 6(O(t))9-((nm l(t))---~(t), dt
(5.3.44)
where M ( t ) is the Heaviside function 9-/'(t)=
1, 0,
t->0, t<0.
At the current instant t, the number of initial links of type II equals the sum of the number of initial links that were of type II at the instant t = 0 and exist at instant t, and the number of initial links of type I which have been transformed into type II within the interval (0, t) and exist at instant t
dt (~')[ 1 - gin(t, ~, 19('))] d~'. (5.3.45) nm,2(t) = nm,2(0)[ 1 - gm(t, O, 19('))] -- foot dnm,1 Differentiating Eq. (5.3.45) with respect to time and using Eqs. (5.3.7) and (5.3.44), we find that
dnm,2 (t) = - ~ @m (t, dt Ot
O, l~('))nm,2(0) - dnm I (t)[ 1 - gm(t, t, 19('))] dt'
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
306
+
t dnm,1 Oqgm dt (~')---~(t, ~', 19(.))dT
fO
dnm,1
{
= -- ~-~ (t) -- Tm(O(t)) nm,2(0)[1 --
-
gm(t, O, O('))]
fot dnm a(r)[1- gm(t, ~', O('))]dT} dt'
dO = - 6(O(t))J-f(nm, l(t))--dT(t)- "gm(O(t))nm,2(t).
(5.3.46)
Under heating, the functions nm,l(t) and nm,2(t) satisfy the differential equations (5.3.44) and (5.3.46) with the initial conditions (5.3.43). The constitutive equation (5.3.24) and the differential equation (5.3.41) for the function nm(t, r) are the same for heating and cooling. 5.3.3
Three-Dimensional
Loading
The constitutive equation (5.3.24) describes uniaxial deformation of a viscoelastic specimen. To derive constitutive equations for three-dimensional loading, we present the stress tensor 6" and the strain tensor ~ in the form 1 6" = ~o-i + ~,
1 ^ ~ = ~ e I + ~,,
(5.3.47)
where I is the unit tensor, o" and e are the first invariants, and ~ and ~ are the deviatoric parts of the stress and strain tensors, respectively. Referring to experimental data discussed in Chapter 2, we assume that the volume deformation is purely elastic. Accounting for the thermal expansion, we write
or(t) = 3K(O(t)){E(t) + 3a[Og - @(t)]},
(5.3.48)
where K(®) is a bulk modulus, and a is a thermal expansion coefficient, which is treated here as a constant. The deviatoric parts of the stress and strain tensors are connected by the constitutive equation similar to Eq. (5.3.24) ~(t) = 2G(®(t))
~M rim / [nm,l(t) + nm,2(t)]~(t) + f0 t nm(t, ~')[~(t) m=l M
= 2G(O(t)) Z
m=l
rim { [nm,a(t)
+ nm,z(t) + Om,~(t)]~'(t) - ~m,Z(t)},
/
~(z)] d~"
(5.3.49)
where G(@) is a shear modulus, and q&,l(t) =
nm(t, ~')d~',
~m,Z(t)=
nm(t, r)~(~')d'r.
(5.3.50)
5.3. Constitutive Models for the Nonisothermal Behavior
307
Differentiating the first equality (5.3.50) and using Eqs. (5.3.41) and (5.3.42), we find that
d~m ,1 (t) = nm(t, t) + fot Onm t, r) dr dt --~--( ~0"t 7m(19(t))nm(t, r)dT
=
"ym(l~(t))[1
-
nm,1(t)]
=
7m(O(t))[ 1
-
nm, 1(t) - qtm,1( t ) ] .
-
(5.3.51)
Similarly,
d ~m,2 (t) = 7m(O(t)){[ 1
27
It follows equations specimen equations
rim,1
(t)l~(t) - ~m,2(t)}.
(5.3.52)
from Eq. (5.3.49) that it is not necessary to solve the partial differential (5.3.41) in order to calculate the mechanical response in a viscoelastic under cooling. Instead, it suffices to integrate the ordinary differential (5.3.51) and (5.3.52) with the initial conditions ~m,l(0)
5.3.4
-
= 0,
I~m,2(0)
The Standard Thermoviscoelastic
= 0.
(5.3.53)
Solid
We now return to uniaxial loading in order to derive an analog of the Zener model (the standard viscoelastic solid) for nonisothermal processes. For a fixed temperature (9, the constitutive equation of the Zener model reads [see Eq. (2.1.5)] d o + or = (El + E2)de e dt T ~ -I- E 1 ~ .
(5.3.54)
Here E 1 = E1((9), E2 = E2(®) are rigidities of the springs, r / = r/(@) is viscosity of the dashpot, and T = rl/Ee is the characteristic time of relaxation. Equation (5.3.54) is equivalent to the integral constitutive equation
[ /0 ] [ ( ')]
o'(t) = E(®) e(t) + Here
Qo(t,®)=-[1-x°(®)]
E(®) =
El(O)
+ E2(O),
Q0(t - r, @)e(r)dr
1-exp
-T(®)
.
(5.3.55)
'
El(®) X ° (19) = E1(19) + E2(®)'
and the superposed dot denotes differentiation with respect to time.
(5.3.56)
308
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
For isothermal relaxation tests with e(t) = eog-E(t),
Eq. (5.3.55) implies that or(t)=E(O)Eo
{ 1 - [ 1 - X o(19)] [1 - e x p ( - T ( ®t ) )1 } "
In particular, o(0) = E(®)¢0 = [El(O) + E2(O)]E0, or(oo) = E(O)X ° (0)~.o = E1 (0)~0, which means that in the Zener model the initial and limiting moduli, as well as the relaxation time, are temperature-dependent. Experimental data for a number of polymeric materials [see, e.g., Nakamura et al. (1986) and Narkis and Tobolsky (1969)] demonstrate that the modulus E2 is temperature-independent E2(®) = E2,
(5.3.57)
whereas the modulus E1 decreases in temperature O and vanishes at the glass transition temperature. We accept the linear dependence
(5.3.58)
El(O) = B(Og - 0 ) ,
which may be treated as a linearization of the function E1 (O) in the vicinity of In the latter case,
Og.
dE1 a
=
- Od ----~-(Og)"
The relaxation time T obeys the Arrhenius dependence [see, e.g., Buckley and Jones (1995) and McCrum (1984)] a(®)-
T(O)T(Og)
exp [ ~ ( 1 -~-
1) Og
(5.3.59)
where A is an apparent activation enthalpy, and R is Boltzmann's constant. To determine parameters of the model, we consider isothermal dynamic test with e(t) = c0 sin tot, where to is the frequency of oscillations. According to Ward (1971), the storage modulus U(to, O), the loss modulus E"(to, O), and the loss tangent tan 8(to, 19) are calculated as E2(Tto)2 E ' = E~ + 1
+ (Too)2'
(5.3.60)
5.3. Constitutive Models for the Nonisothermal Behavior E II ~__
tan 3 =
309
E2 Tto 1 + (Tto) 2'
(5.3.61)
E2 Tto El[1 + (Tto) 2] + E2(Tto) 2"
(5.3.62)
Since E1 vanishes at the glass transition temperature, Eq. (5.3.62) yields T(Og)
=
to tan tS(to, Og)
(5.3.63)
Substituting expression (5.3.63) into Eq. (5.3.61), we obtain
E2 = E'(to, Og)
1 + tan 2 6(to, Og) tan 6(to, Og)
(5.3.64)
Equation (5.3.61) can be presented in the form (Tto) 2 - P(to, O)Tto + 1 = 0,
(5.3.65)
where P(to, O) =
E2 E"(to, 19)
E'(to,
@g) 1 +
E'(to, ®)
tan 2 6(to, ®g) tan t~(to, @g)
Resolving quadratic equation (5.3.65) with respect to Tto, we obtain Tto =
P(to, O) + 4P2(to, O) - 4 2 '
(5.3.66)
which implies that a(O) =
P(to' 19) + v/P2(to, 19) - 4 P(to, Og) + 4e2(to, Og) - 4"
(5.3.67)
Finally, the modulus E1 is found from Eq. (5.3.60) E1 = E~(to, O) -
E2T2(to, O)to 2 1 + TZ(to, O)to2'
(5.3.68)
where Ee and T are determined by Eqs. (5.3.64) and (5.3.66). The shift factor a and the elastic modulus E1 for an epoxy resin are plotted in Figures 5.3.1 and 5.3.2. These figures demonstrate fair agreement between experimental data and assumptions (5.3.58) and (5.3.59). Some discrepancies in Figure 5.3.1 at low temperatures may be explained by the effect of the/3-relaxation transition. The storage modulus U(to, 19) and the loss tangent tan 8(to, 19) at the frequency to = 10 Hz are plotted in Figures 5.3.3 and 5.3.4. These figures demonstrate acceptable agreement between experimental data and their prediction by the model except for a narrow region in the vicinity of the glass transition temperature. Discrepancies
310
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
observed in this region may be explained by the presence of a number of relaxation times that jointly affect the material response near the glass transition temperature. To account for their effect, a series of Zener models should be considered. To derive an analog of the standard thermoviscoelastic medium for three-dimensional deformations, we preserve the "elastic" connection (5.3.48) between the first invariants of the stress and strain tensors and replace the scalar constitutive equation (5.3.54) by the formula d~ + ~ = 2(G1 + G2)d0 ~, dt T - ~ + 2G1-~,
loga
o•O
I 2
I
I
i1/i
I
I
I
~000/o
O
O
I 3
Figure 5.3.1: Logarithm of the temperature shift factor a versus the reciprocate temperature 1000/®K for an epoxy resin. Circles show experimental data obtained by Nakamura et al. (1986). Solid line shows their approximation by the function (5.3.59) with ®g = 140°C and A/R = 2.226. 103K.
311
5.3. Constitutive Models for the Nonisothermal Behavior
El
o
o
o
50
®
150
Figure 5.3.2: Young's modulus E1 GPa versus temperature 19°C for an epoxy resin. Circles show experimental data obtained by Nakamura et al. (1986). Solid line shows their approximation by the function (5.3.58) with Og = 140°C and B = 6.408 MPa/°C.
where G1 (®) and G2(O) are shear moduli. Let Eqs. (5.3.57) and (5.3.58) be satisfied for the shear moduli G1 ( 0 ) = B(Og - O),
G2(O) = G2.
Substitution of these expressions into Eq. (5.3.56) implies that G(O) = G2 + B(Og - 0 ) ,
x°(O) ~(0)
B(Og-~)) =
=
G2 + B(Og - 0 ) ' -
BG2 [G2 + n(Og - ~)]2"
(5.3.69)
312
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
104
E/
10-1
I
I
I
I
I
I
I
50
I
O
140
5.3.3: Logarithm of the storage modulus E ~MPa versus temperature O °C for an epoxy resin. Circles show experimental data obtained by Nakamura et al. (1986). Solid line shows numerical prediction. Figure
The constitutive equation (5.3.49) for the standard thermoviscoelastic medium reads ~(t) = 2G(O(t)){[nl(t) + n2(t) + ~l]O(t) - @2(t)},
where the functions nl, n2,
~1, and ~2 obey Eqs. (5.3.40), (5.3.51), and (5.3.52) B
1 dnl
n2 dt
-
1
n2 dt d~l dt
dO
G2 + B ( O g - 6)) dt '
1 dn2
B
T(O)
T(O)
(1 -
nl(O) = O, dO
n2(0) = 1,
G2 + B(Og - 6)) dt '
1 ~
(5.3.70)
nl -
qtl),
~1 (0)
-
-
0,
313
5.3. Constitutive Models for the Nonisothermal Behavior
100
tan 8
-
o
-
0
9-
10-3
0
0
I
I
I
I
I
I
50
I (9
I
140
Figure 5.3.4: Logarithm of the loss tangent tan 8 versus temperature 19°C for an epoxy resin. Circles show experimental data obtained by Nakamura et al. (1986). Solid line shows numerical prediction.
d~2 1 - --[(1 dt T(@)
- nl)O - ~2],
~2(0) = 0.
(5.3.71)
To demonstrate characteristic features of the standard thermoviscoelastic medium (5.3.70) and (5.3.71), we calculate residual stresses built up in a cylindrical shell under cooling. 5.3.5
Cooling
of a Cylindrical
Pressure
Vessel
A cylindrical pressure vessel is modeled as a two-layered cylinder with length l, inner radius R0, radius of the interface between layers R1, and outer radius R2 (see Figure 5.3.5). The internal layer (a metal mandrel) is treated as an isotropic elastic solid. The external layer models a polymer pressure vessel fabricated on the mandrel.
314
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
Figure 5.3.5: A two-layered cylinder: a metal mandrel (R0 --- r - R1) and a polymeric pressure vessel (R1 --< r --< R2). We suppose that (i) small strains occur in the cylinder, (ii) external body and surface forces are absent, (iii) the cooling process is so slow that inertia forces may be neglected, and (iv) both layers are incompressible. At the initial instant t = 0, the cylinder is in the stress-free state at the glass transition temperature Og for the polymeric material. The cylinder is cooled from the temperature Og to the room temperature (~)r in such a way that the temperature of its boundary surfaces changes according to the law O = O0(t),
(90(0) = Og.
Points of the cylinder refer to cylindrical coordinates {r, 0, z}. Neglecting heat transfer through the edges, we obtain that the temperature depends on time t and polar radius r only, (9 = O(t, r). The temperature t9 in the internal cylinder is uniform in r (9 = O0(t).
(5.3.72)
Assumption (5.3.72) is acceptable provided that thickness of the internal cylinder is small, and its thermal conductivity is large. In the external cylinder, R1 ----- r ----- R2, the temperature 19 obeys the heat conduction equation
0t
- K
\--~-r2
+
10o)
r Or
(5.3.73)
5.3. Constitutive Models for the Nonisothermal Behavior
315
with the initial condition 19(0, r)
=
(5.3.74)
~g
and the boundary conditions
19(t, R1) = ®0(t),
tg(t, R2) = 190(t).
(5.3.75)
The temperature conductivity K is treated here as a constant. Experimental data for several polymers confirm this assumption [see, e.g., Mills (1982)]. We suppose that axial elongation is restricted, and axisymmetrical deformation in the plane (r, 0) occurs in the cylinder. This means that the only nonzero component of the displacement vector is Ur -- u(t, r),
and the nonzero components of the strain tensor equal On
lg
Or'
Er-
E o - r"
(5.3.76)
Combining the incompressibility condition with the formula for the volume deformation E
~" E r -'1-" E0,
we arrive at the relations 0u
Or Ou
+
u -
r
=
3oti(O
-- Og),
< R1,
(5.3.77)
R1 --< r < R2,
(5.3.78)
go
--< r
u
+ - = 3C~e(O - ®g), Or r
where O~i and O~e are coefficients of thermal expansion for the mandrel and the polymeric shell. In sequel, these coefficients are treated as constants. We begin with deformation of the mandrel. It follows from Eqs. (5.3.72) and (5.3.77) that 0 or(Ur)
=
3oq(O0
-
Og)r.
Integration of this equality yields
c(t) 3 u(t, r) = -~oli(l~ 0 -- 19g)r +
(5.3.79)
where C(t) is a function to be found. In particular, 3
u ( t , R1) = -~ ot i ( [~)0 - O g ) R 1
-~
c(t) R1
(5.3.80)
316
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
Equations (5.3.76) and (5.3.80) imply the following nonzero components of the strain tensor 3
C
Er = ~Ogi({~O -- O g ) -- ~-~,
3
C
(5.3.81)
E 0 = ~O~i(O 0 -- O g ) + r-~.
It follows from Eq. (5.3.81) that the nonzero components er, eo, and ez of the deviatoric part ~ of the strain tensor ~ equal 1 C er = - ~ o l i ( O 0 -- O g ) -- - ~ , 1
C
e0 = ~ o ~ i ( O 0 - l~)g) -+- ~-~, ez = - o l i ( ~ ) o
-
(5.3.82)
~)g).
The mandrel is modeled as an incompressible, isotropic, linear elastic medium with the constitutive equation (5.3.83)
?~ = 2Gi~,
where ~ is the deviatoric part of the stress tensor 6-, and Gi is a temperatureindependent shear modulus. This hypothesis is acceptable for metals when the temperature changes within the range of about 100 ° C. Substitution of expressions (5.3.82) into Eq. (5.3.83) implies the following nonzero components of the deviatoric part ~ of the stress tensor 6-: Sr :
Gi
so -- G i
E
2c]
oli(Oo -
Og) -
--~
,
oti(Oo -
~)g) + - - ~
,
Sz = - 2 o t i G i ( O o
-
Og).
(5.3.84)
It follows from Eq. (5.3.84) that Sr -- SO =
-4GiCr -2.
(5.3.85)
The equilibrium equation reads 1
tgO'r + --(O" r -- O'0) = 0,
0r
r
(5.3.86)
where O"r and o'0 are radial and tangential components of the stress tensor 6-. Other equilibrium equations are satisfied identically.
5.3. Constitutive Models for the Nonisothermal Behavior
317
We integrate Eq. (5.3.86) from R0 to R1 and use the boundary condition O'r(t, Ro) = O. As a result, we obtain ¢rr(t, R1) +
[R1
JRo
0" r -- Or0
~ d r
(5.3.87)
= O.
F
Bearing in mind that (5.3.88)
O'r -- 0"0 = Sr -- SO,
and using Eq. (5.3.85), we find that O'r(t, R1) = 4GiC(t)
fR R1 dr o r5 - 2GiC(t)(R°2
-- R12).
(5.3.89)
Equation (5.3.89) expresses the radial stress o-r at the interface between the two layers in terms of the function C(t) that determines the radial displacement of the mandrel. We now consider the external cylinder and present the incompressibility condition (5.3.78) as follows: 0 O---~(ur) = 3C~e(O - Og)r.
(5.3.90)
Integration of Eq. (5.3.90) from R1 to r yields
rl / u(t, R1)R1 + 3O~efRr1[l~(t,O) -
u(t,r) = -
®g]OdO
/•
(5.3.91)
It follows from Eqs. (5.3.76) and (5.3.91) that the only nonzero components of the strain tensor ~ are 1
Er = - - r 2
{ u(t, R1)R1 + 3~e fRr1 [O(t,p) -
EO = - ~ ' u(t, R1)R1 + 3O~e
Og]pdp
/
}
.
[O(t,O ) -Og]lodl9 1
+ 3O~e[O(t,r) - Og]
'
(5.3.92)
Using Eq. (5.3.92), we calculate the nonzero components of the deviatoric part ~ of the strain tensor er = -- r-15
{ u(t, R1)R1 + 3C~efRr [O(t,p) -
1( eo = -~
u(t, R1)R1 + 3C~ef R r [®(t,p) - ~)g]pdp } - ~e[®(t,r) - ®g],
Og]pdp
} + 2~e[O(t,r)
- ®g],
1
ez = -C~e[®(t, r) - ®g].
(5.3.93)
318
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
Combining Eqs. (5.3.80) and (5.3.93), we find that er -- eo = 3ae[O(t, r) _
- Og]
r 2 { u(t, R1)R1
2
+ 3ae
fR r [O(t,p)
- Og]pdp
}
= 3ae[O(t, r) - Og]
2 {C(t)+ ~ai[®0(t) 3
- r--g
- Og]R 2 + 3ae
fRr[O(t, p) 1
Og]p dp
}.
(5.3.94) The shell is modeled as the standard thermoviscoelastic medium (5.3.70) and (5.3.71). Substitution of expressions (5.3.93) into the constitutive equation (5.3.70) implies the following nonzero components of the deviatoric part ~ of the stress tensor: Sr = 2G(O(t, r)){[nl (t, r) + ni(t, r) + ~1 (t, r)]er(t, r) - ~2,r(t, r)}, so = 2G(O(t, r)){[nl (t, r) + nz(t, r) + ~l(t, r)]eo(t, r) - qt2,o(t, r)}, Sz = 2G(®(t, r)){[nl (t, r) + nz(t, r) + ~l(t, r)]ez(t, r) - ~2,z(t, r)}, (5.3.95)
where ~2,k(t, r) =
f0t n(t, "r, r)ek('r, r) dr,
k=
r,O,z.
According to Eq. (5.3.95), we can write sr - so = 2G(O(t, r)){[nl(t, r) + n2(t, r) + ~1 (t, r)] ×[er(t, r) - eo(t, r)] - ~(t, r)},
(5.3.96)
qs(t, r) = ~ 2 , r ( t , r) - ~z,o(t, r).
(5.3.97)
where
It follows from Eqs. (5.3.71) and (5.3.97) that the function ~ satisfies the differential equation
0q,
~(t,r) Ot
= 7(®(t, r)){[1 - n l ( t , r ) ] [ e r ( t , r ) - eo(t,r)] - ql(t, r)} (5.3.98)
with the initial condition ~(0, r) = 0.
(5.3.99)
319
5.3. Constitutive Models for the Nonisothermal Behavior Substitution of expression (5.3.94) into Eq. (5.3.98) implies that
O--~(t'r)
~/(O(t,r))( - ~(t,r) + [ 1 - nl(t,r)]{3~e[O(t,r) - Og]
2 C(/).+. ~ai(Oo(t) 3
fRr (O(t, p)
- r-5
- Og)R 2 + 3O~e
- Og)pdp
] }) .
1 (5.3.100) Let us express the function C(t) in the right-hand side of Eq. (5.3.100) in terms of qJ(t,r). For this purpose, we integrate Eq. (5.3.86) from R1 to R2 and use Eq. (5.3.88) and the boundary condition O'r(t, R2) = 0. As a result, we obtain
trr(t, R1) =
j~
R2 dr [Sr(t,r) - s o ( t , r ) ] ~ .
(5.3.101)
r
1
We assume that no debonding occurs between the polymeric cylinder and the mandrel. This means that the radial stresses O"r in the cylinders coincide at the interface r = R1. Some conditions that ensure contact between the two cylinders will be discussed later. Substitution of expressions (5.3.89) and (5.3.96) into Eq. (5.3.101) yields
R2
1 G(O(t,r)){[nl(t,r) + n2(t,r) + ~l(t,r)] X [er(t, r) -- eo(t, r)] -- ~(t, r)} dr . r
(5.3.102)
Combining Eqs. (5.3.94) and (5.3.102), we find that
= 3
/R2
1) fRl~2 dr) R~ + 2 1 G(O(t,r))[nl(t,r) + n2(t,r) + qq(t,r)]-~- 5-
G(O(t,r))[nl(t,r) + n2(t,r) + ~l(t,r)]
{
O~e[O(t,r)
-- Og]r 2
1 r -- cti[Oo(t) -- Og]R 2 - 2CtefR r [O(t, p) - Og]pdp ) d -~
1
-
a(®(t, r))ql(t, r) ~dro 1
r
(5.3.103)
Given cooling program O0(t), the temperature O(t, r) in the polymeric cylinder is calculated with the use of Eqs. (5.3.73) to (5.3.75). Afterward, the functions nl, n2, and ql 1 a r e found from differential equations (5.3.71). Finally, the functions q~ and C are obtained from Eqs. (5.3.100) and (5.3.103).
320
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
Introducing the dimensionless variables and parameters [~)g -- ®0
Og -- 0 1.~ -~ ~)g __ Or ,
t
00 -- ~)g _ Or ,
t,
T(Og)
r,
r R1
KT(Og) R2
we present Eqs. (5.3.73) to (5.3.75) as
o30 or,
( O20
1 00'~ O(0, r,) = 0, O(t,, 1) = O0(t,), O(t,, R2,) = Oo(t,), r, Or, ] '
-
(5.3.104) where
R2 R1
R2, It follows from Eq. (5.3.59) that T,(O)=exp
(A , I _ co )' cO
(5.3.105)
where T,-
T
A,-
T(Og)'
A
c=
ROg '
~g m ~ r
Og
Equation (5.3.69) implies that (5.3.106)
G((9) = G2(1 + bO), where b = B(Og
-
Or)
G2
Combining Eqs. (5.3.71) and (5.3.106), we obtain b DO 1 an1 (t,, r,) = r,), 1 + bO(t,, r,) ~** (t,, n2(t,,r,) at, 1 On2 (t,, r,) = T,(O(t,,r,)) nz(t,, r,) Ot, ~(t,,
Ot,
r,) =
T,(O(t,,r,))
nl(0, r,) = 0,
b aO (t,,r,), 1 + bO(t,,r,) at,
[1 - n l ( t , , r , ) - qJl(t,, r,)],
n2(0, r,) - 1,
~1(0, r,) = 0.
(5.3.107)
321
5.3. Constitutive Models for the Nonisothermal Behavior
Equations (5.3.100) and (5.3.103) are presented in the dimensionless form as C,(t,) =
G,
Ro
- 1 + 2
[1 + b O ( t , , r ) ]
dr)-1
× [ n l ( t , , r ) + n 2 ( t , , r ) + ~l(t,, r)]~-~-
×
-
~(t,,r,) Ot,
[1 + b O ( t , , r ) ] 2 -k- 2c~,
a, O(t,, r)r
,
= - T,(O(t,,r,))
(
[ n l ( t , , r ) + n z ( t , , r ) + ~l(t,,r)]
fr
O(t,, p)p do
1
ago(t,)
}
- qs,(t,, r)r 2 dr r 3'
~ , ( t , , r , ) + [1 - n l ( t , , r , ) ]
{
a,O(t,,r,)
+ r--~ 1 I2 C , ( t , ) - O o ( t , ) - 2 c ~ ,fl r* O ( t , , p ) p d p
]}/
~,(0, r,) = 0,
,
(5.3.108)
where
C~
C 30li(Og
-
Or)R 2'
~ ~* "- 3oli(Og
Ore -
Or)'
Ol,-
oli ,
Gi G,-
G2
For the numerical simulation, we choose parameters
R2 -
R1
2.0,
R1 -
Ro
1.1,
(5.3.109)
which correspond to data used by Klychnikov et al. (1980). As common practice, cylinders with a smaller ratio R2, are employed. For example, R2, = 1.028 in Cai et al. (1992), R2, = 1.068 in Sala and Cutolo (1996b), R2, = 1.143 in Eduljee and Gillespie (1996), and R2, = 1.171 in Chien and Tzeng (1995). We accept the exponential law for changes in temperature O(t,) = 1 - e x p ( - a t t , ) ,
(5.3.110)
where at is the dimensionless rate of cooling. Equation (5.3.110) corresponds to convective cooling of a heat chamber by air. The dimensionless temperature a9 is plotted in Figures 5.3.6 and 5.3.7 versus the dimensionless time t, for various at values. The temperature grows in time monotonically, and for at ~> 1, the cooling process finishes within the interval t, E [0, 4]. For large K, values, for instance, for K, = 1.0, the difference between the temperature in the center of the polymeric cylinder and the temperature on its boundaries is rather small. This difference increases with a decrease in K, and becomes significant for K, = 0.1.
322
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
la J ' J " 2a
i i
//
lb J'J"
3a
I
2b
J'J"
I
3b
I
I
I
0
I
t,
I
4
5.3.6: The dimensionless temperature O versus the dimensionless time t, at cooling of a polymeric cylinder with K, = 1.0. Curves (a): boundary surfaces of the 1 cylinder r = R1 and r = R2. Curves (b): center of the cylinder, r = i(R1 + R2). Curve 1: at - - 2.0. Curve 2: at - - 1 . 0 . Curve 3: at = 0 . 5 . Figure
The dimensionless parameters n l, n2, and qq are plotted versus the dimensionless time t, in Figures 5.3.8 to 5.3.10. Calculations are carried out for b = 10.0,
A,
=
0.00005,
~g = 333 K,
~r
=
293 K.
(5.3.111)
Parameters (5.3.111) correspond to experimental data for an epoxy resin [see, e.g., Nakamura et al. (1986) and Golub et al. (1986)]. Figure 5.3.8 shows that n l increases in time and tends to its limiting value nl,~ as t, ~ ~. Owing to the difference in temperatures between the center of the polymeric cylinder and its boundary surfaces, the n l value in the center exceeds the n l value on the boundaries. The difference between these amounts is essential for rapid cooling (large at values) and becomes insignificant for slow cooling (small at values).
323
5.3. Constitutive Models for the Nonisothermal Behavior
la
2a lb 2b 3a O 3b
gi g m
•
g
I
0
I
I
I
I
t,
I
I
4
5.3.7: The dimensionless temperature O versus the dimensionless time t, at cooling of a polymeric cylinder with K, = 0.1. Curves (a): boundary surfaces of the cylinder r = R1 and r = R2. C u r v e s (b): center of the cylinder, r = 1(R1 + R2). Curve 1" at = 2.0. Curve 2: at = 1.0. Curve 3: at = 0.5.
Figure
According to Figure 5.3.9, n2 decreases monotonically and vanishes as t, ---, ~. An increase in the rate of cooling at leads to a monotonic decrease in n2. The difference between n2 values in the center and on the boundary surfaces is rather small and increases weakly with the growth of the rate of cooling at. Figure 5.3.10 shows that q~l increases monotonically and tends to its limiting value qtl,~ as t, ---, ~. The qq values decrease with the growth of the rate of cooling at. For a fixed rate of cooling, the qtl value on the boundaries of the cylinder exceeds that in the center. The difference between these amounts is small for slow cooling and increases with the growth of the dimensionless rate of cooling at.
324
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
nl
lb la 2b 2a 3b 3a
I
I
I
I
I
I
0
I
t,
I
I
4
Figure 5.3.8: The dimensionless parameter nl versus the dimensionless time t, at cooling. Curves (a): boundary surfaces of the cylinder r = R1 and r = R2. Curves (b): center of the cylinder, r = l ( g l + g2). C u r v e 1" at -- 2.0. C u r v e 2: at = 1.0. C u r v e 3: at -- 0.5.
The dimensionless parameter q~2 is plotted versus the dimensionless time t, is Figure 5.3.11. Calculations are carried out for parameters (5.3.111) and G, = 10.0,
o¢, = 1.0.
(5.3.112)
The first equality in Eq. (5.3.112) is in good agreement with experimental data provided by Cai et al. (1992) and Eduljee and Gillespie (1996). The behavior of q~2 differs essentially in the center and on the boundary surfaces of the cylinder. The qJ2 value on the boundaries decreases in time, reaches its minimal value, and, afterward, increases monotonically and tends to some positive limiting value. The q~2 value in the center increases in time, reaches its maximum, and, afterward, decreases. With the growth of the dimensionless rate of cooling at, the
325
5.3. Constitutive Models for the Nonisothermal Behavior
n2
Q, I'
Q
la
2a 2b
lb
0
~ t 0
3a, b
t
L ~ J ~
t,
4
Figure 5.3.9: The dimensionless parameter n2 versus the dimensionless time t, at cooling. Curves (a): boundary surfaces of the cylinder r = R1 and r = R2. Curves (b): center of the cylinder, r = l ( g l q- g 2 ) . Curve 1" at = 2.0. Curve 2: at = 1.0. Curve 3: at -- 0 . 5 .
instants when the maximum (minimum) is reached decrease, and the maximum (minimum) values increase. The function C,(t,) characterizing pressure on the mandrel is plotted in Figure 5.3.12. The pressure increases in time, reaches its (positive) maximum value, and, afterward, decreases and tends to some negative limiting value C,(~). Positivity of the function C, means that tension arises between the mandrel and the polymeric shell, which transforms (in time) into compression when C, becomes negative. This phenomenon (successive tension and compression under cooling) is observed when the coefficients of thermal expansion of the mandrel and the shell are close to each other (c~, ~ 1). For Ogi > Oge, only tension occurs, whereas for Ogi < Oge, only compression occurs between the layers (see Figure 5.3.13). The effect of the
326
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
q,1
3a 3b 2a 2b la lb
0
t,
4
F i g u r e 5.3.10: The dimensionless parameter ~1 v e r s u s the dimensionless time t, at cooling. Curves (a): boundary surfaces of the cylinder r = R1 and r = R2. Curves 0a): 1 center of the cylinder, r = ~(R1 + R2). Curve 1: at = 2.0. Curve 2: at = 1.0. Curve 3: at = 0.5.
ratio a , is essentially nonlinear. For a , ~- 1, residual stresses are very small, and they increase significantly with the growth of a,. The dimensionless temperature conductivity r , essentially affects pressure on the mandrel in the process of cooling, but residual stresses are practically independent of this parameter (see Figure 5.3.14). The latter means that the effect of K, may be neglected in the numerical analysis, except for the manufacturing processes, where delamination is important at the interface between the mandrel and the polymeric vessel. Figure 5.3.14 shows that for low rates of cooling, tension occurs between the mandrel and the shell, and the stress intensity grows with a decrease in at. The effect of the material parameters b and A, on pressure on the mandrel is demonstrated in Figures 5.3.15 and 5.3.16. The dimensionless parameter b (which
327
5.3. Constitutive Models for the Nonisothermal Behavior
+0.015 lb
2b
3b
3a
2a--
la
-0.015
I
0
I
I
I
I
I
I
t,
I
I
4
5.3.11: The dimensionless parameter q~2 versus the dimensionless time t, at cooling. Curves (a): boundary surfaces of the cylinder r = R1 and r = R2. Curves (b): 1 center of the cylinder, r = ~(R1 q- e 2 ) . Curve 1: at = 2.0. Curve 2: at = 1.0. Curve 3:
Figure
at - - 0 . 5 .
characterizes the influence of temperature on the shear modulus G of the polymeric medium) affects significantly residual stresses. Its influence is essentially nonlinear: for small b values, residual stresses grow rapidly in b, whereas for large b, the effect of this parameter is rather weak. The parameter A, characterizes temperature shift of relaxation curves according to the time-temperature superposition principle. Surprisingly, its effect on residual stresses is weak and it may be neglected (see Figure 5.3.16).
328
Chapter 5. Constitutive Relationsfor Thermoviscoelastic Media
0.020 • •
• •
•
•
I C~
-/7', g
~o
0.000
-0.005 I 0
I
I
I
I
I
n
I t,
I
I 4
Figure 5.3.12: The dimensionless parameter C, versus the dimensionless time t, at cooling. Curve 1" at 2.0. Curve 2: at 1.0. Curve 3: at = 0.5. -
-
-
-
Bibliography [1] Advani, S. G. (1994). Flow and Rheology in Polymer Composites Manufacturing. Elsevier, Amsterdam. [2] Aklonis, J. J., MacKnight, W. J., and Shen, M. (1972). Introduction to Polymer Viscoelasticity. Wiley-Interscience, New York. [3] Arridge, R. G. C. (1985). An Introduction to Polymer Mechanics. Taylor and Francis, London. [4] Bouche, E (1953). Segmental mobility of polymers near their glass temperature. J. Chem. Phys. 21, 1850-1855.
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329
+0.6
C~
0.0
-0.6
I.
0 Figure
I
I
I
I
I
t,
5.3.13:
cooling with 4: c~. = 3.0.
I
at
I
I
4
The dimensionless parameter C. versus the dimensionless time t. at 2.0. Curve 1: c~, = 0.5. Curve 2: c~. = 1.0. Curve 3: c~. = 2.0. Curve =
[5] Buckley, C. R (1988). Prediction of stress in a linear viscoelastic solid strained while cooling. Rheol. Acta 27, 224-229. [6] Buckley, C. R and Jones, D. C. (1995). Glass-rubber constitutive model for amorphous polymers near the glass transition. Polymer 36, 3301-3312. [7] Buckley, C. P. and Salem, D. R. (1987). High-temperature viscoelasticity and heat-setting of poly(ethylene terephthalate). Polymer 28, 69-85. [8] Cai, Z., Gutowski, T., and Allen, S. (1992). Winding and consolidation analysis for cylindrical composite structures. J. Composite Mater 26, 1374-1399. [9] Chapman, B. M. (1974). Linear superposition of viscoelastic responses in nonequilibrium systems. J. Appl. Polym. Sci. 18, 3523-3536.
Chapter 5. Constitutive Relationsfor Thermoviscoelastic Media
330 +0.6 m
C.
0.0
-0.6
I
0
I
I
I
I
I
I
t,
I
I
4
5.3.14: The dimensionless parameter C, versus the dimensionless time t, under cooling with at = 2.0 and c~, = 2.0. Curve 1: K, = 1.0. Curve 2: K, = 0.25. Curve 3: K, =0.1.
Figure
[10] Chien, L. S. and Tzeng, J. T. (1995). A thermal viscoelastic analysis for thickwalled composite cylinders. J. Composite Mater. 29, 525-548. [ 11 ] De Rosa, M. E. and Winter, H. H., (1994). The effect of entanglements on the rheological behavior of polybutadiene critical gels. Rheol. Acta 33, 220-237. [ 12] Dienes, G. J. (1953). Activation energy for viscous flow and short-range order. J. Appl. Phys. 24, 779-782. [13] Doolittle, A. K. (1951a). Studies in Newtonian flow. 1. The dependence of the viscosity of liquid on temperature. J. Appl. Phys. 22, 1031-1035. [14] Doolittle, A. K. (1951b). Studies in Newtonian flow. 2. The dependence of the viscosity of liquid on free-space. J. Appl. Phys. 22, 1471-1475.
Bibliography
331
+0.6
C,
_
--0.6
I
0
I
I
I
I
I
I
t,
I
I
4
Figure 5.3.15: The dimensionless parameter C, versus the dimensionless time t, during cooling with at - - 2.0, o~, - - 2 . 0 , and K, = 0.1. Curve 1: b = 1.0. Curve 2: b = 5.0. Curve 3: b = 10.0. Curve 4: b = 50.0.
[15] Doolittle, A. K. (1952). Studies in Newtonian flow. 3. The dependence of the viscosity of liquid on molecular weight and free space (in homologous series). J. Appl. Phys. 23,236-239. [16] Drozdov, A. D. (1996). A constitutive model in thermoviscoelasticity. Mech. Research Comm. 23,543-548. [ 17] Drozdov, A. D. (1997a). A model for the non-isothermal behavior of viscoelastic media. In Proc. Int. Symp. "Thermal Stresses '97," Rochester, pp. 337-340. [18] Drozdov, A. D. (1997b). The non-isothermal behavior of polymers. 1. A model of adaptive links. Eur. J. Mech. A/Solids 16, (in press). [ 19] Drozdov, A. D. (1997c). The non-isothermal behavior of polymers. 2. Numerical simulation. Eur. J. Mech. A/Solids 16, (in press).
332
Chapter 5. Constitutive Relations for Thermoviscoelastic Media
+0.6
C,
1,2
--0.6
I 0
I
I
I
I
I
I t,
I
I Z
Figure 5.3.16: The dimensionless parameter C, versus the dimensionless time t, during cooling with at -- 2.0, c¢, = 2.0, K, = 0.1, and b = 10. Curve 1: A, = 5.0. 10 -7. Curve 2: A, = 5.0. 10 3.
[20] Drozdov, A. D. (1997d). A model for the nonisothermal behavior of viscoelastic media. Arch. Appl. Mech. 67, 287-302. [21] Drozdov, A. D. and Kalamkarov, A. L. (1995). A new model for an aging thermoviscoelastic material. Mech. Research Comm. 22, 441-446. [22] Eduljee, R. E and Gillespie, J. W. (1996). Elastic response of post- and in situ consolidated laminated cylinders. Composites 27A, 437-446. [23] Eringen, A. C. (1960). Irreversible thermodynamics and continuum mechanics. Phys. Rev. 117, 1174-1183. [24] Ferry, J. D. (1950). Mechanical properties of substances of high molecular weight. 6. Dispersion of concentrated polymer solutions and its dependence on temperature and concentration. J. Amer. Chem. Soc. 72, 3746-3752.
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[25] Ferry, J. D. (1980). Viscoelastic Properties of Polymers. Wiley, New York. [26] Fox, T. G. and Flory, P. J. (1948). Viscosity-molecular weight and viscositytemperature relationships for polystyrene and polyisobutylene. J. Amer. Chem. Soc. 70, 2384-2395. [27] Fox, T. G. and Flory, P. J. (1950). Second order transition temperatures and related properties of polystyrene. 1. Influence of molecular weight. J. Appl. Phys. 21,581-591. [28] Fox, T. G. and Flow, P. J. (1951). Further studies of the melt viscosity of polyisobutylene. J. Phys. Chem. 55,221-234. [29] Golub, M. A., Lerner, N. R., and Hsu, M. S. (1986). Kinetic study of polymerization/curing of filament-wound composite epoxy resin systems with aromatic diamines. J. Appl. Polym. Sci. 32, 5215-5229. [30] Han, C. D. and Kim, J. K. (1993). On the use of time-temperature superposition in multicomponent/multiphase polymer systems. Polymer 34, 2533-2539. [31] Ilyushin, A. A. and Pobedrya, B. E. (1970). Principles of the Mathematical Theory of Thermoviscoelasticity. Nauka, Moscow [in Russian]. [32] Kenny, J. M. and Opalicki, M. (1996). Processing of short fibre/thermosetting matrix composites. Composites 27A, 229-240. [33] Khristova, Y. and Aniskevich, K. (1995). Prediction of creep in polymer concrete. Mech. Composite Mater. 31, 216-219. [34] Klychnikov, L. V., Davtyan, S. P., Turusov, R. A., Khudayev, S. I., and Enikolopyan, N. S. (1980). Influence of an elastic mandrel on the distribution of residual stresses in the case of frontal hardening of a spherical specimen. Mech. Composite Mater. 16, 226-229. [35] Koltunov, M. A. (1976). Creep and Relaxation. Moscow University Press, Moscow [in Russian]. [36] Kominar, V. (1996). Thermo-mechanical regulation of residual stresses in polymers and polymer composites. J. Composite Mater. 30, 406-415. [37] Lacabanne, C., Chatain, D., Monpagens, J. C., Hiltner, A., and Baer, E. (1978). Compensation temperature in amorphous polyolefins. Solid State Comm. 27, 1055-1057. [38] La Mantia, E P., Titomanlio, G., and Aciemo, D. (1980). The viscoelastic behavior of nylon 6/lithium halides mixtures. Rheol. Acta 19, 88-93. [39] La Mantia, E P., Titomanlio, G., and Acierno, D. (1981). The non-isothermal rheological behavior of molten polymers: shear and elongational stress growth of polyisobutylene under heating. Rheol. Acta 20, 458-462.
334
Chapter 5. Constitutive Relationsfor Thermoviscoelastic Media
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336
Chapter 5. ConstitutiveRelationsfor ThermoviscoelasticMedia
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Chapter 6
Accretion of Aging Viscoelastic Media with Finite Strains This chapter is concerned with continuous growth of viscoelastic media with finite strains. In Section 6.1, we derive a mathematical model for continuous accretion and solve two problems in which the characteristic features of the accretion process are revealed. Section 6.2 deals with winding of a viscoelastic cylinder. This problem is of essential interest for calculating residual stresses in wound rolls of paper and magnetic and videotapes, as well as in composite pressure vessels and pipes manufactured by filament winding. In Section 6.3, we analyze the effect of resin flow on residual stresses in a wound composite shell. Finally, Section 6.4 is concerned with volumetric growth of biological tissues.
6.1
Continuous Accretion of Aging Viscoelastic Media
A mathematical model is derived for the description of continuous surface growth of a viscoelastic medium with finite strains. The growth means a monotonic mass supply to the body from the environment. The process is treated as successive accretion of thin layers on a part of the boundary of a growing body. Since successive layers (built-up portions) are applied to the deformed boundary, final stresses depend on the rate of accretion and on the loading history. The problem of accretion originated in the 1950s and 1960s. It is in the focus of attention owing to a wide range of applications: from building of dams and 337
338
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
embankments [see, e.g., Christiano and Chantranuluck (1974), Dyatlovitskii (1956), Dyatlovitskii and Veinberg (1975), Goodman and Brown (1963), Kharlab (1966), and Rashba (1953)] to creation of self-gravitating planets [see, e.g., Arutyunyan and Drozdov (1984b) and Brown and Goodman (1963)], from manufacturing thin films [see, e.g., Anestiev (1989), Hearn et al. (1986), and Tsai and Dillon (1987)] to winding composite pressure vessels [see, e.g., Drozdov (1994) and Drozdov and Kalamkarov (1995)], from consolidation of metallic droplets [see Mathur et al. (1989)] to snowfalls [see Brown et al. (1972)], from solidification of adhesive layers [see Duong and Knauss (1993a, b)] to manufacturing multilayered cables and wire belts [see, e.g., Kowalskii (1950) and Tomashevskii and Yakovlev (1982)]. In this section, we derive a mathematical model for continuous accretion at finite strains and apply this model to two problems of interest in engineering to demonstrate characteristic features of the accretion process. The exposition follows Arutyunyan and Drozdov (1984a, 1985b), Arutyunyan et al. (1987), and Drozdov (1994).
6.1.1
A Model for Continuous Accretion
A viscoelastic medium in its natural (stress-flee) state occupies a domain 1~° with a boundary F °. At the instant t = 0, external forces are applied to the medium, and continuous accretion begins on a part of its boundary. At an arbitrary instant t --- 0, the accreted medium occupies a domain l~(t) with a boundary F(t) in the actual configuration. The surface F(t) is divided into three connected parts. On a part Fu(t), displacements are prescribed; on a part F~(t), a surface traction is given; and on the part Fa(t) = F(t) \ (Fu(t)U F~(t)) continuous accretion of material occurs in the interval [0, T] (see Figure 6.1.1). Within the interval [t, t + dr], a built-up portion (layer) with volume (thickness) proportional to dt joins the growing body. We assume that the instant when a built-up portion in the vicinity of a point with Lagrangian coordinates ~ = {~i} is manufactured coincides with the instant ~-*(~) when this portion merges with accreted
= D,
Fu(t) l'u(t)
". .~. i. i. i. i i i i : .
Figure 6.1.1: A growing body under loading.
339
6.1. Continuous Accretion of Aging Viscoelastic Media
medium
r,(~) = { o, ~ ~ a(o), t,
~ ~ ra(t).
(6.1.1)
The natural configuration of a built-up portion may differ from the actual (current) configuration of the accretion surface Fa(t). This means that the built-up portion should be previously deformed to merge with the growing body. After joining the accretion surface, any built-up portion is treated as a part of a monolithic medium (see Figure 6.1.2). To describe the accretion process, we introduce three basic configurations. The first is the reference configuration, in which we fix Lagrangian coordinates ~ and postulate a plan (schedule) of accretion. For definiteness, we assume that for any t ~ [0, T], a growing body occupies a domain ~°(t) with a boundary F°(t) = F°(t) U F ° ( t ) ~ F°(t) in the reference configuration. To formulate a plan of growth we determine which points of boundaries of built-up portions and of the accretion surface F°(t) merge with one another at any instant t ~ [0, T]. The second configuration is the natural (stress-free) configuration, where any built-up portion remains until extemal forces are applied. For the initial body at t = 0, the reference configuration coincides with the natural configuration, whereas for built-up portions these configurations may differ from one another. For any accreted layer, its natural configuration is determined either by prescribing an appropriate deformation gradient for transition from the reference configuration to the natural configuration, or by introducing the corresponding stress tensor (preloading). For given constitutive equations, these two approaches are equivalent. For definiteness, we employ the former (geometrical) approach. The third configuration is the actual configuration occupied by an accreted medium at the current instant t under extemal forces. It is determined by solving Natural configuration
Actual configuration /~° (t, ~)
P(t, ~)/~
Reference configuration
Figure 6.1.2: Reference, natural, and actual configurations of a built-up portion.
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
340
governing equations. The actual configuration is characterized by the deformation gradient for transition from the reference configuration to the actual configuration. Continuous accretion is modeled as a limit of the following process of successive layout of built-up portions. We divide the interval [0, T] by points tn = nA (n = 0 , . . . , N), where A = T/N. At instant tn, the accreted medium occupies a domain fl°(tn) in the reference configuration. Within the interval [tn, tn+l], it merges with a built-up portion that occupies a domain Af~°(tn) in the reference configuration, and together they create a new monolithic solid that occupies a domain l~°(tn+ 1). For the built-up portion Al~°(tn), deformation from the reference configuration to the natural configuration is assumed to satisfy the compatibility conditions. The latter means that the deformation gradient is expressed in terms of some displacement vector ~*(~) for ~ E A~°(tn). Continuous accretion is treated as a limit of the process of discrete accretion as N ---, ~, A ~ 0, and volumes of the domains Al20(tn) approach zero. Since the displacement vectors fi*(~) in different built-up portions may differ from one another, the corresponding deformation gradient need not satisfy any compatibility condition for continuous accretion. Let us derive a kinematic formula for the deformation gradient for transition from the natural to actual configuration, which is similar to the multiplicative presentation for the deformation gradient in finite elastoplasticity [see, e.g., Lee (1969)]. We consider a built-up portion in the vicinity of a point ~ and denote by ?0(~), ?*(~), and ?(t, ~) its radius vectors in the reference, natural, and actual configurations. Differentiation of these vectors with respect to ~i implies tangent vectors ~'0 i(~), - , (~)' and gi(t, ~). The dual vectors are denoted as g0(~c), -i ~, i(~), and ~i (t, ~). The gi deformation gradients equal -* VoP = gogi, -
*
-i
~7o? = gogi,
~7"~ =
~*
"
'gi.
(6.1.2)
Here and in the following the argument ~ is omitted for simplicity. Tensors (6.1.2) are connected by the formula ~r* ~ ._ (~70~*)- 1 . ~rO~"
(6.1.3)
Equation (6.1.3) is similar to Eq. (1.1.59) for the relative deformation gradient (7~?(t) for transition from the actual configuration at instant ~"to the actual configuration at instant t V~?(t) = [~'0F(T)] - 1 " Vo?(t).
(6.1.4)
We calculate the Finger tensor F°(t, ~) for transition from the natural configuration to the actual configuration at instant t and the Finger tensor F<>(t, ~', ~) for transition from the actual configuration at instant ~" to the actual configuration at instant t as P°(t) = [~7*?(t)]r. ~7*?(t), where T stands for transpose.
P~(t, ~-) = [~'~?(t)] r . V~?(t),
(6.1.5)
6.1. Continuous Accretion of Aging Viscoelastic Media
341
Expressions (6.1.2) to (6.1.5) determine kinematics of an accreted medium with finite strains. At small strains, it is convenient to use the displacement vectors fi(t, ~) for transition from the reference to actual configuration and fi*(~) for transition from the reference to natural configuration. It follows from Eqs. (6.1.2) and (6.1.3) that
¢o~* = 7 + You,
V*?(t) = I+Vo[fi(t)-fi*], (6.1.6)
¢0~(t) = 7 + ¢0o(t),
where I is the unit tensor. Denote by t*(~), t(t, ~), and t°(t, ~) the infinitesimal strain tensors for transition from the reference to natural configuration, from the reference to actual configuration, and from the natural to actual configuration, respectively. Equations (6.1.6) imply that these tensors are connected by the equality t°(t, ~) = t(t, ~) - t*(~).
(6.1.7)
Denote by U>(t, ~-, ~) the infinitesimal strain tensor for transition from the actual configuration at instant ~-to the actual configuration at instant t. By analogy with Eq. (6.1.7), we write t ~ (t, ~', ~) = t(t, ~) - t(~', ~).
(6.1.8)
We now return to finite strains and discuss constitutive equations for an accreted viscoelastic medium subjected to aging. Denote by I°(t, (;) and Ik~(t, ~-, ~) (k = 1, 2, 3) the principal invariants of the Finger tensors F°(t, ~) a n d / ~ ( t , ~', so). We confine ourselves to isotropic and incompressible viscoelastic materials. It follows from the isotropicity condition that the strain energy density W (per unit volume) depends on the principal invariants of the Finger tensor. For incompressible media
I~(t, ~) = O,
I3<>(t, ~-, ~) = O,
(6.1.9)
and the function W depends on the first two invariants only. In the framework of a model of adaptive links, we treat a viscoelastic medium as a network of parallel elastic springs that replace one another. For a growing medium, it is convenient to introduce two time scales: absolute and relative. The absolute time t is calculated from a fixed origin that is independent of the accretion process. This time is the same for any element of an accreted solid. The relative time tr is calculated from the instant of creation (manufacturing) of a built-up portion. Relative times for built-up elements that merge with a growing body at different instants differ from one another. The absolute time t and the relative time tr are connected by the relationships
t = tr + z*(~),
tr
=
t-
~-*(~),
(6.1.10)
where ~-*(~) is the time (in the absolute scale) when a built-up portion in the vicinity of a point ~ is manufactured. A growing viscoelastic solid provides an important example of a nonhomogeneous medium, where a specific inhomogeneity arises since different elements are manufactured at different instants ~'*, whereas the material response of a material portion (the processes of reformation and breakage for adaptive links) is determined by the time tr elapsed from the instant of its manufacturing.
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
342
We denote by X,(tr, 0) the number of links (per unit volume) that arose at the instant of manufacturing a built-up portion and exist after time tr, and by ~gX, ~(tr, 3rr
Tr) dTr
the number of links (per unit volume) that arose within the interval D, ~ + d~'] and exist at instant tr (in the relative time scale). For the initial links arisen at manufacturing a built-up portion, the specific strain energy W0 (per unit link) depends on the principal invariants of the Finger tensor for transition from the natural to the actual configuration at the current instant t
Wo = Wo(I°(t, so),I°(t, ~)). The natural configuration of links arising in the process of replacement coincides with the actual configuration of a viscoelastic medium at the instant ~" of their creation. The specific strain energy W0 of these links depends on the principal invariants of the Finger tensor for transition from the actual configuration at instant ~"to the actual configuration at instant t
Wo = Wo (l l<>( t , T, (;), 12<>(t,~', ~ ) ) . The total strain energy density (per unit volume) at instant t equals the sum of strain energy densities for all links arising before t and existing at instant t
W(t, ~) = X,(tr, O)Wo(l°(t, ~), I°(t, ~)) +
fO0tr -ff-~Tr(tr, OX, "rr)Wo(I?(t, "r, ~), I2~(t, -r, ~))d'rr.
(6.1.11)
Substitution of expressions (6.1.10) into Eq. (6.1.11) implies that
W(t, ~) = X,(t - C(sc), O)Wo(I°(t, ~),I°(t, ~)) + f~j
cgX,(t - r*(~), ~" - C(~))Wo(II~(t, r, ~),l~(t, ~', ~))d~'. (~) ~
(6.1 12)
For any link, the Cauchy stress 6"0 (per unit link) is expressed in terms of the strain energy density with the use of the Finger formula #o= ~ 2 p o . OWo V/I3 (/~o) 0/~o ,
(6.1.13)
where p0 is the Finger tensor for transition from the natural configuration to the actual configuration at instant t. We should replace p0 in Eq. (6.1.13) by F°(t, ~) for the initial links arisen at the instant when a built-up portion with Lagrangian coordinate is manufactured, and by F<>(t, ~-, ~) for links that replace the initial links at instant ~'.
343
6.1. Continuous Accretion of Aging Viscoelastic Media Bearing in mind the incompressibility condition (6.1.9), we write 6"o(t, ~c) = - p ( t , ~)~1 + 2F°(t, ~c). ~OWo (io(t, (;) io(t ' ~)),
&o(t, r, ~) = - p ( t , ~)I^ + 2F ~(t, r, ~) " -OWo ~ (11<>(t, r, {~),I2<>(t, r, ~)), where p is a pressure. Summing up the stresses 6o for all links that exist at instant t, we find the Cauchy stress tensor in an aging viscoelastic medium
&(t, ~) = - p ( t , ~)7I + 2 IX. (t - r*(~), O)F°(t, ~) " -OWo ~ (io(t, ~), io(t ' ~)) + ~i
(~)
OX. (t - r* (~), r
r*(,~))PO(t, r, ~)
• OP<>O(I~(t, W° r, ~),I2°(t, r, ~))dr I .
(6.1.14)
We calculate the derivative of the specific strain energy Wo with the use of the Finger formula [see, e.g., Drozdov (1996)]
OWo _ ( OWo
OWo ) 7 - OWo po.
(6.1 15)
+ 1°- 2
As a result, we obtain the following constitutive equation: 6"(t, ~) = - p ( t , !~)?_ + 2{X.(t - r*(~), 0)[Wl°(t, !~)P°(t, 1~) + q*°2(t, l~)(P°(t, {~))2]
+ fl
ax*(t - r* (~), r - r* (~))['I'l~(t, r, ~)P~(t,
(!~) Or
T,
~)
+ qrz<>(t,r, ~)(P°(t, r, {~))2]d r } ,
(6.1.16)
where
~o(t ' ~) = &aw°(io(t, ~), zO(t, ~)) + io(t, ~)-~-2aw°(io(t, ~),io(t, ,I,°(t,
¢) =
_
~)),
OWo (io(t, ~), I~(t, ~)),
aWo (i1~ (t, ~-, ~), I2~(t, ~-, ~)), OWo (ilO(t ' r, ¢),12°(t, r, ¢)) + Ii<>(t, r, ¢)--~2 'I'l~(t' ~" ~) = -3771 ~ 2<>( t ' r' ~ ) = - -~2O W° (i I<>( t , r, ~ ) , I2<>( t , r, ~:)).
(6.1.17)
344
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
As a particular case, we consider a neo-Hookean viscoelastic material with the strain energy density IX 0) (I1 -- 3), W0(ll, 12) = 2X.(0,
(6.1.18)
where IX is a generalized shear modulus. Substitution of expression (6.1.18) into Eqs. (6.1.16) and (6.1.17) yields
&(t, ~) = -p(t, ~)l, + Ix [X(t - r* (~), 0)F°(t, ~) +
(~) -~T
- r*(~), r -
(!~))PO(t, r, ~)d
(6.1.19)
where X.(t, r)
X(t, r) = ~ .
(6.1.20)
x,(o, o)
At infinitesimal strains, the constitutive equation (6.1.19) is simplified. Substituting the expression k=?+2~ into Eq. (6.1.19), we find that 6-(t, ~) = -~(t, ~)? + 2IX [X(t - r* (so), O)~°(t, {~) +
°x (t -
(~) ~ r
r*(~), r - r*(~))~°(t, r, ~ ) d r
1
(6.1.21)
where/5 is a new function that is determined from the incompressibility condition (6.1.9). We combine Eq. (6.1.21) with Eqs. (6.1.7) and (6.1.8) to obtain 6-(t, ~) = -/5(t, ~)? + 2IX{X(t - r*(~), 0)~°(t, sc)
+fj <~) -~zax(t_,, (~), r - **(~))[~°(t, ~) -
~°(r, ~)] d r
}.
(6.1.22)
Calculation of the integral in the fight-hand side of Eq. (6.1.22) implies that &(t,
= -~(t, ~)F + 2IX [X(t - r*(~), t - r*(~))~°(t, ~) (e) -O--~r( t - r*(~), r -
¢(~))~°(r, ~) d r .
(6.1.23)
6.1. Continuous Accretion of Aging Viscoelastic Media
345
We set (6.1.24)
laX(t, r) = G(r) + Q(t, r),
where G(t) is the current shear modulus, and Q(t, r) is the relaxation measure that satisfies the condition Q(t, t) = 0. Substitution of expression (6.1.24) into Eq. (6.1.23) results in the constitutive relation for an aging, linear, viscoelastic, incompressible medium
( &(t,
= -~(t, (;)~1 + 2 ~ G(t - r* (~))~°(t, ~)
~[a(r-
r*(~)) + Q ( t - r*(~), r -
(~))]e°(r, ~)dr .
(6.125) We now return to finite strains and extend the constitutive equation (6.1.16) to the case in which M different kinds of links exist with strain energy densities •2). The links replace one another according to the laws that are described by the functions Xm.(t, r). Repeating the preceding calculations, we arrive at the following constitutive equation for an aging, incompressible, viscoelastic medium:
Wo,m(I1,
M
&(t, ~) = -p(t, ~)I + 2
Z (Xm,(t-
m=l
g*({~), O)[XlrlO,m(t, ~)P°(t, ~)
-Jr-XI)'02,m(t,~)(/2"0(t, ~))21 _+_fr I
~OXm, ( t
(0 Or
- r* (~), "I"- r* (~:))
X [XIrl~,m(t, r, ~)[gO(t, T, ~) + xtr2C],m(t,"r, {~)(/w°(t, "r, so))2] d r } , (6.1.26) where
012 (IO(t, ~), IO(t, ~)), xlrOm(t, ¢) -- OWo'm (IO(t' ~)' IO(t' ~)) + IO(t' ~) OW°,m xr~O'm( t' ¢ ) = ~m(t,~',a)
-
aWo,m -~2 oo~t, O, I°~t, 0), aWo,m ~I?~t, ~, ~),I20~t, r, ~)) 011
+ II<>( t' r' ~ ) aWo,m 012 ~I? (t, ~, O, 12° (t, r, ~)), OW°'m (110(t, r, ~), 12<>(t, r, ~)).
~I2
(6.1.27)
346
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
We confine ourselves to relatively slow processes of loading and accretion, in which inertia forces may be neglected. The equilibrium equations for a medium are the same whether or not the mass supply occurs, and can be presented in the standard form
Vtr"(t, ~) + p(t, ~)/~(t, ~) = 0,
~ ~ f~(t),
(6.1.28)
where Vt is the gradient operator in the actual configuration at instant t, p is mass density in the actual configuration, and B is a body force (per unit mass). In applied problems of accretion, the effect of body forces may be neglected, which implies that the equilibrium equation (6.1.28) reads
Vtr"(t, ~) -- 0,
~ ~ l~(t).
(6.1.29)
Boundary conditions on the surfaces Fu(t) and F~(t) are presented in the same form for growing and inert media
~(t, ~) = U(t, ~),
~ ~ Fu(t),
h(t, ~) . 6"(t, !~) = [fit, !~),
!~ ~ F~(t),
(6.1.30)
where (J(t, !~) is a given displacement vector, b(t, ~) is a given surface traction, and h is the unit normal vector to the boundary of the growing body. On the accretion surface Fa(t), the entire tensor P*(~) should be given, and the standard conditions are satisfied that ensure that the accretion surface is traction free h(t, so) • 6"(t, ~) = 0,
(6.1.31)
sc E Fa(t).
The Finger tensor F* may be prescribed explicitly as a function of Lagrangian coordinates ~. As common practice, the tensor P*(~) is determined as a function of the Finger tensor P(r*(~), ~) on the accretion surface Fa(t). In this case, Eq. (6.1.31) either imposes some limitations on the relationship between these tensors, or becomes an identity. For example, growth without preloading is characterized either by the formula 6"(r*(~), sc) = 0,
sc E l~°(T),
(6.1.32)
or by the equivalent condition
P*(~) = P(r*(~), ~),
E 12°(T).
(6.1.33)
It follows from Eqs. (6.1.32) and (6.1.33) that Eq. (6.1.31) is satisfied identically. It is worth also noting other approaches to determining the stress tensor 6" on the accretion surface. Bykovtsev and Lukanov (1985) assumed that a surface traction b was applied to the boundary 1-'a and that the Cauchy stress tensor 6" was an isotropic function of the normal vector h and the surface load b. As a result, a general formula was derived for the tensor 6" that contained only one adjustable scalar function. Trincher (1984) supposed that the equilibrium equation (6.1.28) is satisfied not only in the n
347
6.1. Continuous Accretion of Aging Viscoelastic Media
domain l~(t), but on its boundary F(t) as well, and developed additional restrictions on the stress tensor 6- by differentiating the equilibrium equations with respect to time on the moving accretion surface. Metlov and Turusov (1985) studied accretion of a viscoelastic medium in conditions of frontal solidification and derived additional conditions for the stress tensor on the accretion surface by replacing the solidification front by a narrow zone in which material shrinkage occurs. We do not dwell of those boundary conditions, since they are either based on hypotheses that seem rather questionable (isotropicity of the stress tensor or extension of equilibrium equations to the boundary), or are applied to particular accretion processes. The following conclusions may be drawn: 1. The main feature of governing equations for growing media consists in the presence of an additional Finger tensor F*. This tensor distinguishes the reference configuration where a schedule of accretion is prescribed and the natural (stressfree) configuration. 2. Governing equations for an accreted viscoelastic medium include the kinematic relations (6.1.3) to (6.1.5), the equilibrium equation (6.1.28) or (6.1.29), the constitutive equations (6.1.16) or (6.1.26), and the boundary conditions (6.1.30) and (6.1.31). Unlike problems with free boundary, where location of the boundary is found by solving the governing equations with some additional conditions, the accretion surface is assumed to be known at any instant. 3. Two types of the accretion problems may be distinguished. In problems of the first type, the Finger tensor F* is prescribed as a function of Lagrangian coordinates. In problems of the other type, a relation is introduced between this tensor and the Finger tensor P on the accretion surface Fa(t). To demonstrate characteristic features of the accretion process, we consider continuous growth of an aging viscoelastic cylinder under torsion.
6.1.2
Continuous Accretion of a Viscoelastic Cylinder
Let us consider a circular cylinder with length 1 and radius a l . At the initial instant t = 0, torques M = M(t) and compressive forces P = P(t) are applied to the edges of the cylinder, and continuous accretion of material begins on its traction-free boundary surface. Body forces are absent. Material supply occurs in the interval [0, T]. Owing to the material influx, radius of the growing cylinder a increases in time according to the law a = a(t),
a(O) = al,
a(T) = a2,
(6.1.34)
where al and a2 are given parameters. Volume V(t) of the accreted cylinder is calculated as V(t) = 7raZ(t)l.
(6.1.35)
348
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
Differentiation of Eq. (6.1.35) with respect to time determines the rate of material supply dV da v(t) = --d-~(t)= 27rla(t)--~(t).
(6.1.36)
Points of the cylinder in the reference configuration refer to cylindrical coordinates {R, 19, Z} with unit vectors ~R, ~o, and ~z. At instant t, the accreted cylinder occupies in the reference configuration the domain D.°(t)={0<-R<-a(t),
0<-19 <27r,
O <- Z <- l}.
Within the interval [t,t + dt], a cylindrical shell that occupies, in the reference configuration, the domain dl~°(t) = {a(t) <- R <- a(t + dt),
0<-19 < 2rr,
O <- Z <- l}
is manufactured and immediately merges with the growing body. The instant ~-*(R) when a built-up portion with polar radius R joins the accreted cylinder equals
~'*(R)= { 0, t,
O<--R<--al,
(6.1.37)
R=a(t).
Denote by {r, 0, z} cylindrical coordinates in the actual configuration. Deformation of the cylinder is determined by the formulas ®
r = R,
0 = t9 + a(t)Z,
z = Z,
(6.1.38)
where a = ct(t) is a function to be found (twist angle per unit length). According to Eq. (6.1.38), unit vectors of the cylindrical coordinate frame in the actual configuration equal 6'r = 6'R COS(OgZ) + 6'®
sin(aZ),
~0 = -~R sin(aZ) + ~o cos(aZ),
~'z = ~'Z.
(6.1.39) The radius vectors of an arbitrary point in the reference and actual configurations are ?0 = R~R,
? = rer.
(6.1.40)
Differentiation of Eqs. (6.1.40) with the use of Eqs. (6.1.38) and (6.1.39) implies tangent vectors in the reference and actual configurations g01 = 6'R, gl
--- e r ,
g02 = R~O, g2 = R~O,
g03 = ez, g3 = aRgo + ~z"
(6.1.41)
1
~1 ._ ~'r
,
~2 = -1~ o - a~z,
~3
= ~z.
(6.1.42)
Substituting expressions (6.1.41) and (6.1.42) into Eq. (6.1.2), we find the deformation gradient for transition from the reference to the actual configuration ~'0?(t) =
eRe.r -+- e . o e o + e.ze.z -+- a(t)R~z~o.
(6.1.43)
Denote by {r*, 0", z*} cylindrical coordinates in the natural configuration with unit vectors e-*r , e-*o , and e-*z . We suppose that transition from the reference to natural configuration of any built-up portion dl)(t) corresponds to its torque with some twist angle a*. For a successive accretion of thin-walled cylindrical shells, the angle a* is constant within any shell, but can change from one built-up portion to another. Repeating the preceding calculations, we obtain V0?* = eRe r + e o e o + eze z + cz R e z e o.
(6.1.44)
For continuous accretion, the stepwise function a* is transformed into a continuous function a*(r), whereas formula (6.1.44) remains unchanged. It follows from Eqs. (6.1.43) and (6.1.44) that [~70r(t)]-I
= e r e R -1- eoe.o -+- eze.z -- a(t)R~z~O,
( ~ ' 0 ~ * ) - 1 = ereR - * - + eoeo _,_ _, + ez~z -- a , R e_z, _e o.
(6.1.45)
Substituting expressions (6.1.43) to (6.1.45) into Eqs. (6.1.3) and (6.1.4), we find the deformation gradients for transition from the natural configuration to the actual configuration at instant t and from the actual configuration at instant ~"to the actual configuration at instant t V*?(t)
=
e r-*-er + eoeo-*- + e z-*-ez + [ a ( t ) - a*(R)]R~z~ o,
~ 7 ~ r ( t ) = ere.r + e.oe.o + e.ze.z + [og(t) -- a(r)]R~z~O.
(6.1.46)
Combining these equalities with Eqs. (6.1.5), we obtain the corresponding Finger tensors F ° ( t , R ) = e r e r + {1 + [ o g ( t ) -
+[a(t)-
~*(R)]R(~z~ o + eoez),
P ~ ( t , r , R ) = e.rer + {1 + I o n ( t ) -
+[a(t)-
~*(R)]2R2}~o~ o + ezez
a(r)]2R2}~o~ o + ezez
a(r)]R(~z~O + eoez).
(6.1.47)
The material behavior is assumed to obey the constitutive equation of an incompressible, aging, viscoelastic neo-Hookean medium. We substitute expressions (6.1.47)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
350
into Eq. (6.1.19) and find the nonzero components of the Cauchy stress tensor 6"(t)
O'rr(t,e) -- - p ( t , R ) , ~roo(t, R) = - p ( t , R ) + 1.6e2 {X(l - T*(R),
0)[o~(/)
-- o ~ * ( e ) ] 2
- ~'*(R), 1 " - l"*(R))[c~(t) - o~('/')12 d~"
+
(R) ~ trzz(t, R) = - p ( t , R), ¢roz(t, e) = Crzo(t,R) = txR{X(t - T*(R), 0)[c~(t) - c~*(R)] +
- ~'*(R), ~"- r* (R)) [c~(t) - c~(~')] dl"
(6.1.48)
(R) -~T where p is a function to be found. The equilibrium equation reads
690rrr Or
1
+ --(Orrr -- 0"00) = O. r
It follows from Eq. (6.1.38) that this equality can be presented as
OOrrr 4- --1(Orrr OR
R
_ O'00 ) =
0.
(6.1.49)
We integrate Eq. (6.1.49) from R to a(t) and use the boundary condition
O'rr(t, a(t)) = O. As a result, we find that
Orrr(t,R) -- fR a(t) [Orrr(t, R1) - oroo(t, R1)]R11 dR1.
(6.1.50)
Substitution of expressions (6.1.48) into Eq. (6.1.50) implies that
O'rr(t,e) -- -Id, fR a(t) { X(t - r*(R1), 0)[c~(t)+
i:
o~*(R1)] 2
}
(t - 'r*(R1), 7" - 'r*(R1))[o~(t) - c~('r)] 2 dT R1 dR1.
(R,)
(6.1.51)
Boundary conditions at the edges of the accreted cylinder are written in the integral form
M(t) = 27r fO0a(t) Croz(t,R)R2dR,
P(t) = -27r fO a(t) ~rzz(t,R)R dR.
(6.1.52)
6.1. Continuous Accretion of Aging Viscoelastic Media
351
Substitution of expressions (6.1.48) and (6.1.51) into Eqs. (6.1.52) yields
M(t) = 27rtz +
foa(t)(
X(t - r*(R), 0)[a(t) - a*(R)]
~
ax(t - ~-,(R), 1" _ ~-*(R))[a(t) _ a(~')] dT } R 3 dR,
(6.1.53)
(R) -~T
e(t) = 27rtz foa(t)RdR fRa(t) {x(t - ~'*(R1),O)[a(t) - cx*(R1)]2
+
OX (t - ~'*(R1), r - l"*(R1))[a(t) - a(l")] 2 dr R1 dR1. (R,) ~ (6.1.54)
We confine ourselves to accretion without preloading. The latter means that the natural (stress-free) configuration of the initial cylinder coincides with its initial configuration, whereas the natural configuration of a built-up portion with polar radius R coincides with the actual configuration of a growing cylinder on the accretion surface , { 0, a (R) = a(~'*(R)),
0 <- R _< al, al < R <-- a2.
(6.1.55)
We substitute expression (6.1.55) into Eq. (6.1.53) and present the integral from 0 to a(t) as a sum of two integrals from 0 to al and from al to a(t). Calculating the first integral, we obtain
M(t) - {X(t' O)a(t) + fo t -~r OX (t, ~-)[a(t) - a(~-)] dl" } a~4 27r~ +
/a(t){
X(t - ~'*(R), 0)[a(t) - a(~'* (R))]
,Ja 1
+ ~j
OX ( t - "r*(R), "r- "r*(R))[a(t)- a('r)ldT}R s dR. (R) -~T
Introducing the new variable s = ~'*(R),
(6.1.56)
we arrive at the linear integral equation for the function a(t)
M(t) - {X(t,O)a(t) + fo t OX 27r~
-0--~r(t, ~-)[a(t)
+
J0<
X(t - s, 0)[a(t) - a(s)]
-
a(~')]
d~}--a4 4
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
352
-~r(t Is'°x
+
s,r-
s)[a(t) - a ( r ) ] d r
(s)ds. (6.1.57)
a3(s)
To transform Eq. (6.1.54), we change the order of integration and obtain
P(t) = 2rqx +
foa(t)(X(t
£
t
- r*(R1), 0)[a(t) - a*(R1)] 2
0X
(R1) -~r ( t -
T*
(R1), "r - "r*(R1))[ot(t)
O~(T)]2 dr
} J0" RldR1
R dR.
Calculating the integral, we find that
P(t) = ~'tx
/0a't'{
X(t - r* (R), O)[a(t) - a*(R)] 2
+
(R)~
(t - r*(R), r -
(R))[a(t)
a('r)] 2 dr R 3 dR. (6.1.58)
We substitute expression (6.1.55) into Eq. (6.1.58), present the integral from 0 to a(t) as a sum of two integrals from 0 to al and from al to a(t), calculate the first integral and introduce the notation (6.1.56) in the other. As a result, we obtain
P(t) OX r)[a(t) - a(r)]2dr } -a~4 rr/, - {X(t, O)a2(t) + fo t -~r(t, + +
/o'{
X(t - s, 0)[a(t)
-a-Tr(t - s, r -
-
og(s)] 2
s)[a(t) - a(r)] 2 dr
a3(s)
(s)ds. (6.1.59)
For a given regime of accretion a(t) and a given regime of loading M(t), integral equations (6.1.57) and (6.1.59) permit the twist angle a(t) and the compressive load P(t) to be determined. For a nonaging elastic medium with pX(t, r) = G, Eqs. (6.1.57) and (6.1.59) are simplified
M(t) 2rrG
a4ot(t) + 4
P(t) 7rG
a44a2(t) + fot [a(t) - a(s)]2a3(s)--~s(S)ds. da
/0 t[oe(t)-
a(s)la3(s)
(s)ds, (6.1.60)
Setting t - 0 in Eqs. (6.1.60), we obtain M2(O)
2M(O)
a(O)-
rrGa 4 ,
P(O)-
7rGa41.
(6.1.61)
6.1. Continuous Accretion of Aging Viscoelastic Media
353
Differentiation of the first equality in Eq. (6.1.60) with respect to time t implies that da ~(t) dt
2 dM - ~ ~(t). 7rGa4(t) dt
(6.1.62)
We differentiate the other equality in Eq. (6.1.60) with respect to time and use Eqs. (6.1.60) and (6.1.62). After simple algebra, we find that dP ~(t) dt
1 dM 2 --(t). 7rGa4(t) dt
=
(6.1.63)
Introduce the dimensionless variables a,-
a , a(O)
a,-
a , al
M M(O)'
M,-
P P(O)'
P,-
t, =
t -T'
v,-
vT ~ a 2" (6.1.64)
In the new notation, Eqs. (6.1.36), (6.1.62), and (6.1.63) are written as (asterisks are omitted for simplicity) da dt (t)
v(t) 2a(t)
a(0) = 1
da 1 dM d - T ( t ) - a4(t ) d--t-(t), dP ~(t)dt
a(0) = 1,
2M(t) dM --(t), a4(t) dt
P(0) = 1.
(6.1.65)
We consider the piecewise constant rate of loading v(t) =
vo, 0,
where the parameter v0 equals
v0
T00
O<--t<--To, to < t--< 1,
(6.1.66)
[(a22 ] al
The dimensionless twist angle a and the dimensionless compressive force P are plotted versus the dimensionless time t in Figures 6.1.3 and 6.1.4. The twist angle per unit length a, as well as the compressive force P, increase in time with the growth of torque M. The rate of accretion affects significantly the functions a(t) and P(t). For a given final radius of the cylinder, an increase in the rate of material supply leads to an essential decrease in the a and P values.
6.1.3
Continuous Accretion of an Elastoplastic Bar
Let us consider accretion of an elastoplastic bar by a viscoelastic medium. The bar is loaded by tensile forces, which increase monotonically in time. For relatively
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
354
2.0
1.0 0
t
1
Figure 6.1.3: The dimensionless twist angle a versus the dimensionless time t for an accreted elastic cylinder with a2 = 2al under the dimensionless torque M = 1.0 + t. Curve 1: To = 0.1. Curve 2: To = 0.5. Curve 3: To = 1.0.
small loads, the initial body (the elastoplastic core) is in the elastic state. When the tensile forces grow, the core is transformed into the plastic state, and its material hardens. We study the effect of material accretion and plasticity of the initial body on stresses and displacements in the growing bar. Applications of this problem in industry are connected with reinforcement of engineering structures loaded by forces that may exceed the load-bearing capacity. From a mathematical standpoint, analysis of this problem demonstrates similarity between reference configurations in the elastoplasticity theory and in the theory of accreted media. A rectilinear bar in its natural (stress-free) configuration has length I and a circular cross-section with radius al. At the initial instant t = 0, tensile forces P = P(t) are applied to the ends, and continuous accretion of material begins on the traction-free
355
6.1. Continuous Accretion of Aging Viscoelastic Media
4.0
1.0
l
I
I
I
I
0
I
t
I
1
6.1.4: The dimensionless compressive force P versus the dimensionless time t for an accreted elastic cylinder with a 2 = 2a~ under the dimensionless torque M = 1.0 + t. Curve 1: To = 0.1. Curve 2: To = 0.5. Curve 3: To = 1.0.
Figure
lateral surface of the bar. The function P(t) satisfies the conditions P(0) = 0,
dP -dT(t) > o;
(6.1.67)
body forces are absent. Accretion occurs in the interval [0, T]. Owing to the material supply, the outer radius of the bar in the reference configuration increases according to the law a = a(t),
a(O) = al,
a(T) = a2.
(6.1.68)
We confine ourselves to accretion without preloading, when the natural configuration of a built-up portion coincides with the actual configuration on the accretion surface at instant ~'* when the portion merges with the growing bar.
356
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
The material behavior of the initial body obeys the constitutive equations of an incompressible elastoplastic medium with the strain energy density W(I °, I °) and the von Mizes yield criterion E = E°(X).
(6.1.69)
Here I ° is the kth principal invariant of the Finger tensor for transition from the natural (unloaded) to actual configuration, and E is the stress intensity (6.1.70)
E = (2~" ~)1/2,
where ~ is the deviatoric part of the Cauchy stress tensor 6". The parameter of hardening X equals the work of plastic deformation
X(t) =
#(r) " D*(r)dr,
(6.1.71)
where D* is the rate-of-strain tensor for plastic deformations. In the elastic state, the stress intensity X is less than the yield stress X 0, and the tensor/)* vanishes. In the plastic state, equality (6.1.69) holds, and the rate of strain tensor D* satisfies the associated law D* = A~,
(6.1.72)
where A is a scalar function to be found. The material behavior of built-up portions is governed by the constitutive equations of an aging, incompressible, viscoelastic medium with the function X.(t, T), which determines the reformation process for adaptive links, and the strain energy density Wo(I °, I°), where I ° is the kth principal invariant of the Finger tensor for transition from the natural to actual configuration. Points of the bar in the reference configuration refer to Cartesian coordinates {Xi} with unit vectors ~i. Cartesian coordinates in the actual configuration and in the natural configuration are denoted by {xi} and {x~'}, respectively. For uniaxial extension of a bar, coordinates in the actual configuration xi are expressed in terms of coordinates in the reference configuration Xi as xl = c~(t)X1,
x2 = c~0(t)X2,
x3 = c~0(t)X3,
(6.1.73)
where cz(t) and cz0(t) are functions to be determined. The radius vectors in the reference and actual configurations equal
~o = Xi~i,
? = xiOi.
(6.1.74)
Differentiation of Eq. (6.1.74) with the use of Eq. (6.1.73) implies the tangent vectors in the reference and actual configurations g01 - g'l,
gl
=
o~(t)~'l,
g02 = g'2,
g03 = g'3,
g2 = cz0(t)~'2,
g3 = cz0(t)~'3.
(6.1.75)
6.1. Continuous Accretion of Aging Viscoelastic Media
357
It follows from Eq. (6.1.75) that the deformation gradient for transition from the reference to actual configuration is calculated as ~'o?(t) = o~(t)6'16'1 -k- c~o(8)(6'26'2 -4- 6,36,3).
(6.1.76)
According to Eq. (6.1.76), the Finger tensor for transition from the reference to actual configuration equals F(t) = o~2(t)~'l~'l -k- o~2(t)(~,2~,2 -b ~,3~,3).
(6.1.77)
For an incompressible material with 13(F) = 1,
Eq. (6.1.77) implies that c~0(t) = c~-l/2(t).
(6.1.78)
Substitution of expression (6.1.78) into Eq. (6.1.76) yields ~r0?(t) - tx(t)6'l¢'l -+- 0~-1/2(t)(~'2¢'2 -t- 6,36,3).
(6.1.79)
We suppose that transition from the reference to natural configuration is described by equalities similar to Eq. (6.1.73) X 1 -- o~*Xl,
x 2 = ogoX2,
X3 = aoX3.
(6.1.80)
The amounts c~* and c~ are constant within a built-up portion, but they can change from one built-up layer to another. Repeating the preceding reasoning and employing the incompressibility of accreted elements, we obtain ~g
~
m
c~o = (c~) 1/2.
(6.1.81)
By analogy with Eq. (6.1.79), the deformation gradient for transition from the reference to natural configuration is calculated as (6.1.82)
~r0F* -- O/'6'16' 1 -]- (0g*)-1/2(6'26' 2 + 6'36'3).
Expressions (6.1.79) and (6.1.82) imply the formula for the Finger tensor for transition from the natural to actual configuration
P(t) =
\
-U-
]
6'16'1 "4- ~-~(~'2~' 2 q- 6'3~'3).
(6.1.83)
It follows from Eq. (6.1.83) that the principal invariants of the tensor p0 equal
,0_
+
a'
,o_
~-Z +
-c~
•
(6.1.84)
Let {R, O,Z} be cylindrical coordinates in the reference configuration. For axisymmetrical accretion, the instant ~'* when a built-up portion with polar radius R
358
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
merges with the growing cylinder depends only on R:
r* = r*(R). For accretion without preloading, we write 1, .
c~*(R) =
0_
c~(r (R)), al < R _< a2.
(6.1.85)
By analogy with Eqs. (6.1.83) and (6.1.84), we present the Finger tensor F<>(t, ~') for transition from the actual configuration at instant ~"to the actual configuration at instant t and its principal invariants l~(t, T) and I2<>(t,T) as
P<>(t, ~) = l?(t, ~') =
( a~( t ) ) 2
a('r)
~,~1 + S~-/~(~2~2+ ~3~3),
(~(t>) 2 ~
+ 2 a(~'~)
~(t) '
i2<>(t,,r)=2a(t)+(°t(~')) 2. a(~') -~
(6.1.86)
We substitute expressions (6.1.83), (6.1.84), and (6.1.86) into the constitutive equation of an incompressible, aging, viscoelastic medium (6.1.16) and find that
~(t,R) = Ol(t,R)g'lg'l + o'2(t,R)g'2g'2 + o'3(t,R)e.3e3. Here
{ (a(t) ) O-l(t,R) = -p(t,R) + 2 X.(t - ~'*(R), O) a*(R)
(6.1.87)
e
+ f, t(R) ax*(taT- ~-*(R), ~-- ~-*(R))(~(t)'~ a0")J =
X
*?(t,
r) + *2<>(t, r) ~
dr
,
c~*(R)
o'2(t,R) = o'3(t,R) = -p(t,R) + 2{X.(t - ~'*(R), 0 ) ~ a(t) X [ ~ ° ( t , R ) + ~°(t'R)a*(R)]a(t)
+
f~,(R~ -~r OX,(t -
~'*(R), ~"- ~'*(R)) a(~')
x[ vg~(t'T) + ~l'2~(t' " a(T)l
~(t)
}
(6.1.88)
359
6.1. Continuous Accretion of Aging Viscoelastic Media
where the functions ~ko(t, r) and ~kO(t, ~) are determined by Eqs. (6.1.17) .., OWo (io(t,R),io(t,R)), OWo (lO(t,e),io(t,e)) + io(t,K)_~2 ~ ° ( t ' R ) = -~1 qgo(t,R) = _ OWo (io(t,R),io(t,R)),
012
8Wo (ii~ (t ' r), l~(t, r)), OWo (ii~ (t ' ~.), i2<>(t ' ~.)) + ll<>(t, ~')-~2 • 2~ (t, T) = - OWo (i1~(t, 1"),12~(t, 1")).
(6.1.89)
012
To satisfy the equilibrium equations and the boundary conditions on the lateral surface of the bar, we set 0" 2
--
0" 3 - -
(6.1.90)
0.
We find pressure p from the second equality in Eq. (6.1.88) and Eq. (6.1.90) and substitute the obtained result into the first equality in Eq. (6.1.88). After simple algebra, we arrive at the expression
a*(R)] ~(t) + 2
--~(t - I"*(R), ~- - ~'*(R)) ~l<>(t, 1") \ a(~') J
-
(R)
a(¢)] a(t)
al < R < a2. (6.1.91) Bearing in mind Eq. (6.1.85), we obtain from Eq. (6.1.91)
a(¢*(R))] a(t)
+ 2
OX*(t - ~'*(R), (R) OT
- I"*(R)) ~ ( t ,
~')
-
a(t) 1 ~(T)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
360
_~_~if?(t,T)I(O~(t) __~)-4 ()~_~O~(T)2] /dT,
al <-- R <- a2.
(6.1.92)
Accretion of an Elastic Bar We confine ourselves to active loading of an elastoplastic medium. The latter means that before some instant TO, the initial bar is in the elastic state; at TO it is transformed into the plastic state; and it remains in the plastic state after TO. We begin the study of material accretion assuming that the initial body is in the elastic state. To calculate the nonzero component trl of the stress tensor # in the elastoplastic core, it suffices to set X,(t, ~') = X,(0, 0) in Eq. (6.1.92) and to employ Eq. (6.1.85). As a result, we find that
trl(t) = 2{~l(t)[c~2(t)- c~-l(t)] + xI~2(t)[ct4(t)- ot-2(t)]},
0-
(6.1.93) where W = X.(0, 0)W0, and XI)'1(t)
=
0W 0W -7;-. (I1 (t), I2(t)) + I1 (t)-77-. (I1 (t), I2(t)), ~11
OW
~I2
xI~2(t) = -- ~ ( I 1 (t), I2(t)).
(6.1.94)
Expression (6.1.93) satisfies the equilibrium equations. To find the unknown extension ratio a, we write the boundary condition on the ends of the bar in the integral form f P(t) = [ O"1(t, R) ds (t)
(6.1.95)
dx 2 dx3,
where S(t) is the cross-section area at instant t in the actual configuration. Returning to the reference configuration with the use of Eq. (6.1.73), we find from Eq. (6.1.95) that
P ( t ) - t~(t) 27r
foact>O'l(t,R)RdR.
(6.1.96)
Substitution of expressions (6.1.91) and (6.1.93) into Eq. (6.1.96) implies the nonlinear integral equation for the function a(t)
a(t)P(t) = a2{aIrl(t)[oz2(t)- ~ - l ( t ) ] + xlt2(t)[a4(t)- ct-2(t)]} 2~r + 2
,/a1
X,(t - r*(R),O)
~°(t,R)
a(t) ~(r*(R))
_
a(r*(R))] a(t)
6.1. Continuous Accretion of Aging Viscoelastic Media
+ atr°(t,R)
+2
361
I( ~(T*(R)) C1£ (,))4 - (~(T,(R))) }~(t)2] R dR
R dR f l -~2-_ (t ia(t) aal (R)oX*OT
{
x *l~(t'~) -g~
~'*(R),~'-
T*(R))
,~(t)
Introducing the new variable R = a(s), we find that
a(t)P(t) 2II"
= a2{altl(t)[oz2(t)- ot-l(t)] + xIt2(t)[c~4(t)- o~-2(t)]} +2
So'
+ *°(t,a(s))
+2
{
X,(t - s, O) qt°(t, a(s))
-~
-~
-
a(s) a(t) 1
__(O~(S)2] ds -~) }a(S)~ss(S)
L
t a(s)--d-~s(S) da ds fss t ---~-T(t OX, - s, r - s){~I'~(t, T)
t, <~(-,-))
<~(t)
~
t,-h-777)
d~. (6.1.97)
Finally, changing the order of integration in the third term on the right-hand side of Eq. (6.1.97), we obtain
a(t)P(t) 27r
= a21{Xltl(t)[ot2(t)- a-l(t)] + qt2(t)[a4(t)- a-2(t)]}
+2L Zl(t,~) - ~ + z2(t, ~) \ - ~ /
-,~(t) (6.1.98)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
362 where
Zk(t, w) = ~ ° ( t , a(~'))X.(t - ~', 0)a(r)~-~ar(w) da (S) ds. (t, w) fo ~ ~OX, ( t - s, T - s)a(s)--d-~s
+~
(6.1.99)
Suppose that the mechanical behavior of the initial body and built-up layers is described by the constitutive equations of a neo-Hookean medium with strain energy densities (I ° /'~ Wo = 2X,(0, 0)
-
3)
W =
'
/x ( i o _ 3),
(6.1
2
100)
where /.to and ~ are generalized shear moduli. According to Eqs. (6.1.20) and (6.1.100), equation (6.1.98) reads
a(T)]
--l(t)]q--2"t-~ ~) ~0 Z(/,T) [ / og(t)/
a(t)P(t) = aZ[aZ(t)7r/x
a(t)
d~',
(6.1.101) where
OX - s, ~"- s)a(s)--d~s(S) da (~') + fo ~ -~r(t ds.
Z(t, ~') = X(t - ~', 0)a(~')
(6.1.102) If the initial body and built-up portions are made of the same elastic material, Eqs. (6.1.101) and (6.1.102) imply that
P(t) - aZ[c~(t)- ot-2(t)] + 2 7r/x
~
- \ c~(t)J
~
(~-)d~'. (6.1.103)
We introduce the dimensionless variables t, -
t
T'
a, -
a
al
,
P, -
P
7r/xa2'
(6.1.104)
and present Eq. (6.1.103) as follows (asterisks are omitted for simplicity):
~(t) [l + 2 fot~-2(~')a(~')ff~T(w)d~'l - a-2(t)
1+ 2
~(-r)a(w)
(~')d
= P(t).
(6.1.105)
6.1. Continuous Accretion of Aging ViscoelasticMedia
363
Let yz(t) = 1 + 2 fot c~(~-)a(~-)~--~r(l")dl". da
yl(t) = o~-2(t),
(6.1.106)
Differentiation of the second equality in Eq. (6.1.106) yields dy2 d---T(t) =
2a(t)y~/Z(t) ~-~da (t).
(6.1.107)
To derive a differential equation for the function Yl (t), we write Eq. (6.1.105) in the form c~(t) 1 + 2
/0 t c~-2('r)a('r)
]
(~')d~" -yl(t)y2(t) = P(t).
(6.1.108)
It follows from Eqs. (6.1.106) and (6.1.108) that
1 + 2 fOOt a-2(r)a(T)-~z(T)dT = y I/2(t)[P(t ) + yl(t)y2(t)] . da
(6.1.109)
Differentiating Eq. (6.1.108) with respect to time and using Eqs. (6.1.106) and (6.1.107), we find that
da
,/-7
(t) I "~to+l 2
d a l d y l d ~ -" a-2(l")a(~')~--4T(l")
7i
dP (t)y2(t) = -d-~-(t).
(6.1 110)
According to Eq. (6.1.106),
d~ d-T(t) =
1 dyl 2y~/2(t) dt (t).
(6.1.111)
Substitution of expressions (6.1.109) and (6.1.111) into Eq. (6.1.110) results in
dyld___t(t) _ = -2yl(t)[P(t) + 3yl(t)y2(t)] -1 -d-~dP (t).
(6.1.112)
To derive initial conditions for Eqs. (6.1.107) and (6.1.112), we set t = 0 in Eq. (6.1.105) and employ Eq. (6.1.67). As a result, we obtain y~(0) = 1,
y2(0) = 1.
(6.1.113)
For a given program of loading P(t) and a given program of accretion a(t), the ordinary differential equations (6.1.107) and (6.1.112) with the initial conditions (6.1.113) determine the extension ratio c~(t). When this function is found, the nonzero component ~rl of the stress tensor 6- is calculated by formulas (6.1.92) and (6.1.93). Accretion of an Elastoplastic Bar We now analyze stresses and displacements in an elastoplastic bar when the initial body is transformed into the plastic state. By
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
364
analogy with Eq. (6.1.91), we assume that or 1 is the only nonzero component of the stress tensor in the plastic state. It follows from Eq. (6.1.70) that in this case, 2
(6.1.114)
E = ----~0"1.
V'3 Combining Eqs. (6.1.69) and (6.1.114), we find that 2~3 ~O(x).
O"1 --
(6.1.115)
We suppose that transition from the reference configuration of the initial body to its natural (unloaded) configuration in the plastic state is described by Eq. (6.1.80), where a* is a function of time to be found. This function is independent of spatial coordinates, since homogeneous deformation occurs in the initial body under tension. Employing the incompressibility condition, we arrive at Eq. (6.1.82) for the deformation gradient V0?*(t). The deformation gradient for transition from the natural configuration at instant ~"to the natural configuration at instant t is calculated as -,_,
Vsr (t)
(Og*(T)) 1/2
ct*(t) -
ct*('r)
~'1~'1 +
(~'2~'2 -+- ~'3~'3).
a*(t)
(6.1.116)
The corresponding Finger tensor equals /~*(t' ~') = (a*(t) c~*('r) ) 2 ~'1~'1
c~*(T) +
c~*(t)
(~'2~'2 -+- e3~'3).
(6.1.117)
To find the rate-of-strain tensor for plastic deformations D*, we utilize the formula [see, e.g., Drozdov (1996)]
~---P*(t,r)l,=t. D*(t) = ~1 at
(6.1.118)
It follows from Eqs. (6.1.117) and (6.1.118) that
D*(t)-
[
1(e2e2 W ~'3~'3)1
1 da*(t) e l e l -
~*
d-S-
~
"
(6.1 119) "
Substitution of Eqs. (6.1.115) and (6.1.119) into Eq. (6.1.71) implies that
d-~-X(t)= X/~E°(X(t)) "~*d~-z-(t), dt
2a*(t)
dt
X(~"°) = O.
(6.1.120)
The Finger tensor F°(t) for transition from the natural to actual configuration at instant t is determined by Eq. (6.1.83) with a* = a*(t). Substitution of expression (6.1.83) into the constitutive equation (6.1.91) implies the formula similar to Eq.
6.1. Continuous Accretion of Aging Viscoelastic Media
365
(6.1.93), trl(t) = 2
~l(t)
--Olc~(t) *(t_____2) 1 -I-Xlf2(t)[( a*(t) Ol(t))4--(Ol*(t))21) a(t) "
a*(t)
(6.1.121) Here
XI~l)(=t ~ii(IO(t),IO(t))+ lO(t)~-~2(IO(t),IO(t)), ,I,2(t)
=
OW (iOl(t), lO(t)) '
(6.1.122)
- -ff 2
where
I°(t) = Ik(P°(t)). Combining Eqs. (6.1.115) and (6.1.121), we find that
4 {xi~ I(o~(t))2 o,(t)l
~,°(X(t)) = - ~
l(t)
a*(t)
-
a(t)
[(or(t))4 (ct,(t))2]}
+ xIt2(t)
c~*(t)
-
a(t)
"
(6.1.123) Substitution of expressions (6.1.92) and (6.1.115) into Eq. (6.1.96) yields
~(tlP(t)2~r
-- a21d £°(X(t)) + 2 j0't(Zl(t, r) I\ - ~ / a--~)-(
)~-~
O~(T)1
- c~(t----S
d'r.
(6.1124)
For a given program of loading P(t) and a given program of accretion a(t), the functions a(t), a*(t), and X(t) are determined by the ordinary differential equation (6.1.120), the nonlinear algebraic equation (6.1.123), and the nonlinear integral equation (6.1.124). Numerical Analysis To study the effect of material accretion on stresses and displacements in a growing bar, we solve numerically Eqs. (6.1.123) and (6.1.124) for an ideal (~0 = constant) elastoplastic cylinder. Material of the initial body in the elastic state obeys the constitutive equations of a neo-Hookean medium with a shear modulus/~. The behavior of built-up portions is governed by the constitutive relations of an elastic neo-Hookean medium with a modulus/~. In this case, Eqs. (6.1.123) and (6.1.124) read
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
366
~(t))2
,~*(t) _ X°xfi
cz*(t)
cz(t)
P(t) 7rl~
2/x '
a (r) ] a(r) da X°X~a22/, + 2/'~°/x./o"
(6.1.125)
\ cz(T)
We introduce the dimensionless parameters
,o
-
,o
/2-
T'
/*o,
~ -
/.z
~ ,
(6.1.126)
2/2,
and present Eqs. (6.1.101) and (6.1.125) as follows (asterisks are omitted for simplicity):
P(t) = [c~(t) - c~-2(t)] + 2/2
P(t) = ]~c~-l(t) + 2/2
loll
2
a(~') ] a(~') da a(t) --d-~(~')d~',
c~(t) ' ~ _
c~O'))
Lt I(a(t)) ~
2 - a(~')]a(~.)da a(t) ~-(r)d'r,
0 <- t <- T°,
"r° < t -< 1. (6.1.127)
The dimensionless time T° when the initial body is transformed into the plastic state is found from the algebraic equation O~2(T O) -- cz-l(r O) = 2~.
(6.1.128)
We confine ourselves to a growing cylinder with a2 = 2al under the dimensionless tensile load P(t) = 5t and assume that the dimensionless rate of accretion da
v(t) = 2a(t)-~(t) is a stepwise function,
v(t) =
vo, O,
O<--t<-To, To < t ~ l,
where the constant vo is found from the condition
a 2 - a 2 = voTo. The extension ratio a is plotted versus the dimensionless time t in Figures 6.1.5 and 6.1.6. The function a(t) increases monotonically in time. The rate of increase is rather small for rapid accretion (when the time of accretion To is small), and grows significantly with an increase in the time To. The graphs c~(t) are continuous, but the derivative of a with respect to time suffers a jump of discontinuity at the instant when
6.1. Continuous Accretion of Aging Viscoelastic Media
367
3.0
1.0
~
ld ~ 0
I
I
I
I
I
I
I t
I
3
I 1
Figure 6.1.5: The extension ratio c~ versus the dimensionless time t for a growing elastoplastic bar with/5, = 1.0 and I~, = 0.5. Curve 1: To = 0.1, ~.0 = 0.28. Curve 2: To = 0.5, 1-° = 0.12. Curve 3: To = 1.0, T° = 0.10. the initial body is transformed into the plastic state. The dimensionless yield stress affects essentially the extension ratio: for any instant t, the a value decreases in E. The dimensionless instant T° is plotted versus the dimensionless~ yield stress 1~ in Figure 6.1.7. The time TO increases monotonically in E for small E, reaches its ultimate value TO = 1 at some yield stress E0, and remains equal unity for ~ > ~0 (the latter means that the initial body remains elastic within the interval of accretion). The critical yield stress ~0 decreases monotonically in/5,, i.e., with the growth of the shear modulus of built-up portions. The dimensionless stress o"1, = o1//-to at the final instant T is plotted versus the dimensionless radius R, = R/al in Figure 6.1.8. The results are rather surprising" in the vicinity of the initial cylinder (R, ~ 1), the stress decreases sharply with the growth of the accretion rate, whereas far away from the initial body (R, ~ 2), the stress at rapid accretion essentially exceeds the stress for slow accretion.
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
368
2.0
1.0
•
0
I
I
I
I
I
I
t
1
Figure 6.1.6: The extension ratio a versus the dimensionless time t for a growing elastoplastic bar with/~, = 1.0 and ~ = 2.0. Curve 1" To = 0.1, TO = 1.0. Curve 2: To = 0.5, "r° = 0.77. Curve 3" To = 1.0, TO = 0.44.
Concluding Remarks A mathematical model is derived for continuous accretion of an aging viscoelastic medium. This model differs from the standard models in viscoelasticity by the use of three basic configurations: reference, natural, and actual. Unlike inert media, an additional tensor function should be prescribed for accreted media: the Finger tensor (or an appropriate deformation gradient) for transition from the reference to natural configuration. Two opportunities are distinguished to describe this tensor: explicit, when it is given as a function of Lagrangian coordinates, and implicit, when some relation is prescribed on the accretion surface between this tensor and the Finger tensor for transition from the reference to the actual configuration. The latter approach, corresponding to the engineering demands, will be consistently employed in the book.
6.1. Continuous Accretion of Aging Viscoelastic Media
369
z
31!
21
1
t TO
I
o
I
I
I
I
~
I
I
~o
Figure 6.1.7" The dimensionless instant ~.0 when the initial body is transformed into the plastic state versus the dimensionless yield stress ~, for a growing elastoplastic bar with To = 1.0. Curve 1"/2 = 0.5. Curve 2:/2 = 1.0. Curve 3:/2 = 2.0.
As an example, we consider accretion of an aging, viscoelastic, circular cylinder under torsion. Given torque M(t) and given program of accretion a(t), a linear integral equation is developed for the twist angle a, and a nonlinear integral equation for the compressive load P. These equations are solved numerically for a neo-Hookean elastic material. The rate of material supply affects drastically stresses and displacements in a growing cylinder. An increase in the rate of accretion leads to an essential decrease in the twist angle, as well as in the compressive force. As another example, we analyze continuous accretion of an elastoplastic bar under tension by viscoelastic built-up portions. Stresses and displacements in a growing bar are determined by three functions of time that satisfy differential, integral, and algebraic equations. These equations are solved numerically to study the effect of plasticity and accretion on stress distribution of a growing bar. It is demonstrated that
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
370
10
0"1,
l 1
I
1
1
1
1 ~ R,
2
F i g u r e 6.1.8: The dimensionless stress o"1, versus the dimensionless radius R, for a
growing elastoplastic bar with/~ = 1.0 and ~, = 0.5. Curve 1: To = 0.1. Curve 2: To = 0.5. Curve 3: To = 1.0.
1. The extension ratio a decreases with the growth of the accretion rate and the dimensionless yield stress E. 2. The effect of the rate of material supply on the stress o1 is twofold: an increase in the rate of accretion implies a sharp decrease in the stress in the vicinity of the initial body and an increase in the stress far away from the initial body. 3. The time T° when the initial body is transformed into the plastic state increases with an increase of the dimensionless yield stress ~ and the shear modulus of built-up portions. Three reference configurations in the model of continuous accretion are similar to three configurations in the elastoplasticity theory, in which the unloaded configuration plays the role of the natural configuration. The only difference between these
6.2. Winding of a Cylindrical Pressure Vessel
371
approaches is that the natural configuration for an elastoplastic medium is a function of time, whereas the natural configuration of built-up portions is a function of Lagrangian coordinates.
6.2
Winding of a Cylindrical Pressure Vessel
This section is concerned with modeling the winding process for a viscoelastic cylinder with finite strains. This problem is of special interest for two different areas of applications. The model of an accreted cylinder is widely used to calculate residual stresses built-up in winding and unwinding in rolls of paper [see Altmann (1968), Frye (1967), Hussain and Farrell (1977), Pfeiffer (1966), Rand and Eriksson (1973)], plastic sheets [see Grabovskii et al. (1979), Monk et al. (1975), Yagoda (1980)], rubber [see Wireman (1973)], and magnetic and videotapes [see Grabovskii (1983, 1984), Lin and Westmann (1989), Tramposch (1965, 1967), Wheeler (1985), Willett and Poesch (1988), Zabaras and Liu (1995), Zabaras et al. (1994)]. Distribution of residual stresses in a tape pack characterizes quality of the winding process and determines long-term integrity of the pack. The lack of tension may cause such defects as cinching and spoking, whereas an extensive pressure may lead to the hub instability as well as to crimping or buckling of the tape. Winding of flexible sheetlike materials into rolls is characterized by (i) essential anisotropy of the medium, (ii) creep in successively wound layers, and (iii) significant nonlinearity in the material response affected by the amount of air trapped between layers. To calculate residual stresses in a wound roll, Altmann (1968) treated the winding process as a limit of a discrete process of successive accretion of annular layers on the external surface of a growing disk provided that thickness of the layers tended to zero. The material behavior was governed by the constitutive equations of an orthotropic linear elastic medium. By assuming plane stresses, an integral equation was derived for the stress tensor, and its explicit solution was developed for uniform prestressing. Yagoda (1980) extended the Altmann model by accounting for radial pressure. Explicit formulas for stresses were derived for an arbitrary winding tension by using hypergeometric functions. The obtained solution was presented in the form of an infinite series. Both plane stresses and plane strains were considered. Similar results were derived by Grabovskii et al. (1979) and Grabovskii (1984). To study the effect of entrapped air in reels on residual stresses, Willett and Poesch (1988) employed constitutive equations of an orthotropic elastic medium, in which the radial elastic modulus was a nonlinear (polynomial) function of the radial strain. The tangential elastic modulus, as well as Poisson's ratios, remained constant. Similar nonlinearities in the constitutive equations were suggested also by Hakiel (1987) and Pfeiffer (1979). Pfeiffer (1979) assumed the pressure between successively wound layers to be a nonlinear function of the radial strain. Hakiel (1987) modeled
372
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
the radial elastic modulus as a polynomial function of the pressure. Benson (1995) extended these models to finite strains. Good and Wu (1993) and Good et al. (1994) analyzed the effect of a nip roller on stresses in a wound roll. The effect of the layers viscosity on stress distribution was studied by Tramposch (1965, 1967). Tramposch (1965) used a linear differential constitutive model with four parameters for the description of the viscoelastic response. The effects of the material anisotropy on the stress relaxation in viscoelastic wound rolls were considered in Tramposch (1967). The influence of material viscosity was studied by Grabovskii (1983) for an anisotropic linear viscoelastic medium with the relaxation kernel (2.3.15) and compared with experimental data for a magnetic tape. Lin and Westmann (1989) treated winding layers as isotropic and linearly viscoelastic (with an arbitrary relaxation kernel) and analyzed stresses arising in an accreted cylinder under winding, pausing, and unwinding. An attempt to account for the material nonlinearity in the viscoelastic response was undertaken by Portnov and Beil (1977). An intermediate model between elastic and viscoelastic models, the so-called hypoelastic solid, was proposed by Zabaras et al. (1994) and Zabaras and Liu (1995) for the mechanical behavior of magnetic tapes. To construct that model, (i) stresses and strains in the standard stress-strain relations for a linear anisotropic medium are replaced by their derivatives with respect to time, and (ii) coefficients in the obtained equations are assumed to depend on stresses explicitly. As a result, we arrive at a version of the constitutive relation (2.2.12) for an aging elastic material with small strains. The difference between these models consists in the following: in Eq. (2.2.12) elastic moduli are assumed to depend on time explicitly, whereas in the theory of hypoelastic media they depend on time implicitly through the dependence of the stress tensor on time. A model of a growing medium is employed to determine residual stresses built up in composite pressure vessels and pipes manufactured by filament winding. The winding process consists of wrapping rovings of fibers over a mandrel. Filament winding is carried out with a given helical wind angle th (which serves to increase the carrying capacity of pressure vessels) and with a suitable pretension (which controls the fiber position and provides a pressure necessary for compaction of the entire fiber network). The mandrel serves to resist sag due to the winding tension and the weight of manufacturing structures, and it is removed after curing. There are three main stages of the winding process: 1. Winding, in which resin-impregnated fiber bundles are wounded around a rotating mandrel. 2. Curing, in which the assembly is placed into an oven, where it is subsequently heated, cured and cooled to consolidate the fiber network and to cure the resin. 3. Removal from the mandrel. Two types of the filament winding are distinguished: 1. Wet winding, in which the fiber is saturated with low-viscosity uncured resin during winding onto the mandrel.
6.2. Winding of a Cylindrical Pressure Vessel
373
2. Dry winding, in which preimpregrated tows (the prepreg material) consisting of fibers and partially cured resin are wound. The difference between these types of winding implies the difference in the mathematical models for their description. In the study of the first stage of the dry winding, we concentrate on stresses arising in fabricated composite structures. To analyze the first stage of the wet winding, consolidation of wound layers and the resin flow should be taken into account. The only exception to this rule is the wet winding process, in which the winding time is essentially less than the time necessary for the resin flow, and the rows consolidation may be neglected [see Cai et al. (1992)]. As common practice, the study is confined to wound structures of the simplest form (cylindrical and spherical shells). However, a few exceptions may be mentioned: Dobrovolskii and Kostrov (1970) studied winding of shells of revolution; Mukhambetzhanov et al. (1992) considered winding of conical shells with an arbitrary convex directrix; Loos and Tzeng (1994) analyzed the effect of dome regions of a cylindrical mandrel; and Mazumdar and Hoa (1995a, b) studied kinematics of winding for non-axisymmetrical cylindrical mandrels. Depending on the helical wind angle th, we distinguish the hoop (circumferential) winding with th = 0 and the helical (chord) winding with th 4: 0. In the latter case, the filament is placed along straight lines (chords) on the lateral surface of the wound structure, and the slope angle th can change from layer to layer. Traditionally, the effect of the winding angle is accounted for by replacing the filament stress (in the direction of the fiber) o-, by the circumferential component of the stress, the so-called annular stress [see, e.g., Beil et al. (1983) and Cai et al. (1992)] cr0 = o-, sin 2 ~. However, this formula may be refined provided that the effect of friction between a fiber and a wound structure [see Moorlat et al. (1982)] and slippage of wound fibers [see Beil and Portnov (1973)] are taken into account. Biderman et al. (1969), Dewey and Knight (1969), Fourney (1968), Nikolaev and Indenbaum (1970), Indenbaum and Perevozchikov (1972), Tarnopolskii and Portnov (1966, 1970), Tamopolskii et al. (1972), and Yablonskii (1971) concentrated on the measurement and calculation of residual stresses arising in filament-wound composite structures. The material behavior was governed by the constitutive equations of a linear orthotropic elastic medium. In the framework of linear elasticity, Beil et al. (1980) studied the effect of the winding angle q) on residual stresses. The influence of the winding angle on strength of wound composite structures was analyzed by Bulmanis and Gusev (1983) and Spencer and Hull (1978). A model of a nonlinear orthotropic elastic material with small strains was used by Beil et al. (1983), Portnov and Beil (1977), and Ochan (1977) to calculate stresses built up in a wound cylinder under the action of fiber pretensioning and external pressure. Tarnopolskii and Beil (1983) and Munro (1988) presented surveys of models for stresses built up in wound structures at dry winding.
374
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
Bolotin et al. (1980) proposed a model that took into account changes of stresses caused by filament winding with prestressing, resin flow, solidification, shrinkage of resin under curing, etc. Similar models were recently derived by Cai et al. (1992) and Lee and Springer (1990a). For a bibliographical survey, see Cai et al. (1992), Loos and Tzeng (1994), and Tarnopolskii (1992). A combined model for the filament winding consists of several submodels for filament winding, stresses arising in a wound fiber, resin flow through a porous medium, thermal deformations of a wound structure, and rheokinetics of curing and solidification of the binder [see Lee and Springer (1990a, b), Calius and Springer (1990), and Callius et al. (1990)]. Layers wound on a mandrel are modeled in those works as (one-dimensional) curvilinear beams. This approach permits stress distribution in a wound cylinder to be determined correctly, but it fails to predict stresses in shells of arbitrary shape [see Tomashevskii and Yakovlev (1982)]. Tomashevskii and Yakovlev (1982, 1984) suggested a model in which a built-up element was treated as a thin elastic shell. This model was extended by Obraztsov and Tomashevskii (1987) and Obraztsov et al. (1990). The Tomashevskii and Obraztsov models followed in the main aspects to the models proposed earlier by Brown and Goodman (1963). Arutyunyan (1977) proposed a model for continuous accretion of a linear viscoelastic medium taking into account nonhomogeneous aging of material. Similar approaches to the analysis of stresses built up in elastic accreted media were derived by Trincher (1984) and Zabaras and Liu (1995). The Arutyunyan model was extended to nonlinear viscoelasticity with small strains by Arutyunyan and Metlov (1983), to nonlinear viscoelasticity with finite strains by Arutyunyan and Drozdov (1984a, b, 1985b), and to viscoelastoplasticity with finite strains by Arutyunyan and Drozdov (1985a). Based on the Arutyunyan model, multilayered accretion of a viscoelastic cylinder under tension and compression was analyzed by Arutyunyan and Metlov (1982), winding of a viscoelastic spherical pressure vessel was studied by Arutyunyan and Shoikhet (1981); and winding of a viscoelastoplastic cylindrical pressure vessel was considered in Arutyunyan et al. (1986). Solutions to these problems were derived under the assumption that small strains occured in growing media and the material response was linear, see Arutyunyan et al. (1987). According to the Arutyunyan approach, the material response is homogeneous at the microlevel. The effect of the material inhomogeneity at the microscale on the stress distribution in wound composite structures was analyzed by Paimushin and Sidorov (1990a, b) using an averaging technique. A system of partial differential equations was derived, but no attempt was undertaken to solve the obtained equations numerically. In this section, we study the winding process for a cylindrical pressure vessel under the action of internal pressure and filament pretensioning. Unlike the preceding works, we assume that finite deformations occur in a growing vessel and analyze the effect of (i) the loading history, (ii) the rate of material supply, and (iii) rheological properties of the accreted material on the final stress distribution in a growing body. The exposition follows Arutyunyan and Drozdov (1984a, b) and Drozdov (1994).
375
6.2. Winding of a Cylindrical Pressure Vessel
6.2.1
The Lame Problem for an Accreted Cylinder
Consider a hollow circular cylinder with length l, inner radius a0, and outer radius al. The cylinder is located between two rigid plates, and it does not deform in the axial direction. The friction is neglected between the cylinder and the plates [see Figure 6.2.1 ]. Points of the cylinder refer to cylindrical coordinates {R, (9, Z} in the reference configuration, to cylindrical coordinates {r*, 0", z*} in the natural configuration, and to cylindrical coordinates {r, 0, z} in the actual configuration. The unit vectors of these -* e-*o, ez, -* and 6'r 6'0, 6'z, respectively. coordinate frames are denoted as ~e, ~o, ~z, er, At the initial instant t = 0, a pressure po(t) is applied to the internal surface R - a0, and the material accretion begins on the external surface R = a l. Owing to the mass supply in the interval [0, T], the outer radius of the cylinder in the reference configuration a(t) changes according to the law a = a(t),
a(0) = al,
a ( T ) - a2.
At instant t E [0, T], the growing cylinder occupies the domain {a0<---R<--a(t),
0<--19 < 2 7 r ,
0-----Z-----I}
in the reference configuration. Within the interval [t, t + dt], the cylindrical shell {a(t) <-- R <- a(t) + da(t),
0 <- ~ < 27r,
I.
i ~'
~1
~
a(t)
0 <- Z <- l}
.I
~1
Figure 6.2.1: A growing viscoelastic cylinder under internal pressure.
376
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
joins the cylinder. The built-up portion is previously deformed by an internal pressure so that its inner radius under loading coincides with the outer radius of the growing cylinder. Afterward, the shell is wrapped around the growing cylinder and immediately merges with it. In the absence of body forces, axisymmetrical deformation in the plane Z = const occurs both in the growing body and in built-up portions. Transitions from the reference configuration to the natural and actual configurations obey the equations r = f(t,R),
0 = 19,
r* = f . ( R ) ,
z = Z,
O* = 19,
(6.2.1)
z* = Z,
where f ( t , R ) and f . ( R ) are functions to be found. The radius vectors ?0 and ? in the reference configuration and in the actual configuration equal ?o = R~R + Z~z,
? = f(t,R)G
+ Z~z.
Differentiating these expressions, we find the tangent vectors
g01 "- 6'R,
g0 2 = R ~ o ,
gl(t) = h(t)G,
g03 = 6'Z,
g2(t) = f(t)eo,
g3(t) = g'z,
(6.2.2)
where Of h = -~(t),
and the argument R is omitted for simplicity. Using Eq. (6.2.2), we calculate the dual vectors 1
1
gl(t) = h--~er,
~2 =
1
f--~g'0,
~3
= g~z.
(6.2.3)
Combining Eqs. (6.2.2) and (6.2.3) with Eqs. (1.1.23) and (1.1.42), we find the deformation gradients and the Finger tensor for transition from the reference configuration to the actual configuration at instant t f(t) fTo~(t) = h(t)eRer + --~ eoeo + ezez,
F(t) = h2(t)e.re.r -+-
e.oeo + ~z~z.
The principal invariants Ik(F) of the Finger tensor are calculated as
(6.2.4)
377
6.2. Winding of a Cylindrical Pressure Vessel
(I) 2 I1(/~) = h2 +
+19
I2(F) = h 2 +
k
+
I3(P) =
)-
.
(6.2.5) Expressions (6.2.5) together with the incompressibility condition I3(F) = 0 result in of f(t,R)-~(t,R)
= R.
(6.2.6)
f z ( t , R ) = R 2 + C(t),
(6.2.7)
Integration of Eq. (6.2.6) implies that
where C(t) is a function to be found. Substituting expression (6.2.7) into Eq. (6.2.4), we obtain V0?(t,R) - - - eRR e r
+ ~f (et o, Re )o _ _ + ezez. R
f(t,R)
(6.2.8)
Continuous growth is treated as a limit of a discrete accretion process when builtup portions merge with the growing body at instants tn (n = 0 . . . . . N). Applying this reasoning to transition from the reference configuration to the natural configuration of a built-up portion A~O(tn), w e find that f 2 ( e ) = R 2 + Cn,,
a(tn) <- e < a(tn+l),
(6.2.9)
where Cn, are appropriate constants. By analogy with Eq. (6.2.8), we write R
_
_,
f,(R)_
~'0?*(R) = f,(R)eRer + ~ R
_,
,
e ° e ° + ezez"
(6.2.10)
Introduce the function c(N)(R) =
Co,,
a(to) --< R < a(tl),
C1,,
a(tl) -< R < a(t2),
° ° ° 9
' ' '
9
C(N-1),9 a(tN-1) <- R < a(tN). and denote by C,(R) the limit of c~,N)(R) as N ~ ~. It follows from Eq. (6.2.9) that for the continuous process of accretion f,2(R) = R 2 + C,(R)9
(6.2.11)
whereas the deformation gradient preserves its form (6.2.10). Substitution of expression (6.2.10) into Eq. (6.1.3) implies that
378
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains f , _,_ f ( t ) _,_ ~7*?(t) = - ~ e r e r + --f-, eoeo + ~z~z .
(6.2.12)
Combining Eqs. (6.2.8) and (6.2.12) with Eqs. (6.1.5), we obtain the following formulas for the Finger tensor F°(t) and the relative Finger tensor F<>(t, r): p O ( t ) : ( f ~ t ) ) 2 e.re.r q-
e.oe.o + ezez,
R 2 + C,(R)_ _ R 2 + C(t) erer + R2 + C,(R) e°e° + ezez, R 2 + C(t)
(f(t) ~ 2
F*(t, T) = \ - ~ ] R 2 + C('r)_ _ R 2 + C(t)
erer +
R 2 + C(t)_ _ e o e o + ~z~z.
(6.2.13)
R 2 + C('r)
The principal invariants of the Finger tensors equal R 2 + C(t) + R 2 + C,(R) + 1, l°(t) = I°(t) = R 2 + C,(R) R 2 + C(t)
R 2 + C($)
R 2 + C(t) + + 1. Ii<>(t, z) = I2<>(t,-r) = R2 + C(T) R 2 + C(t)
(6.2.14)
Substitution of expressions (6.2.13) and (6.2.14) into the constitutive equation (6.1.16) implies that (6.2.15)
O" : O'rrerer q- O'OOe.oeo q- O'zze-ze.z,
where Orrr(t,g)=-p(t,R)+2{X,(t-r*(R),O)[~°(t,R) R 2 + C,(R) R 2 + C(t)
+ fcj
R"R2 + C,(R)
r * ( R ) , r - T*(R))[~l<>(t,r,R)
OX, ( t -
(R) -~T
R 2 + C('r)] R 2 + C('r) +~2~(t, r,R)
R 2 + C(t) ] R 2 + C(t)
{
[
R2+ct,]
oroo(t,R ) = - p ( t , R ) + 2 X , ( t - T*(R), O) xrg°(t,R) + ~ ° ( t , R ) R2 + C,(R) R 2 + C(t) R 2 + C,(R)
(g) --~r (t - r*(R), r - r*(R)) ~ ( t ,
r,R)
6.2. Winding of a Cylindrical Pressure Vessel
379
R 2 + C(t)] R 2 + C(t) +~2°(t' z'R)R2 + ~ ) J R 2 + C(r)
¢rzz(t,R) = -p(t,R) + 2 { X , ( t - T*(R),O)[~°(t,R) + ~°(t,R)] +
~j
3X,
(t - ~'*(R) ~"- ~'*(R))[~(t, T,R) + ~2<>(t,z,R)]dz
(R) ~
}
'
(6.2.16) We integrate the equilibrium equation 090"rr
Or
1
(6.2.17)
-[- --(Orrr -- 0"00 ) = 0
r
with the boundary conditions
O'rrlR=ao -- --po(t),
O'rrlR=a(t)= 0
(6.2.18)
°r°° dr = O.
(6.2.19)
and find that f(t,a(t))
po(t) +
Orrr
a f(t,ao)
--
r
We replace the variable r by R with the use of Eqs. (6.2.1) and (6.2.7), substitute expressions (6.2.16) into Eq. (6.2.19), and arrive at the formula
2
X , ( t - T*(R),0) ~°(t,R)
R2 + C,(R)
J a0
((Rz+C(t)) + *°(t'R)
+
f~i
\N
4 C**-~)
OX, (t - ~*(R), T -~-z
2
(Rz+C*(R))2)] -
R 2 + C(t)
T*(R)) [~ ' ~ ( t ,
(n)
+ ~2~(t, ~',R)
-~
+ C('r)
R 2 + C,(R)'~ R 2 + C(t) )
-
"r, R )
R 2 + C(t)
R + C(t) R 2 -k- C ( "r)
dr
R 2 + C('r) "~ R2 +
R2 + C(t)
C(t) ) = po(t). (6.2.20)
Given preloading in accreted layers C.(R) and given pressure p0(t), Eq. (6.2.20) is a nonlinear integral equation for the function C(t), which determines displacements in a growing cylinder [see Eqs. (6.2.1) and (6.2.7)]. Afterward, we calculate pressure p according to Eq. (6.2.19) and the nonzero components of the stress tensor according to Eq. (6.2.16).
380
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
To analyze the effect of material and structural parameters on stresses and displacements in a growing cylinder, we solve Eq. (6.2.20) numerically for the neoHookean viscoelastic material with the strain energy density W0 = 2 X , ( 0 , 0 )
(I1 -- 3),
where/x is a generalized shear modulus. A Nongrowing Elastic Cylinder Setting a(t) = al and
X,(t, ~) = X,(O, 0), we obtain from Eq. (6.2.20)
fafl [ R2 + C(t) l Y ~ Jr U , ~
-
R2 + C,(R)]
RdR
1~2 + c(t~
t~2 + c(t~
_ po(t) ~
(6.2.21) "
Since C,(R) = 0 for R ~ [ao, al], Eq. (6.2.21) reads
faza~[x+C(t) x
x] x+C(t)
dxx+C(t)
2p0(t)
I~ '
(6.2.22)
where x = R 2. For small strains, when C/a~ "~ 1, Eq. (6.2.22) is simplified as
a~ dx po(t) C(t) J a~ x2 - ]d~ .
f
Calculating the integral, we obtain
po(t) C(t) = tZ(aoe _ ale).
(6.2.23)
The values of C for small and large deformations are compared in Figure 6.2.2, where results of numerical calculations are plotted for a l = 2a0. The dimensionless radial displacement c = C/a~ increases monotonically with the growth of the dimensionless internal pressure P0 = po/l~. The rate of growth in the C values at finite strains exceeds that at small strains. For a given internal pressure P0, the radial displacements calculated by employing the nonlinear theory (6.2.22) significantly exceed the displacements found in the framework of linear elasticity (6.2.23). For sufficiently small deformations, results of the linear and nonlinear theories practically coincide. Integrating Eq. (6.2.17) from f(t,R) to f(t, al) and using Eq. (6.2.16), we find that for a nongrowing neo-Hookean cylinder
I.~fa2[x+C(t)_
°rrr(t'R) -- --2 .jR2
X
X
x 1 dx -~-C(t) x + C(t)"
(6.2.24)
381
6.2. Winding of a Cylindrical Pressure Vessel
e °
:::::::::::::::::
-
.::iiiiiiiiiiii
.........
~oO00088s I
I
I
I
0
I
- ........................ ,
I
I
I
Po
0.5
Figure 6.2.2: The dimensionless radial displacement c = C/a 2 versus the dimensionless internal pressure p0 = po/la,. Curve 1: linear theory. Curve 2: nonlinear theory.
Equation (6.2.16) implies that ~roo(t,R) = Orrr(t,R) + ~
[R z + C ( t ) R2
R2
- R 2 + C(t)
1
"
(6.2.25)
At small strains, Eqs. (6.2.24) and (6.2.25) are reduced to the well-known formulas
O'rr(t,R) -" - p o ( t )
1 - (al/R) 2 1 - ( a l / a o ) 2'
1 + (al/R) 2 ~roo(t,R) = - p o ( t ) 1 - ( a l / a o ) 2"
(6.2.26)
For a time-independent internal pressure P0, the dimensionless radial stress Sr -- -Orrr/PO and the dimensionless tangential stress So = ~roo/Po are plotted
versus the dimensionless radial coordinate R. = R / a o in Figure 6.2.3. The radial and tangential stresses in a nonlinear cylinder exceed the corresponding stresses in a linear body. The maximal difference between the radial stresses is about 30%, and it
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
382
|o
Oo •o °:8 O ••
Sr
o le ee ee
° °
°:o
So
°
• • °o°O • °• • • ° o • • • • o• •
•
• o
".
""
e e
°o ° •
"".. 2b • ° ° o °°°°OOoooOoo°
° o
° o
°°Oo
-
°°Ooo°
°°°°°°OOoooo
"'"...lb
la"'".
°ee°eooe° eeoo°o• •
°'"'"'°"'"'""Ooo..oooo..oo....o. e°e°'~'°
°°°eeeee°eoeo°ooooooeooooeoooOOoooooo° eOooooooe °°°°eeooo e
2a
I
1
I
I
:::,.,o
..........
I
"....
• ::: ......
I
I
,.,: .........
I
R,
-
I °" ........ I ::°I8o° °
2
Figure 6.2.3: The dimensionless radial stress Sr "~ --O'rr/t90 (curves 1) and the dimensionless tangential stress So = croo/po (curves 2) versus the dimensionless radial coordinate R, = R/ao. Calculations are carried out for a] = 2a0 and p0 = 0.5~. Curves (a): linear theory. Curves (b)" nonlinear theory. is reached near the middle surface of the cylindrical shell. The tangential stresses in the nonlinear medium are twice the tangential stresses in the linear cylinder, and the difference between them is practically independent of the radial coordinate. A Growing Elastic Cylinder We now analyze the effect of material supply on stress distribution in a growing elastic cylinder for accretion without prestressing. The latter means that the natural configuration of any built up portion coincides with the actual configuration on the accretion surface at the instant when the portion merges with the growing body
C,(R) = ~ O, a0 --- R -- a], L C(r*(R)), al < R < - a 2 .
(6.2.27)
6.2. Winding of a Cylindrical Pressure Vessel
383
Substitution of expression (6.2.27) into Eq. (6.2.20) implies that
po(t) _ f a f ' [R2+C(t) R2 ] RdR I~ R2 - R 2 +-C(t) R 2 + C(t) + Jal
R 2 + C(r*(R)) -
R 2 + C(t)
R 2 + C(t)"
Introducing the new variable x = R 2 in the first integral and s = ~'*(R) in the other integral, we present this equality as
2po(t)_ f a 2 1 [ x + C ( t ) _ Id,
ja 2
X
+ 2
X
x ] dx + C(t) x + C(t)
fot[a2(s)+C(t) _a2(s)+C(s)] a(s) da a2(s) + C(s) aZ(s) + C(t) aZ(s) + C(t) dt (s) ds. (6.2.28)
Differentiation of Eq. (6.2.28) with respect to time implies that
d----~(t) = -i.z Ida2 (x + C(t)) 3 + 2
dpo (t). Jota2 s'+ S'a s,-~(s)ds a ] -I -d? (a2(s) + C(t))3
(6.2.29) Numerical simulation is carried out for al = 2ao, where t, =
po(t) = 0.6/xt,,
a2 = 3ao,
t/T, and for two regimes of accretion: 12.5, 0,
u,(t,) =
0--
(6.2.30)
0 -----t, < 0.6, 0.6
(6.2.31)
and
u,(t,) = f O,
t
12.5,
Here m
u,
uT 7rla~'
where u(t) = 27ra(t)it(t)l is the rate of growth (volume that merges with the growing cylinder per unit time). The regime (6.2.30) is referred to as "rapid" accretion, whereas the regime (6.2.31) is called "slow" accretion. The dependence c = c(t) plotted in Figure 6.2.4 demonstrates that the radial displacements for the "slow" growth exceed the radial displacements corresponding to the "rapid" accretion. The dimensionless radial stress S r = -Orrr/PO(T) and the dimensionless tangential stress So = troo/po(T) are plotted versus the dimensionless radial coordinate
384
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
2.5
m m • o°
• oo
oooooooooooooO °°°°°°°o°°°°°°°°°°
. . -.-"iii.. -."'""'"
::::::::::::::::::
-
......... ~.... '""i t
I
I
I
I
0
l t,
I
I 1
Figure 6.2.4: The dimensionless displacement c = C/a 2 versus the dimensionless time t, = t/T. Curve 1: rapid accretion. Curve 2: slow accretion. R, = R/ao in Figures 6.2.5 and 6.2.6. The function Sr(R,) decreases monotonically from 1 to 0. Radial stresses for the "slow" accretion exceed that for the "rapid" growth. Tangential stresses have jumps of discontinuity at the interface R = a l between the initial cylinder and the region of accretion. In the initial cylinder, tangential stresses for the "slow" growth exceed tangential stresses corresponding to the "rapid" growth, whereas in the accreted region, tangential stresses for the "slow" growth are less than tangential stresses for the "rapid" material supply. A Growing Viscoelastic Cylinder For a nonaging neo-Hookean material, we set X(t, z ) -
1 + Q o ( t - ~'),
where X(t, T) =
x,(t, ~-) x,(o, o)
6.2. Winding of a Cylindrical Pressure Vessel
385
i
p ° e ° o
e0
__ ", I I •
I o
° o °~o • oo • oo • oo
Sl"
:i!!:: °o°o° o °O:o°O °
--
Do °
So
•
° o::::::.
°lo°O
°o,
",°|80o °o
°
oo
"'..
_
88
"'.
:::-..
............
2b
1/~"'..
_
2a
I
I
I
I
_~
I
"'"'...
la
........ :::,.°.
I
"........ "".. ........... ~::::
I
1
...........
I
I
R,
2
F i g u r e 6.2.5: The dimensionless radial stress Sr = - - O ' r r / P o (curves 1) and the dimensionless tangential stress So = croo/po (curves 2) versus the dimensionless radial coordinate R, = R/ao. Curves (a): rapid accretion. Curves (b): slow accretion.
and Qo(t) is a relaxation measure. It follows from Eq. (6.2.20) that the function C(t) satisfies the integral equation
Id'
-
a ao
-
[1 + Q o ( t -
C(R))]
I
R--5 + C , ( R )
-
(1¢)Q o ( t - ~) R2 + C ( r ) - R 2 + C ( t )
R 2 + C(t)
dT
R2 + C ( t ) '
where the superposed dot denotes differentiation with respect to time. We divide the interval [ao, a(t)] into two subintervals: [ao, al] and [al,a(t)]. In the first interval, where C(R) = 0 and C,(R) = 0, we introduce the new variable x = R 2. In the other
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
386 0.2
1.5 la
•
Sr
So
•
• •
°°Ooo
°°°° 2a
°°o o
°°°o o
• •
°Oo
•
o80
:'o
:t°: °. •
• e
•
°e
•
•
°°oo
°°°Oooo °
°Ooo
°°°°OOooooo • e
°ooo
°°Oooo
•
°ooo
•o
2b °o o
eOOooo
°°e
o°
Oo
°o °o o
go °
°oo_ • ....
I
I
I
I
°OOooo
°o °
eo e
lb
°OOoo, °OOoo
°go o
""'.
::::
:::::: ss,
I
2
R.
3
Figure 6.2.6: The dimensionless radial stress S r - - - O ' r r / P 0 (curves 1) and the dimensionless tangential stress So = croo/po (curves 2) versus the dimensionless radial coordinate R. = R/ao. Curves (a): rapid accretion. Curves (b): slow accretion.
interval, we set R = a(s), where s is a new variable. As a result, we arrive at the equality
2po(t) - fal2{ [1 + Qo(t)] x + C(t) x Jao~ -
+2
9_o(t-~')
lot{
+ C(~-)
x
]
x -( -c(t )
J
x+C(T)]dT} x + C(t)
[ a~(~ +
c(t)
[1 + Qo(t - s)] [a2(s) + C,(a(s))
dx
x + C(t)
a2(s) + C,(a(s))]
a2(s) + C(t)
6.2. Winding of a Cylindrical Pressure Vessel
-
387
a2(s) + C(-r) a2(s) + C(t)
Oo(t- r) La2(s) + c(r)
×
a(s)
d~}
da
a2(s) + C(t)-d-[ ~sl ds.
(6.2.32)
Accretion Without Pretensioning To study accretion without preloading in wound layers, we combine Eqs. (6.2.27) and (6.2.32) to obtain the differential equation
dC 2Ao(t)---~(t) = Al(t)
(6.2.33)
with the initial condition
ja2
XJr-C(O) x
2po(O) /_t
Here dx
Ao(t) = fa~{ [1 + Qo(t)]x- ~0.t Q0(t - l")[x + C(~')] d~" [x -q- C(t)] 3 Ja~ + 2
-
Al(t) -
0t{
[1 + Qo(t - s)][a2(s) + C(s)]
ft
} a(s) da Qo(t - "r)[a2(s) + C('r)] d~" [a2(s) + C(t)] 3 d t (s)ds,
2 dpo (t) -
-
f a2
d--i-
Qo(t) 1 -
X
1
x -i -C(t)
x
--fotOO(t--T)[l--lX'~-f(T)121X Jr-C(t) x -jr-MTc(T) I dx -2
-
lot{
Qo(t-s)
[1 - (a2(s)-k-C(s))21 1 aZ(s)+C(t) aZ(s)+C(s)
~st Q.o(t - ~) [(a2(s)+C(T)) 2] a2(s) dT }a(s)da 1a2(s) + C(t) + C(r) ~-(s) ds.
We find the function C(t) from Eq. (6.2.33) and substitute the obtained result into Eq. (6.2.16) and (6.2.19). The problem under consideration is essentially multiparametrical, and we do not intend to provide an exhausting analysis of the effects of material accretion on stress distribution in a growing cylinder. For definiteness, we
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
388
consider the standard viscoelastic m e d i u m with the relaxation measure (2.3.1)
Qo(t) = - x [ 1 - e x p ( - 3 , t ) ] , where X is material viscosity, and 3/is the rate of relaxation, and set al - 2a0,
a2 = 3a0,
P0 = 0.5~.
The dimensionless radial displacement c versus the dimensionless radius of the growing cylinder a , = a/ao is plotted in Figure 6.2.7 for various rates of accretion. Radial displacements decrease monotonically with an increase in the rate of accretion u,. This is quite natural, since at any instant t ~ [0, T], the growth of the rate of accretion leads to an increase of the thickness of the cylinder and, in turn, to a decrease of additional displacements caused by creep.
3.6
•
•
• •
• oo
1.8
•
•
•
•
•
•
•
i
..°°°°°°......
• •
I
2
•
•
•
I
I
I
I
I
I
a,
I
I
3
Figure 6.2.7: The dimensionless displacement c = C/a 2 versus the dimensionless external radius a, = a/ao at X = 0.3 and 3', = 2.0. Curve 1" u, = 2.5. Curve 2: u, = 5.0. C u r v e 3: u, = 10.0.
6.2. Winding of a Cylindrical Pressure Vessel ......
389
'Ollli||, IOooo .... "°OOoooo
•.o..:::::..:I............... oe
°o °
°Oooo
SF
°°°°°Oo °
°°°°°o • • • •• •• • • • °o
°°°°°°°°°°°°°°°°OOooooo go ° •
°°o o
°o °
••
°o ° •• °o
••
•• °°Ooo
go
•
•
°°°e°°OOOooool
I 1
I
I
I
I
I
I R,
I
°°°°OOo
I 2
Figure 6.2.8: The dimensionless radial stress Sr -- -O'rr/t[90 versus the dimensionless radius R, = R/ao at u, = 10.0 and ~/, = 2.0. Curve 1: X = 0.3. Curve 2: X = 0.5. Curve 3: X = 0.8.
The dimensionless radial stress Sr is plotted versus the dimensionless radial coordinate R, = R/ao in Figures 6.2.8 and 6.2.9. Comparison of Figures 6.2.3 and 6.2.5 with Figure 6.2.8 shows that radial stresses in a growing viscoelastic cylinder exceed radial stresses in a nongrowing elastic cylinder, as well as in a growing elastic cylinder. At first sight, this seems paradoxical, since stresses in a nongrowing viscoelastic body should be less than stresses in an appropriate elastic body owing to the stress relaxation. In a growing medium, the stress relaxation is not the only physical phenomenon that determines distribution of stresses. Another phenomenon is an increasing pressure from the accreted part of the body. This pressure increases monotonically in time, which may be explained by a creep flow leading to additional radial displacements of the initial cylinder. These displacements are restrained by built-up portions, which produce an increasing pressure on the initial
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
390
66''eeOOOOe888|o ° °O°Oo o• • o •° •o • •o °o o o °o • o °o •o ° o oO ° o OO • OO
Sr
ee
OO ° o
88 ° o
8 8
8l
•
°o
o• •o
88
88 88
• •
•
~o o• o•
•
•.
o •
"::.
"'...
"::::.. •
oe go oo oo
• • •
°o °
°o:°o °o
°o:°oe °gig
°°o~Ooo ° o
• o
o•
•
° o
go °
I 1
I
I
I
I
I
I R,
I
•o • o
I 2
Figure 6.2.9: The dimensionless radial stress Sr -'- - O r r r / P o v e r s u s t h e dimensionless radius R, = R/ao a t u, = 10.0 a n d X = 0.3. C u r v e 1: 3", = 0.1. Curve 2: 3', = 2.0. Curve 3: 3', = 20.0.
body. Figure 6.2.8 shows that the creep flow plays the key role in this process: with the growth of viscosity X, the dimensionless radial stresses increase. The influence of the characteristic time of relaxation 3/, = ~/T is essentially weaker. The dependence of radial stresses on ~/, is nonmonotonic: for small ~/, values, the stresses increase with the growth of ~/,, whereas for large 3/, values, the stresses decrease (see Figure 6.2.9). This dependence is rather feeble, and it may be neglected in applications. Accretion with Pretensioning To study the effect of the circumferential stress or0 = or0(R) in wound layers on stresses in an accreted cylinder, the function C,(R) should be expressed in terms of the function C(t). For this purpose, we consider a cylindrical shell that merges with the growing body at instant t. The Finger tensor
6.2. Winding of a Cylindrical Pressure Vessel
391
in the built-up portion at transition from the natural to the actual configuration is determined by Eq. (6.2.13), where t is replaced by ~'*(R). Substitution of expression (6.2.13) into the constitutive equation (6.1.19) of a neo-Hookean viscoelastic medium implies that the nonzero components of the Cauchy stress tensor in a built-up portion at instant t = ~'*(R) equal
R 2 + C,(R) O'rr('r*(g),R) = -p('r*(R),R) + la,e2 + C('r*(e))' R 2 + C('r*(R)) oroo('r*(R),R ) = -p('r*(R),R) + t.z R2 + C,(R) ' O'zz('r* (R), R) = -p('r* (R), R) + t-~.
(6.2.34)
We assume that the radial stress O'rr vanishes, and the circumferential stress o'oo equals the prestressing ~ro, and find from Eq. (6.2.34)that
IR 2 + C('r*(R)) R 2 + C,(R) 1 (to(R) = tz L -RT~:(- -C--~,(-~ - R 2 + C('r*(R)) "
(6.2.35)
Using the notation O'o(R)
rl -
2tx '
~'(R) =
R 2 + C,(R)
R 2 + C(r*(R))'
(6.2.36)
we rewrite Eq. (6.2.35) as
if2 -k-2r/~"-
1 = 0.
Choosing the positive root of the quadratic equation, we obtain ~(R) = V/1 + rl2(R)- ~(R). This expression together with Eq. (6.2.36) implies that
C,(R) = [R2 + C(T*(R))] IV/1 + ~2(R) - ~(R)] - R 2.
(6.2.37)
We substitute expression (6.2.37) into Eq. (6.2.32), assume that the pressure Po vanishes, and obtain the nonlinear integral equation for the function C(t)
+
x+
x x
-
Qo(t-~-)
x+C(t)
x + C(r)
+ 2
]
x + C(t)
_ x+C(~-)
x + C(t)
dr
dx x + C(t)
lot(l+Qo(t-s) [a2(s)+C(l) v/1 + r/2(a(s)) - rl(a(s)) a2(s) + C(s)
Chapter 6. Accretionof Aging ViscoelasticMedia with Finite Strains
392
a2(s) + C(s)
--a2(s) + C(t) (X/1 + r/Z(a(s))-
/s t
Qo(t - "g)
~(a(s))) 2] a2(s) + C('r) a2(s) + C(t)
a2(s) + Ea2(s) + C('r) C(t)
a(s) da t ~(s) + a2(s) C(t) - ds = O.
(6.2.38)
In manufacturing composite pressure vessels and pipes, mechanical properties of the initial cylinder (mandrel) and wound layers (resin-impregnated fiber bundles) differ significantly. With reference to winding of cylindrical pressure vessels, we suppose that the mandrel obeys the constitutive equation of an elastic neo-Hookean material with the shear modulus /.~0, and layers are governed by the constitutive relation of a nonaging viscoelastic neo-Hookean medium with the shear modulus/x and the relaxation measure Qo(t). In this case, Eq. (6.2.38) reads
x + C(t) x + 2/x
fot{
x ] dx x -~-C(t)] x + C(t)
a2(s) + C(t) 1 + Qo(t - s) x/,1 + ~2(a(s))_ rl(a(s)) a2(s) + C(s)
a2(S)a2(s)++ C(S)c(t)( V/1 + rl2(a(s))- r/(a(s))) 2 ] a2(s) + C(t) - fs t Qo(t - 7-) a2(s) + C('r)
da × a2(s)a(s) + C(t)-~slds~
a2(s)+C('r)] dr I aZ(s) + C(t)
O.
(6.2.39)
For elastic fibers, Eq. (6.2.39) is simplified as
]d,~/a2 [1 -- I +XC(t) 121 dX x +2 da 2
X
t
fo
1 V/1 + 7/2(a(s))- ~l(a(s))
× I 1 - ( V/1 + r/2(a(s)) - rl(a(s))) 2 (a2(s)+C(s)) 2 ] a 2+( sC(t) ) a(s)
da
a2(s) + C(r)--~sjdst~
. _ . . .
(6.2.40)
0,
where /x
6.3. Winding of a Composite Cylinder with Account for Resin Flow
393
Differentiation of Eq. (6.2.40) with respect to time implies Eq. (6.2.33) with
Ao(t) = ~ f a2
xdx
da 2 (X q-
C(t)) 3
+ 2 foot a2(s)+ C(s)
(aZ(s) + C(t)) 3
da
× IV/1 + r/2(a(s))- r/(a(s))] a(s)-d-[(s)ds,
4rl(a(t))a(t) da Al(t) = - a 2 ( t ) + C(t)--d-[(t)' and the zero initial condition. It follows from Eqs. (6.2.16), (6.2.19), and (6.2.37) that pressure on the mandrel P(t) in a wound elastic cylinder is calculated as
fa(t) F R 2 + C(t) 1 P(t) = I~] e ~ T C(r*(e)) v / l + , o 2 ( e ) - rl(e)
_R 2R2++c(~*(R))c(t)v/1
+
~2(R)- ~(R) R2 + c(t)
= l ~ f o t [ 1 - ( v / l + r l Z ( a ( s ) ) - r l ( a ( s ) ) ) z ( a Z ( s ) + C+ ( sC(t) )) ×
a(s)
da ds (s) . V/1 + rl2(a(s)) - rl(a(s)) dt a2(s) + C(s)
(6.2.41)
The dimensionless radial displacement c = - C / a 2 and the dimensionless pressure on mandrel p, = 2P/tx are plotted in Figures 6.2.10 and 6.2.11 versus the dimensionless radius a, = a/ao of the accreted pressure vessel. The radial displacement c depends essentially on the ratio/2 of shear moduli of the mandrel ~ and of fibers ~ and decreases in/2. Surprisingly, the dimensionless pressure on mandrel p, is practically independent of the ratio/2. The growth of the layers pretensioning rl leads to an increase both in the radial displacements and in the pressure on mandrel uniformly with respect to a,.
6.3
Winding of a Composite Cylinder with Account for Resin Flow
A model is proposed for winding of a composite thick-walled cylinder with finite strains. Continuous growth of a cylinder is treated as a limit of successive accretion of built-up portions (thin-walled shells) consisting of fiber bundles and resin. Owing to preloading fibers, some gradient of pressure arises in the cylinder that causes resin flow. Nonlinear partial differential equations are derived which permit stresses and displacements in a wound cylinder to be determined with account for the material
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
394
0.3
oooooooOOo°°°°°°°° ~
ooooooooooooOOo°°°° ooO°° ° oo°
go ° oo° • ••••
2
ooooooOOoo°
a,
3
Figure 6.2.10: The dimensionless radial displacement c = - C / a 2 versus the dimensionless external radius a, = a/ao of a growing cylinder. Calculations are carried out for al = 2a0, a 2 - - 3a0, u, = 5. Filled dots show t2 = 1.0, unfilled dots show t2 = 5.0. Curve 1:77 = 0.1. Curve 2: r / = 0.2. Curve 3: r / = 0.4.
accretion and resin flow. At small strains, these equations are reduced to a linear Volterra integral equation for the pressure on mandrel. This equation is solved numerically to analyze the effect of material and structural parameters on stresses in a wound cylinder.
6.3.1
Kinematics
of Deformation
A mandrel is modeled as an elastic cylinder with length l and external radius al. Winding of composite layers begins on its external surface at the instant t = 0 and occurs in the interval [0, T]. Accretion of material is treated as a continuous process. At instant t E [0, T], the wound composite cylinder occupies in the reference
6.3. Winding of a Composite Cylinder with Account for Resin Flow
395
0.7
m
I
I
I
I
I
1
I
2
I
I
a,
3
6.2.11: The dimensionless pressure on mandrel p = 2P/Ix versus the dimensionless external radius a, - a/ao of a growing cylinder. Calculations are carried out for al - 2a0, a2 = 3a0, u, = 5. Filled dots show/~ = 1.0; unfilled dots show/~, = 5.0. Curve 1: r / - 0.1. Curve 2: r / = 0.2. Curve 3: r / - 0.4. Figure
configuration the domain fl°(t)={al
<-- R <-- a(t),
0<--19 <27r,
O <- Z <- l},
where {R, (9, Z} are cylindrical coordinates with unit vectors ~n, ~o, and ~z. Within the interval [t, t + dr], a cylindrical shell that occupies in the reference configuration the domain
dl2°(t) = (a(t) < R <- a(t + dt),
0<-19 <27r,
O <- Z <- l},
is wound around the accreted cylinder and immediately merges with it. The law of
396
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
material supply a = a(t),
a ( T ) = a2,
a(O) = al,
(6.3.1)
is assumed to be given. The rate of accretion v(t) is defined as volume of the material that joins the cylinder per unit time da v(t) = 27rla(t)-~(t).
(6.3.2)
Denote by C(R) the instant when a built-up portion with polar radius R joins the growing cylinder. The function C(R) is inverse to the function a(t), T*(a(t)) = t,
(6.3.3)
a(T*(R)) = R.
Differentiation of the first equality in Eq. (6.3.3) implies that --57, u,ad (C(R)) ]-1 •
d C (R ) =
(6.3.4)
Continuous growth of a cylinder is treated as a limit of the following process of successive accretion of thin tings (layers). We divide the interval [0, T] by points tn = nA, where A = T / N and n = 0, 1. . . . . N. At instant t~, the accreted cylinder occupies in the reference configuration the domain ~O(tn) = {al <~ R <-- a(tn),
0 <- 19 < 27r,
0 <- Z <-- l}.
At instant tn+l, a thin-walled cylindrical shell that occupies in the reference configuration the domain Al'~°(tn)={a(tn)<-R<-a(tn+l),
0<-0
<27r,
O <- Z <- l}
merges with the accreted cylinder. Any built-up portion consists of a fiber bundle impregnated by a molten resin. For definiteness, we assume that [in the layer Afl°(tn)] fibers occupy the domain A f ~ ° ( t n ) = {a(tn) <- R <-- b(tn),
0<-0
<27r,
O <- Z <- I},
and resin occupies the domain ArOO(tn) = {b(t~) <- R <_ a(tn+l),
0 <-- 0 < 27r,
0 <-- Z <- 1}.
We suppose that deformation of successive layers for transition from the reference to actual configuration is described by the formulas r = dPn(t,R),
0 = 19,
Z = Z,
a(tn) <- R <- b(tn),
(6.3.5)
where dPn(t,R) is a function to be found, and {r, 0, z} are cylindrical coordinates in the actual configuration with unit vectors G, ~0, and ~z.
397
6.3. Winding of a Composite Cylinder with Account for Resin Flow
The radius vectors in the actual and reference configurations equal (6.3.6)
ro = R~R + Z~z.
? = r er 4- Zez,
Differentiating Eqs. (6.3.6), we obtain tangent vectors in the reference and actual configurations
g01 = 6'R,
g02 = Redo,
g, 1 = - - - ~ ( t , R ) e . r ,
g0 3 = ~'z, g3 = G"
g,2 = f~)n(t,R)eo,
(6.3.7)
It follows from Eq. (6.3.7) that dual vectors are calculated as 1
,~1 = [o3(i)n
---~(t,R)
]-1
1
~2 = ~ g ' 0 ,
e.r,
~3 = ~z.
(6.3.8)
dPn(t, R)
Substituting expressions (6.3.7) and (6.3.8) into Eq. (6.1.2), we find the deformation gradient for transition from the reference to actual configuration O(I~n
dPn(t, R)
fTo~(t) = --ff~(t,R)~RG + ~~O~OR
+ ~z~z,
a(tn) <-- R <- b(tn).
(6.3.9)
The Finger tensor for transition from the reference to actual configuration equals F(t,R) = l OR (t,R)
e.re.r +
'
eoeo + ~z~z,
a(tn) <- R <- b(tn).
(6.3.10) We assume that both fibers and resin are incompressible, I3(F) = 1. It follows from this condition and Eq. (6.3.10) that the function dPn(t, R) satisfies the equation O~f~) n
OR
(t , R ) =
R ~n(t,R)'
a(tn) <-- R <-- bun).
(6.3.11)
Integration of Eq. (6.3.11) implies that • 2(t,R) = R 2 + C(t, tn),
a(tn) <- R <- b(tn),
(6.3.12)
where C(t, tn) is an arbitrary function of time t. Bearing in mind Eq. (6.3.12), we present Eq. (6.3.9) as follows: R Cbn(t,R) V0?(t) = ~n(t,R---------~e.Rer+ --------~-e.oe.o 4- ezez,
a(tn) <-- R <- b(tn).
(6.3.13)
398
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
We suppose that transition of a built-up portion from the reference to natural (stress-free) configuration is determined by the formulas similar to Eq. (6.3.5) r* = dpn(R),
O* = 19,
z* = Z,
a(tn) <- R <- b(tn).
(6.3.14)
Here {r*, 0*,z*} are cylindrical coordinates in the natural configuration with unit -* Co, -* e-*z, and tkn(R) is a function to be determined. Using the incompressvectors er, ibility condition, it is easy to show that the function dpn(g) satisfies Eq. (6.3.11), and, therefore, is presented in the form dp2(R) = R 2 + c(tn),
a(tn) <-- R <-- b(tn),
(6.3.15)
where C(tn) is an arbitrary constant. According to Eq. (6.3.15), the deformation gradient for transition from the reference to natural configuration equals
Vor* - dpn(R)g eger-_. + dpn(g)R eoe° + ezez,
a(tn) <-- R <-- b(tn).
(6.3.16)
The inverse tensor is calculated as
(~,Op,)_l _
~n(R)_,_ R R ereR + dpn(R)e°e° + ezez'
a(tn) <-- R <- b(tn).
(6.3.17)
Substitution of expressions (6.3.13) and (6.3.17) into Eq. (6.1.3) implies the deformation gradient for transition from the natural to actual configuration
dpn(R) ~n(t,R) _,_ ~7*?(t) = ~n(t,R)e';e'r + dpn(R) ~*0~0 + ezez'
a(tn) <-- R <-- b(tn).
(6.3.18)
Combining Eq. (6.3.18) with Eq. (6.1.5), we find the Finger tensor for transition from the natural to actual configuration
F°(t'R) =
I dPn(t,R) ~n(R) 12 e're'r +
~n(t,R) 2 dpn(R) eoeo + ezez,
a(tn) <- R <- b(tn). (6.3.19)
6.3.2 Governing Equations Let us suppose that the response of fibers is governed by the constitutive equation of an isotropic elastic medium with strain energy density W(I °, I°), where 1° is the kth principal invariant of the Finger tensor p0. We employ the Finger formula for the Cauchy stress tensor 6- = - p I +
2 [~1/~0 + ~t2(/~0)2] ,
(6.3.20)
where p is pressure, I is the unit tensor, and the scalar functions qtl and q~2 equal 0W 0W ~1 = ~1 + 1 0 0 1 0 '
*2-
0W
(6.3.21)
399
6.3. Winding of a Composite Cylinder with Account for Resin Flow
Substituting expression (6.3.19) into Eq. (6.3.20), we find the following nonzero components of the stress tensor: O'rr(t,e) -- - p ( t , R ) + 2 dPn(t,R)
[di)n(t,R)
at~l(t,R) + qf2(t,R) dPn(t,R)
12}
2 {qtl(t, R) -[- ~t2(t , R) [ f~n(t' R) ]
¢roo(t,R) = - p ( t , R ) + 2 [ (~n(R)
/ 6n(n)
Orzz(t,r ) = - p ( t , R ) + 2[~l(t,R) + q~2(t,R)],
a(tn) <- R <-- b(tn),
(6.3.22) where I°(t,R) =
qt~(t,R) = qtk(I°(t,R),l°(t,R)),
dPn(t,g) f~n(R)
+n(g) +
dPn(t,R)
]2 +1. (6.3.23)
Let P(t, tn) denote pressure in resin in the domain Arf~°(tn). Integration of the equilibrium equation for the layer A f ~ ° ( t n ) 1
(6.3.24)
OOrrr q- --(O'rr -- 0"00) = 0 Or
r
with the boundary conditions O'rrlR=a(t.) = - P ( t , tn-1),
OrrrlR=b(tn) = - P ( t , t . )
(6.3.25)
implies that dPn(b(tn))
P(t, tn-1) - P ( t , t , ) +
fd~n(a(t.)) (0% -
oroo)r -1 dr = O.
(6.3.26)
According to Eqs. (6.3.5) and (6.3.11), dr _ O ~ n ( t , R ) dR = RdR r OR dPn(t,R) dp2(t,R)
Substitution of this expression and Eqs. (6.3.22) into Eq. (6.3.26) yields P(t, tn) - P(t, tn-1) = 2
+ xIt2(t,R)
aa~t,)
~,,(t, R)
~l(t,R)
dpn(R)
dPn(t,R)
dp2(t, R)
d~n(R)
(6.3.27)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
400
Let r/(tn) be concentration of resin in a built-up portion that merges with the wound cylinder at instant tn
ae(tn+ l) -- b2(tn) rl(tn) = aZ(tn+1) aZ(tn)" -
(6.3.28)
-
It follows from Eq. (6.3.28) that
b(tn) = a(tn)
a(tn)
1 + [1 - r/(t~)]
- 1
.
(6.3.29)
Up to the second-order terms compared to A,
da
a(tn+l) = a(tn) + -d-~(tn)A.
(6.3.30)
Substituting expression (6.3.30) into Eq. (6.3.29) and neglecting terms of the second order compared to A, we obtain
b(tn) = a(tn) + [1 -
da rl(tn)]-~(tn)A.
(6.3.31)
Combining Eqs. (6.3.27) and (6.3.31), we arrive at the formula 1
-~[P(t, tn) - P(t, tn-1) ] --
2[1
-
rl(tn)]a(tn) da
(I)2(t, a(tn)) dt (tn)
{
I(~n(a(tn)))2(dPn(t'a(tn))) 23
× ~1 (t, a(tn)) + ~2(t, a(tn))
dPn(t,a(tn))
--
dpn(a(tn))
[(dpn(a(tn)) ) 4 (d~n(t'a(tn)))4]} d#n(t, a(tn)) -dpn(a(tn)) " (6.3.32)
Continuous accretion is determined by a function of two variables P(t, ~-*) that equals the pressure at instant t in resin in a built-up portion manufactured at instant ~-*, and a function of one variable r/(~-*(R)) that equals concentration of the liquid phase at the instant of accretion at point with polar radius R:
P(t, ~-*(R)) = P(t, tn),
r/(l"*(R))= rl(tn),
a(tn) <--R <-- a(tn+l).
Approaching the limit as N ---, ~ in Eq. (6.3.32), we find that
OP(t,.r*(R)) = 2R[1- rl('r*(R))]da(.r*(R)){~l(t,R)[( dP(R) ) 2 - ( ~ ( t ' R ) ) 2] dt [\~(t, R) \ ~b(R) ~2(t, R)
0r*
-at-alkt2(t,e)I(¢~(e) )4 d#(t,R)
-
((I)(t'e))4] } ok(R)
"
(6.3.33)
6.3. Winding of a Composite Cylinder with Account for Resin Flow
401
The functions ~b(R) and ~(t,R) are determined by analogy with Eqs. (6.3.12) and (6.3.15) ~b2(R) = R 2 + c('r*(R)),
• 2(t,R) = R 2 + C(t, ~-*(R)),
(6.3.34)
where C(t, ~-*) and c(1-*) are limits of the stepwise functions C(t, tn) and C(tn). To transform Eq. (6.3.33), we set II(t, R) = P(t, T*(R)).
(6.3.35)
It follows from Eqs. (6.3.4) and (6.3.35) that
O I I ( t , R ) = OP(t,z*(R))Ida__
oT*
)-7 (~*(R))
1 -1
Combining this equality with Eq. (6.3.33), we find that
2] 4,(R))
dp(t,R)
0II
- - ( t , R) =
4]}
OR
4,(R) )
"
(6.3.36)
Let us now return to successive accretion of thin layers and calculate volume
A V(t, tn) of the part of the built-up portion Al2(tn) occupied by resin at instant t A V(t, tn) = 7rl[dp2+l(t,a(tn+l)) - dp2(t,b(tn))]. Substitution of expression (6.3.12) into this equality implies that
A V(t, tn) = "n'l{[(a2(tn+l) + C(t, tn+l)] - [b2(tn) + C(t, tn)]}. Up to the second-order terms compared to 6t, the increment 6V(t, tn) of the volume A V(t, tn) in the interval [t, t + 6t] equals 0
aV(t, tn) = 7rl-:-7[C(t, tn+ l) - C(t, tn)]6t. dt
(6.3.37)
The volume 6V(t, tn) equals the difference between the amount of resin 6Qn(t) that enters the portion Al~(tn) through the surface R = a(tn) and the amount of resin 3Qn+l(t) that leaves this portion through the surface R = b(tn)
6V(t, tn) - ~Qn(t) - ~Qn+ l (t).
(6.3.38)
The resin flow is assumed to obey Darcy's law, which states that volume of a fluid that passes through a porous wall (per unit area and unit time) is proportional to the difference of pressure and is inversely proportional to the thickness of the wall
402
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
qn(t) = r(t, tn)
P ( t , t n - 1 ) - P(t, tn) dPn(t, b(tn)) - ~n(t, a(tn)) '
P(t, tn) - P(t, tn+ 1) qn+l(t) = K(t, tn+l) d~n+l(t,b(tn+l)) _ d~n+l(t,a(tn+l)) ,
(6.3.39)
where r(t, tn) is a coefficient, which depends on permeability of the fiber bundle manufactured at instant tn and on the fluid viscosity at instant t. It follows from Eq. (6.3.39) that 3Qn(t) = 27rK(t, tn)13t
8an+
l ( t ) --
[P(t, tn-1) - P(t, tn)]On(t, a(tn)) d~n(t, b(tn)) - d~n(t, a(tn))
27rK(t, tn+ 1)lSt [P(t, tn) -- P(t, tn+ 1)](I)n+
l ( t , a(tn+ 1)) d~n+ l (t, b(tn+ l) ) - dPn+l (t, a(tn+ l) )
(6.3.40)
Substitution of expressions (6.3.37) and (6.3.40) into Eq. (6.3.38) implies that
0 O----~[C(t,tn+ 1)
-
{ [dPn(t,b(tn) ) - 1]-1 C(t, tn)] = 2 K(t, tn)[e(t, tn-1) -- P(t, tn)] dPn(t, a(tn)) -- K(t, tn+l)[P(t, tn) -- P(t, tn+l)]
X
E
dPn+l(t'b(tn+l)) dPn+l(t,a(tn+l))
--
1
(6.3.41)
Up to the second-order terms compared to A, the left-hand side of Eq. (6.3.41) equals O O2C ot[C(t, tn+l) - C(t,t,)] = OtOr* (t,t,)A.
(6.3.42)
To transform the right-hand side of Eq. (6.3.41), we use the equalities OP 1 OP P(t, tn-1) - P(t, tn) = - - -3~'* (t'tn)A + 2 0(~'*)2 (t'tn)A2' P(t, t~) - P(t, t.+
1)
--
--
OP 1 3P Or* (t'tn)A - 2 O(~'*)2 (t'tn)A2'
- -
(6.3.43)
which are satisfied up to the third-order terms compared to A. Equations (6.3.12) and (6.3.31) imply that up to the second-order terms compared to A,
d~n(t, b(tn)) Fb (tn! + c(t, tn) dPn(t, a(tn)) - 1 = [a2(tn ) + C(t, tn)
[1+
1/2
-1
]
2a(tn)(1 - rl(tn)) da 1/2 -1 a2(tn) -Or C(t, tn) d t (tn)A
a(t,,)[1 - T/(tn)] da a2(tn) + C(t, tn) -d-[(tn)A"
(6.3.44)
6.3. Winding of a CompositeCylinderwith Accountfor Resin Flow
403
It follows from Eqs. (6.3.43) and (6.3.44) that up to the second-order terms compared to A, the fight-hand side of Eq. (6.3.41) equals
2 ae (t, tn){ K(t, tn+l)[a2(tn+l) 07"*
a(tn+l)[1 --
-+- C(t, tn+l)] r/(tn+l)]
da ] -1 ~-(tn+l)
,%
_K(t, tn)[a2(tn)+C(t, tn)][da ]-1} a(tn)[1 -- Tl(tn)] --dt(tn)
02P (t'tn)A{K(t'tn+l)[a2(tn+l)+C(t'tn+l)][da la(tn+l)[1-- T/(tn+l)] ~-(tn+l)
a(r*) 2
t,n>aatn>+C,,n>lEda1-1} a(tn)[1 --
r/(tn)]
--d7 (tn)
"
With the preceding level of accuracy, this expression reads Or---7
r )O-~-;T*
a(7.*)(1 - ~(r--~)
d-}-(7.*)
a(r*)(1 - r/(r*))
~-(r ))
,1]}
02p O(r*)2 2A
0 0P Or* ----7 Or
r*) K(t, r*)(a2(r *) + C(t, r*)) a(r*)(1 - r/(r*))
da dt-
"r*=tn (6.3.45) "r*= tn
Substituting expressions (6.3.42) and (6.3.45) into Eq. (6.3.41), we obtain
02C 0 IaP OtOr*(t, r*) = 207., O-~r,(t, r*)
K(t,r*)(a2(r*)+C(t,r*))(da a(r*)(1 - r/(r*))
,'X-1] "
d-t ( r ) )
(6.3.46) Combining Eqs. (6.3.33) and (6.3.46) and using Eq. (6.3.34), we find that
02C(t,r.)=4O__O{K(t,r.)[~l(t,R)((oh(R) ) 2 _ ( ~ ( t , R ) ) 2 ) OtOr* Or* dP(t,R) ~(R) +~2(t,R)((t~(R)
4
4)]/.
Finally, employing Eq. (6.3.34), we present Eq. (6.3.47) as follows:
02C (t, 7.*) = 4 0--~ K(t, 7.*) ~l(t,a(7.*)) a2(r,) + C(t, r*) OtOr* 07.*
(6.3.47)
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
404
a2('r *) + c('r*)
-
a2('r *) +
a2(1"*) + c(~-*)
C(t, T*)
"
(6.3.48)
For a given function a(t) that determines the accretion process and for a given function c(~-*) that characterizes preloading, Eq. (6.3.48) is a nonlinear partial differential equation for the function of two variables C(t, ~'*). Denote by ~(tn) preloading in a built-up portion ZXf~(tn) that is wound around the growing cylinder at instant t,,. The stress ~(tn) equals tangential stress o00 in the built-up portion at instant tn when this portion merges with the cylinder, i.e., when its configuration coincides with the actual configuration of the growing cylinder. Neglecting radial stress (which is proportional to thickness of a built-up portion), and equating o'00 to ~(tn), we obtain a nonlinear algebraic equation for the function
O'rr
C(tn)
{[a2(tn)+C(tn, tn) 2 alrl(tn,a(tn) ) a2(tn) + C(tn) + ~2(tn, a(tn))
a2(tn)+C(tn) 1 a2(tn) + C(tn, tn)
[(a2(tn) + C(tn,tn))2 _ (a2(tn)_t_C(tn) )2]} aZ(tn) + C(tn)
aZ(tn) + C(tn, tn)
= ~,(tn).
Calculating the limit as A ~ 0, we find that ~l(~'*,a(~'*))
a2(-r*) + C('r*, 1"*) a2('r *) + c('r*) ] a2(~*) + c(~-*) - a2(~"*) + (7(~~*)
l
+~2(r,,a(~.,))[(a2(~.,)+C(~.,,~.,))2_ + c(4"> -
(a2(,r,)+c(,r,)
1 Y-,(~'*). 2
6.3.3
)2]
+ (6.3.49)
Accretion on a Rigid Mandrel
In applications, rigidity of the mandrel essentially exceeds that of polymer fibers, which implies that the mandrel may be treated as a rigid body It follows from this assumption and the incompressibility condition for polymer fibers and resin that displacements for transition from the reference to actual configuration equal zero on the interface between the mandrel and the composite cylinder and on the outer boundary of the cylinder
C(t, O) = O,
C(~-*, ~'*) = O.
(6.3.50)
405
6.3. Winding of a Composite Cylinder with Account for Resin Flow
For definiteness, we confine ourselves to a neo-Hookean elastic medium with i.t ri 1o - - 3 ) , -~-~,
W--
(6.3.51)
where/, is the generalized shear modulus. Substitution of expression (6.3.51) into Eq. (6.3.21) yields 01--
/x 2'
q'2 = O.
(6.3.52)
Combining equalities (6.3.52) with Eqs. (6.3.49) and (6.3.50), we find that
1 + ~(r*)
- [1 + e(r*)] = £o(r*),
(6.3.53)
where £ o ( r * ) - £(r*) /z "
c(r*) ? ( r * ) - a2(r,),
(6.3.54)
Introducing the notation ff = 1 + e, we transform Eq. (6.3.53) into the quadratic equation ~-2 _+_~ 0 ( T * ) ~ . _
which has the only positive root
1 = 0,
1E÷
C(r*) = ~
+ :~02(, *) - :~0(r*)
(6.3.55)
.
Returning to the initial notation, we obtain from Eqs. (6.3.54) and (6.3.55) c('r*) = a2('r *)
{ 1~I ¢ 4
+
/2~(r*)/2 /* -- ~(r*)l ~ -- 1 } .
We substitute expressions (6.3.52) into Eq. (6.3.48) and integrate the obtained equality from 0 to r*. Bearing in mind Eqs. (6.3.50), we arrive at the formula
( [
l+~(r*)l+(7(t,r*)l
0C0__7(t, r*) = 2Ix K(t, r*) 1 + C(t, r*) -
- K(t, O) [(1 + ~(0))-
f +- -~-~
{ + ~(o)
,
(6.3.56)
where C(t, r*) = ~r*_______~.C(t, a2(,r *)
(6.3.57)
Chapter6. Accretionof Aging ViscoelasticMedia with Finite Strains
406
Equation (6.3.56) together with Eqs. (6.3.53) and (6.3.57) implies that
aZ(~'*)--~-(t,~'*) = 2/zK(t, ~'*)
1 +ff(~.,)(7(t,~-*)] + 2E(0)K(t, 0).
1 + (?(t, ~'*)
(6.3.58) Neglecting changes in the resin viscosity caused by curing, we set (6.3.59)
K(t, "r*) = K0, where K0 is a constant. It follows from Eqs. (6.3.58) and (6.3.59) that
OC'(t,T*)- 2tzK° {~o(0)- [I+C(t'T*)~'(~'*)]}. Ot aZ('r*) ~'(~'*) 1 + C(t, T*)
(6.3.60)
Equations (6.3.50) and (6.3.55) imply that at instant ~'* the fight-hand side of Eq. (6.3.60) equals 2/xK0 [E0(0) - E0(~'*)]. a2(,r *) Therefore, differential equation (6.3.60) with the second boundary condition in Eq. (6.3.50) has a nonzero solution if and only if the dimensionless preload intensity E0(r*) is not constant. For a constant preload, E(~'*) = E(0), fiber bundles do not move with respect to resin. We recall that this rather surprising result is obtained under the assumption that the mandrel is rigid. For a monotonically increasing preload intensity, we integrate Eq. (6.3.60) from ~'* to t, utilize the second boundary condition in Eq. (6.3.50) and find that /(?(t,'r*){ •/0
E0(0) -
[1-~- C ~'(~'*)
-
~(T*)] } -1 1+ C
d C - 2/xs:o (t - r*).
(6.3.61)
a2(r *)
The integral in the left-hand side of Eq. (6.3.61) may be calculated in elementary functions, but we do not dwell on this question.
6.3.4
Accretion with Small Strains
We now concentrate on small strains when the dimensionless ratios (6.3.54) and (6.3.57) are small compared to unity. Bearing in mind Eq. (6.3.52), we find from Eq. (6.3.49) that
1 ~(T*, T*) - ~(T*) = ~2o(T*).
(6.3.62)
Linearization of Eq. (6.3.48) with the use of Eq. (6.3.62) implies that --zT(t, "r*) = - 4 / x - 0"r*
1
K(t, ~'*) C(t, T*) - ~(T*, T*) + ~E0(T*)
]} .
(6.3.63)
407
6.3. Winding of a Composite Cylinder with Account for Resin Flow Integrating this equation from 0 to r*, we obtain
at2
o~
a2(r*)--~(t, r*) + 4/xK(t, r*)[C(t, r*) - (7(r*, r*)] = a 2--~-(t, 0) + 4/xK(t, 0)[(7(t, 0) - C(0, 0)] + 2/z[K(t, 0)£o(0) - K(t, r*)~o(r*)].
(6.3.64)
Substitution of expressions (6.3.54), (6.3.57), and (6.3.62) into Eq. (6.3.33) yields
OP(t,r*)=_21x[1-rl(r*)]da(r*)[~'(t,r*)-~;(r*,z*)+~~,o(r*)l. Or* a(r*) dt
(6.3.65)
We integrate Eq. (6.3.65) from 0 to t and employ the boundary condition
P(t, t) = O. As a result, we find pressure on the mandrel Po(t) = P(t, O)
~(T*) da Po(t) = 2Ix fo t 1 --a(T*) dt (r*) I(7(t, T*) - C'('r*, "r*) + 2Eo(r*)] dr * = 2tx
jo t 1 - rl(r*) --(T*) da [M(t, "r*) + ~Eo(r*)l d~'*, a(r*) dt
(6.3.66)
M(t, r*) = C;(t, "r*) - ff;('r*, r*).
(6.3.67)
where
Equation (6.3.67) implies that (6.3.68)
M(r*, ~*) = O. Let K be rigidity of the mandrel and
C(t, O) = - Po(t__~). K
(6.3.69)
The constant K may be easily expressed in terms of elastic moduli and internal and external radii of the mandrel, provided that it is modeled as a linear elastic cylinder. To avoid tedious calculations, we assume that K is given. It follows from Eqs. (6.3.66) and (6.3.69) that (7(0, 0) = 0. Accounting for this equality and substituting expression (6.3.69) into Eq. (6.3.64), we find that 0/14(t, r*) + Ot where
4/xK(t, r*) M(t, z*) aZ('r*)
F(t, T, ),
(6.3.70)
408
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
F(t, r*) = --K
2 dPo (t) + 4txK(t, ~ P o ( t )O) --~ aZ(r.)
a(r*)
]
2 + a2(r *) [K(t, 0)£(0) - K(t, r*)£(r*)].
(6.3.71)
Neglecting curing of resin [see Eq. (6.3.59)], we integrate Eq. (6.3.71) with the boundary condition (6.3.69) and obtain M(t, r*) =
4/xKo F(s, r*) exp _ aZ(r *) (t - s)] ds. J
(6.3.72)
Equations (6.3.71) and (6.3.72) imply that 1 (a(O))2frldP°'" _ 4/xKo (t - s)] ds M(t, r*) = - -K k, a(r ) ---dT-ts) exp aZ(y*) J 4 tXKO ~ i [ 4/,K0 aZ(r,) K Po(s) exp - aZ(,r,) (t - s)] ds 2Ko f~i [ 4/zKo a2(r,) [£(0) - £(r*)] exp - a2(r,) ( t - s)] ds. We integrate by parts the first term in the right-hand side of this equality and calculate the third term. As a result, we arrive at the formula
1 a,0:) 2
E
M(t, r*) = - -~ (a(r )
4~Ko I a2(r*)K 1 -
a(O) 21f,P°(s)exp[-a2(r, t 4/XK0 (t - s)] ds )
(a(r))
exp[
1
a2(r,)
(6.3.73)
Substitution of expression (6.3.73) into Eq. (6.3.66) leads to the linear Volterra integral equation t
H(t)Po(t) +
fO
L(t, s)Po(s) ds = G(t)
with 21X~o'tl-n('r)(a(O))2da H(t) = 1 + -~a(r) ~ -dr (r) dr' rl(r) da G(t) = foot 1 -a(r) dt (r) {E(r)
(6.3.74)
409
6.3. Winding of a Composite Cylinder with Account for Resin Flow
+ [~(0) - ~('r)] I1 - exp ( - 4/zK° a2(~). (t - ~')) ] } d,~' L(t, s) = L1 (t, s) - L2(t, s), 2tx 1 - rl(s ) -K a(~
Ll(t,s) -
a(O)'~ a(s),l
4txKo
--~'s',t )
exp -aZ(s )
4/xKo foS l - rl('r)(a(O)'~ 2 [ 1 - (a(O))21 da L2(t,s) - 2/x K a2(0) a('r) \ a('r) } a-~ ~-~-('r) × exp - 4~Ko a2(,r) (t - s)] d1". Assuming the parameters r/and ~ to be independent of ~-*, we introduce the dimensionless variables and parameters t,
t T'
a
a(0)'
a,
/x(1 - rl) /2 = - - , K
Po ]~(1 -- r/)'
Po,
v,
vT 7ra2(0)/,
4/,zKoT k - --. a2(0)
(6.3.75)
In the new notation, Eqs. (6.3.2) and (6.3.74) are presented as follows (asterisks are omitted for simplicity)" dA ~(t) dt
= v(t),
H(t)Po(t) + ~
~0"t[L1 (t, s)
- L2(t, s)]Po(s)ds = G(t),
(6.3.76)
where A(t) = a2(t) and H(t) = 1 +
G(t)
l-A-- ~
,
1
= ~ lnA(t),
L l ( t , s ) -L2(t,s) -
(
v(s)
~exp a2(s)
-KA(s
1
A+s,
t--S 1 ~ ( t - s)
'
[exp(- ¢:(t - s))
/
s)
t--s exp - K A(s) -
t--s) exp - K A(s)
° (6.3.77)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
410
0.4
Po
0.0 0
t
1
Figure 6.3.1: The dimensionless pressure on mandrel P0 versus the dimensionless time t for a growing cylinder with a 2 = 1.5al, v(t) = 1.25t, and k = 0.5. Curve 1" /~ = 0.1. Curve 2:/2 = 0.5. Curve 3:/~, = 1.0. Curve 4:/~, = 5.0.
To study the effects of material and structural parameters on the pressure on mandrel P0, we solve numerically Eqs. (6.3.76) for accretion with a constant rate of material supply v. Results of numerical simulation are presented in Figures 6.3.1 to 6.3.4. Figure 6.3.1 demonstrates that the pressure P0 decreases with the growth of ~. This result is quite natural, since/5, is proportional to the shear modulus of fibers ~. With an increase in ~, any built-up layer resists more effectively to its deformation, and the influence of the mandrel weakens. Figure 6.3.2 shows that the pressure P0 increases with the growth of k. Since k is inversely proportional to the resin viscosity, the latter means that an increase in viscosity (e.g., owing to polymerization) leads to a decrease in the pressure on
6.3. Winding of a Composite Cylinder with Account for Resin Flow
411
0.4
Po
0.0 0
t
1
Figure 6.3.2: The dimensionless pressure on mandrel P0 versus the dimensionless time t for a growing cylinder with a 2 = 1.5al, v(t) = 1.25t, and/~, = 1.0. Curve 1: = 0.01. Curve 2: ~ = 1.0. Curve 3: ~ = 10.0. Curve 4: ~ = 50.0.
mandrel. This effect follows from Eqs. (6.3.77), since for small
A2(s ) - ~:
A3(s)
+ ~
A2(s)
•
Combining this equality with Eq. (6.3.76), we find that the function Po(t) increases in k, at least for sufficiently small k values. Figures 6.3.3 and 6.3.4 demonstrate that the pressure P0 increases with the growth in the accretion rate v. This effect is rather strong for large k values, in which the resin viscosity is small, and it is essentially less pronounced for small k values.
Concluding Remarks A model is developed for accretion of a composite cylinder with finite strains when resin flow is taken into account. Continuous material supply
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
412
0.15
P0
0.0 0
t
1
Figure 6.3.3: The dimensionless pressure on mandrel P0 versus the dimensionless time t for a growing cylinder with/~ = 10.0, ~ - 500.0 and a time-independent rate of accretion v. Curve 1:a2 = 1.2al. Curve 2:a2 = 1.5al. Curve 3:a2 = 2.5al.
is treated as a limit of the process of successive merging thin-walled shells (built-up portions) with a growing body, when thicknesses of the shells tend to zero. Preloading of fibers leads to initial stresses in built-up layers and to some gradient of pressure in resin. This gradient implies resin flow through the fiber bundles, which is governed by Darcy's law. The nonlinear partial differential equation (6.3.48) is derived for radial displacements of fibers when the accretion rate and preload intensity are given. An explicit solution of this equation is obtained when the mandrel rigidity essentially exceeds rigidity of fibers. At small strains, two linear integro-differential equations (6.3.63) and (6.3.65) are derived for radial displacements of fibers and pressure on the mandrel. These equa-
413
6.4. Volumetric Growth of a Viscoelastic Tissue
0.15
P0
0.0
I
I
I
0
I
I
I
I
t
1
6.3.4: The dimensionless pressure on mandrel P0 versus the dimensionless time t for a growing cylinder with/~ = 10.0, k = 0.1 and a time-independent rate of accretion v. Curve 1:a2 1.2al. Curve 2:a2 1.5al. Curve 3:a2 = 2.5al.
Figure
=
=
tions are reduced to a linear Volterra equation (6.3.74), which is solved numerically. At any instant t, the pressure on mandrel P0 is determined by three dimensionless parameters ~, K, and v. It is shown that P0 decreases with the growth of the shear modulus of fibers ~ and with the growth of the resin viscosity k-1, and it increases with an increase in the rate of accretion v.
6.4
Volumetric G r o w t h of a Viscoelastic Tissue
A model is developed for volumetric growth of a soft biological tissue with finite strains. We distinguish the mere growth (material production) and forced deformation of a tissue caused by physical and chemical stimuli. The model accounts for an
414
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
"internal" inhomogeneity of a growing tissue, where any elementary (from the standpoint of the mechanics of continua) volume contains portions produced at different instants. We derive constitutive equations for an incompressible, growing, viscoelastic medium, subjected to aging, and apply them to two problems of interest in biomechanics. The first deals with growth of a viscoelastic bar driven by compressive forces, which reflects remodeling of large femoral bones caused by trauma. We analyze numerically the effect of compressive load on the rate of cellular activity and demonstrate that a linear law of growth implies finite material supply. The other problem is concerned with radial deformation of a growing viscoelastic cylinder under internal pressure. The model reflects pathological mass production in large blood arteries and veins. It is shown that a linear law of growth implies some conclusions that contradict experimental data. A more sophisticated equation for the growth rate is suggested that is in qualitative agreement with observations. The exposition follows Drozdov and Khanina (1997).
6.4.1
A Brief Historical Survey
Huxley (1931) and Thompson (1942) proposed first models for volumetric growth of biological objects. However, observations regarding the effect of mechanical stresses on the rate of mass production have been collected even earlier [see, e.g., Glock (1937)]. Detailed historical surveys of experimental data and methods for the analysis of growth of biological tissues can be found in Cowin (1986), Fung (1990), and Regirer and Shtein (1985). Hsu (1968) proposed a model for volumetric growth based on the following hypotheses: 1. Material of a growing body is incompressible. This permits the volume element in the actual configuration at an arbitrary instant t to be calculated by using the mass balance equation. To describe a way for a new material to be supplied, some fictitious mass sources are assumed to be distributed in the growing body. 2. Any volume element (in the sense of the mechanics of continua) of a growing medium consists of some portion existed at the initial instant t = 0 and new portions that join the medium within the interval [0, t]. The initial (reference) configuration is assumed to be the same for all these elements. This means that the only strain tensor and the only rate-of-strain tensor may be used for the entire volume element. 3. Constitutive equations of a growing medium express some corotational derivative of the Cauchy stress tensor in terms of the stress tensor, some strain tensor, and its derivative with respect to time. For the stress-free deformation, the constitutive relations imply homogeneous dilatation of the growing body (hypothesis of the normal growth). 4. The characteristic time of growth essentially exceeds the characteristic time of changes in external loads, whereas the latter significantly exceeds the characteris-
6.4. Volumetric Growth of a Viscoelastic Tissue
415
tic time of natural oscillations. This means that (i) inertia forces can be neglected, and (ii) at any instant t (in the scale of the growth process), a growing body is in its equilibrium state. We do not mention here other assumptions of Hsu (1968), for example, linearity of the constitutive equations, since they do not play the key role in our analysis. What we would like to emphasize is the presence of an additional function (the rate of growth or the mass production per unit current volume per unit time), which is assumed to be prescribed. As mechanical examples, Hsu considered uniaxial elongation of a growing bar and simple shear of a growing medium. Nowinski (1978) replaced the incompressibility condition in the Hsu model by the assumption that the rate of deformation is time-independent. To the best of our knowledge, this hypothesis is not confirmed by experimental data. Nowinski studied homogeneous deformation of a growing bar and the Lame problem for a growing cylinder and employed results of the latter problem to describe stresses in a core of a growing trunk. Entov (1983) applied the Hsu model to describe pathological deformations of vertebrae (scoliosis and kyphosis). Shtein and Logvenkov (1993) and Stein (1995) solved several simple mechanical problems for a growing tissue with small strains based on the Hsu model. Nikitin (1971) suggested distinguishing the growth process, where an additional mass is produced, and the influence of biological stimuli, which lead to muscle shortening in a soft tissue. The latter can cause an effect similar to the effect of growth: elongation of fibers of a soft tissue in one direction together with their thinning in other directions. In the Nikitin model, the deformation tensor is assumed to be proportional to a scalar function (biostimulus), which, in turn, is governed by an ordinary differential equation, where the magnitude of the brain signal is included as an input. The model is applied to calculate residual stresses in the lumina of a blood vessel, which is treated as a composite cylinder consisting of collagenous fibers and elastine. The concept of a biostimulus as an additional stress produced by electric signals in a soft tissue, was developed by Akhundov et al. (1985). Assuming this stress to depend linearly on the entire history of loading, a Volterra integral equation is proposed to describe this dependence. As an example, propagation of steady longitudinal oscillations through a human hand is analyzed. Nikitin (1971) and Akhundov et al. (1985) modeled the effect of a biostimulus by including an additive term into the constitutive equations for an inert medium. Another approach was proposed by Cowin and Hegedus (1976), Cowin and Nachlinger (1978), Cowin and Van Buskirk (1978), Firoozbakhsh and Cowin (1981), Hegedus and Cowin (1976), Goodship et al. (1979) and Lanyon et al. (1982). According to the Cowin model, elastic moduli of a biological material depend on some scalar parameter, the so-called measure of internal remodeling. This parameter treated as a material function obeys some nonlinear differential equation, the fight side of which depends on the stress tensor. The model is applied to calculate stresses in long bones, see
416
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
Hart et al. (1984a,b) and Firoozbakhsh and Aleyaasin (1989). For comparison of experimental data with predictions of the model [see, e.g., Cowin et al. (1985)]. A comprehensive survey of experimental data regarding the effect of stresses on the material properties of bones is presented by Burger et al. (1994). Cowin (1986) generalized the concept of a scalar measure of remodeling by introducing a tensor measure, the so-called fabric tensor for cancellous bone tissue. The tensor measure is employed to model the following phenomenon discovered by J. Wolff in 1872: orientation of trabeculae of a cancellous bone coincides with the direction of a principal stress independently of changes in external loads caused by trauma, pathology, and changes in the life pattern. An extension of the Nikitin model to finite strains was proposed by Rodriguez et al. (1994) based on the kinematic concept developed by Skalak et al. (1982). By analogy with the finite elastoplasticity theory, an additive presentation of the total strain tensor ~ at infinitesimal strains as a sum of an elastic strain tensor ~e and a tensor of growth strains ~g is replaced by a multiplicative presentation for the deformation gradient 21 at finite strains as a product of elastic /~e and growth .21g deformation gradients. An elastic strain measure is connected with the Cauchy stress tensor by constitutive equations of an elastic medium, whereas the rate-of-strain tensor/)g for the growth deformation is prescribed as a function of the current stress tensor. Comparison of the Rodriguez model with the constitutive models in the elastoplasticity theory [see, e.g., Lee (1969) and Nemat-Nasser (1979)] shows that the main difference between them consists in the constitutive equation for the "inelastic" (plastic or growth) rate-of-strain tensor: instead of the associative law for the plastic flow, the Fung equation is employed for growth. As examples, inhomogeneous torsional deformation of a growing circular cylinder and a homogeneous deformation of a growing medium are considered to model the heart ventricular hypertrophy. A discussion of the Rodriguez model can be found in Cowin (1996), where it is shown that the rate-of-strain tensor for growth deformations can depend on the strain tensor instead of the Cauchy stress tensor provided the characteristic time of loading is essentially less than the characteristic time of growth. The preceding models (i) are concentrated on elastic materials and (ii) do not distinguish growth and deformation of a biological tissue caused by a biostimulus. Drozdov (1990) derived a model for the volumetric growth of a viscoelastic soft tissue and applied it to calculate residual stresses in large blood vessels caused by a pathological cellular activity. By analogy with the Hsu model, the Drozdov model presumes that the material is incompressible, and inertia forces may be neglected. However, other hypotheses suggested by Hsu are declined. The growing medium is considered as a composite material consisting of two phases: liquid and solid. The mechanical response in the liquid phase is described by the constitutive equations of an incompressible fluid, whereas the response in the solid phase is governed by the constitutive equations of an incompressible viscoelastic medium. Referring to the model of continuous surface accretion, the natural configuration of a new portion of material in the solid state is assumed to coincide with the actual configuration of the body (accretion without preloading).
6.4. Volumetric Growth of a Viscoelastic Tissue
417
In this section, we extend the Drozdov model and study stresses arising in growing viscoelastic solids owing to the volumetric growth. Unlike Drozdov (1990), we (i) consider a wide class of growing viscoelastic media with an arbitrary strain energy W, and (ii) assume that the rate of growth at the current instant t is determined by the stress intensity. The latter provides an additional differential equation for the mass production and permits stresses, displacements, and the rate of growth to be found for a given program of loading.
6.4.2
Constitutive Equations
We consider a viscoelastic medium, which occupies in the initial configuration a domain f~0 with a piecewise differentiable boundary F0. Points of the medium refer to Lagrangian coordinates ~ = {~i}. Denote by dVo(~) the volume element at the initial instant t = 0. Owing to the material supply, this element is transformed into the volume element dV(t, (;) at instant t > 0. The growth process is determined by the function
d(dV(t,~)) ~(t, ~) = -~ dVo(l~)
(6.4.1)
'
which characterizes the rate of relative changes in the volume element. Integration of Eq. (6.4.1) with the initial condition
dV(O, ~) = dVo(g;) implies that
dV(t, ~) = b(t, ~)dVo(~),
(6.4.2)
where
b(t, ~) = 1 +
f0 t ~(s,
(;)ds.
(6.4.3)
A growing medium is assumed to be incompressible, which means that its mass density p does not change in time. For definiteness, we suppose that the medium is initially homogeneous and p is independent of Lagrangian coordinates ~. It follows from Eq. (6.4.2) that the elementary mass dm(t, ~) in the volume element dV(t, (;) equals
dm (t, ~) = pb(t, (;) dVo(!~).
(6.4.4)
According to Eq. (6.4.4), production of mass within the interval [t, t + dt] equals
db dm (t + dt, ~) - dm (t, ~) = p--;7(t, ~) dVo dr. at
(6.4.5)
418
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
Combining Eqs. (6.4.2) and (6.4.5), we find that the mass production h(t, ~) per unit time per unit current volume is calculated as
h(t, ~) -
p db (t, ~). b(t, ~) dt
(6.4.6)
Equation (6.4.6) expresses the function b(t, (;) that characterizes changes in the volume element in terms of the rate of mass production h(t, ~) introduced by Hsu (1968). Integration of Eq. (6.4.6) implies that
b(t, ~) In b(0, ~)
1 ft -
h(s,
p
~) ds.
~u
Bearing in mind that b(0, ~) = 1 [see Eq. (6.4.3)], we obtain
b(t, ~) = exp
I~f0th(s, ,~)ds 1.
(6.4.7)
Differentiation of Eq. (6.4.7) with respect to time with the use of Eq. (6.4.3) yields
~(t, ~) - h(t' ~) exp [fot h(s'P ~~)dS] We begin with a growing medium that obeys the constitutive equations of an incompressible elastic solid with the strain energy density per unit volume in the initial configuration Wo(I°, I°). Here I ° is kth principal invariant of the Finger tensor p0 for transition from the natural (stress-free) to actual configuration. For the initial material (which exists at the initial instant t = 0), the natural configuration coincides with the initial configuration 12,o = F(t, ~), where F(t, ~) is the Finger tensor for transition from the initial to actual configuration at instant t at point ~. We confine ourselves to material supply without proloading and assume that the natural configuration of an element that merges with the growing medium at instant s coincides with the actual configuration of a growing body at that instant
po = P~(t,s, ~), where P<>(t, s, ~) is the relative Finger tensor for transition from the actual configuration at instant s to the actual configuration at instant t at point ~. The potential energy W(t, ~) of the volume element dV(t, ~) equals the sum of potential energies of the initial volume element dVo(~) and new elementary volumes that join the growing medium in the interval [0, t]. According to Eq. (6.4.2), the volume element
db dt (S, !~)dVo(~) ds = ~(s, !j) dVo(~) ds
6.4. Volumetric Growth of a Viscoelastic Tissue
419
merges with the growing medium within the interval [s, s + ds]. Therefore the function W(t, (;) is calculated as follows: 17¢'(t, ~) - Wo(I1 (t, ~:), I2(t, ~)) dVo(~)
+ =
/o
Wo(Ii<>(t,s,~),I2~(t,s, ~:))13(s, ~)dVo(~)ds
[
Wo(6(t, ~),6(t, ~)) +
Io
]
Wo(I~(t,s, O,l~(t,s, ~))t3(s, ~)ds dVo(O,
(6.4.8) where
Ig(t, ~) = Ig(P(t, ~)),
I~(t, s, ~) = I~(P ~(t, s, O).
To calculate the strain energy density per unit volume of a growing medium W(t, ~), we divide the potential energy ff'(t, ~) by the volume dV(t, ~). Combining Eqs. (6.4.2) and (6.4.8), we arrive at the equality
W(t,~)
-
b(t, ~)
Wo(II(t, ~),I2(t, ~)) +
/ot Wo(Ii<>(t,s,~),I2~(t,s, ~))~(s, ~)ds J . (6.4.9)
To determine the Cauchy stress tensor 6"(t, ~), we employ the Finger formula and Eq. (6.4.9) and find that
&(t, ~) = -p(t, ~)I + ~-----~ b(t, ([~l(t, ~)F(t, ~) + ~2(t, ~)F2(t, ~)]
+
/ot[q~(t, s, ~)P<>(t, s, ~) + q,~(t, s, #)(P~(t, s, 0)2]13(s, ~) as } . (6.4.10)
Here p(t, (;) is pressure, i is the unit tensor, and
OWo
~, OWo
~1 (t, ~) -- --~1 (I1 (t, ~), 12(t , ~)) + 11(t, J - ~ 2 (I1 (t, ~), I2(t, ~)), ~Wo
+2(t, ~) = - - ~ ( I 1 (t, ~), I2(t, ~)), ,?~,s, ~ :
OWo ~Wo~i?~t,~,¢~,i~t,s, ~ + i?~t,s, o-ri d (11<>(t, s, ~), I2<>(t, s, ~)),
, ~ ~t, s, ~ : - -OWo ~ (Ia<>(t,s, ~),i2~(t,s ' ~))
(6.4.11)
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
420
Our purpose now is to derive constitutive equations for a viscoelastic growing medium. Referring to the concept of adaptive links, we treat a viscoelastic material as a network of parallel elastic springs that replace one another. The process of replacement is characterized by a function X.(t, 7) that equals the number of links (per unit volume in the initial configuration) arising before instant ~" and existing at instant t. The potential energy at instant t of the entire system equals the sum of potential energies for all springs that exist at that instant. Potential energy of the initial links (arisen at the instant t = 0) equals X,(t, O)Wo,(ll(t, ~), 12(t, ~)) dVo(~), where Wo,(II(t, ~), I2(t, ~)) = ~ W o ( I i ( t ,
x,(0, 0)
~), I2(t, ~))
(6.4.12)
is the potential energy per link. Potential energy of new links that replace the initial links in the volume element dVo(~) within the interval [0, t] is calculated as a sum of potential energies of links formed in this volume. Within the interval [~', ~" + d~'],
OX, ~(t, 0~
T) dVo((;) dT
new links arise that exist at instant t. The potential energy of any new link is calculated as
Wo,(Im<>(t, ~', !~),12<>(t, ~', ~)). Summing up potential energies for all the links, we find the potential energy of the initial viscoelastic material
(Vo(t, ~) = X.(t, O)Wo.(Ii(t, {~),12(t, ~)) dVo(~) +
fOt--~-(t, °~X* T)Wo,(II<>(t,r, ~),I2°(t, r, ~))dVo(~)dr
IX,(t, O)Wo,(Ii(t, ~),12(t, ~))
+
/o
1
-~r (t, r)wo,(1~(t, r, ~),1~(t, r, ~)>dT dVo(~).
(6.4.13)
By analogy with Eq. (6.4.13), we write the potential energy of a viscoelastic material that merges with the growing body at instant s
6.4. Volumetric Growth of a Viscoelastic Tissue
421
[-
dfV(t,s, sc) = [X.(t - s, O)Wo.(I~(t,s, ~),I2~(t,s, ~)) +
--~r(t - s, ~" - s)Wo,(Ii~(t, ~', sO),I~(t, ~', sC))dl"
×/3(s, s¢) dVo(~) ds.
(6.4.14)
The potential energy ff'(t, ~) of the entire system of links equals the sum of potential energies t
l~r(t, sc) = l~'o(t, ~) +
fo
d I~(t, s, ~).
(6.4.15)
Substitution of expressions (6.4.13) and (6.4.14) into Eq. (6.4.15) implies that ff'(t, ~) = { X,(t, O)Wo,(Ii(t, ~),I2(t, ~))
+ +
/o --d-dr(t,~>Wo,(I~(t,~,¢),I~(t,~,~>>d~] ft IX ~(s, (;) ds
,(t - s, O)Wo,(ll<>(t, s, ~), 12<>(t, s, !~))
+ fss t --~(tOX* _ s, r - s)Wo.(I~(t,~',~),Iz~(t,r,,))drl}dVo(,) To find the strain energy density of a growing viscoelastic medium W(t, ~), we divide the potential energy W(t, ~) by the volume element dV(t, (;) and substitute expression (6.4.12) into the obtained equality. As a result, we arrive at the formula w(t,~) - b(t, ~-----~{IX(t, O)Wo(Ii(t, so),I2(t, so)) +
+
t OX N(t,~)Wo(I?(t,~,~l,I~(t,~,~)>d~
fo Io'
~(s, ~) ds
]
?
(t - s, O)Wo(I1<>(t, s, ~), I2<>(t, s, ~))
+ f t ~OX(t - s, ~" - s)Wo(Ii<>(t, ~', ~),I2~(t, ~', ~)) d~-I } , where
(6.4.16)
422
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
X(t,s) =
X.(t,s) x,(0, 0)
Introducing the notation ~(t,s,~) -
1
OX
b(t, +
~) r)~(r, ~) dr ]
~ ( t , s) + x(t - s, o)t3(s,
fsax( tOs
r, s -
X(t,O) =-o(t, ~) - b(t, ~)'
(6.4.17)
we present Eq. (6.4.16) as follows: W(t, ~) = ~o(t, ~)Wo(I1 (t, ~), I2(t, ~)) +
E(t,~',,~)Wo(Ii~(t,r,~),l~(t,~',,~))d~ ".
(6.4.18)
Combining Eq. (6.4.18) with the Finger formula for the Cauchy stress tensor d'(t, ~), we arrive at the constitutive equation for an incompressible, growing, viscoelastic medium #(t, ~) = --p(t, ~)] + 2 { ~ o ( t , ~ ) [ ~ l ( t , ~ ) F ( t , ~ ) +
+
~2(t, ~)/~2(t, ~)]
~(t, T, ~>[~,l~(t, T, ~)P~(t, r, ~) + q,~(t, T, ~>(P~(t, r, ~))2] dT , (6.4.19)
where the functions qJk(t, ~) and qJ~(t, ~-, ~) satisfy Eq. (6.4.11). When the material viscosity may be neglected, X(t, 1") = 1, Eq. (6.4.19) coincides with the constitutive equation (6.4.10) for a growing elastic medium. For a nongrowing medium with ~(t, ~) = O,
b(t, ~) = 1,
Eq. (6.4.19) coincides with the BKZ-type constitutive equation (2.4.42) with one kind of adaptive links. According to Eq. (6.4.19), stresses in a growing viscoelastic medium are determined by three material functions: the strain energy density W0(I1, I2); the function X(t, T), which characterizes reformation of adaptive links; and the rate of mass production h(t, ~). The functions W0 and X (which determine material properties) can be found in the standard tests [see Chapter 4 for details]. The rate of growth h is presented as a sum h(t, ~) = ho(t, ~) + hi(t, ~), (6.4.20) where ho(t, ~) is a stress-independent component of the rate of growth, and ha (t, ~) is the rate of growth caused by mechanical loads. When the stress-independent growth
423
6.4. Volumetric Growth of a Viscoelastic Tissue
is completed (e.g., for adults), we set ho(t, {~) = 0,
(6.4.21)
whereas several different models are suggested for the function hi (t, ~). Experimental data show that peak stresses under periodic loading affect significantly the rate of mass production [see, e.g., Lanyon and Rubin (1984) and Rubin and Lanyon (1984)]. On the other hand, some experiments demonstrate that the effect of stresses (or strains) on the cell population can be nonlocal owing to the fluid flow through extracellular spaces [see Lanyon (1994)]. We do not intend to discuss these concepts in detail because existing experimental data are not sufficient for their verification. For definiteness, we confine ourselves to the following simple model, which goes back to the elastoplasticity theory. We postulate that the rate of growth is determined by the stress intensity ]~ = (2~ : ~) 1/2,
(6.4.22)
where ~ is the deviatoric part of the stress tensor 6-. Two dependencies are proposed. The first is the linear law with a threshold hi(t, ~) =
K[E(t, ~) - E.], 0,
E(t, {~) -> E., E(t, ~) < E.,
(6.4.23)
where E. is some "equilibrium" stress intensity and K is a material constant. If E exceeds its equilibrium value E., the mass increases, whereas for E < E. no resorption occurs. Equation (6.4.23) qualitatively describes Wolff's law [see Burger et al. (1994)]. An advantage of Eq. (6.4.23) is that it requires only two parameters K and E. to be found in experiments. The main drawback of formula (6.4.23) is that it can lead to unbounded material supply. To avoid nonadequate physical conclusions, we generalize Eq. (6.4.23) and set hi (t, ~) =
Y(E(t, ~)) - .T,, 0,
.T(E(t, ~)) >- Y,, y ( ~ ( t , ~)) < y , ,
(6.4.24)
where .T(~) is a nonmonotonic function with ~T(0) = 0. Referring to the models suggested by Fung (1990) and Pauwels (1980), we assume that the function Y(E) increases in the vicinity of the point E = 0, reaches its maximum at some point E0 > 0 and, afterward decreases and tends to zero as 2£ ---, ~. Our purpose now is to analyze two mechanical problems for growing media.
6.4.3
Compression of a Growing Bar
We apply the constitutive equations (6.4.19) and (6.4.23) to analyze stresses built up in a growing rectilinear viscoelastic bar under compression. The initial body is a bar with length 1 and cross-sectional area So. Its points refer to Cartesian coordinates
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
424
{Xi} with unit vectors 6'i (i = 1, 2, 3). The body is in equilibrium under the action of compressive forces P0 applied to its ends. At the instant t = 0, the load changes and new compressive loads P = P(t) are applied to the ends of the bar. The lateral surface of the bar is traction-free and body forces are absent. An increase in the load causes volumetric growth of the bar. Our objective is to determine stresses and displacements arising in the bar, as well as the rate of mass production. Denote by {xi} Cartesian coordinates in the actual configuration. For uniaxial extension of the bar, coordinates xi are expressed in terms of the Lagrangian coordinates X i as Xl -- o~(t)X1,
X3 = c~0(t)X3,
x2 -- ogo(/)X2,
(6.4.25)
where a(t) and ao(t) are functions to be found. The radius vectors in the initial and actual configurations equal r0 = X16'I + X26'2 q- X36'3,
?(t) = o~(t)X16' 1 q- cto(t)(X2~'2 + X36'3).
Differentiation of these equalities implies tangent vectors in the initial and actual configurations gl 0 = 6'1, gl -- o~(t)e'l,
g20 -- 6'2,
g30 -" 6'3,
g2 = Ct0(t)6'2,
g3 = O~0(t)e'3.
(6.4.26)
It follows from Eq. (6.4.26) that the deformation gradient V0?(t) and the Finger tensor F(t) for transition from the initial to actual configuration are calculated as Vo?(t) = ct(t)elg'l q- cto(t)(g'2g'2 q- g'3e3), F ( t ) = o~2(t)elel q- ct2(t)(e2g,2 q- e3g,3).
(6.4.27)
According to Eq. (6.4.27), 13(F(t)) = ot2(t)ct~(t).
(6.4.28)
d r ( t ) _ l~/2(p(t))" dVo
(6.4.29)
On the other hand,
Substitution of expressions (6.4.28) and (6.4.29) into Eq. (6.4.2) yields
b(t) ~ 1/2 ,~o(t) =
S-~]
"
(6.4.30)
Combining Eqs. (6.4.27) and (6.4.30), we obtain b(t) F ( t ) -- ot2(t)e'le'l q- ----77~.,(e,2e,2 + e,3e,3).
a(t)
(6.4.31)
6.4. Volumetric Growth of a Viscoelastic Tissue
425
By analogy with Eq. (6.4.31), we calculate the Finger tensor for transition from the actual configuration at instant r to the actual configuration at instant t as
(O~(t) '~2
b(t) t~(r) 6'16'1+ b(r) --a(t) (6'26'2+ e'3e'3).
(6.4.32)
We substitute expressions (6.4.31) and (6.4.32) into the constitutive equation (6.4.19) and find that 6-(t)
= O-l(t)e'le'1 -+-o-2(t)e,2e'2 -k-o-3(t)g,3e3,
(6.4.33)
where crl(t) = - p ( t ) + 2~o(t)[qtl(t) + d/z(t)a2(t)]a2(t) + 2
f0t--=(t, r) [61~(t, r) +
q~z<>(t,r)
(og(t) / 2] (og(t) / 2 dr, ~ ~
b(t) tre(t) = tra(t)= - p ( t ) + 2Eo(t) ~ l ( t ) + qf2(t)-~ +2
b(t)
or(t)
b(t___))a(r) ) b(t) o~(r) dr. /ot~(t, r) [qJ~(t, r) + q~(t, r) b(r) a(t) b(r) a(t) (6.4.34)
Equations (6.4.34) obey the equilibrium equations. To satisfy the boundary conditions on the lateral surface of the growing bar, we set or2(t) = o'3(t) = O.
(6.4.35)
It follows from Eqs. (6.4.34) and (6.4.35) that the only nonzero component of the stress tensor equals
+ 2
=(t, r)
+qJ~(t, r)
qq~(t, r)
[(og(t)) 4 -d~
-
c~(r) J
b(r) a(t)
(b(t) Ol(T))2]}dT" b(r) a(t)
(6.4.36)
Boundary conditions on the edges are written in the integral form F P(t) = - ] trl (t) Js (t)
dx2dx3,
(6.4.37)
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
426
where S(t) is the bar cross-section in the actual configuration. It follows from Eq. (6.4.37) together with Eqs. (6.4.25) and (6.4.30) that ~0. P(t) = -trl(t)a2(t) JfSo dX2 dX3 - - t r l ( t ) ~b(t) Here So is the bar cross-section in the initial configuration and So is its area. Substitution of expression (6.4.36) into this equality implies the nonlinear integral equation for the extension ratio a(t)
(>211
2~o(t){d/l(t)Ia2(t)_ b(t)]
+ ~2<>(t,r)
b(t)
I(tx(t))4 (b(t) ct('r)) 2] } P(t)a(t) \~ - b(r) a(t) dr = - b(t)------~o•
(6.4.38)
It follows from Eqs. (6.4.22), (6.4.33), and (6.4.35) that 2
[O'l(t)l.
(6.4.39)
~(t) = 2P(t)a(t) b( t )So x/~ "
(6.4.40)
£(t) = ~
Equations (6.4.37) and (6.4.39) imply that
According to Eqs. (6.4.6), (6.4.20), and (6.4.21), hi(t) -
p db (t). b(t) dt
(6.4.41)
Substitution of expressions (6.4.40) and (6.4.41) into Eq. (6.4.23) yields
1 db(t)= K [2P(t)a(t) ] b(t---)d-t P b(t)Sox/~ - X, , db b(t) dt 1
~~(t)
= 0,
~,(t) >--~,,,
£(t) < £,.
(6.4.42)
Equations (6.4.38) and (6.4.42) allow the functions a(t) and b(t) to be found numerically. Let us suppose that the mechanical behavior of the growing medium obeys the constitutive equation of a neo-Hookean elastic solid
X(t, ~') = 1,
1
W0(I1,/2) = ~/~(I1 - 3),
(6.4.43)
6.4. Volumetric Growth of a Viscoelastic Tissue
427
where/x is the generalized shear modulus. We substitute expressions (6.4.43) into Eq. (6.4.38) and utilize Eqs. (6.4.11) and (6.4.17). As a result, we obtain
c~(t)-
b(t) I jo't Ion(t)b(t)(o~(T))2] db dT
a2(t )
+
a ( r) . b( r) .
-~ .
. ( r) a ( r) -dT
P(t)
p~So . (6.4.44)
First, we calculate the ultimate compressive force P,, which implies no mass production. Setting b(t) = 1, and a(t) = a, in Eq. (6.4.44), we find that
P*
- 2 _ 0~,
--
0~,
~
- - ~
~s0"
The threshold stress intensity is found from Eq. (6.4.40) 2P, a,
Y_,, -
(6.4.45)
So ~/3" Substitution of expression (6.4.45) into Eq. (6.4.42) implies the differential equation
db 2K ~(t) - --[P(t)a(t) dt pSo X/~
- P,a,b(t)],
b(O) = 1.
(6.4.46)
We now consider sudden change in the compressive load from P, to P1 > P,, which gives rise to the growth process. Setting t = 0, b(0) = 1, P(t) = P1, and a(0) = al in Eq. (6.4.44), we obtain P1
- 2 _
t,So Differentiation of Eq. (6.4.44) with respect to time implies that Yl(t)
+
b(t) ] da 1 db 1 dP 2 ot3(t) Y2(t)J -d-7(t) - a2(t ) Y2(t)-dt (t) . . .i~So. dt (t),
where
t
Yl(t) = 1 +
Y2(t) = 1 +
1 db ~a2(r) - - ( r ) dt dr,
fO
~0"t ~a ~( r) ( r )db dr. b(r) dt
We suppose that the compressive load does not change in time
P(t)
= P1,
t > O.
In this case, Eqs. (6.4.46) and (6.4.47) are written in the form
(6.4.47)
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
428
dbdt(t) = 2KP°a° Po ao da dt(t)
=
b(O) = 1,
a(t)Y2(t) db ~(t), a3(t)Yl(t) + 2b(t)Yz(t) dt
a(0) = al.
(6.4.48)
To study the effect of loading on stresses in a growing bar and on the material production, we solve Eqs. (6.4.48) numerically. The extension ratio a and the ratio of v o l u m e elements b are plotted versus the dimensionless time 2KP, a , t, = ~ t
pSo V/3 in Figures 6.4.1 and 6.4.2. Calculations are carried out for P , = 0.5/,~S0 and various ratios p = P 1 / P , .
1.5
0.5
I
0
I
I
I
I
I
I
I
t,
I
10
Figure 6.4.1: The extension ratio for a growing bar a versus the dimensionless time t,. C u r v e 1:P1 = 2P,. Curve 2:P1 = 3P,. Curve 3:P1 = 4P,. Curve 4:P1 = 5P,.
6.4. Volumetric Growth of a Viscoelastic Tissue
I
I
I
429
I
I
I
0
I
t,
I
10
Figure 6.4.2: The dimensionless ratio of volume elements for a growing bar b versus 3P,. Curve 3:P1 = 4P,. the dimensionless time t,. Curve 1:P1 = 2P,. Curve 2:P1 Curve 4:P1 = 5P,. =
The extension ratio a increases monotonically in time and tends to some limiting value a(~). At the initial stage of growth, a decreases with an increase in the load intensity P1. At the stage of steady growth, the effect of initial conditions decays, and the extension ratio increases with an increase in the compressive load P1. The effect of compressive forces on the extension ratio is rather weak, since the material production compensates partially compression of the growing rod. The compressive force affects significantly the ratio of volume elements b(t). For any instant t, the ratio b increases proportionally to the increase in the load intensity P1. For example, when P1 = 2P, compression of the bar causes an increase in the volume element by about 49.7%, whereas for P1 = 5P,, the increase in the volume element reaches 523.4%.
430
6.4.4
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
The Lame Problem for a Growing Cylinder
We consider axisymmetrical deformation of a growing circular cylinder. The initial cylinder is in its natural configuration, and it has length l, inner radius al, and outer radius a2. Points of the cylinder refer to cylindrical coordinates {R, 19, Z} with unit vectors ~R, ~O, and ~z. At the instant t = 0, pressure P(t) is applied to the inner lateral surface of the cylinder and the mass production begins. Edges of the cylinder are located between rigid plates that resist axial deformation of the cylinder. The outer lateral surface is traction-free; body forces are absent. Denote by {r, 0, z} cylindrical coordinates in the actual configuration with unit vectors ~r, ~0, and ~z. Deformation of the growing cylinder is governed by the equations r = dP(t,R),
0 = 6),
(6.4.49)
z = Z,
where ~(t,R) is a function to be found. The radius vectors in the initial and actual configurations equal ?(t) = d P ( t , R ) ~ r + Ze.z.
?o = R e R + Z e z ,
Differentiation of these equalities implies tangent vectors in the initial and actual configurations g20 = R~O,
g l 0 -- 6'R,
O~
gl = --~(t,R)P.r,
g3 0 = ez,
g2 = ~(t,R)~o,
~'3 = ez-
(6.4.50)
According to Eq. (6.4.50), the deformation gradient ~r0?(t,R ) and the Finger tensor F(t, R) for transition from the initial to actual configuration are calculated as O~
-
• (t,R) _ _
V o r ( t ) = -7-~ ( t , R ) e . R e r + O K -
P(t,R) =
-~- (t, R)
It follows from Eq. (6.4.51) that I3(F(t,R)) =
~eoeo
+ ezez,
e.oe.o + ~ z ~ z .
ere'r -+-
(o~(i) )2 --~(t,R)
(~(RR)) 2.
(6.4.51)
(6.4.52)
We substitute expression (6.4.52) into Eq. (6.4.29) and use Eq. (6.4.2). As a result, we obtain the differential equation • (t,R) 8 ~ ~(t,R) R OR
= b(t,R).
(6.4.53)
6.4. Volumetric Growth of a Viscoelastic Tissue
431
Integration of Eq. (6.4.53) implies that
• 2(t,R) = 2
/a1
b(t, w)w dw + C(t),
(6.4.54)
where C(t) is a function to be determined. Equations (6.4.51) and (6.4.53) result in
(Rb(t'.R)) 2 F(t,R) = \ dP(t,R)
e're'r +
( (I)(~R)) 2
#oP.o + ~z#z.
(6.4.55)
Similar to Eq. (6.4.55), we calculate the Finger tensor F°(t, r,R) for transition from the actual configuration at instant r to the actual configuration at instant t
F°(t,r,R) =
(b(t,R) dP(r,R)) 2 (~(/,R)) 2 b(r,R) ~(t,R) e're'r + ~(r,R) ~o~o + ~z~z.
(6.4.56)
We substitute expressions (6.4.55) and (6.4.56) into the constitutive equation (6.4.19) and find the Cauchy stress tensor
&(t,R) = O'r(t,R)e.rer -+- tro(t,R)~o~o + ~rz(t,R)~z~z,
(6.4.57)
where
... R2b2(t,R)] RZb2(t,R) O'r(t,R) = -p(t,R) + 2~o(t,R) ~l(t,R) + ~2(t,K) -~-~,-R-) ~2(t,R) +2
fOt ~(t, r,R)
[~ ( t ,
b2(t'R)~2(r, R)
r,R) + q,2°(t, r,R) b2(r,R ) ~2(t,R )
b2(t,R) ~2(~-, R)
X ~ ~ d r , b2('r, R) ~2(t, R)
R2 R)] ~2(t' R2 R) tro(t,R) = -p(t,R) + 2~,o(t,R) [~l(t,R) q- ~2(t,R) ~2(t' +2
dp2(t'R) ~2(t'R) dr, /ot ~(t, ¢,R) EqJ~(t, ¢,R) + ~2<>(t,r,R) ~2('r, R) ] ~2(7",R)
trz(t,R) = -p(t,R) + 2~o(t,R)[~l(t,R) + q,z(t,R)] + 2
f0t ~(t, ¢, R)[q,1°(t, ¢, R) + q,2<>(t, ¢, R)] dr.
(6.4.58)
We integrate the equilibrium equation 1 O30"r+ --(O" r -- O'0) = 0
Or
r
from r = ~(t, al) to r = ~(t, a2) and use the boundary conditions
OrrlR=a, = -P(t),
OrrlR=a2 = 0.
(6.4.59)
432
Chapter 6. Accretion of Aging ViscoelasticMedia with Finite Strains
As a result, we obtain
~(t,a2)0"0- 0"rdr ddP(t,al) r
P(t) =
=
fa
0"o(t,R) - 0"r(t,R) --OR OdP(t,R) dR. d#(t,R)
Substitution of expressions (6.4.53) and (6.4.58) into this equality implies the nonlinear integral equation fai2{~o(t,R) I~l(t,R)( d#2(t'R)R 2
+~2(t,R) ( ~4(t'R)R4 +
/o
R2b2(t,R)) ~2(t,R)
R4b4(t,R) ~4(t,R) ) ]
[
~(t, 1",R) ~ ( t , -r,R) ~2(1., R)
(~4(t,R)
+
b2(t,R) ~2(1",R)) b2('r, R) ~2(t, R)
b4(t,R) ~4('r, R) ) dr"[ b(t,R)R dR _ P(t) b4('r, R) ~4(t, R) 2 " f • 2(t, R)
-
(6.4.60) We confine ourselves to an elastic neo-Hookean medium (6.4.43) with a shear modulus/x. Combining Eq. (6.4.60) with Eqs. (6.4.11) and (6.4.17), we obtain
e(t) - faa2{ -~1 I1 I~
1
b2(t,R)
I dD(t,R) n / 4]
2 ((I)('/', e))4] fot l IIb(t'e) l + ~2(r,R ) 1 - b(r,R)
0b (1",R) dT I RdR. (6.4.61)
According to Eq. (6.4.58), the nonzero components of the stress tensor equal 0"r(t,R) = -p(t,R) + 0"o(t,R) = -p(t,R) +
I~b(t, R) IR2 + f0 t ~2(~-,R) Ob ~--(~', R) d~'] , • 2(t, R) b2(1",R) Ot
/-z~2(t,R) IR-2 if_ ~oot b(t,R)
1 Ob ~2(T,R) Ot (~',R) dT] ,
0"z(t, R) = -p(t, R) + IX.
(6.4.62)
It follows from Eqs. (6.4.22) and (6.4.57) that
]~2= ~[(0-r 2 -- 0"0)2 -~- (0"r -- o"Z)2 + (0"0 -- 0"Z)2]•
(6.4.63)
6.4. Volumetric Growth of a Viscoelastic Tissue
433
We accept the linear law of material production (6.4.23), which implies the differential equations 1
0b
b(t,R) Ot
(t, R) = ~ [£(t, R) - ~ , ],
~(t, R) >-- £ , ,
p
1 Ob (t,R) = O, b(t,R) Ot
~(t,R) < ~ ,
(6.4.64)
with the initial condition b ( O , R ) = 1.
For a given pressure P(t), Eqs. (6.4.61) to (6.4.64) determine stresses and displacements in a growing elastic cylinder. The governing equations can be simplified provided the cylinder is thin-walled: rt =
d
a0
~1,
where d = a2 - al is thickness, and a0 = (al + a2)/2 is the middle radius. It follows from Eq. (6.4.54) that up to terms of the second order of magnitude compared to r/, • 2(t,R) = C(t).
(6.4.65)
Neglecting terms of the order of r/, we set
b(t,R) = bo(t),
(6.4.66)
where bo(t) = b(t, ao). Substitution of expressions (6.4.65) and (6.4.66) into Eq. (6.4.61) implies that 1 -- F2(t) +
F(r)
1-
F2(t) F2(-r)
1 dbo ~(r) bo(r) dt
dr = Po(t),
(6.4.67)
where
Co(t)-
C(t) a2 ,
F(t)-
bo(t) Co(t)'
Po(t)-
P(t)ao lad
We now introduce the functions
dbo 1 dt (r) dr, Hi(t) = 1 + foot F(r)bo(r)
H2(t) = 1 +
fo t ~F(r) ~ ( r )dbo ddtr , bo(r)
which satisfy the ordinary differential equations
dill (t) = 1 dbo (t), dt F(t)bo(t) dt
dH2 (t) - F(t) dbo --(t) dt bo(t) dt
(6.4.68)
434
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
with the initial conditions H i ( 0 ) - - 1,
H2(0) = 1.
(6.4.69)
Differentiation of Eq. (6.4.67) implies that the function F(t) obeys the ordinary differential equation T 2 F , t ,~H l , ct , ~dcdtE ,t ~ = - dPOd___(t),
F(0) = [1 - P0(0)] 1/2
(6.4.70)
It follows from Eqs. (6.4.62) and (6.4.65) that up to the terms of the order of *7 O'r O'r
--
--
0"0
O" z
--
I~[F(t)Hl(t) - F - l ( t ) H 2 ( t ) ] ,
=
I~[F(t)Hl (t) - 1],
fro - trz = ~ [ F - 1 (t)H2(t) - 1].
We substitute these expressions into Eq. (6.4.63) and obtain E(t) = ~
(t),
(6.4.71)
where +(FH1-
1)2 +
)2],,2
-if-- 1
(6.4.72)
Combining Eqs. (6.4.64) and (6.4.71), we find that dbo dt
(t)
-
-
-
K/.z 1/~2 [H(t) p
m
v 3
~0]+b0(t),
b0(0) = 1,
(6.4.73)
where ~0 =
~/~--,
and for any real x, X, X ~ O,
[x]+=
0, x < 0 .
The linear growth law (6.4.23) implies a bounded material production in a growing elastic bar under compression. Our purpose now is to show that the law (6.4.23) leads to a physically incorrect conclusion that the material production in a growing elastic cylinder under internal pressure is unbounded. The latter implies that the refinement (6.4.24) of the law (6.4.23) is really necessary. We suppose that at the instant t = 0, a time-independent pressure P0 is applied to the inner lateral surface of the cylinder. The load is sufficiently large to give rise to
435
6.4. Volumetric Growth of a Viscoelastic Tissue
the growth process. According to Eq. (6.4.70), for a time-independent P0, the amount F is time-independent as well, F = (1
-
Po)
(6.4.74)
1/2.
We divide the first equality in Eq. (6.4.68) by the other and find that 1 d i l l = - ~ dH2 .
Integration of this equality with the initial conditions (6.4.69) implies that
n2-1
H1=1+
(6.4.75)
F2
Substitution of expression (6.4.75) into Eq. (6.4.72) yields H ( t ) = X/~[(F
-
F - l ) 2 + (F - F - 1 ) Z ( t ) + Z2(t)] 1/2,
(6.4.76)
where Hz(t)
Z(t)-
F
-
(6.4.77)
1.
It follows from Eqs. (6.4.68), (6.4.73), and (6.4.77) that dZ --- (t) dt
1 dH2 -
F dt
1
(t)
-
f~
dbo
(t) = Ktx ~/-~[H(t) - ~0] + . bo(t) dt p V3
Combining this equality with Eq. (6.4.76), we arrive at the differential equation ~d Z( t )
= L{[(F
-
F - l ) 2 + (F - F - 1 ) Z ( t ) + Z2(t)] 1/2 - ~)},
dt
(6.4.78)
where L-
2K/x
Z~_
Zo
It follows from Eqs. (6.4.69) and (6.4.77) that Z(0) = F - 1 -
1.
It is easy to check that the function f ( Z ) = (F - F-1)2 + (F - F-1)Z "+"Z 2
is positive for any real Z. It decreases in the interval (-0% Z0) and increases in the interval (Zo, ~), where
1(1)
Zo-~ ~ - F .
436
Chapter 6. Accretion of Aging Viscoelastic Media with Finite Strains
For any F E (0, 1) [see Eq. (6.4.74)], we have 1(1 z(0)
-
z0 =
~
)2 -
F
->0,
which means that the function in the right-hand side of Eq. (6.4.78) increases in Z and tends to infinity. Therefore, if the growth process begins at the instant t = 0, it does not stop at any instant t > 0, and the function Z(t) tends to infinity. It follows from Eqs. (6.4.75) and (6.4.77) that the functions H1 (t) and H2(t) tend to infinity as well. Finally, we find from Eq. (6.4.73) that the function bo(t) increases monotonically and tends to infinity. Thus, the linear law (6.4.23) implies unbounded growth of an elastic cylinder under steady internal pressure provided the pressure at the initial instant is sufficiently large to initiate the material production. Concluding Remarks A new model is derived for the volumetric growth of soft biological tissues with finite strains, which describes the material supply and accounts for the material inhomogeneity. The inhomogeneity is caused by the difference in the mechanical properties of material portions which join a growing body at different instants. For an aging viscoelastic medium, two types of inhomogeneity are distinguished: (i) that caused by the difference in elastic moduli and relaxation functions, and (ii) that caused by the difference in the natural (stress-free) configurations of joining elements. A new constitutive equation (6.4.19) is derived for a growing viscoelastic medium subjected to aging, and two simple growth laws are suggested [see Eqs. (6.4.23) and (6.4.24)]. To analyze the effect of loading on the growth process, two problems with biomechanical applications are considered. In the first problem, we study growth of a viscoelastic bar under compressive loads, in which a sudden increase in forces gives rise to the material production. This model can describe local cellular activity that causes adaptive remodeling of long bones and cartilages. It is shown that the material production to a large extent compensates axial compression of the bar, and the extension ratio weakly depends on the load intensity. On the other hand, the load significantly affects the rate of material production, which increases sharply with the growth of the load intensity. In the other problem, we analyze radial deformation of a growing viscoelastic cylinder under internal pressure. This problem is of interest for the study of residual stresses built up in arteries, veins, ventricular miocardium, and trachea [see Fung (1990) for experimental data]. It is demonstrated that the linear growth law (6.4.23) leads to incorrect (from the biological standpoint) conclusions, and a new nonlinear equation is suggested for the rate of growth.
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[103] Rodriguez, E. K., Hoger, A., and McCulloch, A. D. (1994). Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27, 455-467. [104] Rubin, C. T. and Lanyon, L. E. (1984). Regulation of bone mass by applied dynamic loads. J. Bone Jt. Surg. 66A, 397402. [105] Shtein, A. A. and Logvenkov, S. A. (1993). Spatial self-organization of a layer of biological material growing on a substrate. Physics--Dokl. 38(2), 75-78. [ 106] Skalak, R., Dasgupta, G., Moss, M., Otten, E., Dullemeijer, P., and Vilmann, H. (1982). Analytical description of growth. J. Theor. Biol. 94, 555-577. [107] Spencer, B. and Hull, D. (1978). Effect of winding angle on the failure of filament-wound pipes. Composites 9, 263-271. [108] Stein, A. A. (1995). The deformation of a rod of growing biological material under longitudinal compression. J. Appl. Math. Mech. 59, 139-146. [109] Tarnopolskii, Y. M. (1992). Problems in the mechanics of winding thick-walled composite structures. Mech. Composite Mater 28, 427-434. [110] Tarnopolskii, Y. M. and Beil, A. I. (1983). Problems of the mechanics of composite winding. In Handbook of Composites (A. Kelly, S. T. Mileiko, eds.), Vol. 4. Fabrication of Composites, pp. 47-108. North-Holland, Amsterdam. [ 111 ] Tarnopolskii, Y. M. and Portnov, G. G. (1966). Variation in tensile force during the winding of fiberglass articles. Mech. Polym. 2(2), 278-284 [in Russian]. [ 112] Tarnopolskii, Y. M. and Portnov, G. G. (1970). Programmed winding of polymer glasses. Mech. Polym. 6(1), 48-53 [in Russian]. [ 113] Tarnopolskii, Y. M., Portnov, G. G., and Spridzans, Y. B. (1972). Compensation of thermal stresses in components made of glass plastics by layer winding. Mech. Polym. 8(4), 640-645 [in Russian]. [114] Thompson, D. W. (1942). On Growth and Form. Cambridge University Press, London. [115] Tomashevskii, V. T. and Yakovlev, V. S. (1982). Generalized model in the winding mechanics of shells of composite polymer materials. Mech. Composite Mater. 18(5), 576-579. [116] Tomashevskii, V. T. and Yakovlev, V. S. (1984). Technological problems in the mechanics of composite materials. Soviet Appl. Mech. 20(11), 3-20 [in Russian].
Bibliography
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[117] Tramposch, H. (1965). Relaxation of internal forces in a wound reel of magnetic tape. Trans. ASME J. Appl. Mech. 32, 865-873. [118] Tramposch, H. (1967). Anisotropic relaxation of internal forces in a wound reel of magnetic tape. Trans. ASME J. Appl. Mech. 34, 888-894. [ 119] Trincher, V. K. (1984). Formulation of the problem of determining the stressstrain state of a growing body. Mech. Solids 19(2), 119-124. [ 120] Tsai, C. T. and Dillon, O. W. (1987). Thermal viscoplastic buckling during the growth of silicon ribbon. Int. J. Solids Structures 23,387-402. [121] Wheeler, J. (1985). Long-term storage of videotape. SMPTE J. 650-654. [122] Willett, M. S. and Poesch, W. L. (1988). Determining the stress distribution in wound reels of magnetic tape using a nonlinear finite-difference approach. Trans. ASME J. Appl. Mech. 55, 365-371. [123] Wireman, J. (1973). Strip winding of tread rubber. Rubber World 169(2), 33-40. [124] Yablonskii, B. (1971). Stresses in a multi-layered structure at winding of a strip on a cylinder. Soviet Appl. Mech. 7(12), 130-133 [in Russian]. [125] Yagoda, H. P. (1980). Resolution of a core problem in wound rolls. Trans. ASME J. Appl. Mech. 47, 847-854. [126] Zabaras, N. and Liu, S. (1995). A theory for small deformation analysis of growing bodies with application to the winding of magnetic tape packs. Acta Mech. 111, 95-110. [127] Zabaras, N., Liu, S., Koppuzha, J., and Donaldson, E. (1994). A hypoelastic model for computing the stresses in center-wound rolls of magnetic tape. Trans. ASME J. Appl. Mech. 61,290-295.
Chapter 7
Accretion of Viscoelastic Media with Small Strains This chapter deals with accretion of viscoelastic and elastoplastic media with small strains. Section 7.1 is concerned with growth of a nonlinear viscoelastic conic pipe. In Section 7.2 we consider growth of a viscoelastic spherical dome. Section 7.3 deals with accretion of viscoelastic beams when debonding occurs either on the interface between two beams or on the contact surface between a beam and a rigid foundation. Torsion of a growing elastoplastic cylinder is analyzed in Section 7.4.
7.1
Accretion of a Viscoelastic Conic Pipe
In this section we analyze stresses built up in a growing conic pipe made of a nonlinear viscoelastic material. Two models are employed for the nonlinear response. Nonlinear Volterra equations are derived for the angle of twist under torsion of an accreted cone. The effects of material and structural parameters on stress distribution are studied numerically.
7.1.1
Formulation of the Problem
A conic pipe is characterized by angles ~b0 and t~l , and distances a and b from its vortex to the edges (see Figure 7.1.1). At the instant t = 0, torques M = M(t) are applied to the edges of the cone. Lateral surfaces of the cone are tension-free; body forces are absent. Under the action of torques M, the cone deforms. Simultaneously with torsion, accretion of material occurs on the outer boundary surface. Owing to the material 446
7.1. Accretion of a Viscoelastic Conic Pipe
447
...-
-...
•o •••• o.
•, 0 2
'... ~
••"
•
•."
•••
O~
•.. ~
bl -
."
°.
'"-.
b2 ,iw
Figure 7.1.1: A conic pipe• influx, the outer angle th increases according to the law
4~ = ~b(t),
~b(0) = ~1,
~(T)--
(~2,
where T is the accretion time. Points of the cone refer to cylindrical coordinates {r, 0, z} with unit vectors G, ~0, and G. At an arbitrary instant t, the growing cone occupies the region f~°(t) = {ztan (h0 -< r --< ztan th(t),
0 --< 0 < 27r,
a --< z <-- b}.
A material portion at a point with angle (h is manufactured at instant ~'*((h), which coincides with the instant when this portion merges with the growing cone,
0, t,
-r*(~b) =
t~O ~ t~ ~ ¢~1, ~b = ~b(t).
(7.1 1)
Confining ourselves to accretion without prestressing, we determine stresses and strains in a growing cone for a given torque M(t) and for a given program of accretion ~b(t). Changes in external forces and the accretion process are assumed to be so slow that inertia forces can be neglected. 7.1.2
Kinematics
of Accretion
We seek the displacement vector fi for transition from the initial to actual configuration in the form = uo(t,r,z)~o.
(7.1.2)
448
Chapter 7. Accretion of Viscoelastic Media with Small Strains
Here
uo = U(t)A(y)r,
(7.1.3)
where U(t) and A(y) are functions to be found, and y = r 2 + z2. Differentiation of Eq. (7.1.2) implies that
fTfi= Ouo__ uo_ _ Ouo_ _ Ouo__ O----~ereo- ~eOerr + - - ~ eoeo + --z--ezeO.oz It follows from this equality that the strain tensor ~ for transition from the initial to actual configuration equals
= -21 OUOor - UOr (OrO0 nt- e'OOr) nt- --~eoeo + -~ oz (OzOO + OOP.z).
(7.1.4)
Combining Eqs. (7.1.3) and (7.1.4), we find that = U(t)At(y)r[r(~.r~0 + e.oe.r) + Z(ezeo + e.oe.z)],
(7.1.5)
where the prime denotes differentiation. The process of continuous accretion is treated as a limit of the following process of discrete accretion. Divide the interval [0, T] by points 0 = to < tl < • • • < tM = T, where tm = mA, and A = T / M . At instant tm, the growing body occupies in the reference configuration the domain
~O(tm)
=
{ztan ~b0 -< r --- ztan
dp(tm), 0 <- 0 < 27r,
a <-- z <-- b}.
Within the interval [tm, tm+ 1], a built-up portion that occupies in the reference configuration the domain Al~°(tm) = {ztan Ck(tm) <-- r <-- ztan ~b(tm+l),
0 --- 0 < 27r,
a -< z -< b}
merges with the growing body. The natural configuration of this portion can differ from its initial configuration. Denote by fi* the displacement vector for transition from the initial to natural configuration. For the accretion process without preloading, the displacement vector fi* in a built-up portion coincides with the displacement vector fi in the growing cone at the instant when this portion merges with the body. Bearing in mind formulas (7.1.2), (7.1.3), and (7.1.5), we obtain that in the domain AlI°(tm)
u-* = U(tm)A(y)r ~o, ~* = U(tm)At(y)r[r(G~o + e.Oe.r) nt- Z(eze.o -t- e.oez)],
(7.1.6)
where ~* is the strain tensor for transition from the initial to natural configuration. When M tends to infinity, Eqs. (7.1.6) imply formulas for the displacement vector fi* and the strain tensor ~* at continuous accretion
u = U(T*(4)))A(y)r~o, ~* = U(~'*(~b))A/(y)r[r(G~0 + eoer) + Z(ezeo + eoez)].
(7.1.7)
7.1. Accretion of a Viscoelastic Conic Pipe
449
Combining Eqs. (7.1.5) and (7.1.7), we obtain the strain tensor 5° = 5 - 5 * for transition from the natural to actual configuration (7.1.8)
5° = [U(t) - U(-r(~b))]A'(y)r[r(P.r#0 + #.oG) + z(P.zeo + P-O#z)]. According to Eq. (7.1.8), the first invariant e ° of the strain tensor 5° vanishes E° = 0,
(7.1.9)
and the only nonzero components of the deviatoric part ~0 of the strain tensor 5° are o = eOr ero
=
[ U ( t ) - U(~'*(¢))]At(y)r 2 ,
e °Oz = ezo o = [ U ( t ) - U( ~.. (dp))]A'(y)rz.
(7.1.10) Let F 0 = (2~0.
~,0)1/2
be the strain intensity for transition from the natural to actual configuration. Substitution of expressions (7.1.10) into this equality yields F ° = 2 l U ( t ) - U(~-*(~b))l [A'(y)lry 1/2.
(7.1.11)
The strain tensor U >(t, ~') for transition from the actual configuration at instant ~" to the actual configuration at instant t equals 5~(t, ~') = 5(t) - 5(z). Substitution of expression (7.1.5) into this equality implies that the first invariant eo(t, ~-) of the tensor 5 <>(t, ~-) vanishes, e°(t, 1-) = 0,
(7.1.12)
whereas the nonzero components of the deviatoric part O°(t, ~-) of the tensor 5°(t, ~-) equal o = [ U ( t ) - U(,r)]a '(y)r 2, e ~ = eor
e Oz ~ = e~ = [ U ( t ) - U(~-)]A'(y)rz. (7.1.13)
The strain intensity F ° ( t , ~') = [20°(t, z)" 0°(t,
'1")]1/2
for transition from the actual configuration at instant ~"to the actual configuration at instant t is calculated as F ° ( t , ~-) = 2 l U ( t ) - U(~')[ {At(y){ry {/2.
(7.1.14)
450
Chapter 7. Accretion of Viscoelastic Media with Small Strains
7.1.3
Constitutive Equations
We consider two constitutive models for the viscoelastic behavior of the cone that can be treated as extensions of constitutive equations in linear viscoelasticity to nonlinear media. The volume deformation is assumed to be linear and merely elastic. For aging media, the latter means that the first invariant or of the stress tensor 6- is connected with the first invariant e ° of the strain tensor 5° by the linear differential equation [see Eq. (2.2.12)] do" d---t-(t) = 3 K ( t - r
, de ° )---d-f(t).
(7.1.15)
Here K = K(t) is the current bulk modulus, and the functions o'(t) and e°(t) obey the initial conditions ~r(r*) = 0,
e°(r *) = 0.
For an aging linear viscoelastic material, the deviatoric part ~ of the stress tensor 6- is expressed in terms of the deviatoric part ~0 of the strain tensor 5° by the formula [cf. Eq. (2.2.1)], ?fit) = 2 f j X(t - r*, r -
(7.1.16)
$ *) -d~ ~ ( °r ) d r .
The relaxation function X(t, r) is presented in the form (7.1.17)
X(t, r) = G(r) + Q(t, r),
where G = G(t) is the current shear modulus, and Q(t, r) is the relaxation measure that satisfies the condition (7.1.18)
Q(t, t) = O.
Integration by parts of the right-hand side in Eq. (7.1.16) with the initial condition O°(r*) = 0 implies that ~(t) = 2
(t - r*,t - r*)g,°(t) -
- r*, r -
r*)~°(r)d
.
(7.1.19)
Equation (7.1.19) can be also presented as ~(t) = 2
X(t - r*, 0)O°(t) +
-0-7r(t -
r-
r*)[O°(t) - O°(r)] d r
. (7.1.20)
Model 1 The first model goes back to Guth et al. (1946) and Rabotnov (1948). According to it, to derive tensor-linear constitutive equations for the nonlinear viscoelastic behavior, it suffices to replace the linear term O°(r) in Eq. (7.1.19) by a nonlinear term ~(F°(t))O°(t), where ~ ( F ) is a material function. As common prac-
451
7.1. Accretion of a Viscoelastic Conic Pipe
tice, the power law • (F) = F ~-1
(7.1.21)
is employed in applications, where a E (0, 1) is a material parameter. As a result, we arrive at the integral constitutive equation for the shear deformation ~(t) = 2
(t - ~'*,t - ~'*)~(F°(t))O°(t) -
~
- ~'*, ~"- "r*)~(F°(~'))~°(~ ") d r
.
(7.1.22t Model 2 The other model is based on the concept of adaptive links that replace one another. The processes of breakage and reformation of links in a linear viscoelastic medium are described by the function X ( t , ~'), whereas the elastic response of a link is characterized by a term linear with respect to the shear strain ~0. Replacing this term in Eq. (7.1.20) by the nonlinear term ~(F°(t))~°(t) and taking into account that ~'°(t) - ~°(1") = ~'(t) - ~(1") = ~'~(t, 1"), we arrive at the constitutive equation for a nonlinear viscoelastic medium [see Eq. (3.3.30)] ~(t) = 2
(t - 1"* 0)~(F°(t))~°(t) + '
O X ( t - ~'*, ~ " - "r*)~(FO(t, ~'))~¢(t, "r) d-r . 0T
(7.{.23> We employ constitutive equations (7.1.22) and (7.1.23) for the analysis of stresses built up in a growing cone. 7.1.4
Governing
E q u a t i o n s ( M o d e l 1)
Substitution of expressions (7.1.9) to (7.1.11) into Eq. (7.1.22) implies that the nonzero components of the stress tensor 6- are calculated by the formulas O'rO --- O'Or
= 2 ~ r ~ + 1 [A'(y)[ ay(a-1)/2
× { X ( t - ~'*,t - ~'*)[U(t) - U(r*)I ~ sign[U(t) - U(r*)]
-
0x
-~(t-
~'*, r -
~'*)IU(I")- U('r*)l ~ sign[U(~')- U(~'*)] d~'},
Ozo = CrOz = 2 ~ r ~ z l A ~ ( y ) l ~ y ( ~ - ' > / 2 × { X ( t - ~'*,t - r*){U(t) - U(~'*)I ~ sign[U(t) - U('r*)]
-
£ -0--~(tox
~'*, ~"- ~'*)lU('r) - U('r*)l ~ sign[U(~-) - U(I"*)] dr}. (7.1.24)
452
Chapter 7. Accretion of Viscoelastic Media with Small Strains
These functions satisfy the equilibrium equation 0 0 --~r (r2 Oro) + -~z(r2 O'zo) = 0.
(7.1.25)
The surface traction vanishes on the boundary surfaces of the cone, which means that O'rO COS6
-- O'zO sin 6 = 0
(6 = 4~0,
6 = 6(0).
(7.1.26)
The boundary conditions on the edges are written in the integral form M(t) =
f027r
dO
[ atan~b(t,
O'zo(t, r, a)r 2 dr
a a tan 4)o =
f027r
dO
[ btan~b(t)
(7.1.27)
Orzo(t, r, b)r 2 dr.
d b tan 4~o We calculate the integrals with respect to 0 and introduce the new variables r = a tan ~b in the first integral and r = b tan ~b in the other integral. After simple algebra, we obtain from Eq. (7.1.27) M(t) = 27ra 3
/~
= 2"n'b3
fa4,0
(t) O'zo(r, a tan qb, a) sin 2 ~b cos -4 ~bd~b
,h(t) Ozo(r, b tan 4~, b) sin 2 ~b COS-4
(~ d~b.
(7.1.28)
Substitution of expressions (7.1.24) into Eq. (7.1.25) yields Or a [r~+31A'(y)I~y(~ -1~/2] + &O [ra+2 z IA'(y)I oty (a-l,/2]
__ 0. (7.1.29)
We assume that the function A(y) increases monotonically, which implies that IA'(y)l = A'(y).
(7.1.30)
Calculating derivatives in the left-hand side of Eq. (7.1.29), we find that (c~ + 3)ra+2[A l(y)]~y(~-1)/2 + 2c~r~+a[A/(y)]~-1A,(y)y(~-l)/2 + (~ - 1)ra+a[At(y)]ay(a-3)/2 + ra+2[At(y)]ay (~-1)/2
+ 2c~r~+2z2[A/(y)]a-1A,(y)y(a-1)/2 + (c~ - 1)ra+2z2[A ~(y)]~y(a-3)/2 = O. (7.1.31) Equation (7.1.31) can be presented in the following form: 2c~yA'(y) + (2c~ + 3)A'(y) = 0.
(7.1.32)
7.1. Accretion of a Viscoelastic Conic Pipe
453
To solve differential equation (7.1.32), we introduce the new variable Z = A t and rewrite Eq. (7.1.32) as
dZ 2 a y - - = - ( 2 a + 3)Z.
de
It follows from this equality that
dZ Z
3)dy -
1+~-
Y
(7.1.33)
Integration of Eq. (7.1.33) yields dA
dy
1 Cly_[l+(3/2a)] 2
(7 1.34)
where C1 is an arbitrary constant. It follows from assumption (7.1.30) that the parameter C1 should be positive. Integration of Eq. (7.1.34) implies that O~ A ( y ) -- - ~ C l y
-3/2a
+ C2,
(7.1.35)
where C2 is an arbitrary constant. The displacement vector fi is characterized by the product of the functions A(y) and U(t). Because the latter function has not yet been determined, we set C1 = 1 in Eq. (7.1.35). As a result, we arrive at the formula A(y) = - 3 y-3/2~ + C 2.
(7.1.36)
Substituting expression (7.1.36) into Eqs. (7.1.24), we find the nonzero components of the stress tensor 6ru+l OrrO -- O'Or
-
(r 2 + Z2)(~+4>/2 {X(t - ~'*,t
ft , -aX ~T(t-
~'*)[U(t) - U('r*)] ~ sign[U(t) - U(~-*)]
~'*, 1"- ~'*)lU(z)- U(~'*) I~ sign[U(z) - U( "r*)]d~'},
raZ trzO = trOz = (r 2 + zZ)(a+4)/2 {X(t - y * , t - z*)lU(t) - U(r*)] a sign[U(t) - U(I"*)]
OX - f j -O-~-(tz*, ~"- r * ) l U ( z ) - U(r*) i~ s i g n [ U ( z ) - U( , )]dr}.
(7.1.37)
It is easy to show that expressions (7.1.37) satisfy conditions (7.1.26) on the boundary surfaces of the cone. Combining Eqs. (7.1.28) and (7.1.37), we obtain
M(t) = 27r
/
~b(t)
sin ~+2 dp{X(t - r*(dp),t - r*(~b))
J q)0
× I U ( t ) - U(r*(~b))l ~ s i g n [ U ( t ) - U(r*(~b))]
454
Chapter 7. Accretion of Viscoelastic Media with Small Strains ( t - r*(+), r - r*(+)) f f, ~ ~OX × {U(~')- U(~-*(qb)){~ s i g n [ U ( r ) - U(C(ck))]dT}drh.
(7.1.38)
The integral from qb0 to qb(t) equals the sum of two integrals: from qb0 to qbl and from qbl to qb(t). Bearing in mind Eq. (7.1.1), we write
M(t) 27r
-
J(ot)
-
Ix(t,t)lU(t){ ~ sign U(t) - fotox -~r(t, ~'){U(~'){c~sign U(r)d T]
,.qb(t) aq~ sin ~+2 dp{X(t -
+/.
r*(d~),t - 7*(4)))
1
× [ U ( t ) - U(r*(4~)){ ~ sign[U(t)- U(C(qb))]
fj
(~
~(t
- T*(4,), T - T*(4,))
× {U(~')- U(C(4~)){ ~ sign[U(~-)- U(C(d~))]dr}dd~,
(7.1.39)
where J(c~) =
f
~l
sin ~+2 ~ d+.
(7.1.40)
We introduce the new variable 4~ = qb(s) and present Eq. (7.1.39) as
M(t) 27r
-
J(c~)
-~r(t, r) l U(~') {orsign U(T)d T] Ix(t,t) I U(t)[ s sign U ( t ) - fotox
dqb + ~0"t sin ~+2 dp(s)--~s(S){X(t - s,t - s){ U(t) - U(s)I s sign[U(t) - U(s)]
f t OX -
-0--~-r( t - s, ~"- s){ U(~') - U(s){~ sign[U(~')- U(s)] dr} ds. (7.1.41)
Given accretion program q) = qb(t), Eq. (7.1.41) is a nonlinear integral equation for the function U(t). After determining this function, we find the displacement vector from Eqs. (7.1.2) and (7.1.3) and the nonzero components of the stress tensor 6" from Eqs. (7.1.37). Substitution of expressions (7.1.37) into the formula for the stress intensity = (2~" implies that
~)1/2
455
7.1. Accretion of a Viscoelastic Conic Pipe ,~1/2 ~(t) = 2(O'2o + ,1.2 ~zO )
2r s (r 2 + z2)(s+3)/2{X(t -
f,j ~°x (t -
~-*, ~-
r*,t
~'*){ U(t)
~ * ) 1 u(~-) -
Ol
U(~'*) I sign[U(t)
u(~-*)I ~ sign[U(r)
U(r*)]
- U(-r*)] d'r}. (7.1.42)
Let us now consider some particular cases. For an aging elastic medium with X(t, 1-) = G(~'), Eq. (7.1.41) reads M(t)
2Ir
- J(a)[G(t)lU(t)I
s sign U ( t ) -
(~') I U('r)I s sign U(~') d-r]
d~b - s){ U(t) - U(s)I s sign[U(t) - U(s)] + f0 t sin s+2 d~(s)--d-~s(S){G(t - f t dG -~--~-T(~"- s){ g('r) - g(s)m s s i g n [ U ( - r ) - g(s)] d'r} ds.
(7.1.43)
For a nonaging elastic material with G(t) = G, Eq. (7.1.43) is simplified and can be presented as M(t) 2 arG
- J ( a ) l U ( t ) I s sign U(t) +
fo'
{ U(t) - U(s)I s sign[U(t) - U(s)] sin s+2 ~b(s)
(s) ds. (7.1.44)
It follows from Eq. (7.1.42) that the stress intensity ~ is calculated as ~, = 2 G r S l U ( t ) - U(~'*)I s sign[U(t) - U(r*)](r 2 +
z2) -(s+3)/2.
Bearing in mind that r = z tan ~b, we find that ~, = 2Gz-3lU(t) - U(r*(~b))l s sign[U(t) - U(r*(~b))] sin s ~b cos 3 ~b.
7.1.5
(7.1.45)
Governing Equations (Model 2)
Let us consider the accretion process for a viscoelastic cone with the constitutive equations (7.1.15) and (7.1.23). Substitution of expressions (7.1.12) to (7.1.14) into Eqs. (7.1.15) and (7.1.23) implies the following nonzero components of the stress tensor 6"
456
Chapter 7. Accretion of Viscoelastic Media with Small Strains
O'rO :
: 2,~r=+llAt(y)lay(a-1)2
O'Or
× {X(t - ~'*, 0)lU(t) - U(~'*)I '~ sign[U(t) - U(~'*)]
+ fj ox(, 0~"
-
,,
~-
~'*)lU(t)
-
U(T)I ~ sign[U(t)
-
U(~-)] d'r},
trzO = trOz = 2 ~ r ~ z l A ~ ( y ) l ~ y < ~ - l ) 2 × {X(t - ~'*, 0)lU(t) - U(~'*)l ~ sign[U(t) - U(~'*)]
+ f ~ j a-O--~r X (t-
~'*, ~" - ~'*)lU(t) - U(~') i~ sign[U(t) - U(~-)] dr}. (7.1.46)
Expressions (7.1.46) obey the equilibrium equation (7.1.25) and the boundary conditions (7.1.26) provided the function A ( y ) has the form (7.1.36). Substitution of expression (7.1.36) into Eq. (7.1.46) yields r a+l O'rO :
O'Or :
(t.2 + Z2)(a+4) 2 {X(t - r*, O)[U(t) - U(I"*)I ~ sign[U(t) - U(~'*)]
o~+ fjax(t
-
~ - - ~'*)lU(t)- U(~')l '~ s i g n [ U ( t ) -
U(~')] d~'},
r~z O'zO :
Oroz :
(r 2 + z2)(,~+4>2 {X(t - ~'*, 0)lU(t) - U(I-*)I ~ sign[U(t) - U(I"*)]
+ f~jOX -O--r-r(t - ~-,, ~- _ ~-*)lU(t) - U(~') i~ sign[U(t)
U(~-)] at}.
(7.1.47)
Combining Eqs. (7.1.47) with the boundary conditions on the edges (7.1.28), we obtain M(t)
27r
-
J(t~){X(t, 0)[U(t)l ~ sign U(t)
°IX (t, r ) l U ( t ) + f0 t -~-T
+
t
lo f
U(r)l ~ s i g n [ U ( t ) -
U(~')] d~'}
d4~ sin ~+2 d~(s)--~s(S){X(t - s, O)lU(t) - U(s){ ~ sign[U(t) - U(s)]
t 0IX
+
-~r(t - s, r - s ) l U ( t ) - U(r)l ~ sign[U(t) - U(r)] dr} ds.
(7.1.48)
7.1. Accretion of a Viscoelastic Conic Pipe
457
The fourth term in the right-hand side of Eq. (7.1.48) is transformed as
t sin ~+2 dp(s)---~s d dp(S)ds fs t -~r(t o3X - s, r - s ) l U ( t ) - U(r)l ~ sign[U(t) - U(r)] d r
fo =
~0 t IU(t)
- U(~')I ~ sign[U(t) - U(~-)]dl" ~0"rolX -~r(t - s, ~" - s)sin ~+2
d~b ch(s)--d-~s(S)ds.
This equality together with Eq. (7.1.48) implies that
M(t) -
27r
J ( c ~ ) x ( t , 0)lu(t)l = sign u ( t )
t
+
fo
®(t, ~ ' ) l U ( t ) - U(T){ ~ s i g n [ U ( t ) -
U(~')] d~',
(7.1.49)
where
OX ddp ®(t, ~-) = -~r(t, ~-)J(a) + X(t - ~-, 0)sin ~+2 ~b(~')-d-~r(~') +
8T
- s, r - s) sin s
oh(s)
(s) ds.
(7.1.50)
After determining the function U(t) from the nonlinear integral equation (7.1.49), the stress intensity X can be calculated by the formula similar to Eq. (7.1.42) X(t) = (r 2 +
2?-c~ + Z2)(a+3)2 [X(t - ~*, 0)lU(t)
- U(~'*)I ~ sign[U(t) - U(~'*)]
~ j ~aX(t - ~-*, 1- - r*)lU(t) - U(~-)I ~ sign[U(t) - U(~')] dl"}.
For an aging elastic cone with
(7.1.51)
X(t, ~-) = G(I"), Eqs. (7.1.49) and (7.1.50) read
M(t) - J(c~)G(O)IU(t)I ~ sign U(t) 27r +
f0 t O ( ~ ' ) l U ( t ) -
U(r){ ~ s i g n [ U ( t ) -
U(~-)] d~',
(7.1.52)
where dG O(~') = -ff-~-r(~')J(c~) + G ( 0 ) s i n ~+2 ~b(-r)
+
(-r)
fo ~ --dgr(~ dG - s) sin s +2 4~(s)--d-~s(S) d~b ds.
It follows from Eq. (7.1.51) that
(7.1.53)
Chapter 7. Accretion of Viscoelastic Media with Small Strains
458
X(t) =
+
2r a
(r 2 -+- Z2)(s+3) 2 {G(O)IU(t)
fldG('r-
- U0"*)[ s sign[U(t) - U0-*)]
~'*)lU(t) - U(~-)I s sign[U(t) - UO')] d~-}.
d'r
(7.1.54)
Finally, for a nonaging elastic medium, G(t) = G, Eqs. (7.1.52) and (7.1.54) are reduced to Eqs. (7.1.44) and (7.1.45) developed for the first constitutive model.
7.1.6
Numerical
Analysis
We confine ourselves to a growing nonaging elastic conic pipe, where the constitutive equations for Model 1 and Model 2 coincide. The torque M(t) is assumed to increase monotonically in time in such a way that the function U(t) increases as well. In this case, Eq. (7.1.44) implies that
J(a)US(t) +
t
fo
ddp
[U(t) - U(s)] '~ sin s+2 dp(s)---~-(s)ds = m(t),
(7.1.55)
where
M(t) m ( t ) - 27rG" For a point at a fixed distance z from the vortex of a growing cone, the dimensionless stress intensity ~z 3
2G is calculated as E, = [U(t) - U(r*(~b))] s sin s ~b cos 3 ~b. We study stresses in a monolithic cone (q)0 = 0 °) by assuming the dimensionless torque m to increase in time linearly
m(t)
= ml + (m2 - m l ) t , ,
where t, = t / T is the dimensionless time, and ml, m2 are given parameters. To solve numerically Eq. (7.1.55), we divide the interval [0, 1] by points t,n = n / N (n = 0 . . . . . N), and replace the nonlinear Volterra equation (7.1.55) by the difference equation
n-1 J(~)u~ + 1 Z ( U n _ Um)S sins + 2 ¢~(t, m ) ~ t ( t , m ) = m(t, n). m--O
(7.1.56)
7.1. Accretion of a Viscoelastic Conic Pipe
459
Since (i) the second term in the left-hand side of Eq. (7.1.56) is nonnegative, and (ii) the function U(t) increases monotonically, the solution Un is located in the interval
[Un-I, U~],
(7.1.57)
where U~ is the only solution of the equation J(a)U~ = m(t, n). At any step n = 0 . . . . , N, Eq. (7.1.56) is treated as a nonlinear algebraic equation for Un. We confine ourselves to two programs of accretion. According to the program I, the angle ~b increases linearly in time from the initial value qbl to the final value ~b2, q,(t)
= 4,~ + (4,2 -
4,~)t,.
(7.1.58)
Let 7r
V(t) = ~-(b 3 - a3)(tan 2 ~b(t) - tan 2 qbo)
,
J
U
0
I 0
I
I
I
I
I
I t,
I
I
Figure 7.1.2: The dimensionless p a r a m e t e r U versus the dimensionless time t, for a cone g r o w i n g with a constant rate of increase in the angle ~b at qbl = 30 °, q~2 = 60 °, ml = 0.1, a n d m2 = 0.2. C u r v e 1: ~ = 0.5. C u r v e 2: a -- 0.7. C u r v e 3: a = 0.9.
460
Chapter 7. Accretion of Viscoelastic Media with Small Strains
be volume of the conic pipe at instant t. According to the program II, the rate of accretion dV/dt (instead of dd~/dt) is constant, which means that tan 2 ~b(t) = tan 2 ~)1
+
(tan 2 ~2 - tan 2 ~1)/,.
(7.1.59)
Results of numerical simulation are plotted in Figures 7.1.2 to 7.1.8. In Figures 7.1.2 and 7.1.3, the dimensionless twist angle U is presented as a function of the dimensionless time t, for regimes of accretion (7.1.58) and (7.1.59), respectively. For regime I, an increase in the torque implies a monotonical growth of U. The function U(t,) increases practically linear for relatively large a values (when the material behavior is close to linear) and demonstrates rapid growth for small a values (when the material response becomes essentially nonlinear). The difference between the twist angles for different a values increases in time. For the regime II of accretion, an increase in the torque leads to an increase in the twist angle as well. However, the
U
3
I 0
I
I
I
I
I
I
I t,
I
I 1
Figure 7.1.3: The dimensionless parameter U versus the dimensionless time t, for a cone growing with a constant rate of material supply dV/dl at ~bl = 30 °, ~ = 60 °, ml = 0.1, and m 2 --- 0.2. Curve 1" a = 0.5. Curve 2: a = 0.7. Curve 3: a = 0.9.
7.1. Accretion of a Viscoelastic Conic Pipe
461
difference between the U(t,) values corresponding to different c~ values is essentially less than for the accretion regime I. With the growth of time, this difference decreases and tends to zero as t, approaches infinity. Figures 7.1.4 to 7.1.8 present results of numerical simulation for the accretion regime II. Figure 7.1.4 demonstrates the effect of the rate of growth of torques m2 on the twist angle U. For any instant t,, the function U increases monotonically in m2. The influence of m2 on the twist angle U is essentially nonlinear: relatively weak for small m2 values and rather strong for large m2 values. In Figures 7.1.5 and 7.1.6, the angle of twist U is plotted versus time t, for different q)2 values, i.e., for different rates of the material influx. For small rates of growth, the function U increases significantly in time, whereas for large rates of growth it remains practically constant. The parameter of nonlinearity a essentially affects the twist angle: for a fixed time t,, the function U decreases in a for any rate
U
I 0
I
I
I
I
I
I t,
I
I 1
7.1.4: The dimensionless parameter U versus the dimensionless time t, for a cone growing with a constant rate of material supply dV/dt at a = 0.7, q~l = 30 °, q~2 = 60 °, and ml = 0.1. Curve 1:m2 -- 0.15. Curve 2:m2 = 0.20. Curve 3:m2 = 0.25.
Figure
Chapter 7. Accretion of Viscoelastic Media with Small Strains
462
7
U
I
0
I
I
I
I
I
t,
1
Figure 7.1.5: The dimensionless parameter U versus the dimensionless time t, for a cone growing with a constant rate of material supply dV/dt at c¢ = 0.7, thl - 30 °, =- 0.1, and m 2 = 0.2. Curve 1:th2 = 40 °. Curve 2:th2 = 60 °. Curve 3:th2 = 80 °.
ml
of accretion. The influence of a is stronger when the rate of the material supply is smaller. Figures 7.1.7 and 7.1.8 demonstrate distribution of the dimensionless stress intensity E, in an accreted part of the cone for a relatively slow accretion (Figure 7.1.7) and for a rapid accretion (Figure 7.1.8). At points located at a fixed distance from the vortex, the stress intensity decreases in th monotonically and vanishes on the outer surface of the growing cone. For slow accretion, the effect of the material parameter a is nonmonotonic: with the growth of cr the stress intensity E, increases in the vicinity of the initial cone (relatively small th values) and decreases far away from the initial cone (relatively large th values) (see Figure 7.1.7). For rapid accretion, the stress intensity E, increases monotonically in a at any point of the growing cone.
Concluding Remarks To calculate stresses in an accreted conic pipe under the action of torques applied to its edges, we derive nonlinear integral equations based on
7.1. Accretion of a Viscoelastic Conic Pipe
463
U
3
t
/ 0
I
I
I
I
I
I
I
I
t,
I
1
Figure 7.1.6: The dimensionless parameter U versus the dimensionless time t, for a cone growing with a constant rate of material supply dV/dt at c~ = 0.5, ~bl = 30 °, ml = 0.1, and
m2 = 0.2. C u r v e
1:(~)2 -- 4 0 ° . C u r v e
2 : ( ~ 2 -- 6 0 °. C u r v e
3:(D2 -- 8 0 ° .
two different constitutive models in nonlinear viscoelasticity. The governing equations are solved numerically for a nonlinear elastic cone and the following conclusions are drawn: 1. For both regimes of the material influx, the twist angle U increases monotonically in time. This increase is, however, more pronounced for regime I of accretion. 2. With the growth of a (which characterizes the material nonlinearity), the twist angle decreases. Divergence in the U values corresponding to different ct values increases in time for regime I of accretion and decreases for regime II. 3. The twist angle U monotonically increases with the growth of torques and decreases with an increase in the rate of accretion. These effects become more pronounced for small ct values, when the material behavior is essentially nonlinear. 4. The dimensionless stress intensity ~ , decreases in ~b and vanishes on the outer surface of the growing cone. For slow accretion, the stress intensity ~ , increases
Chapter 7. Accretion of Viscoelastic Media with Small Strains
464
I
30
I
I
I
I
I
I
I
~b
60
Figure 7.1.7: The dimensionless stress intensity E, versus the angle 4, for a cone growing with a constant rate of material supply dV/dt at 4,1 = 30 °, ~b2 = 60 °, ml = 0.1, and m2, = 0.2. Curve 1: a = 0.4. Curve 2: a = 0.9.
in a in the vicinity of the initial cone and decreases far away from it. For rapid accretion, the stress intensity E, increases monotonically in a at any point of the growing cone.
7.2
Accretion of a Viscoelastic Spherical D o m e
In this section, we analyze stresses and displacements in a spherical dome at continuous accretion. The material behavior is governed by the constitutive equations of a linear nonaging viscoelastic material. Deformation of the structure is described in the framework of the membrane theory for thin-walled shells. Unlike the "onedimensional" accretion problem considered in Section 7.1, some arbitrariness arises in determining the natural configuration of built-up portions. Assuming that transition of an accreted element from the initial to natural configuration corresponds to its deformation under some horizontal load, we derive an ordinary differential equation
7.2. Accretion of a Viscoelastic Spherical Dome
465
0.3
X*
2
• • ° °°°o • ° o ooo °
°°°°Ooooooo° Oo°
•... "'iiiii °°°°°o °°o
I
i
30
4,
80
Figure 7.1.8: The dimensionless stress intensity ~, versus the angle q~ for a cone growing with a constant rate of material supply dV/dt at q~m = 30 °, q~2 = 80 °, mm = 0.1, and m2 = 0.2. Curve 1: a = 0.4. Curve 2: a = 0.9.
for this load. An explicit formula is developed for the horizontal displacement at the upper edge of a dome at accretion with a constant rate of material supply. The effects of material and structural parameters on displacements of the dome are studied numerically. The exposition follows Arutyunyan and Drozdov (1991) and Drozdov (1988).
7.2.1
Formulation
of t h e P r o b l e m
At the instant t = 0, a spherical shell with radius R and thickness h begins to grow on a horizontal plane (see Figure 7.2.1). Points of the shell refer to spherical coordinates {r, 0, q)} with unit vectors G, ~0, and ~ . At points of the middle surface of the shell, these vectors are denoted by G, ~1, and ~2, respectively, where n stands for the normal to the shell, and the subscript indices 1 and 2 denotes curvilinear coordinates ~1 = 0
466
Chapter 7. Accretion of Viscoelastic Media with Small Strains
Z o
•
•
•
o
•
• °
•
o
•
o
Illll
IIII
O
R
Figure 7.2.1: A growing spherical dome• and ~2 = ~b in the middle surface. The material supply is determined by the function O = O(t),
0-----t-----T,
where the angle 0 corresponds to the upper edge of the growing dome, and T is the time of accretion. Continuous accretion is treated as a limit of the following discrete process• Let us divide the interval [0, T] by points t m = mA, where A = T I M and m = 0, 1. . . . . M. At discrete accretion at instant tm the growing body in the reference configuration occupies the domain ~'~0(tm)=
R-~<-r<-R+~,
O(tm)~O~,0 - - < ~ b < 2 7 r
.
A built-up portion that occupies in the reference configuration the domain Af~,°(tm) =
R-
~ <-- r <-- R + ~, O(tm+l) <-- O < O(tm), 0 <-- ~ < 2Ir
,
is manufactured at instant tm and immediately merges with the growing body. This portion is not assumed to be stress-free in the reference configuration. To describe its natural configuration, we suppose that before deformation, any built-up element lies on a horizontal plane. At instant tin, the weight qm and some external pressure Pm are applied to this element. Under these loads, the element deforms while located on the plane without friction. The force qm equals the product of the specific weight q (per unit area of the middle surface) and the arc length R[O(tm) - O(tm+l)]. The pressure Pm is chosen in such a way that the built-up portion can merge with the growing dome without prestressing. The latter means that the horizontal displacements of points at the upper edge 0 = O(tm) of the growing dome coincide with the horizontal displacements of points at the lower edge of the built-up element after its deformation caused by the forces Pm and qm.
467
7.2. Accretion of a Viscoelastic Spherical Dome
Denote by fi, fi* the displacement vectors and by 5, 5" the strain tensors for transition from the initial to actual and natural configurations. At small strains, the strain tensor 5° for transition from the natural to actual configuration equals 5° = 5 - e^*.
(7.2.1)
The material behavior obeys the constitutive equation of a nonaging linear viscoelastic medium [see Eq. (2.2.68)], 5°(t) +
6-(t)- 1 + v v 1 -2v
Q0(t - 1")5°(1") d
e°(t) +
Oo(t - r ) e ° ( T ) d
~I .
(7.2.2)
Here E is Young's modulus, v is Poisson's ratio, Qo(t) is a dimensionless relaxation measure, ~'* is the instant when a built-up portion is manufactured, I is the unit tensor, e ° is the first invariant of the strain tensor, and the superposed dot denotes differentiation. The instant ~'* coincides with the instant when a material portion merges with the growing dome, T*(O(t)) = t.
(7.2.3)
Our objective is to determine stresses and displacements in a growing dome for a given program of accretion O(t) under the following assumtions: 1. Processes of loading and accretion are so slow that inertia forces can be neglected. 2. The Kirchhoff hypotheses may be applied to an accreted dome as well as to any built-up portion. 3. The membrane theory of thin-walled shells can be used.
7.2.2 Governing Equations We assume that axisymmetrical deformation occurs in the growing dome and present the displacement vector fi for transition from the reference to actual configuration in the form tl - - U 1 (t,
0)~1 +
(7.2.4)
Un(t, O)en,
where U 1 and U n are functions to be found. It follows from Eq. (7.2.4) that the nonzero components of the strain tensor ~ at points of the middle surface equal [see Novozhilov (1959)]
l(Oul ) --~ +
ell-- ~
Un
,
1
e22-- ~(U 1 cot 0 +
Un).
(7.2.5)
468
Chapter 7. Accretion of Viscoelastic Media with Small Strains
Suppose that the displacement vector fi* and the strain tensor 5" for transition from the initial to natural configuration are presented similarly to Eqs. (7.2.4) and (7.2.5) U-* : Ul(O)e * - 1 -+- Un(O)en, *
E^* = Ell* (0)~'1~'1 q" E22 * (0)6'26' 2,
(7.2.6)
* Ell * , and E22 * are functions to be determined. where u •1, Un, It follows from Eqs. (7.2.1), (7.2.4), and (7.2.5) that the nonzero components of the strain tensor t ° equal
1(0Ul
)
,
1
E02 -- ~(U 1 cot 0 + U n ) -
EO1 -- ~ l k - - ~ q- Un_ -- Ell,
,
E22.
(7.2.7)
Expressions (7.2.7) determine axisymmetric deformation in the middle surface of the shell at transition from the natural to actual configuration. In the theory of thin shells, a strain along the normal to the middle surface enn 0 is added to these expressions, which is independent of the displacements at points of the middle surface. As a result, we find that o EO = EO1 Jr- E02 -[- Enn°
(7.2.8)
Substitution of expressions (7.2.7) and (7.2.8) into the constitutive equation (7.2.2) implies the nonzero components of the stress tensor 6O'll(t)- 1 + v -+-
°r22(t)-
~nn(t)-
cOl(t) + 1 - 2v cO(t) 00(t-
E
T) E01(T) -I- 1 - u 2v e°(r)
E 1+ v {[e02(t)+
1 - t' 2v e°(t)]
+
e2°i(r)+ 1 - 2vv
Oo(t-r)
e°(r)
I } dr
dr
E {lEOn(t)+ v e0(t)] 1+ v 1 - 2v +
Q o ( t - T) En0n(T) nt- 1 - v 2v e°(r)
dr .
(7.2.9)
According to the static Kirchhoff hypothesis, the normal stress O'n, vanishes (7.2.10)
~rnn(t) = O.
It follows from Eqs. (7.2.9) and (7.2.10) that EO nn __ --
l,'
1-2v
EO,
469
7.2. Accretion of a Viscoelastic Spherical Dome
which implies that
eo = nn
v
1-v
(eo + e ° 2 ) '
co_
1-2V(eOl+e2o)"
(7.2.11)
1-v
Combining Eqs. (7.2.9) and (7.2.11), we obtain 0rll (t) --
or22(t) --
E p 2 [E°l(t)+ pE02(t)] +
1 ~
Oo(t-
T)[E01(T)+
pE02(T)] d'r
,
E 1,2 ([Eo2(t)-+- V,01(t)] -+-fr I 00(t_ T)[E02(T)-~- p'01(T)] aT).
1 ~
(7.2.12) Substituting expressions (7.2.12) into the formula for loads in the middle surface of the shell h/2 Nij --
o'ijdz
(i, j = 1, 2),
(7.2.13)
d -h/2
we calculate the nonzero forces N1 l(t) -
N22(t) -
Eh 1 --
1) 2
Eh 1 ~
lJ 2
[E01(t) q- pE02(t)] +
00(t- T)[E01(T)-+- I~E02(T)]dT ,
([E02(t) q- I~E?I(I)] + ~[00(t--T)[E02(T)-] - ~E01(T)] aT}. (7.2.14)
For axisymmetrical loading, the equilibrium equations for an element of the shell read [see, e.g., Novozhilov (1959)] 8N11 + (Nil - N 2 2 ) c o t 0 + qlR -- 0, 00
(7.2.15)
Nil + N22 - qnR = 0,
(7.2.16)
where ql and qn are the loads applied to the unit area of the middle surface and directed along the vectors ~1 and G, respectively. Suppose that weight of the dome is the only external load. Then ql -- q sin 0,
qn = - q cos 0.
(7.2.17)
We substitute expressions (7.2.17) into Eq. (7.2.16) and find that N22 = - N i l - qRcos 0.
(7.2.18)
Combining Eqs. (7.2.15) and (7.2.18) and using Eq. (7.2.17), we obtain 0Nil qR + 2Nil cot 0 + - 0. 80 sin 0
(7.2.19)
Chapter 7. Accretion of Viscoelastic Media with Small Strains
470
The initial condition for the differential equation (7.2.19) is written as Nil (t, O(t)) = 0,
(7.2.20)
which means that external load vanishes on the upper edge of the growing dome. To solve Eq. (7.2.19), we first integrate the homogeneous equation
ONll + 2N11 c o t 00
0 -- 0
and obtain Nll = Z sin -2 0,
(7.2.21)
where Z is an arbitrary function of time t. Afterward, we seek a solution of the nonhomogeneous equation (7.2.19) in the form (7.2.21), where Z(t, O) is a function to be found. This leads to the differential equation 0Z -
00
- q R sin 0,
which implies that z = qR(cos 0 + C), where C is an arbitrary function of time. This expression together with Eq. (7.2.21) yields
Nll = qR sin -2 0(cos 0 + C). Finally, we determine C from the initial condition (7.2.20) and find that Nil(t, 0) = qR sin -2 0[cos 0 - cos O(t)].
(7.2.22)
Combining Eqs. (7.2.18)and (7.2.22), we obtain
N22(t, 0) = -qR{cos 0 + sin -2 0[cos 0 - cos O(t)]}.
(7.2.23)
Let us present Eqs. (7.2.14) in the operator form
1 ( I - R)(E°I + vE°2) -
-
1
12 2
E~N11,
--
1)2
( I - R ) ( E ° 2 + va°~) - ----E-h---Nz2. (7.2.24)
Here I is the unit operator, and R is the relaxation operator with the kernel -Qo(t). For any integrable function f(t, 0),
I f = f(t, 0),
Rf = -
(o)
Qo(t - "r)f('r, O)d'r.
Denote by I + N the operator inverse to the operator I - R,
I+K
=(I-R)
-~,
471
7.2. Accretion of a Viscoelastic Spherical Dome
and by C0(t) the kernel of the Volterra operator K Kf =
(o)
C'o(t - r ) f ( r , 0) d'r.
It follows from Eqs. (7.2.24) that El01 -+- pE202 __
1-
122
1 -
v2
ve°l + %o2 - ~
(I + K)N~,
(7.2.25)
(I + K)N22.
(7.2.26)
First, we multiply Eq. (7.2.26) by v and subtract from Eq. (7.2.25). As a result, we find that 1
e°l = ~-~(I + K)(N11 - vN22).
(7.2.27)
Afterward, we multiply Eq. (7.2.25) by v and subtract from Eq. (7.2.26) to obtain 1
e°2 = ~--~(I + K ) ( N 2 2 - VNll).
(7.2.28)
Substitution of expressions (7.2.7) into Eqs. (7.2.27) and (7.2.28) implies that 0Ul + 00
,
lg n
=
R
Rell + ~--~(I + K)(Nll
,
Ul cot 0 + Un
-- R e 2 2 -+-
-
(7.2.29)
vN22),
R ~--~(I + K)(N22 - t'Nll).
(7.2.30)
Subtracting Eq. (7.2.30) from Eq. (7.2.29), we obtain Oul 00
Ul cot 0
= R ( E l,l
-
, ) -kE22
R(1 + v) (I + K)(Nll
- N22).
(7.2.31)
Equation (7.2.30) can be rewritten as ,
R
v -- R e 2 2 + 77,(1 + K ) ( N 2 2 sin 0 Ln
- VNll ),
(7.2.32)
where V -- /gl COS 0 q- U n
(7.2.33)
sin 0
is the horizontal displacement at points of the middle surface of the dome. Finally, substitution of expressions (7.2.22) and (7.2.23) into Eqs. (7.2.31) and (7.2.32) yields 0Ul
ao
-- Ul c o t 0 = R ( E l l
-- E22 )
Eh
2 coso_coso ,, 1 '
sin 2 0
472
Chapter 7. Accretion of Viscoelastic Media with Small Strains
sin 0
* -- -qR2 -- RE22 ~ (I + K) [cos 0 + 1 + v (cos 0 - cos O(t))] / sin 2 0
(7.2.34) To solve Eqs. (7.2.34), it is necessary to determine the preloading strains e~l (0) and e~2(0), which characterize transition from the initial to natural configuration.
7.2.3
Determination of Preload
* and e22 * , we consider deformation of a built-up portion that To find the strains ell merges with the growing dome at instant tm. We assume that to transform this portion from its initial to natural configuration, it suffices to apply a horizontal compressive load Pm = p(tm) to its external boundary surface• Since this transformation occurs instantaneously, we can neglect the material viscosity• At this transformation, loads in the middle surface N~ are calculated by formulas (7.2.14), where the strains e° are replaced by eij, and the integral terms vanish Eh
Nil
Eh
1 - /)2 (ell "+- 11•22)'
N22
1 - p2 (e22 -+- /Pe~l)•
(7.2.35)
The forces N~I and N~ satisfy the equilibrium equations (7.2.15) and (7.2.16), where ql and q2 equal projections of the horizontal pressure p(tm) on the unit vectors 01 and On: ql = --p(tm) cos 0,
qn -- --p(tm) sin 0.
Substitution of these expressions into Eqs. (7.2.15) and (7.2.16) implies that dN~l + (Nil - N22) cot 0 - p(tm)R cos 0 = 0, dO
Nil + N22 + p(tm)R sin 0 = 0.
(7.2.36)
Excluding N~2 from Eqs. (7.2.36), we find that
dN~l dO
+ 2Nll cot 0 = 0.
Integration of this differential equation with the initial condition
Nll(O(tm+l) ) = 0 yields N~I(O) = O,
0 • [19(tm+l), l~(tm)].
(7.2.37)
Combining Eqs. (7.2.36) and (7.2.37), we obtain N~2 = - p ( t m ) R sin 0.
(7.2.38)
7.2. Accretion of a Viscoelastic Spherical Dome
473
It follows from Eqs. (7.2.35), (7.2.37), and (7.2.38) that Ell
Rv -~p(tm) sin 0,
R -E-hP(tm)sin 0.
e22 --
Approaching the limit M ~ oo, we find that at continuous accretion
Rv . , ~1(0) = --E-~P(r (0)) sin 0,
,
e22(0 ) =
R --~p(r*(O))sin O.
(7.2.39)
Our aim now is to derive an explicit expression for the function p(t). For this purpose, we return to the discrete accretion and calculate the horizontal displacement Um = V(tm, O(tm) -- 0) at instant tm at the top edge 0 = O(tm - 0) of the dome. It follows from Eq. (7.2.34) that
qR2 cos O(tm) I 1~ f+ Um = sin O(tm) { Re~2(O(tm) - 0) - -ffh--
m Co(tm - r)dr ] } . -1
We substitute expression (7.2.39) into this equality and calculate the integral beating in mind that C0(0) = 0.
(7.2.40)
As a result, we obtain e2 Um
~
m m
Eh
sin O(tm){p(tm-1)sinO(tm) + qcos O(tm)[1 + Co(tin -/m-l)]}. (7.2.41)
On the other hand, the horizontal displacement at the lower edge of the built-up portion at instant tm can be found from Eq. (7.2.34), where the integral term is neglected
{, E
v(tm) = sin O(tm) RE22(O(tm) + O) Eh
cos O(tm) + sin20(tm)
This equality together with Eq. (7.2.39) implies that
{
v(tm) = - Eh sin "O(tm) p(tm)sin O(tm) l+v (COSO(tm)--COSag(tm+l))]}. + q [COS O(tm) + sin20(tm)
(7.2.42) The built-up portion can merge with the growing dome provided the horizontal displacements V(tm) and Um coincide. It follows from this condition together with
Chapter 7. Accretion of ViscoelasticMedia with Small Strains
474
Eqs. (7.2.41) and (7.2.42) that
p(tm-1)sin O(tm) + qcos O(tm)[1 + Co(tin -
tin-l)]
l+u
= p(tm)sin O(tm) + q cos O(tm) + sin20(tm) (COS '0(lm)
-- COS 1.~(/m+l))] .
It is convenient to present this equality as follows:
q p(tm) - p(tm-1) = sin O(tm) {Co(tin -- tm-~) COSO(tm) 1 + u [COSO ( t m + l ) -- COS O ( t m ) ] } . + sin 20(tm)
(7.2.43)
It follows from Eq. (7.2.40) that Co(tm - t i n - l ) = C 0 ( 0 ) A
-au o ( A ) ,
where lim
A-.O
o(a)
- O.
Similarly, COS ~_~(tm+l) -- COS O(tm) -- -- sin
dO O(tm)-d-f(tm)A + o(A).
Substituting these expressions into Eq. (7.2.43) and approaching the limit as M ~ ~, we find that
d tP ( t ) = q [ ~'°(O)c°tag(t)- sinl +2 u10(t--------~ dO d--i-(t) ,
p(0) = 0.
(7.2.44)
For a given accretion program O(t), Eq. (7.2.44) determines the pressure p(t). Afterward, the strains e~'1 and E~2 a r e found from Eqs. (7.2.39), and displacements at the middle surface are calculated according to Eqs. (7.2.34).
7.2.4
Displacements in an Accreted D o m e
Substitution of expressions (7.2.39) into Eq. (7.2.34) implies that 0Ul 00
-- U 1
cot 0 =
R2(1 + u ) (
Eh
p(~'*(O))sin 0
+q{ cos 0 + sin22 0 (cos 0 - cos O(t))] + G(t - r) E cos 0 + 2 (cos0- cos o(T))] dT}). (0) sin 2 0 (7.2.45)
475
7.2. Accretion of a Viscoelastic Spherical Dome The boundary condition for Eq. (7.2.45) is written as
Ul t , ~
= 0,
(7.2.46)
which means that vertical displacements vanish on the horizontal plane. According to Eq. (7.2.45), the function y-
Ul sin 0
satisfies the equation
OY
R2(1 + v ) (
Eh
00
+ q
{I
+
p(T* (0)) sin 0
2
cos 0 + si--~-~(cos 0 - cos O(t))
1
*(0) (?0(t - r) cos 0 + sin 2 0 (cos 0 - cos O(r))
d~"
.
We integrate this equality from 0 to 7r/2, use Eq. (7.2.46), and obtain
Ir/2 / ul(t,O) = - R 2 ( 1 + v) sin0 p(r*(~))sin~ Eh ao + q
/ 2
cos ~ + sin2 ~ (cos ~ - cos O(t))
]
+ fj(~ Co(,--,>loose+ &(cos¢-cos o(,,)>]d'r})dld. (7.2.47) After determining Ul, the function Un is found from Eqs. (7.2.30) and (7.2.39)
Un(t, O) = - u l ( t , + q
+
n2/ p(~'*(O))
0)cot 0 - ~
cos 0 + ~ ( c o s
0 - cosO(t))
*(0) (~0(t - 1-) cos 0 + sin 2 0 (cos 0 - cos O(~')) dl"
. (7.2.48)
7.2.5 Numerical Analysis We study numerically the effect of accretion on the horizontal displacement Vl at the upper edge 0 = O(t). It follows from Eqs. (7.2.34) and (7.2.39) that the dimensionless
Chapter 7. Accretion of Viscoelastic Media with Small Strains
476
horizontal displacement
Vl, equals
Vl,(t) = [p,(t) sin O(t) + cos O(t)] sin O(t),
(7.2.49)
where
Eh
Vl, --
p m
qR 2 V l '
P,
q
•
Denote by
._ dO dVdt - - 2 7rR2h sin O(t)---~-(t)
(7.2.50)
the rate of mass supply. We confine ourselves to accretion with a constant rate of growth, d V / d t = constant. Introducing the dimensionless time t, = t / T , we write Eq. (7.2.50) as sin 0
dO - - cos 19, dt,
(7.2.51)
where the angle 19 = O(T) characterizes position of the upper edge of the dome at instant T. Integration of Eq. (7.2.51) with the initial condition
O(0)-
,'/r 2
implies that cos O(t) = t, cos (9.
(7.2.52)
Equation (7.2.44) can be written in the dimensionless form as
dp, 1 + v dO dt, - X cot O - sin2 0 d t , '
p , ( 0 ) = 0,
(7.2.53)
where X = C0(0)T. Substitution of expression (7.2.52) into Eq. (7.2.53) after simple algebra yields
dp, dt,
Xt, +
cos 19
V/1 - t 2 cos 2 0
,
p , ( 0 ) = 0.
1 - t 2 cos 2 0
Integration of this equation implies that
p , ( t , ) = cos 19
_
j0 t* [
XT
(1 -- T2 COS2 0 ) 1/2
,+v (1 - -r2 cos 2 0 ) 3/2
(1 + v)t, cos 19
X (1 _ V/1 _ t2cos2 ® ) + COS l~ V/1 - t 2 cos 2 0
] d-r (7.2.54)
7.2. Accretion of a Viscoelastic Spherical Dome
477
We combine Eqs. (7.2.49) and (7.2.54) and use formula (7.2.52). As a result, we find that Vl,(t) = p,(t)(1 - t 2 COS2 {~) nt- t, cos 0 V/1 - t 2 COS2 (~ =
X (1- t2cos20)(1cos 19
V/1- t2cos20)
+ (2 + v)t, cos O V/1 - t 2 COS2 0 .
(7.2.55)
According to Eq. (7.2.55), the dimensionless horizontal displacement Vl, at the upper edge of a growing dome is determined by three dimensionless parameters X, v, and 19. Dependencies Vl,(t,) are plotted in Figures 7.2.2 to 7.2.4. Figure 7.2.2 demonstrates the effect of Poisson's ratio on the horizontal displacement of the upper edge of a growing dome. The dimensionless displacement Vl, increases with an increase in Poisson' s ratio v and reaches its maximum for an
4
Vl,
0
t,
1
Figure 7.2.2: The dimensionless horizontal displacement Vl, at the upper edge of an accreted dome versus the dimensionless time t, at O = 30 ° and X = 10.0. Curve 1: v = 0.0. Curve 2: v = 0.3. Curve 3: v = 0.5.
Chapter 7. Accretion of Viscoelastic Media with Small Strains
478
incompressible material, v = 0.5. However, the influence of Poisson's ratio is rather weak and it may be neglected. Figure 7.2.3 shows the effect of the material viscosity on Vl.. The dimensionless horizontal displacement at the upper edge increases monotonically with the growth of X. For small X values (less than 1), the dimensionless displacement is small, and it increases rapidly for relatively large XFigure 7.2.4 demonstrates the effect of the rate of accretion. For a fixed O, the horizontal displacement increases with the growth of the rate of accretion. For large rates of material influx (i.e., for small 19 values), the dependence vl.(t.) is nonmonotonic: the displacement grows, reaches its maximum, and, afterward, decreases. For relatively slow accretion (i.e., for large 19 values) the function Vl.(t,) becomes monotonic. The dimensionless displacement Vl. is relatively small at slow accretion and increases significantly at rapid accretion.
m
ZJ1,
0
t,
1
Figure 7.2.3: The dimensionless horizontal displacement Vl, at the upper edge of an accreted dome versus the dimensionless time t, for v = 0.3 and 19 = 30 °. Curve 1: X = 0.1. Curve 2: X = 1.0. Curve 3: X = 10.0.
7.2. Accretion of a Viscoelastic Spherical Dome
479
Vl,
0
t,
1
Figure 7.2.4: The dimensionless horizontal displacement Vl. at the upper edge of an accreted dome versus the dimensionless time t, for v = 0.3 and X = 10.0. Curve 1: O = 30 °. Curve 2: ® = 60 °. Curve 3: ® = 75 °.
Concluding Remarks Continuous accretion of a linear nonaging viscoelastic dome is studied at small strains. Governing equations are derived in the framework of the membrane theory of thin-walled spherical shells. Stresses in a growing dome under the action of its weight are independent of preloading and are determined by the dome geometry at the current instant. Displacements in a growing dome depend essentially on the accretion program and preloading. The effect of material and geometrical parameters on the horizontal displacement of the upper edge of a growing dome is analyzed numerically. The following conclusions are drawn: 1. The dimensionless horizontal displacement at the upper edge of a growing dome increases in time monotonically for relatively small rates of accretion. For large rates of growth, the horizontal displacement increases in time, reaches its maximum, and afterward decreases.
Chapter 7. Accretion of Viscoelastic Media with Small Strains
480
2. The horizontal displacement at the upper edge of a growing dome increases with an increase in the Poisson's ratio v and reaches its maximum for an incompressible material with v = 0.5. However, the effect of Poisson's ratio is rather weak, and it may be neglected. 3. The horizontal displacement at the upper edge increases monotonically with the growth of the material viscocity X.
7.3
Debonding of Accreted Viscoelastic Beams
Two contact problems of debonding are studied for accreted viscoelastic beams. The first problem deals with two cantilevered beams linked by an adhesive layer. When strains in the layer exceed some critical level, the layer is torn and the beams sever. Motion of the boundary between the region of contact and the debonding zone is analyzed numerically for elastic beams accreted on their outer surfaces. The other problem is concerned with a growing elastic beam connected to a rigid foundation by a nonlinearly elastic adhesive layer. A partial differential equation for the beam deflection is derived with specific boundary conditions in the integral form. Explicit safety estimates are developed that ensure that no debonding occurs. The effects of material and geometrical parameters on the ultimate intensity of prestressing are analyzed numerically. The exposition follows Drozdov (1989). 7.3.1
Accretion of a Two-Layered Beam
Let us consider two cantilever elastic beams with length 1 connected by an adhesive layer (see Figure 7.3.1). The beams have rectangular cross-sections with width b and
~.
a(t)
.I
W
X
(t) .3"
-I
Figure 7.3.1: Two growing beams linked by an adhesive layer.
7.3. Debonding of Accreted Viscoelastic Beams
481
thickness h = h(t). At the initial instant t = 0, the thickness equals h(0), and it increases in time owing to the continuous material supply in the interval [0, T]. Continuous accretion is treated as a limit of the following discrete process. Let us divide the interval [0, T] by points tm = mA, where A = T / M o and m = 0, 1. . . . . M0. At discrete accretion, at instant tm the growing beams in the reference configuration occupy the domains ~ + (tm) = {0 <-- x <-- l, 0 <-- y <-- b, 0 <-- Z <-- h(tm)}, f~-(tm) = { 0 < - - x ~ l ,
O <-- y <- b, - h ( t m ) <- z ~ O},
where {x, y, z} are Cartesian coordinates. Two built-up portions that occupy in the reference configuration the domains AI~ + (tm) = {0 <-- x ~ l, 0 <-- y <-- b, h(tm) <-- z <- h(tm+l)}, A~'~-(tm) -- {0 <-- x <-- I, 0 <-- y <- b, -h(tm+l) <- z <- -h(tm)},
are manufactured and immediately merge with the growing bodies. The natural and reference configurations of the initial beams coincide. We assume that before joining the accreted beams, any built-up portion is elongated by tensile loads applied to its ends. Transition from the natural configuration of a built-up portion to the actual configuration of the growing structure at the instant when the portion joins the beams is characterized by the longitudinal strain e.. For definiteness, the strain e. is assumed to be constant. To describe deformation of accreted beams, we present the longitudinal strain e at transition from the reference to actual configuration as a sum of the strain el owing to axial elongation and the strain e2 owing to bending e ( t , x , z ) = e l ( t , x ) + ez(t,x,z).
(7.3.1)
According to the Euler hypotheses, c92w e2(t, x, z) = -z-Y~x2(t, x),
(7.3.2)
where w(t, x) denotes deflection of the longitudinal axis at instant t at point x. The longitudinal axis of any beam is assumed to be located on the interface between the beam and the adhesive layer. Let us consider a built-up portion with a coordinate z, which merges with the growing beam at instant ~'*(z). On the surface of accretion of the beam, the strain owing to axial elongation equals el (r*(z),x) and the strain owing to bending equals e2(~-*(z), x, z). For accretion without prestressing, the strains in the built-up portion at transition from the reference to natural configuration equal e~(x) = el (l"*(z), x),
(7.3.3)
ez(X,Z) = ez(r*(z),x,z),
(7.3.4)
482
Chapter 7. Accretion of Viscoelastic Media with Small Strains
which implies that the strains from the natural to actual configuration e ° = el - el,
E° = e2 - E2
(7.3.5)
vanish at the instant of accretion ~'*(z). To account for preloading with a given longitudinal strain e,, we replace Eq. (7.3.3) by the formula e I (x, z) = el (1" (z), x) - e,.
(7.3.6)
Combining Eqs. (7.3.1), (7.3.2), (7.3.4), (7.3.5), and (7.3.6), we find that e ° ( t , x , z ) = e, + e l ( t , x ) -
el(~'*(z),x)-z
02W
-~--Zz( t , x ) -
02W *
-ff~x2(~" (z),x)
1. l
(7.3.7)
For uniaxial loading of a linear elastic medium, the stress or is connected with the strain e ° by Hooke's law or = Ee °,
(7.3.8)
where E is Young's modulus. Substitution of expression (7.3.7) into Eq. (7.3.8) implies that or(t,x,z) = E
e, + e l ( t , x ) - el(T*(Z),X ) --Z
(t,x)-
-7--V(~'*(Z),X)
•
(7.3.9) Let N be longitudinal force, and M bending moment: N(t, x) = b
fh(t) or(t, x, z) dz, dO
M(t, x) = b
fh(t) or(t, x, z)z dz. JO (7.3.10)
Substitution of expression (7.3.9) into Eqs. (7.3.10) implies that
[ N(t, x) = Eb h(t)e, + h(t)el(t, x) -
hZ(t) 02W(t, x) + 20X
2
[~ M ( t , x ) = Eb
--
fh(t) el(r* (z), x) dz JO
fo0h(t)~°~2w(l"*(z), x)z dz 1'
/2(t ) E, + - - - - ~ - e l ( t , x ) -
el(r*(z),x)zdz
dO h3(t) t92w(t,x) + ~0"h(t)O2w(r*(Z),X)Z 2 dz ]
Combining Eq. (7.3.11) with the equilibrium equation N(t, x) = O,
(7.3.11)
(7.3.12)
7.3. Debonding of Accreted Viscoelastic Beams
483
we obtain
h(t)e, + h(t)el(t,x)-
h(t) el('r*(z),x)dz dO h(t)c32W(~"•(Z), x)z dz
h2(t) 02w (t, x) + 2 Ox2 ao
= O.
Differentiation of this equality with respect to time yields
0el (t,x) - h(t) O3w (t,x) - l_~ hd (t)e.. Ot 20tOx 2 h(t) dt
(7.3.13)
We now differentiate Eq. (7.3.12) and find that
0114(t, x) Ot
Ebh(t) [~-~-(t)e, + h(t) Oel 20t (t,x)-
h2(t) 03w (t,x)] -~
°t°x 2
(7.3.14)
.
Substitution of expression (7.3.13) into Eq. (7.3.14) results in
OM (t'x) = Ebh(t) [ l ~t (t)e*
h2(t)12OtOX c33W2(t,x)] .
(7.3.15)
At the initial instant t = 0, no external forces are applied to the beams, M(0, x) = 0,
(7.3.16)
w(0, x) = 0.
(7.3.17)
and no deformation occurs
Integration of Eq. (7.3.15) with the initial conditions (7.3.16) and (7.3.17) yields
M(t,x) =
~Eb (
[ha(t)- h2(0)]e, - ~lfo0tO3W 9tOx2(r,x)h3(r)dr
)
=dh ] = - Eb [h3(t) -~x2 (t'x) - 3 fo t O2w-z--xox~(r,x)h2(~')ar(~')dr Eb [h2(t) _ h2(0)]e,"
(7.3.18)
+ 4
The bending moment M satisfies the equilibrium equations
OM - Q, Ox
OQ - bq, Ox
(7.3.19)
where Q(t, x) is the transverse force directed along the z axis, and q = q(t, x) stands for intensity of the distributed load applied to the beams from the adhesive layer and directed against the positive direction of the z axis. The adhesive layer with thickness H is treated as a Winkler foundation, which resists deformation in the z-direction only. For relatively small strains, the mechanical
484
Chapter 7. Accretion of Viscoelastic Media with Small Strains
response in the layer is governed by the constitutive equation of an aging elastic medium [see Eq. (2.2.12)] 0q Oe (t, x), Ot (t,x) = Eo(t) -~
(7.3.20)
where Eo(t) is the current Young's modulus, and 2w(t,x) H
e(t,x) =
(7.3.21)
is the strain. When the strain exceeds its ultimate value e*, the adhesive layer is torn, and the beams sever from each other. After debonding of the layer, its response vanishes; q = 0. An adhesive layer between two cantilever beams begins to tear from the free end x = I. Denote by ~-*(x) the instant when the beams sever at point x, and by a(t) coordinate of the left end of the interval of debonding at instant t (see Figure 7.3.1). We set formally a(t) = 1,
(7.3.22)
0 <-- t <-- T*(I).
The functions z*(x) and a(t) are inverse to each other a(T*(x)) = x,
T*(a(t)) = t,
and they can be found from the equalities w(z*(x),x)-
1
.
-~He ,
1
.
w(t,a(t)) = ~ H e .
(7.3.23)
Substituting expression (7.3.21) into Eq. (7.3.20) and integrating by parts with the use of Eq. (7.3.17), we find that 2 q(t, x) = -~ q(t,x) = O,
Eo(t)w(t, x) - d t aE° dr
x)
0 <-- x <- a(t),
(7.3.24)
a(t) < x <- 1.
Equations (7.3.19) together with Eqs. (7.3.18) and (7.3.24) imply that
3 04W ~x4(Z,x)dz h (t)-~gxa(t,x)- 3 fOth2('r) ~T( r ) -04W E2H4 [ Eo(t)w(t,x)
3 .~04w
h (t)--~(t,x)
ft -~r dEo(r)w(r,x)dTl
t hZ('r)~ (~')-~x4(~',x)d~" °~4W - 3 fOO = O,
O < x <_ a(t) '
(7.3.25)
a(t) < x <- I.
(7.3.26)
485
7.3. Debonding of Accreted Viscoelastic Beams
Our objective now is to integrate Eq. (7.3.26) in the interval [a(t), l] in order to derive boundary conditions for Eq. (7.3.25) at the point a(t). At the free end of a cantilever beam, the transverse force Q and the bending moment M vanish Q(t, l) = 0.
M(t, l) = O,
(7.3.27)
It follows from Eqs. (7.3.19) and (7.3.27) that M ( t , x ) = O,
a(t) < x <-- 1.
Substitution of expression (7.3.18) into this equality implies that
02W °32W h3(t)-ff~xZ(t,x)3 f0 t h2(1") ~-~hT ('r)-~x2('r,x)d'r
= 3[h2(t) - h2(0)]~,. (7.3.28)
We differentiate Eq. (7.3.28) with respect to time t to obtain 03w 6e, dh 8tOx 2 (t, x) - h2(t ) dt (t).
Integration of this equality from 0 to t with the use of the initial condition (7.3.17) yields
02W(t, x) = 6e, OX2
1 1 1 h(O) - h(t) "
(7.3.29)
Differentiating Eq. (7.3.29) with respect to x, we find that
03w Ox 3 (t,x) = 0.
(7.3.30)
When x ~ a(t) + 0 in Eqs. (7.3.29) and (7.3.30), we arrive at the following boundary conditions for Eq. (7.3.25): 02w (t, a(t)) = 6e, [ 1
tgX2
_
Lh(O)
03w (t, a(t)) = 0. 8x 3
1 ] h-(t ) ] '
(7.3.31 )
Two other boundary conditions w(t, 0) = 0,
olw
- - ( t , 0) = 0 Ox
(7.3.32)
mean that the end x = 0 is clamped. To solve integro-differential equation (7.3.25) with a moving boundary, it is convenient to introduce the new unknown function
Ow dt
v(t, x) = --:-7(t, x).
(7.3.33)
486
Chapter 7. Accretion of Viscoelastic Media with Small Strains
Differentiating Eq. (7.3.25) with respect to time, we find that the function v satisfies the equation 03479 24 Eo(t) 03(,4 (t, x) + EH~ ~h3(t) v(t, x) = 0.
(7.3.34)
Differentiation of equalities (7.3.32) implies that
0379
v(t, o) = o,
= - ( t , o) = o.
(7.3.35)
ax
We differentiate Eqs. (7.3.29) and (7.3.30) and obtain
03279 6E, dh 03x2 (t,x) -- hZ(t ) d t (t),
033v 03x3 (t,x) = O,
a(t) < x <<-1.
Approaching the limit as x ~ a(t) + 0, we arrive at the boundary conditions 032v 6E, dh 03x2 (t, a(t)) - h2(t ) d t (t),
033v 03x3 (t, a(t)) = 0.
(7.3.36)
We introduce the dimensionless variables and parameters x,-
E0,
x 1'
E0 E0(0)'
t t, - T '
w w, - h(0)'
24E0(0)l 4 a = EHh3(O),
vT 79*- h(0)'
6e,12 / 3 - h2(0),
h,-
~-
h h(0)'
He* 2h(0)"
The governing equations (7.3.33) to (7.3.36), read (asterisks are omitted for simplicity)
03479 otEo ( t ) 03X4 (t,x) + h3(t ) v ( t , x ) = O, v(t, O) = O,
0379 --z-(t, O) = O, dX
w ( t , x ) = j~0t v ( T , x ) d r ,
0279 [3 dh (t), -ff~x2 (t, a(t)) - h2(t ) d t
0379 -ff~x3 (t, a(t)) = O,
w(t,a(t)) = 6.
(7.3.37)
To study the effect of accretion on deformation of elastic beams linked by an adhesive layer, we integrate Eqs. (7.3.37) numerically by assuming that h(t) = 1 + ht,
Eo(t)-
1 + [ E 0 ( ~ ) - 1 ] [ 1 - exp(-'yt)],
where h is the dimensionless rate of accretion, E0(oo) is the dimensionless Young's modulus of an aged layer, and 31is the dimensionless rate of aging. The dimensionless coordinate a of the fight end of the contact zone is plotted versus the dimensionless time t in Figures 7.3.2 to 7.3.6.
7.3. Debonding of Accreted Viscoelastic Beams
487
1.0
0.6
I
I
I
I
0
I
I
I
t
I
I
1
Figure 7.3.2: The dimensionless coordinate of the right end of an adhesive layer a versus the dimensionless time t for a system of two elastic beams linked by an aging adhesive layer with 13 = 0.5, 3/ = 0.2, ~ = 0.05, h = 0.5, and E0(~) = 2.0. Curve 1: c~ = 0.1. Curve 2: c~ = 1.0. Curve 3: c~ = 10.0.
We draw the following conclusions: 1. At small c~ values, fracture of the adhesive layer is relatively small, and the beams sever only in a small interval. An increase in a leads to an increase in the length of the torn zone. This dependence is rather strong for small c~, and it weakens essentially with the growth of c~. For relatively large c~ values, the function a(t) becomes practically independent of a. 2. Destruction of the adhesive layer increases with the growth of the dimensionless parameter/3. The effect of this parameter is strong for small/3 values. With an increase in/3, its influence on the fracture becomes less pronounced. 3. For relatively large 8 values, the adhesive layer remains undamaged. When decreases, length of the debonding zone increases monotonically.
Chapter 7. Accretion of Viscoelastic Media with Small Strains
488
l
I
0
I
1
I
I
I
I
t
I
I
1
Figure 7.3.3: The dimensionless coordinate of the right end of an adhesive layer a versus the dimensionless time t for a system of two elastic beams linked by an aging adhesive layer with a = 1.0, 3/ = 0.2, 8 = 0.05,/t = 0.5, and E0(~) = 2.0. Curve 1: /3 = 0.5. Curve 2:/3 = 1.0. Curve 3:/3 = 1.5. Curve 4:13 = 2.0.
4. The rate of accretion significantly affects fracture of the adhesive layer. Length of the torn zone monotonically increases with the growth of the dimensionless parameter h. The effect of accretion is essentially strong for relatively small/t values. 5. Surprisingly, aging of the adhesive layer weakly affects its fracture. Even for relatively large 3/values, an increase in E0(~) by an order causes nonsignificant changes in the intervals in which the beams sever from each other. To obtain a more pronounced effect of aging, a more complicated model should be considered, in which the ultimate strain for failure e* is thought of as a function of the time of loading t - 1"* [see, e.g., Drozdov and Gertsbakh (1993)].
7.3. Debonding of Accreted Viscoelastic Beams
I
0
I
i
I
489
I
I
i
I
t
I
1
Figure 7.3.4: The dimensionless coordinate of the right end of an adhesive layer a versus the dimensionless time t for a system of two elastic beams linked by an aging adhesive layer with a = 1.0,/3 = 0.5, 3, = 0.2, h = 0.5, and E0(oo) -- 2.0. Curve 1: /5 = 0.01. Curve 2:/5 = 0.03. Curve 3:/5 = 0.05. Curve 4:/5 = 0.07. 7.3.2
A c c r e t i o n of an Elastic B e a m o n a N o n l i n e a r Winkler Foundation
We consider an elastic b e a m with length 21 linked to a rigid foundation by an adhesive layer (see Figure 7.3.7). The b e a m has a rectangular cross-section with width b and thickness h(0). At the initial instant t = 0, accretion of material begins on its upper surface z = h(0), where {x, y, z} are Cartesian coordinates. Owing to the material supply in the interval [0, T], thickness of the b e a m increases according to the law h = h(t). Continuous accretion is treated as a limit of the following discrete process. Let us divide the interval [0, T] by points tm = m A , where A = T / M o and m = 0, 1 . . . . . M0. At instant tm the growing b e a m in the reference configuration occupies the domain
~O(tm) : { - 1 <---x <-- 1, 0 <-- y <-- b, 0 <- z <-- h(tm)}.
490
Chapter 7. Accretion of Viscoelastic Media with Small Strains
•
l •
I
['
".
o
"~
•
\ I
I
0
I
I
I
I
I
I
I
I
t
1
Figure 7.3.5: The dimensionless coordinate of the right end of an adhesive layer a versus the dimensionless time t for a system of two elastic beams linked by an aging adhesive layer with c~ = 1.0, 13 = 0.5, 3' = 1.0,/5 = 0.01, and E0(~) = 2.0. Curve 1: /t = 0.1. Curve 2: h = 0.5. Curve 3: h = 1.0. Curve 4:/t = 2.0.
A built-up portion that occupies in the reference configuration the domain Al~°(tm) = { - 1 <- x <- 1, 0 <- y <- b, h(tm) <- z ~ h(tm+l)}, is manufactured at instant tm and immediately merges with the growing beam. The initial beam is in its natural (stress-free) state. Before joining the growing beam, a built-up portion is elongated by tensile loads applied to its ends. Transition from its natural configuration to the actual configuration of the growing beam at instant of their joining is characterized by a constant longitudinal strain e,. In a built-up portion arising at instant 7*(z), the strain e ° at transition from the natural to actual configuration at instant t is calculated according to Eq. (7.3.7). By assuming the beam to be linearly elastic with Young's modulus E, we arrive at the following
7.3. Debonding of Accreted Viscoelastic Beams
491
-\ -\ •
I
\ \
I
I
I
I
I
I
0
I t
I
I 1
Figure 7.3.6: The d i m e n s i o n l e s s coordinate of the right e n d of an a d h e s i v e layer a v e r s u s the d i m e n s i o n l e s s time t for a s y s t e m of two elastic b e a m s linked b y an a g i n g a d h e s i v e layer w i t h c~ = 1.0,/3 - 0.5, 3/ = 10.0, 3 = 0.01, a n d h = 0.5. C u r v e 1: Eo(~) = 1.5. C u r v e 2: Eo(~) = 15.0. f o r m u l a for the b e n d i n g m o m e n t
M(t,x) =
h3(t)-~x2(t,x ) - 3
-~x2 ('r, x)hZ ('r)
Eb [hZ(t) _ hZ(0)]e," +
(~')d~"
(7.3.38)
4
Equation (7.3.38) differs from Eq. (7.3.18) by the sign of the first term in the fight-hand side, because we suppose now that positive deflection w corresponds to compression of the adhesive layer. For a beam under the action of its weight, the bending moment M satisfies the e q u i l i b r i u m equations OM Ox
- Q,
OQ Ox
- b(Fh-
q),
(7.3.39)
492
Chapter 7. Accretion of Viscoelastic Media with Small Strains
LZ a(t)
IIIIIIIIIIIIIIIIIIIIIII
I
I
.I
(t)
IIIIIIIII
I
I
I
I
IIIII
x
Figure 7.3.7: An accreted beam on a Winkler foundation.
where Q(t, x) is the transverse force (positive direction of which coincides with the z axis), F is the specific weight, and q = q(t, x) stands for intensity of the distributed load applied to the beam from the adhesive layer and directed along the z axis. The adhesive layer with thickness H is treated as a nonlinear elastic Winkler foundation with the constitutive equation q(t, x) = ~ E°e(t' x), LO,
w(t, x) >-- O, w(t,x) < O,
(7.3.40)
where E0 is Young's modulus of the layer, and the strain e is determined by Eq. (7.3.21). We confine ourselves to monotonic debonding, in which the beam severs from the foundation at point x an some instant ~'*(x), and the foundation response at x is neglected for t > ~'*(x). The function ~'*(x) and the inverse function a(t) are determined from the nonlinear equations w(~'*(x),x) = O,
w(t,a(t)) = 0.
(7.3.41)
Since deformation of the beam is symmetrical with respect to the point x -- 0, it suffices to consider only the fight side of the beam, 0 -< x -< l. Accounting for the weight of the beam, we write the governing equations (7.3.25) and (7.3.26) as follows:
h3 04w(t,x)-3foth2(,r)~(,r) 04w
(t)-~Zx4 -
12[ 2E0 ,t ] ' ~ Fh(t) - --ff-wt ,x)
3 04W
h (t)~x4(t, x) - 3 -
-~x4 (T, x>&
12F h(t), E
0 <- x <- a(t),
(7.3.42)
f0t h2(r) dd_~hT 04w(~', x) dr ('r)-~x4 a(t) < x <- l.
(7.3.43)
Our objective now is to derive boundary conditions for Eq. (7.3.42) with the use of Eq. (7.3.43). For this purpose, we integrate the equilibrium equations (7.3.39) with
493
7.3. Debonding of Accreted Viscoelastic Beams
the boundary conditions M ( t , l) = O,
(7.3.44)
Q(t, l) = 0
in the interval [a(t),/], i.e., in the region where q = 0. As a result, we obtain M(t, x) = ~ F bh(t)(l - x) 2,
Q(t,x) = - F b h ( t ) ( l -
x).
(7.3.45)
We substitute expression (7.3.38) into Eqs. (7.3.45) and use Eq. (7.3.39) to obtain
02w
t
h3(t)-~x2 (t,x ) - 3
6F - --h(t)(l E
-
o33W h3(tl-~x3 (t,x ) - 3
fo
dh 02W h2 ('r) ~-~ ('r) --~x2 ('r, x ) d~
x) 2 - 3[h2(t) - h2(0)]e,, t
fo
12F ---h(t)(l E
dh 03w h2 ('r) ~--T7('r) -~x3 ('r, x ) d~"
- x).
(7.3.46)
Setting x = a(t) in Eq. (7.3.46), we arrive at the formulas
02W
h3(t)-~x2 (t,a(t)) - 3
~0t h2('r) ~T(r)-~-ff 02W(r, a(t)) d~
6F - - - h ( t ) [ l - a(t)] 2 - 3[h2(t) - h2(0)]E,,
03W
h3(t)-~x3 (t, a(t)) - 3
ft
12F -~h(t)[1E
hZ('r)
~
03W
(~')-~-Yx3(~', a(t)) dl-
a(t)].
(7.3.47)
~w - - ( t , 0) = 0. Ox
(7.3.48)
Owing to the symmetry of deformation,
Instead of the fourth boundary condition, we write the integral equilibrium equation
fo a(t>q(t, x) dx = F a(t)h(t),
(7.3.49)
Chapter 7. Accretion of ViscoelasticMedia with Small Strains
494
which means that the foundation response equilibrates weight of the beam. Substituting expressions (7.3.21) and (7.3.40) into Eq. (7.3.49), we find that
fOa(t)w(t, X)dx
= FHa(t)h(t).
(7.3.50)
2E0
Integro-differential equation (7.3.42) with conditions (7.3.47), (7.3.48), and (7.3.50) determines the functions w(t, x) and a(t). It is convenient to reduce Eq. (7.3.42) to a partial differential equation by introducing a new function v(t,x) according to Eq. (7.3.33). Differentiation of Eq. (7.3.42) implies that ~4 V (t, X) if-
OX4
24E0 12F dh EHh3(t-----~ v(t, x) - Eh3(t) -d-~(t).
(7.3.51)
To derive boundary conditions for Eq. (7.3.51), we differentiate Eqs. (7.3.46) with respect to time and set x = a(t). As a result, we find that
02v F 2 (t,a(t)) O x = 6 { ~[I-
a(t)] 2 - h2(t)e*} dh ~-~(), t
03 v (t, a(t)) = 12F dh - Eh3(t)[l - a(t)]--dT(t). Ox3
(7.3.52)
It follows from Eq. (7.3.48) that
Ov
~xx(t, 0) = 0.
(7.3.53)
We differentiate Eq. (7.3.50) with respect to time t, use Eq. (7.3.41), and, finally, obtain
FH d fo a(t) v(t, x) dx - 2E0 dt [a(t)h(t)].
(7.3.54)
At t = 0, Eq. (7.3.42) reads 3 04w 24E0 12F h(0), h (0)~x4 (0,x) + EH w(O,x) = E
0 -< x --- l.
The solution of this equation is independent of x FHh(0)
w(O,x) = ~ .
2E0
(7.3.55)
Integration of Eq. (7.3.33) with the initial condition (7.3.55) yields
w(t, x) =
FHh(0) ~o"t 2Eo + v(r, x) dr.
(7.3.56)
495
7.3. Debonding of Accreted Viscoelastic Beams
Substituting expression (7.3.56) into Eq. (7.3.41), we find that f0 t
FHh(0)
v(T, a(t)) d~- = - 2 ~ "
(7.3.57)
To describe debonding of an accreted beam linked to a rigid foundation by a nonlinear adhesive layer, the linear partial differential equation (7.3.51) should be solved together with the boundary conditions (7.3.52), (7.3.53) and the integral equations (7.3.54), (7.3.57). We do not dwell on the numerical analysis of this problem and confine ourselves to safety estimates, which ensure that tearing of an adhesive layer does not occur. The latter means that for any t ~ [0, T], (7.3.58)
a(t) = l
and ~o t V(T, l) d'r
FHh(0) -
-
(7.3.59)
2E0
We introduce the new variable Yc = l - x ,
and rewrite Eqs. (7.3.51) to (7.3.54) and (7.3.57) accounting for equality (7.3.58) (tilde is omitted for simplicity)
04/3
24E0
12F dh v ( t , x ) - Eh3(t) dt (t),
Ox4 (t,x) + EHh3(t---~ o~2v
fo
03v Ox 3 (t, O) = O,
h2(t) dt (t),
fO I v(t, x) dx - FHI dh (t), 2Eo dt
Ov (t, 1) = O, Ox
t
dh
6e,
o~X2(t, 0) =
v(T, 0) d~ - -
FHh(0) ~ 2Eo
(7.3.60)
Introducing the dimensionless variables and parameters x, -
x l'
t, -
t T'
24E014 a = EHh3(O ),
h, -
h h(0)'
Eh(O)e.
[3 =
2F/2 ,
7[fl,
V vo
12F/4 v0 = ETh2(O ),
we present Eqs. (7.3.60) as follows (asterisks are omitted):
O~4V Ol 1 dh 0x4 (t,x) + h--~(t)v(t,x) - h3(t ) dt (t),
(7.3.61)
496
Chapter 7. Accretion of Viscoelastic Media with Small Strains 02V
[3 dh
o~3v
Ox2 (t, O) = -h2(t----~ d---[(t), o~V --(t,
fo v(t, x) dx = a1 dh --~(t),
1) -- O,
Ox
Ox3 (t, O) = O,
f0 t v('r, O) d'r -> - -1.
Og
We now set
~(t)
v(t, x) = u(t, x) + 1 dh
(7.3.62)
and find that the function u(t, x) satisfies the following equations:
04U Og OX4 (t, X) + ~ u ( t ,
a2u [3 dh Ox2 (t, O) = - h2(t----~ d----~(t), Ou (t, 1) Ox
O,
(7.3.63)
X) = 0, ~3u ~x 3 (t, O) = O,
f01 u(t, x) dx
(7.3.64)
O,
(7.3.65)
[ot u(~', O) dT >-- - h(t)
(7.3.66)
Ol
We introduce the new variables U1 = u,
02 -
03 u
02U
o3U
u3-
Ox'
Ox2 '
u4-
(7.3.67)
OX3'
and present Eq. (7.3.63) in the vector form OU ~ ( t , x) = A(a, h(t))U(t, x), Ox
(7.3.68)
where
U1 U =
U2 U3 U4
A(cz,h) = '
0 0 0 -czh-3(t)
1 0 0 0 1 0 0 0 1 0 0 0
"
Denote by ~ ( a , h, x) the solution of the matrix differential equation 0x
-
A(ct, h)~
with the initial condition • (,~, h, O) - I,
(7.3.69)
497
7.3. Debonding of Accreted Viscoelastic Beams
where I is the unit matrix. For any t >-- 0 and any x E [0, 1], (7.3.70)
U(t, x) = ~ ( ~ , h(t), x)U(t, 0).
Bearing in mind Eqs. (7.3.64), (7.3.65), and (7.3.67), we find that Ul(t, 0) f0 ~ll(OZ,h(t),x)dx + U2(t,0)f0 ~12(ot, h(t), x) dx jB dh
h2(t) dt
(t)
~13(o~, h(t), x) dx,
fo
U 1(t, 0)(I)21 (Or, h(t), 1) + U2(t, 0)(I)22(0~, h(t), 1)
/3 dh (t)~23(a,h(t), 1), h2(t) dt
(7.3.71)
where f~ij are components of the matrix ~ . It follows from Eqs. (7.3.71) that [3 dh
~ dh(t)(J2(a,h(t)), U2(t, 0) - h2(t ) dt
U1 (t, 0) - h2(t ) dt (t)f]l(a, h(t)),
(7.3.72)
where ~]1 and (12 satisfy linear algebraic equations
~]l(Ct,h)
/o 1¢~ll(O~,h,x)dx
+ ~J2(c~,h)
/o 1t~12(o~,h,x)dx
=
/o 1¢~13(o~,h,x)dx, (7.3.73)
Ul(ot,h)dP21(ot,h, 1) + U2(ot,h)dP22(ot,h, 1) = ~23(a,h, 1).
Combining Eqs. (7.3.67) and (7.3.72), we obtain dh u(t, 0) - h2(t ) dt (t)~rl (ct, h(t)).
Substitution of this expression into Eq. (7.3.66) yields t (J1 (c~, h(T)) dh
/3
fo
h2i~
h(t)
d---~('r) d~- -> - ~'c~
(7.3.74)
It follows from inequality (7.3.74) that the adhesive layer remains undamaged provided that for any h E [ 1, h(1)],
Ef
o ~ <_ h -
Ul (a, s)s -2 ds
(7.3.75)
.
Finally, combining Eqs. (7.3.73) and (7.3.75), we arrive at the safety estimate which ensures loading without fracture aCl <_ h
Is
h (I)22(0g S, l) ~1 dPl3(O~,S,x)dx _ (i)23(0g, s ' 1) f l (i)12(O/ s , x ) d x ds "
~ _
"
m
(I)21(0~, S, 1) fO f~12(°l,s,x)dx - (I~22(°~,s, l)fo 1 f ~ l l ( ° l , s , x ) d x $2
]-1 "
(7.3.76)
Chapter 7. Accretion of Viscoelastic Media with Small Strains
498
To study the influence of the material parameter ct and the final thickness h on the ultimate/3 value, we integrate Eq. (7.3.69) numerically and use formula (7.3.76). The critical 13 value is plotted versus h in Figure 7.3.8 for various ct. For small thicknesses h, the critical parameter 13 decreases rapidly with the growth of h. For relatively large final thicknesses h, the critical/3 value becomes practically independent of h. The ultimate 13 value is rather high for small ct values, and it decreases rapidly with an increase in ct.
Concluding Remarks Two contact problems of accretion are studied for growing beams linked by an adhesive layer with account for fracture of the layer. In the accretion problem for two cantilevered beams, a nonlinear integro-differential equation is derived for the dimensionless length of the contact zone a. This equation is integrated, and the influence of material and structural parameters on the debonding process is analyzed numerically (see Figures 7.3.2 to 7.3.6). Results of numerical simulation
30 i
"
-i I
1
h
2
Figure 7.3.8: The dimensionless parameter/3 versus the dimensionless thickness h. Curve 1: c¢ = 0.5. Curve 2: c¢ = 1.0. Curve 3: c~ = 5.0.
499
7.4. Torsion of an Accreted Elastoplastic Cylinder
are formulated and discussed in Section 7.3.1. The most surprising result is a weak effect of aging on fracture of adhesive layers. The other problem deals with an elastic beam lying on a nonlinearly elastic Winkler foundation. A partial differential equation with a moving boundary and specific integral conditions is derived for the beam deflection. An explicit safety estimate is developed [see Eq. (7.3.76)] that ensures that fracture of the adhesive layer does not occur. This estimate is presented as an inequality for the dimensionless intensity of prestressing/3. The ultimate/3 value is determined by two dimensionless parameters: the relative rigidity of the layer a and the final thickness of the beam h. The critical 13 value decreases rapidly with the growth of h. This dependence is rather strong for small h, and becomes less pronounced for large thicknesses. For a fixed h, the ultimate/3 value decreases rapidly in a.
7.4
Torsion of an Accreted Elastoplastic Cylinder
In this section, we study torsion of an accreted circular cylinder made of an ideal elastoplastic medium. In the classical torsion problem for an elastoplastic cylinder, only one plastic region arises on the boundary of the cylinder when the torques exceed some critical value. Our purpose is to show that the material accretion can change the deformation process. Instead of one plastic zone, two plastic regions arise in a growing cylinder under some assumptions regarding the loading regime and the accretion program. Moreover, a plastic region appears inside an accreted cylinder, but not on its boundary surface.
7.4.1
Formulation of the Problem
Let us consider a circular cylinder with length I and radius al. At the instant t = 0, torques M = M ( t ) are applied to its edges, and continuous accretion of material begins on the stress-free lateral surface. Owing to the material supply in the interval [0, T], radius of the cylinder increases according to the law a = a(t),
a(O) = al,
a ( T ) = a2.
The rate of accretion is calculated as v(t) = 27rla(t)~tt (t).
(7.4.1)
Here dV t , v(t) = -d7( )
where V(t) stands for volume of the cylinder at instant t. We suppose that M(0) = 0,
(7.4.2)
Chapter7. Accretionof ViscoelasticMedia with Small Strains
500 the function
M(t) increases
monotonically in time,
dM ~ ( t ) > O, dt
(7.4.3)
d [ l dv--~ M - - ~ (t) 1 -<0.
(7.4.4)
and satisfies the inequality
Equation (7.4.4) means that the torque M increases in time rather slowly. For example, for a constant rate of accretion v and the power law M = MotK, inequality (7.4.4) implies that 0 --< K -< 1. The material behavior is governed by the constitutive equations of an ideal elastoplastic medium. In the elastic region, the stress-strain relations read o- = 3KE °,
~ = 2G0 °,
(7.4.5)
where K is a bulk modulus; G is a shear modulus; and o-, ¢0 are the first invariants and ~, 00 are the deviatoric parts of the stress tensor 6- and the strain tensor 5° at transition from the natural to actual configuration, respectively. We accept the von Mizes yield criterion, which implies that in the elastic region 2£ < ~0,
(7.4.6)
= (2~" ~ ) 1 / 2
(7.4.7)
where
is the current stress intensity, and 2£0 is the ultimate stress intensity. In the plastic region, the strain tensor 5° for transition from the reference to ^0 which obeys the actual configuration equals the sum of the elastic strain tensor G, constitutive equations (7.4.5), and the plastic strain tensor 5° 50
^0 + ,,0
-- E e
Ep.
As common practice, plastic deformation is assumed to be isochoric (the plastic incompressibility), ~°-0. The constitutive equation for the plastic flow reads 0~° = A~,
Ot where A is an unknown function, which is determined from the yield condition = £0.
(7.4.8)
7.4. Torsion of an Accreted Elastoplastic Cylinder
501
At the initial stage of loading, the stress intensity 2~ is relatively small, and the entire cylinder is purely elastic. At some instant T, a plastic region arises in the accreted cylinder and the material accretion stops. Afterward, the plastic region grows in time under the action of torques M(t). We confine ourselves to the active loading, when unloading does not occur. Our objective is to determine locations of elastic and plastic regions for an arbitrary accretion regime a(t) and an arbitrary loading program M(t), and to find stress distribution in a growing elastoplastic cylinder.
7.4.2
S t r e s s e s a n d Strains in a G r o w i n g C y l i n d e r
Points of the cylinder refer to cylindrical coordinates {r, 0, z} with unit vectors 6'r, 6'0, and ~z. Under torsion of a circular cylinder, the displacement vector fi at transition from the reference to actual configuration equals fi = ~(t)rz~o,
(7.4.9)
where c~(t) is an unknown function of time (twist angle per unit length). According to Eq. (7.4.9), the first invariant e of the strain tensor ~ vanishes,
E = 0,
(7.4.10)
and the deviatoric part 0 of the tensor ~ equals
1
= z~(t)r(~o~z + ~z~O).
(7.4.11)
z
For accretion without preloading, the displacement vector fi* and components of the strain tensor 5" for transition from the initial to actual configuration of built-up portions are determined by formulas (7.4.9) to (7.4.11), where the function c~(t) is replaced by the function o~*(r) = c~(~-*(r)).
(7.4.12)
Here ~'*(r) is the instant when a built-up portion with polar radius r merges with the growing cylinder, ~-*(r) = ~0,
Lt,
0 --< r --< al,
(7.4.13)
r = a(t).
Combining Eqs. (7.4.10) to (7.4.12) with the equality ~o = ~ _ ~.,
(7.4.14)
we find that the first invariant of the strain tensor ~o vanishes, and its deviatoric part is calculated as ~o= 1 ~[c~(t)- ~*(r)]r(P.O~z + P.z~O). z
(7.4.15)
502
Chapter 7. Accretion of Viscoelastic Media with Small Strains
It follows from Eqs. (7.4.5), (7.4.7), and (7.4.15) that in the elastic region the stress tensor 6- has the nonzero components Croz = OrzO = G r [ a ( t ) -
a*(r)],
(7.4.16)
and the stress intensity equals E = 2Gr[a(t)-
a*(r)].
(7.4.17)
We assume that in the plastic region CrOz = Crzo are the only nonzero components of the stress tensor 6", which implies that (7.4.18)
E = 2OrOz.
Combining Eqs. (7.4.8) and (7.4.18), we find that in the plastic region
O'Oz-- -~1E°.
(7.4.19)
Expressions (7.4.16) and (7.4.18) satisfy the equilibrium equations and the boundary conditions in stresses on the accretion surface r = a(t). Boundary conditions on the edges of the cylinder are written in the integral form g ( t ) = 2rr
7.4.3
fOa(t)Oroz(t, r ) r 2 dr.
(7.4.20)
Accretion of an Elastic Cylinder
Let us consider the initial stage of accretion, when the torques M ( t ) are small, and the plastic region is absent. It follows from Eqs. (7.4.17) and (7.4.20) that 2zrG
fOa(t)[ a ( t ) -
a * ( r ) ] r 3 dr = M(t).
Substituting expression (7.4.12) into this equality and bearing in mind that a(0) = 0,
(7.4.21)
we find that a(t)
fO0al r 3 dr
+ f a(t) [a(t) - a(T*(r))]r 3 dr -
al
M(t) .
2"rrG
(7.4.22)
We calculate the first integral in the left side of Eq. (7.4.22) and introduce the new variable C(r) = s in the other integral. Accounting for Eq. (7.4.13), we obtain a4(0)
ft
da
+ Jo [a(t) - a(s)]a3(s)--z--(s)ds ds
M(t)
21rG"
7.4. Torsion of an Accreted Elastoplastic Cylinder
503
Differentiation of this equality with respect to time t implies that
da dt
~(t)
2 dM ~(t). 7rGa4(t) dt
=
(7.4.23)
Integration of Eq. (7.4.23) from 0 to t with the use of Eq. (7.4.21) yields c~(t) =
-~2
fot --~-('r)a-4('r) dM d~-.
(7.4.24)
Substituting expression (7.4.24) into Eq. (7.4.17) and employing Eq. (7.4.12), we find that in the elastic region
~(t, r) = m4r f j mdM "n" (r) dt ('r)a-4('r) dT.
(7.4.25)
Let us transform expression (7.4.25). For this purpose, we differentiate the identity
a(~'*(r)) = r,
al <- r <-a2,
with respect to r and obtain d----~-(r) =
Ida 11 ~ - (~'*(r))
.
(7.4.26)
It follows from Eq. (7.4.1) that
da v(~'*(r)) v(~'*(r)) dt (l"*(r)) = 27ra(~'*(r))l 27rrl " Combining this equality with Eq. (7.4.26), we arrive at the formula
d T* dr (r) =
2 7rrl v(T*(r))
al < r < a2.
(7.4.27)
We differentiate Eq. (7.4.25) with respect to r and use Eq. (7.4.27). As a result, we find that
0~, (t, r) = _4
t
O--F-
(r) --~-('r)a-4('r) d r
71"
-
1 F
dM
_
4r
dM~ (r*(r))
(r)
7ra4('r*(r)) dt
[~(t, r) - f ( r ) ] ,
(7.4.28)
where
F(r) =
81 dM ~ (~'*(r)). rv(T*(r)) dt
(7.4.29)
Setting r = a(t) in Eq. (7.4.25), we obtain ~(t, a(t)) = O.
(7.4.30)
Chapter 7. Accretion of Viscoelastic Media with Small Strains
504
To integrate equation (7.4.28) with the boundary condition (7.4.30), we introduce the new function
~i,(t, r) Z(t,r) - - - .
(7.4.31)
r
It follows from Eq. (7.4.28) that the function Z satisfies the equation
OZ
F(r) rE .
0---7(t, r) =
(7.4.32)
Integration of Eq. (7.4.32) from r to a(t) implies that
Z(t, a(t)) - Z(t, r) = -
f r a(t)
F(p)p -2 dp.
Combining this equality with Eq. (7.4.31) and utilizing Eq. (7.4.30), we arrive at the formula
£(t, r) = r
f
a(t)
F(p)p -2 do.
(7.4.33)
Let us calculate the second derivative of the function £(t,r) with respect to r. Differentiation of Eq. (7.4.28) yields
O2£ (t, r) = 1 -~[£(t,r) Or2
1 [0£ dF 1 l dF - F(r)] + -r -ff;-r(t,r) - -~r (r) . . . . r dr (r). (7.4.34)
It follows from Eqs. (7.4.27) and (7.4.29) that t"
dF 81 I d----r(r) = v (T-~r)) I
1 dM r 2 d-t (7.4.35)
Combining Eqs. (7.4.34) and (7.4.35), we obtain
02X (t, r) = Or 2
81
r3v(r*(r))
{ d--~-( M r * ( r ) ) - 2 7 r r 2ld~ [ v -1- ~dM -~(t) 1
t=r*(r) / " (7.4.36)
It follows from Eqs. (7.4.3), (7.4.4), and (7.4.36) that
032~(t, r) > 0, Or 2
al < r < a2. --
_
(7.4.37)
Equations (7.4.3) and (7.4.25) imply that for any t - 0, the function £(t, r) is positive in [al, a(t)]. Beating in mind Eq. (7.4.30), we find that £(t, r) reaches its maximum
7.4. Torsionof an Accreted Elastoplastic Cylinder
505
either (i) at r = al, or (ii) at some internal point R E (al, a(t)). Condition (ii) cannot take place, since at the point of maximum °32~ (t,R) < O, Or 2
which contradicts Eq. (7.4.37). Therefore, the stress intensity in a growing elastic cylinder reaches its maximum at the boundary of the initial cylinder r = a l for any loading program that satisfies inequalities (7.4.3) and (7.4.4). The first instant when a material portion is transformed into the plastic state coincides with the instant T when the accretion process stops. According to Eq. (7.4.8), the instant T is found from the equation ~(T, al) = ~0.
(7.4.38)
Suppose that
F(r) < E °,
al -< r <-a2.
(7.4.39)
Combining Eqs. (7.4.33) and (7.4.39), we find that ]~(T, al) = al
fa a2 F(r)r -2 dr 1
< ~°al
fa a2F(r)r -2dr 1
= ~0
( a1l-)
m
a2
< ~0 .
(7.4.40)
It follows from Eqs. (7.4.38) and (7.4.40) that a plastic region arises in an accreted elastoplastic cylinder if and only if the function F(r) exceeds the yield stress E ° in some subinterval of [al, a2]. Equation (7.4.35) together with Eqs. (7.4.3) and (7.4.4) implies that
dF dr
~(r)
< 0,
(7.4.41)
which means that the function F(r) decreases monotonically in r. Therefore a plastic region arises in an accreted cylinder provided the function F(r) satisfies the inequality
F(r) > E °,
r E (al,b)
(7.4.42)
for some interval [al,b] C [al,a2]. In the following we suppose that inequality (7.4.42) holds and analyze deformation of the cylinder with a plastic region.
7.4.4 An Elastoplastic Cylinder with One Plastic Region At instant T, the material supply stops, while the plastic region begins to grow under the action of the torques M(t). At some instant t > T, material in the plastic state occupies the domain Cl(t) <- r <- ce(t), and material in the elastic state occupies
Chapter 7. Accretion of Viscoelastic Media with Small Strains
506
the domains 0 < r < C 1 (t) and c2(t) < r < a2. Positions of the interfaces between elastic and plastic regions Cl(t), c2(t), as well as the twist angle a(t), are functions to be found. Equation (7.4.20) is written as
troz(t, r)r 2 dr + do
troz(t, r)r 2 dr + dCl(t)
M(t)
troz(t, r)r 2 dr -
27r
(t)
Substitution of expressions (7.4.16) and (7.4.18) into this equality implies that
Gt~(t)
r 3 dr + •] 0
r 2 dr + G --2
,] C 1 (t)
[cz(t) - ot('r*(r))]r 3 dr -
M(t) 27r
(t)
(7.4.43) We differentiate Eq. (7.4.43) with respect to time t and find that 1 4 da 3 d C l 2+ [ Xd °c c2~ ( t ) - ~ ( t ) - c~(t) dcl (t) 1 -~GCl (t)-d-f (t) + G~z(t)Cl(t)-~(t)
-d?
1 G[a4 _ c4(t)]dt~
+~
dc2..
1 dM
--d-~-(t) - G[cz(t) - t~(~'*(cz(t)))]c3(t)--~-q) - 27r --~ (t). (7.4.44)
As common practice in the elastoplasticity theory, the stress tensor # is assumed to be continuous on the interfaces r = Cl (t) and r = c2(t) between elastic and plastic regions. This hypothesis together with Eqs. (7.4.16) and (7.4.18) implies that 1
Ga(t)Cl (t) = _~~ o,
(7.4.45)
leo" G[o~(t)- ot(~'*(c2(t)))]c2(t) = -~
(7.4.46)
Differentiation of Eq. (7.4.45) with respect to time yields
dc1 dt
m
c1 da ~z dt
We find ct from Eq. (7.4.45) and substitute into this equality. As a result, we obtain
dcl (t) = 2Gc2(t ) da dt - ~--6 --d-~( t ) "
(7.4.47)
We now differentiate Eq. (7.4.46) with respect to time t
dc2
[cz(t) - ot('r*(c2(t)))]--~-(t) +
da d , d'r dc2 - ~ ( t ) - -~('r (c2(t)))-~-r (CZ(t))--~-(t) c2(t) = 0.
507
7.4. Torsion of an Accreted Elastoplastic Cylinder Combining this equality with Eqs. (7.4.23), (7.4.27), and (7.4.46), we write
~,° dc2 { da l [ 8 1 d M - - ('r*(c2(t)))] dc2 --~- / c2(t) 2Gc2(t) dt (t) + ---~(t)- 2Gc2(t ) c2(t)v(-~(c2(t))) dt (7.4.48)
=0. It follows from Eqs. (7.4.29) and (7.4.48) that
~0 dc2 (t) + Ida ( t ) - F(c2(t)) dc2 (t) ] c2(t) = 0, 2Gc2(t) dt -~ -~c~(]) dt which implies that
dc2 (t) = 2Gc2(t) da dt F(c2(t)) - ~0 d---~(t)"
(7.4.49)
Substitution of expressions (7.4.47) and (7.4.49) into Eq. (7.4.44) yields
da 2 dM ~(t). d---t(t) = 7rG[a4 + c4(t)- c4(t)l dt
(7.4.50)
The ordinary differential equations (7.4.47), (7.4.49), and (7.4.50) determine deformation of an elastoplastic cylinder and location of elastic and plastic regions. It follows from Eqs. (7.4.8) and (7.4.17) that the stress intensity ~ is calculated as
~,(t, r) = 2Ga(t)r, ]£(t,r) = 22°,
0 <- r
<--
Cl(t),
(7.4.52)
Cl(t) -- r -- c2(t),
~(t, r) = 2G[a(t) - a(~'*(r))]r,
(7.4.51)
c2(t) <- r - a2.
(7.4.53)
Equation (7.4.51) together with Eq. (7.4.45) implies that
E(t,r) = "Z° r
0 <- r <- Cl(t),
c,(t)'
(7.4.54)
which means that for any loading program
E(t,r) < Eo,
0 -< r < Cl(t).
To transform Eq. (7.4.53), we differentiate it with respect to r and employ Eqs. (7.4.23), (7.4.27), and (7.4.29). As a result, we obtain, similar to Eq. (7.4.28), 0~ (t, r)
Or
1 [~(t, r)
r
F(r)],
c2(t) -- r -< a2.
(7.4.55)
508
Chapter 7. Accretion of Viscoelastic Media with Small Strains
It follows from Eq. (7.4.52) that £(t, c2(t)) = ]~0. Integration of Eq. (7.4.55) with this initial condition leads to the formula £(t,r) =
r £o _ c2(t) c2(t)
F(O)O -2 do ,
c2(t) -< r -< a2.
(7.4.56)
(t)
Equation (7.4.42) implies that a plastic region arises in an accreted cylinder if the function F(r) exceeds the yield stress ~0 in some interval [al, b], where F(b) = ~0.
(7.4.57)
b -> a2,
(7.4.58)
First, let us suppose that
which means that F(r) > £0 in the entire interval [al, a2]. Equation (7.4.56) implies that for any r > c2(t) £(t, r ) <
£ ° r 1 -- C2(t) f~[ d ; l __ ]~0. C2(t) (t)
According to this inequality, under the assumption (7.4.58), the plastic region monotonically grows. At some instant r {1), its external radius c2 coincides with radius of the cylinder a2. At that instant, the external elastic region disappears. Afterward, deformation of an accreted elastoplastic cylinder coincides with deformation of an elastoplastic cylinder with a fixed radius a2. We assume now that b < a2,
(7.4.59)
i.e., that the interval [al, b], where F(r) >-- £0, is less than the entire interval [al, a2]. Our purpose now is to prove that the stress intensity £ on the external boundary of the cylinder r = a2 becomes equal to £ 0 before the external boundary of the plastic region c2(t) reaches the point r = b. Let us suppose that this assertion is false, i.e., that at some instant r (2) we have c2(r (2)) = b, while £(r(2), a2) < £o.
(7.4.60)
Since F(r) < £o in the interval [b, a2], it follows from Eq. (7.4.56) that ]~(r (2), a2) = --~ ~0 _ b
/a2
> £°a2 b ( 1-b
F(r)r -2 dr
]
f adr 7g ) =£0,
which contradicts inequality (7.4.60). Therefore, under condition (7.4.59), another plastic region arises at instant r (2) on the boundary surface of the cylinder.
7.4. Torsion of an Accreted Elastoplastic Cylinder
7.4.5
509
An Elastoplastic Cylinder with Two Plastic Regions
Let us study deformation of an elastoplastic cylinder with two plastic regions that occupy the domains Cl (t) -< r ~< c2(t) and c3(t) <- r -< a2. It is easy to show that the unknown functions Cl (t), c2(t), c3(t), and og(t) satisfy the following equations similar to Eqs. (7.4.47), (7.4.49), and (7.4.50):
dog dt
~(t)
=
~C[c4(t)-
c~(t) + c4(t)] '
dCl 2Gc~(t ) dog --d~(t), dt ( t ) = - E0
de2 2G c22(t)dog dt (t)= F(c2(t)) - ~o ---~(t), dc3 2G c~(t) dog dt (t) = F(c3(t))- ~o --~(t).
(7.4.61)
The stress intensity E(t, r) is calculated by analogy with formulas (7.4.52), (7.4.54), and (7.4.56)
~(t,r) = ~° r
c~(t)'
E(t, r) = Eo,
r[
O <-- r <-- Cl(t), Cl(t) <-- r-----c2(t),
E(t, r) = c2it) E° -- c2(t)
(t)
c3(t)----- r ~ a 2 ,
F(p)p -2 do ,
c2(t) --< r --< c3(t). (7.4.62)
Let us show that for an arbitrary loading program M(t), the interfaces r = c2(t) and r = c3(t) reach the point r = b simultaneously at some instant ~.(3). To prove this assertion, we suppose that it is false and demonstrate that our assumption leads to a contradiction. Let us suppose that c2(~-(3)) = b at some instant ~.(3), but c3('r (3)) > b. According to Eq. (7.4.62), in this case 1 - b fb c3(,r(3)) F(r)r-2dr] ~('1"(3)' C3(T(3))) -- C3(7"(3)) b
> EOc3(T(3)) b
1 - b / c30"(3)) dr ] = E ° db
r2
'
which is impossible, since the stress intensity in an ideal elastoplastic solid cannot exceed the yield stress. Using similar reasoning, it can be shown that the assumptions c3('r (3)) -- b and c2(~"(3)) < b are false as well. Thus, under condition (7.4.59), at the third stage of deformation of an accreted elastoplastic cylinder, two plastic regions
510
Chapter 7. Accretion of Viscoelastic Media with Small Strains
grow and, finally, join at the point b. Further deformation of an accreted cylinder coincides with deformation of an elastoplastic cylinder with a fixed radius a2.
Concluding Remarks In the torsion problem for an ideal elastoplastic circular cylinder, an inert cylinder is purely elastic provided the torques are sufficiently small, and a plastic region arises on the boundary of the cylinder if the torques exceed some critical level. An increase in torques enlarges the plastic region until it occupies the entire cylinder. We extend this problem to an accreted elastoplastic cylinder. It is demonstrated that the effect of accretion leads to two new phenomena: 1. In a growing cylinder, a plastic region arises in an internal region, but not on the outer surface of the cylinder. 2. Two different scenarios exist for the development of the plastic region. According to the first, an increase in torques implies growth of the plastic region until it reaches the boundary surface of the cylinder. Afterward, the mechanical behavior of an accreted cylinder coincides with the behavior of an elastoplastic cylinder with a fixed radius. According to the other scenario, two plastic regions (internal and external) arise in the cylinder. With an increase in the torques, the plastic regions grow and, finally, joint at some instant. The choice of one of these scenarios is determined by inequalities (7.4.58) and (7.4.59).
Bibliography [ 1] Arutyunyan, N. K. and Drozdov, A. D. (1991). Contact problem for a growing dome. In: Contact Problems in Mechanics of Growing Solids, pp 57-66. Nauka, Moscow [in Russian]. [2] Drozdov, A. D. (1988). A problem of a growing spherical dome. Dokl. Akad. Nauk UkrSSR A6, 30-32 [in Russian]. [3] Drozdov, A. D. (1989). Contact problems of debonding for growing viscoelastic solids. Mech. Solids 24(3), 95-101. [4] Drozdov, A. D. and Gertsbakh, I. (1993). Detachment of an elastic beam from a viscoelastic support: a variational approach. Int. J. Mech. Sci. 35, 463-478. [5] Guth, E., Wack, P. E., and Anthony, R. L. (1946). Significance of the equation for state for rubber. J. Appl. Phys. 17, 347-351. [6] Novozhilov, V. V. (1959) The Theory of Thin Shells. E Noordhoff, Groningen. [7] Rabotnov, Y. N. (1948). Some problems in the creep theory. Vestn. Mosc. Univ. 10, 81-91 [in Russian].
Chapter 8
Optimization Problems for Growing Viscoelastic Media This chapter is concerned with a new class of optimization problems in the mechanics of growing viscoelastic media subjected to aging. These problems are characterized by the following features: • Owing to mass supply to an accretion surface, the size of a growing medium increases in time. • Preloading in built-up portions causes residual stresses even when external forces are absent. Sections 8.1 and 8.2 are concerned with an optimal rate of accretion for elastic and viscoelastic solids. It is shown that the optimal rate of material supply is a stepwise function of time with one point of discontinuity provided external loads increase monotonically. For nonmonotonic forces, the optimal regime of accretion can have two points of discontinuity. Optimal preload distribution is analyzed for a wound cylindrical pressure vessel in Section 8.3. In Section 8.4, an optimal shape is designed that minimizes the maximal deflection of a growing reinforced beam. Section 8.5 deals with an optimal rate of cooling for a polymeric spherical pressure vessel.
8.1
An Optimal Rate of Accretion for Viscoelastic Solids
An optimal regime of accretion without preloading is analyzed for elastic and viscoelastic media at small and finite strains. We begin with accretion of a linear elastic solid with infinitesimal strains and show that the optimal regime of material supply 511
512
Chapter 8. Optimization Problems for Growing Viscoelastic Media
is a stepwise function with one point of discontinuity. The optimization problem is generalized by accounting for material viscoelasticity and geometrical nonlinearity. Explicit restrictions on the aging function and external loads are derived that ensure that the stepwise rate of accretion would minimize the cost functional. The exposition follows Arutyunyan and Drozdov (1988a). 8.1.1
Torsion of an Accreted Viscoelastic Cylinder with Small Strains
Let us consider a circular cylinder with length I and radius al. At the initial instant t = 0, torques M = M(t) are applied to the edges of the cylinder, and continuous accretion of material begins on its traction-free lateral surface. Body forces are absent, and the torques M(t) change in time so slowly that inertia forces may be neglected. Owing to material supply in the interval [0, T], the radius of the growing cylinder a increases in time according to the law
a = a(t),
a(O) = al,
a(T) = a2,
(8.1.1)
where al and a2 are given parameters. Volume V(t) of the accreted cylinder is calculated as
V(t) = rra2(t)l.
(8.1.2)
Differentiation of Eq. (8.1.2) with respect to time implies the rate of material supply
v(t) = _dT(t)dV = 27rla(t)~t (t).
(8.1.3)
We suppose that the function v(t) is bounded
0
~
V1 ~
v(t)<- v2 < ~,
(8.1.4)
where V 1 and V2 are the minimal and maximal rates of accretion. Integration of Eq. (8.1.3) from 0 to t implies that
lf0t
a2(t) = a 2 + -~
v(s) ds.
(8.1.5)
Combining Eq. (8.1.5) with the third equality in Eq. (8.1.1), we find that
fo ~ v(t)dt = 7r(a 2 - a~)l.
(8.1.6)
Points of the cylinder refer to cylindrical coordinates {r, 0, z} with unit vectors 6'r, ~0, and ~z. At instant t, the accreted cylinder occupies in the reference configuration the domain ~°(t)={O<--r--
0--0<27r,
O --< z --< l}.
513
8.1. A n Optimal Rate of Accretion for Viscoelastic Solids
Within the interval [t,t + dt], a cylindrical shell that occupies in the reference configuration the domain d l 2 ° ( t ) = {a(t) <- r <- a(t + dt),
0<- 0<27r,
O <- z <- l}
merges with the growing body. The instant ~'*(r) when an element with polar radius r joins the accreted cylinder equals ~*(r) = ( 0 , t,
O-< r --
(8.1.7)
The displacement vector fi for transition from the reference to actual configuration equals f i - - c~(t)rz~o,
(8.1.8)
where c~ = c~(t) is a function to be found (the twist angle per unit length). Let 5 be the strain tensor for transition from the reference to actual configuration. According to Eq. (8.1.8), the first invariant E of the tensor 5 vanishes, and the nonzero components of the deviatoric part 0 of the strain tensor are calculated as 1
eoz = ezo = -~a(t)r.
(8.1.9)
We confine ourselves to accretion without preloading. The latter means that the natural (stress-free) configuration of the initial cylinder coincides with its reference configuration, whereas the natural configuration of a built-up portion with polar radius r coincides with the actual configuration on the accretion surface of the growing cylinder. The displacement vector for transition from the reference to natural configuration ~* is written as u
= ~ (r)rzeo,
(8.1.10)
where f0, • ct*(r) = [c~('r (r)),
0 - r -< al, al < r --< a2.
(8.1.11)
The first invariant e* of the strain tensor 5" for transition from the initial to the actual configuration vanishes. The nonzero components of the deviatoric part 0* of the tensor 5" equal 1
eoz = ezo = -~a ( r ) r .
(8.1.12)
5° = 5 - 5"
(8.1.13)
Denote by
the strain tensor for transition from the natural to actual configuration, and by E° and ~0 its first invariant and the deviatoric part. In follows from Eqs. (8.1.9), (8.1.12), and
514
Chapter 8. Optimization Problems for Growing Viscoelastic Media
(8.1.13) that e ° = 0,
(8.1.14)
and the only nonzero components of the tensor ~0 are calculated as 0 1 e°z = ezo = ~ [ a ( t ) - c~*(r)]r.
(8.1.15)
The material behavior of an incompressible linear viscoelastic material obeys the constitutive equations (2.2.72)
6"(t) = - p ( t ) l + 2 {G ( t - ~'*)~'°(t)- f t. ~a [G(~'-~'*) + Q(t-~'* , ~'-~-*)]~°(~-)d~-) , (8.1.16) where & is the stress tensor, I is the unit tensor, p is pressure, G(t) is the current shear modulus, and Q(t, T) is the relaxation measure. We assume that the function G(t) is positive, increases monotonically in time, and tends to some limiting value as t ---*
0 < G(t) < G(oo),
dG
--~(t) > 0.
(8.1.17)
We confine ourselves to the Arutyunyan relaxation measures [cf. Eq. (2.3.29)]
Q(t, ~) = -~b(~')[1 - e x p ( - 7 ( t - ~'))],
(8.1.18)
where 3/is a positive constant, and th(t) is an aging function, which satisfies the conditions
0 <- dp(t) <_ G(t),
,/4, (t) < 0. dt
(8.1.19)
Expression (8.1.18) adequately describes the stress relaxation in aging polymers,
concrete, and soils [see Arutyunyan et al. (1987)]. Substituting expressions (8.1.14) and (8.1.15) into Eq. (8.1.16), we find the nonzero components of the stress tensor & Oroz(t, r) = Orzo(t, r) = r{G(t - T*(r))[a(t) - a*(r)] f l (r> O-~r[G(~"0 ~'*(r)) + Q(t - "r*(r), ~"- T*(r))] × [ a ( l " ) - c~*(r)] d~'}.
(8.1.20)
Expressions (8.1.20) satisfy the equilibrium equations and boundary conditions on the lateral surface of the cylinder. Boundary conditions on the edges are written in the integral form
M(t) = 27r
~0"a(t)~roz(t,r)r 2 dr.
(8.1.21)
515
8.1. An Optimal Rate of Accretion for Viscoelastic Solids
Substituting expression (8.1.20) into Eq. (8.1.21) and using Eq. (8.1.11), we obtain M(t) 2rr
-
foal{ G ( t ) a ( t ) - fOt03~r[G(r) +
a(t){G(t -
Jal
C(r))[a(t)-
+ Q(t - r*(r), r -
+ Q(t, r ) ] a ( r ) d r
a(C(r))] -
} r 3 dr
f l -ff-rr[G(r03 r*(r)) (r)
r*(r))][a(r) - a(r*(r))]dr}r 3 dr.
We calculate the first integral in the right-hand side of this equality and introduce the new variable s = r*(r) in the other integral. After simple algebra, we arrive at the integral equation M(t) 27r
{ G ( t ) a ( t ) - f0t ~rr03[G(r) + Q(t, r ) ] a ( r ) d r / "t~-rr[G(r03 + jf0t{G(t - s)[a(t) - a(s)] - ~S s)
a4
4
+ Q(t - s, r -
s)][a(r) - a(s)]dr
da a3(s)--dT(S)ds.
(8.1.22)
We suppose that the torque M(t) satisfies the conditions M(0) > 0,
dM
~(t) dt
>- 0.
(8.1.23)
The optimization problem consists in determining such a function v ° (t) which (i) is piecewise continuous in the interval [0, T], (ii) satisfies restrictions (8.1.4) and (8.1.6), (iii) minimizes the twist angle = a(T)
(8.1.24)
at the final instant T. It is assumed that VlT < 7 r ( a ~ - a2)l < v2T.
(8.1.25)
According to Eqs. (8.1.4) and (8.1.6), this means that the set of admissible rates of accretion is not empty. Our objective is to derive an explicit formula for the optimal rate of accretion v ° ( t ) under some hypotheses regarding the material behavior (physical linearity and the absence of aftereffect). Afterward, we demonstrate that the optimal rate of accretion preserves its structure when these assumptions are not fulfilled, but the material viscosity and the nonlinearity of response are not very large.
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
516
We consider two particular cases: an aging elastic medium (2.2.12) and an aging viscoelastic medium with a constant shear modulus and the relaxation measure (8.1.18). For an aging elastic medium with
G = G(t),
Q(t, ~-) = 0,
(8.1.26)
Eq. (8.1.22) is equivalent to the ordinary differential equation
a4 + fo t G ( t - s)a3(s) ~ s s(s)ds l d a --~-(t)= --~-(t) dM 27r [a(t)--~
(8.1.27)
with the initial condition 2M(0)
a(O) = 7rG(O)a4.
(8.1.28)
It follows from Eqs. (8.1.2) and (8.1.3) that Eq. (8.1.27) can be presented in the form
d---[(t) = --~-(t)
aeG(t) + ~
'/o
a(t - s)V(s)v(s)ds
l'
.
(8.1.29)
For an aging viscoelastic medium with a time-independent shear modulus G and the relaxation measure (8.1.18), Eq. (8.1.22) implies that
M(t) _ a4 a(t) +
27rG
4
{
+
/o'{
/oo
[~b(~')(1 - e x p ( - v ( t - ~-)))]a(z)d~"
[a(t) - a(s)] +
× [a('r)-
}
Js
G(t) =
}
~-~r[~b(~"- s)(1 - e x p ( - y ( t - 1")))]
a(s)]d~" a3(s)--~s(S)ds.
(8.1.30)
We differentiate Eq. (8.1.30) twice and exclude the integral terms. After simple but tedious calculations, we arrive at the differential equation
a4(t) ddp - s)a4(s)dsJ } ---~-(t) da 4 d2a dt 2 (t) + { a3(t) ~tt (t) + ~7 [ae(t)(1-dp(O))- fo t ---~-(t 1
d2M
dM
]
- 27rG --~-(t) + 7--~-(t)J
(8.1.31)
with the initial conditions (8.1.28) and
da 2 4 [dM d---~-(0) - 7rGa --~-(0) + 7~b(0)M(0) ] . It is convenient to introduce the new variables yl(t) =
a(t),
da y2(t) = ---;7(t), at
(8.1.32)
517
8.1. An Optimal Rate of Accretion for Viscoelastic Solids
and to substitute expressions (8.1.2) and (8.1.3) into Eq. (8.1.31). As a result, we obtain
dyl (t) = y2(t), dt
[
dy2dt(t) = -3/ 1 - dp(O) + 7V(t----~-
( t - s) \ V(t)
dM [ d2M + v--dT-(t GV2(t) [--~-(t)
1
ds y2(t)
2,n'/2
yl(O) =
,
2 [dM
2M(O)
y2(O) - 7rGa4 --~-(0) + 7qb(O)M(O)
7rG(O)a41'
(8.1.33)
Optimal Accretion of an Aging Elastic Cylinder We begin with optimization of the accretion process for an aging elastic cylinder. Denote by A v(t) an admissible increment of the rate of accretion, and by A V(t) and A a(t) the corresponding increments of the volume and the twist angle. It follows from Eqs. (8.1.3) and (8.1.29) that the functions A V(t) and A a(t) satisfy the equations
dAY ~(t) dt
= A v(t),
dAa 1 dM --(t) d----~tt)'" - 7rl2 dt × A V(0) = 0,
~0t G(t -
a4G(t) + - ~
G ( t - s)V(s)v(s)ds
s)[v(s)A V(s) + V(s)A v(s)] ds,
Aa(0) = 0.
(8.1.34)
We integrate the second equality in Eq. (8.1.34) from 0 to T. Using Eq. (8.1.24), we write
1 r dM 7ra4G(t) + l f o t G(t - ~')V(~')v(~')d T] -2 dt [ (t ) -~ A ~ = 7rl2 f o --d~ -~ t
X
fo
G(t - s)[v(s)A V(s) + V(s)A v(s)] ds
l 2 for [v(t)A V(t) + V(t)A v(t)] dt ft T G(s - t)---~s dM (S) "n'l × [2a4G(s) + ~
lfoS
G(s - ~)V(T)v(~) dT
]2
ds.
(8.1.35)
518
Chapter 8. Optimization Problemsfor Growing Viscoelastic Media
Let q~(t) be a continuously differentiable function that satisfies the boundary condition ~(T) = 0.
(8.1.36)
The function qJ(t) will be chosen in the following. We multiply the first equality in Eq. (8.1.34) by q~(t) and integrate from 0 to T
r q~(t)ddtAV (t) dt =
qj(t)A v(t) dt.
We now integrate the left-hand side of this equality by parts and use boundary conditions (8.1.34) and (8.1.36). As a result, we find that --d-~(t)AV(t) + qj(t)A v(t) dt.
0 =
(8.1.37)
Summing up Eqs. (8.1.35) and (8.1.37), we obtain
AdO =
fOT A V(t) { --d-~(t) gilt + -v(t) 4 ) ~ ftT G ( s - t) ~_~Ms (s) ["//" -~alG(S f o T dt+ + ~ l f o S G(s - T)V(r)v(r) d ~ - 2 }ds
X(t)A v(t) dt, (8.1.38)
where
V(t) f T dM X(t) = q~(t) + ~ G ( s - t)--j~s (S)
Ea4Os,+1loS
× ~
-~
G(s - r)V(r)v(r) d
ds.
(8.1.39)
Suppose that the function q~(t) satisfies the equation
dd/ (t) = dt 7rl2
G(s -
ds
(s)
a4G(s)
+ ~ m fOs G(s - r)V(~')v(r) d ,r1-2 ds.
(8.1.4o)
Then the first integral in Eq. (8.1.38) vanishes, and we obtain
A~ =
X(t)A v(t) dt.
(8.1.41)
It follows from Eq. (8.1.41) that the optimal rate of accretion has the form •
o
f Vl,
X(t) > O, < O.
V (t) = 1 702, X(t)
(8.1.42)
8.1. An Optimal Rate of Accretion for Viscoelastic Solids
519
Indeed, Eq. (8.1.42) implies that for any admissible increment of the rate of accretion
Av(t), A v(t) >-- O,
X(t) > O,
A v(t) <-- O,
X(t) < O.
These inequalities together with Eq. (8.1.41) yield A~>__O, which means that the program of accretion (8.1.42) minimizes the functional ~ . Differentiation of Eq. (8.1.39) with respect to time t with the use of Eqs. (8.1.3) and (8.1.40) yields
d---~(t) = - ,n.l2
---~-(t)
a4G(t) + --~
G ( t - ~')V(r)v(~')d~"
+ f T d G-d-~s(S - t)dM --~-s(S) [Tra~G(s)+ -~ ~ 1 foos a(s - r)V(r)v(r) dr ]-2
ds} (8.1.43)
It follows from Eq. (8.1.43) and inequalities (8.1.17) and (8.1.23) that the function X(t) increases monotonically for any admissible accretion program and any admissible loading program. Therefore, three opportunities arise: (i) The function X(t) is positive in the entire interval [0, T]. (ii) There is a constant T O ~ (0, T) such that the function X(t) is negative in [0, T °) and is positive in (T °, T]. (iii) The function X(t) is negative in the interval [0, T]. It follows from Eq. (8.1.42) that in cases (i) and (iii) the optimal rate of accretion equals either Vl or v2 in the entire interval [0, T], which contradicts condition (8.1.25). Therefore, only the case (ii) occurs, and the optimal rate of material supply has the form
vO(t)
= {v2, 0 --< t -Vl,
T °, T o < t --< T.
(8.1.44)
The parameter T o is found from Eqs. (8.1.6) and (8.1.44)
TO = 1r(a~ - a2)l - VlT. V2 - -
(8.1.45)
Vl
Our purpose now is to demonstrate that Eq. (8.1.44) determines the optimal rate of accretion not only for torsion of an aging elastic cylinder with small strains, but also for accretion of viscoelastic solids with finite and small strains under other types
520
Chapter 8. Optimization Problems for Growing Viscoelastic Media
of loads provided that the loading is monotonic in time, and the material viscosity and the load intensity are not very large. Optimal Accretion of a Viscoelastic Cylinder Let us consider torsion of a viscoelastic cylinder with a constant shear modulus and the relaxation measure (8.1.18) under the action of a time-independent torque M. According to Eqs. (8.1.33), the optimal rate of loading v ° (t) minimizes the functional
y2(t) dt
q~ =
(8.1.46)
on solutions of the differential equations
dYZd__t(t) : - 7 I 1 - dp(O) + 7 V ( t ) 2- v ( t ) l f o t d d--d-f(t ds yE(t), ~ ] V E- ( ts)VE(s) ) dV ~(t) dt
= v(t),
y2(0) =
27~b(0)M zrGa 4 ,
V(O) = 7ra21,
V(T) = 7ra21.
(8.1.47)
Denote by A v(t) an admissible increment of the rate of accretion, by h y2(t) and A V(t) the increments of the functions y2(t) and V(t), and by AcP the increment of the functional (8.1.46) caused by the increment of the accretion rate. It follows from Eqs. (8.1.47) that the functions A V(t) and Ay2(t) satisfy the equations
dAV ~(t) dt
= A v(t),
dAy-------~2(t)= - 2 Av(t) - A V ( t ) v ( t ) + 7 dt V(t) V(t----) V2(t)
--~
(t - s)V2(s)ds
- t d"Yd p vfo( t ) ---~(t - s)V(s)A V(s)ds 1 yE(t) - 3/ [ 1 - th(0) + TV(t)2v(t) - VE(t)l foot -dT(t d4~ - s)V2(s)ds ] AyE(t), A V(0) = 0,
A V(T) = 0,
AyE(0) = 0.
(8.1.48)
f0 T A yE(t) dt.
(8.1.49)
Equation (8.1.46) implies that
A~ =
Let ~t1(t) and qJE(t) be continuously differentiable functions, which will be determined later. We multiply the first equality in Eq. (8.1.48) by qq (t), the other equality by q~2(t), integrate from 0 to T, and add the obtained results to Eq. (8.1.49). Integrating
8.1. An Optimal Rate of Accretion for Viscoelastic Solids
521
by parts, we find that A~ =
Jo {
Ay2(t )
+
+7-
d~l (t)A V(t) + ~1 (t)A v(t) - d-~~t2(t)A yz(t)
2q~z(t)y2(t) A v(t) - A V(t) v(t) + 7 A V(t) fot dth (t - s) V2 (s) ds V(t) V(t----~ VZ(t) --~
]
_ vV(t) fot - g2(t)l
2v(t) (t - s)V(s)A V(s)ds + TtO2(t)Ayz(t ) 1 - ~b(0) + "yV(t)
ft
ddp - s) V2 (s)ds Jlldt + d/z(T)A yz(T). ---~(t
(8.1.50)
We chose the function ~2(t) from the condition that all the terms proportional to A yz(t) vanish. As a result, we find that d~2(t) = 1 + 3, [ 1 - dp(O) + 7V(t)2v(t) -
l
dt
---~-(t- s) V2(s) ds 1 ~2(t),
~2(T) -- 0.
(8.1.51)
Equation (8.1.50) is presented in the form A~ =
fo T X(t)A v(t) dt 2d/z(t)yz(t)v2(t) [v(t)
~' ft ~t (t _ s)V2(s)dsl} A V(t) dt
v(t)
ddp - s)V(s)A V(s) ds, - 23' fo r d/z(t)yz(t) VZ(t ) dt ~o"t --d-f(t
(8.1.52)
where
X(t) = q'l(t) + 2
~2(t)y2(t)
v(t)
.
(8.1.53)
We change the order of integration in Eq. (8.1.52) and choose the function qtl(t ) from the condition that all the terms proportional to A V(t) vanish. We arrive at the differential equation
[v(t) d~l (t) = 2~2(t)y2(t) V2
d-7
+ 2vV(t)
7 fot ddp (t - s)VZ(s) ds ]
-37
r --T-(sdch ~2(s)y2(s)ds. t) VZ(s)
ft as
(8.1.54)
Chapter 8. Optimization Problems for Growing Viscoelastic Media
522
Equations (8.1.52) and (8.1.54) imply that the increment of the functional ~ is calculated as T
Adp =
fO
X(t)A v(t) dt.
(8.1.55)
It follows from Eq. (8.1.55) that the optimal rate of accretion has the form (8.1.44) provided the function X(t) increases monotonically for an arbitrary function v(t). Differentiation of Eq. (8.1.53) with respect to time t with the use of Eqs. (8.1.47), (8.1.51), and (8.1.54) yields
dX 2 [ fotd~ (V(s)) dt(t) = ~ - ~ ye(t) + 7Y(t) -d-t-(t- s) ~ - ~ - 3/
---~s(S- t)Y(s) \ ~
2
ds (8.1.56)
cls ,
where
Y (t) = - y2(t)qt2(t ).
(8.1.57)
According to Eqs. (8.1.47), (8.1.51), and (8.1.57), the function Y(t) satisfies the equation
dY = -y2(t), rd --7(t)
Y(T) = O,
which implies that
Y(t) =
f
T
(8.~.58)
y2(s) ds = a(T) - a(t).
It is convenient to rewrite the first equality in Eq. (8.1.47) as follows:
dy2 d--T(t) = -TF(t)y2(t),
y2(0) =
27~b(0)M 7rGa----------~l,
(8.1.59)
where
2v(t) fot~t (V(s)) 2 ( t - s) ds F(t) = 1 - th(0) + 7V(t) k v(t)
d4~ 2v(t) + f0 t -di-(t - s) = 1 -- th(0) + 7V(t)
V(s)
2
ds.
(8.1.60)
Bearing in mind inequalities (8.1.4) and (8.1.19), we find that
2v2 O <-- F(t) <-- 1 - ~(0) + V(O) = 1 - ~b(t) +
2v2
v(o)
fot -~s o+ (t <-- 1 - dp(T) +
s) ds
2~ v(o)
(8.1.61)
8.1. An Optimal Rate of Accretionfor ViscoelasticSolids
523
It follows from Eqs. (8.1.59) and (8.1.61) that
y2(O)exp{-TT[1-dp(T)+
2v2 ]} <-y2(t)<--y2(O).
yv(o)
(8.1.62)
Combining Eqs. (8.1.58) and (8.1.62), we obtain
0 <- Y(t) <- y2(O)T.
(8.1.63)
It follows from Eqs. (8.1.19), (8.1.62), and (8.1.63) that the first and third terms in brackets in Eq. (8.1.56) are positive, whereas the second term is negative. According to Eqs. (8.1.3), (8.1.4), and (8.1.63), the absolute value of the second term does not exceed (8.1.64)
T T [ 6 ( 0 ) - ~b(T)lY2(0).
Substitution of expressions (8.1.62) to (8.1.64) into Eq. (8.1.56) implies that the function X(t) increases monotonically provided that TT[~b(0)- ~b(T)] < exp { - T T [ 1 - ~b(T)+
2v2
yv(0)
.
(8.1.65)
This inequality means that the optimal rate of accretion of a viscoelastic circular cylinder has the form (8.1.44) if the material aging obeys Eq. (8.1.65).
8.1.2
Extension of an Accreted Elastic Bar with Finite Strains
We now demonstrate that the stepwise rate of accretion (8.1.44) remains optimal if the behavior of an accreted medium obeys nonlinear governing equations with finite strains. Since the governing equation for torsion of a growing viscoelastic cylinder with finite strains is linear [see Eq. (6.1.57)] and we intend to analyze the structure of an optimal rate of material supply for nonlinear systems, we consider continuous accretion of a neo-Hookean elastic cylinder under tension. The kinematics of uniaxial extension is determined by the only parameter, the extension ratio a. According to Eqs. (6.1.107) and (6.1.112), the functions
yl(t) = a-2(t),
y2(t) = a 2 + 2
t
fo
da
a(~)a(T)~-~r(T)d~,
(8.1.66)
expressed in terms of the extension ratio a, satisfy the following nonlinear ordinary differential equations:
dyld__F(t) = _ 2y17r/x(t__~)I_P(t)Trlx+ 3yl (t)y2(t) dy2 v(t) dt (t)= 7rly~/2(t)
]-1
dP
-~(t), (8.1.67)
Chapter 8. Optimization Problemsfor Growing Viscoelastic Media
524
with the initial conditions yl(O) = 1,
y2(O) = a 2.
(8.1.68)
Here a(t) is radius of the growing cylinder at instant t and/z is the shear modulus. The tensile force P(t) is assumed to increase monotonically in time
0 = P(O) <-- P(t) <-- P(T) = P1,
dP ~ ( t ) > O. dt
(8.1.69)
Our objective is to find a function v ° (t) piecewise continuous in the interval [0, T], which satisfies restrictions (8.1.4) and (8.1.6) and minimizes the extension ratio at the final instant a(T). According to Eq. (8.1.66), this problem is equivalent to maximizing the functional
dp = yl(T).
(8.1.70)
Let us calculate the increment A~ of the functional • caused by an admissible increment A v of the function v(t). For this purpose, we derive equations for the increments A yl(t) and Ay2(t) of the functions yl(t) and y2(t). It follows from Eqs. (8.1.67) and (8.1.68) that
dAy1 ~ ( t )
dt
-
2 dP (t) I[P(t)'trlx + 3yl (t)y2(t)] -2 {3yl (t)[A Yl (t)y2(t) + Yl (t)A y2(t)] rrlz dt _ [ P(t) + 3yl (t)y2(t)] Ayl(t)}, I. 7r/x
(8.1.71)
J
d A Y2 1 A v(t) - Ayl(t) v(t) ] dt (t)= 7rly~/2(t) ~Yl(--~ ' Ayl(0 ) = 0,
(8.1.72)
Ay2(0) = 0.
(8.1.73)
According to Eq. (8.1.3), the increment A V(t) of the volume V(t) satisfies the equation
dAV ~(t) dt
= A v(t),
A V(O) = O,
A V(T) = 0.
(8.1.74)
Equation (8.1.70) implies that A~ = Ayl(T ).
(8.1.75)
We multiply Eq. (8.1.71) by tPl(t), Eq. (8.1.72) by ~2(t), and Eq. (8.1.74) by ~3(t), where ~1 (t), ~2(t), and qJa(t) are functions to be determined. The obtained equalities are integrated from 0 to T and added to Eq. (8.1.75). As a result, we find that
A ~ = A Yl (T) +
~0"T
I[I1(t)
( d A y 1 (t) . . . . 2 dP (t) P(t)Tr~ + 3yl(t)y2(t)] dt 7rl~ dt
-2
8.1. An Optimal Rate of Accretionfor ViscoelasticSolids × { 3yl(t)[Ayl(t)yz(t)+yl(t)Aye(t)]-[P(t)+3yl(t)yz(t)I 7xif
+
f0 T
qJe(t)
Ayl(t)})dt
1 { dAy 2 [Av(t) - Ayl(t) v(t) ] } dt dt ( t ) - 7rly~/Z(t) -~Yl(-~
~3(t) [ -dt ( t ) - A v ( t )
+
525
dt.
Integration by parts with the use of boundary conditions (8.1.73) and (8.1.74) implies that
foT[Xl(t)A yl(t) + X2(t)A y2(t) + X3(t)A V(t) + X(t)Av(t)] dt
Adp = -
+ [1 + djI(T)]Ayl(T) + qJe(T)Ay2(T),
(8.1.76)
where
d~_~t [P(t) ] -2 v(t) d/z(t) Xl(t) = ------[1(t) - 2~tl(t)'rr/xP(t),rr~dPdt(t) 7rlx + 3yl(t)yz(t) - 21rl yffZ(t)' X2(t) = ~tt" (t) +
6qtl (t)y2(t) dP (t) l[P(t)'tr/x+ 3yl(t)yz(t)] -2 ~rlx dt
X3(t) = dd-~~t3-(t), qt2(t) X(t) = 7rly~/2(t) + q13(t).
(8.1.77)
We choose the functions ~(t) in such a way that the terms with A yl, A y2, and A V in Eq. (8.1.76) vanish. It follows from Eqs. (8.1.76) and (8.1.77) that the functions ~k(t) satisfy the ordinary differential equations dqtl ~(t) dt
dt
dqJ3 dt
~(t)
(t)
-
=
-2 v(t) ~2(t) 2t)l (t) P(t) dP (t) P(t)Trl~ + 3yl(t)y2(t) + 27ri y~/2(t) ' 7rtx 7rlx dt _
6d/l(t)y2(t) dP --(t) L[P(t)rr/x+ 3yl (t)yz(t) 1 -e 7fix dt (8.1.78)
= 0
with the boundary conditions qJl(T) = - 1,
tOz(T) = 0.
(8.1.79)
526
Chapter 8. Optimization Problems for Growing Viscoelastic Media
The increment of the functional ~ equals Adp = -
f0TX(t)A v(t) dt.
(8.1.80)
Equation (8.1.80) implies that the optimal rate of accretion v ° (t) that maximizes the functional ~ , i.e., that ensures that A~ _< 0 for any admissible increment A v(t), is determined by Eq. (8.1.42). It follows from Eq. (8.1.42) that the optimal rate of accretion has the form (8.1.44) provided the function X(t) increases monotonically in time dX ~(t) dt
_ O.
(8.1.81)
Differentiation of Eq. (8.1.77) with the use of Eqs. (8.1.67), (8.1.77), and (8.1.78) implies that
[P(t)
dX 1 dP ]-1 d-T(t) = 7r2~ly~/2(t) d-T(t) 1 7rtx + 3yl(t)y2(t)
-
+ 3yl(t)yz(t) k ~'tx
1-1} .
(8.1.82)
Comparing Eq. (8.1.82) with Eq. (8.1.81) and using Eq. (8.1.69), we find that the optimal rate of accretion has the form (8.1.44) provided that qlz(t) - 6y~(t)O1(t)
[
P(t) "rrtx + 3yl(t)yz(t)
1-1
>-- O.
(8.1.83)
It follows from Eq. (8.1.83) that at finite strains, a monotonic growth of the load P(t) is not sufficient for the accretion rate (8.1.44) to be optimal, and more sophisticated restrictions should be imposed on the load. We derive appropriate estimates under the assumption that
PI<~3 7r/aa2.
(8.1.84)
Obviously, the extension ratio a exceeds unity for any tensile load P(t) >- 0 that satisfies inequalities (8.1.69). According to Eqs. (8.1.66), this means that for any t ~ [0, T], 0 < yl(t) --< 1,
a 2 <-- a2(t) -- y2(t) < ~.
(8.1.85)
It follows from Eqs. (8.1.69) and (8.1.85) that P(t) + 3yl(t)yz(t) "tr/~
]-1 <
1
1
< . -- 3yl (t)yz(t) -- 3a2yl (t)
(8.1.86)
8.1. An Optimal Rate of Accretionfor ViscoelasticSolids
527
Integration of the first equation in Eq. (8.1.67) with initial condition (8.1.68) implies that
Yl (t) = 1 -
2 f0 t dP Yl(~') [P('r) + 3yl('r)y2('r) ]-1 -d-}-('r)d'r'. ~r/~ k "tr/.z
Combining this equality with Eqs. (8.1.69), (8.1.85), and (8.1.86), we obtain
2 fot dP 2P(t) 2P1 yl(t)- 1 - 37rpa2 -d-~-(r)d~"= 1 - 37r/.ta2 --> 1 - 37r/m2. (8.1.87) We integrate the second equality in Eq. (8.1.67) from 0 to t and use Eqs. (8.1.4), (8.1.68), and (8.1.87). As a result we find that
y2(t)=a
2+
/ot ,trly~/2(,r) <--a~ [1+ ~
1-
2P1
(8.1.88)
According to Eq. (8.1.79), there is an interval [tl, T] such that the function ~l(t) is nonpositive
lit1(t) -- 0,
tl <-- t --< T.
(8.1.89)
It follows from Eqs. (8.1.69), (8.1.78), and (8.1.89) that the function q~2(t) increases monotonically in [h, T]. This assertion together with Eq. (8.1.79) leads to the estimate ~2(t) --< 0,
tl <-- t --< T.
(8.1.90)
Inequalities (8.1.4), (8.1.69), (8.1.89), and (8.1.90) together with the first equation in Eq. (8.1.78) imply that
dq,,
--(t)
dt
-----0,
tl -- t --< T.
(8.1.91)
Combining Eqs. (8.1.79), (8.1.89), and (8.1.91), we find that -
1 ----- ~tl(t ) --< 0,
(8.1.92)
tl --< t ----- T.
We integrate the second equality in Eq. (8.1.78) from t to T, use boundary condition (8.1.79), and obtain q~z(t) =
'rr~6 f
r y2('r)~ 1(7") L 7r/x
-2 d P
+ 3yl ('r)yz('r)
-d-~-(~') d~-.
(8.1.93)
It follows from Eqs. (8.1.85), (8.1.86), and (8.1.92) that for any t E [tl, T],
2
Iq,2(t)l <-- 37rtm4
2 2P1 -d-~-(l-)dl- - 37r/xa4 [P(T) - P(t)] -- 37r/xa 4 .
ft r dP
(8.1.94)
528
Chapter 8. Optimization Problems for Growing Viscoelastic Media
Equations (8.1.90) and (8.1.94) yield 2P1 37r/_ta4 -- tO2(t) --< 0,
tl --< t --< T.
(8.1.95)
The first equation in Eq. (8.1.78) together with Eq. (8.1.79) can be presented as follows: dt~l --(t) dt
~(T) = -1,
= A(t)~l(t) + B(t),
where
[
2P(t) dP (t) P(t) + 3yl(t)y2(t) I 7fix A(t) - (Tr/x)2 dt
]2
(8.1.96)
B(t) = v(t)~z(t) 27rly~/2(t) "
,
(8.1.97) The solution of Eq. (8.1.96) reads I/t 1 (t) =
-O
1 (t)
(8.1.98)
+ ®2(t),
where
[/t
t91(t) = exp -
]
O2(t) = - f T
A(I") dl" ,
B(T)exp[- f~a(s)dsl dT. (8.1.99)
Combining Eqs. (8.1.86) and (8.1.87), we obtain
[
P(t) + 3yl(t)y2(t) 7rtx
]1 1E
<--3- ~ 1 -
2P(t) 1 -1 37r/.ta2 •
It follows from this inequality and Eq. (8.1.97) that 0 <- a(t) <-
2P(t)
2P(t)
dP (t) (3~u~2) 2 dt
_
-2
(8.1.100)
_
Integration of Eq. (8.1.100) from t to T yields 0 -----
A(I-) dr -<
2P ae~t) (3~r~2) 2
f0 pl ( 3 ~2P ( ___ ~1) ~ 1-
-2
3~~) 2P
1
3 7rtm 2
) -2
dP
d P = fo e'(3~rua~)-~ 2ydy
(1 - 2y) 2
8.1. An Optimal Rate of Accretion for Viscoelastic Solids 1 ln(1-2y)+ 2
( =ln
529
y=P1(3"n'/.~21)-1 y=0
1 1 - 2y
2Pl ) 1/2 P1 ( 2P1 3~~ +37r/xa~ 1 - 37r~a 2 )
1
-1 (8.1.101)
Equations (8.1.99) and (8.1.101) imply that for any t ~ [0, T],
l~)l(t) -> I 1
2Pl I-1/2exp I- 31rpa Pl 2 I 1 -
31r/xa 2
2Pl /-1]
3~r/xa 2
"
(8.1.102)
To estimate the function 192(t), we employ Eqs. (8.1.97), (8.1.99), (8.1.100) and find that for any t E [tl, T], 0 --< (92(t) --< -
jT
B(~') d~" = - ~27rl
y ~--F-(-~ d ~'.
We use inequalities (8.1.4), (8.1.87), and (8.1.95) to obtain
v2T P1 ( 0 --< O2(t) --< ,tra2113"trga~ 1
2P1 ) 3/2 37r/.ta 2
(8.1.103)
Let us suppose that
/1
2Pl /
I
Pl / 1 - 3 7 r 2Pl /-11 /xa ~
37r/.tal2 exp-37rPa12
v2T Pl
> "tra~137rpzt~" (8.1.104)
It follows from Eqs. (8.1.102) to (8.1.104) that Ol(t) > 02(t) for any t E [tl, T]. The latter together with Eq. (8.1.98) implies that tOl(t) < 0,
tl <-- t --< T.
(8.1.105)
Equations (8.1.89) and (8.1.105) mean that the function ~l(t) is negative in the entire interval [0, T]. We now return to inequality (8.1.83). It follows from Eqs. (8.1.69), (8.1.85), and (8.1.88) that for any t E [0, T],
I
P(t) + 3yl (t)y2(t) 7rtx
]11 [ >--
3a~
1+
,, + 37rl.ta~ ~
(1 -
2,1,1,2]1
37rtxa21
530
Chapter 8. Optimization Problems for Growing Viscoelastic Media
This inequality together with Eq. (8.1.87) implies that
6y l,,, r"`t, + L "n'tx
2 ( 281 )2 [ -- a2
1-
3,rr/.m2
el v2T(
1 + 3,n./.m2 + ~
281 /-1/2]-1 1-
3,n./.ta2
. (8.1.106)
It follows from Eqs. (8.1.95), (8.1.98), (8.1.102), (8.1.103), and (8.1.106) that inequality (8.1.83) holds provided that
v2Z/ 2el /-1/2] / / [ / / -1]
P1 [ P1 3~'l.~a~ 1 + 31rga~ +~--~12/ 1 - 3 ~ 2
{/ X
2P1 1 - 37r/m~
P1 exp - 37r/.ta 2
-<
2P1 1 - 37r/.ta 2
1-3~
2el /1/2 } 2
_ vzT
P1
7raZl 37rtm--~1
"
(8.1.107) Equation (8.1.107) ensures that the optimal rate of accretion has the required form (8.1.44). This inequality is satisfied if the maximal tensile force P1 is sufficiently small and it can fail with the growth of P1. Therefore, formula (8.1.44) provides the optimal rate of accretion when external loads are not very large. To demonstrate the efficiency of the optimal law of accretion, we solve Eqs. (8.1.67) and (8.1.68) numerically for the rate of accretion (8.1.44) and for a constant rate of accretion ~. Introducing the dimensionless variables and parameters t t,
--
T'
a a,
TO T°
-
--
al _
vT
,
v,
VlT
T'
--
~,, Trait _
Yl, = Yl, v2T
l '
Y2,
Y2 --
~, ai
7JT l '
=
l '
we present Eqs. (8.1.67) and (8.1.68) as follows:
dyl, dt,
2yl, dP, P, + 3yl,y2, d t , '
1/2'
dy2,
v,
dt,
Yl,
y2,(0)-- 1.
yl,(0) = 1, (8.1.108)
Calculations are carded out for a cylindrical bar with a2 = 2al loaded by the tensile force P, - 6t,. We set Vl, = 0 and v2, = 5, which implies that ~, = 3 and T° = 0.6. The extension ratio ct is plotted versus the dimensionless time t, in Figure 8.1.1. The
8.1. An Optimal Rate of Accretion for Viscoelastic Solids
0
531
t,
1
Figure 8.1.1: The extension ratio ct versus the dimensionless time t, for an accreted bar under tension. Curve 1: optimal rate of accretion. Curve 2: accretion with a constant rate.
optimal regime of accretion allows the extension ratio a at the final instant T to be reduced by 20% compared to accretion with a constant rate of material supply.
Concluding Remarks A new optimization problem is formulated and solved, where the rate of material supply is chosen to minimize displacements and stresses under prescribed initial and final sizes of the body and pointwise restrictions on the rate of accretion. For an aging elastic body with small strains, the optimal rate of accretion is a stepwise function with one point of discontinuity [see Eq. (8.1.44)]. The optimal rate of accretion is maximal in the initial interval of time and it is minimal at the final stage of accretion. For aging viscoelastic media and for elastic media with finite strains, some explicit restrictions are derived that ensure that the stepwise func-
532
Chapter 8. Optimization Problemsfor Growing Viscoelastic Media
tion (8.1.44) remains the optimal rate of accretion. From the engineering standpoint, these restrictions mean that the material viscosity and the stress intensity should not be very large.
8.2
Optimal Accretion of an Elastic Column
An optimal regime of accretion is analyzed for a linear elastic column. Together with material supply to the column, some equipment is installed on its upper end. Weight of the equipment is the only force that causes axial deformation of the column. At any instant, this weight is unknown, but the final weight, as well as the maximal rate of lifting for the equipment are prescribed. The objective is to determine a rate of accretion that minimizes the maximal displacement of the upper end of the column at the final instant. We derive a solution to the optimization problem and demonstrate that two optimal regimes of accretion occur with one and two points of discontinuity. Explicit conditions are developed for each of the optimal regimes. The exposition follows Drozdov (1988).
8.2.1
F o r m u l a t i o n of t h e P r o b l e m a n d G o v e r n i n g E q u a t i o n s
We consider an elastic column with length I and a circular cross-section. The lower end of the column is fixed, and the upper end is free (see Figure 8.2.1). Continuous accretion of material occurs on the traction-free boundary surface of the column in
oo °o oo °o
°o
o°
°o
° .
°o
o .
°o
o°
°o
° .
o°
o°
oo
oo
o°
o°
oo
° .
°o
° .
° °
o°
oo
oo
oo
oo
o°
o°
°o
° °
o°
°o
o .
°o
o°
oo
° .
. °
° .
oo
o°
°o
o°
oo
oo
°o
oo
°o
° °
o°
°o
° °
o°
°o
oo
o°
°o
° .
°o
°o
oo
°o
oo
°o
oo
°o
oo
°o
oo
. °
oo
° °
° .
o°
oo
o°
oo
oo
IIIIII
I~
oo
I
I
'11111111
Figure 8.2.1: A growing column with equipment on the upper end.
533
8.2. Optimal Accretion of an Elastic Column
the interval [0, T]. Owing to the mass supply, the cross-sectional radius increases according to the law a = a(t),
a(O) = al,
a ( T ) = a2,
(8.2.1)
where a l and a2 are given parameters. Points of the column refer to cylindrical coordinates {r, 0, z}. At instant t, the growing column occupies the domain 12°(t) = {O <_ r <_ a(t),
0_<0<27r,
O <_ z <_ l}
in the reference configuration. Within the interval [t, t + dt], a built-up portion that occupies in the reference configuration the domain d l ) ° ( t ) = {a(t) <- r <- a(t + dt),
0<- 0<27r,
O <- z <- l}
merges with the column. Let ~'* = C(r) be the instant when a built-up portion with polar radius r joins the growing body. The function C(r) is inverse to the function a(t), a ( z * ( r ) ) = r,
C ( a ( t ) ) = t.
The rate of material supply v is determined as dV da v( t) = -d-~(t) = 2 7rla(t)--d-~ (t),
(8.2.2)
where V(t) is volume of the column at instant t. The function v(t) is assumed to be piecewise continuous and bounded O <- vl <<- v(t) <- v2 < ~,
where that
V1
(8.2.3)
and v2 are given parameters. Integration of Eq. (8.2.2) from 0 to t implies
aZ(t)
if0'
= a 2 + --~
v ( s ) ds.
(8.2.4)
Substitution of expression (8.2.4) into the third equality in Eq. (8.2.1) results in the integral restriction on the function v(t) fo T v(s) ds = "rd(a 2 - a2).
(8.2.5)
In the interval [0, T], some devices are installed on the upper end of the column. To account for the effect of equipment on the column deformation, we introduce compressive force p = p(t) applied to the upper end of the column and equal to the weight of the equipment. The weight vanishes at the initial instant, and equals some
534
Chapter 8. Optimization Problems for Growing Viscoelastic Media
value P0 at the final instant T,
p(O) = O,
p(T) = Po.
(8.2.6)
The function p(t) is not assumed to be monotonic: during the installation process, some additional tools are supplied to the upper end of the column and removed after their use. The rate of the equipment lifting
u(t) = ~---Pt(t)
(8.2.7)
l u ( t ) l - u,
(8.2.8)
is bounded
where the maximal rate U is given. At any instant t, the force p(t) >- 0 is less than the Euler critical force for the initial column, which implies that the growing column does not lose its stability. Integrating Eq. (8.2.7) with the first boundary condition in Eq. (8.2.6), we find that
p(t) =
f0t u(s)ds.
(8.2.9)
The other boundary condition in Eq. (8.2.6) implies that
fo r u(s) ds
Po.
(8.2.10)
For uniaxial loading along the z axis, the stress or in a linear elastic medium is connected with the strain E° for transition from the natural (stress-free) to the actual configuration by Hooke's law
or = Ee °,
(8.2.11)
where E is a constant Young's modulus. We confine ourselves to accretion without preloading. The latter means that the natural configuration of the initial column coincides with its reference configuration, whereas the natural configuration of a built-up portion with a polar radius r coincides with the actual configuration on the accretion surface at instant T*(r) when the element merges with the growing column,
e*(r) = {0, e(T*(r)),
0 <- r <- al, al < r <-a2,
(8.2.12)
where e(t) and e*(r) are strains for transitions from the reference to actual and natural configurations, respectively. At small strains, we write
e°(t, r) = e(t) - e*(r).
8.2. Optimal Accretion of an Elastic Column
535
Combining this equality with Eq. (8.2.12) and substituting the obtained result into Eq. (8.2.11), we arrive at the formula 0 <-- r --< al, al < r < - - a 2 .
or(t, r) = (E[e(t) - e('r*(r))],
(8.2.13)
Expression (8.2.13) satisfies the equilibrium equations and boundary conditions at the lateral surface of the column. We write boundary conditions at the ends in the integral form
2rr
~0a(t)or(t, r)r dr
= -p(t).
Substitution of expression (8.2.13) into this equality implies that
2rrE
[e(t) {fOal e(t)r dr + /a(t) Ja1
e(r*(r))]r dr
I
= -p(t).
We calculate the first integral and introduce the new variable s = ~'*(r) in the other integral. As a result, we find that
7rE
{
e(t)a 2 + 2
Jo'
[ ~ ( t ) - e(s)]a(s)
(s) ds
}
= -p(t).
(8.2.14)
Setting t = 0 in Eq. (8.2.14) and using Eq. (8.2.6), we obtain e(O) = O.
(8.2.15)
Differentiation of Eq. (8.2.14) with respect to t yields
dp
"n'Ea2(t)-d-~(t ) = - -d-~-(t).
(8.2.16)
Equations (8.2.4), (8.2.7), and (8.2.16) imply that
de l fot v(s) ds dt (t) = - - -u(t) ~ I a 2 + --~
I -1 .
(8.2.17)
Our objective is to determine a program of accretion v(t) that satisfies conditions (8.2.3) and (8.2.5) and minimizes the maximal axial strain at the final instant T, rain max [e(T)], v(t) u(t)
(8.2.18)
where the maximum is calculated over the set of piecewise continuously differentiable functions u(t) that satisfy inequalities (8.2.8) and (8.2.10).
536
Chapter 8. Optimization Problems for Growing Viscoelastic Media
Introduce the dimensionless variables and parameters t, -
t
a, -
T '
,
vT
v, -
am
_VlT Vl,
a
u, -
7ra21'
v2,
,
UT
,n.a9,,it
e, -
Po
_v2T
,n.a2,,i t
uT
7tEa 2 e Po
['a2"~ 2
Ul, = - - , Po
r / = ~,) - al
- 1.
(8.2.19)
In the new notation, Eqs. (8.2.3), (8.2.5), (8.2.8), (8.2.10), and (8.2.17) are written as follows (asterisks are omitted for simplicity): de
--
dt (t) =
-
u(t) 1 + f o v ( s ) ds
,
0 ~ Vl ~ v ( t ) < - v2 < ~ ,
--U 1 ~
u(t) <-- Ul,
fo
e(O) = O,
(8.2.20)
fO U(S) dS = 1"1,
(8.2.21)
u ( s ) d s = 1.
(8.2.22)
It is assumed that (i) Restrictions (8.2.21) and (8.2.22) are self-consistent:
V1 < '17 < /32,
U1 > 1.
(8.2.23)
(ii) For an arbitrary admissible function v ( t ) , there is an admissible function u ° (t) that maximizes the functional
(P -IE(1)I.
(8.2.24)
(iii) There is an admissible function v o (t) that minimizes the functional max Ie(1)l.
(8.2.25)
u(t)
8.2.2
Optimal Regime of Loading
Let us fix an admissible rate of accretion v ( t ) and determine the function u° (t), which maximizes the functional (8.2.24). Evidently, for any admissible functions u(t) and v ( t ) , the value ¢(1) is nonpositive and
(I) -- -e(1). To account for the isoperimetric condition in Eq. (8.2.22), we introduce the Lagrange multiplier qtl and seek a function u° (t) that satisfies the first inequality in Eq. (8.2.22) and maximizes the functional
(I)l -- --E(1)-+-~I IfO 1 u ( t ) d t -
1].
8.2. Optimal Accretion of an Elastic Column
537
Let Au(t) be an admissible increment of the function u(t). According to Eq. (8.2.20), the increment A e(t) of the strain e(t) caused by the increment Au(t) obeys the differential equation
dAe
~(t)
dt
= -
Au(t)
1 + fo v(s) ds'
A e(0) = 0.
(8.2.26)
We integrate Eq. (8.2.26) from 0 to 1 and find the increment A(I) 1 of the functional (I) 1 caused by the increment Au(t)
A(I)I = f01 X 1(t)Au(t) dt, where
t
(8.2.27)
],
1 + f v(s) ds
Xl(t) = ~1 +
(8.2.28)
It follows from Eqs. (8.2.22) and (8.2.27) that the optimal regime of loading u ° (t) has the form
o
/--Ul,
Xl(t) < 0,
+Ul,
Xl(t) > 0.
u (t) =
(8.2.29)
Indeed, any admissible increment of the function (8.2.29) satisfies the inequalities Au(t) >-- 0,
Xl(t) ~ 0,
Au(t) <-- 0,
Xl(t) ~ 0.
Combining these estimates with Eq. (8.2.27), we find that
A(I)1 ~ 0, which implies that the function u ° (t) maximizes the functional ~1. It follows from Eqs. (8.2.21) and (8.2.28) that X1 (t) is a nonincreasing monotonic function. This means that the following three opportunities arise: (i) The function X1 (t) is positive in the interval [0, 1]. (ii) There is some To E [0, 1] such that X1 (t) > 0 for 0 <- t < To and X1 (t) < 0 for T0
t
(8.2.30)
538
Chapter 8. Optimization Problems for Growing Viscoelastic Media
To find the instant To, we substitute Eq. (8.2.30) into the second equality in Eq. (8.2.22) and obtain
l+ul 2Ul
To - - - .
(8.2.31)
Returning to the dimension variables (8.2.19) and substituting expression (8.2.30) into Eq. (8.2.9), we find that
pO (t) = Ut,
0 < t <
P°(t) = Po + U(T - t),
Po + UT 2U ' Po + UT 2U
< t -- T.
(8.2.32)
It follows from Eq. (8.2.32) that at any instant t, the optimal load p° (t) is positive, which means that the loading program (8.2.30) is admissible. The functional (8.2.24) reaches its maximum on the admissible loading program u° (t), which is independent of the accretion program v(t) and has only one point of discontinuity To.
8.2.3
Optimal Regime of Accretion
We now seek the function v o(t) that satisfies restrictions (8.2.21) and minimizes the functional (8.2.25) with u = u o(t). To account for the isoperimetric restriction in Eq. (8.2.21), we introduce the Lagrange multiplier $2 and look for the function v o(t) that satisfies the first condition in Eq. (8.2.21) and minimizes the functional ~2 = - e ( 1 ) + q~2
[/01
v(t)dt-
1
rl •
(8.2.33)
Integrating Eq. (8.2.20) from 0 to 1, we obtain
e(1) - -
fo
u° (t) dt 1 + fo v(s) ds"
(8.2.34)
Let A v(t) be an admissible increment of the function v(t). It follows from Eq. (8.2.34) that the corresponding increment A e(1) is calculated as
fo t Av(s)ds Ae(1) . .fo . 1 . . .u°.(t) dt [1 + fo v(~') d~']2 = fooI Av(s)ds fs 1
u° (t) dt
(8.2.35)
[1 + fo v('r) d'r] 2"
Equation (8.2.33) together with Eq. (8.2.35) implies that A(I) 2 --
f01X2(t)A v(t) dt,
(8.2.36)
8.2. Optimal Accretion of an Elastic Column
539
where A~2 is the increment of the functional (I) 2 caused by the increment A v(t) of the function v(t), and X2(t) = q~2 - ft
ds d~-]2" [1 + u° fo(s)v(~-)
(8.2.37)
Repeating the same reasoning that was employed to derive the optimal regime of loading (8.2.29), we find from Eqs. (8.2.36) and (8.2.37) that the optimal regime of accretion has the form v ° (t) = [ Vl, X2(t) > 0, ( v2, X2(t) < 0.
(8.2.38)
Let us analyze the behavior of the function Xz(t). First, we show that X2(0) < X2(1).
(8.2.39)
Equation (8.2.31) implies that
1 To=-+ 2
1 1 >-. 2Ul 2
Therefore,
1 1 1 <-
1 mT 0
According to Eqs. (8.2.30) and (8.2.37), X2(1) - X2(0) -
fo
u°(t)dt
[1 + fo v(s) ds] 2
__l~l ( ~o T°
dt [1 + fo v(s) ds] 2
[1 + fo v(s)dsl2
" (8.2.40)
The second term on the right-hand side of Eq. (8.2.40) is estimated as follows:
~rl
~0"l-T°
dt [1 + fo v(s) ds] 2 _
d~" [1 + f:o+~ v(s) dsl 2
f0 l-T°
d'r
/o1-o
[1 + fo v(s)ds]2 <
jo
[1 + fo" v(s)ds]2 "
Substituting this expression into Eq. (8.2.40), we obtain the required inequality (8.2.39).
540
Chapter 8. Optimization Problems for Growing Viscoelastic Media
Differentiation of Eq. (8.2.37) implies that dX2 (t) = u ° (t) dt [1 + fo v('r) dl"] 2"
It follows from this equality and (8.2.30) that the function X2(t) increases in the interval [0, To] and decreases in the interval [To, 1]. This property together with inequality (8.2.39) implies that the following four opportunities may occur: (i) The function X2(t) is positive in the interval [0, 1]. (ii) There is some T1 ~ [0, 1] such that the function X2(t) is negative in the interval [0, T1] and it is positive in the interval [T1, 1]. (iii) There are two constants 0 < T2 < T3 < 1 such that the function X2(t) is negative in the intervals [0, T2] and [T3, 1], and it is positive in the interval [T2, T3].
(iv) The function X2(t) is negative in the interval [0, 1]. In cases (i) and (iv), the optimal rate of accretion equals either 111 or tl 2 in the entire interval [0, 1], which together with Eq. (8.2.23) contradicts the second inequality in Eq. (8.2.21). Therefore, these cases cannot occur. The optimal regime of accretion has the form o
f V2,
0 --< t --< T1,
Vl,
T1 < t --< 1
v ,
O<--t<--T2, T 2 < t - T3, T3 < t - - < 1
v (t) - ],
(8.2.41)
in case (ii) and the form
v ° (t)
=
111,
v2,
(8.2.42)
in case (iii). Let us analyze when the optimal rate of accretion is determined by Eq. (8.2.41). Substituting expression (8.2.41) into the second equality in Eq. (8.2.21), we find that '1~ -- V 1
T1 = ~ .
(8.2.43)
V2 -- Vl
The regime of accretion (8.2.41) is optimal provided X 1(1) > O.
(8.2.44)
It follows from Eq. (8.2.37) and the equality Xz(T1) = 0 that Eq. (8.2.44) can be presented as
[1 + fo v°(l")d~']2
> 0.
(8.2.45)
541
8.2. Optimal Accretion of an Elastic Column
According to Eq. (8.2.30), inequality (8.2.45) holds provided that To > T1.
(8.2.46)
Assuming Eq. (8.2.46) to be satisfied, we substitute expressions (8.2.30) and (8.2.41) into Eq. (8.2.45) and obtain
Ul
1
[1 +
v2T1 + Vl(S-
T1)] 2 -
v2T1 + Vl(S-
[1 +
T1)] 2
Calculation the integrals implies that 2
1 + ( v 2 - Vl)T1 + VlTo
<
1
1 + vzT1
+
1
1 + ( v 2 - vl)T1 + Vl
. (8.2.47)
Substitution of expressions (8.2.31) and (8.2.43) into Eqs. (8.2.46) and (8.2.47) yields U l [ 2 r / - (Vl + / 3 2 ) ] < 132 - Vl, 4Ul 1 v2 - Vl < + . (8.2.48) 2(1 + r/)Ul + Vl(1 - Ul) 1 + r/ (1 + r/)v2 - (1 + vz)vl If inequalities (8.2.48) are not true, the optimal regime of accretion has the form (8.2.42). The parameters T2 and T3 are found from the conditions Xz(T2) = 0,
Xz(T3) = O,
v ° (t) dt = rl.
(8.2.49)
Substitution of expression (8.2.42) into the third equality in Eq. (8.2.49) yields T3 -
T2 --
(8.2.50)
132 - 1"/. V2 V1 -
-
We subtract the second equality in Eq. (8.2.49) from the first and substitute expression (8.2.37) into the obtained result fr] 3
u ° (t) dt = 0. [1 + fo v° (s) ds] 2
(8.2.51)
This equality holds provided T2 < To < T3.
(8.2.52)
Substituting expressions (8.2.30) and (8.2.42) into Eq. (8.2.52) and calculating the integral, we find that 2 1 q- (V2 -- v l ) T 2 +
vlTo
=
1 1 + v2T2
+
1 1 + (V2 -- Vl)T2 -k- VlT3
.
(8.2.53)
Concluding Remarks A problem of optimal accretion is analyzed for a linear elastic column, on the upper end of which some equipment is installed. The weight
542
Chapter 8. Optimization Problems for Growing Viscoelastic Media
of the equipment is the only force that causes axial deformation of the column. The initial and final volumes of the column, as well as pointwise restrictions on the rate of accretion are given. The initial and final weights of the equipment and pointwise restrictions of the rate of its lifting are prescribed, but the weight of installed devices at any instant t is unknown. The optimal rate of accretion is found that minimizes axial displacement of the upper end of the column at the final instant. It is shown that two optimal regimes of accretion occur: with one and with two points of discontinuity. The optimal regime with one point of discontinuity (8.2.41) occurs when inequalities (8.2.48) are satisfied. The point of discontinuity T1 is found from Eq. (8.2.43). The optimal regime with two points of discontinuity (8.2.42) takes place when one of inequalities (8.2.48) is false. The instants T2 and T3 are found from the nonlinear algebraic equations (8.2.50) and (8.2.53).
8.3
Preload Optimization for a Wound Cylindrical Pressure Vessel
This section is concerned with optimization of preload distribution in a wound elastic cylindrical pressure vessel subjected to aging. In manufacturing multilayered composite pressure vessels and pipes by filament winding, resin-impregnated fiber bundles are wound around a rotating mandrel with some pretensioning. Our purpose is to determine the optimal preload distribution that maximizes the load-beating capacity of the vessel under internal pressure. An integral equation is derived for the optimal intensity of preload. This equation is solved explicitly for a nonaging elastic material. For an aging elastic cylinder, an approximate solution is developed, and the effects of material and structural parameters on the optimal prestressing are analyzed numerically. The exposition follows Drozdov and Kalamkarov (1994b, 1995a, b).
8.3.1
Formulation of the Problem and Governing Equations
We consider a hollow cylinder with length l, inner radius a0, and outer radius al. Material accretion occurs on the outer boundary surface of the cylinder in the interval [0, T]. Owing to material supply, the outer radius a changes according to the law a = a(t),
a(O) = al,
a(T) = a2.
Points of the growing cylinder refer to cylindrical coordinates {r, 0, z} in the reference configuration. At instant t, the growing cylinder occupies the domain {ao<-r<-a(t),
0---0<27r,
O<-z<-l}.
Within the interval [t,t + dt], a cylindrical shell that occupies in the reference configuration the domain { a ( t ) < r <-a(t)+da(t),
O - O<2~r,
O~z---l},
543
8.3. Preload Optimization for a Wound Cylindrical Pressure Vessel
joins the cylinder. The built-up shell is previously deformed by an internal pressure so that its inner radius (in the actual configuration) coincides with the outer radius of the growing cylinder. After this operation, the shell is wrapped around the cylinder and immediately merges with it. The instant ~-*(r) when a built-up portion at the point with a polar radius r joins the growing cylinder equals
0, "r*(r) =
t,
ao <-- r -----al, r = a(t).
The mechanical behavior of an incompressible, aging, elastic medium is governed by the constitutive equation 6-(t,r) = -p(t,r)~1 + 2 G(t - r*(r))O°(t,r) -
(3(r-
r*(r))O°(r,r)d
,
(r)
(8.3.1) where 6- is the stress tensor, ~0 is the deviatoric part of the strain tensor for transition from the natural to actual configuration, I is the unit tensor, p is pressure, G(t) is the current shear modulus, and the superposed dot denotes differentiation with respect to time. We assume that external body forces and surface tractions are absent, and plane axisymmetric deformation occurs in the growing cylinder. Let u(t, r) be the radial displacement for transition from the reference to actual configuration. It follows from the incompressibility condition 0u u --+-=0 Or r that u = D(t)r -1,
(8.3.2)
where D = D(t) is a function to be found. According to Eq. (8.3.2), the nonzero components of the deviatoric part of the strain tensor for transition from the initial to actual configuration 0 equal er = - D ( t ) r -2,
eo = D(t)r -2.
(8.3.3)
The radial displacement for transition from the reference to natural configuration u*(r) and the nonzero components of the deviatoric part of the strain tensor for this transformation 0*(r) are determined by Eqs. (8.3.2) and (8.3.3), where the function D(t) is replaced by some function D*(r) that will be found later. We substitute expressions (8.3.2) and (8.3.3) into Eq. (8.3.1) and use the equality g'°(t, r) = g'(t, r) - g'*(r). As a result, we find the following nonzero components of the stress tensor o-:
544
Chapter 8. Optimization Problems for Growing Viscoelastic Media r Orr
= - p ( t , r ) - 2r -2 [G(t - r * ( r ) ) ( D ( t ) - D * ( r ) )
-
G(r-
r*(r))(D(r) - D*(r))dr
,
(r) or 0
+ 2r -2 [G(t - r * ( r ) ) ( D ( t ) - D * ( r ) )
= -p(t,r)
-
G(r-
r*(r))(D(r) - D*(r))dr
,
(r)
(8.3.4)
tr z = - p ( t , r).
For the initial cylinder, the reference and natural configurations coincide D * ( r ) = O,
ao <-- r <- al.
To determine the function D * ( r ) in the region al < r <- a2, we consider deformation of a built-up portion at the instant when it mergers with the growing cylinder. Setting t = r*(r) in Eqs. (8.3.4), we obtain the following stresses in the cylindrical shell: Orr(r ) = - p ( r )
- 2G(O)r-2[D(r*(r))
- O*(r)],
(to(r) = -p(r)
+ 2G(O)r-Z[D(r*(r))
- O*(r)],
(8.3.5)
(rz(r) = - p ( r ) .
We neglect the radial stress in a thin-walled shell, O r r ( r ) = 0,
and introduce the notation ~ ( r ) = tro(r),
where ~(r) is the preload intensity. After simple algebra, we obtain from Eq. (8.3.5) sC(r)r 2 D*(r) = D(r*(r)) -
4G(0) '
al < r <- a2.
(8.3.6)
The function ~(r) is chosen to optimize properties of growing cylindrical pressure vessel. As is well known [see, e.g., Banichuk (1980)], the load-bearing capacity of cylindrical and spherical pressure vessels is maximal provided the stress intensity is uniform across the cross-sectional area. Therefore, we seek a function ~(r) that ensures a uniform stress intensity in all wound layers after manufacturing and application of some internal pressure. We integrate the equilibrium equation 0o" Or
1
+ -(O'r - o'0) = 0 r
(8.3.7)
545
8.3. Preload Optimization for a Wound Cylindrical Pressure Vessel from a0 to a(t), use Eq. (8.3.4) and the boundary conditions Orr(t, ao) = O,
O'r(t, a(t)) = O,
and find that [G(t) fa/ldr -
7g +
fa(t) ,1a0
dr] O(t) G ( t - r*(r))Tg
[fafl -r~ dr ~o t (7,(r)D(r)dr + fa(t) -~ dr fl (7fir- r*(r))O(r)d T1 Jal (r) .a(t) D*
= G(O)
,l a1
dr (r)Tg.
Replacing the integrals with respect to r by integrals with respect to s = r*(r), and using Eq. (8.3.6), we obtain [ ~ ( - ~1a 21 1_) _ = _ 1 4
G(t) + fOt G ( t - s)-==-=., a'(s)it(s) ds ] O(t)
f0t[ ( a12
)
a12 (3(r) +
f0"( 3 ( r - S)a-~5(s a(s)) ds + G(O)a3(r) a(,) 1o(,)a"
/o'
~(a(s))it(s) ds. a(s)
(8.3.8)
Differentiation of Eq. (8.3.8) with respect to time t yields [}(
) jO"t it(S, ] 1 it(t) 1 _ 1 G(t)+ G ( t - S ) a 3 ( s )ds D(t) = - ~(a(t)) D(O) =0. a--~o a 2 -~ -~ , (8.3.9) Given preload intensity {~(r), Eq. (8.3.9) is a linear differential equation for the function D(t). We suppose that the wound pressure vessel is loaded by an internal pressure P immediatelly after manufacturing. Integrating the equilibrium equation (8.3.7) from a0 to a2 = a(T) and using the boundary conditions O'r(t, ao) = - P ,
O'r(t, a2) = 0,
we find that P =
2 O- 0 m
fai
~
Or r
dr.
r
Substitution of expressions (8.3.4) into Eq. (8.3.10) yields
2
(8.3.10)
546
Chapter 8. Optimization Problems for Growing Viscoelastic Media
+ 2
--
G(t - s)(D(t) - D*(a(s)))
~s"t(Tr(~" -- s)(O(~') -
~/(S)s) ds. O*(a(s))) d r ] ~aJI,
(8.3.11)
Differentiation of Eq. (8.3.11) with respect to time implies that
[(~
1 _ a12
)
G(t) + 2
/0'
G(t-
°'s'l
S)a3(s)ds D(t) = O.
(8.3.12)
.
It follows from Eq. (8.3.12) that D(t) = D ( T + 0),
t >- T,
(8.3.13)
where D1 = D ( T + 0) satisfies the equation P _ 2
[( ) {(~ a--~ 1_ a2
1)fo~
+
/0~
1 G(T)+2 a21
o,s, 1
G(T-S)a3(s)dS
D1
i t ( S)) d s [ G ( T - s ) D * ( a ( s ) ) G(I-)D(~-) dr + 2 fo ~ a3(s
/s~
G(~"- s)(D(~') - D*(a(s)))] d r
}
.
(8.3.14)
It follows from Eqs. (8.3.4) that the stress intensity in the cylindrical pressure vessel equals S
=
or 0 -
or r =
4E
-~
G(t - ~'*(r))(D(t) - D*(r))
- foot Gr(s - "r*(r))(D(s) - D*(r)) ds] .
Differentiation of Eq. (8.3.15) with respect to t yields = 4 r - 2 G ( t - ~'*(r))b(t).
It follows from this equality and Eq. (8.3.13) that for any t >- T,
4~
S(t, r) = S(T, r) = -;2 G ( T - z*(r))(D1 - D*(r))
-
(r)
6;(~"- z*(r))(D(z) - D*(r))dz .
(8.3.15)
8.3. Preload Optimization for a Wound Cylindrical Pressure Vessel
547
Integrating by parts, we obtain 4[
S(t, r) = ~-~
- ~-*(r))(D, - D(T/) +
a(r
+
a(O)(D(r*(r)) - D*(r)) (8.3.16)
a(r - r*(r))D(r)dr . (r)
Substitution of Eqs. (8.3.6) and (8.3.9) into Eq. (8.3.16) implies that
4I
S(t, r) = ~(r) + -~ G(T - r*(r))(D1 - D(T)) _ l~r 2
a(s-r*(r))~(a(s))gt(s)a-l(s)ds 1 (r) (ao 2 - a12)G(s) + 2 fo G(s - r ) / t ( r ) a - 3 ( r ) d r "
Transforming the integral term, we find that
4[
S(t, r) = ,~(r) + -~ G(T - r*(r))(D1 - D(T))
l fra
G(r*(P) - r*(r))~(P)p-ldp (ao 2 - a-(2)G(r*(p)) + -2-~aP- ~ - ~ ) -
] r*(/3))/3-3d/3 " (8.3.17)
We set S -- S*,
(8.3.18)
where S* is the ultimate stress intensity permitted by the elastic material without failure. Equation (8.3.17) results in r 2
--[S* - sO(r)] = G(T - r*(r))(D1 - D(T)) 4
G ( r * ( p ) - r * ( r ) ) ~ ( p ) p -1
2
dp
(ao 2 - a-{2)G(r*(p)) + 2 fa~ G ( r * ( p ) - r*(/3))/3 -3 d/3 (8.3.19)
It follows from Eqs. (8.3.4) and (8.3.10) that for any t -> T,
e = 4
fail IG(t)D(t) - fOt
G(r)D(r) dr
] -~dr + faa2 dr 1 S(t, r)--r"
Combining this equality with Eqs. (8.3.13) and (8.3.18), we obtain P=2
1) ( a2
1 [ a2
G(T)D1 - fo r (~(r)D(r) dr I S+ * In a 2m al .
Chapter 8. Optimization Problems for Growing Viscoelastic Media
548
Integrating by parts and employing Eq. (8.3.9), we find that
P - S* ln a2 = 2 ( --a2 - -~1) G ( T ) ( D 1 - D(T))
-if or
G(s)'(a(s))it(s)a-l(s) ds
1
(a02 - al 2)G(s) ~ -2 -~0 - ~ - ~)~-'r)a-3('r) d r
"
(8.3.20)
It follows from Eq. (8.3.20) that
P - S* ln(a2/al) 2(ao 2 -- a12 )
= G(T)(D1 - D(T))
_ 1 fa a~ 2
,
G(r*(p))~(p)p -1 dp a12)G(r*(p)) + 2 laP1G(r*(p)-
(ao2 -
'r*(/3)),8-3d,/3"
(8.3.21)
Excluding D1 - D(T) from Eqs. (8.3.19) and (8.3.21), we arrive at the integral equation for the function ~(r)
d a=
r2 -~-[S* - {~(r)] +
G(T
r*(r)) [(P - S* ln(az/al) G(T) [ ao 2 - al 2 -
fa a2 +
8.3.2
G(r*(p) - r*(r))~(p)p -1 dp
(ao 2 - aTZ)G(r*(p)) + 2 fa°, G(r*(p) - r*(/3))/3 -3 d/3
G(r*(p))~(p)p -{ dp
]
, (ao2 -- al2)G(r*(p)) + 2 faOllG(r*(p) - r*(/3))/3 -3 d/3 "
(8.3.22)
Winding of a Nonaging Cylindrical Pressure Vessel
For a nonaging elastic material with
G(t) = G, Eq. (8.3.22) reads
22( 2a°al S* a2 ) f a r2[S * - ~(r)] = a2 _ a2 P l n ~ a1 + 2a2
,r ~(p)p p2 _ ado2"
Differentiation of this equality with respect to r yields dsc
2r2~
r-d7r +
r 2 m
a 2
=2S*.
(8.3.23)
Setting r = a l, we obtain ~(al) = {~0,
(8.3.24)
549
8.3. Preload Optimization for a Wound Cylindrical Pressure Vessel
where {~0 = S * +
a,~
aO -
S*ln---P al
)
.
(8.3.25)
The solution of Eq. (8.3.23) with the initial condition (8.3.24) has the form 2Pa 2 [ a~ in ( a 2 ) 2 ] s~(r) = S* 1 + r2 _ a-----~ -7- r2 _ a2.
(8.3.26)
• eeeoee e
•e
"'. la
eeeoe
eeeeeeeeeeeeeeeeeeeeeee e 2a
-
_
...........................
2b
,oooooooOO°
eeeeeeeeeeeeeeeeee° oee o"
1
• ''"T ........ I
lb I
J
eeeeeeeoeoOeeeoeeeeeeeeeeeeeeeoeeeeeeeeeoeeeeeee I
I
I r,
I
I 2
Figure 8.3.1: The optimal dimensionless preload intensity ~, versus the dimensionless polar radius r, for a nonaging w o u n d cylinder with ~/2 = 2. Curves (a): P, = 0, Curves (b): P, = 1. Curve 1: a thin initial cylinder 011 = 1.25, a0 = 0.8al). Curve 2: a thick initial cylinder (~/1 = 5, a0 = 0.2al).
Chapter 8. Optimization Problems for Growing Viscoelastic Media
550
Using dimensionless variables and parameters r,-
r al
,
~
~ * - S*'
P*-
P
al
S*'
'171 --
a2 '
1/2-
ao
' al
we present Eq. (8.3.26) as follows: ~,(r,) = 1 +
2 2 171r,
- 1
l n w - P* r,
•
(8.3.27)
The optimal preload intensity ~, is plotted versus the dimensionless radial coordinate r, in Figure 8.3.1. For a thick initial cylinder, the optimal preload intensity ~, is practically independent of the radial coordinate r,, as well as of the internal pressure P,. For a thin initial cylinder and the zero internal pressure P,, the optimal preload intensity ~, decreases in r, and tends to some limiting value as r ---, oo. The latter means that the preload becomes independent of the coordinate in a region far away from the initial cylinder. With the growth of the internal pressure P,, this behavior changes drastically. For P, = 1, the optimal preload intensity ~, increases monotonically in r,. For a fixed r,, it decreases with the growth of the pressure P,. For a thin initial cylinder and a large internal pressure P,, a region is observed where the preload intensity ~, is negative. This region arises at P, ~ 0.95 and grows with an increase in P,. Since negative preloading is impossible in wound layers, this means that the optimization problem has a solution only for sufficiently thick initial cylinders or for sufficiently small intensifies of internal pressure.
8.3.3
W i n d i n g of an A g i n g C y l i n d r i c a l Pressure V e s s e l
For an aging elastic cylinder, differentiation of Eq. (8.3.22) with respect to r yields
d~[r~r(Tr(T-r*(r))
r ~rr(r) + 2sO(r) 1 + ~
(r) G(T - r*(r))
G(O)r
-2
]
(ao 2 -- al2)G(r*(r)) + 2 fa~ G(r*(r) - r*(p))p -3 do
[O(**(p) -- **(r))
2dT*
r d--r-(r)
G(r*(p) - r*(r))
(~(T - z*(r)) ]
G(T - T*(r)) ]
G('r*(p) -- r*(r))~(p)p-ldp (%2 _ al2)G(,r,(p)) + 2 fav, G(r*(p) - "r*(/3))]3-3d/3 r d'r*
(~(T - T*(r))
= 2S* 1 + ~-d-7-r(r) G(T - r*(r))
(8.3.28)
To solve Eq. (8.3.28) explicitly, we suppose that the function G(t) does not change significantly in the winding process, i.e., that a small parameter ~ E (0, 1] exists
8.3. Preload Optimization for a Wound Cylindrical Pressure Vessel
551
such that G(t) = G(0)[1 + #g(t)],
(8.3.29)
where g(t) is a bounded function in [0, T]. Substitution of expression (8.3.29) into Eq. (8.3.28) implies that d~ 2r 2 fa2 r~-rr (r) + r2 _ a2Se(r)[1 + / , A ( r , / , ) ] - tx Jr B ( r ' p ' t x ) ~ ( p ) d p = 2S*[1 + #F(r,/x)],
~(al) = ~0,
(8.3.30)
where A ( r , ~ ) --
r2
-
-
a 2 d~'*(r))
2r
dr
g(T - -r*(r)) 1 + tzg(T - l"*(r))
(ao 2 - ai2)g(~'*(r)) + 2 fa~ g(~'*(r) - ~.,(p))p-3 do r 2 (a02 - r -2) +/z[(a02 - al2)g(~'*(r)) + 2 fa~ g(T*(r) - ~.,(p))p-3 do]' B(r, o, tz ) -
2 dr*(r) r 0 dr
g(1"*(p) - ~'*(r)) 1 + Ixg(~'*(p)- ~'*(r))
g(T
T*(r)) 1 + # g ( T - ~'*(r)) -
T*(r)) p -2) +/x[(ao 2 - a-{2)g(~'*(O)) + 2 fa° g(T*(O ) -- ~'*(13))/3-3dl3] ' 1 + I~g('r*(O ) -
(ao 2 --
F(r,l~)-
r d1"*(r) g ( r - l"*(r)) 2 dr l + lxg(T-~'*(r))"
We seek a solution of Eq. (8.3.30) in the form
~(r,/z) = ~(°)(r) +/z~(1)(r) + " . .
(8.3.31)
Substituting expression (8.3.31) into Eq. (8.3.30), we find that the function ~(°)(r) satisfies Eqs. (8.3.23) and (8.3.24), and, therefore, has the form (8.3.26). The function ~(1)(r) obeys the differential equation d~(1) 2r2 sc(1)(r) = F(1)(r), r dr (r) + r2 _ a2
sc(1)(al) - 0,
where F(1)(r) = 2S* f (r, O) -
2r2~(°)(r)A(r, O)
+
- 4 =r
dr* (r) { dr g(T2a 2 + --~
r*
(r))[S* -
~(o)
f
a2
B(r, p, 0)~(°)(p) do
(r)]
g(~'*(p)-- T*(r))-- g(T -- T*(r)) } 02 -- ag 0~(°)(01 dp
(8.3.32)
552
Chapter 8. Optimization Problems for Growing Viscoelastic Media
+
2r2a~
(r 2 - a2)2
+ 2
~(°)(r) [(ao 2 - a{2)g(r*(r))
g(r*(r)
-
-
]
'r*(p))p -3 dp .
1
(8.3.33)
We accept the exponential dependence (2.3.27) for the shear modulus G(t) of an aging elastic material G(t) = Go + (G~ - Go)[1 - exp(-yt)],
(8.3.34)
where Go is the initial shear modulus, Go~ is the equilibrium shear modulus, and y is the characteristic rate of aging. It follows from Eqs. (8.3.29) and (8.3.34) that /x -
00
Go
- 1,
g(t) = 1 - exp(- ~/t).
We confine ourselves to winding with a constant rate of material supply (8.3.35)
u = 27ra(t)it(t)l.
The constant u is determined from the condition u T = 7rl(a 2 - a2).
(8.3.36)
According to Eqs. (8.3.35) and (8.3.36), a2)t a2 T ,
(a 2 -
a(t) = al
1+
r 2 _ a2 r*(r) = T a 2 _ a------~l.
(8.3.37)
Substitution of expressions (8.3.34) and (8.3.37) into Eq. (8.3.33) implies that F(1)(r) = 2S*F(,1)(r,), where F*(1)(r*) = '0 '022- 1 ( 1 - ~,(°)(r,))exp 1
n2
+ r--7/,
-'0 '02
1
2
p 2 _ r2
"02_ r E
(exp(-rl~/2-l)-exp(-'0'02
r,2&)(r,) r2 - 1 1)) ('012r 2 - 1)2 ( ' 0 2 1) (1 _ exp (_'0'02 r,
+2fl and '0 = yT.
r 2 _ p2 (1-exp(-'0'02_l))~
]'
1))
p~(,O)(p)dp]
~1-~peTiJ
553
8.3. Preload Optimization for a W o u n d Cylindrical Pressure Vessel
Introducing the dimensionless quantity ~,(1)(r,) --
~(1)(r) S
*
we write Eq. (8.3.32) as d~,(1) _ dr,
2 r,
E
22
F(,1)(r, ) _ ~ ¢~lr, ,(1) rlZr 2 - 1
]
~,(1)(1) -- O. '
The optimal dimensionless preload intensity ~, is plotted versus the dimensionless radius r, in Figure 8.3.2. The function ~, decreases monotonically in r,, while the rate of decrease is maximal for a nonaging material. For a fixed time of winding
'8gg,,o
::::::::::::::::::::::::::::::::: 3 • °°°°°°°°°°OOOooooooooOOooo •
- -.
°°°OOooooooooo
"......................... :::::::::::1111211111111
°°o
2
"o.oo.o.o.oooo
°°°OOooo
••o••o••••••o•oo••oo•o•oo•oooo•o••oooooo•oooooo••••••••••••
°
1
I 1
I
I
I
I
I
I r,
I
I 2
Figure 8.3.2: The optimal dimensionless preload intensity ~, versus the dimensionless polar radius r, for an aging wound cylinder with ~/1 -- 1.25, ,72 = 2.0,/z = 0.2 and P, = 0.5. Curve 1: the zero approximation ~,(0).Curves 2 and 3: the first approximation ~,(0) +/z~,(1). Curve 2: ~/= 1. Curve 3: ~/= 100.
554
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
T, this means that an increase in the rate of aging 3t implies an increase in the optimal preload intensity at any point of a growing cylinder. For any r,, the preload intensity in a nonaging cylinder is less than the preload intensity in a cylinder subjected to aging. Concluding Remarks Optimal preload intensity is determined for a wound cylindrical pressure vessel subjected to aging. An explicit formula (8.3.27) is derived for the optimal preload intensity ~ as a function of polar radius r for a nonaging material, and an approximate solution (8.3.31) is developed for an aging material. The effects of material parameters on the optimal preload intensity are studied numerically, and the following conclusions are drawn: 1. For a thick initial cylinder, the optimal preload intensity ~ is practically independent of polar radius r, as well as of internal pressure P. 2. For a thin initial cylinder and a sufficiently small internal pressure P, the optimal preload intensity ~ decreases monotonically in r and tends to some limiting value far away from the initial cylinder. 3. For a fixed duration of winding, the growth of the rate of aging implies an increase in the optimal preload intensity. 4. For a thin initial cylinder and a sufficiently large internal pressure P, the optimal preload intensity ~ increases monotonically in r. For a sufficiently large pressure P, a small region arises in the vicinity of the initial cylinder, where the preload intensity is negative.
8.4 Optimal Design of Growing Beams This section is concerned with optimal design of growing reinforced elastic beams subjected to aging. We derive the optimal shape of a growing beam that minimizes its maximal deflection under various assumptions regarding the accretion process. The effects of material aging and the rate of material supply on the optimal thickness are analyzed numerically. The obtained results are of interest for optimal design of reinforced cantilevers and bridges. The exposition follows Drozdov and Kalamkarov (1994a, 1995c). Optimal design of elastic and elastoplastic thin-walled structural members has been studied in a number of works [see, e.g., Banichuk (1980) for a detailed survey]. Arutyunyan and Kolmanovskii (1983) and Drozdov and Kolmanovskii (1984) analyzed the effect of material viscosity on the optimal thickness of an inert, inhomogeneously aging, viscoelastic beam. The problem of optimal design for an inhomogeneously aging viscoelastic solid under three-dimensional loading was formulated by Zevin (1979). The first attempts to account for the effect of accretion on the optimal shape of a growing viscoelastic beam were undertaken by Drozdov (1983, 1984) and Kolmanovskii and Metlov (1983).
8.4. Optimal Design of Growing Beams 8.4.1
555
Formulation of the Problem and Governing Equations
We consider plane bending of a reinforced elastic beam. The beam has a rectangular cross-section with a constant width b. Thickness of the beam is a piecewise continuous and bounded function h of a longitudinal coordinate x,
0 < hi <- h(x) -< h2 <
~,
(8.4.1)
where h l is the minimal and h2 the maximal admissible thickness. A reinforcement occupies a domain with a constant thickness h0 < hi in the vicinity of the longitudinal axis. The rest of the volume is filled by a main material. The accretion process occurs in the interval [0, T]. It consists in joining of portions of the reinforcement and of the main material with the growing beam. Denote by lo(t) length of the reinforcement and by l(t) length of a part of the beam filled by the main material (see Figure 8.4.1). The functions lo(t) and l(t) increase monotonically and satisfy the inequality
lo(t) >--l(t) and the boundary conditions l(0) = 0,
/0(0)
il ,,IIIIII',', .........................
= lo,
l(T) = lo(T)=
lo(t)
ll.
I IIIIII',IIIII',IIII',IIIII',',,,
m
m
Figure 8.4.1" A growing reinforced cantilevered beam.
(8.4.2)
Chapter 8. Optimization Problems for Growing Viscoelastic Media
556
As an engineering application, we refer to manufacturing of a reinforced steelconcrete column and suppose that one end, x = 0, is clamped, and the other end, x = lo(t), is free. The mechanical response in the main material (concrete) obeys the constitutive equation of an aging elastic medium, and the behavior of the reinforcement (steel) is governed by the constitutive relation of a nonaging elastic material. For uniaxial loading, the stress tr is connected with the strain e by the following formulas: • In the reinforcement,
tr(t,x) = Eoe°(t,x),
(8.4.3)
• In the main material tr(t, x) - E(t - "r*(x))e°(t,x) -
F.('r- "r*(x))e°('r,x)d'r.
(8.4.4)
(x) Here E0 is a constant Young's modulus of the reinforcement, E(t) is a current Young's modulus of the main material, the instant ~'*(x) coincides with the instant when a builtup portion at point x merges with the growing beam, and the superposed dot denotes differentiation with respect to time. For accretion without preloading, the strain in a built-up portion at a point x for transition from the reference to natural configuration coincides with the strain on the accretion surface of the growing beam at instant ~-*(x). Since the strain from the reference to actual configuration vanishes at the free end, x = lo(t), the strain e°(t,x) in Eq. (8.4.3) coincides with the strain e(t,x). For built-up portions of the main material, which join the deformed lateral surface, the strain e°(t, x) in Eq. (8.4.4) equals the difference e(t, x) - e(~'*(x), x). Let w(t, x) be the beam deflection at instant t at point x. It is assumed that 1. The function w and its derivatives are so small that we may neglect the nonlinear terms in the expression for the curvature of the longitudinal axis. 2. The hypothesis regarding plane sections in bending is fulfilled. 3. The function w changes in time rather slowly, and inertia forces can be neglected. According to Eqs. (8.4.3) and (8.4.4), the bending moment M(t, x) is calculated as
M ( t , x ) = ~ [Eoh3w~(t,x) + E(t - ~'*(x))(h3(x) - h3)(w'(t,x) - w'('r*(x),x)) -
L~('r- "r*(x))(h3(x) - h3)(w'('r,x) (x)
-
w'(z*(x),
x))dr],
M(t, x) = 1Eoh3ow'(t, x ), 12
0 --< x <---/(t),
l(t) < x <- lo(t),
where prime stands for the derivative with respect to x.
(8.4.5)
8.4. Optimal Design of Growing Beams
557
We assume that the longitudinal axis is horizontal and that weight is the only force applied to the beam. The latter implies that the bending moment M equals M ( t , x ) = bg
poho(lo(t) - x) 2 + P
0 <- x <- l(t),
(h(s) - ho)(s - x ) d s , dX
1 M(t, x) = -~bgpoho(lo(t)
-
x) 2,
(8.4.6)
l(t) < x <-- lo(t),
where P0 and p are mass densities of the reinforcement and the main material, respectively, and g is the gravity acceleration. It follows from Eqs. (8.4.5) and (8.4.6) that leo h3 + E(t - T* (x))(h 3(x) - h3)] wit(t, x) = (h3(x) - h~)
E ( I " - ~'*(x))w'(r,x)d~" (x)
+6bg{poho[(lo(t)-x)2+~(h3(x)-h3)(lo(~'*(x))-x)
+ 20
(h(s) - ho)(s - x ) d s
2]
(8.4.7)
.
dX
For a cantilevered beam, the boundary conditions for Eq. (8.4.7) read w(t, O) = O,
(8.4.8)
w'(t, O) = O.
Let V(t) be volume of a part filled by the main material l(t)
V(t) = b
(8.4.9)
(h(x) - ho) dx, JO
and dV bl--
dt
the rate of material supply. The parameter u is assumed to be time-independent. We introduce the dimensionless variables and parameters
x, U,
--
=
x ~-1'
uT bholl '
t,
=
V,-
t ~,
h,
V bholl '
-
h , ho T,-*
E, T* T '
-
E Eo'
1o,
[ 3 _p _ - - , Po
l0 = --
11'
l l,
-
ll ,
z =~6bgp°14w', Eoh 3
and rewrite Eq. (8.4.7) as follows (asterisks are omitted for simplicity):
558
Chapter 8. Optimization Problems for Growing Viscoelastic Media
[ 1 + E(t - T*(x))(h 3 (X) - - 1 ) ] z ( t , X) = (h3(x) - 1)
E ( I " - r*(x))z(~',x)d~" + (lo(t) - x) 2 (x)
+ E ( O ) ( h 3 ( x ) - 1)(10(r*(x))- x) 2 + 2/3
fx l(t) ( h ( s ) - 1 ) ( s -
x)ds.
(8.4.10)
Differentiating Eq. (8.4.9) with respect to time and using the boundary conditions (8.4.2), we obtain dl
u
dt
l(O) = O,
1'
h(l(t))-
1(1) = 1.
(8.4.11)
The maximal deflection of the growing beam equals w ( T , ll) = E°h3dp 6bgpol~ '
where =
1 (1 - x ) z ( 1 , x ) d x .
f0
(8.4.12)
Our purpose is to find a function h ° (x) that obeys restrictions (8.4.1) and minimizes the functional (8.4.12) on solutions of Eqs. (8.4.10) and (8.4.11).
8.4.2
Optimal Thickness of a Nonaging Elastic Beam
For a nonaging main material with a constant dimensionless Young modulus E, Eqs. (8.4.10) and (8.4.12) imply that
1
do =
f0
(1 - x)(lo('r* (x))
- x) 2 dx -k- [~f~o,
where fO ~o =
1 -- X 1 + E(h3(x)-
1)
I2yl(X) + (1 -- X)2 -- (/(~'*(X)) - x) 2 dXo 13
The function Yl (x) =
fx
(h(s) - 1)(s - x) ds
satisfies the differential equations Y~ = - Y 2 ,
y~ = 1 - h(x),
yl(1) = O,
y2(1) = O.
(8.4.13)
8.4. Optimal Design of Growing Beams
559
The optimal thickness h ° (x) minimizes the functional ~0 on solutions to Eqs. (8.4.13) and satisfies condition (8.4.1) and the equality
fooI h(x) dx = gl,
(8.4.14)
where V1 = 1 + V. (T). Let us calculate the increments A yl(x ), A y2(x ) of the functions yl(x), y2(x) caused be the increment Ah(x) of the thickness h(x). Equation (8.4.13) implies that Ay~ + Ah(x) = 0,
Ay2(1 ) = 0. (8.4.15) To account for the isoperimetric condition (8.4.14), we use the penalty method [see, e.g., Vanderplaats (1984)] and minimize the new functional Ay~ + Ay 2 = 0,
t~ 1 = t~)0 + tZ
Ayl(1 ) = 0,
h(x) d x - V1
,
where p~ is a positive constant. The increment A~I of the functional ~1 caused by the increment Ah(x) of the function h(x) is calculated as A(I) 1 -- - 3 E
+2
fol ( 1 - x)h2(x)Ah(x) I2yl(x) + ( 1 - x)2 - (lo('r*(x)) - x)21d x [1 + E(h3(x)- 1)] 2 /3 1 (1 --x)Ayl(X) dx + 2/x 1 + E(h3(x)- 1)
f0
[/o
-
Ah(x)dx. (8.4.16)
We introduce the functions ~1 (X) and ~2(x), which satisfy the differential equations 1--x
~[ = 21 -+- E(h3(x)- 1)'
~2 = qq'
qq(0) = 0,
q~2(0) = 0,
(8.4.17)
multiply the first equality in Eq. (8.4.15) by qq(x), the other equality in Eq. (8.4.15) by q~2(x), integrate from 0 to 1 and add to Eq. (8.4.16). Integrating by parts and using Eq. (8.4.17), we obtain
A~ 1 =
/o'
N(x)Ah(x)dx,
(8.4.18)
where N(x) = 2/x If01 h(x) d x - V1] + ~2(x) 3E(1 - x)h2(x) [2yl (x) + [1 + E(h3(x)- 1)] 2 l
(1 - x ) 2 -
(lo(l"*(x))
- x) 2 ]
J
Chapter 8. Optimization Problems for Growing Viscoelastic Media
560
To solve the optimization problem, we employ the gradient projection method [see, e.g., Vanderplaats (1984)]. At the first step, we fix an admissible function h~l~(x). Using some thickness h~k~(x) at the kth step of the iterative procedure, we calculate a thickness h~k+l)(x) at the (k + 1)th step according to the formula:
{
H~k)(x), hi <-- H~k~(x) <-- h2, hi, H(k)(x) < hi, h2, H(k)(x) > hE.
h(k+l)(x) =
(8.4.19)
Here H (k)(x) = h (k)(x) - odV (k)(x),
a is a positive constant, and N~k)(x) is the function N(x) corresponding to the thickness h~k~(x). First, we set c~ = a0, where ct0 is a given constant, and calculate the functional ~1 for h~k~(x) and for h~k+l~(x), where the function h~k+l~(x) is determined by Eq. (8.4.19). If
(I)l(h(k+l)) < (I)l(h(k)), we pass from the kth to the (k + 1)th step of the iterative process. If
(I~l(h (k+l)) > t:I~l(h(k)), we reduce c~ by twice and repeat the same procedure with the new ct value. For the numerical analysis, we consider two regimes of accretion: (a) 10(~'*(x))= 1. (b) /0(~'*(x)) = x. According to regime (a), all reinforcement is installed at the initial instant t = 0. Regime (b) means that at any instant t, length of the domain occupied by the reinforcement coincides with length of the domain filled by the main material. Calculations are carried out for the following parameters: hi = 1.5,
h2 = 4.0,
V1 = 3.0,
ct0 = 0.1,
~ = 0.8.
The iterative algorithm demonstrates rather rapid convergence: 500 iterations ensure the minimum value of the functional ~1 with accuracy higher than 99.8%. The difference between the requested volume V1 and its optimal value is less than 1%. The optimal thickness h ° (x) is plotted in Figure 8.4.2 for regime (a) of accretion and in Figures 8.4.3 and 8.4.4 for regime (b). Figure 8.4.3 corresponds to/3 = 0.3, and Figure 8.4.4 corresponds to/3 = 1.0, where/3 equals the ratio of mass densities for the main material and the reinforcement. The functions h ° (x) are similar for regimes (a) and (b). The optimal thickness of a growing cantilever is maximal in the vicinity (0 --- x -< Xl) of the clamped end,
8.4. Optimal Design of Growing Beams •
561
•
•
•
• •
• •
•
• •
• •
h
•
O •
• •
• •
•
•
• •
• • • •
• •
o
o •
• •
OO O oo
! $ oo
oo o° °
•
•
•
•
•
•
•
•
•
•
1.5
I
I
I
I
{
•
•
1" I
•
2" .I ........
x
• °
"" "I .....
3 :_'__1
.
.
.
.
.
.
.
.
.
1
Figure 8.4.2: The optimal dimensionless thickness h ° of a nonaging growing elastic beam versus the dimensionless longitudinal coordinate x for regime (a) of accretion. Curve 1" E -- 0.1. Curve 2: E = 0.3. Curve 3: E = 0.5.
x = 0, decreases (practically linearly) from the maximal to the minimal value in the middle part of the beam (Xl --- x -< x2), and it is minimal near the free end, x = 1, of the beam. Lengths of the regions where the optimal thickness is maximal or minimal, as well as the slope for the function h ° (x) decrease with an increase in the Young's modulus of the main material (i.e., with the growth of E). Transition from regime (a) to regime (b) of accretion leads to an increase in lengths of the regions where the optimal thickness takes its extreme values. For regime (b) of accretion, lengths of the regions where the optimal thickness is extreme decrease with an increase in mass density of the main material (i.e., with an increase in/3). This dependence is, however, rather weak, so that it may be neglected.
Chapter 8. Optimization Problems for Growing Viscoelastic Media
562
. . . . . . . . . . . . . . . . . . . . . . . . .
i-
- -:-
•
• •
• •
•
h
•
-
•
• •
O
•
• •
• •
•
•
•
•
•
• o • oo
•
oe
8 0 8
°g •
• • o oo oO •
• °
•
•
2""
1.5
I
I
I
I
I
I
• _1 ........
x
3 _'1 . . . . . . . . .
1 ..........
1
Figure 8.4.3: The o p t i m a l d i m e n s i o n l e s s thickness h ° of a n o n a g i n g g r o w i n g elastic b e a m v e r s u s the d i m e n s i o n l e s s l o n g i t u d i n a l coordinate x for r e g i m e (b) of accretion w i t h / 3 = 0.3. C u r v e 1" E = 0.1. C u r v e 2: E - 0.3. C u r v e 3: E = 0.5.
8.4.3
O p t i m a l T h i c k n e s s o f an A g i n g Elastic B e a m
For an aging reinforced elastic beam, we confine ourselves to the case in which all the reinforcement is installed at the initial instant lo(t) = 1,
O < t <- T.
Setting t = ~'*(x) in Eq. (8.4.10), we find that Z(T* (x), x) - (1 - x) 2.
Differentiation of Eq. (8.4.10) with respect to time yields __~gz( t , x ) = c~t
2[3u(l(t) - x) 1 + E ( t - ~'*(x))(h3(x) - 1)
8.4. Optimal Design of Growing Beams
563
•
• •
• • •
•
h
•
O
•
• •
•
•
•
• •
•
e e e •
•
ee eee
$ 0
o
o: o•
o •
•
•
2".'. 1.5
I
i
i
i
I
i
• _1 . . . . .
3
_'___~, ........
x
1 .........
1
Figure 8.4.4: The optimal dimensionless thickness h ° of a nonaging growing elastic b e a m versus the dimensionless longitudinal coordinate x for regime (b) of accretion with 13 = 0.6. Curve 1" E = 0.1. Curve 2: E - 0.3. Curve 3: E = 0.5.
We integrate this equation from ~'*(x) to 1 and find that z(1, x) - (1 - x) 2 + 2/3u
fl
(l(t) - x) dt ¢x) 1 + E ( t -
"r*(x))(h3(x) - 1)
.
(8.4.20)
Differentiation of the identity
"r*(l(t)) = t with respect to time with the use of Eq. (8.4.11) implies the differential equation for the function ~'*(x)
d'r* dx
-
h(x)u
1
(8.4.21)
564
Chapter 8. Optimization Problems for Growing Viscoelastic Media
with the boundary conditions 'r* (0) = O,
'r* (1) = 1.
(8.4.22)
Substituting expression (8.4.20) into Eq. (8.4.12), setting t = ~'*(y), and using Eq. (8.4.21), we arrive at the formula 1
= ~ + 2/3~o, where dPo --
fo I(1 -
x) dx
(y - x)(h(y) - 1) dy
fx 1 1 + E('r*(y)
- "r*(x))(h3(x) -
1)"
The optimal thickness h ° (x) satisfies condition (8.4.1) and minimizes the functional ~0 on solutions to the problem (8.4.20) to (8.4.22). To account for the latter condition in Eq. (8.4.22), we employ the penalty method and introduce the new functional (I) 1 = (I) 0 -Jr" /.e['r*(1)- 1] 2.
Here ~ is a positive constant, and the function ~-*(x) satisfies Eq. (8.4.21) with the initial condition -r*(o) = o.
The increment A ~ 1 of the functional ~1 caused by an increment Ah(x) of the function h(x) is calculated as A ~ 1 = 2/z(-r*(1)- 1)A'r*(1) +
/o
(1 - x ) d x
{Ah(y)[1 + E(~'*(y) - ~'*(x))(h3(x) - 1)]
- (h(y) - 1)[E(I"* (y) - ~'*(x))(h 3 (x) - 1)(A ~-*(y) - A ~-*(x)) + 3E(-r*(y) - ~'*(x))h2(x)Ah(x)]}
(y - x ) d y [1 + E ( ~ ' * ( y ) - ~'*(x))(h3(x)- 1)] 2.
Changing the order of integration, we obtain
A(I) 1 --
2 / x ( 7 " ( 1 ) - 1)A~'*(1)+ - 3(1 - x)h2(x)
-
Jx
R(x)A~-*(x)dx.
fl
Ah(x)dx {fox
y)dy 1 + E ( ~('1*-(yx))(-x - ~'*(y))(ha(y)1)
(h(y) - 1)(y - x)E(~'*(y) - r * ( x ) ) d y [1 + E(r*(y) - ~'*(x))(h3(x) - 1)]2
} (8.4.23)
565
8.4. Optimal Design of Growing Beams
Here R(x)
= (h(x) - 1) f x
Jo
(x - y)(1 - y ) ( h 3 ( y ) - 1)/)(~-*(x) - T * ( y ) ) d y [1 + E(~'*(x) - ~'*(y))(h3(y) - 1)] 2
- (1 - x ) ( h 3 ( x ) - 1)
fx
(y - x ) ( h ( y ) - 1)E(~'*(y) - T * ( x ) ) d y [1 + E(~'*(y) - ~'*(x))(h3(x) - 1)] 2 "
Equation (8.4.21) implies that dA~'* dx
Ah(x) u
A'r*(0) = 0.
(8.4.24)
We introduce a function q~(x) that obeys the differential equation qJ(1) = 2/x('r* (1) - 1),
q~' = R ( x ) ,
(8.4.25)
multiply Eq. (8.4.24) by q~(x), integrate from 0 to 1, and add to Eq. (8.4.23). The obtained expression is integrated by parts with the use of Eq. (8.4.25). As a result, we obtain Eq. (8.4.18) with N(x) -
q,(x)
fo x +
(1 - y ) ( x - y ) d y 1 + E(T*(x) - T*(y))(h3(y) - 1)
- 3(1 - x ) h 2 ( x )
fx
( h ( y ) - 1)(y - x ) E ( r * ( y ) - T * ( x ) ) d y [1 + E(~'*(y) - ~'*(x))(h3(x) - 1)] 2 "
To find the optimal shape of a growing beam, we employ the gradient projection method based on the algorithm (8.4.19). Numerical simulation is carried out for the following parameters: hi = 1.5,
h2 = 4.0,
u = 2.0.
The dependence E ( t ) is accepted in the form (2.3.27), E ( t ) = E(0) + [E(~) - E(0)]
[
1 - exp
-Ta
'
(8.4.26)
where E(0) is the initial Young's modulus, E(~) is the equilibrium Young's modulus, and Ta is the characteristic time of aging. The dimensionless optimal thickness of an aging growing beam is plotted in Figures 8.4.5 and 8.4.6. Data presented in Figure 8.4.5 are typical of polymer composite beams armored by steel, and data presented in Figure 8.4.6 are typical of reinforced concrete. The coefficient/x ensures that the difference between the desired value ~-*(1) = 1 and its optimal value is less than 1%. The penalty coefficient equals 0.8 for Figure 8.4.5 and 4.0 for Figure 8.4.6. The optimal shape of an aging beam is similar to the optimal shape of a nonaging beam. The function h ° (x) equals the maximal value h2 in the vicinity of the clamped end, decreases linearly to the minimal value hi, and equals hi in the vicinity of the
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
566
•
•
• •
• •
•
• •
• •
•
• •
• •
• •
•
• •
° o
h
• •
•
•
•
O
•
•
oO •
go go
°gO ° o
o • go
• •
Oo •
• o ° e
| oo °gO
"!
! °8 o Ooo
oo° °
.:.?..
gO
•
•
• •
• •
• •
• •
• •
•
• •
•
• •
•
o •
•
•
• •
•
•
• •
•
3"
•
" •
1.5
I
I
I
I
I
0
I
I x
2"". ° o
1 °°Oo
".1__~" ..... l .... ~',__
1
F i g u r e 8.4.5: The optimal dimensionless thickness h ° of an aging growing elastic beam with E ( 0 ) = 0.1 and E ( ~ ) = 0.5 versus the dimensionless longitudinal coordinate x. Curve 1" Ta 0.5. Curve 2: Ta 2.0. Curve 3: Ta -- 10.0. - -
=
free end,
f h2, h " ( x ) = I hi,h2_ ~h2x2 __-- x l ( x
0 ~ x ~
-
Xl,
Xl), x2Xl~x~-< x --< x2,1.
(8.4.27)
With an increase in the characteristic time of aging Ta, lengths of the regions where the optimal thickness is extreme, decrease, and for large Ta values, the region with h ° (x) = hi disappears (see Figure 8.4.5). This dependence is stronger for polymeric composites than for reinforced concrete. The effect of the rate of accretion on the optimal thickness of a growing beam is depicted in Figure 8.4.7 for a polymeric beam reinforced by steel. The penalty coefficient/x equals 4.0, which implies that the difference between the desired value
567
8.4. Optimal Design of Growing Beams
•
•
•
•
•
•
h
•
O oo
•
•
8
| |
°I
oo
•
•
•
•
e
•
2
"" """~1 1.5
I
I
I
I
I
J
• --i-:::
x
.....
_1.........
_1..........
1
Figure 8.4.6: The optimal dimensionless thickness h° of an aging growing elastic beam with E(0) = 0.01 and E(~) = 0.14 versus the dimensionless longitudinal coordinate x. Curve 1" Ta 0.5. Curve 2: Ta 2.0. Curve 3: Ta 10.0. -
-
-
-
-
~'*(1) = 1 and its optimal value is less than 1%. For large rates of accretion, the optimal shape of a growing beam is determined by formula (8.4.27). With an increase in the rate of material supply u, or, which is equivalent, with an increase in the desired volume of the beam, the optimal function h ° (x) "slides" as a rigid body along the x axis without significant changes in its shape. For small u values, the dependence h ° (x) becomes essentially nonlinear, which may lead to some difficulties in manufacturing beams with the optimal thickness distribution.
Concluding Remarks The optimal shape is analyzed for a growing cantilevered reinforced beam loaded by its weight. A semianalytical technique is employed to derive the optimal thickness of a beam. The following conclusions are drawn:
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
568
•
• •
• •
• •
•
•
h
O
1.5
_
•
•
".
I
]
•
. 2
I
.........
1_. . . . . . . . .
l .........
3 •
l .........
_1. . . . . . . . . x
_1. . . . . . . . .
l ......... 1
Figure 8.4.7: The optimal dimensionless thickness h ° of an aging growing elastic beam with E(0) = 0.01, E(~) = 0.14, and Ta = 2 versus the dimensionless longitudinal coordinate x. Curve 1: u = 1. Curve 2: u = 2. Curve 3: u = 3. 1. The optimal thickness of a growing cantilevered beam is a piecewise continuously differentiable function of the longitudinal coordinate. In the vicinity of the clamped end, the optimal thickness is uniform, and it equals the maximal admissible value. In the middle of the beam, the function h ° (x) decreases (practically linearly). In the vicinity of the free end, it is uniform and takes the minimal admissible value. 2. For a nonaging material, the optimal thickness distributions are similar for various regimes of accretion. Lengths of the regions in which the optimal thickness is maximal or minimal, as well as the slope of the function h ° (x), decrease with an increase in Young's modulus of the main material. Transition from regime (a) to regime (b) of accretion leads to an increase in lengths of the regions, where the optimal thickness takes its extreme values. For regime (b), an increase in mass density of the main material leads to a decrease in lengths of the regions in which the optimal thickness is extreme.
8.5. Optimal Solidification of a Spherical Pressure Vessel
569
3. Aging of the main material essentially affects the optimal thickness of a growing beam. With an increase of the characteristic time of aging, lengths of the regions in which the optimal thickness is extreme decrease, and for large Ta values, the region with h ° (x) = hi disappears entirely. For large rates of accretion, the optimal shape of a growing beam is determined by Eq. (8.4.27). With an increase in the rate of material supply, the optimal function h ° (x) "slides" as a rigid body along the abscissa axis.
8.5
Optimal Solidification of a Spherical Pressure Vessel
An optimal regime of cooling is found for a spherical polymeric pressure vessel. The problem consists in determining the temperature of the outer boundary of a mold during the solidification process that would maximize the load-bearing capacity of the vessel after its manufacturing. An explicit expression for the temperature is derived as a function of position of the interface between the solid polymer and the melt. To determine the interface location at an arbitrary instant, a nonlinear ordinary differential equation is developed and solved. The exposition follows Arutyunyan and Drozdov (1985, 1988b). Solidification of polymers is a complicated physical and chemical process, in which several subprocesses may be distinguished. Models for this process are quite sophisticated, and they require a large number of parameters to be found experimentally [see, e.g., Flemings (1974), Isayev (1987), and Kurz and Fisher (1986)]. Thus, it is quite natural to simplify the mathematical description as much as possible to account only for basic phenomena in design of the manufacturing process. As common practice, the following hypotheses are introduced: • The material in the liquid phase is modeled as an ideal fluid, the material in the solid state is treated as an elastic solid, and the effects of viscosity and plasticity are neglected [see, e.g., Aleong and Munro (1991), Cai et al. (1992), Eduljee and Gillespie (1995), Gutowski et al. (1987), Hyer et al. (1986), Hyer and Rousseau (1987), and Zakhvatov et al. (1990)]. • The activation energy of a polymer is rather high, which implies that the interface between solid and liquid phases may be treated as a solidification front [see, e.g., K i m et al. (1995), Klychnikov et al. (1980), Tien and Richmond (1982), and Turusov et al. (1979)]. • Motion of the interface between the solid polymer and the melt is determined from the standard Stefan problem for the heat conduction equation [see, e.g., Carslaw and Jager (1959)]. This motion is assumed to be independent of stresses built-up in the solidifying shell [see, e.g., Hojjati and Hoa (1995), Kim et al. (1995), and Liu (1996)].
Chapter 8. Optimization Problems for Growing Viscoelastic Media
570
Based on these assumptions, we calculate residual stresses built-up in a thick-walled polymeric pressure vessel during its solidification, as well as stresses arising in the vessel after loading. Afterward, we derive a cooling program that maximizes the load-bearing capacity of the pressure vessel. The latter is an essential novelty of the present analysis, since the optimality criteria suggested in previous studies [see, e.g., Afanasiev (1981), Egorov et al. (1982), Harper (1985), Lee and Weitsman (1994), Protasov et al. (1983), Weitsman and Harper (1982), and the bibliography therein] concentrated on the cooling process and did not account for successive loading of a solidified pressure vessel.
8.5.1
F o r m u l a t i o n of t h e P r o b l e m
We study solidification of a polymeric melt in a spherical mold consisting of two parts. The inner part is a sphere with radius a l, the outer part is treated as a solid space containing a spherical opening with radius a2. Centers of the inner and outer parts of the mold coincide (see Figure 8.5.1). At the initial instant t = 0, the mold is heated to the melting temperature 0 , , and the melt at the temperature (9, is poured into the mold. Afterward, the external
a(t)
./.ii!!!ii
,
i.
2
. . . . .
•
.
.
.
.
.
]:: 3 ::'|
4
.
r"
.
.
/b_ .
.
. .
. .
. .
. .
. .
. .
! .
.
!.
. .
. .
. .
. .
. .
. .
. .
.
.
.
.
.
.
.
.
.
.al
a2
Figure 8.5.1: A solidified thick-walled spherical shell in a mold. 1" inner part of the mold; 2: polymeric melt; 3: solid polymer; 4: outer part of the mold.
8.5. Optimal Solidification of a Spherical Pressure Vessel
571
boundary of the mold is quenched to some temperature O0 < (9,, and the melt begins to solidify. Solidification occurs in the interval [0, T] due to cooling of the external boundary of the mold according to the law 0 = O(t),
O(0) = O0.
(8.5.1)
At instant t ~ [0, T], the substance occupies a region al <- r <- a(t) in the liquid state and a region a(t) <- r <- a2 in the solid state. Here {r, 0, ~b} are spherical coordinates, and a(t) is a function to be found. Denote by ~-*(r) the instant when a material portion with polar radius r is transformed from the liquid to the solid state. The function ~-*(t) is inverse to the function a(t): "r*(a(t)) = t,
a("r*(r)) = r.
The natural (stress-free) configuration of the molten polymer coincides with the reference configuration of the heated mold. The natural configuration of the solid polymer is obtained from the actual configuration of the melt at the instant of solidification ~'* after homogeneous volume deformation with some shrinkage strain A. For simplicity, the parameter A is assumed to be constant. The polymeric medium is incompressible in both liquid and solid states. Assuming the mechanical behavior of the polymer to be linear and accounting for the temperature deformation, we write the following constitutive equations: • In the liquid state 6-(t) = - p ( t ) ) l ,
e(t) = 3at [O(t) - O , ] .
(8.5.2)
• In the solid state 6-(t) = - p ( t ) I
+ 2 G [0(t) - ~,(r*)],
E(t) = ~(~-*) - )t + 3C~s [O(t) - 19,].
(8.5.3) Here 6- is the stress tensor; ~ is the strain tensor for transition from the initial to the actual configuration; I is the unit tensor; p is pressure; E is the first invariant and is the deviatoric part of the strain tensor 5; G is a shear modulus; and C~l, as are coefficients of thermal expansion in the liquid and solid states, respectively. The mold is treated as a linear elastic medium that obeys the constitutive equation 6-(t) = K0 { e(t) - 3a0 IO(t) - 19,] } I + 2G0O(t),
(8.5.4)
where K0 is a bulk modulus, Go is a shear modulus, and a0 is a coefficient of thermal expansion. At instant T, solidification finishes, and the pressure vessel remains in the mold until its temperature reduces to the room temperature O r . Afterward, it is removed from the mold and a pressure P is applied to its external surface r = a2. The vessel deforms, and some stresses arise in it. As is well known, the load-bearing capacity
572
Chapter 8. Optimization Problems for Growing Viscoelastic Media
of a thick-walled spherical shell is maximal provided the stress intensity is uniform across its cross-section. Our objective is to derive such a program of cooling 0 = O(t), which ensures a uniform distribution of the stress intensity in the pressure vessel.
8.5.2
Temperature Distribution
To calculate temperature in a solidifying melt, we assume that the function O(t) changes so slowly that at any instant t thermodynamic equilibrium is established in the system. Owing to the spherical symmetry, the equilibrium temperature field @(t, r) satisfies the Laplace equation
020
2 019 Or 2 (t, r) + r --~r (t' r) = 0
(8.5.5)
in the mold and in the solid polymer. Temperature conductivity of the melt is determined by convective flow, which implies that temperature of the melt is independent of the polar radius. Since temperature at the interface r = a(t) equals the melting temperature ®,, we obtain O ( t , r ) = ®,,
al <-- r <-a(t).
(8.5.6)
It follows from Eq. (8.5.6) that heat flux from the melt to the boundaries r = al and r = a(t) vanishes. The boundary condition on the surface r = a l implies that temperature of the inner part of the mold remains constant as well @(t, r) = ®,,
0-
r -< al.
(8.5.7)
Neglecting heat flux from the melt, we write the Stefan conditions at the interface r = a(t) as O(t, a(t)) = O,,
Pl-~-~(t) = K
(t, a(t)).
(8.5.8)
Here P is mass density, K is temperature conductivity of the solid polymer, and/.L is latent heat of solidification. In the outer part of the mold, the temperature @(t, r) satisfies Eq. (8.5.5) with the boundary condition (8.5.1) @(t, oo) = O(t).
(8.5.9)
On the interface between the solid polymer and the outer part of the mold, temperatures and heat fluxes coincide O(t, a2 -- 0) = O(t, a2 + 0),
~O ~O n: O-----~(t,a2 -- 0) = K0--~-r(t,a 2 + 0),
where K0 is temperature conductivity of the outer part of the mold.
(8.5.10)
573
8.5. Optimal Solidification of a Spherical Pressure Vessel
The general solution of Eq. (8.5.5) has the form O(t,r) = Al(t)r -1 + A2(t),
a(t) <-- r <-- a2,
O(t,r) = Bl(t)r -1 + B2(t),
a2 --< r < ~,
(8.5.11)
where the unknown functions Ak(t) and Bk(t) are determined from boundary conditions. Substitution of expressions (8.5.11) into Eqs. (8.5.8) to (8.5.10) implies that A1
A1 B1 ~ + A 2 = ~ + B2, a2 a2
+ A2 = ~),, a
KA1 = KoB1,
da A1 p l ~ - ~ = -- K a2 .
B2 = O,
(8.5.12)
Solving Eqs. (8.5.12), we find that Al(t) = [19,-O(t)] Ia-~t) + (~00 - 1 ) 1 1 - 1 A2(t)=O(t)+
(f-Ko - 1 )
IO*-a2 1 -O(t)] [a-~t)+ ( K1 J --l 1- -) K 0--a2
1 (~ - 1 )111 --
a--~+
Bl(t) = K [19, - O(t)] K0
K0
a2
(8.5.13)
B2(t) = O(t). Radius of the interface a(t) obeys the ordinary differential equation (K )llda K a2[ 1 a-~ + --K0- 1 ~- ( t ) - - p-/x [ ® , - O(t)],
8.5.3
Stresses
a(0) = a2.
and Displacements
Owing to the symmetry of deformation, the displacement vector from the initial to actual configuration equals (8.5.14)
fi = u(t, r)e.r,
where u(t,r) is a function to be found. It follows from Eq. (8.5.14) that the only nonzero components of the strain tensor ~ are 0u Err
--
Or'
u
eoo= e66 - r"
The first invariant • of the strain tensor ~ is calculated as E =
Ou + 2u Or r
(8.5.15)
574
Chapter 8. Optimization Problems for Growing Viscoelastic Media
and the nonzero components of the deviatoric part ~ of the strain tensor ~ equal 2(On err = -3 Or
u)
-r
,
e00 = e 6 ~ = - ~
l(0UrU - _u) . r
(8.516)
We begin with stresses in the mold. Combining expressions (8.5.15) and (8.5.16) and the constitutive equation (8.5.4), we find the nonzero components of the stress tensor &
Orrr = go
E
] 4oo(o u) 1 oo(o u)
o3rOu+ ~2Ur -3c~°(19 - 19,) + - ~
troo =o"4, 4) = Ko
Or
r
Or
-r
- --~
'
-~r - r
"
(8.5.17) Substitution of expressions (8.5.17) into the equilibrium equation 030"rr Or
"k- --2( O.r r
_
0"00) ="
(8.5.18)
0
r
implies the differential equation
032U
20u 2u Or 2 + r Or - r2 -
9c~oKo 019 3Ko + 4Go Or .
(8.5.19)
Temperature 19 in the inner part of the mold is independent of r, and Eq. (8.5.19) reads 02u Or 2
+
20u
2u
r Or
r2
- 0,
0 -< r -< al.
(8.5.20)
The general solution of Eq. (8.5.20) is (8.5.21)
u ( t , r ) = C l ( t ) r + C2(t)r -2,
where C1(t) and C2(t) are functions to be found. Since u(t, 0) = 0, it follows from Eqs. (8.5.17) and (8.5.21) that
Orrr = Oroo = ~r4~4~ = 3KoC(t),
u = C(t)r,
0 <- r <- al,
(8.5.22)
where C(t) is an unknown function. Combining Eqs. (8.5.11) and (8.5.19), we find that in the outer part of the mold 02U Or 2
20u +
2u
r Or
--
r2
--
--
9aoKoBl(t) (3Ko + 4Go)r 2'
a2 -< r < ~.
(8.5.23)
A particular solution of Eq. (8.5.23) reads 9 c~oKoB1(t)
9ao KKoA1(t)
uo(t) = 2(3Ko + 4Go) - 2Ko(3Ko + 4Go)"
(8.5.24)
575
8.5. Optimal Solidification of a Spherical Pressure Vessel
The general solution of the nonhomogeneous equation (8.5.23) equals a sum of its particular solution (8.5.24) and the general solution (8.5.21) of the homogeneous equation (8.5.20) a2 --< r < ~.
u(t, r) = uo(t) + C l ( t ) r + C2(t)r -2,
(8.5.25)
Substitution of expression (8.5.25) into Eq. (8.5.17) implies that Orr r =
g 0 3Cl(t) + 2u0(t)r - 3c~°(19(t'r) - 19,)
I
2uo(t)
troo = tr66 = Ko 3 C l ( t ) + 2G0 3r
2C2(t) r2
r
~
r2
+ uo(t) ,
- 3c~o(O(t, r) - (9,)]
1
(8.5.26)
a2~r
+ uo(t) ,
The outer boundary of the mold is stress-free 6rrr(t,
~ ) = 0.
(8.5.27)
We substitute expressions (8.5.26) into Eq. (8.5.27), use condition (8.5.9), and obtain Cl(t) = ao [O(t) - 19,].
(8.5.28)
We now combine expressions (8.5.25) and (8.5.26) with Eq. (8.5.28), set r = a2, and arrive at the formulas
1
u(t, a2) = uo(t) - ~o [i~, - O(t)] a2 + a--~2C2(t), O'rr(t, a2) = -3c~0K0 [l~(t, a 2 ) - O(t)]
8Go C2(t). 2 (3Ko - 2 G o ) u o ( t ) - -~a32 + 3a----7
(8.5.29)
It follows from Eqs. (8.5.2), (8.5.6) and (8.5.15) that • =
Ou Or
+
2u r
= O,
al <-- r <- a(t).
(8.5.30)
Integration of Eq. (8.5.30) yields u(t, r) = 13o(t)r -2,
(8.5.31)
where/30(t) is a function to be found. Since radial displacements in the inner part of the mold and in the melt coincide at the boundary r = al, Eqs. (8.5.22) and (8.5.31) imply that/30 = Ca~. We substitute this expression into Eq. (8.5.31) and obtain
u(t,r)= C(t)al ( ~ ) 2
al <-- r <-a(t).
(8.5.32)
576
Chapter 8. Optimization Problems for Growing Viscoelastic Media
This equality together with Eq. (8.5.16) implies that err(t,r)
=
-2C(t) (~-~)3
eoo(t,r) = erk4(t,r) = C(t) ( a l )3 F
(8.5.33)
al <-- r <-- a(t).
Substitution of expression (8.5.2) into the equilibrium equation (8.5.18) yields 030"rr
Or
-
al <-- r <-- a ( t ) .
O,
Integrating this equality from r = al to r = a(t) and using Eq. (8.5.22), we find that (8.5.34)
O'rr(t, a(t)) = 3KoC(t).
It follows from Eqs. (8.5.3), (8.5.15), and (8.5.30) that in the region occupied by the polymer in the solid state Ou
2u
Or
r
- - + - - = 3C~s [ ® ( t , r ) - 0,] - A. Using Eq. (8.5.11), we rewrite this equality as follows: 1 0 (rZu) = 3 a ~A1 + 3as(A2 - O , ) -
r 2 Or
A.
r
(8.5.35)
Integration of Eq. (8.5.35) results in
3
u ( t , r ) = -~otsAl(t) + -- a S A l ( t ) + 2
EOts(A2(t) -
0,)-
r +
/31(t)
[as(O(t,r)-O,)-Alr+
1,2
[31(t) r2 '
a(t) <- r <- a2,
(8.5.36) where/31(t) is an unknown function. Equating radial displacements at the interface r = a(t), we find from Eqs. (8.5.8), (8.5.32), and (8.5.36) that
1[
os
C(t) = a---{1 /31(t)+ - ~ A l ( t ) a 2 ( t ) - - ~ a
(t)
.
(8.5.37)
Equating radial displacements in the mold and in the solid polymer at the boundary r = a2, we obtain from Eqs. (8.5.29) and (8.5.36) C2(t) =/31(t) + [ 2 A l ( t ) +
u0(t)] a~
as(O(t, a2) - 0 , ) + a o ( O , - O(t)) - -~ a 3.
(8.5.38)
8.5. Optimal Solidification of a Spherical Pressure Vessel
577
Equations (8.5.29) and (8.5.38) imply that
-
( 3 K o + ~8G o )
ao [19. - O(t)] + ~ Go
-
os
- ~2a2 Al(t)J
2 + ~ ( 3 K o + 2Go)uo(t). 3a2
(8.5.39)
Substituting expression (8.5.36) into Eq. (8.5.16), we find that
~31(t)] err(t, r) = --_2 r [as -~-Al(t) + F2 '
eoo(t, r) = e4~4,(t,r) = 1 F
A 1(t) + ]31(t) r2 1 .
(8.5.40)
It follows from Eqs. (8.5.3) and (8.5.40) that the nonzero components of the stress tensor 6- in the solidified polymer are calculated as 4 G [ /31(t)- C( T*(r))a~ + -~Al(t)r as 2] , --~-
trrr(t,r) = - p ( t , r ) -
troo(t, r) = o66(t, r) = -p(t, r) +-~-2G [~1 ( t ) - C('r*(r))a~ + 2 A1 (t) r2] •
(8.5.41)
We integrate the equilibrium equation (8.5.18) from r = a(t) to r = a2 and substitute expressions (8.5.41) into the obtained equality. As a result, we find that
Orrr(t, a2) - trrr(t,a(t)) - 12G
fa'2[ Os t) ~31(t)- C(T*(r))a3 + -2Al(t)r2
-~ = O. (8.5.42)
The integrand is transformed with the use of Eq. (8.5.37) OgS
/31(t)- C(r*(r))a~ + -~Al(t)r 2 1
as [Al(t)- Al('r*(r))] r 2 + ~hr 3 "-- /31 (t) -- ~31(T* (r)) + -~-
(8.5.43)
Substitution of expressions (8.5.34), (8.5.37), (8.5.39), and (8.5.43) into Eq. (8.5.42) implies that
\---~-1 + -~a32 i l l ( t ) -
3°/°K° q- '~OlsaO [19(t,a 2 ) - 19,]
Chapter8. OptimizationProblemsfor GrowingViscoelasticMedia
578
( -
)
C~s[3K°(a(t)) 2 8 G °
3Ko+~Go
c~o
8
2Go)uo(t) +
3a2 (3Ko +
= lZG
-
[19, - O(t)]
-2- t.~ al
~ al
+
/
Al(t)
3a2
[ (a(t) >+ ooOlna l a23 3Ko
fa( 2 { [/31(t)-/31(~'*(r))]
\ al
+ -OgS ~-[al(t)-
al(~'*(r))] r 2 }
t)
dr
r4 "
(8.5.44) It follows from Eqs. (8.5.11) and (8.5.13) that
O(t) =
19(t, a2) -
-
=
1
~- Bl(t) + B2(t) - O(t) = ~ A l ( t ) , Koa2 a2 +
~ -
1
KO
(8.5.45)
Al(t).
Substitution of expressions (8.5.24) and (8.5.45) into Eq. (8.5.44) yields Mo~31(t) + -
12G
M(a(t))A1(t) = N(a(t))
/ai~{
[/31(0
-- it31(~'*(r))]
os
+ -~- [ A l ( t )
-
Al('r*(r))]
r 2
> dr
t)
r4, (8.5.46)
where
Mo-
M(r)
=
3Ko 8Go a ~1 + 3a 3' 3aoKo + as + -2-
N(r)
-~asGo
Koa2 +
r [3Koa---~- (~l) ~+ ~o 3a2
= -~ 3Ko
+ ~Go-
(,~)
3Ko+~-Go
ao
E~ ( ~ ), +
~-1 KO
-a2
_ 3C~oKKo 3Ko + 2Go Koa2 3Ko + 4Go'
12Gln
.
(8.5.47)
We set t = ~-*(r) in Eq. (8.5.46) and introduce the functions /]l(r) = Al('r*(r)),
~l(r) = fll('r*(r)).
(8.5.48)
8.5. Optimal Solidification of a Spherical Pressure Vessel
579
In the new notation, Eq. (8.5.46) reads Mo/31(r) + M(r)Al(r) = N ( r ) - 12G
fra2/
as [Al(r) + 5-
[/3~(r) - /31(s)]
Al(S)] S2 } d s
(8.5.49)
-~" s
Setting r = a2 in Eq. (8.5.49), we obtain ~1(a2) = M001IN(a2) - M(a2)~,,l(a2)] .
(8.5.50)
Differentiation of Eq. (8.5.49) with respect to r implies that
d~l dM d_~ Mo---~-r (r ) + -~-r (r)Al(r) + M(r) (r) _ dN(r) - 12G frr a2 [d/31 (r) + ~as dA1 (r)s 2 ds dr [~ 2 dr 7 -
d N ( F ) - 4 G ( -/g
a 31) d/31 (1 1)dA1 - -- ---~r ( r ) - 6otsG - a2 ~ ( r ) "
It follows from this equality that . d[31 d.Ttl, , dM dN mo(r)---~-r(r) + m(r)---d-~-rtr) + - d T ( r ) A l ( r ) - -~rr(r) = 0,
(8.5.51)
where 1), mo(r) = Mo + 46 ( 1r3 _ a32
re(r)
M ( r ) + 6 a s G ( r1
1) a2 (8.5.52)
Given program of cooling O(t), the ordinary differential equation (8.5.51) with the boundary condition (8.5.50) determines the function/31(t). Afterward, displacements in the solidifying polymer are found by formulas (8.5.32) and (8.5.36), and stresses are calculated according to Eqs. (8.5.22) and (8.5.41). 8.5.4
S t r e s s e s in a P r e s s u r e V e s s e l after C o o l i n g
After solidification and cooling of the polymeric pressure vessel to the room temperature Or, it is removed from the mold and loaded by a constant external pressure P. Owing to the symmetry of loading, the displacement vector ~0 for transition from the initial to actual configuration at the final instant To has the form similar to Eq. (8.5.14) fi(To, r) = U(r)e.r,
Chapter 8. Optimization Problems for Growing Viscoelastic Media
580
where the function U(r) is determined from the incompressibility condition (8.5.3). Bearing in mind Eqs. (8.5.2) and (8.5.8), which implies that e(~-*) = 0 during the solidification process, we obtain from Eqs. (8.5.3) and (8.5.15)
dU 2U + = 3~s(Or dr r
--
l~,)
--
h.
(8.5.53)
Integration of Eq. (8.5.53) yields
U(r)
[
=
r + r--2,
Ols(~ r -1~),)-
(8.5.54)
where the constant/3 is to be found. Combining Eqs. (8.5.16) and (8.5.54), we find the nonzero components of the deviatoric part ~ of the strain tensor
2/3 err
=
r3'
--
e00 = e+,~ -
/3 r3.
(8.5.55)
Substituting expressions (8.5.33) and (8.5.55) into the constitutive equation (8.5.3), we obtain the nonzero components of the stress tensor 4G
O'rr(r) -- - p ( r ) -
--~ [~ - C(~'*(r))a~] ,
2G oroo(r) = o-4,6(r) = - p ( r ) + --~ [~ - C(~'*(r))a~] . According to Eq. (8.5.37), these equalities can be presented as
Orrr(F)
-~
-p(r)-
4G[
-~
~
-
~I(F)-
as h 3] -~-Al(r)r 2 + ~ r ,
ash 31 . 2 + -~r ~roo(r) = cr+4~(r) = - p ( r ) + ~2 G [ [3 - ~l(r) - -~Al(r)r (8.5.56) It follows from Eq. (8.5.56) that the stress intensity E = (2~"
~)1/2
is calculated as = -'~3(O'rr -- 0"00).
(8.5.57)
Substitution of expressions (8.5.56) into Eq. (8.5.57) yields ~(r) -- 4G~//3 r 3 [ ~l(r) + z-x-Al(r)r as.. 2 - 3h r3 - ~ 1
(8.5.58)
581
8.5. Optimal Solidification of a Spherical Pressure Vessel
The load-bearing capacity of a spherical pressure vessel is maximal provided the stress intensity ~ is uniform ~(r) = ~0.
(8.5.59)
To calculate the constant ~0, we write the equilibrium equation (8.5.18), where the second term on the fight-hand side is transformed with the use of Eqs. (8.5.57) and (8.5.59)
d°'rr + ~ 0 ~ dr
r
-0.
Integrating this equality from al to a2 and using the boundary conditions
Orrr(t,a l )
Orrr(t,a 2 )
= 0,
--
-P,
we find that P
E0 =
•
(8.5.60)
V ~ ln a2/al
It follows from Eqs. (8.5.58) and (8.5.59) that ~° + A) r3 - Tc~s~ ( r ) r 2 +/3. /31(r) = ( 4Gv/~
(8.5.61)
We differentiate Eq. (8.5.61) with respect to r and substitute the obtained expression into Eq. (8.5.51). As a result, we arrive at the equation dA1 (r) + f(r)741(r) = F(r), dr
(8.5.62)
where f(r) =
--~-r( r ) - Olsrmo(r)
r e ( r ) - -~rZmo(r)
~ \ + ~.) rZmo(r) F(r) = [dN --~r (r) - ]{ ~ o4G
-1 . Ira(r)- -2-°~Sr(r)] 2mo
(8.5.63) It follows from Eq. (8.5.13) that Al(0) = K°a2(®, - O0), K
which implies the boundary condition for Eq. (8.5.62) Al(a2) = K°a2(®, - ~0). K
(8.5.64)
582
Chapter 8. Optimization Problems for Growing Viscoelastic Media
The solution of the ordinary differential equation (8.5.62) with the boundary condition (8.5.64) reads Al(r) = K°a2(O, - O0)exp
K
[~r a2f ( s ) d s ]
fr a2F(p)exp [fP
-
f(s)ds
] dp. (8.5.65)
Returning to the initial notationl we find that A l ( a ( t ) ) = K°a2(O, - Oo)exp
K
t)
f(s)ds
-
t)
F(p)exp
t)
f(s)ds
dp.
(8.5.66) Combining Eqs. (8.5.13) and (8.5.66), we arrive at the implicit solution to the optimization problem: the optimal temperature of cooling O is determined as a function of position a of the interface between the solid polymer and melt =
-
+
- 1
(8.5.67)
a l (a(t)).
To develop an explicit formula for the optimal cooling program, it suffices to integrate the Stefan equation (8.5.12)
da
aZ(t)--77(t) = -
ar
K___a p/x
l(a(t)),
(8.5.68)
with the function (8.5.66) in the fight side. We confine ourselves to determining the optimal time of cooling T. For this purpose, we rewrite Eq. (8.5.68) as aZda
K -
-
--~
/]1 (a)
dt,
p/z
and integrate the obtained equality from t = 0 to t = T. Bearing in mind that a(O) = a2 and a ( T ) = al, we find that T = p---~
K
fa2 ~o r 2 dr L1
(8.5.69)
Al(r)
8.5.5 Numerical Analysis We introduce the dimensionless variables and parameters r,M1,-
r
al
,
A1, =
a2 d M c~oGo d r '
C~o ~
~a2
A1,
a3
mo, = --~2mo, GO
m,-
a2 d N
N1, - GoA d r '
f, = azf,
a2
a0Go
m,
ao F, = -~F,
583
8.5. Optimal Solidification of a Spherical Pressure Vessel
a,
w
a2 al
:~,
K, -
G
G, -
Go
:~ox~ -
Ko
4Gh
a, -
Go
K,
n
hK
zX O, = aoKo hK (®*
'
as ao
~ T .
r~
--
0o),
K KO (8.5.70)
aoptza 2
It follows from Eqs. (8.5.47), (8.5.52), and (8.5.70) that
8 ( 1 )
77- 1 ,
m o , ( r , ) = -5 +3K-a3* + 4 G *
m,(r,)=
K, 3 K , + ~ a ,
l
(3 K , +
( 3K, a3,r,2 + 5 8 )
+ ~a,
Ml,(r,) = -
+
3K, +
r, + K , -
~
3K,+2
-3K*K*~-7,+4
1)
_,_
(1_r,,-1),
r--f, + 3a*K*a3*r*'
2 1 Nl,(r,) = 3K, a3,r, + 4 G , - - .
(8.5.71)
r,
In the new notation, Eq. (8.5.62) reads dA1, dr,
AI,(1) = A O,,
+ f , ( r , ) A 1 , = F,(r,),
(8.5.72)
where M1, - a, mo, r, f* = m , - ~1 a,mo, r 2'
N1, - (1 + £ , )mo, r 2
F, =
m,-
l a, mo, r2
.
(8.5.73)
The optimal time of cooling is calculated as
T, =
fal
r 2 dr
1 Al,(r)"
(8.5.74)
To study the effects of material and geometrical parameters on the optimal time for cooling T,, we integrate Eq. (8.5.72) numerically and calculate the integral in Eq. (8.5.74). To transform the dimensionless parameter K,, it is convenient to introduce Young's modulus E0 and Poisson's ratio 1'0 of the mold. Since eo Ko = 3 ( 1 - 2vo)'
Eo Go = 2(1 + vo)'
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
584
we find that 2(1 + vo)
K~
3(1 - 2vo)"
We assume that solidification of a polymeric melt occurs in a metal mold and set vo = 0.3, which is The perature pressure The
c~, = 2.0,
typical of polymers and metals used in applications. optimal time of solidification T, is plotted versus the initial jump in temof the mold in Figures 8.5.2 and 8.5.3, and versus the final thickness of the vessel in Figures 8.5.4 and 8.5.5. results are rather surprising.
0.3
,.......... ~ "-... " \ ....-
I -2
I
I
I
I
I
I I log A O,
I 2
Figure 8.5.2: The dimensionless time T, for optimal cooling of a polymeric pressure vessel versus the dimensionless jump in the initial temperature A O, for K, = 0.01, ~, = 0.5, and a, = 2.0. Curve 1: G, = 0.1. Curve 2: G, = 0.5. Curve 3: G, = 1.0.
8.5. Optimal Solidification of a Spherical Pressure Vessel
585
"3
2
0
I
-2
I
i
I
i
i
a
log A O,
2
Figure 8.5.3: The dimensionless time T, for optimal cooling of a polymeric pressure vessel versus the dimensionless jump in the initial temperature A O, for G, = 0.1, ~, = 0.5, and a, = 2.0. Curve 1: K, = 0.01. Curve 2: K, = 0.1. Curve 3: K, = 0.5.
When the ratio of thermal conductivities for the polymer and the mold is small (which is typical of conventional materials), the optimal time of cooling depends extremely weakly on the initial jump in the mold temperature. For example, an increase in A O, by three orders of magnitude leads to a decrease in T, by several percent. Only for very large A O, values (about 100, which implies that 19, - O0 equals hundreds of degrees), the effect of the initial jump in temperature becomes pronounced. For a fixed A O,, the optimal time of cooling increases with the growth of the dimensionless shear modulus G, of the polymer. However, this dependence is significant only for relatively small initial jumps in the mold temperature, and it becomes negligible for large A O, values. The dependence of the optimal time of cooling on the ratio K, of temperature conductivity is essentially nonmonotonic. For small jumps in the initial temperature
Chapter 8. Optimization Problems for Growing Viscoelastic Media
586
/
"....
,,
", % %
g
/
L 0.1
2
-
al/a2
0.9
Figure 8.5.4: The dimensionless time T, for optimal cooling of a polymeric pressure vessel versus the dimensionless ratio al/a2 for r, = 0.01, G, = 0.1, and A O, = 1. Curve 1" ~, = 0.1. Curve 2: ~, = 0.5. Curve 3: ~, = 1.0.
(which correspond to large times of cooling), the dimensionless time T, sharply increases with the growth of K,. For large jumps in the initial temperature, an inverse dependence can be observed: the optimal time of solidification decreases in K,. This may be explained as follows: for large A O, and small K, values, huge thermal gradients arise in a solidifying vessel, which produce large residual strains. To diminish these strains, which reduce the load-bearing capacity, the optimal regime of cooling should be retarded and an appropriate time of solidification T, should increase in r,. In contrast, for small jumps in the initial temperature A O, and large ratios K,, the solidification process is slow and thermal gradients are extremely small. Residual stresses built up in a spherical vessel are not sufficient to equilibrate nonuniform stresses arising under external pressure P. This implies that the optimal rate of solidification should be increased, and the corresponding time of cooling should decrease in K,.
8.5. Optimal Solidification of a Spherical Pressure Vessel
_
g-"
587
2
',
-
'....
I 0.1
al/a2
0.9
Figure 8.5.5: The dimensionless time T, for optimal cooling of a polymeric pressure vessel versus the dimensionless internal ratio al/a2 for G, = 0.1, ~, = 0.1, and A O, = 1.0. Curve 1: K, = 0.01. Curve 2: K, = 0.1. Curve 3: K, = 1.0.
The optimal time of cooling is plotted versus the ratio a l/a2 of the internal and external radii of the spherical vessel in Figures 8.5.4 and 8.5.5. The dependence T,(a,) is nonmonotonic: the time T, increases for small ratios (very thick shells), reaches its maximum for pressure vessels in which the internal radius equals about half the external radius, and decreases sharply for large ratios (very thin shells). Such a behavior can be explained by an interaction between residual stresses built up in the solidification process and stresses arising in a spherical shell loaded by external pressure P. For small al/a2 values, the vessel may be treated as an elastic space with a small opening. As is well known, large stresses arise in the vicinity of an opening under loading. To equilibrate these stresses, large residual stresses should be produced when the shell is manufactured. These stresses are results of large temperature gradients in the solidifying medium. Since the magnitude of thermal
588
Chapter 8. Optimization Problemsfor Growing ViscoelasticMedia
gradients is proportional to the rate of cooling, the optimal time of cooling should decrease when the radius of opening a l decreases. In contrast, for large al/a2 values, thickness of the pressure vessel is very small and the stress intensity is practically uniform across its cross-section. This implies that residual stresses, as well as the temperature gradients, should be rather small. On the other hand, for a thin-walled shell, the total amount of material subjected to phase transition is extremely small, and as a result the time of solidification is small as well. The dimensionless time of cooling T, provides the optimal ratio between the "smallness" of thermal gradients and the "smallness" of the volume to be solidified. The optimal time of cooling T, decreases with the growth of the dimensionless parameter ~,. According to Eqs. (8.5.60) and (8.5.70), ~, is proportional to external pressure P, which confirms the preceding explanation. Indeed, the stresses arising in a spherical shell are proportional to external pressure. Appropriate residual stresses that equilibrate these stresses should be proportional to P as well. Since residual stresses are proportional to the temperature gradients, the optimal thermal gradients should increase with the growth of external pressure. The latter means that the rate of cooling should increase in P, because the thermal gradients are proportional to the rate of solidification. As a result, an increase in P should lead to a decrease in the optimal time for cooling, which is demonstrated in Figure 8.5.4. Figure 8.5.5 shows a nonmonotonic dependence of the dimensionless time of cooling T, on the dimensionless ratio K,. For very thick pressure vessels, the optimal time increases in K,, whereas for relatively thin-walled shells it decreases in r,. An explanation of this phenomenon was given earlier. It is worth mentioning the effect of the parameters ~, and K, on the point of maximum of the function T,(al/a2). This point moves to large al/a2 values with the growth of ~, (rather weakly) and to small al/a2 values with an increase in K,. Concluding Remarks An optimal regime of solidification is studied that ensures the maximum load-bearing capacity of a spherical polymeric pressure vessel. An explicit formula is derived for the optimal regime of cooling and for the optimal time of solidification. The effects of material and geometrical parameters are analyzed numerically. The following conclusions are drawn: 1. The optimal time of cooling decreases with the growth of the initial jump of temperature A ag, and increases with an increase in the dimensionless shear modulus G,. 2. The dependence of the optimal time of solidification on the ratio of heat conductivities of the polymer and the mold is essentially nonmonotonic: T, increases in K, for small A O, and small a l/a2 values and decreases otherwise. 3. The dependence of the optimal time of cooling on thickness of the polymeric pressure vessel is also nonmonotonic: T, increases in the ratio al/a2 for very thick shells and decreases for thin-walled ones. Some physical explanations are suggected for the nonmonotonic dependencies of the optimal time for cooling on the material and structural parameters.
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This Page Intentionally Left Blank
Index Abel kernel, 28 Accretion, 337 continuous, 338 discrete, 340 Activation area, 111 energy, 112 enthalpy, 267 volume, 111 Adaptive link, 41 Adhesive layer, 480 Aftereffect, 515 Aging, 35 Annealing, 81 Biological tissue, 413 Biostimulus, 415 Boltzmann's constant, 267 Built-up portion, 337 Chain, 43 rule, 72 Column, 532 Condition compatibility, 340 Euler-Lagrange, 218 Legendre-Hadamard, 219 self-similarity, 238 Stefan, 572 Cone, 446
Configuration actual, 338 intermediate, 174 natural, 338 reference, 339 unloaded, 364 Crack, 2, 212 Creep function, 38 kernel, 38 measure, 37 Crosslink, 43 strong, 279 weak, 279 Curing, 276 Dashpot, 26 Briant, 108 Eyring, 108 fractional, 32 Newtonian, 32 power-law, 109 Debonding, 480 Deformation gradient, 6 relative, 11 Derivative corotational, 20 generalized, 22 Jaumann, 20 Oldroyd, 21 covariant, 6 593
594 Derivative (cont.) of the Dirac function, 70 fractional, 28, 179 material, 179 Dilatation, 72 Dislocation, 112 Dome, 464
Index memory, 208 Mittag-Leffler, 34 of positive type, 71 strong, 71 stepwise, 512 Wright, 34 Gel, 276
Energy free (Helmholtz), 72 internal, 73 Entanglement, 43 Entropy, 73 Equation Andrade, 268 Arrhenius, 267 Doolittle, 113 equilibrium, 75, 21 heat conduction, 314 Hooke-Eyring, 111 Hooke-Norton, 111 hybrid, 269 Laplace, 572 multiple-integral, 124 single-integral, 117 WLF, 268 Fading memory, 72 Fiber bundle, 542 Formula Finger, 206 Kohlrausch-William-Watts, 57 Stokes, 75, 218 Function aging, 61 chain-breakage, 44 chain-distribution, 44 chain-reformation, 44 damping, 208 Dirac (delta), 70 Euler (gamma), 28 fractional-exponential, 58 generalized, 59 Heaviside, 305
Hardening, 356 Heat capacity, 72 flux, 572 latent, 572 Humidity, 39 Hypotheses Kirchhoff, 467 mapping, 302 Interface, 317 Lame
parameter, 221 problem, 375 Laplace transform, 56, 71 Law associated, 356 Hooke's, 26 Newton's, 26 of thermodynamics first, 77 second, 77 Wolff's, 423 Load-bearing capacity, 542 Loss tangent, 96, 283 Lyapunov functional, 134 Mandrel, 313, 542 Mass density, 72 Melting, 572 Membrane theory, 467 Mixture, 173 Modulus bulk, 49
595
Index
complex, 281 dynamic, 96 loss, 96, 292 shear, 49 storage, 283 Young's, 35 Motion micro-Brownian, 43 random, 43 rigid, 12 Network, 43 Operator fractional, 30 nabla, 5 Overstress, 111 Plastic work, 356 Poisson's ratio, 48 Polymer amorphous, 82 crosslinked, 130 noncrosslinked, 145 semicrystalline, 82 Preloading, 339 Pressure vessel, 542 Principal stretch, 11 Principle correspondence, 143 Gibbs, 76 Lagrange, 213,222 of minimum free energy, 73 separability, 145,207 superposition Boltzmann, 35 time-temperature, 81, 263 Process adiabatic, 73 isothermal, 76 thermally activated, 111 Pseudo-time, 81
Quenching, 83 Radiation, 39 Relaxation function, 35 kernel, 35 measure, 35 regular, 54 singular, 54 weakly singular, 55, 58 spectrum, 48 Resin flow, 393 Rubber-glass transition, 81,276 Rupture, 302 Safety estimate, 495 Shock wave, 2 Slippage, 373 Spring, 26 linear, 26 nonlinear, 107 power-law, 108 Solidification, 572 Standard thermoviscoelastic medium, 307 Strain energy density, 72 Knowles, 234 Mooney-Rivlin, 185 neo-Hookean, 208 intensity, 121,127, 449 Strength, 278 Stress annular, 373 filament, 373 intensity, 114 equilibrium, 111,423 reduced, 114 viscous, 115 Temperature conductivity, 315,572 Tensor deformation, 7 Almansi, 8
596 Tensor (cont.) Cauchy, 8 Finger, 8 Hencky, 9 Piola, relative, 11 objective, 18 overstress, 174 rate-of-strain, 18 elastic, 174 fractional, 180 inelastic, 111 plastic, 174, 356 viscous, 174 Rivlin-Ericksen, 22 spin, 19 strain, 9 generalized, 13 infinitesimal, 8 stretch, 10 vorticity, 18 White-Metzner, 23 Theorem Alfrey, 143 Bernstein, 71 Riesz, 35 Weierstrass, 135
Index
Thermal expansion, 263 shift, 264 Thermodynamic stability, 219 Thermorheologically simple media, 262 Vector Burgers, 111 dual, 4 tangent, 3 velocity, 18 Viscosity, 26 Volume Eyring, 108 free, 113, 268 Volumetric growth, 413 Winding, 371 angle, 373 dry, 373 filament, 372, 542 wet, 372 Winkler foundation, 492 Yield criterion, 356
