Mesomechanical constitutive modeling

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MESOMECHANICAL CONSTITUTIVE MODELING

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Series on Advances in Mathematics for Applied Sciences - Vol. 55

MESOMECHANICAL CONSTITUTIVE MODELING

Vratislav Kafka Institute of Theoretical and Applied Mechanics Academy of Sciences Czech Republic

World Scientific Singapore • New Jersey 'London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Kafka, Vratislav. Mesomechanical constitutive modeling / Vratislav Kafka. p. cm. -- (Series on advances in mathematics for applied sciences ; vol. 55) Includes bibliographical references and index ISBN 9810244851 (alk. paper) 1. Strength of materials - Mathematical models. 2. Strains and stresses - Mathematical models. 1. Title. II. Series. TA405.K315 2001 620.1'12'015118-dc21

00-051360

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by World Scientific Printers

Contents Introduction

l

Symbolic notation

3

Chapter I: General mesomechanical model of heterogeneous, statistically homogeneous materials

5

Chapter II: Models of materials with statistically isotropic structure

9

n. 1 General features of the model Model description of strain- and stress-distribution; specific stress power; basic relations among internal stress components in the A-Model and strain components in the B-Model; constitutive equations of material constituents; the complete set of basic equations; the complete system of combinations of continuous and discontinuous infrastructures of the material constituents.

9

n.2 General model of two-phase materials Both substructures continuous; both substructures discontinuous; inclusions of one material constituent in the matrix of the other; homogeneous stress model; homogeneous strain model.

26

n.3 Discussion of separability of deviatoric and isotropic responses of the mesoscopic stress- and strain-components n.4 Demonstrative stress-strain diagrams for two-phase materials Elastic and viscous material constituents; elastic and elastic-plastic material constituents - elastic moduli homogeneous; stress-strain diagrams [flow-curves] of two-phase composites with one phase elastic and the other one rigid-plastic [viscous] with different types of

v

40 42

VI

Contents

continuity of the substructures and under the assumption of homogeneous stress or homogeneous strain. n.5 Relation between our two-phase model and the exact bounds for macroscopic elastic moduli Relations to the bounds; some special cases; structural parameters independence of Young's moduli; structures corresponding to the bounds; conclusions.

46

n.6 Confrontation with some theoretical solutions for two-phase materials 57 Rigid spherical inclusions in a viscous matrix; spherical bubbles in a viscous matrix; elastic spheres in a viscous matrix; viscous spheres in an elastic matrix; elastic spheres in an elastic matrix.

Chapter III: Plasticity of polycrystalline metals

64

m.l Small deformations

65

///. 1.1 Solution to the identification problem 66 66 a) Yield condition in the simple Mises form; the special case of elastic homogeneity. P) Yield condition taking into account the influence of fluctuations. 74 74 III. 1.2 Metallic polycrystalline materials under complex loading Numerical examples: a.) Aluminum alloy: (i) combination of tension and torsion, (ii) ratcheting. P) Mild steel: combination of two modes of tension in two perpendicular directions. Conclusion. III. 1.3 The influence of the rate of loading upon the stress-strain curve III. 1.4 The yield-point jog

85 85 8989

///. 1.5 Residual internal stresses and stored energy III. 1.6 Cyclic hading

90 90 97 97

and upon the strength

m.2 Finite deformations, continuum damage and localization

100

Contents III.2.1 The process of necking Short specimens; long specimens. UI.2.2 Continuum damage and material softening without change of Young's modulus Relation to the elastic material with cracks; material softening.

vii 107

109

III.2.3 Application to tension tests of mild steel and aluminum alloy Sorbitic steel; aluminum alloy AlMg3; relation of the observed microstructural changes to our model description of continuum damage; calculated engineering stress-strain diagrams for different lengths of the AlMg3-specimens.

112

Chapter IV: Time-Dependent Deformation

121

Burgers model; Lethersich model. IV. 1 Application to creep of concrete

Chapter V: Fracturing V. 1 Quasihomogeneous stable microfracturing Model of the matrix; mesomechanical description of an elastic material with quasihomogeneous distribution of cracks with zero volume fraction; quasihomogeneous stable microfracturing; deviatoric loading; uniaxial compressive and tensile loading; constitutive equation of the matrix; model of the inclusions; model of the whole composite; example of application to structural concrete; conclusions.

124

131 131

V.2 Localized cracking in concrete

143

Chapter VI: Shape memory

151

VI. 1 Shape memory resulting from heterogeneity on the atomic scale Applicability of our concept to heterogeneity on the atomic scale; assumptions of our mathematical model; general form of the mathematical model.

151

VI.2 Pseudoelasticity Determination of the material parameters; incomplete pseudoelastic transformations in binary alloys.

166

Vlll

Contents

VI.3 One-way shape memory effect Numerical example of a unidirectional one-way shape memory.

172

VI.4 Two-way shape memory effect

179

VI.5 Discussion of our model for shape memory

184

Chapter VII: Transversely isotropic materials

186

VII. 1 Transversely isotropic two-phase model VH.l.1. Solution to the identification problem based on the flow-curve

189 194

VH.2. Transversely isotropic materials with unidirectional continuous fibers 197 Macroscopic yield condition of materials with unidirectional continuous fibers; solution to the identification problem; description of a concrete fiber-reinforced material.

Chapter VHI: Appendices

205

Appendix v m . 1 - On the validity of Hill's equation

205

Appendix Vm.2 - Proof of Eq. (II. 1.29)

207

Appendix vm.3 - Deduction of Eqs. (n.2.10) to (n.2.24)

208

Appendix vm.4 - Deduction of Eqs. (H.2.33) to (D.2.43)

209

Appendix vm.5 - Determination of the input data for the solution to the identification problem based on the elastic-plastic stress-strain diagram

210

Appendix Vffl.6 - Discussion of Eqs. (V.2.8) and (V.2.9)

212

Appendix vm.7 - Structural parameters of transversely isotropic materials

214

Appendix vm.8 - Deduction of Eq. (VII. 1.1.2)

215

References

219

Subject Index

225

Introduction Most practically important materials are dispersed systems consisting of two or more Theologically different materials or phases. Therefore, their constitutive modeling calls for understanding and describing the deformation processes in their structure. There are different ways of attacking this problem and this monograph deals with one of them. The basic ideas of this approach were outlined in a previous author's monograph - Kafka (1987). Since then, a number of new results and new applications in this line have been worked out and the aim of this monograph is to summarize them. The fundaments of our approach can be characterized as follows: a) It is not macroscopic (not working with macroscopic stresses and strains only), but it does not go into microscopic analysis so far that it would describe the geometry of individual microelements. Instead, it describes the influence of the structure of the material constituents in a specific statistical way by structural parameters that are defined as integral forms in the distribution functions describing the influence of the structure upon the distribution of stresses and strains. The structural parameters are determined by a mathematical analysis of the stress-strain diagrams or flow-curves acquired by simple macroscopic tests. The distribution functions themselves need not be determined, the user of the method works only with the structural parameters. b) It is not limited to 'composite materials', in which the structure is usually rather coarse, it can be applied even to very fine structures, even to heterogeneities on the atomic scale (e.g. in the application to the shape memory materials). c) The model is very general: In a two-phase material, specific values of structural parameters cover the cases of: (i) Inclusions of one material constituent in the matrix of the other constituent, (ii) Both constituents having continuous substructures, (iii) Both constituents having discontinuous substructures, (iv) Homogeneous stress field (Reuss' solution), (v) Homogeneous strain field (Voigt's solution). This means that our model surmounts in a qualitative way the classical 'series' or 'parallel' arrangement schemes.

1

2

Introduction

d) The model works not only with the averaged values of stresses and strains in the individual material constituents, but also with tensorial values that describe the influence of fluctuations - of space deviations from these averages. Under uniform loading or uniform constraint of a representative volume of the heterogeneous material, the stress- and strain-fields in one material constituent are described generally as heterogeneous. Only in the case that the constituent in question forms inclusions, the fields inside the inclusions are described as homogeneous. e) Our model is a special case of the internal variables concept, which means that the current state of the material is described by a finite number of variables, some of them being common apparent variables, the others internal variables. Our internal variables have clear physical meaning, they are of tensorial character and their dimension is stress. f) The mechanical properties of the modeled heterogeneous material are identified with the properties of its representative volume, loaded at its surface S either by displacements «,<5) = e^Xy or by tractions r,(5) = o y ? j , derived, respectively, from uniform overall strain e,y or stress Gy. Although such identification is implicitly generally used and without using it investigation of material properties on specimens would be senseless, it has its limitations. These limitations are discussed and specified. Basically the model is formulated for materials with statistically isotropic structure, but generalization to materials with statistically transversely isotropic structure is shown too. Applications of the model, presented in the monograph, cover composite materials, polycrystalline metals and their alloys, concrete, shape memory materials, fiber-reinforced materials, and there are referenced publications dealing with biological tissues. I would like to gratefully acknowledge the significant help of my colleagues Ing. David Vokoun, PhD, Ing. Miroslav Hlavacek, PhD and Dr. Miroslav Karlflc, my co-authors of the papers dealing with the problems of shape memory, elastic bounds and localization, respectively. The respective chapters were based on these papers. I would also like to express my thanks to Ms. Marta Buckovi for her help with the preparation of the graphical parts. I acknowledge also the support of the work concerning shape memory problems by the Grant Agency of the Academy of Sciences of the Czech Republic under grant No. A2071801, and of the work concerning reology of transversely isotropic media by the Grant Agency of the Czech Republic under grant No. 103/00/008.

Symbolic Notation Sjj {Ejj} a

macroscopic stress (strain}

a

£

£

ije I ijn [ ije I ijn) mesoscopic stress {strain} in the elastic /inelastic

subvolumes - averaged values in the respective material constituent dip I djjn {ijjg I ijjn} stresses {strains} that correspond in the e-/n-material constituent to the strain (stress) history identical with the macroscopic strain {stress} history deviatoric stress - macroscopic {mesoscopic}

Sy {Sjjg, Sjjn} e^j {e^, e^}

deviatoric strain - macroscopic {mesoscopic}

8jja{8ijGe, 8jjOn] isotropic part of the stress tensor - macroscopic {mesoscopic} SjjE {SjjEg, 8jjEn} isotropic part of the strain tensor - macroscopic (mesoscopic) Kronecker's delta

8y £

jje{e'ijn)

=e

ijei£ijn}~£~ij

= e

£

£

m me £

»e { ijn} ~ ije i qn}

e

'ije>e'ijn

A-variant of the model ' n m e *-variant of the model

deviatoric parts of e^.e^,

8jjEg, 8jj£'n

isotropic parts of CF

Cijetyn} =<7ije{ ijn}-°ije{Vijn} = a

ije \aijn) ~ <*ij

s

'ije>s'ijn

m

in

e'^.e^ the A-variant of the model

th e fl-variant of the model

deviatoric parts of

c'^o'^

Sjjo'g, Sjja'n

isotropic parts of a'ije,a'ijn

i,j,k,l [<x,P }

indices the repetition of which means (does not mean) summation

3

4

Notation

vg,vn(vg + vn= 1) volumefractionsof the respective material constituents T7e,r/n{T7°,T/°}

non-negative'structural parameters'for the deviatoric {isotropic} parts of stress and strain in the A-variant of the model Xe • Xn iXe < Xn) non-negative 'structural parameters' for the deviatoric {isotropic} parts of stress and strain in the B-variant of the model E Young's modulus v Poisson's ratio H = (l+v)/E deviatoric elastic compliance p = (l-2v)/E isotropic elastic compliance c deviatoric elastic limit X scalar measure of plastic deformation rate y12 (= 2 e 12 ) technical measure of shear deformation in the xrx2 directions Numbering of equations and figures: In every section or subsection that has its own running head the numbering of equations and figures starts from number 1. Quotations of equations and figures inside one such section or subsection is by these short simple numbers only. Quotation of an equation or a figure appearing in another section or subsection is preceded by the number of the respective section or subsection (of the running head). So, e.g. quotation in subsection (m.l) of equation (2) appearing in the same subsection (m.l) is only Eq.(2), but quotation of the same equation in another subsection will be Eq.(IQ.1.2).

Chapter I General Mesomechanical Model of Heterogeneous, Statistically Homogeneous Materials The type of material that we are going to analyze is supposed to be homogeneous on the macroscale and heterogeneous on the microscale. It is composed of several material constituents that are defined by their specific material properties and by the geometry of their substructure that is statistically regular. This means that the substructure can be different in different macroscopic volumes, but the respective statistical description is the same up to the degree of accuracy that is important for macroscopic response. The substructures of all the material constituents form the structure of the material. The sum of the subvolumes of the material constituents equals the volume of the body and the volume fraction of any material constituents is the same in any macroscopic volume. By one mathematical model only such process can be described, for which - to a sufficient degree of accuracy - the volume fractions of the material constituents remain unchanged. In our analysis we use three scales for the description of stress- and strain-fields: Thefirstscale is the macroscopic with the macroscopic stress GjjfX.t) and strain £g(X,t),

which depend on the macroscopic coordinate X and time

t. It is commonly used and does not need any explanation. (In our approach time t is to be understood as the measure of progress of the deformation process. In the case of viscosity it is suitable to identify t with the current time. In some other cases some special "intrinsic" time may be preferable). Further.we introduce the notion of a small spherical macroscopic neighbourhood N(X) of the macroscopic point X . N(X) is small with

5

I. General Model

6

regard to the body and the space-variations of Oy and e^ , but it is macroscopic, i.e. its diameter is at least one order higher than the characteristic length of the structure. For the description of the internal phenomena in N(X) we use microscopic coordinates X . We can use them for the description of the infrastructures of the material constituents in N(X) and for the evaluation of the average values of stress and strain in individual material constituents. These average values are different for different material constituents, but for one material constituent comprised in one N(X) they are unique. Therefore, they are functions of X , but not of X. For the n-th material constituent they are described as a^(X,t), e^(X,t). This is the second scale, called mesoscale. However, the stress-field and strain-field in the n-th material constituent in one N(X) need not be homogeneous and generally they vary with X . Hence, they must be described as a^(X,x,t)n , Ejj(X,x,t)n. This is the third scale, called microscale. From the above definitions it follows: (at(X,x,t)n)n

= oIJn(X,t)

(et(X,x,t)n)n

= £ljn(X,t)

where (■■■)„ means the volume average in the n-th material constituent. The mesoscopic variables are basic for the analysis, the influence of the microscopic variables will be expressed in terms of fluctuations and the relations between the macroscopic variables (i.e. the macroscopic constitutive equations) represent the result of the analysis. The conditions of equilibrium and of continuity of displacements lead to the well known equations: £vn°ljn=Glj

(1)

n=1

S X £#,=£,;

(2)

n=1

where Vn is the volume fraction of the n-th material constituent and

£*„ - 1 .

n=1

Furthermore, the following equation holds true:

(3)

7

I. General Model Oiieij=Oij£lj-

W

For thefirsttime the last equation was deduced by R. Hill(1963) and - as already mentioned in the Introduction - it can rigorously be proven (cf. Kafka 1983) for a volume loaded at its surface S either by displacements K,(S) = e^Xy or by tractions t,{S) = "o^j, derived, respectively, from uniform overall strain Cyy or stress Oy. The possibilities of applications to more general cases are discussed in Appendix vm.l. In what follows we accept the fundamental theorem of the theory of internal variables saying that any current state of the material can be described by a finite number of variables. Hence, we can coasider /7(S p J = rj /y £ /y -C7 /y £0=O

(p = \...,P)

(5)

as a function of variables Sp that include also internal variables. This function equals zero throughout any process. P is some finite number of variables. Consequently any variation of TJ(Sp) equals zero:

sn(sD)=t^-5S p = o. p=idS

(6)

n

Furthermore, let us suppose that we use all the constraints that are valid among variables Sp and that we eliminate some of them and restrict their number to the independent variables only. In this way function n(Sp) another form that we express as

n'(Sp)

will change into

(p = \...,Q)

where Q(< P) is the number of independent variables. Then it holds true for any variation of 77'( Sp ) :

sn'(Sp)=i^fssq=o

(?)

I. General Model

8

for any set of independent variations 5Sq

dn' dSq

= 0.

and therefore (8)

The number of the constraints was P-Q and with Q equations (8) the total number of equations is P, equalling the number of unknowns Sp .

Chapter II Models of Materials with Statistically Isotropic Structure II. 1 General Features of the Model In our model representation the heterogeneous material in question is described by (0 the constitutive equations of the material constituents, (li) the volume fractions of the material constituents, (iii) specific characteristics describing the structure of the material. Now we are going to deduce structural characteristics: Without any loss of generality the distribution of strain and stress in the n-th material constituent of the neighborhood of one particular macroscopic point X can be described alternatively in the following two dual ways:

A)

e^(X,x,0n=ea/3(X,f) + ^(X,x,0 n e^„(X,0

(1)

e

e(X,x,t)„ = e(X,t) + f {X,x,t)n e'n(X,t)

or: B)

VX,x 1 0 n =^ n (X 1 0+(3,(X,x,f) fl «WX,0 a(X,x,on =<Wo+HX,x,oX(X.O s^(X,x,O n = S^(X,0 + ^ ( X , x , 0 „ ^ , n ( X , 0 a(X, x, t)n = o(X, t) + <pa (X, x, O X (X, t)

eap (X, x, t)„ = e^n (X, t) + <PaP (X, x, t)„ e^n (X, t) e(X,x,t)n=in(X,t) +
9

(2)

10

II. 1 hotropic Structure - Generalities

where e^ / d^e (S^ / 8„p<j) is the deviatoric / isotropic part of the strain (stress) tensor;

e^, / 5 ^ e (S^ / 6,^0)

is the macroscopic deviatoric /

isotropic part of the strain (stress) tensor; Sa^/S^d^eo^/Sape,,) is the deviatoric / isotropic part of the stress (strain) tensor that corresponds in the /i-th material constituent to the strain (stress) history identical with the macroscopic strain (stress) history; the distribution functions f and (p are the only expressions on the right-hand sides of the above equations that depend on X and by definition they have unit norm, i.e. for any of the four expressions f or

„=1

(3)

<

n='\.

There is no summation with the repeated indices a, /? . The meaning of the remaining variables becomes clear after performing the averaging procedure <•••>„ in all the equations: A

)

B

)

e

'afin = eapn ~ ^ap

S

'afin ~ Sapn ~ Sap

(4)

(5)

The above equations are quite general, descriptive of any arbitrarily complicated reality. It is necessary to simplify them to create an operative mathematical model. The fundamental theorem that forms our concept is that in the simplified model representation the distribution functions f and (p can be assumed independent of t. From the mathematical point of view this means that the second addends on the right-hand sides of Eqs. (1) and (2) - functions (representing fluctuations) of two variables X and t for a fixed X - are described as products of two

n.l Isotropic Structure - Generalities

11

functions, one depending on x, the other on / . Such a step - often used in mathematical analysis - means of course loss of generality, but it simplifies the analysis substantially. From the physical point of view this step means that the structure of the neighborhood of one particular macroscopic point X is characterized in a specific way by functions /(X,x) and (p(X,X), that it cannot arbitrarily change in the course of the deformation process. In the following sections we will show that even with this simplification the model is general enough: (i) It covers as special cases all possible combinations of continuous and discontinuous substructures (continuity means such property of the substructure that rigidity of the respective material constituent leads to rigidity of the composite; an infrastructure that is not continuous is called discontinuous) of the material constituents in a two-phase material: (1) both substructures continuous, (2) both substructures discontinuous, (3) inclusions of the first material constituent in the matrix of the other, (4) matrix of the first material constituent with inclusions of the other. Furthermore, it spans the interval between the natural limits resulting from the hypothesis of homogeneous stress (Reuss' solution) and that of homogeneous strain (Voigt's solution). Both the last named models result as limiting special cases of our general model. (it) It takes into account not only different average values of stress and strain in individual material constituents (mesoscale), but also the influence of the lower scales in terms of fluctuations. Only in the case of inclusions in a continuous matrix the description of stress and strain in the inclusions turns out as homogeneous in one N(X) . (iii) It admits very complicated structures and a broad variety of constitutive equations of the material constituents. (iv) It enables relatively easy solutions to the identification problems based on simple macroscopic experiments. (v) It leads to results that are compatible exactly or very closely with a number of other mathematical solutions based on quite different approaches. (w) A number of applications is shown to agree with experimental evidence. (v«) As far as we know, the model is the simplest with such properties. Model Description of Strain- and Stress-Distribution Using our fundamental theorem specified above we can simplify the description substantially. All the functions f and (p art now independent of t , which leads to:

n.l Isotropic Structure - Generalities

12

eap(X,x,t)n = eafl(X,t) + f0al)(X,x)ne^n(X,t)

A)

(6)

e(X,x,t)n=e{X,t) + f*(X,x)ne'n(X,t) Safi(X,X,t)„=Sapn(X,t) + flp(X,X)nS^n(X,t) o(X,x,t)n=dn(X,t)

+ f<>(X,x)no'n(X,t)

and similarly for the B-model, where in Eq. (2) there appear n

(7)

= <s/y(X,x,0„e//(X>x,0n+3(T(X,x,0„e(X,x,0n>n =<

E

{[^n(X,0 + ^(X,x) n s^ n (X,f)]

a,/}=1,2,3

®[^(X,0+^(X,x)ne^n(X,0]} + 3{dn{X,t)+f<>(X,x)n<j'n(X,t)] *[t(X,t)

+

f°{x)n6XX,t)]>n

= Sljn(X,t)e,n(X,t) + 3on(X,t)£n(X,t) +-Ls^(X,0e^(X,0+4^(X.0^(X>0

where we have used the relations (3) and (4) and new definitions: ^ = <[^(X,x) n -1][f*,(X,x) n -1]> n =

n^

(8)

13

n.l Isotropic Structure - Generalities 4 - = <[r(X,x) n -1][^(X,x) n -1]> n Vn = „-i.

The newly defined quantities T/„ and 77° are called structural parameters. They are independent of the macroscopic coordinate X due to the assumed statistical homogeneity and independent of the tensorial orientation a/3 due to the assumed statistical isotropy. They are scalar quantities characterizing the structure of the material, the user of the model works only with them (without the need of the distribution functions), they are determined from simple macroscopic experiments. The specific stress power in a unit volume of the composite material is: V ^ = I ^ [ % ( X , 0 ^ ( X , 0 + 3a n (X,0e n (X,0 +—

(9)

s'ijn(X,t)e;jn(X,t)+-^o'n(X,t)en(X,t)].

Thus, the specific stress power is expressed in terms of functions of macroscopic coordinates only. In any material constituent the expression for the specific stress power is composed of one part that is related to the average values of stress and strain and of the other part that describes the influence of fluctuations. In the case of elasticity the last expression can be integrated and we arrive at the formula for elastic energy: 1 N el=^LVn[Mn{SjjnS!in+

W

2 n=1

Hn

1 1 — S~nS'ijn)+3Pn(
Tin

Pn

(10)

^n

where Hn,pn are elastic constants: Hn=(\+vn)IEn

,

p„=(1-2v„)/En,

vn ,En being respectively Poisson's ratio and Young's modulus of the n-th material constituent.

n.l Isotropic Structure - Generalities

14

In the case of the dual model B the resulting formulae of this paragraph can be rewritten with the only change that instead of T)n,t]° we have to write Xn> X° *•* the definition: ^-=<[„ Xn

(11)

=<^(X,x)n9)^(X,x)n>n-1 -l=<[ ? , n Xn = ^ (For X , xthe ) „B-model ^ ( X , x ) nwe > „will - 1 . need also another equation that can easily be the B-model weway willasneed deduced For in a quite analogous Eq. also (9): another equation that can easily be deduced in a quite analogous way as Eq. (9): 1 , ^ . 3 ,^,, c tf # # = L v n [ e ^ s ^ +3e„
Xn

Xn

Basic Relations among Internal Stress Components in the A-Model and Strain Components in the B-Model Let us now combine equations G.5) and (9). In (9) we consider infinitesimal variations instead of rates of strain and in (1.5) the deformations are also considered to be infinitesimal. In this way we can write: ffjj dEjj - a j dEjj = Sy 5eri + 3&8E N

(13)

1

3

Vn

Vn

L vn[sanfeijn+ 3
On the strength of Eqs. (1.1) and (4)i>2 the last equation can be rewritten as:

I V f l [ ( S * + ^ ) * ^ + 3 K + 4 > & n ] = 0. n=1

Vn

(14)

Tin

The variations 6e^m and 8e'n are not independent, as Eqs. (1.2) and (4)j give:

n.l Isotropic Structure - Generalities N

15

N

L^^=o, n=1

£^&;=o.

as)

n=1

Let de'ip, and 5 £ „ be arbitrarily chosen variations that are expressed from the above equations (15) and used in (14). This leads to: I

vn[(sijn-siim^-^)8^jn

n=1,2,..,m-1,m+1r..,W

'In

(16)

T

lm

+3(<jn-<jm+?f-?f)8en]

= 0.

In 1m Furthermore, the variations 8e'q„(n±m)

are bound by the relation

resulting from their deviatoric character:

V4,=0

(17)

( 8jj = Kronecker's delta). However, the expression in the brackets preceding 5e^n in Eq.(16) is also deviatoric and therefore it holds:

V * > - W — " — >=°-

(18)

Tin Vm Let us multiply the last expression by vn £„ , where %n are random multipliers, different for different n . Then we add the resulting expression to (16) and arrive at: I

vn[(sin-sijm+^-^)(Se^+^) +3(on-om+ZL-?f)8e'n}

In

<

= 0.

(19)

n.l Isotropic Structure - Generalities

16

In the last equation the variations 8e'n{n*m) are independent - arbitrary scalars. The expression ( Se^ + £„ 8g) h an arbitrary tensor, as 5e^ n is an arbitrary deviator and £„ is an arbitrary scalar. Therefore, the following equations hold true: fy-Sfr+ir-n0

' °n-°m + -f-^

=0 ■

(20a)

It is clear that the number of independent equations (20a) is N-l for the deviatoric parts and N-l for the isotropic parts. Equations (20a) are die basic relations among internal stress components for model A. By a quite analogous procedure the respective basic relations can be deduced for the dual B-model. In this case it is necessary to work with strains and with variations of stresses. Instead of Eqs. (9), 0.1), (8)i,2 we use Eqs. (12), 0-2), (5)i,2 and arrive at: e

e

ijn

e

ijm _

in-e^^-f—f-

£„

Eg

n =0 en-e ^L-^- = 0 . -w ., *„ e m+ m+ 0 c

(20b)

These are the basic relations among internal strain components for model B. Compared with (20a) we see that stresses and structural parameters rj are replaced by strains and structural parameters %■ Constitutive Equations of Material Constituents It is assumed that for any n-th material constituent the respective constitutive equation can be expressed as a special case of the following general formula: Df?)e#(X1x1f)n + 4%<X,x,f) f l = 43)S/y(X,x,0„ + 4 % ( X . x , f ) „

+/
(21)

n.l Isotropic Structure - Generalities

17

where T is temperature and T„ some other parameter that is specific for the n-th material constituent. Equation (21) includes as special cases e.g. thermoelasticity (D^, D(n3>, l(nl),l(n3),lfl5> are constants, the other material parameters have zero values), viscosity, viscoelasticity, plasticity etc. Accordingly, the parameters D and / have then different meanings. They can be constants or functions of X, t or T. It is a restriction of our model that they cannot be functions of X . (Thus e.g. - for fixed X and t - in a phenomenological theory the scalar measure of plastic deformation dX is unique. In our model it can be different in different material constituents, but the fluctuations of plastic deformation are described by the fluctuations of Sg only, not by variations of c/A.) If we perform the averaging procedure <...>n in Eqs.(21), we get:

/#> e* + Cn2) e^ = 4 3 ) sin + # > sijn

(22)

I2>ea + I?ia = t*o„ + £>*„ + A„5> 7 + £> U tf >T„ + £>*„ where all the variables are functions of X and t only. From the definitions of s^, and an it follows:

$% + &?%= WsiJn+Wkijn

(23)

«>* + £>#= Ptn+Wn ++FT + W U Wx„+W*n. Subtraction of (22) and (23) gives (according to (4)):

D ^ +^

= Z#>4, + 0 £ %

(24)

With the use of Eq.(6), Eq.(21) can be rewritten as: ^e^D^^

+

fe(x)n[^e'in

+

^e^n)

(25)

18

n.l Isotropic Structure - Generalities

= tf an+44) *n + r (x)n [ff o'n+44) (t;i Comparison of (23), (24) and (25) leads to: f'(x)n=f*(x)n=f(x)n

(26)

e

f (x)n=f"{x)n=f°(x)n where we have newly defined f(x)n as the distribution function for deviatoric parts and f°{x)n as the distribution function for isotropic parts. From (25) and (8) we conclude: — = < [f(x)n]2 >n ~ 1 = < [f(X)„ ~ 1]2 > n > 0

(27)

n„

^= n -i=<[f o (x) n -i] 2 > n ^o. Tin

The non-negative character of the structural parameters 7/n, T]° following from the above equations - is their very important property and their vanishing and infinite values have special physical meaning. In the case of the B-model the reasoning is quite similar, Eqs.(22) and (24) are formally the same and structural parameters Xn • X°n a r e a 8 a i n n o n _ negative. The respective formulae follow from (27) writing %,

II. 1 Isotropic Structure - Generalities

19

variables to be determined. The rates of the remaining variables are: < *ijn><*tin>£ijn>£'ijn and ^i/ • Hence, their number is 4N+1 . For model A the set of basic equations is: equations (LI) (L2) (4) ll2 (20a) (22) (24) -

number 1 1 N N-l N N

Thus, the number of basic equations is also 4N+1. Equations (1.1), (1.2), (4)i,2and (20a) are written for tensors and not for their rates, but this is only richer information, the analogous equations for rates can be obtained by a trivial derivation. For model B the set of basic equation is quite similar, instead of (4)1>2 and (20a) it is necessary to introduce Eqs. (5)i,2 and (20b). From the structure of the basic equations we see that there appear neither the distribution functions f, (p , nor the variables d^n , ejin. They served only for the deduction, the user of the model does not need to know them. Hence, in the final form of the model the restriction imposed is the time-independence only of the structural parameters, which are integral forms of the distributions. Such a restriction is weaker than the time-independence of the distribution functions themselves. With concrete forms of the constitutive equations of the material coastituents in (22) and (24) it is possible to derive the macroscopic constitutive equation. It will be characterized by the parameters of the constitutive equations of the material constituents, their volume fractions, the structural parameters t] or % and generally by internal variables represented by O^ , (7^, or e^ , e^, . The Complete System of Combinations of Continuous and Discontinuous Infrastructures of the Material Constituents In what follows we will show that with different values of T] and X it is possible to model compact and loose infrastructures of the material constituents. The analysis will be limited to such cases only, where the following quadratic forms (appearing in Eq.(10)) are finite:

n.l Isotropic Structure - Generalities

20

1 'In

^-2 , 1 _/2

Q

V°n Qen

=

e

/Jh eljn

1

+

~~eijneijn

*ln

In the case of the B-model quite analogous quadratic forms with X instead of 7] are considered to be finite too. Successively we will investigate model A with different values of the structural parameters and then model B. A 1: Finite parameters Tf : 0 < 77n < <» , 0 < T7„ < °° We are going to prove that this case corresponds to continuous infrastructures of the material constituents, i.e. rigidity of one material constituent leads to rigidity of the composite. Rigidity of the n-th material constituent in deviatoric deformation can be described by the following special forms of Eqs. (22) j and (24) i:

where fin has vanishing value. As the Q-forms are finite, the values of S^, , S^, must be finite, too. Therefore, eljn =e'ljn = 0 , which - compared with (4)j means that ©^ = 0 . A quite analogous procedure leads to the conclusion that rigidity in the volumetric deformation leads to £ = 0 . This shows that rigidity of the n-th material constituent results in rigidity of the composite, i.e. the n-th material constituent has a compact infrastructure. A 2: Infinite values of one pair of structural parameters r\ : In this case Eq. (20a) degenerates to:

n.l Isotropic Structure - Generalities

S * . - S * , - — = 0. Vm

21

(28)

Let the resistance of the m-th material be vanishing. Then its deviatoric properties can be described as follows:

1

s

-~ei"> H-m

jjm

«'

-

1

o'

Mm where \lm is infinite. From the value of the Q^, -form being limited the values of e

#n • e#m follow to be also limited. Therefore, S/J>n = S^, = 0 and this results -

with regard to (28) - in S^ = 0 . Thus, the average value S^, in the

n-th

material constituent is zero. However, according to (27) j it holds for infinite 7]n :

<[f(x)n]2>n=lBy definition n=1 and therefore f(x)„=1 (cf. Appendix Vffl.2.). (29) This means that the vanishing modulus in the m-th material constituent results in vanishing stress not only in the w-th constituent, but also in the M-th constituent. Therefore, the n-th material constituent is not continuous, it forms inclusions. The deduction for the isotropic component would be quite similar. The m-th material constituent is continuous, which can be proved in the same way as in A I. In a two-phase material, which is the main area of application of our concept, the infinite values of T/„ and rfn represent the model of inclusions of the n-th material in the compact matrix of the /n-th material. A 3: Zero values of one pair of structural parameters 77 : T]n-T]° = 0 If the structural parameter 77„ is zero, then - for the Qsn form to be finite the value of s L must also be zero. This is valid throughout the deformation

n.l Isotropic Structure - Generalities

22

process and therefore it also holds s'^n = 0 . With regard to Eq. (24), this means that e^=0

(30)

throughout the process (supposing the existence of the virgin state without deformation). Further, from (30) and (6) i we see that

es(X,x,t)n=eln=eil i.e. deviatoric deformation in the n-th material constituent equals the macroscopic deviatoric deformation. The same is valid for the isotropic part of deformation the deduction is quite analogous. In the case of a two-phase material it results from (1.2) that and from (4) 12 and (6)i, 2 :

Cj \X ,X,t)m = £q„, = £y . Thus, in a two-phase material the zero values of one pair of structural parameters t)n,r]^ are sufficient for the deformation to be homogeneous throughout the composite (Voigt's model). A 4: Infinite values of all structural parameters i) : r/n = r/° = rj m = r/£, = °° In such a case all equations (20a) degenerate to: S

lin

=

S

tm

'

a

n=am

and by a reasoning analogous to that used in A2 we can see that the stress state in all material constituents is also homogeneous. With regard to (1) we conclude

al{X,x,t)n =al(X,x,t)m =... = a# i.e. homogeneous stress state throughout the composite (Reuss' solution). A5: Zero values of all structural parameters TJ :

7]n = rfn = r\m = rj°m = 0

By a deduction quite similar to that used in A3 we arrive in this case at Voigt's solution, i.e. at homogeneous deformation throughout the composite. If there are N material constituents, it is sufficient if N-l pairs of structural parameters are zero to arrive at the Voigt solution. The reason was shown in A3 and for a two-phase material models A5 and A3 are equivalent

n.l Isotropic Structure - Generalities

23

51: Finite parameters % : 0<xn <°°> Q<Xn
where fln is infinite. Finite Q w implies finite e ^ and e ^ . Therefore, s /jh = s'an — 0 - which - compared with (5)] - means that S^ = 0 . A quite analogous procedure leads to (7 = 0 . This proves that the substructures described by model 5 with finite structural parameters % are discontinuous. 52: Infinite values of one pair of structural parameters %'• Xn=X°n=°°

• 0<^m
,

0<*°
In this case Eq. (20b) degenerates to:

Am

Let the m-th material be quite rigid, i.e. with infinite Young's modulus. Then (cf. Al), e^, = e'^, = 0 and therefore Using the same reasoning as that used in A2 for S^ = 0 we can easily prove that

et{X,x,t)n

=Sf, =el(X,x,t)m

a0.

For a two-phase material this also means "e^ = 0 and therefore, the

m-

th material constituent has a continuous infrastructure in this case. On the other hand, the n-th material constituent has a discontinuous infrastructure, which can be proved in the same way as in 51; here we let the mth constituent be without resistance and show that this leads to

n.l Isotropic Structure - Generalities

24

a^X.x, t)n =
=

0

We can proceed in a way that is dual to A3, which leads to e

ijn = eijn ~ Oi

s

S

qn = 0

l \X • x> On = Sfn

= S

H

and similarly for isotropic parts. In the case of a two-phase material this results in: (7, ( X , X, t)m =
Afl

Am

A/tfl

By a procedure that is dual to that used in A4 we can easily show that in this case: MX.X, On = M X - X ' f)/n =... - ffj i.e. Voigt's solution. Model B4 is therefore equivalent to AS and in the case of a two-phase material also to A3. B5: Zero values of all structural parameters X '• An ~ Xn ~ Xm ~ Xm ~ ^

Similarly as in B3 and A3 we can prove that in this case it holds:

al(X,x,t)a=al(X,x,t)m=...sal i.e. the Reuss solution. Model B5 is therefore equivalent to A4.

n.l Isotropic Structure - Generalities

25

Concluding this paragraph let us mention that not all of possible abstract combinations have physical meaning. The situation is simple and clear in the case of a two-phase material. Models Al, B\ and two equivalent models A2 and B2 have physical meaning of continuous infrastructures, discontinuous infrastructures and of inclusions in a matrix. The remaining models correspond to the Voigt and Reuss solutions, which are mathematical fictions only, because there does not exist any three-dimensional structure that would lead to homogeneous stress or strain. In spite of it they are used sometimes as simple approximations. In the case of three material constituents the situation is more complicated. From the preceding deduction it is easy to see that for all r\ finite we will get a model for three continuous infrastructures, for all % finite a model for three discontinuous infrastructures. For two pairs of J] finite and one infinite we get two infrastructures continuous and one discontinuous (with contact surfaces with both the continuous constituents), for one pair of r\ finite and two pairs infinite one infrastructure continuous and two discontinuous with equal homogeneous stress in the two discontinuous constituents, for one pair of % finite and two pairs infinite one infrastructure continuous and two discontinuous with equal homogeneous strain in the two discontinuous constituents. Other combinations can be derived similarly. From the above examples we see that in the case of three constituents there arise restrictions and our scheme is not so general and straightforward as in a two-phase material. In what follows the analyses are limited to two-phase models only.

II.2 General Two-Phase Model

26

II.2 General Model of Two-Phase Materials This two-phase variant of our multiphase model has high practical importance for description of a number of materials. It covers as special cases different constitutive relations of material constituents and different types of the substructures. These are separately discussed below. Both Substructures Continuous It was shown in the preceding section that the case of all substructures continuous is described by the A-model. The special variant of the complete set of basic equations for two-phase materials reads in this case: veoijB +

vnaijn=aij —

" o ^ije ■*" ^n Ejjn

&ije ~ Mo &ije

e

jfo

=

+

(1) (2)

^ij

£ e = Pe °e + <*e Pe + « o T+(be

Sije "o >

e

ije ~ eij »

®ifO = M© Sjffl + Sjjg fig

,

e

ijn ~ Mn Sijn + sijn fyj >

&ijn ~ ®#n

—

3jf '

®ijn ~ Hn &ijn + §ijn " n

»

(3)

t'e=ee -t

(4)

£e=PeK+0ePe

(5)

*n = Pn °n +<*n Pn +<*„ t + <»„

(6)

£'n=£n - t

(7)

&'n=Pn<*n+<'ni>n

(8)

where e0B/S/yo[<5,yeo/<5,y<7o] are the deviatoric [ isotropic ] parts of the average strain/stress tensors in the e-constituent and similarly with the n-constituent, and

n.2 General Two-Phase Model

27

the macroscopic values are indicated by the overbars. The symbols ju [=(l+v)/£] and p [=(l-2v)/£] mean deviatoric and isotropic elastic compliances, v means Poisson's ratio and E Young's modulus. The symbols with primes - defined by equations (4) and (7) - characterize the influence of fluctuations, of the heterogeneity of strain- and stress-fields, equations (5) and (8) were deduced in Section D. 1, aB and an are coefficients of thermal dilatation. Symbols hg and hn can have different meanings in different applications: • In the case of plasticity they can represent the rate of scalar measure of plastic deformation (usually denoted by Xg or Xn). • In the case of viscosity they can represent the inverse value of the constant coefficient of viscosity (usually denoted by 1/(2He) or 1/(2Hn)), or the coefficient of Bingham-type viscous deformation (denoted $g or fin). • In the case of continuous fracturing they represent the rates of deviatoric compliances (i.e. (i„ or fi„). Symbols
=j{Ve

lMe(Sije Sije + — S'^ S^,) + 3p a (<7* + — O * )] + Vn \Vn(Sijn Sijn + — S^nS~n) + 3p„(<7* +—
where r}e, rjn are structural parameters for the deviatoric parts of stress- and straintensors, t]g and r)° for the isotropic parts. The relation between the deviatoric and isotropic average stresses and the symbols with primes are - according to (II. 1.20a):

n.2 General Two-Phase Model

28

s ..

_c.. + £ » ! _ £ » i = o

a -a

+^2—^2- = 0

(9)

In the general form of the model all symbols present in equation (9) can be variables - even structural changes are admitted - and differentiation gives: e.. _c.. +_Lc:. _£2Lri _ _ L s - + - ^ - r i le Ve Vn %

=0

(9*) Ve

We)

*ln

(in)

From equations (1) to (90 the general form of macroscopic constitutive equation with tensorial internal variables results (see Appendix vm.3) as follows: e,- = PS/ + vg(Mesijg+

* ( s ^ + v „ ( M n s ^ + Ufn s'ijn)hn

(10) Ve

Vn

t = P&+BT+v.(M:<xe+M?
)pe + vn(M°on+M'n°o'n

){>„

Tin

where P=Ve M„ [ Ve \iene + Vn Hn nn +TJeTJn (Va H„ +Vn Hn )\IR MeHVeHnlMn+HnlVeHene

+ VnUnlnWR

Mn =[ MaMnV?n +He ( W 7 « + Me=VnHn(He-Vn)n„'FI

(11)

Wifla)lfR (12)

n.2 General Two-Phase Model

P = PePn[VePJ)°e+ " n p„r?° + T ? X Kpe+Vnpn)]IR0

29

(13)

a=[(veae+vnan)pepn-q°en0n

(14)

+{vgaeP„ + vrficnpg)(vepgn° +vnpnr]° )]/R(

o

M°e=\pePnT}°7)° K

+P„(«Wtf

+V„Plfl^)]/R0

=\PePjfe ?7° +Pe(VePent+VnPnVn

)V^o

M'B°=vnpn(pg-pn)T)°IR0

(15)

Mf =VePe(Pn~ PeW ' Flo +^ePn+Vnpe)(VgpeT]0a+Vnpn7]0n)

Fto=PePnn^n

■

The evolution equations of the internal variables are: sije = Metij

+

vn[-(NsijeUnln -^s'iie)hB R

HNsijn-^S:jn)hn R R

Ve

(16)

In

6B=M° & + vn [N0 {ccn -cce)T + N0 (a>„ -d) e ) o

-(N0ce-^o'e R0 Ft0

ne

)p. + (N0on-^-c'n Flo

)p„

(17)

n°

(18) H

H

vn

ne

30

n.2 General Two-Phase Model

an=Ko

+ ve[N0 (<xe -an)T + N0 (cdg-a>n)

-{Noan-P^LG'n

H

o

Vn

)Pn + {Noae-££!a>g

)Pg

We

where N = {venene + vniin7]n)IR N0 =

(VePeWe'+VnPnnZ)/Ro

Vje^iVnUnTlellniHe-MSij+llelVnVnllnSije -(Vn1n

+ v

e(veHn

+ vn J"e))&'& 1 K

-VnVe{MnVnSij„+(VBHn+VnHg)S^]hn

+ Vn lln (Vetin + Vn He)( ^ % Ae ~ ~ 4 > An)} Ve

6'e =—

{vn Pnlel°n[.{Pe-

Pn)& +

Vn

(<Xe-<Xn)t]

H

o

+VeWn Pnn°n °e ~(PnIn

+ Vg(Vg p„ +Vn pg))<7e

-vnTfg\pnTfnon+{ygpn+vnpg)a'n

]pn

+vnp„T]°7]0n((be-(bn) ^nPn(VgPn

+ Vnpg)(^ael)°-^0'nrn)} Ve

An

■

]pe

n.2 General Two-Phase Model

4,=—

31

{veVeWn&n-He)*!

+nn[VaileneSijn-^e'ne +

V„(VeHn+VnHe))^jn\hn

, ■ - V. Vn\Me *le % +(Vg jU„ + V„ M e ) ^ ] ^

<*'n = — {ve PeVlVnKPn"o

(23)

Pe)& + (<Xn-<*e)f]

-r)°[Ve Pe if* °n ~(Pen°e + "#,(". pn + Vn pg))o'n ]pn

-verfn\perf0oe+{vepn

+ vnpe)oe\pe

(24)

0

+vepej]°ei] n((bn-
+

vnpe)(^o'n i ) » - 4 a i #>} •

The number of internal variables could be reduced, as o^ can be expressed by o*n and
32

H.2 General Two-Phase Model

^=■517-+4 +A. + A,-

(25)

From experience Hn, Xn, $n as well as fin are known to be nonnegative (Un non-negative means that fracturing cannot cause rise of the shear modulus). This knowledge follows also from the second law of thermodynamics: The deviatoric stress-power in a unit volume of the n-th material constituent under homogeneous strain is: ( K )hom = Sf6f = /*„ Sf Sf + /)„ S,~ s f

and the rate of the respective elastic energy (Wf )hom = — ( - H n Sf Sij ) = HnSi/ Sf +-fln Sf Sf .

The difference of the two expressions represents the rate of energy dissipation that - according to the second law of thermodynamics - cannot be negative: (K)uom " ( W ^ J h c , = Sij Sf (-±-+Xn

+

$n

+

U

) ?> fj

(26)

and as Hn,Xn, $n and fin are independent, it holds true: Hn7>0,Xn>0,$n>0,(inZ0,hnZ0.

(27)

Furthermore, according to Eq.(K.1.9) the deviatoric stress power in the nth material constituent in a unit volume of the composite is expressed as follows: Wn = V„ {Sf,, e/j,, +-—Sfn 6fn ) n

"

(28)

=«**.*, ^s^-L^s^

In

and the respective rate of energy dissipation:

+ 4,4A)]

n.2 General Two-Phase Model

W„-W? =vn(SijnSijn + — S ^ 4 , )fln .

33

(29)

In Even in this case of microscopically heterogeneous stress- and strain-field as described by our model, hn must be non-negative, as T]n is non-negative by definition. Both Substructures Discontinuous The discontinuous substructures are described by the B-model. In this model equations (1),(2),(3),(5),(6) and (8) keep their formal applicability, but the meaning of a'iiB, a~n, e ^ ande,^, is different here. Instead of equations (4), (7), (9) and (9') the following relations must be applied: °'ije = °ije ~ &$

(30) (31)

<*'ijn = 0fn ~ °ij

eae-eijn + efijelxe-

Xe

'

"

X°e

e'ijnlxn=^

\Xe)

(Xlf^

Xn

X°n

(32)

Wn)

(X°nfXn

Again, after a successive elimination procedure (cf. Appendix vm.4) the respective macroscopic constitutive equation can easily be derived: * f = P*i

1r e' + —\Vg Vn {»„ -HB)Xe Xn i-T^TXn S (Xn)

e' -—^jt,] (Xe)

+ VelVn (/*e -Vn)Xn »# + UniXe Xn + VB Xe + ^n Xn)sIJe]hB + Vn[VB{H„-fle)xeS9

+ He(Xe Xn + Vg Xe + Vn Xn)Slln\h„)

( 3 3)

34

n.2 General Two-Phase Model

p&+at+±[vgVn(pn-pe)[Z&X0n-£^-t0e] Xn Xe

i = S

o

V

+ Ve(VgpnXe+ nPeX°„

+ P„Xe X°„)
+ Vn(VnPoX0„+VePnXe+PeXeX°n)„

(34)

+

Ve[Vn(Pe-Pn)X°&+Pn{VeXe+VnXn+XZXn)Ce]fie

+

VnlVe(Pn-Pe)X0eV+Pe(VeX°e+VnXn+Xe'Xn)On]Pn}

where s^ and on are internal variables with the following evolution equations: Sijn = M^

4- ^[

Xni-Sy + 0 + * 8 ) S » > e (35)

r—

,A

,.

IL

Xe&Hn . Xn

*n=MOn*

+ %-{xOn[-V+(l

Xn^ie

. 1

Xe

+ X°o)°o]Pe+XttV-V

+

X°n)0n]Pn

+x!Xnia>e-(0„+(ag-an)t] . Xe £n *.o Xn vo

(36)

Xn ^e +o \ Xel 0 Aa

A,n

and p, Mn, S, p, 8, M°, S0 are scalars: P = \VeHnXeXn

+ (VeHe + WnHVeHnXe

+ WeXnWS

(37)

M„ = {Vg Hn Xe + V„ He Xn + Pe Xe Xn VS

(38)

S = (VeHn + VnVe)XeXn

(39)

P=[PePnXeXn+(VePe

+ WnXe

+ VnH*Zn

+ VnPn)(VeP„Xe+VnPeXnWS0

^=[(veaepn+vnanp0)XeX°n+(vgag-i-vnan)(vepnxl+vnpeXn)VS0

(40)

(41)

n.2 General Two-Phase Model M°n ={VePnX0e+VnpeX°n+PeX°eXn)IS0 So = (Va Pn + Vn pe)X°

Xn + "• Pn X°e + Vn Pe X°n ■

35 (42) (43)

Let us shortly comment on the duality of models A and B: (i) To underline the duality of the two models, the same symbols for analogous quantities a'ij,e^,p,p,S are used, although their definitions in the two models are different. (ii) It can easily be seen that the substructures in model A (with finite structural parameters) are really continuous, and those in model B discontinuous. To this purpose let either fit, pe or pn , pn in equations (11) and (13) be vanishing (which means rigidity of one of the material constituents). This gives macroscopic elastic compliances P and p vanishing too, i.e. rigidity of the composite corresponding to the A-model. On the other hand, if the same substitution is done in equations (37) and (40) - valid for the B-model, p and p are not vanishing, the composite is not rigid. Similarly, let either / i a , pg, or p.n, pn go to infinity (which means absolute lack of resistance of one of the material constituents). This leads to absolute lack of resistance in the composite described by model B (infinite values of p and p resulting from equations (37) and (40) ), but not in the composite described by model A (finite values of p and p resulting from equations (11) and (13)). An analogous result is obtained for the macroscopic coefficient of linear thermal expansion. For aB and pg or an and pn vanishing (meaning no volumetric changes in one of the material constituents) the macroscopic volumetric changes are vanishing in the composite described by the A model (vanishing value of 8 resulting from equation (14)), but not in the composite described by the B model (finite value of 8 resulting from equation (41)). (Hi) There are two tensorial internal variables in the A-model, whereas only one in the B-model. The choice of a^ for the internal variable and elimination of the others is not the only possibility; it seems to be advantageous in the case of plastic deformation because of its adequacy for the formulation of yield condition in the n-th material constituent, but in other cases other internal variables can be preferable. Inclusions of One Material Constituent in the Matrix of the Other Let the material constituent that forms inclusions be the ^-constituent. According to Section n . l , the model of such structure can be arrived at either from

n.2 General Two-Phase Model

36

the A-model by attributing infinite values to structural parameters i]g, vfe, or from the B-model by attributing infinite values to structural parameters %e> X%Let us start with the A-model: For infinite values of r\g, r]g, equations (10) to (22) take on substantially simpler form. There is no need for the evolution equation for a'IJn, as this quantity can be simply expressed from equation (9): s

ijn

=

Tin (sije ~sijn)

=

~~(Sln

~ s 0)

(44)

on=7]°{oe-on)

=

-^-{on-3).

With the use of these equations we arrive at: °l=P8t+-fi[flnfre+1ln)(8l-V„8t,)h.

+ V„ lri„(H„ ~»e )8, +Veiye +»?„ )«*,] *n

+

(45)

vnnnUin-ne){srsljn)tin}

t = pd + 3t + M°a(3-vnon)t>e + ^-M{Pn-Pe)V+Pe(Ve+ri0n)°n)f>n

(46)

n0 *n Pn£jh^S-V-*n)n0n "o

Sijn =MnSjj+

—{Vg(Sij-Vn

+ tin Sij ~(4

+ VgM°g
+

Vn M°n (bn

S?n)/7e

+r

ln)Sijn] K + Hn(S~i Sijn)fln}

<*n= M°nV+-Z-{Ve(3-VnOn)pe

+[r}°3-(vl+T)°)On\pn

(48)

+ Pn(^-On)T)°n +v^[cbg-(bn+{ae-an)t]}

.

It is clear from the above expressions that the only internal variable here is <7g„. The constants in the above expressions result from Eqs.(ll) to (15) for infinite values of TJ8 and r\l, and from Eqs.(44):

n.2 General Two-Phase Model ^VniVeHe+VniVeVe

+ VnVnWR

37

(49)

Mn=(VeHe+Vn1nVFI

(50)

P = Pn[VePe + Vn(vepe+Vnpn)\/R0

(51)

S=[(veae + vnan)pX

(52)

+vg(veagpn^-vnanpe)\IR0

Ml=Pniy*+rinWo MnHVePe+Pnn°n)/R0

(53)

Ro=Pntf+Veiv*Pn+V„Pe\

If letting ixe, p e or ju„, p n go to infinity, or to vanishing values, it can easily be shown - similarly as in Section n. 1 - that the e-material forms a discontinuous substructure (inclusions) and the n-material a continuous substructure (matrix). According to Section n.l the case of inclusions of the e-constituent in the matrix of the n-constituent can also be described by the B-model with Xe and X° going to infinity. It can easily be shown - by an analogous procedure as with the A-model - that it is really so. This analogous procedure leads to formulae that are equivalent to equations (45) to (53): If the symbols r\n and r\° in equations (45) to (53) are replaced by ve2lxn and ve2IXn° respectively, we receive formulae that result from the B-model; and as r]n, r]°, xn • X° are free parameters that are to be determined from experiments, the two variants derived from the A-model and the B-model have exactly the same meaning. This formal transition from the Amodel to the B-model in the case of inclusions is without problems if the changes of structure are excluded - i.e. if f\° = i)n = 0, x° = Xn = 0 • I f ^se; changes are taken into account, this transition works only in some special cases. Homogeneous Stress Model Supposition of homogeneous stress in a heterogeneous material - often called Reuss* solution - corresponds to a series arrangement of rheological models in one-dimensional simple schemes. However, there does not exist any real threedimensional structure that would lead to homogeneous stress, and therefore, this

n.2 General Two-Phase Model

38

scheme is only fictitious. Nevertheless, this fictitious scheme can be received from our model too, as its trivial oversimplified variant. According to Section II. 1 the respective macroscopic constitutive equation is arrived at either from the A-model with all structural parameters infinite, or from thefl-modelwith vanishing values of its structural parameters. This constitutive equation reads: % = (VgHe + VnHn)Sij + VeSq He + Vn S/j Hn

(54)

+ vgffpg + vnopn + ve(bg + vn(bn a

ijn =Gjjo =<*ij ■

(55)

Homogeneous Strain Model Supposition of homogeneous strain in a heterogeneous material - often called Voigt's solution - corresponds to a parallel arrangement of rheological models in one-dimensional simple schemes. Similarly as homogeneous stress, this scheme is only fictitious in a three-dimensional medium. According to Section n. 1 the respective macroscopic constitutive equation is arrived at either from the A-model with vanishing values of structural parameters, or from the B-model with structural parameters infinite. From both these ways the same unique result is: 5|=*» = « » ! = ■

*

*

v

Uie^n%l+VeHnSijehe -\PePn& +

+ VnHeSijnhn)

(56)

(vgp„ae+vnpean)t

ePn + VnPe + Ve P„Oe pe + V„ p„On pn + Vg Pn(&e + Vn pgQ)n]

Sijn=———— ( M f + VeSijehe - VgSijnhn) v ern "'" 'nr*e a

n = ————[Pe^+Ve(ae-CC„)t+Vg<7gpe-Veanpn V ePn + vnPe

+vg((bg-d)n)].

(57)

n.2 General Two-Phase Model

39

In the derivation of the above expressions the changes of structural parameters must be excluded, they do not have sense, as these expressions do not depend on structural parameters. The expressions for S/^,
40

II.3 Separability of Deviatoric and Isotropic Responses

II.3 Discussion of Separability of Deviatoric and Isotropic Responses of the Mesoscopic Stress- and StrainComponents It follows from the preceding sections that in our model representation, the mesoscopic stress tensors (averages in individual material constituents - e.g. Oyn) are related to the macroscopic stress tensors <7,y (by which the representative volume under consideration is loaded on its surface) in a specific way that deserves comments: If the investigated heterogeneous material was in the virgin state (stressfree and strain-free on the macroscale as well as on the microscale) and then it is loaded by some moderate macroscopic stress 3^ causing only elastic response, then the deviatoric part of

isotropic mesoscopic response 5tjOn. This is no model approximation, this is an exact result that can be rigorously proved (under the assumption of statistical homogeneity and isotropy): Proof: Let all components of the deviator s^ be vanishing with the exception of S|2

=

S^i. According to our model the only non-vanishing response on the

mesoscale will be s 12n = S21n (i.e. S^n = 0 •

for i/jtl2,21) and StjOn = 0 .

In the first step we are going to prove that 8^on s 0:

Let us turn the coordinate axes X;, X2 by angle 45°, the axis of rotation being X3. In the rotated system X*h x'2 and X*s = X3 the only non-vanishing macroscopic stress components will be <5vi = Sj 2 , &zz = - S12 • Due to the additiveness of any elastic response we can consider separately the response to c7^ on the mesoscale and denominate it ( c J i . Due to the assumed statistical isotropy, the response to 022 must be

II.3 Separability ofDeviatoric and Isotropic Responses

41

(OA>2 = -(<7
In the second step we will prove that Syn = 0

for (/* 12,21:

=

s, 3n

Let us first admit that S)2 S21 could cause non-vanishing values of e.g. =S3 1n . These mesoscopic components can be described in a rotated system

of axes X"i, X*'2 = X2 and x"3, with X2 the axis of rotation and the angle of rotation 45°. The respective normal components are: a

11n

=S

13/J'°33n- - S 13/J-

From the point of view of normal stress components the plus and minus sense of the axes indicating the direction of stressing is irrelevant and therefore, the angles between X'I and x"i, X*i and x"3, X*2 and X"t, X*2 and x"3 are the same (60°). Accordingly, if the response to (Jii is (CT11*n)l=(s13n)l

the response to a^2 must be (°11 n )2 =( s 13n )2 =-{&\ 3/i )l

and consequently the response to s, 2 = s^ is A similar procedure can be performed with other components to show that sijn=Ofor ij± 12,21 q.e.d. It is very clear that what was proved above for the mesoscopic values for averages in the material constituents - is not true for the local microscopic values. In our model we do not work with microscopic values, but we describe their influence by tensorial variables - those labeled with primes. Of course that such a description cannot be but approximate, and in our approximation the elastic response in these values with primes has a similar character (discussed above) as the mesoscopic values. Thus, e.g. the elastic response to macroscopic stressing s^2= s ^ in the n-th material constituent is described by S)2n=S2in anf^ Si2n= s2in • T n e reason for such a scheme can be seen in the fact that under the assumed stressing the effect of fluctuations S^2n will clearly dominate over e.g. S|3n •

As for the inelastic processes, they are described in the material constituents similarly as in a macroscopic analysis, but they result from the just existing stress state in the material constituent

42

II.4 Demonstrative Diagrams

II.4 Demonstrative Stress-Strain Diagrams for Two-Phase Materials Elastic and Viscous Material Constituents To clarify the difference among different types of continuity of two substructures in a two-phase material, the respective flow-curves will be presented for one phase elastic (labeled with index e), the other phase viscous (labeled with index n ). Both materials are assumed incompressible, the process is assumed isothermal. In such a model the constitutive relations (n.2.3) to (Q.2.8) are simplified by the following relations: he = Pe=(>e=e=0> hn=—,

nn = pn=pn=(bn=0

and

T=o.

Then it is straightforward to rewrite the simplified macroscopic constitutive equations and evolution equations of Section H.2. They read: • For both infrastructures continuous: e

j=-£jj\sijn-sgn)

c.. - 1 it * • vn(S" S

"° v}

v Si n s

° i ~ '*>) 2Hn„ }

m (1)

S

'> + 2Hne{S"°

v0ne * » H -

• For both infrastructures discontinuous: 1

,

»# =T-T-\V»

-

1

r/

Ve Sj +^77—"IK Zo + Vn Xn + Xe Zn)**

-V,Z»SfJi Sijn =

Vn+Xe

(2) t, +

^

[S tf -(1 + Xn)Sjn] ■

43

II.4 Demonstrative Diagrams • For elastic inclusions in the viscous matrix: °iJ =-^T^V°

+T

>n)Sijn -Tin *ij] (3)

S

>"V*

h

2Hneve

• For viscous inclusions in the elastic matrix: -

1

/

-

V +T

n le

v

(4) Si,n=— 'tin'

fi+Ve

V Vn

[(Vn+Ve)t >l o u" „ 2Hn

e

S

ii°}-

With the use of these relations the respective macroscopic responses to uniaxial step-function loading for randomly chosen values of finite structural parameters were plotted in Figs.l to 4.

5

supr.s,

0

Fig. 1 Continuous substructures

k"'

Fig.2 Discontinuous substructures

II.4 Demonstrative Diagrams

44

e

7

Y

1 ■

—

/s

1

V

^

supr §'

t

s

*^ "^ --—n .

C

s 0

\ \t \

t

Fig. 3 Elastic inclusions - viscous matrix

Fig. 4 Viscous inclusions - elastic matrix

It can be seen in the above schemes that continuous viscous substructure prevents macroscopic immediate deformation response to the loading steps, continuous elastic substructure causes asymptotic approach of macroscopic strain to zero value after unloading. Also the courses of other stress- and straincomponents agree with what is expected in such a kind of composite. Elastic and Elastic-plastic Material Constituents - Elastic Moduli Homogeneous Another important special case is that with homogeneous elastic properties in the whole medium, but with differing inelastic response of the two material constituents. The respective model for continuous substructures follows from equations (II.2.1) to (II.2.24) that are simplified by assumptions:

he=Pe = Pn =^ 9 =0 , hn=Xn, nn=ne=n,

pn=pe=p,

ti)n>0

and 7 = 0 .

Stress-strain Diagrams [Flow-curves] of Two-phase Composites with One Phase Elastic and the Other One Rigid-plastic [Viscous] with Different Types of Continuity of the Substructures, and Under the Assumption of Homogeneous Stress or Homogeneous Strain The influence of different types of continuity of the substructures is further demonstrated on the respective stress-strain diagrams and flow-curves in Fig.5. Similarly as in the preceding case the isotropic response is assumed homogeneous, and therefore, only deviatoric parts are taken into account. To

II.4 Demonstrative Diagrams

45

complete this scheme, also the simple hypothetical cases of assumed homogeneous stress (Reuss' model) and homogeneous strain (Voigt's model) are demonstrated, in spite of die fact that no real structure can be represented by these schemes. The respective formulae of the constitutive equations are not given explicitly, they follow from the general form in a straightforward way.

5ct>eme of the infra structures

(3

0< if < o . 0 < rf -c ««,

®

0 < n < Mj n

O S ■

Stress - strain diagrams ■ elastic a rigid-plastic 5 ^^*

Structural parameters

m

oo

0 < X
0 < X'<

oo

c". c°. E

X°.Z'.

Infrastructures

Si

e

0

6

|2C ,

K , K:, e L

S

. t

-* €

S

7*Z ,

oo

•■ o continuous,

,

m £

x°. x". o

f-rf-

^Z

s

5

•J

o -

cr -cr -cr

e

v-.

* • - OO j 0 < A < O o

0< X°<

Flow curves m elastic a viscous

•discontinuous

Fig. 5 Schemes of stress-strain diagrams of elastic - rigid plastic composites, and flow- curves of elastic - viscous composites, with different types of continuity of substructures.

46

II.5 Elastic Bounds

II.5 Relation between Our Two-Phase Model and the Exact Bounds for Macroscopic Elastic Moduli Generally speaking, the aim of our approach is to determine the structural parameters from the inelastic response of the material samples and then to predict the response of the material in other inelastic processes. This approach is not designated for applications in elasticity, as it is not possible to determine the structural parameters from the elastic response alone, but it is also not necessary to determine them, as for prediction of elastic responses macroscopic Young's modulus and Poisson's ratio are sufficient Nevertheless, description of the elastic part of deformation is comprised in our model and the intent in this section is to show: (i) Whether this description is in agreement with the exact upper and lower bounds derived by other authors for elastic moduli of heterogeneous two-phase materials, (ii) How are the values of our structural parameters limited if they are to agree with these bounds, (iii) What is the relation of our concept to special cases of the bounds. This section summarizes results of the paper published by Kafka and Hlav&ek (1995). The forthcoming analysis is based on equations derived in Section n.2 that are valid for macroscopic compliances and coefficients of thermal dilatation. To understand that in this section we have in mind two material constituents with elastic properties and not with elastic and inelastic properties as in Section n.2, we use here indexes a and b instead of e and n used in Section n.2. Let us further recall the relations of our macroscopic compliances to macroscopic Young's modulus E, Poisson's ratio v , shear modulus G and bulk modulus K: fi=(UV)/E=V(2G),

p=tf-2v)/E=V(3K).

(1)

It was explained in Section II. 1 that model A with non-zero finite values of all structural parameters r] describes composites with continuous infrastructures of both the material constituents, and model B with non-zero finite values of all structural parameters % describes composites with discontinuous infrastructures. The case of inclusions of one of the material constituent (a) in the matrix of the other constituent (b) can equivalently be described either by model A

n.5 Elastic Bounds

47

with infinite structural parameters r\a and T]° , and finite parameters rjb and ■q%, or by model B with infinite structural parameters Xa and Xa" and finite parameters Xb and Xt" ■ Now let us confront the statements presented above with our formulae: Model A with TjgJiZflbj]% positive finite is to correspond to the case of two continuous infrastructures. In such kind of a two-phase structure it must hold: • If one of the two material constituents was absolutely rigid (e.g. \ia = pa - aa = 0), the composite should also be absolutely rigid (ju = p = a = 0 ) and the stresses in the other constituent ( s ^ , cb) caused by loading should be vanishing (not so the isotropic stresses caused by temperature change). • If one of the two material constituents was absolutely compliant (e.g. Ha=pa=cca=°° )• m e composite should not be absolutely compliant (P*°°,p±°°,a*°° ) and the stresses in this constituent ( s ^ , aa ) should be vanishing. It is very easy to see from equations (n.2.15), (n.2.17) and (II.2.19) that it is really so - the infinite values must be understood in the sense of limits. Model B with Xa, Xa . Xb < Xb" positive finite is to correspond to the case of two discontinuous infrastructures. In such kind of a two-phase structure it must hold: • If one of the two material constituents was absolutely rigid (e.g. Ha=pa=cca=0), the composite should not be absolutely rigid (p*0,p*0,&*0) and the stresses in the other constituent (S^b,(Tb) should not be vanishing. • If one of the two material constituents was absolutely compliant (e.g. lia=pa=cca=BO), the composite should also be absolutely compliant (jff = p = ff = °°) and the stresses in this constituent ( s ^ ,aa ) should not be vanishing. Again, it is easy to see from equations (II.2.30), (II.2.32) and (E.2.34) that it is really the case. Inclusions of one of the material constituents (e.g. a) in the matrix of the other constituent (e.g. b) can be modeled - according to Section II. 1 - by letting parameters J]a ,ria in the A-model grow over all limits. In such kind of a twophase structure it must hold:

n.5 Elastic Bounds

48

Absolute rigidity [absolute compliance] of the matrix (b-material) leads to absolute rigidity [absolute compliance] of the composite, whereas for the inclusions (a-material) it is not the case. Furthermore, without going into details, the values of stresses for these limit cases of moduli must have the values that correspond to the structure of matrix and inclusions. It can be seen from equations CQ.2.37), (n.2.39) and (n.2.41) that our model is in agreement also with these requirements. Relations to the Bounds Hashin and Shtrikman (1963) obtained bounds for the macroscopic elastic bulk and shear moduli, K and G , in the case of the elastic biphasic macroscopically isotropic composites with volume fractions va and phase moduli Ka , Ga fixed. Later, these bounds were generalized for any relations between Ka , Ga by Walpole (1966). The bounds for the macroscopic thermal dilatation were presented by Levin (1967). If using our symbols [i.e. p a =1/(3K a ),/x a =1/(2G a )] the generalized Hashin - Shtrikman bounds for macroscopic compliances p , p take on the following forms ^ U

v (Pi,/P«)-1 *(2p b 7,0 + 1

Vs u

v

(Pb/P)-1

(P/,/pa)"1

*

, * , _ > <

U v

(Pfr/Pa)-1 {2Pbln') +\

<J&Mzl<.

(^,/M.)-1 ( / l b / ^ a ) - 1 (Hblii*) + ^

1+ v

(2)

b

b

(^//^a)-1 (nb/n**) + 1

(3)

where p' = Hb

, »." = Ha

H* = fibb, »** =

(4) fiaa

(5)

for the cose, in which (Pb-P*)(Hb-Ha)*0

(6)

n.5 Elastic Bounds

49

and (7) (8)

for the case with (9)

(Pb-Pa)(Hb-»a)^0The symbol p.^ is defined as follows:

flop =2/1,,

tip + 3pa 3/*/j + 4p„

cc,f}=aj).

(10)

The bounds (2)-(3) for p,fl can be given another more explicit form. For example, if pb< pa , fib
(11)

2(/fb + 3p b )(y a na + vb yb) + na (3fib + 4p b ) (3jU6 + 4pb)(va nb + vb nm) + 2nb (nb + 3pb)

(12)

Similar equalities can be deduced for the lower bounds, and for other relations among the elastic compliances of the phases. The bounds for the macroscopic coefficient of thermal dilatation read according to Levin (1967): fib + 2pb ■ < - ^ - ^ " »b + 2(vapb + vbPa) <xa-ccb

„ a

Mfl+2Pi, Ha+2(vaPb + vbPa)

(13)

for the case aa>ab

, pb>pa

,

p.b>Ha

(14)

n.5 Elastic Bounds

50

For the case where the relations in (14) are opposite, the relations in (13) are opposite too. Now it is a straightforward (but laborious) way to come to the bounds for our structural parameters using the expressions (II.2.15), (n.2.30), (n.2.17), (H.2.32),(n.2.19), (H.2.34) for JI.p.B : Model A: Bounds for T]a" and rjb0: r n i n ( — , — ) < ± ( - ^ +-^-)<max (—,—). Ma Mb 2 P a J 7 ° p X /i a /*„

(15)

Bounds for T}a and r/fc: for the case where relation (6) is valid: min( 1 Maa

- L ) < - ^ +-^^max( fibb

1

Ha !a

Mb

1

-!-)

(16)

Maa P-bb

and for the case of relation (9): m i n ( J - , ^ - ) < - ^ - +-^_<max(-L,-^-). Mab

Mba

Mb7?*)

A*a '/a

Mab

(17)

Pba

It is very interesting that relation (13) - valid for thermal dilatation coefficients - does not lead to some new bounds for our structural parameters, but exactly to the bounds given by inequalities (15). Model B Bounds for Xa and min(jua .Mb)^2(- *bPa , X°a

Xb°:

V

aPb )<,m!a(n ,fi ). a b Xb

Bounds for %a and %b: for the case where relation (6) is valid:

(18)

n.5 Elastic Bounds

min 02 M , /*«,) < ^ Xa

+^ Xb

51

< max (/2M , fibb)

(19)

^^-<max(fiat>,fiba)

(20)

and for the case of relation (9): Tmn(fiab,ftba)<^^-

+ Xa

Xb

Again the bounds (13) for the macroscopic thermal dilatation coefficients do not lead to some new bounds for our structural parameters, but exactly to inequalities (IS). Some Special Cases Through a variational principle, Hashin (1962) obtained bounds for the macroscopic bulk and shear moduli in the case of a special macroscopically isotropic arrangement of spherical shell and kernel formations called "composite spheres assemblage". The bounds for the macroscopic bulk modulus merged and coincided with the respective Hashin-Shtrikman bound (1963) for general microstructure, thus yielding the exact value for the bulk modulus of the composite spheres assemblage. It follows that the general Hashin-Shtrikman bounds for the bulk modulus are the best bounds and cannot be improved, if volume fractions va are fixed. However, the bounds for the macroscopic shear modulus of the "composite spheres assemblage" did not merge and the respective HashinShtrikman bound lies between them. Now, let us consider the following special cases: • First, let J"a=M6 . Then, bounds (15) of model A coincide to give: _2(1-2vB) 1 + Va

v

a

CaVa

>

<■•bVt

• Second, let fiaa = fibb (16) coincide t oyi
1.

I

with (pb - pa){Hb - Ha) > 0

y

"

7-5va 2(4-5va) "

(21) hold. Then, bounds

(22)

n.5 Elastic Bounds

52

• Third, let && = (!■& with (pb - p„)(nb - / i a ) < 0 hold. Now, bounds (17) coincide to yield: =1,

e

eaJ7a

ea=tpA{vayb),

eb=(p2(vayb).

(23)

b1b

Here, the functions q>u
. Pa = Pb = P

In such cases, inequality 16) transforms into an equation:

v^

na

+

^

nb

=

Jz^L

(24)

8-iov

which means that one of the structural parameters r/a , r\b need not be determined from experiments, but calculated from equation (24). Structural parameters Va • Vb are °f n 0 interest in such a case. Structural Parameters Independence of Young's Moduli It is clear from the above considerations that our structural parameters depend not only on the geometry of structure of the composite, but also on Poisson's ratios. On the other hand it was implicitly assumed that they are independent of Young's moduli. This hypothesis is in agreement with equations (21)-(24) derived above. It can be corroborated also by the following considerations: Let us assume that in some composite the volume fractions, structure and Poisson's ratios are fixed, but the compliances jua, \ib in equations (Q.2.15) or (n.2.30) change with some parameter p (due to some change of E a , Eb ):

n.5 Elastic Bounds

53

A = PHa . Pi = PVb ■ (25) It is clear that if the compliances of both constituents in some composite increase or decrease in the same way, i.e. with the same parameter, the macroscopic compliance must change in the same way too, i.e. p(M'a .lil) = P(PHa,PHb) = PP(Pa,Hb)(26) It is easy to see from equations (H.2.15) and (H.2.30) that with constant structural parameters equation (26) is really fulfilled, i.e. the assumption of independence of the structural parameters of Young's moduli is not contradicted. Quite similarly the same can be demonstrated with equations (n.2.17), (II.2.32), (H.2.37) and (H.2.39). Now to the case where not both compliances, but only one is changing. Let one of the compliances, e.g. \xa , be infinite. This means that the a-constituent corresponds to continuous pores. For this case equation (II.2.1S) transforms as follows: P=^Pb-

(27)

"b

It is clear that if the compliance of the only material constituent of such a spongy material increases or decreases with some parameter p , the macroscopic compliance must change in the same way, i.e. P(Pfib) = pP(llb). (28) Again it is easy to see from Eq.(27) that with unchanged structural parameter t]b , Eq.(28) is really fulfilled and hence the assumption of independence of the structural parameters of Young's moduli is not contradicted, not even in the case in which only one material constituent changes its modulus. Quite similarly it is possible to proceed with equations (TI.2.17), (H.2.37) and (n.2.39) with the same result. The above considerations do not present a rigorous proof that the structural parameters are independent of Young's moduli, but this model hypothesis of independence does not lead to contradictions; neither in the above considerations, nor in all other applications. Structures Corresponding to the Bounds Let us assume that the volume fractions and elastic constants in some two-phase composite are fixed. Let as further assume that there exists some structure - or some family of structures - by which the upper or lower bound of the

54

n.5 Elastic Bounds

bulk or shear modulus is realized. Intuitively it is natural to expect that the structures corresponding to the upper bounds will be formed by some type of matrix of the stiffer material constituent and by some form of inclusions of the material constituent that is more compliant. If we want to measure stiffness and speak of structure as of something what is independent of stiffness, it is inevitable to measure stiffness by Young's moduli only, not by Poisson's ratios, which are not independent of our structural parameters. Such a view is the only possible in general, as on changing both Young's moduli and Poisson's ratios, it would be at all impossible to distinguish between stiffer or more compliant materials. With this viewpoint accepted let us scrutinize formulae (11) and (12). Let us put aside for a while the fact that they correspond (under some given assumptions) to the upper bounds, let us concentrate on the question to what kind of structure they correspond. This can be clarified letting the structure and Poisson's ratios befixedand letting Young's moduli undergo some virtual changes. Thus, e.g. if the virtual change of Eb is increasing to infinity, the corresponding virtual change of both nb and pb is decreasing to zero values, which according to (26) and (27) - leads to zero values also of p and p (i.e. to absolute rigidity of the composite). On the other hand, if the virtual change of E a is increasing to infinity, the corresponding virtual change of both /xa and p a is decreasing to zero values, but the resulting values of p and Jl are finite. Similarly absolute compliance of the ^-constituent (Eb=0) leads to absolute compliance of the composite, but absolute compliance of the a-constituent {Eg = 0) does not lead to absolute compliance of the composite. This means that the infrastructure of the ^-constituent is continuous (matrix), but the infrastructure of the a-constituent is discontinuous (inclusions). For the cases of relations other than ftb < fia , pb < \ia the reasoning can be performed quite analogously. This is in agreement with our intuitive assumption, but also with the Hashin's result arrived at with the "composite spheres assemblage" in the case of bulk modulus (the stiffer shells forming the matrix and the more compliant kernels forming inclusions). Now let us turn our attention to the bounds for our structural parameters, equations (15) and (16). Assuming again that \lb<\ia ,pb< pa the upper bounds are realized by the following equations: PX

Pbifb Vb

(29)

n.5 Elastic Bounds

^

+

J^L.=

3 * + 4 ft.

.

55 (30)

According to our hypothesis substantiated in the preceding paragraph, the structural parameters do not depend on the variations of Young's moduli. Thus, we can assume fixed volume fractions, Poisson's ratios and structural parameters, but varying modulus Ea to Ea/p , which gives variation of p a to ppa. Hence, we can write:

-Jfc_ + - ! i — L . PPaTa

PbTb

(3i)

l*b

Subtraction of equations (29) and (31) leads to T]a=°° and the same procedure performed with equation (30) leads to t]a =°°. This means that the aconstituent forms inclusions, and thus, what results from our model is in accordance with the preceding reasoning. A similar procedure can start from relations (18) and (19). It leads to Xa =°° and Xa —°° ■ T n ' s a gai n says that the a-constituent forms inclusions. Hence, we see that the conclusions following from our concept are in full agreement with the results that can be deduced from Hashin's bounds. But with our approach we are able to achieve something more: Using the infinite values of t]a and 77° in equations (29) and (30) gives:

„? = v lb

1+v

a

"— 2(1-2v„)

Vb=2va^P/

(32)

;

(33

>

-ovh

which are the finite structural parameters corresponding to the family of structures by which the upper bound is realized (all of them having a matrix formed by the ^-material and inclusions of the a-material). With these structural parameters specified, it is possible to use them in equations (II.2.37) to (II.2.41) and model the thermomechanical behavior of the composites with structures by which the bounds are realized.

56

n.5 Elastic Bounds

Conclusions a) By their defining equations structural parameters r/ and X are bound to be nonnegative and this bound spans the area between the limits given by Reuss' and Voigt' s solutions, i.e.. between homogeneous stress and homogeneous strain models. The bounds given in the preceding paragraphs make narrower the extent in which these parameters can correspond to real three-dimensional macroscopically isotropic structures with fixed volume fractions. These bounds are not at variance with the structural parameters being non-negative (cf. Eqs. (IS) to (20)). b) The fact that the Reuss model (corresponding to infinite rj-parameters or zero ^-parameters) does not correspond to some real structure, is clearly seen from relations (15) - (17), (18) - (20), where the finite bounds are surpassed with such values of parameters. The same holds true for Voigt's model. c) It is shown that the structural parameters of our model depend not only on the geometry of structure, but also on Poisson's ratios of the material constituents. d) All evidence indicates that the structural parameters do not depend on Young's moduli. e) Using Hashin's formulae, it was shown that the structures corresponding to the bounds are formed by matrix and inclusions, and that this finding is in full agreement with our mesomechanical model. f) In some special cases the confrontation of our model with the bounds can be used for a reduction of the number of structural parameters that are to be determined from experiments (cf. Eq. (24)). g) Our model makes it possible to express the structural parameters corresponding to the family of structures that realize the bounds - in terms of volume fractions and Poisson's ratios (equations (32) and (33)). With their values known it is possible to model the thermomechanical properties of such composites, among others to calculate average stresses in their material constituents. h) Kafka's concept is designated for modeling inelastic processes, not for elasticity, but it turned out that confrontation with the bounds valid in elasticity offers important pieces of information even for practical use of the model.

II.6 Confrontation with Other Models

57

II.6 Confrontation with Some Theoretical Solutions for Two-Phase Materials The 'classical' solutions to the problems of quasihomogeneous materials with discontinuous heterogeneity consider the geometry of composition as the exactly known input information, apart from the exact knowledge of the volume fractions and the constitutive equations of the material constituents. The aim is to derive the complete forms of the macroscopic constitutive equations (which can be achieved usually only with some simplifying assumptions). In our approach we suppose only the a priori knowledge of the type of the infrastructures, i.e. whether they are continuous or discontinuous, and in the resulting forms of the macroscopic constitutive equations there appear some free parameters (structural parameters and possibly others) that are to be determined by a mathematical analysis of the experimental macroscopic stress-strain diagrams or flow-curves. Hence, if we want to confront our models with some 'classical' solutions, the exact correspondence means that it is possible to find such non-negative structural parameters 7/ or x . for which the respective macroscopic constitutive equations are identical. Our approach is based upon simplifying assumptions (cf.n.l) and some simplifying assumptions are generally used in the classical solutions too. Therefore, the correspondence may not be in some cases exact, but it should be at least approximate for the range that is specified by the assumptions of the classical solutions, which means very often for small volume fractions of the inclusions. A number of classical theoretical solutions that are presented for confrontation were taken from Reiner's monograph (1958): Rigid Spherical Inclusions in a Viscous Matrix Under the supposition that the volume fraction of the inclusions is very small in relation to unity, A. Einstein presented the respective solution showing that the resulting macroscopic constitutive equation is of the same type as the matrix, i.e. linearly viscous, with the following macroscopic coefficient of viscosity:

H = H(\ + 2.Sve)

(1)

n.6 Confrontation with Other Models

58

where H is the coefficient of the viscous matrix and vg the volume fraction of the inclusions. In our concept the respective macroscopic constitutive equation follows from Eq.Cn.4.30 with fXe = 0 , i.e.:

=

'> i^fer>

(2)

which means that in agreement with Einstein our resulting macroscopic constitutive equation is also linearly viscous and H=

°+7]n H .

(3)

V„T]n

The two expressions for H are identical for

*T«L5- '

(4)

The structural parameter r/n is non-negative only for vg < 0.6, which is not surprising with regard to the supposition under which Einstein's solution was derived. It is interesting to note that Reiner in his monograph (p.S31) states that according to his and Amstein's experiments with mortar, Einstein's equation was approximately valid up to the volume fraction of sand 0.6. Spherical Bubbles in a Viscous Matrix Under a similar supposition Guth and Mark derived an analogous solution for bubbles instead of inclusions. The resulting macroscopic equation was again linearly viscous with the macroscopic coefficient of viscosity:

H = HV-ve)

(5)

where ve is the volume fraction of the bubbles. In our concept the respective macroscopic equation follows again from Eq.(n.4.3i), this time with fl0 = °° , i.e.:

n.6 Confrontation with Other Models

j 6,1

1+T

?" c 2HvnS"

59 (6) {)

which means that our macroscopic equation agrees in the type (linear viscosity) and

- 1-v H =2-H

(7)

The two expressions for H are identical for non-negative value J]n = 0 . Elastic Spheres in a Viscous Matrix Again under the supposition that the volume fraction of the spheres is very small compared with unity, Froehlich and Sack derived the following macroscopic constitutive equation:

8t + T,89=k(^ + T2ei)

(8)

with 7; = 3H/i e (1 + fv e )

(9)

T2 = 3AV/U1-fv e ). The respective equation in our concept is identical with Eq.(n.4.3i), which can be rewritten in the form: 2

57 + T; £ = 2AY V° +T?" (e, + % %) v

(10)

nVn

with

T;=2HnBve^^vnri„

(11)

n.6 Confrontation with Other Models

60

The types of the two macroscopic equations are again identical, but for the determination of one parameter rjn we have two equations: '1

=

'1

•

'2

=

'2 •

From these two equations we get: (Vnh=Vg—^j .

„ "

(T? )2=y

(12)

2-5ve+7.5v? 3-7.5*. •

For very small vg we get (Inh = {Tt„h±2v./3

.

(13)

For v e =0.1 the difference in the two values for t]n is about 3%. Viscous Spheres in an Elastic Matrix This problem with the same supposition as in the preceding cases was solved by Oldroyd with the resulting equation: ^ + 7" 1 %=M^ + 7"25,) .

(14)

Explicit analytical expression was given only for k and only under the assumption that the medium is incompressible:

*=r-0-fv/.>M,

(15)

■e

In our concept the respective equation is identical with Eq.(HA.4{), which can be rewritten in the form:

n.6 Confrontation with Other Models

% + 2 H

" - ^

l s =

* '

(

^

+ 2 H M

- ^

l s )

61

<16)

with 1/

^e(

1+J

7e)

Identification of k with k* results in:

3-5v,n which is positive for vn < 0.6.

Thus, the types of the equatioas as well as the limitation for the structural parameter again agree with the specification of the problem. Elastic Spheres in an Elastic Matrix In this case the agreement in the type of the resulting rheological equation is clear - it must be elastic. Only confrontation of the expressions that we get for the structural parameters are interesting. For the inclusions we use index 6 , for the matrix index rt . For a special 'composite spheres assemblage' Hashin(1962) gave an exact solution for the bulk moduli. If using our symbols, his result has the following form:

3(1-v„)(^-1K -^. = 1+ f-2 P 2(1-2vn)+(1+vn)[^-(^-1K]

.

(18)

For the shear moduli, the exact solution - presented in the same paper is known only for the case of very low volume fractions of the inclusions:

n.6 Confrontation with Other Models

62

15(1-v„)(1-^_) — =1

— vg . 7-5v„+2(4-5v„)^-

P

(19)

P-e

In our concept the respective equations can be arrived at from Eqs.(n.2.51) and (H.2.49). We get: Pn _Pntf+Ve(VeP„

P

+ VnPe)

(20)

vepe+tj°(vePg + v„pn)

Pn _»n1n

+ Ve(VeVn+VnHe)

(21)

Comparison of Eqs. (18) and (19) with (20) and (21) yields: o=Ml1V£L) " 2(1-2v„)

2M4-5VJ ,n

7-15v. + 5v„(3v.-1) It is easy to see that ifn is positive for the natural interval of Vn :

0 < v„ < 0.5 . With the same interval for v„ , rjn is positive only for vg < 0.46, which is not surprising, as Eq.(19) was derived only for very small values of vg . From practical point of view the application of our concept to purely elastic bodies does not make much sense. The macroscopic moduli can be easily measured and it is of no use to calculate the respective structural parameters, which - on top of it - is not always possible to do with such poor input data. However, Eq.(22) is important as it is used for an approximate determination of ifn even in cases of inelastic deformation. Usually, inelastic deformation has only deviatoric character, and therefore, it may be difficult to determine the structural parameters for the isotropic parts, which remain elastic.

n.6 Confrontation with Other Models

63

Then we can use Eq.(22) even if the inclusions are not exactly spherical (Eq.(18) is exactly valid for any structure if the shear moduli of the constituents are equal). Structural parameter rj° can also be calculated directly from (20) , but we need more information. Concluding this section, we can state that in all cases an exact correspondence was found in the types of the rheological equations and it was possible to find such non-negative structural parameters that represented either an exact or a very near correspondence in the coefficients. It would be possible to proceed with other solutioas, but it is of no substantial interest, the above examples, taken from Reiner's monograph, were presented only for a short demonstration.

Chapter III Plasticity of Poly crystalline Metals The structures of polycrystalline metallic materials can be very different, but their elastic-plastic responses (their stress-strain diagrams) are qualitatively very similar. This indicates that the mechanism that is responsible for macroscopic elastic-plastic response is similar in spite of the differences in details of the structures. Our mesomechanical approach is aimed at describing the main factors representing the background of this mechanism. Process of plastic deformation in polycrystalline metals is described in our concept in such a way that one of the two substructures is attributed to areas with regular atomic lattice with easy plastic glide (n-substructure), and the other substructure to barriers resisting this glide (e-substructure). The forms and nature of the barriers can be different in different materials, but it is assumed that they exist and have specific statistically homogeneous features. Application of the general model presented in the preceding chapters leads to a special kind of constitutive modeling with tensorial and scalar internal variables. The tensorial variables describe distribution of internal stresses on the mesoscale, the scalar variables describe the changes in the structure of the material. These variables have clear physical meaning: The tensorial variables are (i) average stresses in the two substructures, and (ii) tensors describing the influence of fluctuations of stresses. The scalar variables are structural parameters (deduced in Section n.l as integral forms in the distribution functions that describe distribution of stresses and strains under the influence of specific features of material structure). There are several basic variants of our model for several different deformation processes: • For small deformations the values of structural parameters are assumed to remain unchanged and the changes of state of the material are described only by changes of internal stresses. • For finite deformations the values of structural parameters are assumed to change, which models the violation of continuity of the substructure of barriers.

64

HI. 1 Plasticity - Small Deformations

65

• For short loading paths (typically: tensile test) the use of tensorial internal variables is limited to deviatoric quantities, i.e. the changes of the state of the material are described by only deviatoric stresses, the influence of the isotropic parts of stress tensors is assumed to be negligible. • For long loading paths (typically: cyclic loading) the influence of the isotropic parts of internal stresses is assumed to play an important role: due to plastic deformation the atomic lattice in the n-substructure is violated, which causes increase of volume; the e-substructure (the barriers) resists this increase, which results in compressive isotropic stresses in the n-substructure and tensile isotropic stresses in the e-substructure.

m . l Small Deformations In the extent of small plastic deformations the structural parameters are assumed constant, the changes of state of the material are described by changes of internal stresses only, and in the case of short loading path only by deviatoric stresses. The respective modification of the general formulae presented in Section n.2 is received by the following restrictions: [ 1 ] hg=0 , pg = 0 , (bg=0 [2] hn=Xn , pn =0 , eon =0 [3] f = 0 [4] i)g =0 , r)„ =0

(e-substructure elastic, no change of elastic constants, no change of volume) (n-substructure elastic-plastic, no change of elastic constants, no change of volume) (isothermal process) (structural parameters related to deviatoric processes do not change).

The isotropic (volumetric) response is elastic in the whole material, and therefore, the structural parameters r/,,T/° are not important for the inelastic response in such model. If necessary, they could be determined approximately from the average between the Reuss and the Voigt solutions, or - if the bulk moduli of the two material constituents do not differ substantially - the fields of isotropic parts of stress- and strain-tensors can be approximately modeled as homogeneous.

66

m.1.1 Plasticity - Identification of Parameters

From the rates of scalar quantities in the set of equations (n.2.1-24) the only one appearing here is hn =k„. To determine it, one scalar condition must be formulated: the yield condition valid for the n-substructure. 777.2.1 Solution to the Identification Problem Solution to the identification problem, i.e. determination of the material parameters, will be shown for such cases, where the experimental isothermal elastic-plastic stress-strain diagram in the extent of small deformations is available. The respective mathematical model follows from Eqs. (II.2.10) to (TI.2.18) with the restrictive relations [1] to [4] formulated above. a) Yield Condition in the Simple Mises Form Let us at first assume that for some family of two-phase materials it is possible to formulate the yield condition - for the material constituent that deforms plastically - in the simplest Mises form that takes into account only the average deviatoric stresses Sf„ : <

3

2

(1)

Then our macroscopic constitutive equation follows from Eqs. (n.2.10) to (n.2.18) with the restrictions specified above, and with Xn following from Eq.(l), which gives: 5^,^=0. (2) For Sjj„ we use the respective expression from (H.2.18) and arrive at: 4=0

for

S^S^fc2,

V* [(Ve Men e + V„ HnT]n ) | C2. - HeTle S k l n S ^ ] for

sijnsijn=^

(3)

IH. 1.1 Plasticity - Identification of Parameters

67

where the meaning of the brackets « » i s given as follows:

(4)

« X » = -1(X + | X | ) .

In the case of an increasing uniaxial tension the deviatoric parts of stresses and strains can be expressed as follows:

' 1 0 0 ^ 0 Sij=S 0 -0.5 0 0 -0.5 )

( 1 0 eH=e 0 -0.5 0 0

0 } 0 -0.5

(5)

and similarly for other components of stress and strain. The meaning of the scalar quantities S and e is clear from Eq.(5). Our constitutive equation can be integrated in this case, which gives:

X=

-IV A(w-ws)-B\r\ 1 1--ws

Y=

A(w-ws)-C\n

(6)

1- w 1-Mfc

where X = e - (e)s C„Me

(7)

s-(s)s

£l*

A=

(8)

T\n(Vn+T]e)

B= v,

^n„

{V„+T]ef

68

°

m.1.1 Plasticity - Identification of Parameters

—

v

e vn

.?

.

W=^^(s'n/Cn).

(9)

Subscript s characterizes the values at the start of plastic deformation (see Fig. 1). The identification problem is solved as follows: From Eq.(6) it is easy to deduce: y=dY_ =

dX

AQ-w) + C A(\-w) + B

ds_ *Bde

From Eqs.(6) and (10) expressions W and ln(1 — w) which leads to: A0-ws)(C-B)-CX

+ BY + (C-B)

,=0.

are excluded,

(11)

On a stress-strain diagram resulting from a very slow tensile test we select three suitable points on its non-linear part (P1? P2 and P3 in Fig.l) and determine in these points the values of Xp,Yp,V'p(p=1,2,3) (see Appendix VIII.5). We insert these three sets of values in Eq.(l 1) and get three equations for three constants A(]-WL),B,C to be determined. These equations are not linear, but in spite of it they are easy to solve. Subtraction of two of them gives: B(Y, -Yz-X, +{C-Bf

+ X2)-(C-B) *-Y*

=0

(X, - X2) (12)

and a similar equation with indices e.g. 2,3 instead of 1,2. Thus, Eq.(12) represents a set of two independent equations for two constants B ,(C- B) . The first of these constants - B - can be excluded and we get a quadratic equation for (C-B) . It has a trivial solution C = B , which has no physical meaning, and a non-trivial solution:

m.1.1 Plasticity - Identification of Parameters

69

Fig.l Scheme of the elastic-plastic stress-strain diagram C-B=

( 1 - YO(1- V 2 ')(1- Yi)[X, (Y2 - Y3) + X2 (Y3 - YJ + *3 (Vi " V2)]/[ ( K - >2) (1 - V3') (X3 - Y3)

(13)

+ (v 2 '-v 3 ')(i-vr)(x,-v^) + (vj-VT){1-v$)(x 2 -y 2 )] . Now, it is a straightforward matter to calculate the values of B, C and >4(1-ivs) fromEqs.(12)and(ll). With A(\-ws),B and C known we are able to calculate r\g , rj„ and another constant, e.g. vn . The expressions for A, B, C are given in Eqs. (9), the expression for ws follows from (10) and (H.2.12):

"S

~~~

Vele

UnVeln

7

C„

~

VeT)g

'In ~77 M

+ VeHeVe + VnUnVn

(14)

70

IE. 1.1 Plasticity - Identification of Parameters

After some calculations a cubic equation for Vn can be derived from (8) and (14): K^l+K^+K.v^K^Q

(15)

where: K3 = Bnn(nn-ne) K2 = (C-B)\Mn (H„-Ve) + Bne(2nn-He)] K, = (C-Bf ng {{2nn-ne) + ne[B+ AV-ws)]} K0 =

(16)

(C-Bfn20.

Only the solution in the interval < 0,1 > has physical meaning. With Vn and vg found the structural parameters % , r\n can be determined from Eqs.(8). This solves the identification problem for the case that vnye are unknown and Hn ,He are known. In the case that the unknown value is either jJ.n or fie, Eq. (IS) can easily be transformed into a quadratic equation for fJ.n or jJ.e using expressions (16). The Special Case of Elastic Homogeneity In many applications it is possible to simplify the model assuming that the elastic properties are homogeneous, i.e. J"e=Mn = J". Pe = Pn = P

(17)

In such cases the model simplifies substantially: If assuming that in the virgin unloaded state all the stress components have zero values, then at the start of plastic deformation (s'^s = ws= 0, Eq.(lS) is reduced to a quadratic form and the whole analysis is reduced to deviatoric quantities alone, as the isotropic parts of stress and strain turn out to be homogeneous. The character of the curve described by Eq.(6) is then such that at point S the slope and the curvature of the stress-strain diagram are maximum and with increasing deformation and stress the slope of the curve approaches some limit value and the curvature approaches zero value. At point S the slope of the curve corresponds to:

m. 1.1 Plasticity - Identification of Parameters (w) s =0

,

V 7 e + VV7n

(V)s =^%=vg A+B

71

(18)

vei]e + vnT)n + vnt}er}n

and the limit values are: (w),=1

,

(Y\=^

=^ s - -

(19)

Such a course of the stress-strain diagram agrees with what is experimentally observed, but only in some limits of small deformations. If these limits are exceeded - i.e. if the e-substructure starts loosing its continuity or deforming in an inelastic way - the curvature of the stress-strain diagram starts increasing again and the slope decreasing from the 'limit' value. Therefore, the model parameters can be determined only by taking into account the segment of the diagram, in which the general theoretical character of the curve is maintained (see Fig. 1). It is clear from the above relations that it is possible to determine not only the values of the structural parameters, but also the volume fractioas of the two substructures. As shown in Kafka(1988) the determined volume fraction of the n-constituent agreed very well with the volume fraction of aluminum in the case of aluminum alloys. The reason for such result was evidently the fact that aluminum is very easily plastically deformed whereas the non-aluminum ingredients were more resistant to plastic deformation. In the next Section EH. 1.2 it will be shown that in the case of mild steel the situation is different, the volume fraction of the subvolumes remaining elastic does not correspond to the volume fraction of the non-ferrous elements . The case of inclusions of one of the material constituents in the matrix of the other one (cf. Section n.2) makes the analysis even easier. p) Yield Condition Taking into Account the Influence of Fluctuations In the analysis discussed above the yield condition was formulated in terms of solely average stresses in the plastically deforming substructure. This led to very simple solution of the identification problem, but the yield condition can be dependent also on the effect of fluctuations, which are described in our model by the terms with primes. In polycrystalline metals the structure is very fine, and especially if describing complex loading the effect of fluctuations cannot be omitted. After having tested a number of different formulae we arrived at the following form that gives good results - even for complex loading as is shown in Section m . 1.3:

72

HI. 1.1 Plasticity - Identification of Parameters s 3 2

1 / /

(20)

In the case of elastic homogeneity the term s'^s'^ has zero value at the beginning of plastic deformation, then - in the case of monotonic uniaxial tension - it increases and approaches some finite value. This means that - contrary to the case of Eq.(l) - there exists strain-hardening even in die ^-substructure, but it is limited. Equations (2) and (3) must be replaced by the following forms: §ijn Sjn ~~~Sijn §ijn = 0 "n

An=0

"1)

for

ty,fy,
(22)

"•(»V7. + "/flu fyityi " K + »J.)*#i 3jh r

'or

3

% J % J =~2Cn

2

+ S

'

~ ijn

/

/

s

ijn •

Even these formulae can be integrated, but the result is substantially more complicated; instead of Eq.(6) it is received:

X-

<W - fl .(-1)*(1-.L«

♦ F,,

n

«£l

where X and V have the same meaning as in Eq.(7), the other terms are defined as follows:

') There is unfortunately a misprint in this equation in (Kafka, 1996)

ID. 1.1 Plasticity - Identification of Parameters

a = [sn+4E=)/c„

73

(24)

(jx =

(26)

p=Wnln+VeVn+VeTloylln)

Fx=fa vlVe -rfVn

Q

Y

F

y=>fhi

'Vein

2(vnr}n+t]er]n+vgrig>ln~n) WniVn

+ *7e)( V / e + V ? n

+Wn) ,3/2

[VniVn+Vef-virf]

These relations are substantially more complicated than in the case of the simpler yield criterion, and solution to the identification problem is not so easy. Fortunately, what remains valid even in this case, are the expressions for V" at the beginning of plastic deformation, and for the respective limit value:

"o7?0 + V ? "

(<»)s=1 •

(Y% = v.

{
(nt=-| =T^-. B 1 + r/e

,

(27)

(28)

One of the possibilities how the identification problem can be solved, is: • First Step: Approximation of the experimental stress-strain diagram with the use of the simple criterion (2), and determination of the respective values of (YOs and ( V ) t .

74

HI. 1.2 Complex Loading

• Second Step: Using the values (Y" )s and (Y" )i determined in the Firsf Step, estimating the first approximation of vn (the value of vn determined in the First Step can be used as this first approximation in the Second Step), calculating the remaining parameters ve (=1- vn) , T]t and r\K from Eqs.(27) and (28) and comparing the theoretical stress-strain diagram that is based on these values with the experimental diagram. • Third Step: Improving successively the estimated value for vn and repeating the Second Step so long, as the best approximation of the theoretical curve is achieved. This procedure has been used in a number of applications (some of them will be shown in the forthcoming sections) and it gave good results.

III. 1.2. Metallic PolycrystaUine Materials under Complex Loading In this section our concept is applied to small isothermal plastic deformations with short loading paths (no cycling), but under complex loading. The basic aim is to show that our model is able to describe processes of this kind and to cope with input data acquired from simple tension test These input data are determined by a mathematical analysis of the respective stress-strain diagram as shown in the preceding section. In the former Kafka's monograph (1987) the problem of elastic-plastic deformation under complex loading was discussed in a qualitative way only, and it was hinted at the paper by Inoue and Yamamoto(1980), where on the basis of older Kafka's papers the authors were able to model relatively well the elastic-plastic complex loading process, experimentally performed with thinwalled tubular specimens of an aluminum alloy. But to achieve a fairly good correspondence between the experiment and the model, it was necessary to accept some new hypotheses that were outside the scope of Kafka's concept, and to introduce a new parameter for whose determination the simple tensile test data were not sufficient it had to be determined by fitting the complex-loading data. After many attempts and after testing a number of hypotheses such variant of our model was found that is fully in the framework of Kafka's concept a simple tension test is sufficient for the determination of the input data and the coincidence with experiments is very good. The main point was to find the proper form of the yield criterion for the n-substructure that would take into account the influence of fluctuation of stresses. This criterion (IH.1.1.20) and the way of working with it were shown in the preceding section m . 1.1. In the variant of our model used here the elastic moduli of both the substructures are assumed identical, there are no changes of volume fractions of

m.1.2 Complex Loading

75

the substructures and of structural parameters, and no inelastic changes of volume. The changes of state of the material are described by the changes of internal stresses only. Then the macroscopic constitutive equation and the respective evolution equations of tensorial internal variables take on the following form:

e^LiSq + VnSjjniX} , e=po

8*-i,-v.P8»-**'

^

[l]

(1)

(2)

vb*.-K + * ) * t f )

(3)

where

p=Wn+vn%

. q=p+nen„-

(4)

The macroscopic constitutive equation (1) with the evolution equations for the deviatoric latent variables (2) and (3) must be completed by the expression for { k J. This can be done with the use of the yield condition valid for the ^-constituent The yield condition is assumed in the form (HI. 1.20): s

ijnsijn

— ~2Cn

+

~Sqnsijn

■

(*)

Vn

Without the second addend on the right-hand side, Eq.(S) would mean Mises' criterion written for average values of deviatoric stresses in the nconstituent. Such criterion was used in some former papers and it gave good results for some problems, but not for the case of complex loading. It has always been understood that - to be precise - the yield criterion should be dependent not only on the average mesoscopic values, but also on fluctuations. The next step is that Eq.(S) is differentiated and expressions (2) and (3) are used for S/jh and s^,. In this way it is arrived at:

X = MQ

S !i

f

x,

,

(6)

76

{X}=0

m.1.2 Complex Loading

for

sijnsijn < f c * + — Sp^

(7)

'In

{X} = (X +1 X\) / 2

for

s*. «„ = f c2n + - 1 - s ^ s ^ . 'In

If the parameters of the material are known, Eqs.(l) to (7) will be sufficient for the calculation of the deformation response to any loading path. The variables that are not explicitly comprised in Eqs.(l) to (7) can be calculated - if necessary - from the set of Eqs.(II.2.1) to (n.2.9) . In general, the solution to the identification problem was shown in Section m.1.1. From the above set of equations it can easily be found that at the starting point S of plastic deformation it holds: (S,1„)S=C,

(Si'm)s=0

V

f"l°

(8)

) = [,._

VeJ)a + VnX]n

f . - , -p]/2 .

(9)

(dd^/dSiOa

Similarly for the asymptotic values it turns out that:

Wln)-=l7^-(Sl1n)« V

n

+

(10)

T

\e

(de l1 /ds, 1 ) as =^-(1 + 7,e) = [—-f—

p]/2.

(11)

The solution to the identification problem then proceeds as follows: a) Oetermination of the values of v and E and of n and p is trivial and will not be discussed. b) First estimate of the value of v* As the first estimate the value of va determined with the use of the simpler Mises' criterion as shown in Section m.1.1 can be used. c) With the use of this value of ve (and of course of v„ - l-v„), t]t is calculated from Eq.( 11), where (dff11 / de 11 )as is measured on the experimental curve.

HI. 1.2 Complex Loading

77

d) Then r\n can be calculated from Eq.(9), where ( d s ^ / d e ^ J g is again measured on the experimental curve. e) Now with the first set of values ve ( = 1- v„), r]e and t]n the theoretical stressstrain diagram is plotted on the screen of our calculator and compared with a set of experimental points characterizing the experimental tensile stress-strain diagram (the coordinates of these points are measured on it). Of course that in this first approximation we can expect a strong disagreement Then the value of ve is successively changed and the procedure repeated to achieve the best possible coincidence of the theoretical curve with the experimental one. The values of ve ( = l-v„), T]e and r/„ with which this best coincidence is achieved, are the sought parameters of the material. It is clear enough that for plotting the theoretical diagram it is to be worked not with the rates (e.g. e, 1 ), but with differentials (e.g. de, •,) that must be replaced by finite very small differences in the calculation. The measuring of the values of (ffu)s and (d^,■)/dSi 1 ) s is not straightforward in some cases, but these problems will be discussed in the next paragraph.

Numerical Examples A. Aluminum alloy: (i) Combination of tension and torsion (ii) Ratcheting The first example is based on the experimental results presented by Inoue and Yamamoto(1980). The material under study was an aluminum alloy (Al with Mg), the exact composition was not given. Thin walled tubes made of this material were loaded by a combination of tension and torsion in the first experiment and by ratcheting in the other. In Figure 1 the experimental stress-strain diagram in simple tension (direction xO is demonstrated with the respective values of ( ^ n ) s , (dio^ / de 11; J s and (dcx^ / d£AA)as . Let us comment on the position of the point S. If there appears the yield-point jog in the diagram, the structure of the material changes and the process that is described by our model with fixed structural parameters starts in the point S as depicted. Furthermore, to the value of (dau / dien)„: In our model the material constituent that is labeled by e is assumed to remain elastic and continuous throughout the described process. In such a case the curvature of the theoretical diagram decreases with increasing deformation and the value of dcr^ / d e ^ approaches some asymptotic value.

m.1.2 Complex Loading

78

But of course this type of the process has its limits. If the deformation surpasses these limits, the e-substructure starts loosing its continuity (cf. Subsection m.3) and as a result the stress-strain diagram starts to bend and the value of d a ^ / d c ^ severely decreasing. That is the reason why the value of (da M / de^Jgj must be determined in the way that is demonstrated in Fig. 1. By the method described in the preceding paragraph the parameters of the material were determined as follows: p = 5.410'6 MPa1 |i = 2.28 10 s MPa"1 C„ = 63.66 MPa, ( d a 1 1 / d c 1 1 ) s = 3099MPa , (do,, / d e 1 1 ) „ = 98l MPa , ve = 0.056, vn = 0.944 , T}e =2.7433125 and 7], = 0.011429242 . With the parameters of the material known it is possible to calculate the deformation response for a given loading path. 7^n

s%

m

1 day, \

\ dC„ L

A " is

L^-~

dc

^—

/

Hf '50

^^"^

S/^

% p

\£)

t. mo.

experiment

¥(S) 50

0

2

4

-

6

8

Strain c„ (%) Fig.l Experimental stress-strain diagram in uniaxial tension of the investigated aluminum alloy

10

ni.1.2 Complex Loading

79

(i) Combination of tension and torsion: The loading path for this case is given in Figure 2. The values of ( d a ^ / dcf12) for the respective segments were read from the figure as follows: Segment

(dtr^/dcr^)

0-A A-B B-C C-D D-E

1/0 -0.87911/1 -1.664/1 0/1 1.236/1

These values together with the data given in Figure 2 comprise all the necessary input information for the calculation so that anybody can use them to check the respective program before using it for application to other problems.

^ ^

j2.trt(b(cy

5 •** c

12. 7 tlPa

h9

c1

1 SO

Tens tie stress

—

ojf

| 100

1£
n

0

(tlPa)

Fig.2 The loading path of the aluminum alloy tubular specimen

The resulting calculated deformation response in comparison with the experimental results is given in Figures 3 and 4.

80

m.1.2 Complex Loading

(ii) Ratcheting: Another loading path performed experimentally by Inoue and Yamamoto (1980) was ratcheting with a constant tensile stress a^= 58.84 MPa combined with torsional deformation changing in cycles from zero to Yi2=2c12=0.0l. In this case no yield point jog was reported and therefore the determination of the input data from the tensile test was not straightforward. There was a slight uncertainty how to describe the beginning of plastic deformation. In the end it was decided to use the same parameters as in the preceding case with the only exception that the plastic process was assumed to start in point (S) instead of S (see Figure 1). This meant for the input data that the value of c was not 63.66 MPa, but 49.82 MPa. All other input parameters were introduced in the same values as before. The resulting increase of deformation with the number of cycles N is described and compared with experiments in Figure 5.

Si H? 150 ■

®

1°

100

/°

P 50 ■

J

-*1t 1° sfe)-®

o experiment — theory

/ C

1

i 2

5i |

3%

a.

Fig.3 The deformation response in the axial direction of the aluminum alloy tubular specimen subjected to loading according to Pig.2

m.1.2 Complex Loading

81

1& oo

__©

60

ZQ_2—•

~%

10 o experiment — theory

20

rW Ct

'>

1

Ji*-2*i2 4 %

J

Fig.4 The torsional deformation response of the aluminum alloy tubular specimen subjected to loading according to Fig.2

>?

k?

4

3

o o

2 C

o experiment — theory

j

o

1

N C7

2

<•

6

12

Fig.S The deformation response in the axial direction of the aluminum alloy tubular specimen subjected to ratcheting; N = number of cycles

HI. 1.2 Complex Loading

82

B. Mild steel: Combination of two modes of tension in two perpendicular directions The second example is based on the experimental results presented by Marin and Wu(1956). The material under study was mild steel designated as SAE 1020 (0.19% carbon, 0.48% manganese, 0.24% silicon, 0.013% phosphorus and 0.043% sulfur). Thin walled tubes made of this material were subjected to axial tension and internal pressure. In Figure 6 the experimental stress-strain diagrams for simple tension are demonstrated for uniaxial tensions in the tangential direction ((7^) and in the axial direction (<722 )• I I c a n be seen that there is a slight anisotropy in the plastic segments of the diagrams.

600

Strain C„ , Zn {'/.) Fig.6 Experimental stress-strain diagrams of the investigated mild steel in two orthogonal directions of uniaxial tension (axial tension and tangential tension caused by inner pressure) with theoretical diagram averaging the experimental data There was no problem in this case with the determination of the value of (dau / de^)^ = (dCT22 ^ d £22 )as- O" toe o m e r nan( l m e determination of

IE. 1.2 Complex Loading

83

(^ii)s = (CT22)s a™1 ( d ^ i i / d e 1 1 ) s = (d(T22/de 2 2; s was not straightforward. Several attempts were necessary for achieving the best possible fitting of the tensile experimental curve. The values of the elastic constants n and p were determined from the elastic parts of the diagrams given in the paper. Surprisingly, they differ slightly from the values given numerically by the authors (it is not mentioned in the paper how these numerical values were determined). In the way described above the parameters of the material were determined as follows: 1 p = 2.617 10"* MPa

fi = 5.854 10"6 MPa_l

C„= 229.19 MPa,

(dff 1 1 /de 1 1 ) s = 105240MPa, (dff^ /d£„)as =5993MPa, v. = 0.758, vn= 0.242, 77, = 31.24036, 77„= 2.16488. The very complicated loading path is given in Figure 7. The respective theoretical deformation response in comparison with the experimental results as presented by Marin and Wu (1956) are demonstrated in Figures 8 and 9.

SCO

1 ® iys too

^

^

1 ioo

1

)

0-■(g)

200

1

'•OO (J)

60 0

Axial stress ff-l2 [tIPaj

Fig.7 The loading path of the mild steel tubular specimen

Discussion The most important new point that opens the possibility of modeling complex loading processes is the form (5) of the yield condition. Although the way to finding it was long, the result is simple. The second addend on the right hand

84

ID. 1.2 Complex Loading

side of Eq.(S) means strain-hardening in the n-material constituent. The value of S^S'^ is zero in the elastic range, in the case of a monotonic uniaxial loading it increases, but the rate of the increase is decreasing and the value of S^S'^ approaches asymptotically some finite limit. The value of S^S^ / T]n depends on the heterogeneity of the stress field in the n-material and in the case of J]n infinite (which corresponds to inclusions of the n-material with homogeneous stress field inside) it is vanishing. The agreement in Figures 3 and 4 is very good and does not call for comments. On the other hand the agreement in Figure 5 is a bit worse. The reason is probably in the assumed start of plastic deformation that is - for this mode of loading - described by the curve that begins at point (S) (Figure 1). In reality, the plastic deformations for the first cycles are higher than the calculated values, which means that reality is after all a bit nearer to the yield-point jog scheme.

1 -«?

600

0.1 0.2 1— ' o experiment - theory

iOO

-

X

/

f

/ (

/

1——./

/

0.1V*

//

4

F

/d/?°

' ■'

/

/20/°

/

200

I7

0.3y.E„ / / l ~ "" / ' \r^_ / •• ^ ^ ^ / /

/ / /

/

/°®

o / o

f

W©,

O-—-«-u—» >

0

0.1

'

0.2

0.3

f>*

ctv%

Fig.8 The deformation response in the tangential direction of the mild steel tubular specimen subjected to loading according to Fig.7

IH. 1.3 - Influence of Rate of Loading

•# 600

85

experiment theory

£ 22

OB

%

Fig.9 The deformation response in the axial direction of the mild steel tubular specimen subjected to loading according to Fig.7 In the case of mild steel (Figures 8 and 9) the agreement is worse than it was for the aluminum alloy. The reason can be seen in the slight anisotropy of the material and in the fact that the stress state was not exactly two-dimensional because of the inner pressure in the tubular specimen. Conclusion The demonstrated ability of our relatively simple model to describe plastic deformation under complex loading with the use of input data from tension test is important for this specific problem, but also for the corroboration of our general concept

III. 1.3 The Influence of the Rate of Loading upon the Stress-Strain Curve and upon the Strength It is a well-known property of the metallic as well as other materials that the inelastic part of the stress-strain curve is steeper at a higher rate of loading. This phenomenon can be described by our model supposing that there exist two infrastructures in the material: One with elastic properties and the other with elastoviscoplastic properties.

86

in. 1.3 - Influence of Rate of Loading

The model that we use for the elastoviscoplastic properties is a special case of that described in Section E.2. Expression (II.2.2S) takes on the form: K = Pn

(1)

where $n=0

for

1 JSjj„SIJn-j2Kn 6 Pn= — ,[ 2Hn yJS^Sjj,

for

SijnSijn<2K2n

(2)

SinSiin>2Kl

It can easily be shown that this model corresponds to the so called Bingham body. Thus, e.g. for Si2n=%in' t 0

• % , = 0 for J/V12

Eq.(II.2.6) takes on the form: ei2n=MnSl2n+-r7r(S12n-K,n)

(3)

which is the well known Bingham equation. Equation (n.2.10) transforms in this case as follows: %=p%j + vn (Mn sIJn + Af; Sp) $n .

(4)

There are three possibilities: If the loading rate sj/ >s verY high • the second addend on the right-hand side of Eq.(4) is negligible and the process is purely elastic. If ify is very low, the stress-strain curve corresponds to a sequence of states with ended viscous flow, i.e. to an elastic-plastic process. For a finite rate of loading the process is elastoviscoplastic, depending on the rate. The stress-strain curves following from these equations are plotted in Fig.l: At a very slow (quasistatic) loading the process is merely elastic-plastic. There is enough time for the deviatoric stress in the n-constituent to relax at any stage to the value K„ , under which there is no more flow (line O-A in Figs.l and

m . 1.3 - Influence of Rate of Loading

87

2). At a higher rate of loading the values of $n will depend on the given rate of loading, the value of the deviatoric stress in the n-constituent will not be constant as in the preceding case, it will increase, and the macroscopic stress-strain curve will be steeper (line O-B in Figs.l and 2). If the increase of loading proceeding at a finite rate is stopped at point C (Fig.l), the inelastic strain will increase and after some time (theoretically infinite) it will reach point D. If the reaching of point C is followed by a very quick unloading and reloading to point C and then by the increase of loading with the original finite rate, it will correspond to the course O-C-G-C-B , i.e. to a scheme that is similar to the time-independent plasticity. A slow unloading from point C and slow reloading up to the level of (7C leads to the trace O-C-H-E-D. These characteristic features of the mathematical model fully agree with experimental findings as published by Phillips, Tang and Ricciniti (1974) and Phillips and Das (1985).

0 GHJ

f

Fig. 1 - Stress-strain curves at different rates of loading

Another generally known property is the higher strength appearing at higher rate of loading. Our mathematical model explains and quantitatively describes also this phenomenon. To be concise let us discuss here only the simplest case, where rupture of the e-substructure (substructure of barriers) causes the macroscopic failure, and maximum tensile stress in the e-substructure is decisive for its strength. The elastic properties of the two substructures are supposed equal, the volumetric deformations only elastic, and therefore the

88

rn.1.3 - Influence of Rate of Loading

difference in the internal stresses for different rates of loading will be only in die deviatoric parts. It is possible to use the same mathematical model as in the preceding considerations. Again at a very slow loading the deviatoric stress in the n-substructure will not increase above the plastic limit Kn and the process will have the character of an elastic-plastic deformation - line O-A in Fig.2. At a higher rate of loading the deviatoric stress in the n-substructure will increase and the macroscopic stress-strain curve will be steeper - line O-B in Fig.2. The respective courses of deviatoric stress in the e-substructure and the n-substructure, as plotted in Fig.2, show that a certain limit in the e-substructure is reached at a higher macroscopic stress and a lower macroscopic deformation if the rate of loading is higher. The isotropic component of stress in the e- substructure does not depend on the rate of loading and therefore, the tensile stress in the e-substructure, and - according to our hypothesis - the resulting macroscopic failure is to appear at a higher macroscopic stress and a lower macroscopic deformation if the rate of loading is higher. This agrees with experimental evidence.

(S)2 >0

{S

*

{S J

>

>

(**l'2W%S

(*nh

'^nh

o

Fig.2 - Strength at different rates of loading

^

m.1.4 Yield-Point Jog m.1.4

89

The Yield-Point Jog

The phenomenon of yield-point jog is modeled in our concept as transition from one kind of structure to another kind: In agreement with the findings arrived at by J.N8mec(1996) our description assumes that before this transition the n-substructure forms inclusion that are surrounded by thin shells of the barriers - by the ^-substructure. This means in our model that the structural parameters corresponding to the n-substructure are infinite and those of the ^-substructure Gnite. In the course of the yield-point jog the barriers are broken through, which is modeled in our concept by changes of structural parameters: those corresponding to the n-substructure become finite and those of the e-substructure remain finite, but change their values. In Fig.l the courses of macroscopic deviatoric stress and the average stresses in the two substructures are plotted as they follow from our model for randomly chosen parameters. The elastic limit in the n-substructure is reached prior to the yield-point jog, which means that stress in the n-substructure is limited and stress in the e-substructure grows as much more and in the end leads to breaking of this substructure - of the barriers.

s

y

r Fig.l The yield-point jog and the respective mesoscopic stresses

This course of stresses resulting from our model agrees with the observed phenomena in several points:

90

m.1.5 Stored Energy

• The slope of the macroscopic stress-strain diagram is slightly curved prior to reaching the yield-point jog, which is a generally known phenomenon. • The mesoscopic stress in the e-substructure drops due to the yield-point jog, but remains higher than that in the n-substructure. This agrees with the experimental findings (special X-ray diffraction method) by Vasilyev and Kozevnikova (1959). • Our conceit that the barriers are broken through in the course of the yield point jog agrees also with the observed concentration of acoustic emission in the course of the jog, as reported e.g. by Peter and Fehervary(1986). 7/7.7.5 Residual Internal Stresses and Stored Energy M\&T plastic deformation and unloading there remain in the material residual microstresses that are described in our model on the mesoscopic scale by specific latent variables representing the factor that controls changes of inelastic deformation properties. In the extent of small deformations our concept models the changes of state of the material by changes of mesoscopic stresses only, the changes of structure and the corresponding changes of structural parameters that involve continuum damage, are assumed to be specific for finite deformations. Using the solution to the identification problem as presented in section III. 1.1, it is possible to calculate residual internal stresses and stored energy that remain in the material after unloading from any stage of isothermal plastic deformation. The process of unloading is supposed to be elastic and thus, the values of the residual deviatoric internal stresses are given - according to equations (n.2.1, n.2.5, n.2.11 - n.2.14) - by the formulae: sfjn =sIJn-Ms,j

, arn Cn-M0a (1)

i/

■-^Ol

« # = S;jn - T]n M' Stj ,

C'nr =o'n -T)°M'0

O

^ . ( 4 3 ) , o-^i^aL). *•

".

fin

V.

*l°n

In the case of homogeneous elastic properties (/*„ =fle,pn=pe) get:

(2)

we

HI. 1.5 Stored Energy

91

* # ! = « # . - » # - °n=On-V

(3)

«£=«#».

(4)

o'H'=o'„

which means that the values with primes do not change in the course of unloading. In what follows we will limit our discussion to the simplest variant of the model assuming isothermal plastic deformation, elastic homogeneity, constant structural parameters and elastic moduli, no increase of volume in the e-substructure, but existing increase of volume in the n-substructure due to plastic deformation that violates regularity of the atomic lattice. This means that equations (n.2.18), (n.2.19), (n.2.23) and (II.2.24) take on the form:

Sijn= -^(PSijn-Ve^n)K

(5)

*'„=-?*£-<S>„

(6)

pq°

s£ = % ^ e % , - iv„ + »?.)4,tf»
(7) (8)

n

P
p = veT]e+vn7]n p°=veT]%+vnri°

are positive values defined as follows:

, ,

q=veT]e+vniln+rlerln

(9)

0

q =v,TJZ+v„ifn+Ti°if„.

The respective values of S ^ , ore, s'^, c'er follow from Eqs.(l) and (2).

IE. 1.5 Stored Energy

92

• In the case of short loading paths the effect of din is assumed to be negligible and residual energy (Wr) is described by only deviatoric values. In accordance with the general expression (II. 1.10) it holds in this case:

Wf = in[ve (sj, sfj, + -J- s'fe s£) +vn (sj, s£„ + - 1 - s# s£)].

(10)

For a tension test the value of Wr can easily be calculated for any current point of the stress-strain diagram and the course of stored energy plotted and compared with the experimental values arrived at by calorimetric measurements. Such comparisons were shown by Kafka(1979) for several materials: for two kinds of aluminum 04/ /, Al II) and for copper (CM). The respective diagrams are demonstrated in Figs. 1-3. The theoretical calculations assumed that at the beginning of the deformation process there was no stored energy. However, in reality there can exist some stored energy due to solidification and machining of the sample that can cause internal stresses. This effect is strongly pronounced in Fig.2. With the start of plastic deformation these internal stresses are relaxed, which causes drop of stored energy. Apart from the agreement with calorimetric measurements another important agreement must be emphasized. With the X-ray diffraction method the second-order internal stresses can well be determined and it was observed (cf. Titchener and Bever (1958) - pp.331 and 316, Bolshanina and Panin (1957) p.229) that the energy represented by these second-order microstresses is only about 10% of the total stored energy. This is in an excellent accord with our calculations, as the energy resulting from the expression

is only about 10 % of the total value, which means that about 90 % of the stored energy are due to the effect of fluctuations represented by the formula: ~2P(~~Sij« Ue

S

lje + ~ 'In

S

#n s/n ) •

m.1.5 Stored Energy

93

Fig. 1 Calculated and measured course of stored energy in aluminum (Al I) (exp.data from Bui(1965), theor. course from Kafka(1979))

Hence, our approach opens a possibility of calculating stored energy and the comparisons with the results of calorimetric measurements and X-ray diffraction method mean another verification of the adequacy of our model for application to polycrystalline materials. The details of the calculations are given in Kafka(1979). • In the case of long loading paths the effect of (&n cannot be neglected, it can play an important role by causing isotropic residual stresses. Let us first verify the reality of the picture of residual stresses following from our scheme: After uniaxial loading the residual deviatoric stress sTjn is - according to Eq.(5) - of the opposite sign than the macroscopic stress in the course of loading (it can easily be proved that the expression in the parentheses is of the same sign as S^ and this is of the same sign as Sy). This results from the fact that the increase

94

1H. 1.5 Stored Energy

of stress in the n-substructure is limited by plastic deformation, in the esubstructure the stress increases as much more, and the elastic decrease of stresses in the course of unloading is the same in both substructures.

Fig.2 Calculated and measured course of stored energy in aluminum (Al If) (exp.datafromBui(1965), theor. course from Kafka(1979)) On the other hand the sign of the isotropic residual stresses does not depend on the direction and sign of loading. The value of (bn is always positive as a consequence of any plastic deformation that violates regularity of the atomic lattice, which is modeled as a>„ being a positive function of Xn. Then - according to Eq.(6) - the rate of the isotropic residual stress a'n in the n-constituent is always negative and consequently 6'e in the e-constituent is always positive tensile. ( The existence of this tensile stress follows from the condition of

m.1.5 Stored Energy

95

Fig.3 Calculated and measured course of stored energy in copper (CM) (exp.data from Kunin et al.(1964), theor. course from Kafka(1979))

equilibrium of isotropic parts of the respective stress tensors: from multiplication of Eq.(n.2.1) with zero right-hand side by Kronecker's 8y.) This picture is fully corroborated by experimental data arrived at by Vasilyev(1958,1959) and Vasilyev and Kozevnikova(1959) by a special variant of X-ray diffraction method for a number of materials (several kinds of steel, iron, aluminum, nickel, copper and molybdenum). In these papers the following relations for residual stresses were found: <*11/i < G'lln < 0 > <7l1e > ^228 > 0

for tension

°22n < O'lm < 0 , or22« > O'lis

for compression an

>

0

'11

>0 < 0.

HI. 1.5 Stored Energy

96

If splitting the total stress components in deviatoric and isotropic parts, it is possible to write: tflta = Sfl/> + °n ■ <7l*> = « U +

ff

e

and s(i„ + 2s'22n = 0 , s ^ + 2 s ^ 9 = 0 . From the above relations it results: Sfm < 0 , sf,, > 0

for tension

S(1n > 0 , s( t e < 0

for compression ff,, < 0

Cr„ < 0 , <7re > 0

a^ > 0

for tension as well as for compression.

These experimental findings fully agree with the output of our model. Another corroboration is the experimentally observed start of fracturing process at the grain boundaries, where the barriers resisting plastic deformation are usually concentrated, and where - according to our scheme - the isotropic tensile stresses are acting. Let us summarize the influence of residual stresses as it results from our model: that of the deviatoric residual stress and that of the isotropic one: • The deviatoric residual stresses resulting from some inelastic loading path influence the macroscopic plastic limit and the course of plastic deformation of the succeeding process, they are tensorial internal variables representative for the memory that is significant for plastic deformation. They are changed and can even be erased by plastic deformation in a differing direction. This agrees with the wellknown effect of fading memory.

HI. 1.6 Cyclic Loading

97

• The isotropic residual stresses cannot be diminished, they increase due to any plastic deformation of any direction and sign. It is important for the description of processes with long loading paths, as e.g. cyclic loading, where deformation can be small, but in spite of it the strength can be exhausted. III.1.6 Cyclic Loading This section refers to the problem of modeling cycle fatigue with small plastic deformations. In the course of such loading conditions, deformations are not large enough to cause changes of structure in the way described in Section rn.2, but in spite of it the process leads to rupture. In our approach this process is modeled by taking into account the effect that was neglected in the description of short loading path: Creation of a high number of vacancies and irregularities in the inner parts of grains (in their atomic lattice) and the ensuing trend to increase volume. The barriers resist this increase, which results in compressive isotropic stresses in the inner parts of grains and consequently in tensile isotropic stresses in the barriers at the grain boundaries. Theoretical description of these stresses as well as experimental verification of their existence was shown in the preceding Section HI. 1.5. After a number of cycles this mechanism causes violation of the barriers that results in the end in rupture. Such general scheme is corroborated by experimental observations showing that fracturing starts at the grain boundaries, where the barriers resisting plastic deformation are concentrated, and where according to our scheme - the isotropic tensile stresses are in play. The rates of the isotropic residual stresses are expressed by formulae (ffl.1.5.6), (m.1.5.8), (IH.I.5.I4) a (m.l.5.24), where n = A*n

0)

where i2 is a positive constant. With regard to Eq.(m. 1.5.6) it then holds o'n = - ^aXn

(2)

HI. 1.6 Cyclic Loading

98

and if cyclic loading is realized in the ^-direction, Xn can be expressed from Eq.(m. 1.2.1) as follows:

K =-£uwhere eft

(3) is the rate of the plastic part of the imposed strain. If replacing rates

by differentials it finally leads to:

COJ.-J^Q^I.

(4)

If assuming that the absolute value of the increment of plastic deformation |4eft| in one cycle is constant, and the value of | S 1 1 n | can be considered approximately also constant in one cycle, it turns out with regard to Eq.(ra.l.5.1 4 ):

^^^-Mi^lflinA

(5)

where N is the number of cycles. According to our scheme some critical value of 0~re leads to rupture. • For the case, in which | S11n ^ = I S11n (, , i.e. if there is no strain-hardening in the /j-substructure, Eq.(5) leads to

PQ° N| 4ef,l = — I s,1n|, K W = const.

(6)

i.e. to a special case of the 'Manson-Coffin' equation valid for higher temperatures (cf. PuSkir and Golovin(1981), Manson(1953)). The case of zero strain-hardening in the n-substructure corresponds to criterion (ELI.1.1). • For the case, in which | S1 ^ I* > | S., 1n h , i.e. if there is strain-hardening in the n-substructure, Eq.(5) leads to

ni.1.6 Cyclic Loading

99

0(N)|4efi| = const. where 0(N) is a function of N that is smaller than N for N>1. This agrees with the 'Manson-Coffin' equation valid for room temperature, in which case this function is approximately described by N^ where /3 <1. The increase of S11n with plastic deformation corresponds to criterion (m. 1.1.20). Hence, even in the case of cyclic loading our general concept leads to qualitative results that agree with observed phenomena. For concrete materials the values of the respective parameters must be determined to give quantitative correspondence.

100

III.2 Finite Deformations

III.2 Finite Deformations, Continuum Damage and Localization In the preceding sections polycrystalline materials were modeled as media with two substructures, one corresponding to the inner parts of grains, the other to barriers resisting plastic deformation. In the extent of small deformations both these substructures were assumed continuous, with their specific degrees of continuity, and — if there is no yield-point jog — the respective structural parameters that represent in our model statistical description of structure were assumed constant. (In the case of existing yield-point jog the structural parameters undergo jump-change and after the jog there begins the segment of small deformations with newly created constant parameters.) In the small-deformationsegment the changes of state of the material are described by changes of microstresses only. In the extent of finite deformations structural parameters can change. These changes step in if the strength of the barriers is exhausted, the barriers start being broken through by plastic deformation, there arise plastic bridges. This means that continuity of the substructure of barriers decreases and continuity of the other substructure increases. The decrease of continuity is reflected in our model by an increase of the respective structural parameter, the increase of continuity by a decrease of the respective parameter. If the structural parameter is infinite, the respective substructure forms inclusions. The respective variant of the model that we are going to present is largely based on the paper by Kafka and Karlik (in print), where some more details and relations to other approaches to the problem of continuum damage are clarified. The basic set of equations takes on the following form: The relations between the average values of Cauchy stress and true strain in the two substructures and the respective macroscopic values are:

".*#.+*!•«*=«#•

(2)

The constitutive equations of the two material constituents are: ^ln=/i6s9„+8tndXa,

£n=pon=pa

=e

(3)

m.2 Finite Deformations fe'iin=teljn

-detf ,

101

e'n=en ~ «

(4)

fe'an = nte'm+ s;jn d\n,

e'n=0

(5)

deJe = ii ds^ ,

e

deje=de />9 - d e t f ,

e'e=ee - e

(7)

de;e=/xds£9,

e'e=0.

(8)

=e

e=P°e=l

(6)

All our calculations are performed in differential form, in numerical calculations differentials are replaced by small finite increments and after every step the respective change of form is taken into account. In this way every step can be calculated using the formalism of small deformations. The relation between the deviatoric average stresses and the symbols with primes are - according to (TI.2.9): Slln

Sg,

**-«i.+-e—^ =0 In *7e

(9)

and in its differential form:

ds„ - ds* + - U s , * - ^g-dTj, - -J- ds>>, + %- di/„ = 0 . "o

■/«

"n

(9')

'/n

By making use of equations (1) to (90 it is derived: dsiJn =dSij -^-(psijn

-tigSjn)can + ^ - s ^ d 7 j n -HsHjLsfaHri.

ds^=ds#+^(ps^-ij^l)d^I-^-4Idi/fI+^-4,d^

ds'ijn -^[VeVeSijn

" K + »?.)4.](H* + ^ - 8 ^ 1 / , , - ^ . ^ . d l ? .

(10)

(11)

(12)

102

m.2 Finite Deformations

^je=-^(nnSijn+s^n)6Xn-^s;jndrln+^h.s;jedr1e

(13)

where P = v„ri„+v9ri9

(14)

For the determination of 9 tensorial variables c r ^ c r ^ C T ^ . a ^ . e ^ , Eyn, e'jfr.E'jjn-Gij there are 9 tensorial equations (1) to (9) available. For the determination of three scalar quantities drj e ,dr/ n ,dA„ three scalar conditions must be specified. But to do it, it is necessary to distinguish among the different segments of the deformation and damage process. For this purpose a concrete deformation process must be specified. As a simple example let us choose the extension-controlled tension test of a ductile metal. The respective typical stressstrain diagram is demonstrated in Fig.l with engineering stress- and strainmeasures <7uand e u (i.e. stress measured by force divided by the original crosssection and strain measured by the elongation divided by the original length) as coordinates. In this case the different segments are specified as follows: Segment 0-A: Elastic deformation - modeled as homogeneous in the whole body, i.e. d7fc= dJ7„ = dA„= 0, which leads to: a^ = 0^ = atj , £yn = e,^ - ~e,j . Segment A-B: Such a change of the slope of the diagram is observed with some materials prior to the yield-point jog - corresponding to very small inelastic deformation. In our concept this is understood (similarly as by NSmec (1966)) as elastic-plastic deformation of the n-substructure that forms inclusions (77„=°°) encompassed by thin shell matrix of the e-substructure of barriers (0<7]e
103

ni.2 Finite Deformations

%'

Ir

F

/ small

j quaslhomogeneous

strain

\ finite deformation

cr

necking

£//

Fig. 1 Typical stress-strain diagram of an extension-controlled tension test

Segment C-D: Elastic-plastic deformation of the body with no damage, both the substructures are continuous and their structural parameters finite and not changing in the extent of this segment Plastic deformation proceeds in the nsubstructure only, the e-substructure deforms elastically without violation. The parameters t]n*, t]* corresponding to this segment are determined by a mathematical analysis of this part of the stress-strain diagram. The method to do so (for the specific variant of the model applied in what follows) is given in Subsection III. 1.1. Hence, in this segment drjn = drj, = 0, r\n = TJ„*, 77, = 77/. The scalar condition for determining dX„ is the yield condition. In our concept it is based on the well known Mises' formula, but it is assumed that due to the microscopic heterogeneity of the stress field there appears strain-hardening - even in the ^-substructure itself. This is expressed as follows: (15) In

This inequality transforms into equality in the investigated segment CD. The second addend on the right-hand side has zero value at point C, where the value of s'ijn is assumed to be vanishing. In a uniaxial monotonically increasing elastic-plastic deformation the value of S^S^

increases, but this increase is

m.2 Finite Deformations

104

limited, it approaches asymptotically some fixed value. It was shown in Section HI. 1.2 that with this criterion the model is able to describe - in the extent of small deformations - the elastic-plastic deformation processes even under complex very complicated loading paths. The set of the above equations together with the use of this criterion lead to:

V5*

fJA„ = 0

cUn = ±(dXn+| 6ln |)

, , ,

(16) SijnSijn < f C2n + —

for

for

sfafa,

sijnsijn = \
Segment D-E : Finite macroscopically homogeneous elastic-plastic deformation. In the n-substructure elastic-plastic deformation, in the e-substructure elastic deformation and from some point continuum damage. In the model continuum damage is described by an increase of the parameter r\t. It means that the whole structure is changing and the change of parameter t]e is accompanied with a change of parameter r]K. Whereas the c-substructure becomes successively less continuous, the continuity of the n-substructure is increasing and thus the r/„ parameter decreasing. This continuous damage starts when the deviatoric stresses in the esubstructure reach some critical state. In some materials this continuous damage before macroscopic localization is accompanied with the appearance of coarse slip bands (cf. Dao and Asaro 1996). Beginning of the continuous damage does not mean creation of cracks, but only changes in the structure (fragmentation) and creation of vacancies. Young's modulus does not change. This kind of continuous damage is discussed separately in Subsection m.2.2. It is assumed - similarly as in the case of the yield condition for the n-substructure - that the criterion for the e-substructure depends not only on the mesoscopic average values of stress, but also on the quantities with primes, i.e. on the microscopic heterogeneity of strain and stress. The form of this criterion was chosen analogous to equation (IS) and it gave good results - good agreement of the theoretical and experimental diagrams, as will be shown later. Hence, it is assumed that

ni.2 Finite Deformations

**«#. ^ f

C

e

+

— "a

S

ij*S'iie

■

105

(17)

The meaning of the influence of the second addend on the right-hand side (of the fluctuations) is understood to be similar to the so-called 'crack shielding effect', in which the presence of stresses - not directly responsible for the cracking process - resists this process. From the beginning of continuum damage inequality (17) starts to be equality and it gives one scalar equation for two remaining unknown variables At]n and dry,. But these two values are not independent. It was shown by Kafka and Havacek (1995 - Eq.33) that they are bound by the following relation:

d*7„ = - ^ r d r / e .

(18)

Hence, we have the sought three scalar relatioas (15), (17) and (18) for the three scalar quantities dX„, dr/„ and Ar\t. Differentiation of relation (15) leads to:

siin <*sljn — s\n ds;jn + s;Jn s;p —^-d^ <, 0 Vn and similarly relation (17) gives: S

V*

6S

U* ~ — S'ii* d s / * 'to

+ Slje

(19)

277„

S

f . 7 T d 1 . ^Ve

* °•

( 20 >

Substitution of expressions (10) to (13) into relations (19) and (20) leads to: [VePSijnSjjn ~(Vn + * 7 e ) S ^ n ] d A n

2

VT„ +

1T

(r? S

" '" ~S*>n)s'>e(irie ^ f l 1 V %

m.2 Finite Deformations

106

V„[~ PSjjnSiJg -TlnSpSfc +TleSije^jn - S^S^CiX,,

(22)

Vnll

'

-(neSiie-s.ije)sfijn6nn-^[vnr]nr]esijg

nn

n,

0

H-vnn„ + - | )s^]s^d7je > n qsije6sir

For a uniaxial tension in the ^-direction, for which increments of deformation are prescribed, relations (18), (21) and (22) and lead to a set of two linear relations for determining dA„ and drje : A dA„ + A, d7/. ;> A, s11n de„ B,dA„+S 2 d77 e ^ f i g s ^ d e n

(23) (24)

where

A = ( P + v n T?„ ne) s*,n - (vn + le) s;i *2 = —{VeiVn^nVe +

We

^n)^e -^-[VeWn^e-iVele-^Mln] 2

A=<7 ^1 =

v

n (Vn Ve S11n S11« ~ Vn sUn

B

2=-r{[-vnTlnrle^e Te ~~~

Bi=q.

+

(.rles\te~S\'\e)S\\nj

S

lle

+

Ve S11a S11n ~ S11n S11e )

(Vnrln-%)^e]^e 2

(25)

m.2.1 Necking

107

However, it must be taken into account that the values of &X„ and dT^ cannot be negative. If the calculations lead to negative values, they are defined as zeros. The reason why d^, cannot be negative was discussed in Section n.2 (Eq.(n.2.27)). Negative value of drje would mean increase of continuity of the e-substructure, i.e. healing of microcracks. Under very special conditions such process can exist, but such conditions are not considered in our model. Segment E-F: Process of necking - is discussed separately in Subsection m.2.1.

7/7.2.7 The Process of Necking If the increasing parameter T]e reaches some critical value (r]e)a, the continuity of the e-substructure (of the barriers) becomes so low that the process is becoming unstable and necking starts (point E). We assume that the development of necking can be influenced by several factors: (i) by the constraint at the ends of the specimen, where deviatoric plastic deformation would need not only prolongation, but also contraction in the transverse directions, which is prevented by the thick heads; (ii) by the highest temperature in the center of the specimen; (iii) by the length of the specimen; (iv) by a weak spot in the specimen. It seems that different approaches are suitable for short specimens and for long specimens: Short specimens In homogeneous well-machined short specimens the main factors will be those named sub (i) and (ii). In such cases it is better to speak about the start of stiffening than about the start of necking. The stiffening (absence of plastic deformation) starts at the constrained ends of the specimen where temperature is minimal, and proceeds to the center with maximal temperature and minimal constraint. This point of view seems to be natural, as prior to necking small plastic deformation proceeds in the whole sample and what really starts is stiffening at the ends. The two stiffening ends of the sample keep growing, their increasing parts are successively unloaded and from the moment their unloading starts, their structure and structural parameters cease to change. It agrees with the observed phenomenon that 'the growth of an unloading zone after bifurcation is common to all problems of plastic instability' (Tvergaard 1980). It also agrees with our experimental observations showing that the cross-section of a specimen with neck decreases from the heads to the neck. In the neck severe plastic deformation and continuum damage proceed and finally the decrease of continuity of the ^-substructure reaches such state - and

108

m.2.1 Necking

the respective increase of the r\e parameter reaches such value (TJ^ - that macroscopic slip or other form of disintegration of the sample ends the process. The process in the neck is described in our model equally as in the segment D-E, the stiffening parts undergo elastic unloading (in some earlier papers of ours it was assumed that these parts became rigid - neglecting the elastic deformation due to their unloading - but in some cases this factor can be important). In a current state, the relation of the volume of the stiff part to the whole volume of the sample is named V,. It is assumed that V, increases with increasing continuum damage, approaches unity and cannot overpass it. A simple function with this property is to be found to fit the course of the segment E-F measured for the specific material under study. For sorbitic mild steel specimens with circular cross-section good results were found for the following form of this function: dV, =K(l-V,)4[J7,-0?,)cx]dTk.

(1)

For aluminum alloy AlMg3 specimens good results were received for the form: dV, = *(1-V,) 6 i? f dij. . (2) Long specimens In the case of long specimens the effect of long ends of the specimen can be very strong, and in this case the elastic unloading of these parts plays a very important role. It can hardly be expected that in a very long specimen the effect of the constraint of the ends of the specimen will be as important as in the case of short specimens. For the central part of the specimen the situation will be more or less the same as in the hypothetical case in which the ends of the specimen are not constrained, but loaded by uniformly distributed traction. There exists opinion that in such a case there will be no necking (Mattos and Chimisso 1997). However, this hypothesis that has not been experimentally verified and can hardly be, does not seem realistic, as the distribution of loading at very distant ends cannot play any role in the deformation process in the central part of a long specimen - according to general knowledge and experience. In our scheme we assume that in the undeformed state there exist some specific length Ln of an active zone connected with necking in the central part of the long specimen and necking is influenced in this case by factors mentioned above sub (ii), (iii) and (iv). The process of stiffening is assumed to proceed from the ends of this active zone to the center of neck similarly as from the ends of a short specimen with length L
m.2.2 Continuum Damage

109

Lo> Ln of some rod of a specific cross-section made of some specific material is available, it is possible to use our analysis for a prediction of the response of longer specimens of the same cross-section and material. From the stress-strain diagram of the specimen with length U the material parameters are determined and from them the stress-strain diagram of a longer specimen of length L>U can easily be calculated. The passive zone of length L-U> will be unloaded due to the formation of the neck. This unloading leads to elastic contraction, which in turn affects the process of necking. In Subsection III.2.3 this is analyzed for different lengths of specimens made of AlMg3 alloy. The results agree with the general conclusions presented in the papers by Schreyer and Chen(1986): 'All of the softening curves will be steeper if a longer bar is analyzed' and by Larsson and Runesson(1996): 'If the bar is too large, it is not possible to control equilibrium states in the postlocalized regime even with displacement control'. This completes the general formulation of the mathematical model of the whole course of the stress-strain diagram with necking. Examples are presented in Subsection III.2.3.

III.2.2 Continuum Damage and Material Softening without Change of Young's Modulus In the preceding two subsections continuum damage was modeled as a decrease of continuity of the substructure of barriers resisting plastic deformation. In the course of increasing deformation these barriers are more and more broken through by plastic bridges. In our mathematical formalism this process is described by an increase of structural parameter r\t . If the elastic moduli of the two substructures were different, the change of the structural parameter would mean change of macroscopic elastic moduli. But if the elastic moduli of the two substructures are assumed identical, then any change of continuous structural configuration does not influence the macroscopic moduli, which are equal to the moduli of both constituents in any configuration. What does change however, are the inelastic properties (plastic response and strength), as these properties are different in the /^substructure and the ^-substructure. It follows from the considerations presented in the preceding sections that structural parameter r\e increases with increasing continuum damage from some original value 77/ and grows without limits, this growth being ended by rupture. However, it is usual to measure continuum damage by a parameter - denominated usually by D - that grows from zero to unity. Taking account of this usage let us define D as follows:

HI.2.2 Continuum Damage

110

D= ^

**»

(1)

which clearly fulfills the required property. Relation to the Elastic Material with Cracks It can be shown that our definition of damage given by Eq.(l) is in close relation with the usual definition of D used for materials, in which continuum damage can well be measured by the change of Young's modulus: D = 1 - -|*-

(2)

where Ed is Young's modulus after some damaging deformation and E the virgin modulus. If the Young modulus of an elastic material with cracks is to be described by our model, it is necessary to go back to Eq.(IL2.11). If - in this equation - index n is attributed to the cracks, then E„ = 0, /4 = °°, which leads to:

71 = / * , ^ -

(3)

where by definition _

1+v

1 + v-

""T' "-"-ifIf we assume that the values of Poisson's ratios v and vd are approximately identical and that the volume fraction of cracks vn is zero (which gives ve = 1), it turns out from Eq.(3): E = - ^ 1 + 1e

(4)

where Ee is the modulus of the undamaged material and E the modulus of the

m.2.2 Continuum Damage

111

damaged material. If these expressions are used in Eq.(2), it turns out: D = -2*-.

(5)

Expression D in Eq.(5) is clearly a special case of D in Eq.(l) for rfe = 0 . Such zero value of r/* in the case of voids with nearly zero volume fraction corresponds to very small spherical bubbles that have no influence upon the macroscopic modulus. This can be seen from Eq.(n.6.23), where zero volume fraction of the elastic spheres leads to zero values of structural parameters of the matrix (in the last named equation indexes are used differently). On the other hand if - still with zero volume fraction of the cracks - the value of r/e in Eqs.(4) and (5) increases, Young's modulus decreases and damage D increases. The infinite value of t]e means - according to Section H I , paragraph A2 - that the cracks form continuous substructure enveloping discontinuous inclusions of the elastic material. This of course means zero macroscopic modulus and complete damage (0=1). Material Softening Let us go back to our model in which one of the substructures if formed by easily plastically deforming material and the other by barriers whose continuity is more and more decreasing with increasing deformation. Sometimes the existence of softening as a material property is questioned and we are going to discuss this matter from the point of view of our concept. The basis of the doubts consists in the fact that softening appears usually together with localization, and therefore, it can be assumed that macroscopic softening appears only due to localization. According to our model material softening and localization have the same background: Both the possibility of increase of stress with increasing deformation and the macroscopic homogeneity of deformation are conditioned by some degrees of continuity of the substructure of barriers resisting plastic deformation. These degrees corresponding to these two phenomena are usually near, but need not be necessarily identical. Therefore, according to our model strain-softening as a material property can exist in some materials and bodies, whereas in other cases localization precedes the development of strain-softening in the material.

112

m.2.3 Application to Mild Steel and Aluminum Alloy

III.2.3 Application to Tension Tests of Mild Steel and Aluminum Alloy The general considerations presented in the preceding subsections are used for modeling the processes of plastic deformation, continuum damage and necking in tension tests of two technically important materials: sorbitic mild steel and aluminum alloy AlMg3. Sorbitic Steel To clarify some details of the modeling procedure, let us describe at first the necking process in a sorbitic steel specimen (15230: 0.32%C, 0.46%Mn, 0.33%Si, 0.02%P, 0.02%S, 2.46%Cr, 0.08%Cu, 0.16%Ni, 0.10%V) with circular cross-section and 7.5mm diameter (stress-strain diagram taken from GajdoS-1973). Using the identification methods mentioned above the parameters of the material were determined as follows: E=210GPa, v=0.33, vn = 0.9867, ve =0.0133, C„ = 3.2939MPa, C8 = 4546.26MPa, 77/ = 5.6006* 10-4, TJ/=3.52115, (T]e)a = 16.70391, (T?e)d, = 382.5079, R = 0.006 (cf. Eq.m.2.1.1). The complete theoretical stress-strain diagram recalculated for engineering stress <xn and strain eu in comparison with experimental points - together with the course of other quantities in the central necking part of the sample - are given in Fig.l. The start o continuum damage is labeled by SD, the start of necking by SN. In this figure the theoretical courses of the values of i], and of D in the center of neck are also depicted. It can be seen that the theoretical value of D in the moment of rupture is really very near to unity. With the material parameters known, it is easy to calculate also the process of changing form of the specimen in the course of necking. The resulting change of the form in the moment of disintegration is demonstrated in Fig.2 by the longitudinal axial cross-section of the sample. The resulting form agrees well with what is generally observed. Apart from the change of form also the theoretical longitudinal distribution of strain due to necking is depicted in this figure. Aluminum alloy AlMg3 The process of necking in aluminum alloy AlMg3 specimens (2.63%Mg, 0.28%Mn, 0.26%Si, 0.24%Fe, 0.10%Cu, 0.03%Zn, 0.0016%Ti, <0.002%Sb) was studied in detail. Five specimens with circular cross-section 0 10mm, gauge length 50mm were deformed in tension test up to rupture. Four of them were cut by longitudinal axial sections, and by transversal sections in several distances (2mm, 4mm, 6mm and 10mm) from the center of neck. The surfaces of these sections were ground, electrolytically etched and photographed in polarization light One of

m.2.3 Application to Mild Steel and Aluminum Alloy

500

D 1.0 0.5 0.1

0.2

0.3

OA

0

Eng. Strain Fig.l Theoretical and experimental stress-strain diagram up to rupture

Fig.2 Distribution of strain and of continuum damage and change of form of the sample in the moment of rupture

113

114

m.2.3 Application to Mild Steel and Aluminum Alloy

the axial longitudinal sections is shown below in Fig.3, two cross-sections in distances 2mm and 10mm are illustrated in Fig.4. The material parameters were determined in the following values: £ = 76.61GPa, V = 0.34, v„ = 0.97312, ve = 0.02688, C„ = 64.9094MPa, c e = 2375MPa, T}„' = 0.001, r//= (j]e)a = 37.4, (r;,)^ = 266.8371, R = 15 (cf. Eq.m.2.1.2). There was an important difference in the behavior of the sorbitic steel specimen and that of this aluminum alloy. Whereas the stress-strain diagram of the sorbitic steel (Fig.l) showed a significant part (segment SD-SN), in which the process of CD proceeded with macroscopic deformation homogeneous, in the case of aluminum alloy the begin of CD led immediately to necking (which is reflected in the above values by r\e* = (j]e)a )• This behavior of our aluminum alloy specimen resulted if the applied model assumed elastic contraction of the unloaded end parts in the process of necking (see Subsection 1H.2.2). The complete theoretical stress-strain diagram recalculated for engineering stress an and strain £„ with experimental data and with theoretical courses of the values of r/, and of D in the center of neck are shown in Fig.5, the change of form and distribution of strain and of D in the moment of rupture in Fig.6. If these diagrams are compared with those for sorbitic steel presented above, substantial qualitative differences can be detected: • First of all, the segment of the stress-strain diagram with proceeding continuum damage and with macroscopically homogeneous deformation is vanishing in the case of aluminum alloy, but not in the case of steel. This agrees with the generally observed phenomenon that deformation and damage of aluminum alloy is sensitive to local defects, but such sensitivity is substantially lower in steel. In our concept this is understood so that the substructure of barriers is strong in steel, but weak in aluminum alloy, where slight inception of damage leads immediately to necking. • Another difference is the distribution of the value of D in the moment of rupture. Whereas in steel the area with significant value of D is large, in the aluminum alloy it is very localized. It agrees with the preceding point.

m.2.3 Application to Mild Steel and Aluminum Alloy

- 9 ^S. Fig.3 Longitudianal axial section of specimen No. 1

2mm 10mm Fig.4 Cross-sections in distances 2mm and 10mm from the center of the neck

115

116

m.2.3 Application to Mild Steel and Aluminum Alloy Stress-Strain Diagram 350 [-> For the canter of neck

300

o

/ < Real stress

* Exp. points

250 B(Start of CD artjl necking)

-»—*—» %

Fig.5 Theoretical and experimental stress-strain diagram up to rupture

Relation of the observed microstructural changes to our model description of continuum damage Let us turn our attention to the observed changes in the structure as seen in Figs.3 and 4 and to our description of proceeding damage. The basic characteristic of damage in our model is the change of the structural parameter r\e. Quantity D was artificially defined to match the usage of describing damage by a quantity changing from zero to unity (but there does not exist any physical reason for exactly such description). Therefore, in our procedure that follows we are going to compare the change of r\t, i.e. the value of (i]e -r/,*) - where rj/ is the value that corresponds to the undamaged material - with the observed change in the structure that is denominated by h . The value of h is determined from Fig.3 and other similar figures obtained from other samples as the average of the values received from individual samples. In Fig.3 we can see straight lines that are perpendicular to the axis of the sample, drawn in specific distances from the center of the neck. Number N of cross-sections with frontiers between black and white microareas observed in the figure is assumed to be proportional to the number of

m.2.3 Application to Mild Steel and Aluminum Alloy

117

t.0

£

n JTI'ITIV EnnnmnMJjfflJJil3i *

0.5

mmi

measured points

Fig.6 Change of form of the sample and distribution of strain and of the value of D after necking and rupture

cross-sections with grain boundaries. Number N is divided by the length of the straight abscissa La and the resulting value H = N/La is related to H°, which is the value of H corresponding to the not-damaged part of the specimen. In this way, h is defined as h = H/H°. This quantity takes on unit value in the not-damaged part of the specimen and increases with decreasing distance from the center of the neck. It is not possible to calculate the course of h from the known course of (r/ 8 -rfg), but there should exist general qualitative agreement between the courses of these two quantities if the description by (r/ e -rjg) was rational. In Fig.7 both these courses are depicted and in the limits that can reasonably be expected the agreement is good.

118

m.2.3 Application to Mild Steel and Aluminum Alloy

I! "~t

JA r

f'-*

i

Fig.7 Comparison of theoretical description of continuum damage with continuum damage measured in the microstructure The use of the quantity A as a characteristic of continuum damage measured in the microstructure deserves explanation. Such an approach is related to our understanding of the process: It is assumed that in the virgin state there exist two substructures - that of microvolumes with easy glide represented by inner areas of grains, and that of barriers concentrated at grain boundaries. Both these substructures are assumed to have some degrees of continuity characterized by the respective values of structural parameters r\t and TJ„ . At the beginning of macroscopic inelastic deformation these values do not change, the deformation of the n-substructure is elastic-plastic, whereas deformation of the e-substructure only elastic. At this stage of the deformation process the form of grains can be changed - in the case of uniaxial tension the grains are elongated in the direction of tension and grown slimmer in the transverse direction - the magnitudes of deformations in the two substructure can differ, but their difference is limited. According to equations (m.2.7), (m.2.2) (HL2.8) and (m.2.9) the difference in the respective deviatoric parts of deformation in the case of uniaxial tension in the ^-direction is:

III.2.3 Application to Mild Steel and Aluminum Alloy

011/? ~ e l 1 e

-

~ v( e 1 1 n

-e

11a)

=

~ ~vS 1 1 » -'"'''lefato v n n

119

~ s 11n ~ ~ — ) • "n

The value of Sn„ is limited by the yield criterion valid in the plastically deforming n-substructure, the value of s^n is - according to Eq.(m.2.12) limited by the expression

and therefore, the deviatoric deformation of grains is limited as follows: eUn

< MS11e + r / 9 - ^ - [ s 1 1 e - ( 1 + - ^ 2 _ ) s 1 1 n ] Vn

Vn+rle

where the value of Sue is limited by the material strength of barriers. Hence, the decrease of average transverse dimension of grains is limited by the above expression if there is no damage. Only after the start of continuum damage - expressed in our model by the increase of the value of t]t - the decrease of average transverse dimension can proceed. This justifies the use of the measured decrease of average transverse dimension of grains as the measure of continuum damage. Calculated engineering stress-strain diagrams for different lengths oftheAMg3specimens Using the approach to the analysis of long specimens discussed at the end of Subsection m.2.1, and the data for specimens of AlMg3 alloy presented above, the engineering stress-strain diagrams for different lengths of specimens were calculated and plotted in Fig.8. In agreement with the discussion presented in Subsection m.2.1 it can be seen that the diagrams become steeper with increasing length.

120

m.2.3 Application to Mild Steel and Aluminum Alloy

*

— L-LO L-1.5L0 - - U6L0

Exp. points

-*—*—*

0.05

0.1

0.15 0.2 Eng. Strain

%

0.25

0.3

Fig.8 Engineering stress-strain diagrams for different lengths of rods

Chapter IV Time-Dependent Deformation In the case of time-dependent processes the experimental input-data are usually represented by flow-curves under constant load acting in a limited timeinterval. The differential equation that describes the theoretical macroscopic flowcurve for a material with constant structural parameters, with one substructure continuous elastic, and the other substructure continuous elastic-viscous, follows from Eqs.(n.2.10) to (n.2.24), where hg = a>g -f\g =i\n = 0 . Due to the elasticity of the ^-material, it is possible to eliminate internal tensorial variable a'jjn and introduce macroscopic strain Ey instead. From Eqs.(n.2.1) to (II.2.8) it can easily be derived for this case:

(1)

°n = -T(——<* - ^ r 2 - ^ — - * - ) • Ve

v

e

v

e

Pe

With the above expressions and restrictions hg = *«(1 inu-B \- + »?e)fiff-»'.e#] 10 / -IJ -a-iji «# = ^ *#+-^-{(/

+ tf (VeVn + V / e + Vn1)n )Sn}Hn

121

,,»..

IV. Time-Dependent Deformation

122

t = p& +

^-{(Pn-pe)t1°n[PeV+T1°g)ff-Vg(E-aeT)] Flo

Sijn=MnSij+

an

(3)

— {lln[HeC\ +

Tlg)Sil-Vgeij]

=K*+^r[r\°[peV+r]°e)
H

o

~ \Pe Ve V°n + Vl Pe Ve + "n (Pe + *e Pn Kfrn)

Pn

(5)

+ N0(ae-a„)t-N0
= const.

(6)

and the analysis can be limited to deviatoric deformation only. By making second derivatives in Eqs.(2) and (4) and excluding Sy„ and Sjjn it is easy to arrive at the following rheological equation that does not comprise internal variables: e^pSij

with

+ ASij + BSjj + Ceij + Deg

(7)

IV. Time-Dependent Deformation

123

(8)

e=

-rS5 L ( 1 + r 7 « )

(9)

4r/„n

o=—^-^[^enenn+vl^r]e+vn{vnn9^2ve^n)nn\ D =

(io)

_vevni]n 4H*R

If the values of v0 = 1 - vn, \i0, nn are known, solution to the identification problem consists in determining Hn, rie, i]n. In the case of constant macroscopic stress the two first addends on the right-hand side of Eq.(7) vanish. The value of Sy is known. At three selected points on the experimental flow curve the values of e,e,e can be measured and used in Eq.(7). In this way we get three linear equations for three unknowns B,C, D. With these values known, Hn, T]g, t]n can easily be calculated from Eqs.(9) to (11). This is only demonstration of one possibility of solution to the identification problem. In different applications other ways of determination of the material parameters can be more convenient. A number of important special cases can be derived from the model presented above. These special cases can be specified in particular by infinite values of some structural parameters (which means that the respective material constituent forms inclusions), or by zero value of some elastic deviatoric compliance (which means that the elastic response of the respective material constituent is vanishing). Let us bring out two such special models that are particularly important: A) Burgers model: It is well known (cf. Reiner (1958)) that creep of hard-set concrete can satisfactorily be described by the so-called Burgers model that is characterized by the following macroscopic Theological equation:

124

rV.l Creep of Concrete

6| = p l j + ASIJ + Bstj +Cetj .

(12)

This equation was arrived at in a phenomenological way, as a mere fitting description of observed behavior. In our concept the same equation follows from the insight into the structure: The structure of concrete can approximately be modeled as elastic inclusions in a matrix with the properties of Maxwell material, i.e. with immediate elastic response followed by linear creep. Therefore, creep of concrete is expected to be describable by Eqs.(2) to (11) with /7„=1/H„, T]9 =i)l =°°,pn =T = T = d)n = 0, i.e. by isothermal deformation of a composite composed of Maxwell matrix with elastic inclusions, without changes of volume and of elastic moduli. This really leads to Eq.(12), as the coefficient D in Eq.(7) takes on zero value (cf. Eq.(l 1) and definition (n.2.125) of R). B) Lethersich model: It is known as well (cf. Reiner (1958)) that the behavior of colloidal solutions (particles in viscous liquid) can satisfactorily be described by the so-called Lethersich model that is characterized by the following macroscopic rheological equation: W^ASf+BSi+C&f.

(13)

Again, this equation was arrived at in a phenomenological way, but in our concept it follows from the insight into the structure: Matrix is purely viscous, which means that nn = 0 , elastic material forms inclusions, which means that r\e = °°. This gives ji = D = 0 in Eq.(7).

TV.l Application to Creep of Concrete A detailed analysis of creep of concrete mentioned in the preceding subsection starts from Eqs.(TV.2) to (TV.5). Here, it seems advantageous to use internal variable a,^ instead of a^ (cf.Eq.(n.2.1)) and with the restrictions hn ="\/Hn, rje=ri2 = oo, pn = T = 7 = (bn = 0 assuming generally pe * pn , it is received:

mentioned

above, but

IV.l Creep of Concrete

6/;=/^ +

S

E

'"

fl

1 Ve+ln S//)-v9M, "*" 2HnR ~IJe 2HnH„

^nS*

2HnS,l)

2HnR Siie

125

(1)

(2)

(3)

= pa

- „ y e +T ?/>^ ^e=P/,

(4)

where ju, p,R,/? 0 are defined by Eqs.(n.2.49) to (H.2.53). To identify the model parameters from an experimental creep curve we will start with the deviatoric parts. The sample of our material is supposed to be subjected to instantaneous compression (T^ = const. It follows from our formulae that in this case all deviatoric quantities are similar to deviator 5 ^ , i.e. Sg - sTy

6jj - eTjj

s^ - SeTjj

etc.

where s = s11=fa11 ' 1 0 0 ' and Tg = 0 -0.5 0 ,0 0 - 0 . 5 , In the course of creep Sn = 0 and Eqs.(l) and (2) can be rewritten in the form: e =As-Bse se=Cs-Dse where

(5) (6)

rv.l Creep of Concrete

126 A

1 2Hnfi

Vefi(Ve+TJn) 2Hn[rln + ve(ve + vnfi)] '

B

'

Ve+1n

P

C=

B

VefiPP

(7)

Hn

According to Eq.(l), in the moment of applying the load, e and S9 change over zero values to -o+ _ rrs- < . « _ „

Ve+Tln

Integration of Eq.(6) gives: se = j-[Cs-{Cs-Ds?)exp(-Dt)]

.

(9)

Substitution of (9) in (5) and integration results in: e-e0+=s(A-^-)t--^-(C$-Ds°+)(eM-Dt)-^.

(10)

The expression exp(-Df) can be eliminated from Eq.(lO) with the use of Eqs.(9) and (5), and after rearrangement it is received: D(e/s-p) = M+Nt-e/s

(11)

where V M=A-B^ = n^^efi) S 2Hnfi[Tln + Ve(V. + Vnfl)]

N=AD-BC

=

5 ^ 2 . 4Hznnfl[T}n + vg(ve + vnfl)]

(13)

121

IV.l Creep of Concrete

From Eq.(ll) a system of three linear equations can be formulated for determining the values of D, M, N: The values of /Z and s are known, three points on the experimental creep diagram are selected and the values of t,e,B are measured at them. Inserting these values into Eq.(ll) gives the three linear equations. With the values of D,M,N known, a quadratic equation for Ji can be derived in the following form:

~2 M

jU + —

1+K

1+

1-

MK

K

HD)

1+K 1 1+ MK K t

=0

(14)

V

where K

-M + DJI±J(M- Dp)2 + 4JI(DM - N)

2(DM-N)

(15)

Tln=V*(VnUK-Ve)

(16)

2"„ =

(17)

V

nVn *HSNJI[Tln + Ve[Ve + Vnfl)]

(18)

Mn=MM

(19)

Me = fin V

(20)

which solves the identification problem, supposing that ve,vn are known. This, however, is a rather complicated question in the case of concrete. There can be grains of very different size present, and their respective stress, strain and mechanical influence can differ so much that they can hardly be described as one phase, as one material constituent. The reason for this difference lies in the overall heterogeneity of the stress field and strain field. One grain in a homogeneous and

rv.l Creep of Concrete

128

homogeneously stresses unlimited medium has some corresponding stress- and strain-field inside, which does not depend on the size of the grain. However, if the stress field in the surrounding medium is heterogeneous, then those grains, which are small with regard to the characteristic wavelength of the surrounding heterogeneous stress field, are stressed approximately in the same way as in a homogeneous surrounding stress field. On the other hand, the grains that are large also resist the heterogeneous fluctuations of the field, and therefore, their specific elastic energy content is higher. Hence, it seems to be adequate to consider only the coarse fraction of the filler as the elastic inclusions, and the remainder - the fine fractions - as a part of the viscoelastic matrix. Of course, it is not easy to decide where is the boundary. We could select ve = 1 - vn arbitrarily in the interval given by the complete volume of allfillerfractions, but a) the solution need not necessarily exist (and it was found that for high values of v0 it really did not exist in some cases); b) if the solution exists, it describes all right the experimental flow-curve according to Eq.(lO) and the experimental value of ju , but the resulting values of He,Hn can be quite false. It means that the input information is not sufficient, ve cannot be arbitrary. The information that is reliable and relatively easy to arrive at is the value of jua corresponding to elastic inclusions. We proceed in such a way that different values of ve are subsequently tried in Eqs.(14) to (20) until such value of ve is found, for which the resulting value of ne (from Eq.(20)) equals the known value of / i 9 . For a numerical example the experimental data published by Kruml(1978) were used: E = 40000MPa , 5

v = 0.15

/Z = 2.88*1(r (MPa)- , p = 1.75*10-5(MPa)-1. The mean values of the constants D, M and N were obtained from two triplets of arbitrarily chosen points on the creep curve. The measured values of t, e, 8 at these points were used in Eq.(l 1), which resulted in: D = 0.09409(day)_1,

1

N = 1715 * 10-8 (MPa)-1 (day)-1

M = 3.965 * 10"8(MPa)_1(day)-1.

rv.l Creep of Concrete

129

The properties of the elastic inclusions were: E e =49GPa,

ve =0.079,

He = ^ 2 . = 2.2 * 10_5(MPa)-1.

The corresponding value of va according to Eqs.(14) to (20) is v9 = 0.309 and the other resulting material constants are: //„ = 3.32 * 10_5(MPa)-1, Hn = 6.87(MPa)(day), TJ„ = 0.003583. For the determination of oe, o„ it was assumed that the volume changes were merely elastic, which meant that ae, on did not change with time. The structural parameter for isotropic parts J]° was calculated approximately from Eqs.(n.6.22) and (TI.2.51) that were looked upon as two equations for two unknowns pn,rj° . Solution gave: 7^ = 0.2908, p„=1.76*10- 5 (MPa)-\ v„ =0.185. With these parameters determined, it was easy to plot the respective stress- and strain-diagrams as shown in Fig.l, using Eqs.(II.2.1), (n.2.2) and (1) to (10).

130

IV.l Creep of Concrete

sfi

c

Of

+ exp. ■

<•>

\

"Si

C

tOO

50

150

200

t (days)

/Mi,."" 200

t(days)

1

I 1 1

1

1

At

{ 0

V 50

100

i ' Fig. 1 Creep of concrete - internal stresses and strains

Chapter V Fracturing V.l Quasihomogeneous Stable Microfracturing Mesomechanical modeling of stable microfracturing in concrete and similar materials - as a type of cumulative damage leading to localization of strain - differs in some important aspects from analogous mesomechanical modeling formulated for ductile polycrystalline metals in Section m.2, but in its substance it is similar. Ductile metals were modeled as two-phase continua consisting of two substructures: microdomains with easy plastic glide, and resistant elastic barriers with changing continuity - decreasing in the course of advanced deformation that involves damage. If this continuity sufficiently decreased with proceeding deformation, the form of the specimen is not more stable, as local macroscopic concentration of plastic deformation requires a lower supply of external mechanical work than a macroscopically homogeneous deformation. The degree of continuity of the substructure of barriers was measured by the value of the structural parameter corresponding to this substructure. Another feature of the model for polycrystalline metals was in the emphasis laid on deviatoric components of stress and strain tensors as the only important variables in some applications. In the case of concrete and similar materials our mesomechanical model describes the structure on two scales: on the first-step mesoscale (unchanging two substructures: that of resistant inclusions < labeled by e > and that of compliant matrix < labeled by m > ), and on the second-step mesoscale related to the matrix that is represented as a system of two variable sub-substructures: sub-substructure of the basic material of the matrix < labeled by M > and sub-substructure of microcracks < labeled c >. According to this description localization steps in if the sub-substructure of microcracks in the matrix reached some specific degree of continuity, which means that the sub-substructure of the basic material of the matrix reached some specific degree of discontinuity. In such a state the form of

131

132

V.l Quasihomogeneous Fracturing

the specimen is not more stable, as localization of inelastic deformation and of quasihomogeneous fracturing requires a lower supply of external mechanical work than a macroscopically homogeneous deformation. The discontinuity of the subsubstructure of the basic material of the matrix is measured by the respective structural parameter. Hence, the basic idea is similar to the model of ductile metals, where the adequacy of measuring the degree of CD by a structural parameter was corroborated by comparisons with direct scanning of changes in the microstructure. However, contrary to metals, in the case of concrete isotropic parts of stress and strain tensors play an important role. There is a substantial difference between positive (tensile) and negative (compressive) isotropic parts. This implies among others that the formulae for concrete cannot be as general, as those for metals. The respective constitutive equations will depend not only on the sign of the isotropic parts, but also on the relation between the magnitudes of isotropic and deviatoric parts. Therefore, it will be necessary to formulate constitutive models separately for different types of loading. To arrive at our model for concrete, it is necessary to discuss at first the model for the matrix and for the inclusions separately. Model of the Matrix Matrix is assumed to be a fracturing material. In real materials, the quasihomogeneous fracturing is always accompanied by plastic deformation (fracturing alone would be very unstable leading to immediate brittle fracture), but atfirstthe fracturing effect will be discussed separately. Mesomechanical description of an elastic material with quasihomogeneous distribution of cracks with zero volume fraction Usual definition of the degree of cumulative damage D in materials, in which such damage can well be measured by the change of Young's modulus, is:

D= 1 - fa-

(1)

where Em is Young's modulus after some damaging deformation and Eu modulus of the virgin material (of the undamaged matrix). If the Young modulus of an elastic material with cracks is to be described by our model, it is necessary to go back to Eq.(n.2.11), according to which the macroscopic elastic deviatoric compliance \im of a two-phase material is expressed as follows:

V.l Quasihomogeneous Fracturing

133

where vc, vu are volume fractions of the two material constituents labeled by indexes c and M respectively, 1+v m

1 + vr

t/n

fc

c

1 + vM t

M

v m being macroscopic Poisson's ratio, Em macroscopic Young's modulus, and vc,vu, Ec, EM analogous values for the two material constituents. If index C in Eq.(2) is attributed to cracks, then £c = 0, Hc=°° , which leads to: Hm=^-^*-.

(4)

If we assume that the values of Poisson's ratios v m and vM are approximately identical and that the volume fraction of cracks vc is nearly zero (which gives vM = 1), it turns out from Eqs.(l), (3) and (4):

It follows from the deduction of our structural parameters that r\M can vary from vanishing value to infinity. The vanishing value of r\M corresponds to very small spherical bubbles that have no influence upon the macroscopic modulus. This can be seen from the relations valid for elastic spheres (index c) in an elastic matrix (index M), as they were derived in Section n.6, Eq.(TI.6.23):

2^4-^) '** 7-15v c + 5v M (3v c -1) For vc = 0 the respective value of r/M is zero. This agrees also with the generally known finding that vacancies do not influence Young's modulus.

V.l Quasihomogeneous Fracturing

134

Infinite value of r\M means (cf. Section n.2) that the substructure of the basic material labeled by index M is discontinuous, it forms inclusions surrounded by very thin shell matrix of the cracks. This means zero macroscopic modulus and complete damage (D= 1). In this way it is possible to characterize the degree of damage by a dimensionless quantity r\M being in the interval from zero to infinity, or according to Eq.(53) - by D being in the interval from zero to unity. Quasihomogeneous stable microfracturing In the preceding paragraph, CD was related to one structural parameter T]M. Now we are going to discuss the relation of CD to the process of deformation. We will start from the so-called elastic capacity concept. In this concept, there is postulated that in an isothermal process free elastic energy (FEE) in any material is limited, it cannot overpass some limit. There exist just three basic models of materials with limited elastic capacity: • Perfectly plastic material • Perfectly locking material • Perfectly fracturing material. Their stress-strain diagrams for simple deviatoric loading are shown in Fig.l. The perfectly plastic material has been discussed by many authors and it need not be commented. The perfectly locking material is less important, but also well known. In detail, it was described e.g. by Prager (1966). In the description of the m-material as a perfectly fracturing material, no plastic deformation and no locking stresses would be present, which leads to: FEE=

$(s$ne$„+3o*e*)= =

e?' e?'

re> ee>

Hm

Pm

$(s?miimsfm+3o%

I ( 5d3CL + 3£2ii2L)^c t F

pmoeJ,) (7)

where CLP depends on the loading path, having generally higher values for compressive loading. In an active monotonically increasing deformation, from the moment of reaching the respective limit CLP the following differential form of the above equation is in validity:

135

V.l Quasihomogeneous Fracturing

0= 2 /'mi m"/ym< ""^m d 4 , +**/ym < 4**/ym , <"-"A^m K+ = 2

''um UCiim "/ym '-"-'4F/7J

C'ltm ■^ym U»//m "p/ym

Mm

Mm

6p m a£d<+3<7«'
„ t

dMm m + 6-

m

t,

Pm

'm

(8)

dp„

which leads to:

■A

(2

"/ym'-'^ym

+ 6-

-3

e*1

E*1

(9) dpn

Pm

Deviatoric loading The simplest case of the model will be that without isotropic parts of stress, with only deviatoric loading. It is the case of simple shear or twist Here, Eqs.(7) and (9) will reduce to: et

S iim

EEE - jiS/jn, e,ym) - ^(s/yVn nm Sjjm) - j( —

<,»/ f l e e / >ym u a f lvm

a

du =-2u u Mm

'''"'' ~ " = 2u £

' C m °((m gi °/ym »/

Qet

"1/7

e?' de?'m '"* '

M

) < CD

(10)

(ID

where CD is the limit for purely deviatoric loading. For a uniaxial stressing s^2m = $?.\m E q - 0 1 ) c a n be integrated, giving

£» Mm

^S|2«l )

(pe,

^ m

f (12)

. ^ m

where the superscript stars mean values at the beginning of the fracturing process. The relation between S%'2m and ef2m is hyperbolic. By definition Mm - e 1 2 m y ' s 1 2 m

136

V.l Quasihomogeneous Fracturing

and with this definition Eq.(12) leads to: S^m

-

Vm e12/n = 0 .

(13)

"— = 0 -

61201

M/n

This means that if there is no plastic part of deformation, unloading returns stress as well as strain to zero values. In Fig.l the hyperbolic loading curve and the straight unloading lines demonstrate qualitative agreement of our model with generally observed properties of fracturing materials with cumulative damage. In real materials some permanent deformation always remains after unloading - due to plastic part of deformation. It is clear from the above discussion that at the beginning of the fracturing process ^ = 7/^ = 0, and that the value of nM does not change in the course of the deformation process. Therefore, combination of Eqs.(50 and (12) gives: s

nu =

12m

of'

Pl2m

-1 =

-1

(14)

J

f

1

per/, plastic

\ X /

/ / *

/

\ ^ ^

^-brittle

pert,

fracturing

fracture

e

Fig. 1. Stress-strain diagrams of basic models of materials with limited elastic capacity.

V.l Quasihomogeneous Fracturing

137

Uniaxial compressive and tensile loading Let us further assume that the uniaxial loading is compressive, applied in the ory-direction. Let the respective value of CLP be denoted Cuc. A new supposition is accepted:

Pm

Vm

which means that the relative changes of jxm and pm are identical, as they result from the same cause - from the process of fracturing, of violation of elastic bonds. This supposition is equivalent to the assumption that the elastic Poisson's ratio remains unchanged. In this case Eq.(9) takes on the form: ri„

^mSTl/mdS,<,(m+2pm
__?„

Pm I s ! 1m) = 2ll

+,i

Pm\am)

P m ei1mdei1 m +2/i m e^dg^ Pm(e,°L) 2 +2/i m (e£) 2

and from Eq.(5i): d'?M=^2L.

(17)

PM

These relations are valid after reaching the elastic limit and beginning of the fracturing process for compressive as well as tensile loading. However, for these two cases the elastic limits will be substantially different. Constitutive equation of the matrix In what follows it will be assumed that no locking stresses exist and therefore, the superscripts el can be omitted at the symbols for stresses, i.e.:

The deviatoric part of strain increment in the matrix is assumed to consist of the elastic response involving the change of deviatoric elastic compliance due to

138

V.l Quasihomogeneous Fracturing

fracturing, and of the plastic part. In mathematical terms this is expressed as follows: teijm =Mm ds/Jm +s Jm d/i m +sijm d ^ =nm ds Jm +sijm ( l + x r ^ d ^

(19)

where we have newly introduced the assumption

(20)

i.e. the change of the elastic compliance resulting from the fracturing process is assumed to be proportional to the increment of plastic strain, the positive coefficient of proportionality Kp being assumed constant The isotropic part of strain increment in the matrix is assumed to consist of the elastic response involving the change of isotropic elastic compliance, and of the change of volume resulting from the fracturing process. This results in: de m = Pm do-m +om dpm +dm=K<0dXm , ( Ka>0) .

+KjdXm

(21)

(22)

Assumption (22) relates the increase of volume to the increase of plastic deformations, and with regard to Eq.(20) to the change of elastic compliance as well. Model of the Inclusions Inclusions are assumed to be elastic isotropic, their constitutive equation is used in the form: de Se = ju0ds^e, de e = p e 6oe .

(23)

Model of the Whole Composite Our aim is to describe a composite material with elastic inclusions, and fracturing and plastically deforming matrix. With the inclusions assumed elastic, and the matrix assumed with properties described in the preceding section, this model will have the following form:

.

, ■um.-a.iy*".«■.>» n t s W ,

(24,

V.l Quasihomogeneous Fracturing

de

= pda+vm{[r1Z,(Pm-Pe)a

+

Pe(Ve+Vm)0m]—Kti /l

(VePe+TlmPm)Kj

~0 —^ PmVm+Ve(Vepm

(25)

'"

dx

+

139

T + VmPg)

with the macroscopic deviatoric and isotropic elastic compliances and the macroscopic Young's modulus respectively: v

e»e+1lm(Ve»e+VmVm)

P=fl

p

(26)

V =

E =

ePe+TlmKP*+VmPn,)

p

0

_

3

(27)

_

(28)

and the evolution equations He

_fan/*™W » ) d S t f +[t]m S, -{vl +

torn Pm+ve Pe)6a+ {[77° c-(vl+ri^a^^-K^

d«r ffl =

Pm1m+Ve(VePm+Vmpg)

In the above relations ve,vm,ne, constants, fxm, p m , J7M, ju, p

if

-vl Ka}6X„,

. oo)

pe, ju M , pM, rjm, rfm, K^ , Ka are

are variables, their values at the elastic limit

beingn'm = p.u, p'm = pM,J]*M = 0, ft, p*, the last two symbols being given by Eqs.(26) and (27) if Hm,pm are replaced by jXM,pM. For making the model operative, it is necessary to express the increments of dXm. With the use of Eqs.(20), (16) and (17) it is received for uniaxial compression or teasion in the xt direction:

V.l Quasihomogeneous Fracturing

140 dA

-

d

^

-^MdT7m -

K

n

2ju

K

m

v

^mgl1mdSiim+2pm(TmdO-m

^{jim^m

(J1)

+ 2pma2m)

Generally, the experiments with concrete specimens are deformation controlled, and therefore it is necessary to express dXm in terms of de^1 . The first step is expressed in terms of dc^ 1 : dAm = Mdffv

(32)

where __gA M = - 3M

A =

BC+FG ..'.

KU+A(BH+FJ)

-5-^*2 5Mm^1m + 2pm
B =

Emillm

C = lmHm+veng F =

(33)

(34)

(35) (36)

^ ^ VmPm +

(37)

Ve(VePm+VmPe)

G = r)°mPm + vepe

(38)

W = [ifm5l1-(v2+i?#Il)s11m](1 + r p )

(39)

J = hmfl,-(v5+»?m)ffm]—*„-«£*.

(40)

and the second step

V.l Quasihomogeneous Fracturing

6
L

-

=

v

= dfn \ + ME(K+L)

141

(41)

r]m(nm-^e)^+ne(ve+T]m)^m

m

;

:

*n [lm(Pm ~ Pe)
(42)

+ J

lm)^m]Pm I Vm + KjtJm Pm + VePe)

TlmPm+Vg(Vgpm + Vmpe)

(43) Example of Application to Structural Concrete The merits of our model are demonstrated by its application to one kind of structural concrete experimentally investigated by Spooner and Dougill (1975). The gravel used in this concrete was flint. From the data given in the paper it was possible to determine directly: v o =0.6 , vm=0A,

E*= 30.305 GPa.

Poisson's ratio was not given in the paper, it was taken according to known literature data in the value v"* = 0.1515. For the gravel - i.e. for flint Young's modulus was taken also from literature data in the value Ee = 51.751 GPa and its Poisson's ratio was assumed to be the same as that of concrete. Other parameters were determined by a curve-fitting procedure in the values: i\m =0.023, r\°m =2r/m, Kfl =0.35, v„ =0.08, (au)ce, =-30MPa where (<Ju)ei means elastic limit in compression. All other quantities are then determined from the model equations. In Fig.2 the possibility of describing the typical course of stress-strain diagram of concrete is clearly demonstrated. Not only strain-softening, but also the unloading and reloading courses agree well with what is generally observed, and with the experimental data that Spooner and Dougill (1975) presented. In Fig.3 the respective courses of our two variables describing the progress of CD are demonstrated:

142

V.l Quasihomogeneous Fracturing 1

—r

50

1

1

r

1

1

1

exp.points exp. points of unloading and reloading

*

o 45 40

-

-

35

/

V\

oo

po

/

20 15

/

10

/

o 'o

/

/

/ / o'o/

/

/

0.5

A

/

/

/

o

o' o

/

*v

/

o // o

/

^

/

/

O

1 1

.

**■

^ ^6

o / O

/

/ /'O XO&

\ *

\

/ CO

1 /

I25

0

/ \

/

/ A

I 30

5

*

/

'

1.5

•

1—

* - » J

2.5 Strain

'

3.5

. * —

■

4.5

5 X10 - 3

Fig. 2. Comparison of experimental and theoretical courses of strain-softening stress-strain diagram of concrete and of interrupting unloading and reloading (experimental data taken from Spooner and Dougill (1975)). (i) that of r\M starting from zero (completely undamaged state), and growing without limit to infinity (complete damage, from the point of view of the general model corresponding to the state in which the material of the matrix forms discontinuous inclusions surrounded by continuous substructure of cracks); (i'0 that of D (defined by Eq.(53)) that starts again from zero value (completely undamaged state), and its least upper bound is unity (corresponding to complete damage). Conclusions Our mesomechanical model of deformation behavior of concrete is based on assumptions that reflect really observed processes in the structure of concrete in the course of deformation and continuum damage. It is able to describe the typical courses of stress-strain diagrams and of interrupting unloading and reloading. The measure of damage is characterized by a specific parameter t]M (changing from zero to infinity) that follows directly from the model, or by a

V.2 Localized Cracking

143

Fig. 3 Theoretical courses of our parameters measuring continuum damage: T]u (growing from zero without limit) and D (limits zero and unity). derived parameter D that changes from zero to unity - as it is usual in other approaches to measuring continuum damage. It is assumed that the beginning of localization of deformation and damage is closely related to the degree of damage, at which localization requires a lower supply of external mechanical work than a macroscopically homogeneous deformation. Our model with its quantitative description of damage by one scalar parameter is suitable for such analysis.

V.2 Localized Cracking in Concrete Compressive loading of concrete and similar materials leads to the creation of macrocracks that are parallel with the direction of the acting compression, or - if the compression acts in two directions - with these two directions. This has been experimentally ascertained e.g. by Stroeven (1980). The understanding and mathematical modeling of this phenomenon is not trivial. It is possible to formulate a criterion of rupture in planes that are normal to the maximum stretch, this, however, is a mere phenomenological description and it does not explain the phenomenon. A rupture is always caused by forces, and in

144

V.2 Localized Cracking

this case the macroscopic forces - or stresses - acting on the plane of the crack are vanishing. It is but natural to seek the explanation on a lower dimensional scale, i.e. on the scale of micromechanics. Even this, however, is not straightforward: The first approach that can be taken into consideration is the elastic solution for a plane with a circular hole under uniaxial compression. There appears tensile stress at the edges of holes that are perpendicular to the direction of compression. It is really possible to explain in this way the cracks parallel with the direction of the compression in some materials with big pores or with soft inclusions (e.g. light concretes), but not in those with hard inclusions (e.g. structural concrete). The mentioned effect arises at holes, not at flat fissures. In a structural concrete there can exist small holes, but they are rare, their stress fields do not add and because of their small dimensions the energy concentrated at one hole is not sufficient for the creation of a crack. Experimental evidence shows (Lusche(1972), Zaitsev and Wittmann(1977), Stroeven(1980)) that cracking begins at the contact surfaces at large boulders, where the surfaces are parallel to the direction of the acting compression. There is no reason for the holes to be concentrated in such places, but even if they were present, the mentioned effect would not arise, as the boulder has in this case higher rigidity, and therefore it takes on the main flux of forces and the neighborhood of the hole would be relaxed. Another possible approach is the elastic solution for a plane with a circular inclusion under uniaxial compression. In the most dangerous case of an absolutely rigid inclusion, this solution (Muschelishvili(1954), p. 211) yields: 9 = P (1 - v ) (1 - 4 v ) / ( 3 - 4v)

(1)

where 6 has the meaning of the normal stress acting on the contact surface where the surface is parallel to the direction of macroscopic compression p , v is the Poisson ratio of the material. For v = 0.2 , which corresponds to the cement stone, the value of 6 turns out to be 0.073p, i.e. a very low compressive stress. The analogous solution in three dimensions does not lead to significant values either (Lusche(1972)). For inclusions that differ in their shape from a circle the stresses can be a bit higher (Edvards(1951)), but the difference is not substantial and experimental evidence does not show (Zaitsev and Wittmann(1977), Thomas et al.(1963). Shah and Slate(1965)) that the cracks would be concentrated at elongated shapes of grains. It does not seem therefore that the microstresses arising in the elastic range could be the cause of the cracks under consideration.

V.2 Localized Cracking

145

In Stroeven's monograph (1980) some papers are quoted, where the cause of the cracks parallel to the direction of the acting compression is seen in the interaction of spheres (particles of the aggregates). This explanation does not seem to be satisfactory either. It is possible to imagine two spheres and another one that is pressed between them and in this way they are separated. But it is also possible to imagine another configuration by which they are brought nearer. If we imagine a regular arrangement of spheres that are smooth and lubricated, and boundaries that are parallel with the direction of the acting compression and free of stress, then the spheres in the planes normal to the direction of the compression will be separated. However, if there existed cohesion among the spheres, this effect would be a boundary effect, maximum at the free boundaries and diminishing with the distance from them. According to this scheme the cracking should appear at free boundaries by breaking out the boundary boulders. This, however, is not observed as a rule. Another argument against this explanation is the fact that observations of slices of cracked concrete do not reveal concentration of cracking in places, where one boulder is pressed between two others. The key for understanding the real cause of cracking in the planes parallel to the acting compression is after all Eq.(l). For Poisson's ratio 0.5 the value of the stress resulting from it is -0.5p, i.e. a relatively high tensile stress. Poisson's ratio of the cement stone is 0.2, and therefore this result is of no meaning for the elastic range, but the value 0.5 is characteristic for a process, in which the volumetric changes are zero and the deformation has only deviatoric character. This is the characteristic feature of a plastic or viscous process. In the case of a quasihomogeneous microfracturing of the matrix, volumetric changes do appear, but they are of an opposite sign - dilatancy, which leads to an opposite effect rise of the tensile stress. On the basis of the considerations outline above we are going to formulate the basic hypotheses of our explanation: a) In the case of a uniaxial compressive loading the cracks originate at the contact surfaces of large boulders of the hard aggregates in places, where these surfaces are parallel to the direction of the compression. b) The cause of appearance of these cracks is tensile stress arising as a consequence of an inelastic deformation, which can be plastic, viscous, or can have the character of quasihomogeneous stable microfracturing. c) In the case of a compressive loading in two directions the process is quite analogous, the cracks are parallel to the plane given by the two directions of the acting compressions. These basic hypotheses are strongly corroborated by Stroeven's experimental results. According to his findings (Stroeven 1980) uniaxial

V.2 Localized Cracking

146

compression leads to the separation of biconical forms with a boulder of the aggregates in their center - cf. Fig.l. Cracking begins at the surface of the boulder (this evidence is also supported by a number of other experimental works - e.g. Lusche (1972), Zaitsev and Wittmann (1977), Thomas et al. (1963), Shah and Slate(1965)) in places, where the surface is parallel to the direction of the acting compression. It is natural to model these forms as geometric forms with rotational symmetry and then the stress causing the beginning of cracking can only be tensile stress, as due to the symmetry the shear stresses are zero. It was shown in the preceding considerations that these tensile stresses can have significant values only as a consequence of inelastic deformation. In this context we will model structural concrete - or any similar material - as a two-phase composite of elastic inclusions and a matrix with elastic deformation and some kind of inelastic deformation - plastic, viscous, or quasihomogeneous stable microfracturing. The adequate mathematical model is

1 compressive

i

p

tr~7p

loading

*1

X

3

*2

i Fig.l A grain with two conical formations separated from the matrix by the fracturing process under uniaxial compression represented by Eqs. (II.2.45) to (II.2.53), where h0=pe =ae =i]n =r/£ = f = 0 . It follows from these equations that for a space-independent macroscopic stress in the course of the whole loading history the stress-field in the inclusions is also space-independent (which is not the case in the matrix). The knowledge of the

V.2 Localized Cracking

147

homogeneous stress tensor in the inclusion is sufficient for determination of the contact vector stresses t,: t/^ijePj

(2)

where Hj is the outward unit normal of the surface of the inclusion and o^ is the tensor of the homogeneous stressfieldin the inclusion. In accordance with Fig.l the inclusion is modeled as a sphere and the contact stress at all the equatorial points P will be equal. Let us choose a point />(2\ where the normal is parallel to the coordinate axis X2. The contact stress at this point will be:

tf^rf'^i*,

(3)

as the only non-vanishing component of n*j2> is n(
(4)

where T = 5l1 and the tensor Ty has the form:

ft T

l =

0

0

0 -0.5 0 0 0 -0.5

It is a consequence of Eqs. (II.2.45) to (TL2.48) that in this case all the deviatoric components of stresses and strains with unequal indices (i * j) are vanishing and those with equal indices can be expressed as products of some scalars with the tensor T^ ,e.g. Sye=seTg etc. Hence, the only non-vanishing component in Eq. (3) will be ^=a22e=s22e+
(5)

V.2 Localized Cracking

148

and due to the form of the tensor 7^ it holds: 1

1

where se can be expressed with the use of Eq.(II.2.1) in the following way: s,=—(S-vnSn).

(7)

In Eqs.(TI.2.45) to (n.2.48) we can clearly distinguish between the response resulting from the elastic deformation of the matrix (in what follows labeled by the subscript el) and that resulting from the inelastic deformation of the matrix (labeled by subscript nt). It can be shown (Appendix vm.6) that for a uniaxial monotonic macroscopic loading the sign of the response (s 0 ) n/ is the same as that of the response (S8 )# and that of s . This means: Sa=(s,)a/+(sA/ s»gn(se)n/ = sign(se)e, = sign(s„) = sign(j).

(8)

In the case of plastic or viscous deformation the isotropic stress components will arise only due to the elastic deformation of the matrix. In the case of quasihomogeneous microfracturing there are two addends in the part of o„ that result from the inelastic deformation of the matrix (Eqs. (n.2.48) and (n.2.1)). For the first addend, connected with p„, the signs will be the same as in the case of Sa. But the second addend, connected with tfr„ usually outweighs, which manifests itself by the fact that the volume of the matrix increases in spite of the compressive loading. The sign of the second addend is always positive (cf. Appendix VIII.6) and therefore, we have: * . = (*.)•»+(*.)n»

(9)

For a uniaxial compressive monotonic macroscopic loading Eq.(S) can now be rewritten as follows:

V.2 Localized Cracking

t(22) =(S22.)*+(S22.)n/+(
149

00)

with -^Se)nl>0.

(°e)nl*>

which means that the contact stresses at the equatorial points P are changed by the inelastic deformation of the matrix, and this change has the sign of tension. The numerical example that follows shows that the influence of the inelastic deformation upon the contact stresses is significant. This numerical example describes the changes of internal stresses due to creep described partly in Section IV.l. Whereas in IV.l the analysis was aimed at the course of the average stresses in the material constituents, here we are interested in the contact stresses according to Eq.(5). For the deviatoric elastic response of S11a we get from Eqs.(7) and (H.2.47): (Sii»)«r =(«•)* =A*»

T^

rSii.

(11)

Furthermore, it can be shown (cf. Appendix vm.6) that after a long time (theoretically infinite time) the value of S119 will change to:

(Sile)-=(S 8 )- = - ¥ ^ - & i i -

(12)

The response of the isotropic part (7e will be only elastic. From Eqs. (n.2.48) and (II.2.1) we easily arrive at: Pn(ve+V„) PnV°+ve(v9pn+vnPe)

o.

For the uniaxial compression (T^ it holds: -

2 _

_

1 _

and by using the numerical values from Section IV.l we get:

(13)

150

V.2 Localized Cracking

(*.)* =(ff.L=1.00738a=0.3358a 11 (Sn*)* =(«*)«/ =1-2958s11 = 0 . 8 6 3 9 ^ (Sn.)-=(*.)- -3.1580 5^=2.10530^ and for the contact stresses at the equatorial points P according to Eqs.(5) and (6): (42))a/ = - 0 . 0 9 6 2 ^ , (f)L=-0-7168a11 . Hence, we can see that in the elastic range the contact stresses are unimportant, but the inelastic deformation leads to considerable values of tensile stress. The presented micromechanical model has meaning for such cases, where the redistribution of internal stresses due to inelastic deformation is possible, i.e. where it is not preceded by brittle fracture. For concrete and similar materials this is the case only in compression. However, for metals with resistant inclusions the same model and the same equations can be used even for tension, where the cracks will be perpendicular to the direction of the acting loading. In our model there does not appear any parameter characterizing the dimensions of the grains. The fact that the cracks appear preferably at large grains can easily be explained by the 'non-local criterion of strength' presented in the third part of our former monograph - Kafka(1987).

Chapter VI Shape Memory VI. 1 Shape Memory Resulting from Heterogeneity on the Atomic Scale The phenomenon of shape memory (SM) in some binary or ternary alloys is well known and the respective martensitic transformations are studied in detail. What is not so clear is the micromechanical causality of the process. It does not seem that this question could be solved on the basis of only thermodynamic considerations, which can give some limits by which any model is bound, but not the micromechanical structure of the model. The very complicated behavior of SM materials can hardly be described by apparent variables only, it calls for the use of latent tensorial variables. In what follows a model with these properties is presented, based on our mesomechanical concept Recovery of original shape due to temperature change is observed typically in some binary or ternary alloys, in some non-metallic materials with heterogeneous microstructure and in bimetallic actuators composed of two metals with different thermal expansions. The common feature in all these cases is heterogeneity. Let us emphasize that we use the term heterogeneity in a broader meaning than it is common. In the common use a binary alloy is seen as homogeneous, but from our point of view it is heterogeneous, composed of different atoms. The fact that heterogeneity is a common feature of materials with shape memory is assumed in our concept to be essential. Here, it is important to emphasize that the kind of heterogeneity we have in mind is heterogeneity consisting in the presence of different atoms, and not heterogeneity consisting in the presence of martensitic and austenitic subvolumes. Generally, heterogeneity alone need not be sufficient for the SM to exist, but in some cases it can be sufficient. Let us consider two demonstrative schemes: (i) Let an elastic sponge with continuous pores be filled with e.g. paraffin wax. At a lower temperature, the wax will be in a solid state and any sufficiently large deformation will be permanent due to plastic deformation of the wax filling. A

151

152

VI. 1 Shape Memory

rise of temperature will change the state of the wax to liquid and the original shape of the sponge will recover. In this case, the energy causing this recovery is the elastic energy of residual stresses in the elastic sponge that goes after heating to its minimum - to zero value. (ii) Let the scheme described above be complemented by a property of thermal heterogeneity: let thermal dilatation of the wax be higher than that of the elastic sponge. Rise of temperature after deformation will cause not only softening of the wax, but also additional internal stresses, as the elastic sponge prevents the wax from increasing its volume to the extent corresponding to its higher thermal expansion. Such internal stresses can be high and if the elastic sponge is not resistant enough, it can be torn. But if it is able to resist, the shape of the composite will change to such configuration, in which the elastic energy of the internal stresses will reach its minimum, i.e. to its original undeformed shape. (This time the minimum of energy is not zero.) In this case heterogeneity is not sufficient, the ability of the elastic substructure to resist the internal stresses is another unavoidable condition. Here, the forces that the composite can exert on the surrounding medium can be substantially higher than in the preceding scheme, the main part of the energy that causes the recovery of shape is the energy of heat that is transformed into mechanical work by the mechanism of thermal dilatation. In what follows the main ideas of the last scheme are used for the description of SM phenomena in binary alloys. Heterogeneous systems composed of substructures with different thermomechanical properties can be described on the basis of our mesomechanical concept. Application to heterogeneity on the atomic scale is not straightforward, but our preceding papers (Kafka 1990, 1992, 1994, 1999a, Kafka and Vokoun 1998, Vokoun and Kafka 1996, 1998) have shown that it is possible. Applicability of Our Concept to Heterogeneity on the Atomic Scale Our reasoning will start from the scheme of martensitic transformation and detwinning as shown in Fig.l. In this two-dimensional scheme the austenitic and martensitic structures are visualized by squares and rhombi, similarly as they were by Stoeckel and Waram (1991). What we consider to be essential in this scheme, is the different changing of distances between pairs of the first and the second atomic neighbors (no change and substantial change respectively). In a composite material we work with stresses in material constituents (which are some averages of interatomic forces), with elastic moduli, strength, and coefficients of thermal expansion. All these quantities can be related to the model description of changes in an atomic lattice of a SM binary alloy in the courses of martensitic transformation, detwinning and reverse martensitic transformation. This is shown in Figs.(2a) to (2d). Instead of stresses there appear interatomic

VI. 1 Shape Memory

153

forces, instead of moduli slopes of interatomic potentials, instead of strength maxima of interatomic forces in the course of increasing distances of atomic pairs, instead of coefficients of thermal expansion rates of thermal changes of force-free interatomic distances.

deforming - detuinning

Fig. 1 Scheme of the process of martensitic transformation due to cooling and of detwinning due to uniaxial monotonic loading

In what follows the NiTi binary alloy - called nitinol - will be discussed as a typical representative of SM alloys. The first neighbors of Ni atoms are Ti atoms and vice versa. The bonds between these first neighbors are very strong and consequently it is assumed in our scheme that the distances between these first neighbors do not change due to martensitic transformation (and reverse transformation). The bonds between the second neighbors (Ti-Ti pairs and Ni-Ni pairs) are weaker, but they are not negligible (it is characteristic for the binary alloys with strong SM effect that their austenitic structure is BCC and in such structures the energy of the interatomic forces of the second neighbors is not negligible - cf. Johnson and Wilson (1983)). Therefore, martensitic transformation

154

VI. 1 Shape Memory

is connected with the changes of interatomic distances between our second neighbors Ti-Ti and Ni-Ni - in accordance with Fig.l. It is not assumed that the interatomic potentials Ti-Ti and Ni-Ni are the same in the NiTi alloy as they are in pure Ti or Ni metals, but it is assumed, that the qualitative relations of their characteristics (slopes of interatomic potentials, maxima of interatomic forces, rates of thermal changes of force-free interatomic distances) are the same (if some of these characteristics are higher in pure Ti metal than in pure Ni metal, it is assumed to be higher also in the alloy). Martensitic transformation can well be explained on the basis of the above assumptions. This is schematically represented in Figs.2a to 2d: - In Fig.2a the ideal stress-free austenitic atomic configuration is represented at temperature TE , at which the atomic distances of Ti-Ti pairs and Ni-Ni pairs are equal. Equality of these distances corresponds to the square scheme in Fig.l - the left upper scheme. The courses of the atomic potentials in Fig.2a reflect the higher Young's modulus of Nickel and its lower strength compared with Titanium. Two characteristic pairs of second atomic neighbors of every element are distinguished: (Ni), with (NiX and (Ti), with (Ti),,. In this configuration, their distances are equal and their interatomic forces F" and F 7 are all vanishing. - In Fig.2b the configuration at a lower temperature Ti < TE in a fictitious constraint-free state shows that due to the higher thermal dilatation rate of Ni the distances of the Ni pairs are smaller than those of the Ti pairs. In this fictitious configuration no constraint is assumed, and therefore, the interatomic forces F" and FT are again vanishing, the distances of the Ni-Ni pairs are smaller than those of the Ti-Ti pairs, there is no difference between the characteristic (Ni), and (Ni)b pairs, and between the (Ti), and (Ti)b pairs. - In Fig.2c the configuration at the same temperature Ti < TE is shown for the case of constraint, by which the distances of the Ni-Ni pairs are forced to be the same as those of the Ti-Ti pairs. Here again, the distances between the characteristic (Ni), and (Ni)b pairs are the same, and such are also the distances between the (Ti), and (TiH pairs. Due to the constraint, the interatomic forces are not vanishing, the interatomic forces F," between the Ni-Ni pairs are positive (tensile), and those between the Ti-Ti pairs - F,r - negative (compressive). This configuration simulates austenitic structure at a temperature that is lower that TE, but not low enough to cause martensitic transformation. The energy of the interatomic forces increases with decreasing temperature up to the state, in which the forces F," reach their maximum on the curve representing Ni-interatomic potential. In this state the equilibrium of the interatomic forces is metastable, with

155

VI. 1 Shape Memory

a small further decrease of temperature the structure will change to a configuration with lower energy of interatomic forces - to martensitic structure.

0}

equidistance

'

0

^

/,'

^<^__

yy\ /
Distance

! m ~

m

•

(M

1 • r^

V^j'a

(N /v \

N

■r

l i;/) • '

3

_4 (7j.;a

>

$

:

^r

(7-)^ : ^

n ■■■■ n

"

= 0

= 0

1

Fig.2a Scheme of the interatomic forces between pairs of atoms of the same element, of Ni and Ti, at the temperature TE , at which the distances of the pairs are the same in the unconstrained state

- In Fig.2d this newly created martensitic configuration is visualized. One part of the Ni-Ni pairs and Ti-Ti pairs - labeled (Ni), and (Ti). respectively - has smaller distances, the other part - labeled (Ni),, and (Ti)b - has larger distances. This agrees with the representation of martensitic configuration as shown in Fig.l. Such transformation leads to a decrease of energy of interatomic forces. This agrees with the experimentally observed exothermic character of martensitic transformation.

156

VI. 1 Shape Memory

Distance

F' FN = 0

- f (rt)a f

= o

Fig.2b Scheme of the interatomic forces at a lower temperature 7 r , again without any constraint To summarize for the distances and interatomic forces of the atomic pairs: (0 The distances of the Ni-Ti pairs remain nearly unchanged, their changes, as well as changes of the respective interatomic forces, are neglected. 00 The distances of the Ti-Ti pairs do change, but their bonds remain elastic, i.e. their interatomic forces do not get over the respective maxima, there is no dissipation of energy of these interatomic forces. (iii) The distances of the Ni-Ni pairs do change, the bonds of some of these pairs do not remain elastic, i.e. their interatomic forces get over the respective maxima and the energy of these interatomic forces is partly dissipated. Detwinning due to uniaxial monotonic mechanical loading is explicable with this scheme also. The number of the atomic pairs with short and long distances, as mentioned above, remains unchanged, but some individual concrete

VI. 1 Shape Memory

.i-""

austenif

0

k

/

1 /

►*

-4

Wa -^ > 0

<►~

- •

{Ni)b ■■ F," > 0

<>-

- 6 (r,)e : f/
iH-

157

Distance

1

Fig.2c Scheme of the interatomic forces at the same temperature 7", if the distances are constrained to be the same in the austenitic configuration pairs with short distances change to pairs with long distances and vice versa - as visualized in Fig.l. The energy of interatomic forces is the same before and after the process, the energy supplied by the mechanical loading is dissipated in the course of Ni-Ni pairs getting over their energetic barriers (over the maxima of NiNi interatomic forces), and falling into nearby energetic wells. This process is similar to plastic deformation in its dissipative character , but contrary to it the presence of the inviolate Ni-Ti and Ti-Ti bonds that remain elastic ensures the diffusion-less character of the process, and the shape memory. The reverse transformation to austenite can again be explained using the same scheme. The increase of temperature leads to an increase of the zerostress distances between the atoms, and this increase tends to be higher between the Ni-Ni atoms than between the Ti-Ti atoms due to the difference in thermal dilatation.

158

VI. 1 Shape Memory

(*a

^)

&

(^/)a> 0

C/\ > o

U )6 —f (5

Fig.2d Scheme of the interatomic forces at the temperature T2 that is lower than T1, the configuration is martensitic, there are two different distances between atoms of the same element This results in strong dilatoric forces between the atomic pairs Ni-Ni that are in the martensitic structure nearer, and in contractile forces between the atomic pairs Ni-Ni that are in the martensitic structure more distant. This can clearly be seen in Fig.2d, if the Ni-curve is strongly shifted to the right to such an extent that the forces (F*)a become negative. Such interatomic forces tend to such atomic configuration, in which the distances are equal, i.e. to austenitic structure. In this way the heat energy is transformed into an increase in the energy of the interatomic forces. This agrees with the well-known finding that the reverse transformation is strongly endothermic. The pseudoelastic process can well be understood from Fig.2c. It starts in the austenetic state. There is no change of temperature, and therefore, there is no change of the zero-stress distances either. But the distances between the atoms do

VI. 1 Shape Memory

159

change - due to the applied stress. The increase of distances between the atomic pairs Ni-Ni leads to surpassing their energetic barriers and falling down into other energetic wells. This means dissipation of energy and inelastic deformation. This is not the case with the Ti-Ti pairs that remain on the slopes of their original wells, i.e. in the extent of elastic bonds. Unloading means - due to the elastic bonds and the respective interatomic forces between the Ti-Ti pairs - a reverse process and recovery of the shape. Contrary to a real elastic process, however, a part of the energy supplied by the surrounding medium is dissipated. (In a real elastic process all the atomic pairs remain on the slopes of their original energetic wells and therefore no dissipation of energy appears). The low Young's modulus ofmartensite compared with austenite follows from our scheme also in a straightforward way. In austenitic configuration as visualized in Fig.2c a small variation of atomic distances due to mechanical loading causes variation of elastic energy in the way that corresponds to normal elasticity (the respective change of all interatomic forces is given by one slope of the respective potential). In martensitic configuration as visualized in Fig.2d a small mechanical variation of atomic distances of the Ti-Ti pairs is quite similar to the preceding case of austenite, but with the Ni-Ni pairs the situation is quite different: The change of the interatomic forces of the (Ni), pairs that are nearer causes an opposite change of the interatomic forces of the (Ni)b pairs that are more distant. This means that the change of elastic energy of the interatomic forces of the (Ni), pairs and (NiX pairs together will be very low, if any. This explains the low Young's modulus of martensite. In the paper by Kafka(1994) nine important points of agreement between the above explained scheme and experimental evidence are discussed in detail. Assumptions of Our Mathematical Model In the above reasoning the interatomic forces and distances between the pairs of the first and the second atomic neighbors were considered to be the basic factors in the shape memory phenomena. Whereas the distances between the first neighbors were considered unchanged in the course of the respective processes, the distances between the second neighbors were assumed to change and play the main role. Let us now clarify that processes of this kind are really possible. In Fig.3 the scheme of a characteristic cell of atomic configuration is visualized for austenitic configuration {(Ni)-(Ti)-Ni-Ti) and for two variants of martensite: {(Ni)-(Ti)-Ni'-Ti') and {(Ni)-(Ti)-Ni"-Ti"J. Transitions among these configurations can really be fulfilled without any change of the distances between the first neighbors Ni-Ti. If these distances are maintained, the areas of the rhombi {(Ni)-(Ti)-Ni'-Ti') or |(Ni)-(Ti)-Ni"-Ti") will be slightly smaller than that of the

160

VI. 1 Shape Memory

square {(Ni)-(Ti)-Ni-Ti), but this agrees with the fact that martensitic configurations correspond to a lower temperature. If a mechanical loading causes transition from martensitic configuration {(Ni)-(Ti)-Ni'-Ti'J to martensitic configuration {(Ni)-(Ti)-Ni"-Ti"}, the work of

M

1r i

'i

»'

\ \

> /

H \

,/l«1

\

\ \

/ /

ki

(N

\ \

—

I

Fig.3 Scheme of the atomic cell in austenitic configuration and in two variants of martensitic configurations external forces must equal the work of interatomic forces among the second neighbors, i.e. the work realized by changes of distances (Ni)-Ni' and (Ti)-Ti' to (Ni)-Ni" and (Ti)-Ti". The external forces must be in equilibrium with the interatomic forces caused by these changes of distances. According to the preceding subsection the changes of distances among the Ni-Ni atoms are connected with dissipation of energy, whereas the changes of distances among the Ti-Ti atoms are not, they preserve the character of elastic changes. The processes of pseudoelasticity and of reverse transformation can similarly be presented as processes without any change of the distances among first atomic neighbors.

VI. 1 Shape Memory

161

These considerations inspired us to model the SM-processes in binary alloys on the basis of the following assumptions: (i) The binary SM alloy is modeled as a two-phase heterogeneous continuum with one substructure representing the set of the Ti-atoms, the other substructure representing the set of the Ni-atoms. (ii) The Ti-substructure is assumed elastic, the Ni-substructure is assumed elasticplastic. (Hi) In this model description the elastic moduli are described as homogeneous and constant, not differing in the two substructures, having the macroscopic values that correspond to the state in which the described process begins. The two substructures differ only in their inelastic properties. It is obvious that the last assumption that neglects the changes of macroscopic elastic moduli with temperature and the differences in the elastic moduli between the Ti-set and the Ni-set, differs from reality. But the main object of the model description are the macroscopic inelastic deformations that are of a higher order than the elastic ones, and the main mechanism that is seen as responsible for these inelastic deformations are the different properties of the Ti-Ti pairs and the Ni-Ni pairs. These are in our model taken into account. The cases, where elastic changes are neglected, are usually modeled by a rigid-plastic scheme, but our assumptions are slightly nearer to reality. The differences that arise from our simplified assumption of elastic homogeneity are corrected by determining some of the model parameters by fitting the respective theoretical curves to the experimental ones. Our preceding works showed that the model based on these simple assumptions was able to describe the main SM phenomena satisfactorily - even for complex loading paths. General Form of the Mathematical Model On the basis of the assumptions presented above the respective mathematical formulae are - according to Section n.2 - as follows: vn = ^

(1)

"nty + " r e | r = c #

(2)

where vn, v, are volume fractions of the material constituents Ni and Ti. The constitutive equations of the two material constituents are expressed as follows:

de^,= /idSfc+auCMn,

En = pan = pff = e

(3)

162

VI. 1 Shape Memory

de^=de^-de? ,

e'n

de^,=/idSj n +Sj n dA n ,

e'n=0

de^ = n dsm

e, = pa,

,

(4)

=£n-e

(5) = pa = e

(6)

66^ = d e ^ - d e ^ ,

e, = et-e~

(7)

de'IJt=fi6sl

e't=0

(8)

,

where e$n / Sj„ [ 5ije„ / 5ijO"„] are the deviatoric [ isotropic ] parts of the average strain/stress tensors in the n-material constituent and similarly with the {-constituent, and the macroscopic values are indicated again by overbars. The symbols /z[=(1+v)/£] and p[=(1-2v)/E] mean elastic compliances, v meaning Poisson's ratio and E Young's modulus. The symbols with primes - defined by Eqs.(4) and (7) - characterize the influence of the heterogeneity of strain- and stress-fields. Equations (1) to (8) represent a special case of general Eqs. (n.2.1) to (II.2.8) and if only mechanical loading is taken into account, the respective form of Eq.(TJ.2.9i) for deviatoric parts reads:

S

S"

s

ijt

+

—

—

~

u

•

The analogous equation for isotropic parts (n.2.92) has no meaning in the case of our special assumptions (1) to (8). In our SM analysis we consider apart from the mechanical loading effect also a thermal effect and therefore, instead of the above relation it is necessary to write: (S#„)M " ( 5 / ^ + ^ - ^

In

= 0

(9)

Vt

where ( s ^ ^ , (S^)M mean the values resulting from mechanical loading. In our concept, the effect of temperature changes is connected with the differences in thermal expansions of the two elements present in the binary alloy. Due to temperature changes and the difference in thermal dilatation of the two substructures there appear self-equilibrated stresses that are labeled by subscript T: V/,(^ n > T +V f (
(10)

VI. 1 Shape Memory

163

and the total stresses are: a

ljn =(°ijn)M + (T.

+ (Vijth ■

(H)

The self-equilibrated stresses (cr^h" > (aijthhave the"- isotropic parts and their deviatoric parts. The primary effect of uneven dilatations are isotropic parts, but it turns out that the isotropic deformations that result from them (changes of volume) are unimportant What is important is their indirect secondary effect on the deviatoric stresses and the resulting deviatoric deformations. (To clarify this point let us imagine a ball with elastic skin, filled with a plastic material with higher thermal dilatation. If the ball with spherical shape is deformed at lower temperature and then heated, its volume change will be unimportant, but its deviatoric reverse deformation to the spherical shape will be significant.) Therefore, in our analysis we will work only with deviatoric stresses and deformations. Due to temperature changes, it was necessary to introduce new variables and it is necessary to find new equations for the set of equations to be complete and solvable. For deviatoric stresses (s # n ) T , {s^)j, one equation can easily be obtained from Eq.(lO), if it is rewritten for the respective deviatoric parts. For finding the other necessary equation, let us mention that according to Eqs.(3) and (11) the deviatoric stress component (Syn)j causes the deviatoric strains by which the shape memory effect is realized. General experimental experience says that these SM strains do appear if and only if two factors are present: (i) If temperature is increased above some specific temperature T0, i.e. z =

T-To>0.

(ii) If the current state of the material was preceded by mechanical inelastic deformation. In the extent of deformations where the SM effect is observed, macroscopic inelastic deformation implies inelastic deformation of the n-substructure, whereas the /-substructure - that remains elastic throughout the process under study - is the memory-carrier. Inelastic deformation of the n-substructure alone means that the deformations of the n-substructure and that of the f-substructure do differ (as the elastic parameters are assumed to be identical). The measure of the difference in these deformations is S^, as according to Eqs.(4) and (8):

VI. 1 Shape Memory

164

d s ^ = - ( d % - d e # ) = -^(de0-de ?n ) .

(12)

Hence, it is straightforward to connect the value of (S^fr with rand S^ in such a way that if the value of any of the two last named quantities is vanishing, the value of ( S ^ ^ will also be vanishing. The simplest form with this property is: (Sijn)J=
(13)

or - in a differential form: d(S|n)T = ©rdSfl + as^ d r .

(14)

Whereas the integral form (13) can be expected to hold only in some neighborhood of the 'virgin state', the state with zero value of (S^)T, the differential form (14) is more general. It is used in what follows, as we want to describe also physically different processes of heating and cooling with and without inelastic deformation, where (0 can take on different values. From the above set of equations the macroscopic constitutive equation with the respective evolution equations for internal variables is derived in the following form: 6ef=n689+vn8t„dkn dsijn =dsv - " '

r

, "

de=pdff

(15)

' * . ''~T, q+vn(orxlvt

*

(16)

<*s'ijn = <*sijn -dSij +(vtsSn-s~n)dXn/iJ.

(17)

d^r = ^ [ ( d ^ - ds^)/ vt - siJn dk„ In)

(18)

where P = Wn + vti)t,

q =vnt]n + vtT)t +r/ n t) t ,

r = T)ni)t .

(19)

For the inelastic deformation of the n-substructure, we assume validity of Mises' criterion valid for average stresses, which leads to: dA„=0

for

s#,s|h
(20)

VI. 1 Shape Memory dX„= l ( d ^ „ + | d ^ | )

for

165

S^S^fc*

with dX„ following from the above equations in the form: dXn=n——-

^-^—*

»" *

.

(21)

(vtp+vn(orz)silnsijn-vtr]tsljnsHn

The above set of equations means that the value of s^„ S^, is limited. If the limit c„ is reached, deviatoric deformation of the n-substructure can increase without any increase of s^„ s^„ . Similar constraint is assumed to be valid for the difference between the deviatoric deformation of the two substructures and the respective quantity s^s^ that is connected with it by Eq.(12), and the change of (S#/))T(S/J»I)T-

Up to some limit value of s'^s'^ this difference increases with increasing (s^n)T(S/yn)T, from this limit the difference of deformations and the value of s'jjt s'n increase without further change of (S^, )j (SJ„)T • In mathematical terms this reads: co = &(>0) ffl = 0

for for

s'ijts'ijl<\c'2 s^4>|c'2.

(22) (23)

The above set of Eqs. (15) to (21) has some important properties: • The inelastic part of macroscopic deformation (the second addend on the righthand side of Eq.(15i)) is connected with the non-zero value of dA„. This non zero value of dA„ can be caused - according to Eq.(21) - either by the change of macroscopic stress dSy, or by the change of temperature d r . The first named case describes processes of mere inelastic deformation accompanied by detwinning for T vanishing or low, and pseudoelastic processes for higher values of r . The last named case (dr > 0 ) is the case of shape memory in loading-free conditions. • It is clear from Eqs.(16) to (18) that internal stresses - that describe the state of the material - can change not only in cases of changing macroscopic stress or/and inelastic deformation, but also without it due to the change of temperature only. Such process appears e.g. if dsjy= 0 and dA„=0, but d r * 0 . This describes

166

VI.2 Pseudoelasticity

changes of the state of the material, of its internal stresses, without any inelastic deformation. Concluding Subsection VI. 1 let us summarize that a three-dimensional mathematical model with tensorial latent variables was formulated on the basis of qualitative knowledge resulting from heterogeneity on the atomic scale, and on the basis of a number of simplifying assumptions. This model is relatively simple, and it is able to describe the basic SM phenomena: pseudoelasticity, one-way shape memory after uniaxial and complex loading paths, and two-way shape memory. The parameters of the model are to be determined from macroscopic experiments. In what follows some basic applications will be discussed in detail.

VI.2 Pseudoelasticity In the case of pseudoelasticity, temperature corresponds to an austenitic state of the material and it remains constant in the course of the loading and unloading process. Hence, in the set of equations (VI. 1.14) to (VI. 1.21) it is necessary to assume i » 0 , dr= 0. The loading in question will be uniaxial tension in the ^/-direction, and with the above restriction it follows from Eqs. (VI.1.14)and(VI.1.18): d(Sn„)T =a>Tdsf1f= (oxvn[(6^-6^n)lvt-^n6Xnlit]

.

(1)

Equations (VI.I.16) and (VI.I.21) can now be rewritten in the following forms: PSim-ASi'v, d s ^ d S n - v , " - " " . - " " dA„

(2)

where e=—*

(4)

p=vnn„ + vte

(5)

q = p+t]„e.

(6)

VI.2 Pseudoelasticity

167

For a = 0 or x = 0 it clearly holds: 9=rjt, p-p, q = q. The set of equations (VI.1.15) to (VI. 1.23) with the special forms presented above describes isothermal stress-strain diagrams in martensitic as well as austenitic states. The material parameters remain unchanged, only the value of T differs. In Fig. 1 the possibility of describing different stress-strain diagrams for different temperatures is presented in comparison with experimental data received by Miyazaki, Otsuka and Suzuki (1981). The parameters of our mathematical model in this case are: E= 7.4xl0 4 MPa, H = 1.797xl0~5 MPa"1, p = 0.4594xlO"5MPa"', V, = 0.61 , v„ = 0.39 , C„= 50 MPa, rj, = 139.39 , T]n = 373.54, & = 0.01882 K"1, T0 = 212 K, c'= 243.32MPa. For a better understanding let us comment on individual segments of the stress-strain diagrams in Fig.l: - Segments OP and RS : dAn = 0 , elastic processes, ds, 1 n =015^ = d^n. -Segments PQ and UV : d A n > 0 , co-d> , | s 1 1 n | = c „ , inelastic processes with changing value of (s 11n )r , the value of S^n constant and positive at PQ and negative at

UV.

-Segments QR and

SU : dA„>0, | S n „ | = c n , fi) = 0 ,

inelastic processes with constant values of s 11n and

(SII/))T<

d(s11n)T=0, Sun positive at

QR and negative at SU. The limiting points P, Q, S, L/ are specified as follows: - Point P : &|1n has reached the value c„ . - Point Q: s^, has fallen from the zero value at the beginning to the value - c'. - Point S : s, 1n has fallen to the value - c„. - Point U: $'AV has reached the value - c'. - Point R is the start of unloading, chosen at the end of martensitic transformation in the extent of deformation in which the pseudoelastic effect can be observed, i.e. in the extent that can be realized without violation of the elastic bonds between the Ti-Ti atomic pairs.

VI.2 Pseudoelasticity

168

T~ 232.5K T-223,7 K

tlPa

tlPa • etp. •

200 100

&„

300

Jo P

0 V'l

£3.

-T

200-

"1

100

2 3

JF

■»—I

V I

4 -/.

' exp.

1

2

1

1

3

I—m.

4- 5 ' / .

T.

263,4K

f

•

tlPa

T-251K

500 WO 300200 >exp. T 0\V !

2

3 4

—-A

ft-^-l\ U ^

too 5

"J - 0

6'A

V } 2 3 4 5 £ V.

Fig. 1 Theoretical pseudoelastic stress-strain diagrams for different temperatures compared with experimental data taken from Miyazaki, Otsuka and Suzuki (1981)

Let us point out that at points P in Fig.l the theoretical diagrams display breaks, whereas the experimental diagrams of Miyazaki, Otsuka and Suzuki (1981) do not However, it is worth mentioning that in another paper by Miyazaki and Otsuka (1986) such breaks were reported.

VI.2 Pseudoelasticity

169

Determination of the Material Parameters For determination of the material parameters it is necessary to have pseudoelastic experimental stress-strain diagrams for two different temperatures. a) Determination of elastic constants E, n and p does not need comments, according to the assumptions of our model these constants are determined from the properties of the material at the beginning of the process under study. b) The volume fractions v„ and vt are determined from the known composition of the alloy and the relations valid among atomic fractions, mass fractions, atomic weights and volume fractions. c) The value of c„ can be determined from the unloading segment RS in Fig.l: C„-i[(s,i) f l -(S 1 1 )s] = i[(5ii)fl-(a r ii)s].

(7)

d) The value of r\t can be determined from the expression for the gradient (ds11 /den)^ _>„ of the curve describing the segment QR. From Eqs.(VI.1.17) and (VI.1.21) it is possible to deduce that for s^, eji approaching infinity, s,'1n approaches the following value: (Si'mh^e,,-*-. =

cn

v +r

(8)

n lt and if using this expression in Eqs.(VI. 1.150 and (6) it is derived: 'dsi^

{^k,A,~

v,

nV+it)

(9)

We identify approximately the gradient of the segment QR at point R with the asymptotic value given by Eq.(12) and from the measured gradient at R~ and from Eq.(12), fy can easily be evaluated. e) For determination of r\n we will use the expression for (ds^/de^) in the case that sj 1n is vanishing. From Eqs.(VI.1.17) and (6) we get: ^ l } ,

d5

l 1 Js,'1n=0

Vt(Vn1n + VtTlt)

(10)

P (Wn + mt + Vrflrint) '

If we know from experiment the respective gradient, r]n can be calculated from Eq.(13). This gradient can exactly be measured if we have the experimental

170

V1.2 Pseudoelasticity

stress-strain diagram for T - 7 0 . If such diagram is not available, we can use approximately the gradient at the beginning of the segment QR, i.e. at point CT . Here, the value of S)'1n is not quite vanishing, but the differences in the gradients are - according to the experimental results presented by Miyazaki, Otsuka and Suzuki (1981) - not substantial. f) For determination of ti), T0 and c' one experimental stress-strain diagram is not sufficient - as it was in the preceding cases, it is necessary to have some information also from another stress-strain diagram corresponding to another temperature. Atfirstlet us define and determine the value of §1=fflT1=ffl(71-70) .

(11)

For this purpose, we will use the gradient of the segment PQ at point Q~ measured from the stress-strain diagram at temperature 71. Similarly as in the case of the gradient at FT, we will assume that it is near to the asymptotic value corresponding to § l1 and e !1 growing without limits. Similarly again as in the preceding case it is easy to deduce the expression for the limit value of the gradient that is analogous to expression (12) where r/f must be replaced by 9: fM-F

dgii

\ /r=7"i.5ii.eii-»~

M(1+*i)

(12)

The value of
(13)

VI.2 Pseudoelasticity

171

Similarly as with plotting the segments OP and PQ for Tj, we proceed with plotting these segments for T2 . For T2 >T^ the segment OP will be steeper. We do not know the value %2=&r2=d>(T2-T0)

(14)

a priori and therefore, we will try several values of £2 (higher than ^) and the plotting will always be stopped when s^f reaches the value —ti. The respective theoretical value of ffn will always be compared with the experimental value (^'n)oxp and toe right value of | 2 will be that for which these values of o ^ coincide. With ^ and £2 known it is finally received: T0^}-^

(15)

Vl ~ S2

<5> = T ^ M

-

r '0

(16)

which completes solution to the identification problem. Incomplete Pseudoelastic Transformations in Binary Alloys In Fig.l the unloading at points R started when the stress-induced martensitic transformation was complete. In real applications of SM materials the transformation is often incomplete, and therefore, the behavior in such circumstances is important. In our paper - Vokoun and Kafka (1996) - our theoretical model was applied to incomplete transformations, unloading and reloading of NiTi alloy. Without going into detail, the respective theoretical diagrams are demonstrated in Fig.2. It was shown that such prediction of the courses of the diagrams agree with experimental data reported by Tanaka, Nishimura and Tobushi (1995). Let us mention that ternary alloys can display different behavior.

172

VI.3 One-Way Shape Memory

- rvr

JOO-i 250200-

1 2

-- 1 LL ^ -

150.

H ft

'C 100 -

50 ■

0- ' C1

/ i

1

2

l

l

3 e, ("•'•)

4

5

e

Fig.2 Computed pseudoelastic stress-strain diagrams in an incomplete transformation process, unloading and reloading

VI.3 One-way Shape Memory Effect One-way shape memory effect is realized by deformation of the sample at a low temperature ('predeformation'), and by heating it after this predeformation and unloading. Due to this heating the sample returns to its original shape. On the atomic scale, our qualitative explanation of this process is given in Section VI. 1.1. The respective mathematical description results from the general form of our model given in Section VI.1.3. The first phase of the process - mechanical deformation at low constant temperature - is described by a special form of Eqs. (VI.1.15) to (VI.1.21). In the ideal case, when T = 0 , corresponding to perfect martensitic state, the basic

VI.3 One-Way Shape Memory

173

Eqs.(VI.1.16) and (VI.1.21) acquire the following form for uniaxial loading in the x, -direction:

ds 1 1 n =ds 1 1 -^ p s ' 1 "- T ? f ^"dA n nq nq

dXn =

(1) (2)

d*n

^(P«l1n-»7fS|l«)

where p and q are defined by Eqs.(VI.1.19). Together with Eqs.(VI.1.15) and (VI. 1.17) to (VI. 1.20) our special variant of the model is complete. It is assumed that the extent of the predeformation does not exceed the range in which it can be completely recovered, i.e. the range in which the elastic bonds of the Ti-Ti atomic pairs are maintained and no real plastic deformation occurs. After unloading, there remain in the material residual microscopic stresses that are described in our mesomechanical analysis by the following tensorial quantities:

Si'm 0

0

•\

o-K,„ ° 0

«Mf =

0 ^

O-^n

0

o o-^„J

0

0-;Kl, 0 o o -isf,,

0-isfJ

0

r«Mf

0

0

0--H1, o o o -Wv

K

(3)

Hi o--K1n o 0

(4)

0 -is,r1n

where the last relation follows from Eq.(VI.1.9), as in the above loading process (Sfah = S#T7 and (sijt)M = Sp. It is clear from the symmetry of the above matrices that it is sufficient to work only with s[ 1n or sf1f, and ^ n or s,",. The remaining components follow from these values in a straightforward way. The following inequalities result from our formulae:

174

VI.3 One-Way Shape Memory

- c n < s [ 1 n < 0 , s f 1 t > 0 , sfi„>0. s,'^<0 .

(5)

The inequalities (5) together with Eqs. (3) and (4) mean that

sfi„=o => <1f = o, s;;n = o, sft f =o.

(6)

Now let us turn our attention to the process of heating without macroscopic stress. The respective special forms of Eqs. (VI. 1.16) and (VI. 1.21) will be:

q+vna>rT/vt dX

^ 1

^.

(8)

Together with Eqs.(VI.1.15) and (VI. 1.17) to (VI. 1.20) the course of recovery of strain with increasing temperature can easily be described. It is possible to prove that our model describes really the process of recovery that can be - in the extreme case - nearly complete. The maximum recovery of strain is achieved if s[i„ = -C„. In this case, recovery begins immediately with the start of heating and in the whole course of the process it holds: s,i„ = - c „ . The process starts from the state in which (s, 1 n ) r =0and therefore, it is possible to use the integral form (VI.I.13). From Eqs.(VI.I.9), (VI.I.ll) and (VI.I.13) it is derived: S,'ln =^(Sl'lf/^+C n /V f +ft)TS,' 1f //,)

(9)

and from Eqs.(9), (8), (VI.I.15) and (VI.I.18) it can be deduced:

_^a5a.d^=dsf1,=-ld5i1 d^-7

-"*«">»** (vtp+r + vna>rT)cn-nn(vt + T}t
(10) x

V1.3 One- Way Shape Memory

175

It is possible to prove that according to the differential equation (11), s(if approaches zero value withr increasing without limits (Kafka 1990). (For this purpose it is suitable to use substitutions £ = S,'1f - Vn cn , £ = T + Vt l{(Ot]t).) Let us denote the change of S,'1f due to heating As,'1f and the respective change of f^ be denoted Ae^ . From the above relations we receive: As,'n=-S^ ,

Ae^/iS^.

(12)

Now let us investigate the relation between Ae, 1 and e^ - the residual value after unloading from the preceding isothermal process - from the predeformation. Let the starting point of the predeformation be denoted by 0, and the end point after unloading by R. Then it holds according to the relations of Section VLI: sTir = J—(denf-de,,)

(13)

0 "

J"rJe,i=e;ri o R " " u Jden, =[/*dS| 1f = f V -?-(dsu-vndsUn) 0

0

0

t

B

Jd^i=o 0 f?

|ds, 1 n =s[ 1 n . o Hence, it is received: Ae,, = nsfr, = -e(, -Pty^n.

(14)

Eq.(14) says that the maximum strain recovery will be - in its absolute value - smaller than the residual strain after the predeformation. However, this difference is negligible, as the order of /i is 10s MPa'1, the order of (vn/vt) is

176

VI.3 One-Way Shape Memory

1, the order of s[1n is 10 MPa and therefore, the order of (ju s(1n vn I vt) is 10^ , whereas the order of e^ is 10"2. It is possible to use this model not only for a unidirectional loading and deformation, but also for a complex loading (Kafka 1999a). In this paper, it was shown that in accordance with experimental findings our model predicts the following features of the shape memory phenomenon under complex loading consisting of tension and twist: a) Anticipatory extension and torsion performed in arbitrary succession leads always to contraction and detwist owing to subsequent heating. P) Recovery of strain proceeds in the way of simultaneous contraction and detwist.

7) The path of the recovery of strain is independent of the path of the anticipatory deformation and does not copy this path. S) After complex anticipatory deformation at constant temperature the recovery of strain follows the shortest path in the space of deformations, this shortest path being straight line. Numerical Example of a Unidirectional One-Way Shape Memory Let an application of the above discussed model be demonstrated by comparing it with the results of experimental investigation of a TiNi wire (Vokoun and Kafka 1999). Composition of the material was 49.95 at % Ti, 50.05 at % Ni. The wire with circular cross-section of diameter 1 mm was annealed at temperature 883 K for 20 minutes. After annealing, it was water quenched at room temperature. Transformation temperatures determined by a DuPond DSC calorimeter were Ms=280K, Mf = 286 K, As=310K and Af=318K. A specimen of a gauge length 107 mm was cut from the wire. The specimen was predeformed by tensile strain up to 6.2%. After unloading the residual strain was 5.7%. The respective stress-strain diagram is shown in Fig.l. Then the specimen was heated above Af, which led to recovery of the original length. The respective straintemperature diagram is shown in Fig.2. The material parameters were determined by a curve-fitting procedure according to Section m. 1. l.a, and had the following values: v„ = 0.39, v,= 0.61, T/„ = 3.68, t], = 30.54, C„ = 45.31, ju = 5.31xl05 MPa1 , p = 1.44x10-* MPa"1 , 0), = 0.00083K1 , Oh = IK 1 ,

r 0 =52°C

VI.3 One-Way Shape Memory

177

where a>i is the value corresponding to a process without inelastic deformation (segment M-N in Fig.4), and (O2 the value corresponding to a process with inelastic deformation (segment N-P in Fig.4).

150 120 To" | 90

I 60 30 0 C

2

4

6

Strain (%]

Fig.l Experimental stress-stress diagram of predeformation

Fig.2 Experimental strain-temperature diagram heating after predeformation

178

VI.3 One-Way Shape Memory

150 120 75" ^ 90

- _ _ ^ _ ^ - — -

1 * ° ■f^^ 30 0

I C

1

2

3

4

5

Strain [%]

Fig.3 Theoretical stress-strain diagram compared with experimental points

6 M •

♦

♦ ♦

»

„4 c

I

W

*

2

\^

P

• "", #

0 3

20

40

60

» 80

Temperature fC]

Fig.4 Theoretical strain-temperature diagram compared with experimental points It is seen from Fig.4 that our theoretical model is able to describe the course of the one-way shape memory effect, especially that therecoveryof strain is predicted very closely.

VI.4 Two-Way Shape Memory Effect

179

VI.4 Two-way Shape Memory Effect For practical applications the two-way shape memory effect (TWSME) is at least as important as the one-way effect. It consists in repeated changes of form due to repeated changes of temperature, and is achieved by a special procedure called training. There exist different ways of training, which are all understood in our concept as transformations of self-equilibrated microstresses that are originally randomly orientated, and training transforms them into microstresses with preferred orientation - given by the direction of training. One such experimental procedure of training was shown in (Vokoun and Kafka 1998) and in another closely related theoretical paper (Kafka and Vokoun 1998) a qualitative demonstration of application af our model to training was demonstrated. Up to now we have not been able to describe the process of training in a quantitative way, as a quantitative description of the randomly orientated microstresses is very difficult, and on top of that training leads to a partial relaxation of microstresses. But from our qualitative description the following theoretical conclusions were arrived at (Kafka and Vokoun 1998) that agree with experimental findings: • The effect ofuniaxial tensile hading upon the creation of TWSME is inefficient, but not vanishing. • The effect of our training process, consisting in cycles of repeated loading-free heating and cooling under load, is very efficient. • Every cycle in the modeled training process leads to an increase of the TWSME, every subsequent cycle contributes to the TWSME less than the preceding cycle. • In the process of training, the energy of the randomly orientated self-equilibrated microstresses decreases, the energy of the microstresses with one preferred orientation increases. On the other hand, it is not a problem to quantitatively describe the properties )f the already trained material. Let us base our model on the experimental results )resented in (Vokoun and Kafka 1998). Here we will use an alternative variant of the nodel, in which the structural parameters r/nand rjt are assumed equal. A reason for iuch a supposition can be seen in the fact that the configurations of the sets of atoms

VI.4 Two-Way Shape Memory Effect

180

of Ti and Ni in the alloy are similar. Another assumption in this variant will be that the room temperature, in which the experiments were performed, does not correspond to perfectly martensitic state, and then the stress-strain diagram is described similarly as the segments OPQR in Fig.(VI.2.1). The respective stress-strain diagram with experimental points and with unloading is demonstrated in Fig.l.

Stress-Strain Diagram 140

0.01

0.03 0.04 Eng. Strain

0.05

0.06

Fig.l Theoretical stress-strain diagram with the assumption of equal structural parameters

0.07

VL4 Two-Way Shape Memory Effect

181

The investigated material is the same as in Section VI.3, but with our new assumption that the parameters of our mathematical model were now determined by a curve-fitting procedure in the following values: v„ = 0.39, v,= 0.61, 71„ = Vi = 42.3764, C„ = 45.3112, /l = 5.31xlO"5MPa"1, p = 1.44xlO"6MPa:1. After unloading the sample was heated up to 75°C and the respective straintemperature diagram is demonstrated in Fig.2. Strain-Temperature Diagram - Heating 0.06

40

50 60 Temperature T("C)

70

90

Fig.2 Strain-temperature diagram heating of the untrained material after preloading by tension

For receiving a good agreement with experimental data the following parameters had to be used in the model:

VI.4 Two-Way Shape Memory Effect

182

7 0 =22.6°C for 7"<52°C r 0 =52°C for 7 > 5 2 ° C ft) = 1 in the range of only thermoelastic process (O = 0.001 in the range of inelastic process for 7"< 52 °C a) = 0.1 in the range of inelastic process for T> 52 °C. Then, training began by cooling under tensile load 90 MPa. The first cooling process under tension is demonstrated in Fig.3. Strain-Temperature Diagram - Cooling 0.06

1

'

l<

*

0.05-

0.04--

.. *

\*

>

-

* exp. points - heating | 0 . 0 3 r-

0.02

>

° exp. points - cooling

- <J

0.01 -

*

-

*

-

*

o\ ^-~a_

.

o O

20

30

-

40

,

50 Temperature T(°C)

|

0

60

70

80

Fig.3 Strain-temperature diagram - first cooling of the untrained material under tensile load 90MPa after preloading and heating In this cooling process, for the model description the following parameters were used:

VI.4 Two-Way Shape Memory Effect

183

r 0 =22.6°C (0 = 1 in the range of only thermoelastic process co = 0.1 in the range of inelastic process. After this start, 60 cycles of training followed that changed the properties of the material, and for the description of the properties of the trained material some of the parameters in the model description had to be changed. The strain-temperature diagram of the trained material is shown in Fig.4. Trained Material - Heating and Cooling 0.055

o.os •'

* exp. points - heating

0.045

o exp. points - cooling

0.04

— theor. curve - heating - theor. curve - cooling

0.035 c |

55

0.03 0.025 0.02 0.015 0.01 0.005

-o 10

20

o— 30 40 50 Temperature T(°C)

60

70

80

Fig.4 Strain-temperature diagrams (heating and cooling) of the trained material The parameters of the trained material were determined in the values: 7"n=10.6°C for 7<40 °C

184

VI.5 Discussion of Our Model for Shape Memory

7"0=40°C for 7 > 4 0 °C for the segment of cooling: 0) = 100 in the range of the only thermoelastic process (0 = 0.5 in the range of the inelastic process for the segment of heating: (0 = 0.0054 in the range of the only thermoelastic process m = 0.004 in the range of the inelastic process for T< 40 °C © = 0.1 in the range of the inelastic process for T > 40 ° C. The process of heating is stopped at temperature 75°C, the deformation caused by cooling is assumed to be finished when the value of s,'1( falls to (Si'ifJo = -818.35 MPa, which is the value at the beginning of heating of the trained material. This is about 85% of the value prior to training. The 15% decrease is assumed to be due to relaxation of microstresses.

VI.5 Discussion of Our Model for Shape Memory The presented use of our general mesomechanical model for creation of constitutive equations of binary shape memory materials verifies the general possibility of such an application. The essential idea of this application consists in the demonstration that the shape memory effect connected with martensitic transformation can be explained on the same basis as the shape memory effect of elastic bodies that return to their original shape after unloading. The common mechanical factor of both these phenomena can be seen in the existence of an overall elasticity of the material, or of a continuous elastic substructure existing in the material. In the case of binary shape memory alloys this continuous elastic substructure exists on the atomic scale and corresponds to the substructure of the set of atoms of one of the present elements, whose interatomic bonds remain elastic in the extent of diffusionless shape memory processes. This explains also why the shape memory phenomena are characteristic for materials that are heterogeneous on the atomic scale. In our model we work only with deviatoric variables, the influence of the isotropic part of the stress tensor is not taken into account According to Gall and Sehitoglu (1999) the difference between the behavior of NiTi in tension and in

VI.5 Discussion of Our Modelfor Shape Memory

185

compression is negligible if the material is without texture, which corroborates this approach. The model presented above is the simplest variant of this general approach and is assumed to be refined for special materials and loading conditions in further work.

Chapter VII Transversely Isotropic Materials The materials that we are going to describe in this chapter are assumed to be composed of isotropic material constituents, but due to the geometry of their composition they are anisotropic - even in their virgin state defined by zero values of the macroscopic as well as microscopic stresses and strains. Such macroscopic anisotropy must be differentiated from the anisotropy resulting from the material properties of the constituents or from a preceding deformation that resulted in oriented microstresses, microstrains or anisotropic damage. In the case of isotropic materials described in preceding chapters the distribution functions and the respective structural parameters were related separately to deviatoric and isotropic tensorial parts, but this cannot be done in the case of the anisotropy. Here, the distribution functions and structural parameters are related to the total components of the stress- and strain-tensors. This could have been done also in the case of isotropic materials and the resulting model would be simpler, but some important possibilities of the model would be lost. The basic set of equations for our anisotropic model is quite analogous to that of the isotropic case. Equations (H.2.1) and (n.2.2) remain unchanged, instead of Eqs.(TI.l.l) it must be written: e^(X,x,0„ =e„/,(X,0 + ^(X,x,f) n £^ n (X,f)

(I)

< V X , x , O n = < W X , 0 + $(X,x,f) n CT^ n (X,0 (no summation on a,j8, n=< f£p >„=1). The averaging procedure leads to equations that are analogous to (II. 1.4): £

a/)n ~ Safin ~ ^ap

°afln

=

(2)

°afin ~ &afin •

186

VH. Transversely Isotropic Model

187

The fundamental theorem of our approach (cf. Section n.l) leads to analogues of Eqs.(n.l.6) - to the model description of the stress- and straindistribution: Bap (X,x,t) n = eap (X,0 + £ (x)„ e'apn(X, f) ^ ( X , x , 0 „ = ^ „ ( X , f ) + %(x)n

(3)

(T^(X,0

(no summation on a, p). The specific stress power in a unit volume of the composite is expressed by the respective analogues of Eq.(II.1.9): a

HtH = L M f f * i f y i + Zijnt'ijn)

(4)

where

U = ^ 1

(5)

= <[^(X,x)„-1][^(X,x) n -1]>„

^ n

(6)

= <^(X,x)n^(X,x)n>n-1. Instead of Eq.(D. 1.20a) it can be quite analogously deduced: °aBn Capn-Cal)e+—

a

aSe — = 0 •

(7)

Equations (H.1.21) and (n.1.24) remain unchanged and thus, the basic set of equations forming the starting point for the user of the model is represented by Eqs.(H.2.1), (H.2.2), (D.1.21), (n.1.24), (2) and (7), i.e. equally as in the isotropic case by4N+l equations for4N+l unknown rates 6^, 6yn, e^n, e~n, 2jy. For the dual model (B-model), stresses and strains must be interchanged in Eqs.(2) and (7) and symbols r/^,,, r/^must be replaced by symbols Xafln< Xafle •

Similarly as in the case of isotropic materials it can be shown that for a two-phase material the following holds:

188

VII. Transversely Isotropic Model 0
0<Xapn<°°.

describes discontinuous infrastructures of both material constituents, r

lapn=00<

°r

0<7?o/Ja
Xafin = °° , 0 < Xape < °°

describes discontinuous infrastructure of the n-constituent and continuous infrastructure of the m-constituents Vapn

or

= 00

Xapn = 0

Vape

= 00

Xafie = 0

describes the homogeneous stress model (Reuss' solution), Vapn = 0

Wapo = 0

or Xafin = 00 1 Xafie = 00 describes the homogeneous strain model (Voigt's solution). Contrary to the case of the isotropic model the structural parameters Vafin' Xapn &£ not independent of the tensorial directions. This complicates the model substantially, and it seems that if we want to remain in the limits of a reasonably simple operative model, it is necessary to restrict the analysis to the case of transverse isotropy. Then - with the Xj-axis taken for the axis of symmetry - Eq.(7) can be rewritten in the following form (for the respective proof see Appendix Vffl.7): Giin

Can

or - for the B-model:

where

VII.l Transversely Isotropic Two-Phase Model

bn]=Vn.[Xn]

= Xn

for

ij = 11

[in] =nn\x„\

= Xn

hn\=nn\Xn\

= X®n for 0 = 12,13

189

(10)

for ij = 22,23,33

and equally for index e.

VII.l Transversely Isotropic Two-Phase Model Similarly as in the case of isotropic models, special attention is paid to the case of a two-phase model. Equations (E.2.1), (n.2.2), (n.2.4), (n.2.7), (H.2.6) and (n.2.8) remain unchanged, Eqs.(H.2.3) and (H.2.5) are simplified by assuming pe = 0, coe = 0, /7e = 0 , instead of Eq.(H.2.9) we use Eq.(VH.8). In the case of the dual B-model Eqs.(n.2.4), (H.2.7) and (VH.8) must be replaced by Eqs.(n.2.30), (H.2.31) and (Vn.9). To formulate the macroscopic constitutive equation of the A-model we deduce from Eqs.(n.2.2), (H.2.6), (H.2.3) and (H.2.1): *# = Me Sj, + Vn(Hn-He)Sijn + V„fy,/>„ +Sf[pe&+ + vna„ pn+{veae

+

Vn(pn - pg)an

vnan)t+vnti)n].

To arrive at the final form it is necessary to replace the expressions Sjj,,, an by terms comprising the macro-stress rates. To this purpose we deduce at first formulae for d ^ , 6'^ in terms of <7,y, <7,yn using Eqs.(II.2.8), (n.2.4), (H.2.6), (D.2.2), (H.2.3), (H.2.1) and (H.2.5):

tf

*"lf.(<J' e -5..^

ljn)

He re fnPn + (an-«B)T

' "

Pe

He *» "

Me + (i)n Mn

Me

( 2 )

Pn ,&

Pe

j

190

VII. 1 Transversely Isotropic Two-Phase Model

6'dn =

["/*•<*#

+(VeHn+VnHe)<>ijn+(VeSijn-S~n)hn]

M

,

" ^e^n-OPn

s

. + Ve(an-CCa)T+Ve(i)n

"

Pn

(3) ^fl, Mn

pe)^

y

tf

^

Pn

In relation to the inelastic deformations that are the main phenomenon to be described the influence of the differing elastic Poisson's ratios is often negligible and therefore, in what follows only the simpler formulae for equal Poisson's ratios are given. It is assumed: Ve=v n =v

(4)

and with this equality the last addends on the right-hand sides of Eqs.(2) and (3) are vanishing. These two equations, modified in this way, are then used in Eq.(VII.8), which leads to:

*#,=[M]*#-([/V]s,-[Ar]4I)/iII - 5 * ~ K[NK ~lN'K) Pn +M(an -ag)t+[N]
(5)

Pe

with [M] = [Me Mn MlVnh

^Ve

Me [*.] + *„ M„ [*„])]

/[«]

lN]=v^r=*•
= VeMe[Tle]'[R]

(6)

(7) (8)

[ " ] = Me Mn [Ve][Vn] + (Ve Mn + *n Me)(Ve Me [»?.] + Vn Mn [Vn])

(9)

where [M] = Mhe,Tin)

for o0 = 11

[M] = «•(!/?,Vn) for afi = 22,23,33 [ M ] = A^(IJ?,IJ*)

for 00 = 12,13

and analogous meanings have the symbols [/V], [W] and [/?].

(10)

VII.l Transversely Isotropic Two-Phase

Model

191

In the case of the B-model Eq.(l) remains unchanged, and from Eqs.(H.2.8), (D.2.30), (H.2.31), (H.2.5) and (D.2.1) it is received: **! =»n(Sijn -Sij) + (SiJn -Sij)hn+Sij[pn{an ^^^-[MeCSij

-&) + (
~Sijn) + dijpe(&-an)]

(11) (12)

and from Eqs.(9), (D.2.2), (n.2.6), (n.2.3), (H.2.1), (11) and (12) we finally get: S^+5..^ Mo

={p](|..+5..£2.^)_|[Q]{[(1+|[^])S„n

_S ? ]rt„

t^e

(13)

+ ^[(1 + [^ n ])(T f l -a]p„+5,,[(a„-a e )f + „][^„]} with

[P] = [M*.][*«]+V.M*.]+VnM*nJ/['?]

(14)

[Q] = Ve[X.V[R]

(15)

["l = KMn + VnM.)U.]Un] + «'.^n[^] + ^^[Zn] •

(16)

The meaning of the square brackets is analogous to the case of [ A f ] . Due to the assumed equality of Poisson's ratios it holds: Vn^n -He)Sijn + Sf Vn(Pn i

w

e Pe ■ s

pg)6„ (I 7 )

which simplifies the analysis. In this way we have presented the macroscopic constitutive equations for the A-model as well as for the B-model, with tensonal internal variables (7,y„,(7^,and their evolution equations. Similarly as in the case of the isotropic material it is possible to deduce from A-model and B-model their special variants for inclusions and matrix, for homogeneous stress model and homogeneous strain model. Let us clarify these transitions for the case of an elastic isothermal process. From Eqs.(l) and (5) (i.e. for the A-model) we get:

192

VII.l Transversely Isotropic Two-Phase Model

«# = \Me + Vn (Pn ~ Pe)[M]]Sij + 8j[pe + Vn(}ln + % Vn(pn -pe +Hn -He)i[M&„

+MV

2 2

Pe)[M\p

(18)

+^33)1

and from Eqs.(l) and (13) (i.e. for the B-model): ^=[He

+ Vn^n-He)[Ppi+^[Pe

+ Vn(lln-He)^-lP]]&.

(19)

If we denote [P] = p = =1=£"~e22

= jZ

for uniaxial loading

»=.|l=25g-5i-g33 S22

(20)

uniaxial loading a^

2^22

= rr® - ?P- = _2L %3

for

ff„

for uniaxial loading d ^

^23

= ju® = -=2- = -^for uniaxial loading <x12 S,2 a 12 we can easily deduce from the preceding equations: for the A-model:

[p] = Lie + vn(nn-ne)lM] *e Pe [Ve] + Vn Vn [*?n] + ( l fl, +"„ Pn )[%][ln\

=

°

" PePn[>le]M

+ (VePn+VnPe)(VeHe[lle] +

< 21 >

VnPnM)

for the B-model:

[P]=Pe + vn(nn-fig)[P] JVeHe+VnPn)(VePn[Xe]+VnPe[X„])+PePnUe]lX„]

<22)

The above expressions are quite similar to those defined by Eqs.(II.2.11) and (II.2.37) for the isotropic model.

VH.l Transversely Isotropic Two-Phase Model

193

The qualitative difference between the A-model and B-model can be demonstrated by putting ne=<*>, which means that the e-constituent is absolutely compliant For the case of the B-model (discontinuous substructures) this leads to infinite value of [JX\ (corresponding to absolutely compliant composite), whereas for the case of the A-model (continuous substructures) this leads to:

[p] = —^-He

(23)

(i.e. to a finite value of the macroscopic compliance of the composite that depends on the concrete form of the geometry of continuous composition). Due to the assumed equality of Poisson's ratios similar relations follow for the macroscopic Young's moduli for the cases of uniaxial loading either by ffu

or CT22:

for the A-model: E]■

E.E. e *-n

En +

vn{M](Eg-En)

(24)

_ Eg EnMM + JVp Ee+V„En )(veEn{tle]+vn Eg[??„]) {vgEn + vn Ee )[J7e][77n]+ve En [77,,]+vn Ee [r]n] where for loading 9U the meaning of

| E ] , [TJ^], [T7„]

is E, t]g,T]n, for loading

a22 it is E?*,Tif,ii%; for the B-model:

EeEn

: +V [P](E -E ) W=T^. n

n

e

(V0Ee

+v

n

(25)

E

n n)lXo]lX„] + VeEe[xe]+V„Enlx„]

EeEn[Xe]lx„]+(veEn

+ vnEg)(veEelxe]

^

+ vnEn[xn]

£

-e *-n

If one of the moduli - e.g. En - is vanishing, we get from Eq.(24) for the A-model:

ra-iT1*.]

Ee

(26)

194

VH. 1.1 Identification of Parameters from a Flow-Curve

and for the B-model the resulting macroscopic moduli following from Eq.(25) are vanishing. According to our general considerations the model corresponding to inclusions, e.g. of the e-material in the matrix of the n-material, is described as a special case of the A-model with infinite parameters [?7e], or as a special case of the B-model with infinite parameters \xe\ • From Eqs.(24) and (25) we get for such cases:

I i (v.En + vnE.)[r,tt]+veEn [E]JV'f'+;^l)M 1

J

+V

'^En.

(B)

(28)

En[Xn]+ve(veEn + vnEe)

These two expressions are equivalent, as [»7n]i \Xn\ a r e ffee parameters to be determined from experiments and Eq.(28) can be received from Eq.(27) by a simple substitution

similarly as it is the case with Eqs.(n.2.45) to (n.2.53) in the isotropic model. VII.1.1. Solution to the Identification Problem Based on the Flow-Curve Let us assume that the material in question is composed of two material constituents with continuous infrastructure, one of the constituents being purely elastic, the other with elastic response and linear viscosity. The process is assumed to be isothermal. The adequate mathematical model is a special variant of the A-model with hn = 1/(2H„), pn = 0, t = 0,
VII. 1.1 Identification of Parameters from a Flow-Curve

195

Furthermore, let us have at our disposal the flow curves under constant loads (T-n and o22 with measurements of deformation in the longitudinal as well as transverse directions. Let us denote by e^, E£ , e^ the values of deformation that are asymptotically approached with time passing. In the course of the deformation process the deviatoric stresses S^, s,y„ asymptotically approach zero values, and the deformation state approaches an elastic state with /i„ infinite (shear modulus of the n-material vanishing). Then - according to Eq.(VH.1.20)- it results: OO

|_HJ

J I

jjoo _ £11 ~ £ 22 ^11

00

OQ

OQ

j j 3 ~ _ ^ £ 2 2 ~ ^ 1 ~ £ 33 '

25

/jx

22

and using these experimentally determined values in Eq.(VTI.1.23) we have two equations for t]e,Vj® . Hence, together we have four equations for four structural parameters T]g,rif,rin,ri%.

If necessary, the remaining structural parameters

Ve , J?® can be determined quite similarly from a flow-curve underCT12. So far, we have assumed only deviatoric viscosity. But there are cases, in which the n-substructure is formed by interconnected pores with liquid, and under hydrostatic pressure or tension this liquid can leak out or inside. On the mesoscale, this phenomenon can be modeled as volume viscosity. In our model this can be described such that p„ is replaced by a constant - by a coefficient of volume viscosity. If a specimen of such material is loaded by a fixed value of ff:! or o22, the stresses that relax in the asymptotic state are not only deviatoric stresses, but also the respective isotropic components. In this case, we can use formula (VII. 1.26) for determination of r/e, r/f . The advantage is that we do not need the measurements of the transverse deformations as in the case when using Eqs.(VH.1.23) and (VII. 1.20). Such model has meaning among others for biological materials, e.g. for trabecular bone. It was used e.g. by Deligianni, Missirlis and Kafka (1994). In reality, the knowledge of all material constants except for the structural parameters is not very probable, and therefore, we must seek other sources of information. If we introduce the second-order derivatives, we are able to eliminate the internal variables from the macroscopic constitutive equation. To simplify the analysis, only a special case with v e = vn = 0.5 (i.e. pg = pn = 0 ) will be considered in what follows. The resulting rheological equation for uniaxial constant loading a,, (for derivation see Appendix Vm.8) is:

196

Vn.1.1 Identification of Parameters from a Flow-Curve

en + ^ + B e ^ C S n + DSn+Fs,,

(2)

where A = [UnnN-(venn

+ vnne)fr\l(2Hnnn)

B = {N-vefiT)H4H2nnn) C = Ve + Vn{Hn-He)M

(3) (4) (5)

D = [fi. + vtt{2nn-fi0)M+nenn(N-N^/(2Hn^n)

(6)

F = [vnM + ne{N-fr))l(4Hn2nn)

(7)

N = (2N+N*)/3

(8)

N' = (2N' + rf*)/3.

(9)

For a flow curve corresponding to constant stress o^ the derivatives S, S vanish, Eq.(2) simplifies and comprises only three parameters A, B and F. Such differential equation can be identified with the experimental flow-curve, and the values of A, B and F determined by a curve-fitting procedure. The simplest way of doing it is to measure the values of e"n, eii, e"^, S)i in three arbitrarily chosen points, which leads to three linear equations for A, B and F . Supposing that M is known from Eq.(VII.1.24), it is easy to calculate Hn, N and fif . The concrete procedure depends upon what is known a priori. The main object of interest is the value of Hn, which is difficult to determine or estimate in another way. Deligianni, Missirlis and Kafka (1994) used this model for the description of rheological behavior of trabecular bone. For the case of elastic inclusions in viscoelastic matrix the model simplifies substantially. In this case, the structural parameter [rje] is infinite, [r/„] can be determined from macroscopic moduli using Eq.(VII.1.27), Shas zero value, with increasing time if approaches zero value, and t approaches a value that can be determined from Eqs.(2), (3) and (7): {e)t

~-TAj^^-2H;^

(10)

VII.2 Materials with Unidirectional Continuous Fibers

197

with ^=^^n^ln^n+Ve(3veHn+3vnHg-fle)T1n ^2=3^„rinrif

+ vg(3vgnn+2vnng)Tin

+

+ Vgfig^ + ve{3venn + vnne)T]^

(11)

3vt(vg^n+2vn^ig).

Hence, from the asymptotic direction of the flow-curve we can easily determine Hn. In the case of the isotropic material r/„ =77® and Eq.(lO) simplifies in the following way:

i.e. to an expression that can be arrived at from Eq.(TV.7) for deviatoric strain and stress with T]g infinite.

VII.2. Transversely Isotropic Materials with Unidirectional Continuous Fibers A special and important example of transversely isotropic materials is a composite reinforced with unidirectional continuous fibers that are randomly distributed. Similarly as in the preceding paragraphs we assume validity of Eqs.(n.2.1), (D.2.2), (H.2.4), (H2.7), (E.2.6), (H.2.8), (H.2.3) and (D.2.5), but instead of Eq.(VII.8) we write: 5

11e - c 1 1 n

_c

(1)

11

S i J n S ^ ^ - ^ 0

(2)

where M=Vn [ln] =

for tf

^

ij = 22,23,33 # = 12,13.

(3)

198

VH.2 Materials with Unidirectional Continuous Fibers

The meaning of Eq.(l) is clear without comments, Eq.(2) follows from Eq.(VII.8) if [ij e ] is infinite, which corresponds to fibers that represent a special case of inclusions. Equations (VII.1.1) to (VH.1.3) are valid for all indices, as the relations from which they were derived hold true without any limitation. Again we assume For /)' * 11 Eq.(VII.1.5) is also valid, but the definitions of [Af], [A/], [AT] are different due to the infinite value of [77,,] : lM] = (Hn{rinhvgne)]/[R]

(4)

{N] = 4/[R]

(5)

["! = *./[*]

(6)

where [M] = M*(ii%)

for

ij = 22,23,33

(8)

[M] = M8(77®)

for

/y = 12,13

(9)

and similarly for [A/], [AT], [R] . For /y = 11 Eq.(VII.1.5) is not valid, instead Eq.(l) with (H.2.3) and (n.2.6) leads to:

-Ve<7n f>n-Ve(«n

-<*.)?-Vg(On].

This equation can directly be used in Eq.(VII.l.l), as for the assumed equality of Poisson's ratios it holds: Vn (Mn " / l . ) S , 1 n +Vn(P„

-Pe)<*n

= *n (V„ - / * . ) ( « ! 1n + ^-<*n)

This completes the macroscopic constitutive equation.

•

VII.2 Materials with Unidirectional Continuous Fibers

199

Macroscopic Yield Condition of Materials with Unidirectional Continuous Fibers Let the fibers (e-material) be elastic and the matrix (n-material) elasticplastic with the simplest form of the local yield condition (Mises' criterion): SijnSijn <2k2.

(11)

Then it is straightforward to deduce the macroscopic yield condition. Before loading the material is assumed to be without internal stresses, and up to the state in which the limit given by criterion (11) is reached the deformation process is elastic. Therefore, it holds according to Eqs.(lO), (VH.1.5) and (4): Sii„+—
(12) for

/y-22,33,23,12,13

(13)

where [/W] is given by Eq.(4). From Eqs.(12) and (13) all deviatoric components s(y„ can be expressed in terms of a^ and if these expressions are used in criterion (11) it is received: -2 6? + (Z2 -2$ -2)(M*f

+ 6^ M®

+(^+^ 2 3 + 2^)(A^) 2 +2(4+s; 2 3)(A0 2 ^1

2+Z

(14)

(2 + 0

where we have introduced new dimensionless constants: £=

^ = £2 = ^2 V*Vn + Vnl*e ^ePn + ^nPe VeEg + VnEn

S=£± He

=

£s. = h*L. Vn

(15)

(16)

1+V

Criterion (14) with definitions (15) and (16) represents the macroscopic transversely isotropic yield criterion for the special case of equal Poisson's ratios vg = vn = v. In the more general case of unequal Poisson's ratios the overall

200

VII.2 Materials with Unidirectional Continuous Fibers

form with regard to the parts of
/ "

g

(17)

i.e. E = v„Eg + vnEn.

(18)

Thus, in the case of equal Poisson's ratios the macroscopic modulus in the direction of fibers is given by the 'rule of mixtures' and is independent of the common Poissons's ratio and of the structural parameters. Another situation turns out if the composite is loaded by Ozz. Again, we can start from Eq.fVTI.1.1), but this time it is necessary to express dj n from fVTJ.1.5), where M® if defined by Eq.(Vn.2.4). This leads to: (19)

VII.2 Materials with Unidirectional Continuous Fibers

201

(For derivation of the above expression the relation pn = pg nnl ne was used to eliminate pn and reduce the number of parameters.) Furthermore, let us mention the case of loading by 6*12 . Starting from Eqs.fVTLl.l), (VII. 1.5) and (4) we receive for the elastic domain: ?*=■¥- = He + Vn(Hn-He)M°

(20)

with M9 defined by Eq.(4). In the simplest case of identification procedure we suppose that the unknown parameters are En, r/®, r/®, k. The input data are the measured values of E, E9, ju® and e.g. O22, the last symbol meaning the plastic limit under a uniaxial loading o22 • From Eq.(18) it is possible to determine En , from (19) and (4) M* and 77®, from (20) and (4) M® and ju®, and from (14) k. With these parameters determined we can calculate the course of macroscopic strain and mesoscopoic stresses and strains for any given program of CT,y. Description of a Concrete Fiber-Reinforced Material For a numerical example we will use the experimental data published by Prewo and Kreider (1972). The described material is a composite with aluminum matrix and silicon carbide coated boron fibers (BORSIC). In our notation the experimental data are: En = 70160MPa,

Eg = 406928MPa, £® = 149441 M P a , vn = 0.48

and the experimental stress-strain diagram for tension ^22 in the direction normal to the fibers, as depicted in Fig.l by separate points. The common value of Poisson's ratio for both material constituents was assumed 0.33. Then it results by definition: He = —

pe - ^-^-

= 3.2x10 - 6 MPa"1, fin = —

= 0.8x1 or6 MPa"1

and from (19) and (4):

= 18.9x10" 6 MPa" 1 ,

202

VII.2 Materials with Unidirectional Continuous Fibers

M®= 0.80427 ,

77® =0.82985.

From the stress-strain diagram we read: ff22 = 84.2 MPa andfromEq.(14): =L

k=j

^ >/(M®)2(f2 + C + 1) + ^ C ( 1 - C ) + K 2 ( 1 - 0 2 = 35.7MP3

where §, £ are defined by Eqs.(15) and (16). In this way all the material parameters were determined and it is possible to calculate the courses of macroscopic deformation and internal stresses. Equations (VII. 1.1), (VII. 1.5) and (10) lead to:

deb = -^-+vn(^n-fie)ds22n

+ vn(pn-pe)don

+ vns22ndXn

(21)

da22n = M^dcx, - (A^S22n - N*£2n) dXn

(22)

dff33n=-(A'eS!13fl-^^3n)d^

(23)

dan =

**• {(M»+±

Pe Ma

"

)dg22 (24)

da 11n = - d a ^ - d a 3 3 n +3d<7„ .

(25)

With these expressions known we can calculate d e ^ d o ^ , , , da^, d033„ from Eqs.(21) and (VII.1.3). The only unknown that remains to be determined is dXn . This can be evaluated with the use of Eq.(ll), where we assume strain-hardening in the form: k = kQ+akn . After differentiation this equation yields:

(26)

VII.2 Materials with Unidirectional Continuous Fibers

203

Fig. 1 Internal stresses in a fiber-reinforced material under uniaxial tension in the direction normal to the fibers - elastic fibers and elastic-plastic matrix

«i m dSi 1#I + %>„ d s ^ n + Sj3„ dSja,, = 2 k adXn (27) - 2 a-^Sf1n + ^ 2 n + S) inS22n dAn and with the use of Eqs.(22) to (25) it can be finally derived:

204

VII.2 Materials with Unidirectional Continuous Fibers

l

2ne + pe[venn

^n +

+ vnng

s

22n)

- A / ' e [ s 1 ' 1 n ( s 1 1 n ^ ^ + s 2 2 n ) + S 2 2 n ( s 1 1 / J + 2s2;2n)

+ 2 a y s f 1 n + S&n + Si 1n%2n J = d<722 :

•A 11n

M

Me

- A ^ J + fifea.M*

2^e + P 0 VeMn + %Ae

With these expressions it is possible to plot the courses of macroscopic and internal stresses, increasing the value of
Chapter VIII Appendices Appendix VIII. 1 - On the Validity of Hill's Equation Equation (1.4):

is often called Hill's equation, and as mentioned in Chapter I it can rigorously be proven for a volume loaded at its surface 5 either by displacements u,{S) = e^Xy or by tractions t,{S) =o^Hj, derived, respectively, from uniform overall strain e/y or stress (7,y (Kafka 1983). However, the approximate validity of this equation is implicitly generally assumed for more general cases, as without it the use of experimental data, received with specimens of some material (data that specify the respective constitutive equations), for designs of constructions would not be possible (Kafka 1983). Hill himself anticipated the use of his equation generally for constitutive equations (not only for specimens with homogeneous boundary conditions), which is clear even from the title of one of his papers: 'The essential structure of constitutive laws for metal composites and polycrystals' (Hill 1967). Nevertheless, the use of his equation for cases other than those with homogeneous boundary conditions is an approximation. In what follows the limits of such an applicability are discussed, the limits in which such an approximation is acceptable. In Fig.l two characteristic deviations from homogeneous boundary conditions are visualized. The left-hand figure A shows the case of homogeneous loading of a sample where Hill's equation is exactly valid. Some macroscopic subvolume in this sample - called RVE (representative volume element) - has the same macroscopic stressing, but on its surface there are fluctuations of microstress, connected with the microstructure. Validity of Hill's equation in such a RVE was discussed in a number of papers: by Hill (1963, 1967), by Havner (1971), on the basis of a probabilistic

205

206

Vm.l Hill's equation

approach by Kreher and Pompe (1989), from the point of view of plastic deformation by Majumdar and McLaughlin (1975), and by Kafka (1972, 1983). It is not possible to go into detail, but the conclusions can be characterized so that Hill's equation is applicable if the characteristic dimensions of the microstructure are small enough in relation to the dimensions of the volume element in question.

Fig. 1 Two kinds of deviations from homogeneous boundary conditions

Vm.2 Proof ofEq.(II. 1.29)

207

Another question is applicability of Hill's equation for the case demonstrated in part B of Fig.l, i.e. in the case with a gradient of macroscopic stress. It is assumed that for such cases Hill's equation is applicable in the investigated volume element if the variation of macroscopic stress due to the gradient is small in relation to the average macroscopic stress. Such an assumption is a natural application of the general principle saying that small deviations from the input data for which the investigated output is valid will cause small deviations from the validity of the output. Our assumption is corroborated also by the fact that constitutive equations that do not take into account gradients of stress are widely used and are able to describe real behavior of materials and constructions. Rigorous mathematicians sometimes question the use of Hill's equation, but rejection of it would imply rejection of the use of constitutive equations and their specification by experiments with material specimens. The use of Hill's equation is an approximation, but this approximation leads to results verified by experience.

Appendix VIII.2 - Proof of Eq.(IU.29) Proof of the statement: "If

<[f(x)„f >„=1 and „=\,

then

f(x)„«1."

Let f(x)n=-\ + F(x)n . Then n=-f-\V + F(x)n](iV = -\+±\F{x)niiV n K " v,

=1

and therefore j F{x)n6V =0. v. <[f(x)nf>n=±\[\

+

F(x)n]2
V„

n V„

= 1+-l/F 2 (x)„dW' = 1 rn w

and therefore F(x)„s0 and f(x)n = 1 q.e.d.

208

VIQ.3 Deduction ofEqs.(II.2.10) to {11.2.24)

Appendix V m 3 - Deduction of Eqs. (II.2.10) to (IL2.24) The first step will be deduction of Eq.(H.2.102) from Eqs.(H.2.1) to (H.2.9'). Let us note that Eqs.(n.2.1) and (n.2.2) can be rewritten in quite similar forms separately for deviatoric and isotropic parts. Furthermore, Eqs.(n.2.1), (n.2.2), (n.2.4) and (n.2.7) are valid throughout the deformation process and therefore they are valid in the same form also for the rates of stress and strain. Having this in mind we can easily proceed as follows:

*e = Pe<*e + °e Pe +<*« t+Cbg E„=Pn6n+ont>n+anT+<&n ? = Ve (Pe<*e + PeOe + « * T +
+ pnOn + CCnT + (bn)

= ve pe &e + v„ pn an + ve ae pe + vn a„ pn + (vyxe + vnan )t + vgd)e + vn
1°e

Wf

ti

Wf

from which:

2k. d =& + v (-£■

d °'ee ri° ri° "" i

" ti°)

Equations (H.2.5) and (H.2.8) lead to: <*n = — (£n - Pn<^n) = —(£/, -e-p„on) Pn Pn 6'e =— (£e - P e O Pe

=

=—[ve(en Pn

(
-ig)-a'npn\

Wn i^e ~ *n) ~ o'ePe] Pe

Vm.4 Deduction ofEqs.(II.2.33) to {11.2.43) = Pe°e+aef>e

ee-£n

=

+ <X0T +

£±ff-{^L£s. +

209

G)e-pn6n-Onpn-anT-6)n

pn)an+(ae-an)T+(TePe-anpn+(i)B-(bn.

If the above expressions are used in the formula for
Appendix Vm.4 - Deduction of Eqs. (II.2.33) to (II.2.43) The procedure is similar to that in Appendix Vin.3. We will start with the same basic equations: t =ve£e+ vnen £e = Pe <*e + Oe

Pe+agT+
e„=Pn^n+^nPn+ant+n

* = va (pe6e + pe(Tg +ae f+
e=— v

W + Vgti>e + Vn(b„

(
e

but instead of using Eqs.(n.2.4), (n.2.7) and (H.2.9") we will use Eqs.(n.2.30), (H.2.31) and (H.2.32'). This leads to: *n = * + * * = * + —

Pn

(t'n-G'nPn)

210

VIH.5 Identification ofparameters for plasticity

e 0 - *n = Pe°e ~ Pn °n +°e

Pe ~ °n Pn + ( « • -<*n)T + K

Ce = P . < * i + <*i/>. = Pe (<*e ~ &) + (<*« "

-*«)

°)Pe

*n = Pnt'n + C'n Pn = Pn (<*n ~ &) + {<7n ~ &)Pn ■

If the above expressions are used in the formula for 6n, we arrive at Eq.(n.2.36), and with the use of this equation to Eq.(n.2.34). There is no need to repeat the same procedure for the deviatoric parts, as it is quite analogous.

Appendix VIII.5 - Determination of the Input Data for the Solution to the Identification Problem Based on the ElasticPlastic Stress-Strain Diagram At first let us discuss the case of homogeneous elastic constants (Ve=»n=P>Pe=Pn

= P)-

The experimental basis for determining the constants C„, vn = 1 - vg, rig, T]n is the stress-strain diagram of a specimen in simple tension or in simple shear. Determination of the constant cn is straightforward, it is the deviatoric plastic limit in simple tension (c„ =(S|I)JL = § (<7II)L)- However, in the case of existing 'yield point jog' in the experimental diagram, the specification of 'plastic limit' is not quite unambiguous. One possibility is to take for plastic limit the value of ffn corresponding to point Sin Figs. m. 1.1.1 and m. 1.2.1. The other possibility is to take for plastic limit the value of (?,, corresponding to point (S) in Fig. m. 1.2.1, i.e. to the point that is received by a backward extrapolation of the smooth segment of the diagram. It is shown in Section HI. 1.2 that the first choice gave better results for the case of tension and torsion, the other choice for the case of ratcheting. For the determination of vn,t]n,r\g we must know the values of X.Y.Y' in three points of a stress-strain diagram in simple tension (Eqs. (HI. 1.1.7) to

Vm.5 Identification ofparameters for plasticity

211

(H3.L1.16)). Let us measure at such three points their coordinates e^.ff^ and the respective derivatives cfd^ 1 / de^ ^ . Then according to Eqs. (m. 1.1.6) and (HI. 1.1.10): Y_e-(e)s

_^i-(e,i)s

CnHe

C„/* e

51 = £ „ - £ == «n-" 1 ^ 1 1 Y=

s-(s)s

«I1

= 1*11

V-

_.. d§ii

_«ii -(s,i)s

2/i 0

If the known experimental stress-strain diagram corresponds to simple shear, the above relations are quite analogous, but: e- = e, 2 -

£12

s == SJ2 =

#12

r

= Me

^12 Qf£ 12

and instead of cn it is necessary to use kn=yJ3cn/2. Equations (IQ.1.1.6) to (m.1.1.16) are valid also with kn instead of cn. In the case that the elastic constants of the two material constituents are different, the procedure complicates. All the above equations are valid, the values of ne(± nn * p) and of p are supposed to be known, but it is difficult to determine Cn or kn. We must use an iterative procedure in this case. In the first step we proceed as if the material was elastically homogeneous and determine cn, vn ,T]g,T}n. Then - using the values of vn ,770,T/„ SO determined - new value of Cn is calculated from Eq.(n.2.18):

Vm.6 Discussion ofEqs. (V.2.8) and (V.2.9)

212 C„=Ms

=

Mn(su)s (5,1)6

HeVnWn+KHn

+ Vnflg)(VgHgT1g + V„HnVn)

This new value of cn is used for a new calculation of vn, r\g, 77„ and the procedure goes on until the differences become small.

Appendix Vffl.6 - Discussion of Eqs. (V.2.8) and (V.2.9) a) It is easy to show that for a monotonic uniaxial loading (i.e. for a loading Sy =sTjj, where T^ is constant and s is non-decreasing or non-increasing from a zero original state) the inelastic part of sg (s^ = sg Tg) has the same sign as Sg itself and as the elastic part of Sg: Let us start with sn. From (II.2.47) and (n.2.50) we get:

<*,)-="„*-

^ ^ ^

Vn1n + Ve(VeHn + Vn»e)

*

The coefficient of s is positive and therefore, the sign of (s,,)^ is the same as that of s . From (II.2.47) and (V.2.7) it is easy to derive the expression for (sn)g, and show that it is also of the same sign as s . It is possible to imagine that any inelastic process is composed of small steps and every step begins with the immediate elastic response that is followed by the inelastic process that proceeds in some time-interval (which is really so). The first elastic response leads to the following value of —

——"

" in

Eq. (H.2.47):

R

-

W'—tf-8-

In Eq. (H.2.47), the sign of h and of R is always positive and therefore, the sign of the inelastic part of sn, i.e. of (s„ ) n , , is opposite to that of S and sn at the

Vm.6 Discussion ofEqs. (V.2.8) and (V.2.9)

213

first elastic response. This means that the absolute value of Sn diminishes due to the inelastic part of deformation. If the value of S is held fixed, the process of diminishing s„ can proceed, but sn cannot overpass the value proceed, but sn cannot overpass the value Vn

s -

c

Vt+Vn at which the coefficient of h in (H.2.47) becomes zero. If the next rise of S steps in, the elastic response leads to the rise of the 77 ~S—[v~¥TI )S

absolute values of sn and of —-

|-—-——, but the sign of the elastic

change of sn coincides with that of s, whereas the sign of the elastic change of 77 S — (v

—

+77 )S

——

" is opposite. Hence, the sign of the resulting inelastic change

H

of sn, i.e. of (Sn)ni will again be opposite to that of S and the absolute value of of sn, i.e. of (Sn)ni will again be opposite to that of S and the absolute value of S„ will be diminished by the inelastic response throughout the process. The above considerations lead to the conclusion that the signs of Sn and of

n "Q —(V

(S„)e/ will coincide with that of S , whereas the signs of —

+77 )S

^—^-

i

H

and of (sn)ni will be opposite to that of s . Furthermore, if S is held fixed and the process is viscous (not limited by some yield condition or some criterion of fracturing) sn will reach after an infinite time the value

(s„L = -fi>—s. Now it is straightforward to draw conclusions for sg . From Eqs. (V.2.7) and (II.2.47) we see: (se)nl

=

v

~ — "e

(Sn)nl

(se)„ =—[s-vn

(snU = Z%±2s- s.

214

vm.7 Parameters of Transversely Isotropic Materials

Hence, the signs of Sg,(sg)gi and (s e ) n/ coincide with that of S and the last equation gives the value of Sg in a viscous process at fixed S after an infinite time. b) Now we will turn our attention to the isotropic parts of the stress- and straintensors. Without the addends containing (i>g and (bn the situation would be quite analogous. But experimental evidence says that either the response of the isotropic parts is only elastic or there are both the inelastic addends related to the nconstituent - that comprising pn as well as that comprising
no

and because R0 >0 by definition, (dn)nl is negative for d>„ high enough, and

(<*»)„/= - — (*„)* is positive.

Appendix VHI.7 - Structural Parameters of Transversely Isotropic Materials If we suppose that our model is to be descriptive of a transversely isotropic material with the axis of symmetry x,, then all directions perpendicular to x, must be equivalent. Let us label such an arbitrary direction by x2 and suppose that it deviates from the fixed x2 direction anti-clockwise by an angle 8. Any stress component O22 o r °i2 ' S m e n 8>ven by the transformation relations: CT|2 =
215

Vm.8 DeductionofEq.(VIU.1.2)

^23 = (CT33 - <*22 ) Sin^COS 0 +CT23( C 0 ^

d

~^

d

)

that can be written for o&n , o&m , a&n , a£2m . o'zin » az3n > ^ m and The structural parameters if&n > ^ m identical for any value of 6 , i.e.:

or

^ n • IMBI

must

^

a

a* -a* , °22n °22n a 2 2 m + - J r 722n

22m _ n U ~ ^ m

r

^23n

l23m

If we use the transformation relations in the above two formulae and have in mind that these equations with fixed structural parameters must be valid for all values of 6 , it turns out: ^220

= 7

?23n> 1hzm

=

V23m

which leads to Eq.(VII.8). The procedure to show X22n

—

X23n> X?2m ~ Xzim

would be quite the same. The validity of the relations *?12ii = , ?13/|i

T

/l2m=T/l3m

an

^

X\2n=X\2n>

X\2m=X\Zm

follows from the assumed transverse isotropy immediately.

Appendix Vffl.8 - Deduction of Eq. (VII.1.1.2) For the deduction of Eq.(VII.1.1.2) with the respective definitions (Vn.1.1.3) to (VII.1.1.9) of the coefficients comprised in it, it is necessary to start from Eqs. (vn.l.l), (VII.1.3) and (\n.l.5). In our simplified variant we assume: pn -1 - ©„ = 0 and ve - vn - 0.5, which gives pg = pn = 0 . This leads to:

Vm.8 Deduction ofEq. (VII. 1.1.2)

216

«# = »e Sij + V„Ul„ -He)Sjn

<*ijn = — [ " / * • * #

+

+

^"S«h

(1)

("• ^n + "n Me)*Jh + ^ H " .

«'* = M*i -^lNK

S

*" ~ S*")J

(2)

- WK)

o)

and because of the considered type of loading 0*22 = #33 = 0, it is possible to deduce from Eq.(3): Si 1n = ^11/7 — 3-(<*11n + <*22/j + <*33n) = f (<*11n ~<*22n) W

= MS M --i-(Ws M l l -W'sf 1 l l )

where the symbols Wand A?are defined by Eqs.(VH.1.1.8) and (Vn.1.1.9). The symmetry of the problem and the property of deviatoric quantities mean that *h.2n = ^33n %2n

=

S33/J

= —

=

-

2 ^11"

'2'S|1n

%2 ~ $ 3 3 = — 2 511 •

From Eq.(2) it then follows: [ - A I . $ l + K / i n + V„/l.)S 11 „+—T- (V # «nn-S|1n)]-

SM„= Mn

(5)

^"n

Equation (VII. 1.1.2) is then received by simple algebraic linear operations that are only symbolically indicated: (4),(5)=*s^1#I(sl1lSM#,lsf1n)

(6)

(4),(1)^S,,„($,, 5,,«[„,)

(7)

(6),(1),(7) => ^'^(in. ^ L S,1rT)

(8)

(8)=>^n(^,eu,^n)

(9)

Vm.8 Deduction ofEq. (VII. 1.1.2) (7),(9) => s ^ S n , tj,, s^e^Sn,,) (IMIOZ^^H^L^LEH.S,^)

(lOMH^Sn^.s^en.s,^) (\-\) =*£„(%„%, ju,s„n) (13), (12), (11) => lj 1 (gl 1 l s M ,sl 1 l gi 1 l g; 1 ).

217 (10) (11)

(12) (13) (14)

The last functional relation (14) is a symbolic reading of Eq. (VII. 1.1.2).

This page is intentionally left blank

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Thomas T.C. et al.(1963): Microcracking of plain concrete and the shape of the stress-strain curve, J. of the American Concrete Institute, Feb., 209-224. Titchener A.L. and Baver M.B. (1958): The stored energy of cold work, Progress in Metal Physics 7,247-338. Tvergaard V. (1980): Bifurcation and imperfection-sensitivity at necking instabilities, ZAMM 60, T26-T34. Vasilyev D.M (1958): On the microstresses arising in polycrystalline specimens subjected to plastic deformation (in Russian), Zurn.Tekhn.Fiz. 28,2527-2542. Vasilyev D.M. (1959): On the microstresses arising in metals subjected to plastic deformation (in Russian), Fiz.Tverdovo Tela 1,1736-1746. Vasilyev D.M. and Kozevnikova L.V. (1959): On the nature of yield-point jog in pure iron and carbon steel (in Russian), Fiz.Tverdovo Tela 1,1316-1319. Vokoun D.and Kafka V. (1996): On Mathematical Modelling of the Incomplete Transformations in Pseudoelastic Processes in Binary Alloys, Proc. '3rd International Conference on Intelligent Materials', Lyon, France, June 3-5, 1996, eds.P.F.Gobin and J.Taubouet, SPIE vol.2779, pp.541-545. Vokoun D. and Kafka V. (1998): Training of two-way shape memory effect in NiTi wire: One experimental procedure, Acta Technica CSAV 43,5,507-513. Vokoun D. and Kafka V. (1999): Mesomechanical modeling of shape memory effect, Proc. Symposium Smart Structures and Materials 1999 - Mathematics and Control in Smart Structures, ed. V.V.Varadan, Newport Beach California, 1^1* March 1999, SPIE - The International Society of Optical Engineering, Vol. 3667,596-601. Walpole L.J. (1966): On bounds for the overall elastic moduli of inhomogeneous systems, J. Mech. Phys. Solids 14,151-162. Zaitsev J.W. and Wittmann F.H. (1977): Crack propagation in a two-phase material such as concrete. Fracture 1977, vol.3, ICF4, Waterloo, Canada, June 19-24.

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Subject Index A-model, 10-25 Atomic scale heterogeneity, 152 B-model, 10-25 Burgers model, 123 Confrontation with other theoretical solutions, 57 Continuum damage, 100,109 Creep of concrete, 124 Demonstrative diagrams, 42 Detwinning, 153,156 Discussion of the SM model, 184 Distribution functions, 10 Elastic bounds, 46 Elastic capacity concept, 134,136 Elastic energy, 13 Fiber-reinforced materials, 197 -, yield condition, 199 Finite deformations, 100 Fracturing, 131 Fracturing of concrete, 141 Incomplete transformations, 171 Interatomic potentials, 154 Isotropic model, 9 Hill's equation, 1,205 Lethersich model, 124 Localization of deformation, 100 Localized cracking, 143 Macroscale, 5 Macroscopic coordinates, 5 Macroscopic neighborhood, 5

Macroscopic point, 5 Macroscopic stress, 5 Macroscopic strain, 5 Martensitic transformation, 154 Mesoscale, 6 Microfracturing, 131 Microscale, 6 Microscopic coordinates, 6 Necking, 107 Nitinol, 153 One-way shape memory effect, 172 Perfectly fracturing material, 134 Perfectly locking material, 134 Perfectly plastic material, 134 Plasticity of metals, 64 -, complex loading, 74 -, elastic homogeneity, 70 -, strain softening, 109 Pseudoelasticity, 166 Separability of deviatoric and isotropic responses, 40 Shape memory, 151 Small deformations, 65 Specific stress power, 12 Structural parameters, 13 Time-dependent deformation, 121 Transversely isotropic materials, 186 -, two-phase model, 189 Two-phase model, 26 Two-way shape memory effect, 179

225

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Mesomechanical Constitutive Modeling by V. Kafka

The Mechanics of Constitutive Modeling

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Husserl's Constitutive Phenomenology: Its Problem and Promise (Studies in Philosophy)

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Modeling Students' Mathematical Modeling Competencies: ICTMA 13

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Time Dependent Constitutive Behavior and Fracture Failure Processes, Volume 3

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Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation

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