Elastoplasticity of micro-inhomogeneous metals at large strains

International Journal of Plasticity, Vol. 5, pp. 149-172, 1989 0749-6419/89 $3.00 + .00 Printed in the U.S.A. Copyright © 1989 Pergamon Press plc

E L A S T O P L A S T I C I T Y O F M I C R O - I N H O M O G E N E O U S M E T A L S A T L A R G E S T R A I N S

P. LIPL~SKI and M. BERVEILLER

Ecole Natiouale d'lng(~nieurs de METZ

(Communicated by Jean-Paul Boehler, Universit~ Scientifique et Medicale de Grenoble)

Abstract-The aim of this paper is to develop a general approach to the problem of determination of elastoplastic properties of metallic polycrystals at finite transformations. First, the various physical parameters describing the internal structure of the polycrystal are discussed. Next, transition relations between local and overall levels are reviewed. Here, the framework developed by Hill is mainly used. Following, a new integral equation, linking the local and overall velocity gradients is presented. An analogous integral equation was proposed by Dederichs and Zeller for inhomogeneous elastic media characterized by symmetric tensors of elastic moduli and during small strain transformations. The Green technique has been applied to obtain the integral equation. The fundamental properties of the Green tensor are discussed from the point of view of its application at large transformations.

Different methods of resolution of such an equation are presented here. Special attention is given to granular media for which the elastoplastic tangent moduli may be considered piece- wise constant. A new "quasi" self-consistent model is given, the development and applications of which will be treated elsewhere.

!. INTRODUCTION

The determination of the effective properties of heterogeneous materials from those of each of its constituents is a field of research which is closely related to the development of modern materials.

As far as linear elastic properties are concerned, many simple models have been proposed (VOIGT [1889]; P~uss [1929]; HILL [1951]) and the recent development of systematic statistical theories (KxOsEX [1980a,1980b]; DEDE~CHS a, ZEU.EX [1973]) can be considered as a complete solution to the averaging problem of elastic heterogeneous solids.

Until very recently, studies on plastic or elastoplastic properties of inhomogeneous materials have been limited to the applications of SACHS [1928] or TA~'LOX [1938] models and eventually to the different extensions of these.

The self-consistent model initially formulated by I~6m~R [1958,1961] was next developed by Hn.L [1965a]; Btn~t~sEY ~ Wu [1962]; HtrrCal~SOS [1964,1970]; BER- v E n . ~ ~ ZAotn [1979]; WEso [1980]; Hmx et al. [1985]; BEP,*a~a et al. [1987] within the framework of small strains.

It is obvious that forming processes generally lead to finite elastoplastic deformations which simultaneously modify the internal structure and mechanical state of the de- formed material. Four main families of physical parameters can be distinguished, the evolution of which during plastic deformation has an important effect on the further mechanical behaviour of the material.

(1) At a microscopic (intracrystalline) scale, the multiplication of dislocations and the evolution of their spatial distribution are responsible for intracrystalline harden-

149

150 P. LIPINSKI and M. BERVEILLER

(2)

(3)

ing. In mechanical terms, these phenomena can be described, at least partially, with the help of a hardening matrix (FRANCIOSl et al. [1980]; FRA.'~CIOSI ~ ZAOt:I [1982]), whose role is to reflect the interactions between glide systems. At an intermediate (intercrystalline) level, the relative misorientations of crystalline lattices are sources of plastic incompatibilities which generate internal stresses. These stresses can be partially relaxed by additional plastic strain but they always contribute to modify the macroscopic elastoplastic response of materials and cannot be neglected. At the same level, the changes in crystallographic orientations of constituents due to the plastic strain generate a crystallographic texture. On the other hand, the same plastic strain modifies the shape and geometrical orientation of the crystals which, in turn, leads to the formation of the so-called morphological textures. These last two aspects constitute the most important sources of induced anisotropy in the plasticity of metallic polycrystals.

The macroscopic behaviour of the aggregate depends, for a large part, on the behaviour of the single crystal. Its modelling, for finite transformations, was developed by MANDEL [1981]; Hn.L ~ RICE [1972]; ASARO [1979]; NEMAT-NASSER [1979] on the basis of the additive decomposition of the strain rate on elastic and plastic parts. This decomposition was first introduced by KR6SER [1958].

The general framework of transition relations between microscopic and overall variables was discussed by MAIqDEL [1973] and STOLZ [1982]. Hr~L [1972] suggested the useful choice of the velocity gradient and the nominal stress rate in order to establish the transition relationship between local and overall stress and strain measures.

Based on these two formulations concerning single crystal behaviour and transition relations, IWAKUMA ~ NEMAT-NASSER [1984], proposed a self-consistent approach to the polycrystal plasticity at finite transformations, using the Green tensor technique and the properties of Eshelby solution to the problem of the inhomogeneous inclusion in infinite medium, the effective properties of which are those sought for application. This approach was next used for plane polycrystals modelling, for instance by imposing two active slip systems.

This self-consistent method is, in fact, only a particular case of the concentration problem whose aim is to connect local and overall mechanical measures. Indeed, the solution to the problem of concentration is necessary for any method of averaging. Starting with these concepts, developed by KgONER [1980a] and DEDERICI-/S & ZELLER [1973] among others, for linear elastic materials, it is possible to spread the statistical methods to elastoplastic behaviour at finite strains with the help of an integral equation similar to that proposed by DEDERICHS • ZELLER [1973] in the case of elasticity and by BErtV~n.rER a ZAotrl [1984] in the case of elastoplasticity at infinitesimal transformations. This is what is undertaken in this paper. First of all, the framework introduced by HrtL [1972] is reviewed for which the transition operations are characterized by a particularly simple form. Next, an integral formulation of the problem of concentration is proposed, where the lack of symmetries of the instantaneous elastoplastic moduli tensor are taken into consideration. The proposed integral relation is then used to find the effective elastoplastic properties of the polyerystal in the form of tangent pseudo-moduli tensor linking the overall nominal stress rate tensor and the overall velocity gradient.

The evolution laws of internal parameters such as critical shear stress, shape of grains, crystallographic and morphological orientations of grains and others are also given. The knowledge of these is necessary to take into account the evolution of the internal struc-

Elastoplasticity of micro-inhomogeneous metals at large strains 151

ture of the polycrystal along with the macroscopic plastic strain. The constitutive relation of the single crystal used here is that proposed by HrtL [19721, As,~ato [1979]; Nrt~,T-NAssEx [1979]. It is simply recalled in Appendix A and adapted to the problem considered here.

11. BASIC CONCEPTS

In this chapter the basis of averaging procedures at large strain are reviewed. The choice of strain and stress measures, which are convenient for this goal is justified. Next, the averaging operations over a homogeneous macro-element are recalled referring us to Hill's considerations and results (Hn.L [1972]). But first the used notations are summarized.

Bold symbols denote tensors. All tensors are expressed in a fixed rectangular Carte- sian frame. Repeated indices are summed, and comma (), denotes partial spatial dif- ferentiation of the enclosed quantity in the reference coordinate system. Timelike derivatives are denoted by superposed dots. Local variables are denoted by lower case letters and corresponding overall quantities by capital letters.

1. Description o f motion

The motion of the body B is defined by a one-to-one mapping x(X, t) such that the current position x of a material point X at the time t is given by:

x = x(X, t ) X E B, t E R. (1)

Let r(X, t) be some particular configuration of the body called from now on the reference configuration.

X = r (X) . (2)

The motion of the body may now be described with respect to the reference configuration as follows:

x = x(X, t ) = X(¢ - t (X) , t ) = X(X,t). (3)

The choice of the reference configuration is arbitrary but the form of the obtained description strongly depends on it. In the case of the theory of plastic flow, the necessity to describe the evolution of the internal structure of polycrystal imposes in a nat- ural manner the current configuration to be chosen as the reference one.

Let ~ = x (X, r ) and x = x(X, t ) be the positions occupied by a material point X at times r and t, respectively. The deformation of the body with respect to the current configuration X (X, t) is given by:

/ / = x(X,~') = x [ ( x - ~ ( x , t ) , r ] = x , (x , r ) . (4)

Now consider the gradient of the relative deformation

f, = a ~ x , ( x , O = a ~ ( x , t ) (5)


and its instantaneous time derivative:

g = f t ( t ) = O r f t l . = t . (6)

Two fundamental tensors may be derived from (6). First, the strain rate tensor being a symmetrical part of g:

I d = ~ ( g + 'g) (7)

and, the spin being its antisymmetrical part,

1 w = ~ (g - 'g). (8)

It is possible to demonstrate that g may also be considered as the gradient of the velocity vector v =

g = axv. (9)

Expression (9) shows that the velocity gradient g is independent of the reference configuration.

Moreover, a very convenient additive decomposition of g into elastic and plastic parts is valid (NI/~tAT-NASSER [1979]):

g = ge + gp. (10)

Combining eqns (7), (8), and (10), the following expression is obtained:

g = (d e + d p) + ( w e + wP). ( l 1)

2. Stresses. Equilibrium equation

Let d t be a current tension vector acting on a current surface element J, dS, where y stands for an external normal vector. The Cauchy true stress tensor u is defined by:

dt = o .pdS . (12)

If the surface element is reported to the reference configuration, the following relation can be written:

dt = tn .p , dS, (13)

which defines the so called nominal or first Piola-Kirchoff unsymmetric stress tensor n. If, in addition, the force is referred to the reference configuration, a new expression

holds:

dt , = *r.p,.dSK (14)

which introduces the symmetric second Piola-Kirchoff stress tensor at.


Expression (3) describing the transformation from reference configuration to the current configuration enables us to write:

det(f)a = f .n = f . l r . t f , 0 5 )

where

t~x f = m

OX"

If the current configuration is considered as the reference one, expression (15) becomes

a = n = lr ( 1 6 )

but much more complicated relations are obtained when dealing with the stress rates. Indeed deriving the equivalent of (15) for current configuration with respect to T, one has:

= a,¢IT=, = n + g - a - ¢ t r (g)

= i" + g-o" + o ' - tg -- o" tr(g). (17)

The equilibrium equation has a particularly simple form when expressed using the Cauchy stress rate tensor or the nominal stress rate tensor:

div # + pb = O,

div n + pl) = O, (18)

where b is a body force rate and p the mass density in the current configuration.

3. Averaging operations

The aim of this paragraph is to give a partial answer to the question concerning the form of an overall constitutive law which is imposed via the choice of appropriate stress and strain measures.

A transition from the constituent to the macro-element constitutive law is provided via the averaging theorems established by Htu. [1972]. Let V be a volume of aggregate with external surface S. The average value of a certain quantity q over the reference configuration of an aggregate is calculated by:

Q = (q> = -~ qdV~. f19)

Hn.L [1972] demonstrated that: "reference volume averages o f deformation gradient, velocity gradient, nominal stress and nominal stress-rate are uniquely dependent on surface data" if both equilibrium equation and compatibility condition are satisfied throughout the sample.


Denoting the overall quantities by capital letters

F = (f>, G = ( g ) ,

N = (n>, N = <n>, (20)

and specializing the data over the reference surface ~,S, in the form:

o r

v = < g > . X = G - X 1 o n S~

tft.p~dS~ = (tli).p~d,S, = tN.p, dS, (21)

Hill demonstrated the fundamental result which can be expressed as: "The averages o f products f . n, g. n and g. it decompose into the products o f corresponding averages"

( f . n ) = ( f ) . ( n ) = F-N,

<g-n) = ( g ) . ( n ) = G . N , (22)

(g . f t ) = (g) - ( f t ) = G .N .

It is easy to conclude from (22)2 that the total work-rate per unit reference volume is equal to:

( n : g ) = N : G . (23)

No similar result is available for any other conjugate stress and strain measures. According to AsARo [1979], and IWAKtIM_A ~, NEMAX-NASSER [1984], the convenient

choice o f conjugate variables for description o f the polycrystal behavior is the velocity gradient g and the nominal stress-rate ft.

These two measures are of course unsymmetric. ASARO [1979] and then N r ~ T - NASSER [1979] have established the constitutive law for single crystals relating these two tensors in the form:

ft = I: g. (24)

The same approach is adopted in this paper and summarized in Appendix A. It is clear, f rom eqn (AI6), that the obtained local tensor of tangent moduli must also be unsymmetric. Strictly speaking, expression (24) cannot be called a constitutive law since it couples nominal stress rate with material spin w which is included in velocity gradient g. So, eqn (24) does not satisfy the principle of material indifference.

Hn.L [1972] has shown that if the local constitutive relation possesses some symmetries, these symmetries are preserved at the macroscopic level.

Let L be the overall pseudomoduli tensor such that:

N = L : G (25)

f rom the above, no symmetries o f L can be expected.


This fact greatly complicates the construction of a Green tensor necessary for an integral formulation of the concentration problem presented in the next chapter.

i!!. INTEGRAL EQUATION FOR THE LOCAL VELOCITY GRADIENT

The object of this paragraph is to establish an integral relation between the local velocity gradient of the inhomogeneous medium and the kinematic or static conditions imposed on the solid's external surface.

This paper is limited to macrohomogeneous and microheterogeneous media. To sim- plify the general discussion, the solid is supposed to be infinite and only the case of imposed velocities on the external surface will be considered. Extensions of this to other boundary conditions are possible.

All expressions have been obtained in the current reference configuration and have been expressed in the fixed rectangular coordinate system. This was adapted to make all relations more explicit.

Thus, the considered problem may be described by the following set of equations:

* equilibrium equation in the absence of body forces (eqn (18)):

n i j . i ( x ) = 0 (26)

• constitutive relation (see Appendix A):

n,-j(x) = l~jkAx)vt.~(x) (27)

• boundary condition

vi(x) = Gij(x)x i on St.

Eliminating ti o from eqn (26) with the help of eqn (27), the following relation is obtained:

[ l i jkt(X)Vl.k(X)] ,~ = O. (28)

Just as for media with linear properties, it is useful to introduce a fictitious homogeneous medium such that:

IukAx) = L~jkt° + 61ijk~(x) (29)

in which case, eqn (28) can be rewritten as:

L%tvl.~Ax) + [~lu~Ax)vt.~(x)]. = O. (30)

Equation (30) may be rewritten in another form:

a j w d x ) + ~ ( x ) = 0


where

Ajl = L°~.lOkOt

~ = [661kAx)vt, k(x)],~ (31)

are, respectively, the Lam6's operator and volume force. For the moment, decomposition (29) is formally similar to the one applied by

DEDERICHS & ZELLER [1973], but the tensors L ° and 61 do not necessarily have the usual symmetry properties which are well known in the case of elastic constants tensor.

Equation (30) can be transformed into the integral equation by means of the Green tensor g/,m (x - x') for the infinite medium characterized by L °. This Green tensor couples velocity components vt(x) at x with a rate of force i applied in the direction m at the position x'.

0 t i j k l~ lm, ki(X -- X') + ~jm~(X -- X') = 0 (32)

including the requirement: gtm -- 0 when x - , oo. In Appendix B several properties of G are presented and the KNEER'S [1963] method

is recalled enabling the computation of G for anisotropic infinite media. The properties of the Dirac distribution 8(x - x') and the Kronecker symbol 6j,,

imply

v~(x) = f v 6~y6(x - x ' )v j (x ' ) d V '

and taking eqn (31) into account, one has

/tim(X) f V 0 = -- Lij~l~tm.ki(X -- X')Vj(X') d V ' . (33)

Using the property:

g l , , , , k = a g l m = _ a g t m = - g l , , , , k ,

axk ax~

and, after integration by parts, integral (33) becomes:

v,,,(x) fv ° ' ' f , = -- Lijkl[~lrn.k 'Uj(X ) ] , c d V + COkl[~lm(X -- x ' ) v j , i ( x ' ) ] ' k ' d V "

L ° - Loklgl,,,(X -- X')Vz~,HX') d V ' .

(34)

Making use of the divergence theorem, the first two volume integrals are transformed into surface integrals:


Vm(X ) = - f s L ° k l ~ t m , k, vj(x') dS[ + f s L°kl~lm(x - x')uj, i(x') dS~ (35) t~

-- iv" LOkl~lm(X -- X')Vj, ik(X') d V ' .

Generally the Lam~'s operator introduced by (31)1 is not necessarily self-adjoint due to the lack of symmetry of the L ° tensor. In such a case it is still possible to obtain the corresponding integral equation using the adjoint operator of the Green tensor defined by eqn (32) as suggested by Wn.us (private communication). However, if one intends to derive the self-consistent scheme from the integral equation, it is more convenient to choose the A operator as a self-adjoint one and consequently L ° must show the following symmetry.

o Li#! = L°k j .

In this case, the Green tensor also has the symmetry property demonstrated in Appendix B:

~rnl(X -- X ' ) ---~ ~lm(X -- X ' ) .

As a consequence the following identity is valid

LOkl~im(X -- X')~,~/.ik(X' ) = LOkt~mj(X -- X')Vl, ik(X') .

Now, using eqn (30) one obtains:

0 t • t Li jk l~ lm(X -- X )Uj, ik(X ) ~--. - -~mj (X -- X ' ) [~[ijkl(xt)ul, k(X )] , i o ( 3 6 )

Because of eqn (36), eqn (35) becomes:

v,,,(x) fs ° fs ° Lektg lm, k, Vj(X') d S / + = -- Lijkl~lm(X X')Oj.i(X') dS~

(37) P

+ I ~mJ (x -- X ' ) [~lijkl(Xt)Ui, k(Xt)] , i d V ' . ,Iv

After an integration by parts, the last integral may be changed to a volume and a surface integrals and eqn (37) is replaced by

fs ° fs = -- Lukl~lm, k,(X -- X')V./(X') dS[ + •LOktglm(X -- X')V.~,i(X') dS[~

f gmytx - x')tS/et, t x ' ) v , . t ( x ' ) d S / - f ~mj.,(x - x') ' /ek,(x')v, . t(x' , dV'. + .Is • JIf•

(38)

The second and third surface integrals disappear because of Green's tensor property for x - , oo. The first surface integral presents the solution to the homogeneous problem

158 P. LIPINSKI and M. BER'v'EILLER

(homogeneous material of 0 Lokz instantaneous tangent moduli which is submitted to the real boundary conditions). Defining this solution as v ° (x) one may write:

vm(x) = V0m(X) + f gmj.i(X -- X')61Ok~(X')Vt, k(X') d V ' . (39) dV

The velocity gradient may now be calculated as,

grnn(X) = Um.n(X ) = Um0,n(X) + f ~mj.in (x -- X')~)IijkI(X')UI.k(X') dV ' . (40) d~

This integral equation for the velocity gradient is analogous to that of DEDERICHS & ZELLER [1973], but, for the sake of establishing it, it was necessary to impose a symmetry condition on the tangent moduli L°kt of the reference homogeneous medium defined by eqn (29). This is reflected by both the Green's tensor (which is computed for a medium with L ° property) and by the 61 tensor.

Using the notation established in section II, eqn (40) may be rewritten as:

gO(x) = g°(x) + f , F,jlk(X -- X')t~lktmn(X')gnm(X') d V ' (41) ,Iv

where

F0tk(x -- x') --= gu,~j(x -- x'). (42)

Relation (41) is used in the next section to determine the evolution laws of the internal structure of the polycrystal and the overall elastoplastic tangent moduli.

IV. EFFECTIVE ELASTOPLASTIC PROPERTIES OF THE POLYCRYSTAL

On the basis of integral equation (41) and using the averaging relations (20) and (23), it is possible to obtain the pseudo-tangent moduli L elf linking the overall nominal stress rate 1~1 to the global velocity gradient G. The determination of L eft in the general case is formally feasable. In practice, in order to obtain results, various approximations are necessary, leading to different simple models. The procedure adapted here is similar to the one used for elastic heterogeneous media. Nevertheless, there remain two problems relating to the particular form of the constitutive taw described here.

The first difficulty results from the unsymmetry of local tangent moduli. According to Hint [1972], the overall tangent moduli L erf cannot be expected therefore to be symmetric either. To obtain the integral equation in section lII, it has been assumed that

o 0 the moduli of the reference homogeneous medium possess the symmetry: Lijkl = Lukj. Obviously, the tangent properties deduced from the local moduli ! by a simple averaging operation cannot be chosen for L °, and neither can the effective moduli L at. In the case of systematic statistical theories (I~61,mg [1980b]) or in usual self-consistent formulation ( I w A K t ~ & N~MAT-NASSER [1984]), the average value of ! and the effective moduli L eft are used for L °, respectively. It is clear that these models must be reviewed in order to take into consideration the unsymmetry of (I) o r L eft.

The second difficulty stems from the structure of the elastoplastic constitutive law of heterogeneous medium itself. This constitutive relation is incremental in its nature as


it has been justified in sections I and II. Therefore, it is necessary to know the internal structure of the polycrystal and its evolution in a function of applied loading, at each instant. A new question of evolution laws of the physical parameters describing this internal structure arises. The determination of evolution laws constitutes a necessity which is fundamental for the calculation of L en. In the next paragraph the gradient g is supposed to be a priori known. The evolution laws of the critical resolved shear stress on all glide systems, crystalline orientations as well as the internal stresses may be deduced.

In the second paragraph different methods of calculation of g are presented based on the formal approach called Born approximation, or based on various simplifications of the integral equation.

In the case of granular medium, for which the assumption of homogeneous intraganular behaviour seems to be justified, two methods of calculation of the mean value of g inside each grain are developed. The evolution rules of the internal structure for the granular solids are reviewed.

Finally, the methods of determination of L elf are recalled at paragraph 3.

1. Evolution rules f o r internal structure

As has been mentioned in the Introduction, the internal structure of the polycrystal may be essentially characterized by:

• the state of hardening of grains associated with the density and spatial repartition of dislocations. Globally, the effect of dislocations is reflected by an increase of the critical shear stress on different glide systems;

• the internal stresses associated with the intergranular incompatibilities and the inhomogeneities of intra-granular plastic strain;

• the modification of crystallographic orientations of grains; • the evolution of the shape and geometrical orientation of grain axes.

All these physical parameters describing the internal structure may be calculated knowing the velocity gradient g(x). The considerations undertaken in this paragraph are general and not necessarily limited to the granular structures. In the following g(x) is supposed to be known.

Starting from eqn (7) the strain rate is equal:

I dii= ~ (gij + gj~),

or using eqn (41) one can write:

fv 1 du(x) =d?+ . ~ (~:~k + l ) . , )~ l~ l , . . (x ' )g. . , (x ' ) dV ' . (43)

Plastic slips on the active glide systems are determined substituting eqn (43) in relation (AIS):

"~h(x) = ~ MSZ(Ri~Cijktdk1(x) - R~aij(x)dkk(x)). (44) g

160 P. Lu , lssr~I a n d M. BERVE1LLER

One may now calculate the rate of critical shear stress on a glide system g using (A10):

h,f (45)

The above relation describes the hardening phenomenon on the system g due to the plastic slip on all the active systems of a given constituent, interactions between the dislocations on various glide systems being modelled by the matrix H gh.

It is currently accepted that rotations of the crystalline lattice are due to the elastic part of the total spin w. These rotations lead to the development of crystallographic texture, which in turn can highly modify the overall behaviour of material under consideration, reflecting the so-called crystallographic anisotropy. Again, starting from the integral equation (41) the total spin at some point x of the body is given as the antisymmetric part of velocity gradient g, and may be written in the form:

f v 1 wiitx) = wi ° + . ~ (F, jtk -- ~'ak)5lkm,.g.,,,(X') d V ' . (46)

On the other hand, the plastic part of this spin denoted by w p is associated to the plastic slips 5, ~ by the relation (A4):

wiPtx) = Z QijM (RpqCpqkldk l -- Rhpq~Tpq dkk) . g.h

(47)

According to eqn (I 1), the elastic spin is obtained simply as:

W e ----- W ~ W p .

Using eqns (46) and (47) the rate of rotation of crystalline lattice takes the following form.

W/~(X) = W 0 -t- ,, ~ (F/y /k - - l"j i lk)~lktmn(X')gnm(X') d V '

- - Z g gh h m Q i i M (RvqCt~tktdkt R~qtruq dkk). g,h

(48)

Rotations of the crystalline lattice, which is orientated with respect to a fixed reference frame by three Euler angles ~ , ~ , ~ 2 , may be calculated using the well-known relations between rates of change of Euler angles and elastic spin w e, see for instance BrJscr~ [1969]. In this manner, the induced crystallographic texture creation and evolution may be modelled. The evolution rule for the local stresses, and by this way, the evolution of internal stresses, is directly derived from eqns (17), (A16), and (41)

Oij = li.iklglk "4" gik akj -- aij dkk. (49)

Finally, shape and geometrical orientation of the constituent grains may be analysed knowing the velocity gradient gij (x). The internal structure being characterized by the shape, relative position of the constituents, and their interface orientations, the changes


of these parameters are closely related to the symmetric and antisymmetric part of g, respectively. So, the evolution of these parameters, along the loading path, will enable us to simulate the morphological texture development. The complexity of the proposed approach becomes more visible when compared with the analogous problem of the linear elastic medium. In the linear case the evolution problem does not exist, and the internal structure of the material is directly simulated by correlation functions of elasticity constants (KR6NER [1980b]). When dealing with elastoplastic materials, a coupling phenomenon plays an essential role. The instantaneous tangent properties of the material depend on the loading history by means of the internal structure state. On the other hand this internal structure evolution is given by the local, instantaneous velocity gradient, which in turn is determined as a function of instantaneous tangent moduli.

2. Resolution methods of the integral equation

Various classes of resolution methods of the integral equation (41) may be developed starting from different degrees of simplification. First of all, a formal method based on Born's (DEDERICHS #, ZFtI.ER [1973]) approximation is presented, together with its different variants which can be deduced from making appropriate hypotheses. The second part of this paragraph deals with a simplified approach physically reasonable for a granular medium.

A. Born approximation and its simplified versions. The formal solution of the integral equation (41) is obtained in the form of a development on multiple integrals. The approximation of zero range is obtained neglecting the integral term in eqn (41). This leads to the so-called TAYLOR [1938], and LIS [1957] approximation:

g(x) --- g0(=G). (50)

Strictly speaking this approximation is correct only for the homogeneous media and may be considered as satisfactory for the media weakly heterogeneous.

The approximation of first range is accomplished substituting g(x) in eqn (41) by gO. In this case one can write:

g(x) = gO + fv /'(x - x') :61(x') :g°dV' . (51)

Finally, substituting g(x) under integrals successively by new and more and more pre- cise approximations, one has:

g(x) = gO + fv: F(x - x') : 81(x') : g°dV'

+ r(x - x') : :r(x' - x-) - 61(x-): zOdv'dv -

+ fv: f.: fv.r(x- x',:'l(x'):r(,,'- x'):'l(x','r(x" - x ")

: g°dV'dV"dV" +...

(52)

162 P. LIPINSK! and M. BERVEILLER

In this equation the integrals to be calculated depend only on L ° (through r and 61) and on the internal structure of the polycrystal which is supposed to be known.

From each of these expressions, it is possible to deduce the overall velocity gradient G by the averaging operation over all volume of material. According to (20) the following holds:

l fv, [ fv, f r(x-x'):61(x')avav' G = Vtt g(x) d V = I+ -~t ",

l f ,fv fv, r (x - x'):6l(x'):r(x' - xO):6l(x")aVdV'dV" + ..lg° + V ; .

(53)

or in other words eqns (52) and (53) may be written as:

where:

g(x) = a(x) : gO,

G = A : g °,

I is the identity operator defined by Iiikl = 6ik6je

(54)

and the operators a and A are defined by (52) and (53). Equations (52)-(54) remain valid no matter the current internal structure o f the in-

homogeneous material. In particular, if a model of the intragranular plastic inhomogeneities exists, eqn (52) permits taking it into account. This complex aspect of plasticity is still not solved. Various simplifications of eqn (52) neglecting the intragranular inhomogeneities phenomenon are currently used. In fact, the Taylor-Lin model may be iden- tified from eqn (52) by putting down a = A = I.

Finally, another approximation may be proposed starting from the decomposition of 1, on a local part proportional to 61 x - x' I and a part proportional to the term (x - x')-3 depending only on the orientation of the vector (x - x ') . This decomposition introduced by DEDERICnS • ZELLER [1973] can be written in the form:

1 , ( x - x ' ) = . r Z 6 ( x - x ' ) + 1 r2(x x)

( x - x ' ) 3 Ix x ' l " ( 5 5 )

When the nonlocal term may be neglected with respect to the local one, the following results are obtained:

g(x) = gO + fv . 1,1:61(x):g(x)6(x - x ' ) d V ' , (56)

and using the property of Dirac delta:

g(x) = gO + 1,1 : 61(x) : g(x). (57)

Now, supposing the existence of the inverse operator one has:

g(x) = [I +/- , l :61(x)]-I :gO. (58)


The operator A takes a simple form

A=lfv[l-l'~:al(x)]-~dV. (59)

The various approaches presented above are either too general and formal (Born approximation) or too simple (Taylor-Lin model) to produce satisfactory results in the case of inhomogeneous materials.

Restricting our considerations to the case of granular materials, such as metallic polycrystals, it is possible to put the solution (41) into the self-consistent scheme. It was shown (BERADAI et al. [1987]) that the self-consistent approach, at least for small elastoplastic strain, constitutes an acceptable approximation of the real behaviour of metallic polycrystals.

B. Simplified solution for granular media. Consider a medium composed of grains of various crystallographic orientations (eventually of various crystallographic structure). The first approximation may be obtained assuming a uniform elastoplastic behaviour within each constituent. In consequence, the velocity gradient g must also be uniform at the grain level. Accepting this approximation, the formation of intragranular plastic inhomogeneities during the loading process is neglected. The evoked phenomenon of the intragranular inhomogeneities goes beyond the frame of interest of the paper.

The deviation part of instantaneous moduli can now be written as:

al(x ' ) = ~41~O~(x ') I

where

01(x ' )= ( ~ ifx'~Vlifx'E I// (60)

and Vt is the volume of grain with the label L Similarly, noting the average value of the velocity gradient over a grain I as:

gt _ ~ g(x) dV (61)

the field of this velocity gradient can be expressed in the form:

g(x) = ~ gt0t(x). (62) 1

In consequence, the quantity/il(x') : g(x') is simply equal to:

61(x') :g(x') = ~,:llt:giOt(x'). (63) I

Substituting eqn (63) in the integral equation (41), and after the simple algebra one can find:

164 P. Lwt.~sKi and M. BERVEILLER

Introducing:

1 f v ~ f p(x_x,):AiJ:gJdVidV~.

T t y = l f~ f F(x -x ' )dV~dVj , v~ ; .1

(64)

(65)

the above equation becomes:

gl = gO + ~TtJ:AiJ:gj" (66) J

For ellipsoidal inclusions and when I = J, the tensor T u and the tensor of Eshelby S (Esam.nY [1957]) are linked by the expression:

S = T/t: L °. (67)

For I , J, the tensor T u describes the interactions between two inclusions I and J. This tensor as given by eqn (65) was introduced by BERVEILLER et al. [371 where a method of calculation of T IJ in the case of a pair of ellipsoidal inclusions embedded within an anisotropic matrix is also given.

Expression (66) constitutes a linear system of equations for gt, the range of the system being equal to the number of grains of the considered polycrystal.

An application of eqn (66) was proposed by BERVEO.L~R et al. [1986] for the case of a two-phase elastic material in order to model the influence of reinforcements (second phase) on the anisotropic properties of the composite material.

The linear system of eqn (66) can also be rewritten in the form where the terms of interaction between the inclusions are separated:

gt=gO + Tn:Ai1:gt + ~ T t J : A i J : g j . I :¢: J

(68)

The tensors T zJ are composed of two terms: a first one proportional to p-3 and a second one proportional to p-5, where p is a distance between the two inclusions I and J.

The convergence of eqn (68) is weak if L ° is arbitrary. On the other hand, if L ° is lightly different or equal to L eft, the contribution of all the terms TtJ: AIr: gs can be neglected. The interactions between the inclusion I and its surrounding is reflected only by Ttt being dependent on AIt. The choice of L ° = L eft defines the so-called self- consistent approach to the resolution of eqn (41).

C. One-site "quasi" self-consistent method. In the case of the averaging theories for elastic media, the self-consistent approach is obtained choosing the elastic properties of the reference medium being equal to those of the effective one and considering only the interactions between a given grain and a matrix characterized by the L "ft- unknown effective properties of polycrystal. K_~toNmx [19801 showed that this approach gives correct results when dealing with a perfectly disordered medium.

For elastoplastic solids and when the large deformations are considered, the similar choice is not possible, because the tensor of the effective properties of the medium does not possess the symmetries required by L °.

Elastoplasticity o f mic ro- inhomogeneous metals at large s trains 165

Accordingly, we propose to choose for L ° the symmetrical part of L eft at least with respect to second and fourth indices.

1 / ? e f t /" eff '~

When the one-site self-consistent method is considered we obtain:

gl = gO + Tn:Al~:g /

with

] t i r e f f ± l r e f fx

(69)

(70)

The tensor T u is calculated using L ° given by eqn (69), and the velocity gradient for a given inclusion may be found:

gl = (I - Tn:AIt ) - t :g° . (71)

So, the gradient gO is linked to G by the operator A which now becomes

A = ~ ( I - T U : A l l ) - l O l ( x ) d V = ~ f Z ( l - Tn :Ai l ) -I (72) I

where f i = VI/V indicates the volume fraction of grains characterized by the moduli I I. Finally, the evolution rules of the internal structure of the polycrystal can be refor-

mulated in the case of the one-site self-consistent approach. Knowing all active systems of a given grain the velocity gradient may be determined

using eqn (71). From eqns (7) and (8), the strain rate and the total spin may be calculated:

1 d I = 2 (gI+ tgl),

(73) wi = 1 ,gl).

The rate of the critical resolved shear stress on the gth system of the grain I is now:

i.Sc = ~ eh hp p / R p - ! dlkk). (74) H M (RuCimdJa- uo~j h,p

The instantaneous tangent moduli can be determined using (AI6), and the rate of local stresses is directly given by the local constitutive relation (A16):

nl = i ! : gl. (75)


The internal stresses are simply evaluated by the difference n - N. The crystallographic texture evolution is related to the elastic part of the local spin wr:

te 1 = ~,, QijM ( e p q C p q k l d ~ l - Rpqopqh 1 gh

(76)

The change of the shape and geometrical orientation of grain may be determined knowing d I and w t.

A computer code based on the method presented here was developed. The applied algorithm as well as various numerical results are discussed elsewhere (LwtNsm ~ BER- VEILLER [19881).

3. Average properties of the aggregate Whatever is the hypothesis adopted to solve the integral equation (41) for the local

velocity gradient as a function of the reference medium velocity gradient gO, the solu' tion may always be presented in the form of a concentration equation:

g(x) = a(x) :A -1 :G. (77)

Naturally, the operations a(x) and A depend on the used method and, in the general case, take the form of Born's equations (52) and (53).

Starting from the local constitutive relation (A16) and substituting g(x) by eqn (77), the following expression holds for the rate of nominal stresses

h = l(x) :g = l(x): (a(x) :A -I) :G. (78)

Equation (78) enables us to find the instantaneous elastoplastic pseudo moduli of the considered macrohomogeneous aggregate using the averaging operation (20).

Left= --VI fvl(X):a(x):A_ldV. (79)

When the granular medium is treated and the quasi-self-consistent method presented at the previous paragraph is applied, we obtain after some algebra:

L eft = ~ f t l t : (I -- TH: A l t ) -1 : A -l. (80) /

This is the implicit relation for L eft where T t:, Ai t, and A depend on the unknown L ©ff as usually is the case for the self-consistent model.

V. CONCLUSIONS

Starting from the additive decomposition of the velocity gradient on elastic and plastic parts, the local constitutive law for the single crystal, and applying the averaging relationships linking the local (microscopic) and overall (macroscopic) measures, an integral equation has been established for the local velocity gradient. A reference homoge-


neous medium has been used, whose motion under the same boundary conditions is described by the velocity gradient gO. Formally, the solution of the integral equation can be presented, introducing the operator a given by eqn (52) as:

g = a : g °.

Knowing the velocity gradient g, all the evolution laws of the internal structure of the aggregate may be determined.

The various hypotheses adopted in order to obtain the solution of the integral equation classify all the exterior models with respect to the systematic statistical approach developed here. Comparing this approach to the systematic statistical methods in elasticity, the complexity of the elastoplastic problem, due to the incremental character of the constitutive relation and the evolution of internal (physical) parameters of the structure, is manifested.

A new quasi-self-consistent method has been proposed to solve the integral equation. The details concerning this formulation as well as an algorithm and obtained numerical results make the object of interest of another work.

Acknowledgement-The authors wish to express their appreciation to the Direction des Recherches et Etudes Techniques for support of this work under Contract #85/093.

1889

1928 1929

1938a 1938b

195 l 1957

1957

1958a

1958b 1961 1962

1963

1964

1~5 1969 1970

1972a

1972b

1973

REFERENCES

VoiGr, W., "Uber die Beziehung zwischen den beiden Elastizit~tskonstanten isotroper K6rper, ~ Wied. Ann., 38, 573. SACHS, A., "Zur Ableitung einer Fliessbedingung," Z. der V.D.I., 72, 739. Reuss, A., Berechnung der Fliessgrenze yon Mischkristallen anf Grund der Plastizititsbedingung fiir Einkristaile," Z. Angew. Math. Mech., 9, 49. TAS~tOR, G.I., "Plastic Strain in Metals," J. Inst. Metals, 62, 307. TAvtOtt, G.I., "The Mechanism of Plastic Deformation of Crystals," Proc. Roy. Soc. London, A145, 362. H~L, R., "The Elastic Behaviour of a Crystalline Aggregate," Proc. Phy. Soc., 6~A, 349. ESHELnY, J.D., "The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems," Proc. Roy. Soc. London, A241, 376. Ln~, T.H., "Analysis of Elastic and Plastic Strains of a F.C.C. Crystal," J. Mech. Phys. Solids, 5, 143. IQtoNett, E., "Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls," Z. Phys., 151, 504. KRONER, E., "Kontinuumthenrie der Versetzungen und Eigenspannungen," Berlin, Springer-Verlag. KRONER, E., "Zur plastischen Verformung des Vielkristails," Acta Met., 9, 155. BU-D~SEY, B. and Wu, T.T., "Theoretical Prediction of Plastic Strains of Polycrystals," Proc. 4th U.S. Nat. Congr. Appl. Mech., p. 1175. IOCeF.R, G., "Die elastischen Konstanten quasi isotroper Vielkristallaggregate," Phys. Stat. Solid., 3, 331. HUTCHINSON, J.W., "Plastic Stress-Strain Relations of F.C.C. Polycrystalfine Metals Hardening According to Taylor's Rule" and "Plastic Deformation of B.C.C. Polycrystais," J. Mech. Phys. Solids, 12, 11 and 25. Ha.t, R., "Continuum Micro-Mechanics of Elastoplastic Polycrystals," J. Mech. Phys. Solids, 13, 89. Btm~, H.J., "Mathematische Methoden der Texturanaiyse," Akademie Verlag, Berfin. H u ' r c ~ , J.W., "Elastic-Plastic Behaviour of Polycrystalline Metals and Composites," Proc. Roy. Soc., London, A319, 247. Hilt, R. and RICE, J.R., "Constitutive Analysis of Elastic-Plastic Crystals at Arbitrary Strain," J. Mech. Phys. Solids, 20, 401. Hn.L, R., "On Constitutive Macro-Variables for Heterogeneous Solids at Finite Strain," Proc. Roy. SOc. London, A326, 131. DmDinucnx, P.H. and Z~_IJ:~, R., "Variational Treatment of the Elastic Constants of Disordered Materials," Z. Physik, 259, 103.


1973 MANDEL, ~I., "Foundation of Continuum Thermodynamics," in Domingos, J.J.D., Nina, H.N.R., and Whitelaw, J.H. (ed.), Macmillan, New York, p. 283.

1974 TRUESDr:LL, C., "Introduction/l la m~canique rationnelle des milieux continus," Masson et Cie. 1979 As.,~to, R.J., "Geometrical Effects in the Inhomogeneous Deformation of Ductile Single Crystals,"

Acta Met., 27, 445. 1979 BERVErLLER, M. and ZAotrl, A., "An Extension of the Self Consistent Scheme to plastically flowing

Polycrystals," J. Mech. Phys. Sol., 26, 325. 1979 NEr, tAr-NASSER, S., "Decomposition of Strain Measures and Their Rates in Finite Deformation

Elastoplasticity," Int. J. Solids and Structures, 15, 155. 1980 FItANCIOSl, P., BERVEIttER, M. and ZAoul, A., "Latent Hardening in Copper and Aluminum Sin-

gle Crystals," Acta Metall., 28, 273. 1980a KRONEg, E., "Graded and Perfect Disorder in Random Medium Elasticity," J. Eng. Mech. Div.,

ASCE, 106, 889. 1980b KRoNErt, E., "Linear Properties of Random Media: The Systematic Theory," in Cahiers du Groupe

Franqals de Rh~ologie, 15~me Colloque Annuel du Groupe Franqais de Rh6ologie, Paris, pp. 15- 40, D~cembre.

1980 WENG, G.J., "Constitutive Equations of Single Crystals and Polycrystalline Aggregates Under Cyclic Loading," Int. J. Engng. Sci., 18, 1385-1397.

1981 MANDEt, J., "Sur la d6finition de la vitesse de d~formation 61astique et sa relation avec la vitesse de contrainte," Int. J. Solids and Structures, 17, 873.

1982 Fm~_~clost, P. and ZAOtn, A., "Multislip in F.C.C. Crystals. A Theoretical Approach Compared with Experimental Data," Acta Metall., 30, 1627.

1982 SrOLZ, C., "Contribution A 1'6tude des grandes transformations en 61astoplasticit6," Th6se Doct., Ing ENPC Paris.

1984 BERVEILLER, M. and Z^ouI, A., "Modeling of the Plastic Behaviour of lnhomogeneous Media," J. Eng. Mat. Tech., 106, 295.

1984 IWAKUMA, T. and NEMAT-NASsER, S., "Finite Elastic Plastic Deformation of Polycrystalline Metals and Composites," Proc. Roy. Soc., A394, 87.

1985 Hrm, A., BERVEILt~R, M. and Z^o~,% A., "Une nouvelle formulation de la modetisation autoco- h~rente de la plasticit6 des Polycristaux m&alliques," J. Mec. Th6orique et Appliqu6e, 4, 201.

1986 BI~RVEII.LER, M., FASsI-FEHRI, O. and Hm~, A., "D~termination du comportement ~lastique effec- tif d'un mat6riau composite h partir d'un mod61e autocoh6rent/t plusieurs sites," C.R. 5e J. Nat. Com- posites, Paris, Editions Pluralis.

1987 BERAD~, C.H., BERV~rtL~R, M., and Ln'~SKL P., "Plasticity of Metallic Polycrystals Under Complex Loading Paths," Int. J. Plasticity, 3, 162.

1987 BERVEILLER, M., FASst-FEHRL O,, and Him, A., "The Problem of Two Plastic and Heterogeneous Inclusions in an Anisotropic Medium," Int. J. Engng. Sci., 25, 6, 691.

1988 Lwtrcsrl, P. and BERVEILLER, M., "Elasto Plastic Behavior of Polycrystallines Metals under Large Transformations by a New Self Consistent Model" (in preparation).

1988 WItI.IS, J. (private communication).

APPENDIX A CONSTITUTIVE RELATION FOR SINGLE CRYSTAL

The single crystal velocity gradient gij is decomposed on the s t ra in rate d a n d spin w

(see (7) and (8))

g,j = dij + wi i (A1)

which in t u r n may be split on elastic and plastic parts (MAr~EL [1981]; HILL & RXCE

[1972a]; A s , ~ o [1979]):

g~: = (d~ + di~) + twO. + w~). (A2)

The plastic part o f g results f rom the c o m b i n a t i o n of the rate of plastic slips "i: on d i f ferent active glide systems. Let n ~ be the n o r m a l vector to a gl ide p lane and m g the

slip d i rec t ion in this p lane. Thus


1 I J i j ! , g

= = Q ~ 7 . ,, -~ ( m [ 4 - m ~ n f ).~ " s . ,

(A3)

(A4)

For metals, the scalar product m s. n s is generally null. It simply implies that d~'k = 0 and dtk = d[k.

If a represents the Cauchy stress and ~* is the rate of true stress with respect to the crystal lattice, the elastic constitutive relation of single crystal may be written as:

~i~ = C,jk, e~ ~, - ~ij d~, (A5)

where Cok t are the components of the elasticity tensor and:

V s ou =ai i - w ~ o + W~Ok~. (A6)

The nominal stress rate ttij is related to b o by the relation (17) which for convenience is repeated here

nij = 0u - (dik + Wik)~kj + aud[k. (A7)

Thus, introducing (A5) and (A6) into (A7)

flij = Cijk, dk l - d i t a k j - OkiWt7 -- ~_j (C/jkIR[l + Q~atj - Q~/aki)~ z. (A8) g

Now one only needs to clarify the single crystal plastic behaviour that is the relation-

ship between ~g and the resolved shear stress rate ( ~ ) . Only the case of classical plasticity will he studied here, for which the Schmid's law is applicable.

In this case, a slip system g can be active when:

R ~ a i j = CSc, (A9)

where ~-~ represents the current critical shear stress on a considered system. The rate of resolved shear stress depends on both the rate of stress bij and on the rotation of the crystalline lattice Rij. Therefore:

this relation becomes:

U U j~

( ~ ) = R~j~i~ (A10)

by deriving (A3) with respect to time and taking into account (A6). The hardening of the single crystal is described by the hardening matrix connecting

the rate of the critical shear stress on the system g to the plastic sfip rate ~s on systems h. Thus:

~'c s = ~'], H*a'Y s. (AI I) h


The single crystal behaviour is then described by the following relations:

• for an active system which remains active

R ~ oij = rc ~

and

: ( R i j a i j ) , h

for an active system which ceases to be so:

R g v , if R~.aij = 7 g and . . i ja , j < O,

for an inactive system

"~g = 0 ,

"~g = 0 if R g oij < r ff

(AI2)

whatever the value of ~ ' g ~ "~tJ v tJ "

Finally, for the active system and which remains so, the following relation is obtained:

Rg ~* = ~_~Hgh'~ h. (A13) i j a i j h

Using (A5) the above relation may be rewritten:

g h .h g ~_~ ( H gh d" RijCi jk tRkl )" Y = Ri j Cijkl dkt - Rijtrijdkk.g ( A I 4 ) h

Introducing the following notation:

R g r , R h x-I Mgh = ( ngh q" ij',.'ijkl kl!

the slip rate on a system g is given by:

~__~M (RijCiykldkl- R~aijdkk). (A15) ,~g ~ gh h

h

Then, relation (A8) becomes:

flij = Cijkldkl - - dikOkj - - OkiWkj

-- -- QrnyOmi)M (RmlCpqkldkt -- R h Opqdkk )" E ( C i j m n R g m n h" Q~momj g gh h g ,h


Factoring gtk the above expression can be rewritten in the form

ilij = liflagtk

where

1 1

g O - - g gh h -- ~" ( CijmnRgmn + Qim mj Qmyomi)g (RpqCpqkl - Rhom6,¢). gh

(A16)

The tensor I does not possess the usual elastic constants symmetries and particularly the following does not hold:

l i jkl :/: li lkj.

This lack of symmetry is directly due to the presence of the cr~y and Q~ terms in the expression of I. These terms are present in (AI6) simultaneously with the elastic constants Cijkt. In the case of highly ductile materials for which the stresses are negligible with respect to elastic moduli the relation (A16) may be simplified

Iukt = Cukt ~_, • eh s -- CumnRmnM RpcCpckt. gh

Furthermore, if C is isotropic with the Lam~ moduli A and #, then:

lqta = ASijSkt + #(~ik6jI + 8aSia) - 4l~ 2 ~_j R~MehRht. hg

APPENDIX B

CALCULATION OF THE GREEN TENSOR

The explicit calculation of the Green tensor from the equation

o Lijkl~lm, ki(X -- X') -I- 8jm~(X -- X') = 0 (B1)

can be accomplished by Fourier's transforms of K ~ [1963]. If ~ij(k) represents the Fourier transform of gij(x), then

~ i j ( k ) = fv , ~ij(x)e-ikxdVx

and

I fv,~ij(k)ei~dVk.

172 P. LIPINSKI and M. BERVE1LLER

Taking twice the derivation of gij with respect to x one obtains:

1 f~ gij(k)ktkke ikx dVk. g,j.kAx) = 8rrs ~

Next, knowing that $(x - x') = 1, eqn (BI) becomes:

0 ~ Lijkl~lmkk](i = 5jm.

Introducing

(B2)

(83)

Laboratoire de Physique et Mecanique des Materiaux Greco GDE Unit* associ~ au CNRS n ° 1215 Ecole Nationale d'lngtinieurs de METZ Ile du Saulcy 57045 Metz Cedex France

(Received 20 November 1987; in the final revised version 28 January 1988)

then G = tG.

Bit = L °ktkk ki, (B4)

a linear system of equations is obtained

Bjt~tm = ~jm,

the solution of which can be formally written as:

if B,7,t l exists. So ~ m ( x ) -- 9u, .a(x) may be calculated directly from (82) by substituting ~iy by

(85). From the definition (84) of B, Green tensor 9 possesses the symmetry ~ij = gji when it is so for B, in other words if

L Okl = Li°kj

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Elastoplasticity of micro-inhomogeneous metals at large strains