Viscoelastic Structures: Mechanics of Growth and Aging

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Viscoelastic Structures Mechanics of Growth and Aging

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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


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


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




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


~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)


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


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)


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


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)


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_ 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
- 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


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)


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)


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


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)


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~


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,


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,


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


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


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


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

-


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)


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


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


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


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)


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.


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


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


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.


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)


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


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,


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)


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.


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


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 < ~', ~- 0 and a deformation

history up to this instant, {o(r, ~), 0 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 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


(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


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


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


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


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


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)


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


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


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, tO,

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


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


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 = #- 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


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


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


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).


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


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].


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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[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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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


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


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


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)


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)


~ } '(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


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


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


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


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


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


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


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


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


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)


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


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.


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


= 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


0

0.4

0 0

Po

o H °°." -

~'$"

o

_

0 o

d;

Ho

....

OZ. [~'"

oO,O

o..
- 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


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


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


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)


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 - 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)


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


~ 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)


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


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.


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


©

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


. 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)


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


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


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.


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


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


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


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


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.


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


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


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


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.


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


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


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)


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


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)


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 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


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


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


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


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


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


+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.

Bibliography

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.


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


+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.

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[40] Leaderman, H. (1943). Elastic and Creep Properties of Filamentous Materials. Textile Foundation, Washington, D.C. [41] Losi, G. U. and Knauss, W. G. (1992a). Thermal stresses in nonlinearly viscoelastic solids. Trans. ASME J. Appl. Mech. 59, $43-$49. [42] Losi, G. U. and Knauss, W. G. (1992b). Free volume theory and nonlinear thermoviscoelasticity. Polym. Eng. Sci. 32, 542-557. [43] Makhmutov, I. M., Sorina, T. G., Suvorova, Y. V., and Surgucheva, A. I. (1983). Failure of composites taking into account the effects of temperature and moisture. Mech. Composite Mater. 19, 175-180. [44] McCrum, N. G. (1984). The kinetics of the a relaxation in an amorphous polymer at temperatures close to the glass transition. Polymer 25, 309-317. [45] McCrum, N. G. and Morris, E. L. (1964). On the measurement of the activation energies for creep and stress relaxation. Proc. Roy. Soc. London 281A, 258-273. [46] McCrum, N. G., Pizzoli, M., Chai, C. K., Treurnicht, I., and Hutchinson, J. M. (1982). The validity of the compensation rule. Polymer 23,473-475. [47] McCrum, N. G., Read, B. E., and Williams, G. (1967). Anelastic and Dielectric Effects in Polymeric Solids. Wiley, London. [48] Mills, N. J. (1982). Residual stresses in plastics, rapidly cooled from the melt, and their relief by sectioning. J. Mater Sci. 17, 558-574. [49] Morland, L. W. and Lee, E. H. (1960). Stress analysis for linear viscoelastic materials with temperature variation. Trans. Soc. Rheol. 4, 233-263. [50] Muki, R. and Sternberg, E. (1961). On transient thermal stresses in viscoelastic materials with temperature dependent properties. Trans. ASME J. Appl. Mech. 28, 193-207. [51 ] Nakamura, Y., Tabata, H., Suzuki, H., Iko, K., Okubo, M., and Matsumoto, T. (1986). Internal stress of epoxy resin modified with acrylic core-shell particles prepared by seeded emulsion polymerization. J. Appl. Polym. Sci. 32, 48654871. [52] Narkis, M. and Tobolsky, A. V. (1969). Chemically crosslinked polyethylene: modulus-temperature relations and heat stability. J. Appl. Polym. Sci. 13, 22572263. [53] Read, B. E. (1981). Influence of stress state and temperature on secondary relaxations in polymeric glasses. Polymer 22, 1580-1586. [54] Rouse, E E. (1953). A theory for the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1272-1280.

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[55] Sala, G. and Cutolo, D. (1996a). Heated chamber winding of thermoplastic powder-impregnated composites: 1. Technology and basic thermochemical aspects. Composites 27A, 387-392. [56] Sala, G. and Cutolo, D. (1996b). Heated chamber winding of thermoplastic powder-impregnated composites: 2. Influence of degree of impregnation on mechanical properties. Composites 27A, 393-399. [57] Schapery, R. A. (1964). Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. J. Appl. Phys. 35, 1451-1465. [58] Schapery, R. A. (1966). An engineering theory of nonlinear viscoelasticity with applications. Int. J. Solids Structures 2, 407-425. [59] Schwarzl, E and Staverman, A. J. (1952). Time-temperature dependence of linear viscoelastic behavior. J. Appl. Phys. 23, 838-843. [60] Sharko, K. K. and Yanson, Y. O. (1980). Special features of the application of the analogy method in accelerated evaluation of the viscoelastic properties in the nonlinear region. Mech. Composite Mater 16, 548-552. [61] Stouffer, D. C. (1972). A thermal-hereditary theory for linear viscoelasticity. J. Appl. Math. Phys. (ZAMP) 23, 845-851. [62] Stouffer, D. C. and Wineman, A. S. (1971). Linear viscoelastic materials with environmental dependent properties. Int. J. Engng. Sci. 9, 193-212. [63] Suvorova, Y. V. (1977a). Account for temperature in the hereditary theory of elastoplastic media. Problemy Prochnosty 2, 43-48 [in Russian]. [64] Suvorova, Y. V. (1977b). Nonlinear effects in deformation of hereditary media. Mech. Polym. 3,976-980 [in Russian]. [65] Suvorova, Y. V., Viktorova, I. V., Mashinskaya, G. P., Finogenov, T. N., and Vasiliev, A. E. (1980). Study of the behavior of organic plastics under various regimes of temperature and loading. Mashinovedenie 2, 67-71 [in Russian]. [66] Talybly, L. K. (1983). Nonlinear theory of thermal stresses in viscoelastic bodies. Mech. Composite Mater 19, 419-425. [67] Tobolsky, A. V. (1960). Properties and Structure of Polymers. Wiley, New York. [68] Unger, W. J. and Hansen, J. S. (1993a). The effect of thermal processing on residual strain development in unidirectional graphite fibre reinforced PEEK. J. Composite Mater 27, 59-82. [69] Unger, W. J. and Hansen, J. S. (1993b). The effect of cooling rate and annealing on residual strain development in graphite fibre reinforced PEEK laminates. J. Composite Mater 27, 108-137.

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Chapter 5. ConstitutiveRelationsfor ThermoviscoelasticMedia

[70] Urzhumtsev, Y.S. (1975). Time-temperature analogy: a survey. Mech. Polym. 11 (1), 66-83 [in Russian]. [71] Urzhumtsev, Y. S. (1982). Prediction of Long-Term Strength of Polymer Materials. Nauka, Moscow [in Russian]. [72] Viktorova, I. V. (1983). Description of the delayed fracture of inelastic materials taking temperature into account. Mech. Composite Mater. 19, 35-38. [73] Ward, I.M. (1971). Mechanical Properties of Solid Polymers. WileyInterscience, London. [74] Williams, M. L., Landel, R. E, and Ferry, J. D. (1955). The temperature dependence of relaxation mechanisms in amorphous polymers and other glassforming liquids. J. Amer. Chem. Soc. 77, 3701-3707. [75] Yanson, Y. O., Dmitrienko, I. E, and Zelin, V. I. (1983). Prediction of the creep strain of a unidirectionally reinforced organic plastic from the results of quasistatic tests. Mech. Composite Mater 19, 440-444.

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.


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.


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, ~)

• OPO(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


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 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


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)


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


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


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


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


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,


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)


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) = ~


According to Eqs. (6.4.6), (6.4.20), and (6.4.21), hi(t) -

p db (t). b(t) dt

(6.4.41)


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)


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)


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,.


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


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)


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


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)


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


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


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


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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[89] Nikolaev, V. E and Indenbaum, V. M. (1970). Calculation of residual stresses in wound fiberglass articles. Mech. Polym. 6(6), 1026-1030 [in Russian]. [90] Nowinski, J. L. (1978). Mechanics of growing materials. Int. J. Mech. Sci. 20, 493-504. [91] Obraztsov, I. E, Paimushin, V. N., and Sidorov, I.N. (1990). Formulation of problem of continuous growth of elastic solids. Sov. Phys. Dokl. 35(10), 874875. [92] Obraztsov, I. E and Tomashevskii, V. T. (1987). Scientific foundations and problems of the technological mechanics of structures made of composite materials. Mech. Composite Mater. 23, 671-699 [in Russian]. [93] Ochan, M. Y. (1977). Programmed winding of composite articles nonlinearlyelastic in the transversal direction. Mech. Polym. 13(6), 987-993 [in Russian]. [94] Paimushin, V. N. and Sidorov, I. N. (1990a). Mathematical modeling of processes of devising fibrous composite materials and thin-walled structural elements by forced winding. 1. Three-dimensional equations. Mech. Composite Mater 26, 386-398. [95] Paimushin, V. N. and Sidorov, I. N. (1990b). Mathematical modeling of processes of devising fibrous composite materials and thin-walled structural elements by forced winding. 2. Algorithm for determining effective moduli of elasticity and model problems. Mech. Composite Mater. 26, 543-550. [96] Pauwels, E (1980). Biomechanics of the Locomotor Apparatus. SpringerVerlag, New York. [97] Pfeiffer, J. D. (1966). Internal pressures in a wound roll of paper. TAPP149(8), 342-347. [98] Pfeiffer, J. D. (1979). Prediction of roll defects from roll structure formulas. TAPP162(10), 83-88. [99] Portnov, G. G. and Beil, A. I. (1977). A model accounting for nonlinearities in the material response in the stress analysis for wound composite articles. Mech. Polym. 13(2), 231-240 [in Russian]. [100] Rand, T. and Eriksson, L. G. (1973). Physical properties of newsprint rolls during winding. TAPP156(6), 153-156. [ 101 ] Rashba, E. I. (1953). Effect of the construction schedule on stresses in gravity dams. Proc. Inst. Civil Eng. (UkrSSR) 18, 23-27 [in Russian]. [ 102] Regirer, S. A. and Shtein, A. A. (1985). Mechanical aspects of the processes of growth, development and reconstruction of biological tissues. In Summaries

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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].

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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 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)


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)


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


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.


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


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


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


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


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-~]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


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)


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.


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.


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.


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 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)


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


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


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 ~ 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 )
-- £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) 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, 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)


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 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) 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 --

3a~

1+

,, + 37rl.ta~ ~

(1 -

2,1,1,2]1

37rtxa21

530


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

-
- 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 .


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)


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


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).


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


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


+

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 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,


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


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


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


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)'


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.


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


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,.


_

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


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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589

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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

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