Nemat-Nasser -- Finite Elastic-Plastic Deformation of Polycrystalline Metals.pdf

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Finite Elastic-Plastic Deformation of Polycrystalline Metals

T. Iwakuma; S. Nemat-Nasser

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol.394, No. 1806. (Jul. 9, 1984), pp. 87-119.

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Proc. R. Soc. Lond. A 394, 87-119 (1984)Prin ted in Great Br i ta i n

Finite elastic-plastic defo rmation of polycrystalline metalsB Y T. IWAKUMA S. NEMAT-NASSERND

T he Technological Inst itute , Department of Civil Engineering, Northwestern U nivers ity, Evan ston, Illinois 60201, U . S . A .

(Communicated by Rodney H ill, F.R.S. - Received 29 J u l y 1 9 8 3 )Applying Hill's self-consistent method to finite elastic-plastic deforma-tions, we estimate the overall moduli of polycrystalline solids. The modelpredicts a Bauschinger effect, hardening, and formation of vertex orcorner on the yield surface for both microscopically non-hardening andhardening crystals. The changes in the instantaneous moduli with defor-mation are examined, and their asymptotic behaviour, especially inrelation to possible localization of deformations, is discussed. An interest-ing conclusion is tha t small second-order quantities, such as shape changesof grains and residual stresses (measured relative to the crystal elasticmoduli), have a first-order effect on the overall response, as they lead to aloss of the overall stability by localized deformation. The predictedincipience of localization for a uniaxial deformation in two dimensionsdepends on the initial yield strain, but the orientation of localization isslightly less than 45" with respect to the tensile direction, although thenumerical instability makes it very difficult to estimate this directionaccurately.

Our objective is to estimate the overall instantaneous moduli of polycrystallinesolids or composites from the mechanical behaviour of their constituents. Manymodels have been developed for this purpose (Voigt 1889; Reuss 1929; Boas &Schmid 1934; Bruggeman 1934; Huber & Schmid 1934; Boas 1935; Taylor 1934,1938). Taylor (1938), in a pioneering effort, proposed a rigid-plastic model forpolycrystals, which was later generalized by Bishop & Hill (1951). For elasticcomposites, Voigt (1889) and Reuss (1929) each developed a model, and theirresults were examined and expanded by Hill (1952).

An essential requirement is an averaging process whereby, for polycrystallinesolids, for example, the rate stress-strain relations are deduced from the 'elastic 'lattice distortion, and from the 'plastic' deformation by slip on crystallographicslip planes, of the single crystals in the aggregate. Averaging methods that havebeen used seek to account for the interaction of a grain and its surroundings, whichmust result in a compatible overall deformation. The early work of Batdorf &Budiansky (1949) addressed deformation regimes with elastic and plastic strainsof comparable magnitudes. A significant step has been the development of 'self-consistent ' approaches originated by Hershey (1954), Kroner (1958, 1961), and[ 87 1


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88 T. Iwakuma and S. Nemat-NasserBudiansky & Wu (1962) and, with considerably greater bearing on later develop-ments, by Budiansky (1965) and Hill (1965a,b). A somewhat different techniquehas been used by Lin (1957, 1971)who generalized Taylor's work to include elasticeffects. Hutchinson (1970) has given a detailed illustrative account of three majoraveraging techniques att ributed to Lin, Budiansky-Kroner-Wu, and Hill, and hasrevealed that Hill's self-consistent model provides less stiff overall responses.

Another requirement for the estimate of the overall properties of polycrystallinesolids is the description of the behaviour of the single crystal constituent. This hasbeen pioneered by Taylor & Elam (1923, I 925, 1926) and examined extensively byothers, both theoretically and experimentally, as reviewed by Kocks (1970) (seealso Kocks 1958). Various hardening rules have been discussed by Hill (1966),whoproposed a general type of cross-coupling between slip and hardening on differentslip systems, and studied the relation between the hardening rule and the mannerin which the stress rates bear on the corresponding slip rates.A key role in most of these contributions is played by Eshelby's solution of thestate of stress, strain, and energy of a single ellipsoidal inclusion that undergoes auniform transformation strain in an infinitely extended linearly elastic homogeneoussolid (Eshelby 1957, 1959). With the aid of this solution, or an equivalent methoddue to Hill (1965a), or other corresponding solutions for anisotropic materials (see,for example, Bhargava & Radhakrishna 1964; Willis 1964; Chen 1967), the localstresses and strains are related t o the overall applied stresses and strains. In theself-consistent approach of Budiansky-Hill-Kroner, the grain is assumed to bewithin a matrix that possesses the overall average macroscopic moduli, and inthis manner the interaction and compatibility of deformation are taken intoaccount to a certain extent.

For elastic composites the estimate obtained by this method seems to becomeless accurate as the volume fraction of the inclusions (or voids) increases. In arecent paper, Kemat-Nasser, Iwakuma & Hejazi have shown that an excellentprediction of the experimental results is obtained if a periodic structure is assumed,in which case the transformation strains are no longer uniform within each inclu-sion (Nemat-Nasser et al. 1982).The mathematical problem for the periodic struc-ture can be solved completely and as accurately as desired. The non-uniformtransformation strains, although they complicate the calculation, do not presentan insurmountable mathematical task (Asaro & Barnet t 1975); see Mura (1982)for a detailed discussion and references.

All these developments pertain to infinitesimal deformation theories. Little oralmost no progress has been made in the calculation of the overall elasto-plasticbehaviour of composites and polycrystalline solids when strains are finite, and thedomiilallt components of the overall elasto-plastic instantailcous moduli are of'the order of magnitude of the applied stresses; the fundamental work of Hill(1972), however, does provide a general framework and basic guidance. This isnotwithstanding the considerable progress that has been made in quantifying thebehaviour of single crystals at finite strains by Alandel (1965), Hill (1966), Hill &


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89inite elastic-plastic deformation of polycrystalsRice (1972), Zarka (1973), Asaro & Rice (1977), Havner & Shalaby (1977), Asaro(1979), Nemat-Nasser, Mehrabadi & Iwakuma (1980), and Havner (1982a).

At finite elasto-plastic deformations, the shapes of individual grains are alteredduring the course of flow and lattice orientation changed because of finite rota-tions; this leads to 'texture'. The plasticity of textured solids recently has beenaddressed from a phenomenological point of view by Bassani (1977)and Hill (1979).But no micromechanically based calculations with due account of the aspects offinite deformation seem to exist.

The main purpose of the present work is to calculate the instantaneous moduliof polycrystalline solids (and composites?) which undergo (quasi-statically andisothermally) finite deformations, and cause residual stresses of differing magnitudesto develop within grains or constituents of different orientations. The grains mayattain different geometries and orientations in the course of plastic flow. Thesethen may lead to overall instability by localized deformation. The analysis is basedon Hill's averaging theorems, a general formulation of the finite elasto-plasticdeformation of single crystals by lattice distortion and crystallographic slip, andthe calculation of Green's function for incremental deformations superimposed onan initially uniformly (but not isotropically) and arbitrarily stressed solid. The rateproblem is cast in terms of the nominal stress rate. The corresponding instantaneousmoduli that relate the overall uniform nominal stress rate to the correspondinguniform overall velocity gradient are estimated.

Detailed numerical results for uniaxial loading of polycrystals are presented fora two-dimensional illustrative example, where two slip systems within each grainare assumed. This is the simplest case which involves all aspects of the consideredproblem and, in particular, brings into focus factors tha t are inherent in a completefinite deformation formulation of the problem. Since the overall loading is uniaxialand the overall deformation is planar, eight out of the total sixteen global moduliremain non-zero. An interesting feature of the solution is that, even though thechanges in the shapes and the orientations of grains are very small, and the residualstresses in grains are of second order relative to the crystal elastic moduli, theireffect on the overall response is of the first order, as it is precisely these factors thatcause the loss of overall stability by localized deformations. Indeed, if the changesin shape and orientation, and the residual stresses in the crystals are neglected, noloss of stability is predicted. Notation

Throughout, componentswithrespect to a fixed rectangular Cartesian coordinatesystem with coordinate axes xiand unit base vectors e , , i = 1 , 2 , 3 , are used. Localfields (within grains) are denoted by lower case letters, and the overall (uniform)quantities are designated by the corresponding capital letters; for example velocity,nominal stress rate, deformation rate, and spin are represented by vi,ni j ,d i j , andzuiirespectively over typical grains, and by 4, Ti j ,D i j ,and yjfor the overall solid.Various moduli are denoted by capital letters, with a superimposed i2 when the

t To limit t he size of th is paper, results for co:nposites ~vil l e reported elsewhere.


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90 T. Iwakuma and S. Nemat-Nasserquantity refers to the typical grain R ; for example the instantaneous moduli inthe relations ri, = FS,,v,, ,and iy j= jk,V,,,,where repeated indices are summed.Averages taken over all crystal orientations and shapes are represented by (. . ;)e.g. = (n i j ) .

Features essential for the description of single crystals which flow by slip oncrystallographic planes, accompanied by the accommodating elastic lattice dis-tortion (both a t finite strains) have been disussed by Mandel (1965),Hill (1966),Hill & Rice (1972), Zarka (1973),Asaro & Rice (1977), Havner & Shalaby (1977),Asaro (1979), Havner (1979), and Nemat-Nasser, Mehrabadi & Iwakuma (1980)whose studies follow early developments by Taylor & Elam (1923, 1925, 1926)and Kocks (1958) (see also Kocks 1970). Single crystal relations for polycrystalcalculations are briefly reviewed in this section. The presentation follows recentwork by Nemat-Nasser, Mehrabadi & Iwakuma (1980).

I n the microscopic model of elasto-plastic flow of a single crystal, the deformationrate, d i j , and the spin, wij, associated with the velocity gradient,

are each thought to consist of a plastic (denoted by a superscript p) and an elastic(denoted by a superscript e) contribution,

di j=d7j+diP j and 1.3 23 (2.2a,b). .=U.'?.+U.'?.23 2where the additive decomposition of rates follows rigorously from the choice ofvariables and the structure of the theory (Hill 1950, 1958; Hill & Rice 1972; Ifandel1974, 1981 Nemat-Nasser 1979, 1982, 1983). In (2.1) the comma followed by theindex signifies partial differentiation with respect to the corresponding coordinate.

Consider a deformed crystal and let sz be the unit vector in the glide direction onthe a th slip plane of the unit normal n?. Let j" be the slip rate of the a th system.Then dB = p Q y , wFj = wri y , (2.3)where a is summed over all active slip systems, and the notation

p a = &(s;nj"+ ST,:), wzj = &(s;ng- sj"n;), with no sum over a , (2.4a, b)is used.

Traditionally, plastic volume changes have been ignored, as they are oftennegligibly small. Recent experimental observations on super-alloys and other high-strength metals reveal that small voids are often generated at the intersection ofslip bands and non-metallic inclusioils on grain boundaries, which result in afinite, though small, plastic volumetric expansion (see Dyson, Loveday & Rodgers1976; Kikuchi & Weertman 1980; Saegusa, Uemura & Weertman 1980). To accountfor this and, at the same time, to i)rovide potential application to non-metallics,i.e. jointed rocks and granular materials, terms n%y tan v, +sf sj" tan v, +S i j tan v,


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Fi ni te elmtic-plastic defor matio n of polycrystals 91(no sum over a )may be added to the right side of (2.4a),where Sij is the Kroneckerdelta (Nemar-Nasser I 983).

Let ui j denote the Cauchy stress.It is natural to employ stress rates co-rotationalwith elastic lattice distortion for the local variation of tractions, as observed by Hill(1966) and used by Asaro (1979) to describe the local elastic response,

The local stress variation is then expressed by

where LGklrepresents the elastic moduli of the crystal with the usual symmetriesand with, at most, 21 independent components (see, however, Hill 1975) whichmay be reduced further as a result of crystal symmetries.

The dynamics of slip must be in accord with assumed plastic rules (Bishop & Hill1951). A simple model emerges from the Schmid law, and results in normality (Hill1966). According to this model, a slip system is inactive in unloading or when theshear stress falls below the current yield stress. Then, for a typical active slip,

where ra = uijs$ nja (no sum over a ) is the resolved shear stress in the slip direction,h a p is the hardening parameter on the ath system produced by plastic flow of thePth system, and /3 in (2.7) is summed over all active systems. To obtain f a , variousassumptions may be used to calculate the rate of change of vectors na and s"(Asaro & Rice 1977). The calculation based on the lattice elastic spin, i.e.

i-L - w;j sj", h$ = 23 (2.8)e . m ?3 3 vsimplifies the results and has natural appeal. Indeed, then ia= uijpFj, and, with7% denoting the current yield stress in shear, one arrives a t the following flowconditions : j" = if 2 3 23 < I -$ , 0 < u..p"

v= 0 if 0 < a . z~ = r$ and uijpFj < h a p y a ,3. pa . (2.9)

vp > o if 0 < uijp$ = r$ and uijpz7= h a p j p ,1here p$j is defined by (2.4a).If potential deviation from the Schmid law, inherent in frictional or pressure-

sensitive materials, is to be included, then one definesqij - 1- (8%nja +sjan$) + n$ nja tan 7 ,+6ij an r 2 , (2.10)vand sets ia= uijq$ so that pZj everywhere in (2.9) is now replaced by q2j. For

application to geomaterials, tan^, in (2.10) may be viewed as the coefficient of thesliding friction? (being zero when normal stress is tensile, i.e. when uijn$nja > 0).

t For a macroscopic formulation of plastic flow of frictional materials, see Nemat-Nasser &Shokooh (1980)(see also Nem ~t- Nas ser 983).


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92 T. Iwakuma and S. Nemat-NasserFor metals, the last term in ( 2 . 1 0 )is in accord with the suggestion that hydrostaticpressure affects the motion of dislocations and hence the corresponding plastic slip.

To render the final equations more general, in what follows the tensor q t j is usedin place of prj to characterize the flow conditions (2.9).The relation between a prescribed crystal strain rate and the associated slip ratesdepends on the structure of the hardening matrix ha@and, particularly, on themagnitude of the off-diagonal elements relative to the diagonal ones (Hill 1 9 6 6 ) .Here, the possibility of a non-unique relation is noted where other considerationsmust govern a specific selection among all possible ones (Havner 1 9 8 2 b ) ; seeAppendix A.

Substitute from ( 2 . 3 ) and ( 2 . 6 ) into the right side of ( 2 . 2 a ) , and use the flowconditions ( 2 . 9 )with pTj replaced by qFj to arrive a t

Denote the inverse of N a p by Map, and from ( 2 . 2 )to ( 2 . 1 1 ) obtain

where the Jaumann rate of the Cauchy stress is

and a and /3 are summed over all active slip systems.Since the nominal stress rate nij is?nij= $ i j+u i j d k k -d i k u k j+~ j ku k i , ( 2 . 1 4 )i t follows tha t

n . .3 = F c k l ~ k , l , (2 .1 5 )wherep$kz = L g k l - $(Sik&n t8 i18km) umj + $(8j;.k81m-8jlSkm) umi

- ( L $ p g ~ $ ~ - ~ { i & p f ~ ? p - ak l ) ( 2 . 1 6 )upj+u& u p + )MaPq!nn(L%nkl urn,and where some symmetries of LCklare used.$

Within a continuum approximation, micro-heterogeneities at grain level areaveraged to arrive at 'material neighbourhoods' which, to some linear scale, areregarded as locally homogeneous. The linear scale then depends on the dimensions

t n,, is the jt11 component of the tractions transmitted across an elementary area whichIn the reference state 1s normal to the tth coordinate direction. n,, is not symmetric, Ingeneral. When t he reference stat e colncldes with the current state , then the nomlnal stressequals the Cauchy stress but their rates a re not the same, as is evldent from (2.14).$ Note tha t p,4, 1s zero ~f ( 2 . 4 ~ )s used, but ~tis non-zero when dllatancy 1s included; seethe comments that follow (2.4).


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93in ite elastic-plastic deformatio n of polycrystalsof the constituent grains at the microscale. For elasto-plastic polycrystals the matterhas been discussed by Bishop & Hill (1951);Hill (1972); Havner (1974),and , morerecently, Havner ( 1 98 2~ ).n what follows, a representative volume subjected onit s boundaries either to uniform t'raction rates or to a velocity field compatible withuniform overall velocity gradients, is considered. Theri the overall nominal stressrate qjmeasured with respect to the current configuration as reference is given bywhere ( 4 . .) denotes volume average taken over the reference volume.

Let K, be the overall velocity gradient defined byand write

The aim is to estimate the instantaneous modulus tensor Fijkln terms of that ofsingle crystals Fz.kl,equation (2.16), by suitable averaging over all crystal orien-tations and shapes. To this end, Hill's self-consistent method is used, which requiresan examination of the following auxiliary problem.

Consider an extended solid,D, with overall moduliFijk,nder the overall nominalstress rate &j, containing a single ellipsoidal crystal, Q, of moduli, Fsk l .The stressrate is

and across the boundary aQ of the crystal, traction rates must remain continuous.It turns out that , as in linear elasticity, in the present general setting the velocitygradient in the crystal is uniform for ellipsoidal shapes ( $ 4 ) .

In view of this, one seeks to calculate the concentration tensor,st$kl,which relates the uniform velocity gradient in the crystal to the far-field uniformdata.

Prom (3.6), the nominal stress rate in the crystal is

and, therefore, (3.1) and (3.3) require

in accordance with self-consistency.It remains to express the concentration tensorin terms of the crystal and the overall moduli.


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94 T. Iwakuma and S. Nemat-NasserI n view of the linearity of the boundary-value problem described by (3.4) and

(3.5), t is expedient to consider a solution uniform everywhere,

and superimpose on it a perturbation solution,

where v&(x) is the Eshelby-type transformation velocity gradient , identically zerooutside of Q, and, as remarked before, turning out constant within R or ellipsoidalregions.

Let %&(x- x') be the Green function in an unbounded uniform medium, satisfyingwhere 8(x- x') is the Dirac delta function, and observe th at the solution to the fieldequations, n$!!i= 0, is given in view of (3.10) bywhere

It is convenient to introduce a Hill-type constraint tensor, Fz k l ,by

and observe that the nominal stress rate within the crystal is, by superposition,

Substitution into ( 3 . 4 ~ )ields (3.6),where the concentration tensor is

The concentration tensor is affected by the overall and the local moduli, as wellas by the constraint tensor FGkZwhich itself depends on the overall moduli and onthe aspect ratios and the orientation of the ellipsoid fi : equations (3.13) and (3.14).On the other hand, from (3.15), he self-consistency conditions (3.1)and (3.2) equirethat

{ F Z k l v k , ) = W z k l ) %,l ,which is satisfied when 9& is common to all crystals. Observe th at this require-ment is also shared by the small-deformation, self-consistent account of the poly-crystal, where, as pointed out by Hill (1965a, p. 93), single crystals are regardedas spheres or as ellipsoids with corresponding axes aligned. Fortunately, in thepresent finite deformation case, the changes of shape and orientation induced bydeformation of initially spherical grains are rather small compared with variationsin the moduli from grain to grain, and therefore the use of an average (in somesense) concentration tensor may be viewed as justified. At any rate, this or an equiva-lent requirement is an integral part of the self-consistent method.


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95inite elastic-plastic deformation of polycrystalsBased on these comments, the concentration tensor is redefined as

d $ k ~= g $ m n +F t . m n ) - ' ( g % n k l + Snlnkl), (3.17)where we choose to identifyg$,,byDirect calculation now shows that (4yk l ) = Iijkl, he identity tensor.

An alternative approach is to normalize directly the concentration tensor.01$kl, -

d $ k l = d$mn(dmfikl)-l, (3.19)-and to useStklo calculate the local velocity gradient. This method has been usedby Walpole (1969) to ensure self-consistency in estimating the overall moduli of alinearly elastic polycrystal of non-aligned ellipsoidal constituents.Equations (3.17) and (3.8) (or (3.16), (3.19), and (3.8), if we employ Walpole'smethod) complete the sought objective. The actual calculations, however, willrequire numerical effort, even in the simplest case, since the unknown overallmoduli, Fi jk l, also occur in the right side of (3.8) through the specification of theconcentration tensor by (3.17) (or (3.16)). In the self-consistent method used byBudiansky & Wu (1962) and Hutchinson (1964), the concentration tensor is cal-culated on the basis of an inclusion embedded in a solid whose moduli coincidewith the elastic moduli of the matrix. I n this approach the concentration tensor isgiven in terms of Hill's constraint tensor, calculated on the basis of the elasticmoduli of the polycrystal or the composite. This means that, in place of FntnkZn(3.13) and the similar quanti ty in (3.14), the elastic moduli of the polycrystal areused. Note that in the present case, and in general, the overall instantaneous elasticmoduli do not equal those of the single crystal.

I n the phenomenological approach, the elasto-plastic constitutive relations aregenerally stated in terms of a relation between the co-rotational (or Jaumann) rateof change of the Kirchhoff stress and the deformation ra te tensor,

where Tijs the Kirchhoff stress, and where

where LCij is the Cauchy stress,

andqjkZs a homogeneous function of degree zero in D ij (Hill 1959).Whereas (3.22)directly follows from (3.2),equation (3.21), in general, does not hold for the averageof the corresponding local quantities. This is clear from the expression for thelocal material rate of change of the Kirchhoff and Cauchy stresses,


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96 T. Iwakuma and S. Nemat-NasserSimilarly, the material and Jaumann rates of the overall Cauchy stress, iijndi i j ,when defined globally, are not t he averages of th e corresponding local quantities.These stress measures must be deJined in terms of the overall nominal stress rateand the overall velocity gradient.Guided by (3.23), one may introduce the material rate of the overall Kirchhoffstress by

T~~= iijzijDkk= $(aijflji+K,kzkj (3.24)+5,,Jki),which ensures the symmetry of both the Kirchhoff and the Cauchy stresses. Thesestresses are then calculated incrementally from definition (3.24). For example,with t as the load parameter and the Cauchy stress, Zij(t),known, after an incrementof loading At, one obtains

Zij(t+At) = &(t) +&(t) A t +O(At2). (3.25)From (3.3), (3.20), and (3.24), i t follows that

% jk l = 8 ( 6 j k l +Fjikl + zkj &il+ zkiajl) K,l. (3.26)Since the left side is independent of the overall spin, (3.26) yields the followingrestriction on the overall moduli Sijkl:

&jk l +FjiklZkjail+Z;ci 8jl = +Fjilk (3.27)Zliaik+Zli Bjk.Then is defined by

Note tha t these moduli are not necessarily symmetric with respect to the exchangeof the leading and the terminal pairs of indices. This is expected, because the corres-ponding local moduli also lack this symmetry unless a special stress state is involved.With reference to (2.12), observe that terms involving the stress in the right-handtensor of the instantaneous modulus render this modulus non-symmetric withrespect to the exchange of the leading (ij)and the terminal (kl) pairs of indices.Hill (1972)has shown that if the symmetry exists locally, then it will also existglobally.

Since the current configuration is used as the reference one, 4. = Tii = Zii andnij = 1I =.. cij.Consistent with uniform tractions prescribed on the boundary of arepresentative macro-sample, we thus have

This suggests that in place of (3.25)one may first compute efj(t+ A t ) fromefj(t+ At) = cij(t)+ $ij(t) At +O(At2)and then calculate Zii(t+ At ) = (cii(t+At)).

Consistent with this, the overall Cauchy stress rate may be defined by


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97in it e elastic-plastic deform ation of polycrystalswhich, in general, does not satisfy (3.24). Hence the overall Cauchy stress and itsrate obtained from (3.25) and (3.24),respectively, will not in general coincide withthose calculated by using (3.29) and (3.31), although the differences ought t o berather small. Note th at operations (3.25) and (3.30) are valid for t he Cauchystress but not, for example, for the nominal stress. The q uanti ty nij(t)+ni,(t)At+O(At2) s the nominal stress a t time t +At, but referred to th e configuration ofthe solid a t time t.

The numerical results reported in 7 are all based on definitions (3.24)and (3.25).

4. GREEN'SF U N C T I O NEquation (3.12)requires explicit knowledge of the quantity

J..umn = J %% ,m i ( $ - xl)dx',n

which, as will be shown below, is independent of x when Q is ellipsoidal. AlthoughGreen's functions for unbounded media have been calculated for isotropic as wellas for various anisotropic solids, th e case in question presents considerably greateranalytical difficulties, as the number of moduli involved depends on the state ofstress and varies in the course of deformation. None the less, the analytical procedureoutlined for anisotropic elasticity by Stroh (1958, 1962), l'illis (1961), nd Kinoshita& Mura (1971) (see also Mura 1982) provides guidance in the present context.

With th e aid of a Fourier transform, from (3.11) t follows that

whereNij = $ E ~ ~ ~ , E ~ ~ ~ K ~ ~ K ~ ~ofactor (Ki j) , ( 4 . 3 ~ )=D = r,6 ikmejln Kk lK n r n = det ( K f j ) , (4.3b)

K 4 j = %f j l 6 k & > ( 4 . 3 ~ )where eijk is the permutation symbol.

Equations (4.3) suggest a natural contact between the structure of the Greenfunction and the possibility of loss of stability of th e uniformly stressed overallpolycrystal, because the nature of the roots of the determinant D in (4.3b) definesthe ellipticity, parabolicity, or hyperbolicity of the operator in the left side of (3.1 ) .Necessary conditions for the possibility of jumps of certain components of thevelocity gradient across a discchtinuity plane coincide with the existence of thereal roots of D = 0. This rather interesting observation emerges as an inherentpa rt of the present finite deformation theory. It has no counterpart in previousworks on polycrystals and composites, which have been based on infinitesimaldeformation. As discussed in the next section, if the changes in the grains' geo-metries and orientations and their residual stresses are included in the calculation,then indeed at a certain strain level in uniaxial extension the overall moduli tensor


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98 T. Iwakuma and S. Nemat-Nasserceases to correspond to an elliptic operator, and the overall uniform deformationstate ceases to be stable, leading to possible localized deformation. (For generaldiscussions of discontinuity and localized deformation, see Thomas (1961) andHill (1961, 1962)).

Substitution from (4.2) into (4.1) yields

where fJ(4) is a unit sphere in the


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99in ite elmtic-plastic deform ation of polycrystalspresented by Hill ( I 9 6 2 ) one seeks conditions under which the governing equationsfor the rate of deformation from a given initially stressed state cease to be elliptic(Hill& Hutchinson 1 9 7 5 ;Storen & Rice 1 9 7 5 ;Rice 1 9 7 6 ;Needleman & Tvergaard1 9 7 7 ; Hutchinson & Neale 1 9 7 8 ; Nemat-Nasser, Mehrabadi & Iwakuma 1 9 8 0 ;Iwakuma & Nemat-Nasser 1 9 8 2 ) . I n the present work this has a fundamentalbearing on the calculation of the overall moduli by means of Hill's self-consistentmethod, because such loss in ellipticity prevents further calculation of Green'sfunctions and the corresponding concentration tensors.

Locally, the instability condition is attained a t individual grains during thecourse of plastic flow, before the loss of overall stability. However, the constraintimposed by the surrounding grains renders the overall response stable. As thedeformation proceeds, more grains become locally unstable, and the constrainingeffect of the surrounding grains diminishes accordingly, leading eventually to globalinstability.This global instability is closely related t o the location of the poles that enter thecalculation of the concentration tensor. Let v be the unit normal to the surfaceacross which certain components of the velocity gradient may admit jumps. Thecontinued equilibrium condition necessarily requires the continuity of the tractionrates across such discontinuity surfaces, which in the present case is guaranteed by

where [. . -1denotes the jump of the enclosed quantity across the discontinuitysurface. If q is the magnitude of the jump,

[v,,2 1 = r k 2 , ( 5 . 2 )then ( 5 . 1 )yields %Fiiklvi lTk = 0. ( 5 . 3 )For non-trivial values of q, the determinant of the coefficients in ( 5 . 3 )must vanish,

The necessary condition for localization therefore is that ( 5 . 4 )yields real orienta-tions v. This is precisely equivalent to the existence of the real roots of D = 0 inthe calculation of the Green function, ( 4 . 3 6 ,c ) . Localized deformations becomepossible, therefore, when a pole in either the complex c-plane or the complex-{-plane is located on the real axis; see ( 4 . 4 ) - ( 4 . 9 ) .

For the two-dimensional case, the necessary condition ( 5 . 4 )becomesel vt - (e l l + c 18) f v2- (c13+ c17)V: V ; + ( c g+ c 14) 1V$ + c2V: = 0, ( 5 . 5 )

where the coefficients,c i , are given by ( B 3 ) of Appendix B . As is evident from ( B 5 )and ( B 6), if 2; is a root of Do = 0, then the characteristic equation ( 5 . 5 ) admitsreal roots for ( v l / v 2 ) ,when lzEl -t 1.

Since el is expected to remain non-zero during the course of deformation, ( 5 . 5 )may be regarded as quartic in ( v 1 / v 2 ) .f the instantaneous moduli change smoothlywith the deformation, then the condition for the existence of a double real root for


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100 T. Iwakuma and S. Nemat-Nasser( 5 . 5 ) defines the incipience of localization. Indeed, for the two-dimensional illus-trative example of uniaxial loading considered in 5 7, the symmetry reduces ( 5 . 5 )to

c1(v l / v2 )4- (el3+ e l ,) ( v 1 / v J 2+ c2 = 0. ( 5 . 6 )If the instantaneous moduli change smoothly as the axial loading continues, thenthe inception of localization is marked by the vanishing of the discriminant

d = (c13+ - 4c1c2, ( 5 . 7 )associated with the existence of a double root for ( 5 . 6 ) .In terms of the location ofpoles for the calculation of Green's function, the vanishing of d corresponds to thestate when two poles in the z2-plane, one inside and the other outside of the unitcircle, coalesce on the circle.

6. S HAP EC H A N G E A N D R O T A T I O N O F G R A I N SSince the grains are assumed t o be ellipsoidal in shape, their orientation can be

defined by the orientation of the ellipsoid's principal axes. The method does notinclude the size effect in the absolute sense. However, the effect of a change in shapecan be included by means of the change in the relative dimensions of the ellipsoid.

The self-consistent method assumes uniform deformation of each grain. Thissuggests that the principal axes of the ellipsoidal grain can be assumed to coincidewith the principal axes of the Eulerian triad associated with the corresponding localdeformation gradient.

Initially, all grains are assumed to be spherical, and the deformation gradient isX i , A = S ~ A , ( 6 . 1 )

where x measures the current position of a particle which initially is a t X; xi arecomponents of x, and X, are those of X, i ,A = 1,2,3.

Let the deformation gradient be x i , , ( t ) a t a certain time t for a given crystal. Thedeformation gradient after incremental loading a t time t + At then is

xi,A ( t+ A t) [Sij+ vi,j A t] xj,A ( t ) . ( 6 . 2 )With the aid of polar decomposition, define the rotation R and the right-stretchtensor U as

xi,^ = R ~ BU ~ ~ , ( 6 . 3 )R - l = RT, detIRI = $ 1 . ( 6 . 4 )

The current orientation of the principal axes of the ellipsoid is given by

where N(a' , a = 1,2,3,are the principal directions of U ,

here A(,) are the corresponding principal values. Thus, the current orientation ofthe grain is given by the Eulerian triad d a ) ,nd the corresponding shape is givenby the ratios of the principal stretches, e.g. h( , ) / A ( , )and h(.,)/h(,).As deformation


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101in ite elastic-plastic deform ation of polycrystalsproceeds, grains that are assumed to be spherical initially deform by differentamounts and at tain different orientations.

Observe that in the context of the present rather general formulation, such effectsas initial anisotropy or initial texture can be included by considering suitablyorientated grains of suitable relative dimensions.

IFIGURE. Definitions of kinematic quantities in a typical grain.

It should be emphasized that the assumption of an ellipsoidal grain with a uniformdeformation gradient used in the self-consistent calculation is appropriate, as themost general finite deformation (6.3) maps materials within an initially sphericalneighbourhood into an ellipsoid. In this manner the continuum dimension of thelocal neighbourhood' is defined by the average grain size of the polycrystal.

Explicit results are presented in this section for plane deformation of face-centredcubic (f.c.c.) polycrystals. For all crystals in the aggregate, the plane of deformationis defined by the local (100)- and (011)-orientationswith two slip lines (and hencefour slip systems) symmetrical with respect to the (100)-axis (denoted by itl) a tan angle close to $ = 35" which follows from the fact tha t the angle between twoslip planes is approximately 70"; see figure 1. It has been suggested (Asaro 1979)that this may be a reasonable two-dimensional model for f.c.c, single crystals. I nwhat follows we first summarize expressions for this two-dimensional problem, andthen present numerical results, where the effect of deviation from $ = 35" is alsoconsidered.

t It wlll be ~l lu str ate d y numerical examples th at Q slightly larger (smaller) than 35"leads to a 'stiffer' ( 'sof ter ') polycryst~11111~esponse; see figure 2 b .


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102 T. Iwakuma and S. Nemat-Nasser7.1. Constitutive equations for single c~ystals

Let + be the angle between the local ?,-axis and the global xl-direction. The unitvectors defining the slip systems are

- sin ($- $1s1 = - s3 =(7.1)

s 2 = - s4 =Subst itution into (2.4) yields

The spin induced by elastic lattice distortion is

which follows from kinematic assumption (2.8)and 4 E 0. In the course of deforma-tion, the individual crystal orientations are calculated by updating the angle $,similar to (3.25),as

$(t +At) 2 +(t) + $ ( t ) At. (7.4)The elastic moduli of a cubic crystal are usually defined in terms of three elastic

constants Cll, C,,, and C4, (in Voigt's notat ion). Let ZRbe the elasticity tensor ofa typical crystal in the usual crystal directions in three dimensions. Then

Since the local El, E,-coordinate system is defined by rotating th e crystal coordinatesabout the (100)-direction by 45", the elasticity tensor in the local ?{-coordinatesystem, En, is expressed as

Then L" in (2.6) and (2.16) s the crystal elasticity tensor in th e global xi-coordinate.It is related to z" byLz'kl = ail,a js aknt @q?nn, (7.7)


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103inite elmtic-plastic deformation of polycrystalscos y9 -sin $where sin y9 cos y9I.For the isotropic crystal,

wherep and h are the Lam6 constants. The difference, Cll- (C12+2C4,), characterizesthe degree of elastic anisotropy in cubic crystals.

For small deformations, where the residual stresses are ignored, the overall elasticparameters of the polycrystal can be calculated directly from the elasticity of thesingle crystal constituents (Hershey 1954; Kroner 1958).For finite plastic deforma-tions, where residual stresses of differing magnitudes accumulate in crystals ofdifferent orientations and shapes, the estimate of the effective overall elasticityrequires an incremental elastic unloading. The overall elasticity tensor changes inthe course of deformation and, in fact, becomes anisotropic (formation of texture),even though it is initially isotropic because of the random distribution of theorientation of the crystals.

To render the governing equations dimensionless, it is convenient to introducethe yield shear strain,

Yo ~$/c44, (7.9)and set -S i j = f l i j /~$ , 5" = ?'"/Yo, dij = dij/yo, (7.10)Gj= dzj/yo, %,pj = dFj/yo, kaP= haP/C4,,I

where 7% is the common initial yield shear stress. All second-order terms will nowbe proportional to the parameter yo. These terms are not present in the smallstrain theory of crystal plasticity (Hutchinson 1970). Hence to retrieve results forthe case where residual stresses as well as the effects of crystal rotation and shapechange are ignored, simply exclude terms which are proportional t o yo. From (2.16),the non-dimensional instantaneous moduli are expressed as follows:

where the fact tha t pTi = 0 is used and wheregals the inverse of m"P defined by

Note that we are excluding possible pressure sensitivity and plastic dilatancy.I n the case of elastically isotropic single crystals,


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104 T. Iwakuma and S. Nemat-Nasserwhere v is the Poisson ratio and g a p s the inverse of

f l a p = kaP+2 p f jp f j .The flow conditions ( 2 . 9 ) ,with 7% = 7%+ haPyB, become

p = 0 if 0 < s i j p f j < I +kaP[P,v& = 0 if 0 < s i j p$ = 1+ kaP


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105inite elastic-plastic deformation of polycrystalstion of the major principal axis of the local Eulerian ellipse (in two dimensions)relative to the global xl-direction; figure 1. Then

where a$ is obtained from (7.7) by replacing y? with P and by taking its transpose.Finally, the condition for the objectivity of the global instantaneous moduli,

(3.27),becomesF1212 +F2112 +7oX1l = F1221 +F2121f 7 0 8 2 2 9 (7.23)where Sij- Zii/r;. (7.24)

7.3. LocalizationThe necessary condition for the inception of localized deformations is given in

two dimensions by (5.5).If we can estimate the limiting values of x i , the poles forthe integrand in (4.9)on the unit circle, then the orientation of localization relativeto the local x,-axis, 0, becomes

tan 0 = v1/v2 = Im [zi]/a(l -Re [xi]) (7.25)(compare (5.5)with Do= 0 in Appendix B),where a is the aspect ratio of the grain,(B 1).Since the evaluation of the integral (4.9) s done in the local coordinates, thelocalization orientation relative to the global x,-axis, 0 , s given by

7.4. ResultsNumerical results are obtained for uniaxial tension. All quantities are referred

to the current configuration and are then updated incrementally after each incre-mental loading. For each crystal, local deformation and stresses are calculated stepby step, and then the overall instantaneous moduli are estimated by (3.8).

First, typical results are discussed below for the case in which the residual stressesand rotations and shape changes of grains are ignored, to compare the numericalresults with those presented by Hutchinson (1970). Then the general case isexamined.To neglect residual stress and grain rotation and shape changes, exclude second-order terms; these are proportional t o yo. For isotrbpic single crystals, the Poissonratio v = 4 s considered. For the anisotropic case, the elastic parameters associatedwith copper,

C12/Cll = 0.722, C4,/Cll = 0.447, (7.27)are used. Figure 2 shows the stress-strain curves for the locally non-hardeningmodel and compares results with the three-dimensional calculations by Hutchinson(1 97 0) Table 1 shows some examples of local quantit ies and several cases of theoverall histories. In these tables, ,ITdenotes the number of grain orientations chosen


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T. Iwakuma and S. Nemat-Nasser

Hutchinson (1970)---- Bishop & Hill (195 )

1 2 3 4

%I70FIGURE curves; N 46, second-order termsa, 6 . Tensile stress-strain =

are excluded, no hardening.

between 0" and 90, and 1 and 6, are the elongations in the xl- and x,-directions,evaluated by

= n( l+Dl lAt ) -1 , 6, = n ( l+DZ2At)-1, (7.28)where II denotes th e product taken over all preceding (variablel-) loading steps,At; the parameter 7, is the effective yield shear strain defined by

where ,Z is the initial overall shear modulus of the polycrystal calculated in a self-consistent manner, as will be discussed later. I n the case of elastic isotropy, 7, isequivalent to yoof (7.9).

t The 'loading' steps are adjusted to ensure accuracy to within a prescribed limit.


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107in ite elastic-plastic deform ation of polycrystals(Second-order terms a re excluded ; elastically isotropic grains.)

TABLEb. GLOBALQUANTITIES ;N = 46(Second-order terms are excluded; elastically isotropic grains, no hardening.)

no. of grains%/Yo %/YO z11/7; Flll l F121, , , , in plastic stat e0.673 - 0.337 2.020 4.000 1.OOO 00.678 - 0.339 2.035 3.539 0.994 100.683 -0.343 2.044 3.411 0.981 140.698 - 0.354 2.066 3.308 0.956 180.723 - 0.374 2.093 3.238 0.858 220.758 -0.404 2.124 3.202 0.827 240.808 - 0.448 2.162 3.170 0.785 260.878 -0.510 2.207 3.143 0.733 280.968 - 0.592 2.256 3.119 0.668 301.028 -0.648 2.284 3.107 0.538 301.098 - 0.713 2.313 3.086 0.466 321.293 -0.897 2.378 3.067 0.386 341.60 - 1.19 2.458 3.051 0.300 362.04 - 1.61 2.546 3.036 0.212 382.32 - 1.89 2.587 3.022 0.117 383.02 - 2.58 2.650 3.017 0.0885 40

TABLEC. GLOBALQUANTITIES; N = 46(Second-order term s a re excluded; elastically anisotropic g rains, no hardening.)

no. of grainsin plastic stat e

01014182224262830323436384041


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108 T.Iwakumn and S. Nemat-NasserAs is seen from figure 2 a , the two-dimensional results are in good accord with the

three-dimensional ones. Figure 2 b illustrates the effect of varying the angle 4 nthe overall response. Hence it is possible to choose an appropriate 4 to make thetwo-dimensional results analogous to the three-dimensional ones. The choice of4 35" seems to be reasonable. The volumetric change and the change in theglobal instantaneous moduli are shown in figures 3 and 4, respectively.

I isotropiccl / rO

FIGURE = 46, second-order terms a re excluded, no hardening.. Volume changes; N

FIGURE. Changes of global moduli for elastically isotropic case;N = 46, second-order terms are excluded, no hardening.

For the elastically isotropic single crystals, the global instantaneous moduliinitially satisfy

The first equation in ( 7 . 30 )holds throughout the deformation history, whereas


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109inite elastic-plastic deformation of polycrystalsas el becomes large; see table I . Hence the apparent Poisson ratio tends t o the value+,as el increases, and thus the overall response becomes incompressible. Locally,however, the moduli no longer correspond to an isotropic tensor, as illustrated intable I = 4 . 1 2 . The asymptotic results ( 7 . 3 0 ) and ( 7 . 3 1 ) conform toor ~ ~ / y ~expected macroscopic observations.

Observe that, even if individual grains are elastically anisotropic, the globalinitial modulus tensor is isotropic, because of the random distribution of the grains.Indeed, with ji, i?, and ti denoting the overall shear modulus, bulk modulus, andPoisson ratio, respectively, results shown in table 1 c indicate that

These estimates deviate slightly from those obtained by Hutchinson (1970),mainly because of the difference between bhe three-dimensional and two-dimensionalmodels used.From ( 7 . 2 7 )and ( 7 . 3 2 ) ,the effective yield shear strain becomes

As has been mentioned before, no localization of deformation is predicted sincethe effects of residual stresses and of crystal rotations and shape changes areignored in the calculations above. However ( 7 . 3 0 )and ( 7 . 3 1 )indicate tha t

Fllll l / ( l - 2 v )+ a,, q12, l / ( l - 2 v )- a,) F 1 2 1 2 a,) ( 7 . 3 4 )where a, and a, are positive parameters th at become very small a t larger axialstrains. Moreover, figure 4 suggests that a, approaches zero much faster than a,.Therefore by putting

a1 elfrn, a, 6, rn > 0 , ( 7 . 3 5 )and substituting ( 7 . 3 4 )and ( 7 . 3 5 )into the necessary condition ( 5 . 5 ) ,we find

which implies that instability may occur as ellyo -t oo,and tha t, from ( 7 . 3 6 ) ,theorientation of the localization, 0, equals 45". As is indicated by ( 7 . 3 5 ) ,the changein the shearing components ofFij,,s more gradual than th at of the other compo-nents. This effect must be included in the phenomenological constitutive models,otherwise these models will not predict localized deformations even for very largeaxial strains.

Since only a finite number of crystal orientations is used in the calculation, thestress-strain curves are actually piecewise linear, and for large strains especiallythe behaviour depends on the number N of the orientations used.

Figure 5 shows results for the Taylor hardening and elastically isotropic grains,where the 'plastic ' part of the elongation is defined by

9 = 1 - { ( I - v ) / ~ , U )Cll. ( 7 . 3 7 )The yield surface of the polycrystal can be obtained by unloading from various


25/34

110 T. Iwakuma and S. Nemat-Nasserstress states. The evaluation is simplified if elastic isotropy is assumed and theresidual stresses and the crystal rotations and shape changes are neglected. Locallythe stress state in each grain is not uniaxial, and, therefore, the overall yield surface,as an envelope of the local yield surfaces, exhibits a corner and a Bauschinger effectconsistent with the changes of the global instantaneous moduli. A typical result isshown in figure 6.

-.0

--- Hutchinson (1970)0 1 2 3

4'/7OFIGURE curve for elastically isotropic case with Taylor hardening;. Stress-strain

N = 46, second-order term s are excluded, ka@ = 0 . 0 2 .

211 /7$FIGURE . Yield surfaces for elastically is ot ~o pi c ase at s,/y, = 0, 2.02, and 4.12;

A' = 46, second-order term s a re excluded, no hardening.

Consider now the case when residual stresses and crystal rotations and shapechanges are included, i.e. when second-order terms (proportional to yo)are included.The calculation follows the same procedure as before, but close to the inception oflocalization the evaluation of the Green function breaks down, its the operatorPijkla2/axiax,ends to lose its ellipticity. This is manifested by the loss of con-


26/34

111in ite e l as t iq la st ic de formation of polycrystalsvergence of the iterative scheme necessary for the evaluation of the overall instan-taneous moduli. However, the critical axial stress and the orientation of localizeddeformation can be estimated. To illustrate this and related points, consider severalvalues for yo ,e.g.

y o= 0.1,0.05,0.01. (7.38)These are rather large, but serve to identify the main features of the problem.

FIGURE curves for indicated values of y o ( y o 0 when second-order. Tensile stress-strain =terms are excluded); N = 46, no hardening. ( a ) elastically isotropic case; (b ) elasticallyanisotropic case.

Figures 7a, b show the corresponding stress-strain curves for both elasticallyisotropic and anisotropic crystals. These results are based on the concentrationtensor normalized according to (7.19a). The numerical calculation becomes un-stable when e,/yo atta ins the values given in tables 2-4. This instability is associatedwith the finite movement of the poles in the complex z2-planeand with one or moreadditional grains becoming plastic during the iteration procedure. The erraticvariation in the location of poles stems in part from the fact that only a finitenumber of crystal directions, N, are used. Before instability, however, the poles,for the case of elastically isotropic crystals, move continuously along a straight lineemanating from the origin of the complex plane, i.e, on

- (7.39)Values of this angle are given in tables 2-4 for indicated values of yo .Based on this ,we may estimate the localization angle from (7.25) and (7.26) as

@ = arctan [sinB/a(1- cos B)]- p. (7.40)Note t hat (7.40) yields O = 45" for y o = 0, i.e. when residual stresses and crystalrotations and shape changes are ignored; in (7.40),put a = 1 ,P = O0 , and 8 = 90".

const.


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112 T. Iwakuma and S. Nemat-NasserSince the effects of rotation and residual stresses are taken into account in local

constitutive relations, the overall instantaneous moduli show non-symmetry buttend to have forms similar to the local ones.

Note t hat the objectivity condition (3.27) or (7.23) is always satisfied (within theaccuracy of the numerical estimates).

TABLE. GLOBALAN D LOCAL QUANTITIES AT LOCALIZATION, AND INSTANTANEOUS AND ELASTIC MODULI

elastically isotropic anisotropicglobal0.7562.09

8644 local

0 14.0 16.00 13.5 15.61.113 1.125 1.1240 0.61 0.722.10 2.06 2.07

.0.0107 0.0376 0.0300 0 -0.0121 -0.0137

instantaneous and elastic modulinear yield a t in near yield a t in

point localization unloading point localization unloadingpl l l l 3.80 3.17 3.79 2.53 1.99 2.49F z z z z 4.00 3.23 4.01 2.73 2.04 2.74P I 1 2 2 2.00 2.77 2.01 1.39 2.06 1.39FZZll 2.00 2.63 1.99 1.39 1.84 1.39Flzlz 0.901 0.851 0.896 0.569 0.461 0.551p 1 2 2 1 1.10 1.06 1.10 0.783 0.716 0.797

The strong (induced) anisotropy of the instantaneous moduli, as well as that ofthe elastic moduli a t the bifurcation point, is also shown in tables 2-4. The elasticmoduli are obtained by unloading. Also the typical order of magnitude character-izing shape changes and grain rotations is given in these tables.Observe that the axial strain in figures 7 a ,b is normalized with respect to 7,(to yo in the isotropic case and to 1.48y, in the anisotropic case). Hence, the esti-mated axial strains at localization for y, = 0.1, 0.05, and 0.01, respectively, aree, = 7.6, 5.1, and 2.2% in the anisotropic case.

Data presented in figures 7 a , b and tables 2 to 4 are based on (7.19a).Since, inprinciple, (2Zkl)ust equal the identity tensor, deviation from this may beregarded as a measure of the numerical inaccuracies. For the elastically isotropiccrystal, the value of e l l y o to localization increases as yo (th e yield shear strain) isdecreased, especially when hardening is included. Calculations reveal greater


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113inite elastic-plastic deformation of polycrystalsI N S T A N T A N E O U S A N D E L A S T I C M O D U L I

elastically isotropic anisotropicglobal1.012.25

8 7 44

local0 14.0 22.00 12.6 20.91.098 1.104 1.0920 1.77 4.002.19 2.14 2.240.0744 0.123 0.03610 -0.0375 - 0.0654

i ns t an t ane ous and e las ti c modul inear yield a t in near yield at inpoint localization unloading point localization unloading

Fl l l l 3.90 3.06 3.89 2.64 2.11 2.63F z z z z 4.00 3.08 4.01 2.74 2.18 2.75E;l22 2.00 2.92 2.00 1.39 1.94 1.39FZZll 2.00 2.83 1.99 1.39 1.84 1.39F ~ z ~ 2 0.950 0.525 0.944 0.622 0.299 0.616Flzzl 1.05 0.638 1.06 0.730 0.416 0.734

TABLE . GLOBALAND LOCAL QUANTITIES AT J ,OCALIZATION, AND I N S T A N T A N E O U S A N D E L A ST IC h l O D U L I

(S= 46, yo = 0.01.)elastically isotropic anisotropic

globale l i r o 2.15 1.56z l l / 7 0 , 2.54 2.648/deg 89 87O/deg 44 45localIlr,,,ti,,/deg 0 14.0 30.0 0 12.0 20.0$Id% 0 12.8 29.2 0 11.0 29.5a 1.053 1.054 1.038 1.052 1.056 1.035PId% 0 2.53 11.8 0 2.82 12.4&I 2.36 2.29 2.70 2.32 2.25 2.808 2 2 0.217 0.261 -0.132 0.173 0.241 -0.2348 1 2 0 - 0.0799 - 0.174 0 - 0.161 - 0.286

i ns t an t ane ous and e las ti c modul inear yield a t in near yield at inpoint localization unloading point localization unloading

F l l l l 3.98 3.02 3.97 2.72 2.06 2.71Fzzzz 4.00 3.03 4.01 2.74 2.06 2.75F1122 2.00 2.97 2.00 1.39 2.02 1.39FZZll 2.00 2.95 2.00 1.39 1.99 1.39&,lz 0.990 0.201 0.985 0.665 0.169 0.662F,,.,, 0.101 0.227 1.02 0.686 0.197 0.689


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T. Iwakuma and S. Nemat-Nasser

I I I I I I I1 2 3

cllroFIGURE8a. Tensile stress-strain curves for indicated values of yo (upper dot-dash curve is

for the case where second-order terms are excluded) ; N = 46, no hardening, elasticallyisotropic case.

Ii

o* I 4-

I I I I I 10 10 20 30%/To

FIGURE8 b . Tensile stress-strain curves for indicated values of y o ;LV= 46, Taylor hardening with k f f b= 0.01, elastically isotropic case.

numerical stability when the concentration tensor is normalized in accordancewith Walpole's method, i.e. when (7.19b) is used. Figures 8a, b are obtained in thismanner. Table 5 compares some of the data estimated on the basis of the twonormalization methods, ( 7 . 1 9 ~ )nd (7.19b), for yo = 0.01 and no hardening.

Results presented in figures 7 a ,b and 8a, b show that instability by localized


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115in ite elastic-plastic deforma tion of polycrystalsTABLE. EFFECT COMPAREDF NORMALIZATION METHOD, ( 7 . 1 9 ~ )

WITH ( 7 . 1 9 b ) ,ON RESULTS AT LOCALIZATION(N = 46, yo = 0.01, elastically isotropic crystals.)

deformation may s ta rt while the sample exhibits overall positive hardening; in thecase of figure 8 b , instability occurs in the presence of local (and hence a strongglobal) positive hardening.

This work has been supported by the National Aeronautics and Space Adminis-tration under grant NAG 3-134 to Northwestern University.

R E F E R E N C E SAsaro, R. J. 1979 Acta metall. 27 , 445-453.Asaro, R. J . & Barnett, D. M. 1975 J. Mech. Phys. Solids 23 , 77-83.Asaro, R. J. & Rice, J. R. 1977 J. Mech. Phys. Solids 25 , 309-338.Bassani, J. L. 1977 Int. J . mech. Sci. 19, 651-660.Batdorf, S. B. & Budiansky, B. 1949 A mathematical theory of plasticity based on the concept

of slip. Tech. Note no. 1871, Na t. Adv. Comm. Aeronaut.Bhargava, R. D . & Radhakrishna, H. C. 1964 J. phys. Soc. Japan 19, 396-405.Bishop, J . F. W. & Hill, R. 1951 Phil. Mag. 42 , 414-427; 1298-1307.Boas, W. 1935 Helv. phys. Acta 8, 674-681.Boas, W. & Schmid, E. 1934 Helv. phys. Acta 7, 628-632.Bruggeman, D. A. G. 1934 Z. Phys. 92, 561-588.Budiansky, B. 1965 J . Mech. Phys. Solids 13, 223-227.Budiansky, B. & Wu, T. T. 1962 I n Proc. 4th U.S. National Congress of Applied Mechanics,pp. 1175-1185. New York: A.S.M.E.Chen, W. T . 1967 Q. JI Mech. appl. Math. 20, 307-313.Dyson, B. F., Loveday, M. S. & Rodgers, M. J. 1976 Proc. R . Soc. Lond. A 349, 245-259.Eshelby, J . D. 1957 Proc. R. Soc. Lond. A 241, 376-396.Eshelby, J. D. 1959 Proc. R. Soc. Lond. A 252, 561-569.Havner, K. S. 1974 2. angew. iWath. Phys. 25 , 765-781.Havner, I


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T. Iwakuma and S. Nemat-NasserHill, R. 1972 Proc. R . Soc. Lond. A 326, 131-147.Hill, R. 1975 Math. Proc. Camb. phil. Soc. 77,225-240 .Hill, R. 1979 Math . Proc. Camb. phil. Soc. 85, 179-191.Hill, R. & Hutchinson, J. W. 1975 J . Mech. Phys. Solids 23, 239-264 .Hill, R. 8E Rice, J . R. 1972 J. Mech. Phys. Solids 20,401-413 .Huber , A. & Schmid, E. 1934 Helv. phys. Acta 7,620-627.Hutchinson, J. W. 1964 J . Mech. Phys. Solids 12,11-24; 25-33 .Hutchinson, J . W. 1970 Proc. R. Soc. Lond. A 319,247-272.Hutchinson, J.W. & Neale, K. W . 1978 I n Mechanics of sheet metal forming: material be-haviour and deformation analysis (ed.D. P . Koistinen & N.-M. Wang), pp. 127-153. New

York, London: Plenum Press.Iwakuma, T . & Nemat-Nasser, S. 1982 Int . J .Solids Struct. 18, 69-83.Kikuchi, M.& Weertman, J . R. 1980 Scr. metall. 14,797-799 .Kinoshita, N. & Mura, T. 1971 Physica Status Solidi A 5,759-768 .Kocks, U. F. 1958 Acta metall. 6, 85-94 .Kocks, U. F. 1970 Metall. Trans. 1, 1121-1143 .Kroner, E. 1958 2 .Phys. 151,504-518 .Kroner, E. 1961 Acta metall. 9, 155-161.Lin, T. H . 1957 J.Mech. Phys. Solids 5, 143-149.Lin, T. H . 1971 In Advances i n applied mechanics (ed. C.-S. Yih), vol. 11 , pp. 255-31 1. NewYork: Academic Press.Mandel, J . 1965 Int. J . Solids Struct. 1, 273-295 .Mandel, J. 1974 In Foundations of continuum thermodynamics (ed.J . J . D. Domingos,M. N. R.

Nina & J . H. Whitelaw), pp. 283-304. London: MacMillan Press L td .Mandel, J . 1981 Int . J . Solids Struct. 17,873-878 .Mura, T. 1982 ~Uicromechanics f defects in solids. Boston, The Hague : Martinus Nijhoff.Needleman, A. & Tvergaard, V. 1977 J . Mech. Phys. Solids 25, 159-183 .Nemat-Nasser, S. 1979 Int . J . Solids Struct. 15, 155-166.Nemat-Nasser, S. I 982 Int . J. Solids Struct. 18,857-872 .Nemat-Nasser, S. 1983 J . appl. Mech. 50, 1114-1126 .Nemat-Nasser, S., Iwakuma, T. & Hejazi, M. 1982 Mech. Mater. 1, 239-267 .Nemat-Nasser, S., Mehrabadi, M. M. & Iwakuma, T. 1980 In Three-dimensional constitutiverelations and ductile fracture (ed. S. Nemat-Nasser), pp . 157-1 72. Amsterdam : North-

Holland.Nemat-Nasser, S. & Shokooh, A. 1980 Int . J . Solids Struct. 16,495-514 .Reuss, A. 1929 2 . angew. Math. Mech. 9,49-58.Rice, J.R. 1976 In Proc. 14th In t. Congress of Theoretical and Applied Mechanics, vol. 1 , pp.207-220 . Amsterdam: North-Holland.Saegusa, T., Uemura, M. & Weertman, J . R. 1980 Metall. Trans. A 11 , 1453-1458 .Storen, S. & Rice, J . R . 1975 J . dlech. Phys. Solids 23,421-441.Stroh, A. N. 1958 Phil. Mag. 3, 625-646 .Stroh, A. N. 1962 J . math. Phys. 41, 77-103.Taylor, G.I . 1934 Proc. R. Soc. Lond. A 145, 362-404.Taylor, G.I. 1938 J . Inst. Metals 62,307-324.Taylor, G.I . & Elam, C. F. 1923 Proc. R . Soc. Lond. A 102,643-667 .Taylor, G.I. & Elam, C. F. 1925 Proc. R . Soc. Lond. A lb8,28-51.Taylor, G.I . & Elam, C. F. 1926 Proc. R. Soc. Lond. A 112,337-361.Thomas, T . Y. 1961 Plastic $ow and fracture i n solids. New York: Academic Press.Voigt, W. 1889 Wied. Ann . 38,573-587 .Walpole, L. J . 1969 J . Mech. Phys. Solids 17,235-251 .Willis, J.R. 1964 Q . Jl Mech. appl. Math. 17, 157-174.Zarka, J. 1973 J . Me'c. 12,275-318 .


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117in it e elastic-plastic defo rma tion of polycrystals

Depending on the hardening rule considered, Nap in (2.11)may become singular,which would lead to possible non-uniqueness in the choice of the active slip systems.For example, the case of Taylor's hardening, hl1= h12, with 4 = 45", renders Napsingular. Hence there are an infinite number of solutions for ya , and all combina-tions give the same rate of plastic work on the active slip systems.

Another example is the case where

in which the matrix Nap is not singular, but there are several alternatives arisingfrom flow rules (2.9).Usually there are three possibilities: either one of the two slipsystems is inactive or both are active. The latter case yields a smaller rate of plasticwork than the former. Havner (1982b) discusses these and related cases and sug-gests that the choice should be made on the basis of the minimum rate of plasticwork.

[For Appendix B see overleaf.]


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118 T. Iwakuma and S. Nemat-NasserI N T W O D I M E N S I O N S

For simplicity in notation, let the principal axes of a typical elliptical graincoincide with the global coordinate directions. The results for other cases are thenobtained by the usual tensor transformation; see $ 7 . Define

a = a,/a,, ( B 1 )where a , and a , are the principal radii of the elliptical grain, and set

1 = ' 1 1 1 1 ' 2 = '2111, ' 3 = 1211, '4 = '2211,' 5 = '1121, '6 = 2121, '7 = 1221, '8 = '2221, 1 ( 1 1 2 )' 9 = '1112, '10 = 2112, '11 = 1212, '12 = '2212,

'1 3 = 1122, '14 = '2122, '15 = '1222, '16 = 2222'From ( B 2 ) ,define the following additional non-dimensional coefficients:

c1 = Fl F, -F3F5, ' 2 = 10 '16 - '12 '14, \C3 = F7F13-F'F15, C4 = '6'16-'8'14,C5 = ' 4 ' 1 0 - ' 2 ' 1 2 , ' 6 = ' 1 ' 11 - ' 3 ' 9,C 7 = F 5 F 1 6 + F 6 F 1 5 - F 7 F 1 4 - F 8 F 1 3 , C 8 = F 4 F 9 + F 3 F . F ; O - F 2 F 1 1 - F 1 F 1 2 ,Cg = 2'16 - ' 4 ' 1 4 , = 3'13 - ' 1 ' 1 5,l ~

c11 = F5F;l - ' 7% '12 = ' 8 '10 - '6 '129 ' 13 = F 5 F 1 2 + ? l l ( F 6 + F 1 3 ) - F 7 F 1 0 - F 9 ~ F 8 + F 1 5 ) ~ (B 3 )' 14 = 'g Fl 6+ '10('8+ '1 5) - '1 1 '1 4- '12('6+ F13) , ' 15 = '5'12+ '6('4+ Fll )- F7F 10- F8( F2+ '9 )y '16 = F 3 F 6 + F 5 ( F 4 + F l l ) - F l F ~ - F 7 ( F 2 + F 9 ) , '17 = F3 ?14+ F4(F 6+ F13) -F 1 '16-'2('8+'15), Cl8 = '4'5+'3('6+?l3)-'2'7-'1('8+'15), ' 19 = 3'14+ '13('4+ ' 1 1 ) - '1'16- '15('2+ ' S ) , '2 0 = '10'15 + 16('2 + 9 ) - '12 '13 - '14('4+ '1 1 ) . I

The calculation of the concentration tensor requires aijkln ( 3 .13 ) or J i j k l in ( 4 . 1 ) ,where J i j k , in two dimelisions can be calculated directly by complex integration ( 4 . 9 )which involves the following line integrals on the unit circle y :'

where D, = Do( z )= D l z s + D 2 z 6 + D 3 z 4 + D ~ z 2 + D ~ . ( B 5 )


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119inite elastic-plastic deformation of polycrystalsHere th e super imposed s tands for the complex conjuga te , and

Dl = (e l + a 2 ( c 1 3+ c , , ) + a 4 c 2 )+ i a ( c l l + c1 8+ a 2 ( c g+ c 1 4 ) ) , D2= 4 ( c 1- a 4 c 2 )+ 2 ia { c l l + c1 8- a 2 ( c g+ c 1 4 ) ) , D, = 6c1- 2a2( c 13+e l , )+ 6a4c , .

Using (B 2 ) a n d ( B 4 ) , we express Jijklexplic i tly a s

Therefore, from ( 3 . 1 3 ) ,the components of .%7.jk,re :

As i s clea r f rom th e co ntext , gijklorresponds to th e E she lby tensor Si j k lin theinfini tesimal deform ation theory .

For the numerical calculat ion, from ( 3 . 1 3 ) , 3 . 1 4 ) ,a n d ( 4 . 1 ) ,the concent ra t iontensor ( 3 . 1 6 )is expressed a s

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