continuum micro-mechanics of elastoplastic polycrystals.pdf

7/27/2019 Continuum micro-mechanics of elastoplastic polycrystals.pdf

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J. MC&. Phys. Solids, 1965,Vol. 13,pp. 89 to 101. PergamonPress Ltd. Printed n Great Britain.

CONTINUUM MICRO-MECHANICS OF ELASTOPLASTICPOLYCRYSTALSBy R. HILL

Department of Applied Mathematics and Theoretical Physics, IJniversity of Cambridge,Cambridge

(Recei ved 16th Janumy, 1905)

SUMMARYTHE internal inhomogeneities of stress and strain in an arbitrarily deformed aggregate of elasto-plastic crystals are evaluated theoretically. A tensor constitutive law of a general kind is assumedfor the individual crystals. The implied mechanical properties of the aggregate as a whole areestimated by means of a self-consistent model akin to one used by HERSHEY (1954), KR~NER(1958, 1961) and BUDIANSKY and WV (1962), but differing in significant respects.

1. INTRODUCTIOI~IN CONTINUUM mechanics of solids a basic problem of long standing is to correlatepolycrystal and monocrystal behaviour under plastic strain. Specifically, supposingthe shapes, orientations, and mechanical states of all grains in an aggregate knownat some stage, at least in a statistical sense, it is required to derive the isothermalconstitutive relations for the aggregate as a whole. These are the tensor connexionsbetween arbitrary differential increments of overall stress and quasi-static strain,formed as space averages of the field variables. Once the incremental fields havebeen determined somehow, the accompanying changes in the geometry and stateof all grains follow from the given monocrystal properties. In this way the totaleffect of a continuing process of overall deformation can in principle be analysedstep by step.

The incremental behaviour at a generic stage, however, is the sole object ofinterest here. Neither the preceding strain path nor the existing pattern of localstress and lattice orientation are particularized. Correspondingly, a constitutiverelation of sufficiently general type is adopted for a typical grain, capable of re-presenting any actual mechanism of plastic flow. The aim is to construct a broadtheoretical framework for rationalizing observations on metal polycrystals atordinary temperatures. Computations for particular aggregates are deferred andthe present conclusions are mainly qualitative.

It is regarded as being of the essence of the problem to take due account of theinhomogeneity of the distortion within a polycrystal. In fact, the primary taskof the analysis wil l be to determine the average strain concentration factor ingrains with a given lattice orientation. For this purpose we follow in spirit the self-consistent method originated independently by HERSHEY (19.54) and KRGNER(19%) for elastically deformed aggregates. However, it has been found necessaryto re-appraise the method and to systematize its details. An extension to plastically

89


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so R. HILLdeformed aggregates is then formulated. This differs fund~entally from one alsodue to KR~NER (1961), which was developed by &JDI~NSKY and Wu (1962) andHUTCITINEON (X964).

2. SYMBOLIC NOTATIONIn order to let the essential structure of subsequent formulae appear more

clearly, a familiar symbolic notation is adopted for curvil inear tensors.Associated tensors of second order are denoted simply by their kernel letter,

u say, set in lower case boldface as if for a vector. Correspondingly, their tensorcomponents in any representation are considered to be arranged in some definitesequence as a 9 x 1 cohunn. Associated tensors of fourth order are denoted byan ordinary capital, A say, and are regarded as Q x 9 matrices. More precisely,the leading pair of indices is set in correspondence with rows and the terminal pairwith columns, both in the chosen sequence, so that the second-order inner productof tensors A and u can be written as the matrix product Au. Similarly, AB canstand for the fourth-order inner product of A and B. Subsequent formulae can be:interpreted in terms of tensor components in any representation, so long as it bedone consistently.

A special word is needed in regard to inverses of fourth-order tensors A, whichare here always symmetric with respect to interchange of the leading pair of indicesand also of the terminal pair. The representative matrices are thus singular, withrank < 6; for instance, As vanishes identically when s is the column for anyskew second-order tensor. Nevertheless, equations of type u = Av are compatiblewhen u and u are any ~~~e~r~c second-order tensors and matrix A has rank 6.In this sense we can define a unique inverse A-l as the solution of

AA-1 = I or A-l A = I ,where I is a unit matrix with the symmetries of A and with mixed tensorcomponents

f (6x Sd + V SK9in terms of the Kronecker delta. Then

/$-~=A-~A~=Iv=vas required, for any A, u and v with the stated properties.

3. AN AUXIU~~RY PROBL EM(i) A gl y t 1,in e cr s a arbitrarily ellipsoidal in shape, is imagined to be embeddedin a finite homogeneous mass of some different material. Neither phase is necessarily

elastic or isotropic, but in each the invariant relationship between an objectivestress-rate i and the strain-rate c is supposed to be one-one and linear. The tensorsof instantaneous moduli are denoted by L o and L , respectively, and their inversecompliances by M, and M. In addition to the symmetries mentioned already,the representative matrices are required to have full diagonal symmetry so that thecross-mod& and compliances are pairwise equal. A further minimal restrictionis that the respective quadratic forms, u (Lu) etc., in any second-order tensor uof rank 1 are positive.


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~oRtinu~~ mice-mech~i~ of eiastoplastic polycrystals 91On the external surface the veIocity is prescribed to correspond to a uniform

overall strain-rate F. Primarily, we seek the deformation induced within thecrystal, given that the velocity and traction-rate are continuous across the inter-face. To be definite, stress-rate is taken to be the convected time-derivative ofcontravariant K irchhoff stress, based on the current configuration. In the absenceof body forces this satisfies the equilibrium equations

where vi is velocity, 3r~k/3& = 0 since the existing stress is self-equilibrated,and x6 is a rectangular coordinate. The convective term would vanish identicallyin a field of uniform straining and will in any event be disregarded, presumingthe existing stress to be a sufieiently smatl fraction of the dominant moduli.This approximation is permissibie since Hadamard instability, which would involvelocal spins large compared with the strain-rates, is prevented by the minimalrestrictions placed on the moduli (HILL 1962, 5 4 (ii)). Moreover, these same re-strictions also ensure a unique solution (op. cit. $4 (iii)), which can therefore beexpected to have the character of a uniform field locally perturbed in the neigh-bourhood of the crystal. In particular the overall, or macroscopic, stress-rate +is then equal to Lz, since the contribution from the crystal becomes vanishinglysmall when the outer phase is sufficiently extensive. Furthermore, f and E arealso approached asymptotically by the local field values at the external surface.

It is plain that the solution is formally identical with that of the analogo~~sdisplacement problem in linearized elasticity when both materials are Green-elastic.In that context the sohition is well known in outline when the outer phase can betreated as unbounded. I ts principal feature is that the ehipsoid is strained uni-formly, though not necessarily coaxially (ESIIELRY 1957; 1961).

This property prompts the introduction of an overall constraint tensorL* for the outer phase, with inverse M*, in respect of loading over the interfaceby any distribution of traction-rate compatible with a uniform field of stress-rate,i* say. That is, if E* is the accompanying uniform rate of straining of the ellipsoid,

+* = - L* c*, (2)(A* + L,) 66 = (L + L) T, (Al* + MC) ic = (n-l* + M) f, (3)

which furnish the required fields in the crystal in terms of the macroscopicquantities. I t seems that this attractively direct approach has not been adoptedby other writers, with apparently the single exception of HERSHEY (1954), who,however, did not emphasize its advantages.


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92 FL HILL(ii) Instead, following ESI~ELBY (1957), it has become customary to focus

attention on the entire two-phase system, starting from a certain transformationproblem for an infinite homogeneous elastic continuum with stiffness tensor L t.In this, an ell ipsoidal region would undergo an infinitesimal transformation straine if free, but attains only the strain Se i t r s i tu . The tensor S is determinable uni-quely, by virtue of the minimal restriction on L , and it obviously possesses aninverse S-r since e vanishes with Se (no stress being then induced outside thetransformed region nor therefore within it). The xf components of S, being dimen-sionless, are functions of the mod&i ratios and of the aspect ratios of the ellipsoidand its orientation in the frame of reference. When L is isotropic, explicit formulaefor the components on the ellipsoid axes have been given by Eshelby (op. cit.).Formulae for the two-dimensional analogue have been given recently by BHARGAVAand RADHAKRISIINA (1964) hen the medium is orthotropic and by WILLIS (1964)when it has cubic symmetry. For general anisotropy the basis for a tbree-dimen-sional solution was sketched by ESHELBY (1951, . 105)$.

The general connexion with L* or M* is most easily obtained by imaginingthe transformation problem solved from the viewpoint of (l),nterpreted in anelastic context. That is, we substitute

e* = Se, T* = L (e* -e) in T* = -L * e* .Then, since these hold for all e,

L * S = L ( I -S) , ( I -S ) M* = SM, (1)where I is the unit tensor defined in 5 2. These are equivalent formulae for L *or iI f* in terms of S. Or they can be put inverseiy as

s = ( L * + L ) - 1 L = .M* (M* + Myfor S in terms of L * or M*.

Another dimensionless tensor l, the dual of S, could just as well be admittedon this footing. Set

M* T = S i t I = Y , say7>

(5)?I , = .L* S = Q, say,so that M I = 111 I - T), (I - T) L = TL ,and

1(6)I zz L * (L * + L ) -1 = (M * + M) - M .

The significance of 2 is that the stress T* in the transformed region can be writtenas T r , where 7 is the stress that would remove the strain e. Separate symbolsP and Q have been introduced for the products in (5) since these appear frequentlyin the sequel. We note the further connexions

t&e-oceupetion with this standpoint cnn lead to needlessly devious derivations. An extreme example is to befound in several papers by BHARCAVA and RADHAKRISH~A1063a, 11; 1964) on the iw~-~mension~ auxiliary problem.The elastic Reid outside the ellipse is actually obtained by exact analysis at the outset, from which L*, M and theeomp&bIe Internal field oould forthwith be read off, but are not. Instead, the total potential energy of both phasesis Iabm+~Iy oomputcd for any final internal &rain, and afterwards minimized 89 a means of satisfying the remainingrrquirement of traction continuity at tire interface.

~W~LUS (1~64, private communication) has developrd this into a feasible numerical procedure by reducing theintegrals.


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continua micro-mechaniai f el8stoplast.k olycrystals 93PL+MQ=2,

P=M(I -T), Q=L (I-S), (7)and F- l=L* - j -L , Q - l=M* +M,From the latter pair one sees that matrices P and Q have the diagonal symmetrystipulated for the mod& and compliances (while S and T generally do not). Thiscan of course also be established purely within the context of the transformationproblem by means of Bettis reciprocal theorem.

(iii) We return to the original auxil iary problem. As in a related situation(HILL 1963) it is advantageous to define concentration-fac~r tensors A, and fz,such that

ic = AE Z, ic = Bc Gt (8)with the consequent inter-relations

LcAo = B,L, 111, , = A, M . (9)From (8),A, = (L* + L,)- (L* + L), B, = (M * + MC)-1 (M* + M), (1~)

in terms of the overall constraint of the outer phase. Eliminating this with thehelp of (7), we have the variants

AC-l = I + P (L, - I ;), Be-l =I +Q(M , ---& I ), (11)in terms of the expbcit differences between the phase properties, and involvingquantities directly to do with the transformation problem.

8. POLYCRYSTALELASTICITY(i) We consider polycrystals whose geometry is such that the grains can be

treated, on average, either as variously-sized spheres or as similar ell ipsoids (inparticular elliptic cylinders) with their corresponding axes aligned. On the otherhand, the lattice orientation in the Cartesian frame of reference may vary fromgrain to grain, not necessarily rtdi0mly.

Tensors L, and MC will now denote the stiffness and compliance of a typicalcrystal, in respect of incremental elastic strain (not an essential inte~retation,as remarked in 5 3 (i)). In the auxiliary problem L and M become the overalltensors for the poiycrystaf itself, while L , M *, S and T specificahy relate to thesphere or ellipsoid representing the average grain shape. Consequently, in acommon frame of reference, the components of L, M, L*, M*, S, T, P, Q areconstants, whereas the components of L et MC, A,, B, depend on local latticeorientation.

In the self-consistent method due to HERSHEY (195k), and independentlyto KRSNER (1958), the stress field around a grain in the auxiliary problem is takento be representative of the actual neighbouring field in the polycrystal, averagedfor all grains of that orientation. Further, and more explicitly, it is postub*-4that a certain macroscopic tensor quantity should be obtained as the average,over the relevant range of lattice orientations, of the corresponding local tensorfor a grain in the auxiliary problem. But which quantity should be singled out


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94 R. HILLfor this treatment has perhaps seemed arbitrary hitherto : Kroner examined twopossibil ities, only one of which was tried by Hershey, while ESHELBY (1961, 6)proposed yet a third. Still other choices are equally natural. By means of (2),however, it is a simple matter to established their equivalence, thus finally justifyingthe description self-consistent.

To begin with, suppose that the macroscopic strain-rate is taken as the averageof the crystal strain-rate in the family of auxiliary problems. Then, from (8)and afterwards (9),

(A$=1 +cf +&II ,} (12)where orientation averages are indicated by enclosure within curly brackets.Alternatively, dual operations with stress-rate give

(BC} = I 3 L = {L, A,}. (13)That such dual approaches are equivalent can be seen at a glance from (2) : theleading equation states an invariant linear relation betwren deviations from therespective means, and so the average deviations necessarily vanish together,

I f, now, one wished to transform equations (12) y means of (lo),hey couldfirst be put as

((A-, - I ) AC) = 0, ((MC - MB,--) Be> == 0,from which

P ((Lc - L) A,) = 0, S ((MC - M) Bc) = 0.Treating (13) imilarly,

Q((M,-M)B,}=O, T((L,-L)A,)=o.The matrix factors P, Q, S, T are non-singular and can be dropped. I t is thenapparent that not only are (12) nd (13) equivalent to one another but also toeach of

{WC - L) AC} = o, ((Me - M) B,) = 0. (14)I n fact, as they stand, these are just the averages of the dual polarizationtensors : (ic - Let) = (L * + L){Z - EC) = 0,

{EC M i,) = (M* + M ){* - ic} = 0,obtained directly from (2).Returning again to (12)and (13), nd combining them in turn with (11)

([f + P (L, - L)]- } = I , { [ I + Q (M, -M)]-} = 1. (15)Alternatively, by combining (14) nd (II),

([P + (L, - L)-1-l) = 0, {[Q + (M, - M)-l]-I ) = 0. (16)While these four variants are equivalent, as already proved, it may in practice beslightly ore convenient to proceed from one in preference to the others. In anyevent the final result of carrying out the appropriate averaging is a set of algebraicequations just sufficient to determine the separate components of L or M, in termsof matrices P and Q furnished by the solution of the transformation problem.


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Continuum micro-mechanka of ela&opla6tic polywystals 95On the other hand, if the primary datum is the constraint tensor of the outer

phase in the auxiliary probiem, a possible starting-point would be either of((L, - L) (A* + &)-I) = 0, ((i& - M) (BP + M,)--) = 0, (17)

from (IO) with (14). The equivalence with (16) can be checked at once by meansof (7). More attractive variants are

(L* + Q-1 = {(L* + L)-} (M* + M)- = ((nl* + W-l>, (18)from (10) with (12) and (IS) respectively. Or these can be solved as

L = ((L* + Le)-j -1 - L* , M = {(M* + M& 1)-l - & I* .(ii) I t appears that the self-consistent method has so fas been applied numeri-

cally only when the grains are spheres and the lattice orientation is random.Tensors L and &f are then isotropic and can be written in the symbolic notation as

L = 3~ i + 2~ (I - i i ) ,M -2& ji +2-211 I - i ),

or still more shortlv asL S(3K, 2p), M z &, $ ( )

where K and p are the usual bulk and shear mod&. Column i is the unit vectm representing 6$,$/B, and the row i is its transpose, so that f i = 1, Clearly,matrices i i and I - i represent isotropic operators that decompose a second-order tensor, such as stress or strain, into its hydrostatic and deviatoric parts,They also have the properties

(i i )* =11 i, i i (1 - i ) = 0, (I - i )* = I - i .I t follows that the product of any pair of isotropic fourth-order tensors is isotropicand commutative, and that the coefficients in its.decomposition are just the respec-tive products of the original components. For instance, the order of factors inany product in (4), (5) and (6) can be reversed, with the consequence that T=I--Sand S S (a, 8) say?, T EE I - Q, 1 - /I ), 7

& = (3K (1 - a), 2P (1 - B)), I

Equally, the average of any orientation-dependent fourth-order quantity,with matrix Z say, is isotropic and can be similarly decomposed :(2) z (I, q) where 6 = tr (2X), 6 + 5~ = tr 2,

tThe actual values awea = 8 - W = K/(U + 4 M) [me, for exampIe. ES-Y (IWR)].


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96 R. EIILLwhere tr stands for trace (i.e. sum of elements in the principal diagonal). Coe-fficients 4, T)are invariants and can therefore be evaluated from the Cartesian com-ponents of Z for any convenient lattice orientation. In this way each of theequivalent relations (15), 16), 1'7)r (18),when averaged in form (2) = 0, isreduced to a pair of simultaneous algebraic equations 5 = 0 = v in the overallmoduli K and I*.

Krijner and Hershey, starting in effect from the second of (la), arrived re-spectively at the second of (15) and (18),oth of which yield a quartic equation7 = 0 for p alone when the lattice has cubic symmetry (in which case .$ = 0 statesthe obvious fact that K is just tbe single crystal bulk modulus). The quartic containsa non-vanishing factor 9K + 8p, the numerator of the fraction 1 - p correspondingprecisely to the factor T noticed above, and so can be reduced at sight to a cubic?.The cubic, aa such, was also obtained by K rijner essentially from the first of (12),which leads to the first of (IS),nd again by Eshelby from the first of (14) phrasedin terms of energies& But the final coincidence of these approaches, here shownto be inevitable, went unexplained.

5. AUXILIARY PROBLEM WITH PLASTIC FLOWThe previous formulation and analysis of the auxiliary problem already covers

certain inelastic behaviour (for instance hypoelastic), but not elastic/plasticbehaviour as typefied by the common metals at not too high temperatures. Thisis characterized by a non-linear relation between stress-rate and strain-rate (stillhomogeneous of degree one). With polycrystall ine metals mainly in mind, wetherefore attempt an appropriate extension of the analysis in f 3.

(i) Suppose, first, that only the crystal is elastic/plastic. Then equations (2)still hold and the connexion between local and overall quantities, replacing (a),is clearly determinable in principle. To derive it explicitly, suppose the operatorM, for a crystal in situ to be piecewise linear. I f necessary, this can always beassumed as an approximation that can be made as close as one wishes. Then

M, - -UC = rl F + . . . . ,c (21)

where &, is the elastic compliance tensor of a crystal, and each 7 is 1 or 0 accordingas its associated m, ic is positive or negative. Succeeding terms, up to any finitenumber, are similar in type to the one displayed. This law is recognizably akinto what is often premissed for multislip in a metal cubic crystal. Without neces-sarily intending this interpretation, one may say that the strain-rate tensor in acontributory mode of plastic deformation always has a representative vector ina specific direction, typically m, where m, m, = 1; furthermore, workharden-ing in this mode is controlled by a modulus he, whose value may conceivablydepend both on the mode in question and on which others are simultaneouslyactivated. Naturally, every m, and h , s a function also of the current stress in

+Namely82 + (SC,, + 4%) 2 - c,+ %I - 44 M - eM (Cl, - cl*) (Cl, + 24 - 0

with the usual notation for crystal moduli. The version given by ESHELBY (1961) contains some misprints.fKr6nera second method, essentially in the form of the first of (15), has recently been adopted by KNEER (1968)

to compute the bulk and shear moduli when the lattice is hexagonal.


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Contimuun micro-mechanics of elastoplsstic polycrystals 97the crystal and of its mechanical state, while the cartesian components of anym , depend as well on lattice orientation.

It can now be seen that M, is constant in each of the pyramidal regions createdby the dissection of stress-rate space by the set of hyperplanes m , i c = 0 , etc.In particular, it is equal to AC in the region whose boundary corresponds to thelocal vertex on the yield surface of the crystal. Within.any pair of adjoining re-gions the families of active modes differ by just one member whose direction is normal to the mutual interface. Moreover, this member is not activated byany stress-rate lying in the mutual surfade; consequently, strain-rate varies con-tinuously with stress-rate through the entire space. Finally, because of the assumednormality, matrix M, automatically has the required diagonal symmetry when thisis stipulated for *KC. A linear equation, identical with the second of (3), is thereforevalid in respect of each pyramidal region in ie space or its equivalent in f space.

When the yield surface is locally regular, its unique normal defines the directionof the possible plastic strain-rate. Expression (21) terminates with the modeshown and inverts to

wheregip, is the tensor of elastic moduli, reciprocal to -R,, and we assume that ge > 0.By evaluating in analogous fashion the matrix reciprocals in (lo), the concentrationfactors are obtained as

(L* + LTi,)- I, I,___gc - I, ( L * + .Epc)-1c,I L * $-E,)-1 (L* + L) (23)where 17 s 1 or 0 according as 1 , L* +_cT,$ ~ L* + L) E s positive or negative,and

(M* + yUc)-l m , m , 1, + m , (M* + -r YJL m, (&f*A,)-Al*tI)24)where 7 is 1 or 0 according as m , (M* + J AY,)-l M* + M) f is positive ornegative. Denominators of the quotients here are identically equal and are assumedpositive (for which an amply sufficient condition would be that the various matricesare positive definite). The plastic part of the crystal strain-rate is

~~=(~,-~,)B,f=~m,m~~M*+~c)-l(M*+M)~,h, + m, (M* + A,)-1 m ,in the direction of the yield surface normal.

In the event that both phases have the same elastic moduli, so that .!Z, = L ,-UC = M, these formulae reduce to

&=I+77 PI, I ,g, - l e PI, Bc=I-- Q me whc +mQm,


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98 R. HILLwhere P and Q are as in (7). When, in addition, the phases are elastically isotropicthe decomposition (20) is available, Thus, supposing also that the plastic modeinvolves no di~a~tion, as is appropriate for metals,

with

If, further, the mode is a simple shear without hardening, we recover the prototypeformulae of BUDIANSKY, HASHIN and SANDERS [1960, (12) and (13), obtainedvia the transformation probIem outlined in $3 (ii) here].

More to the present purpose, suppose that the crystal is rigid/plastic, withoutspecializing in other directions. Then, by putting A, = 0 in (%), and using (t),

where v is X or 0 according as (S-r me) ? is positive or negative, ands-1 =I -j-ML, Is = (I + .L* M)-,

as in (4), a prime signifying the transpose. Equally, by a limiting operation on(23) or directly from A, P = MCB&.3which is the second of (9) with the first of(5) in the form 6 = LPI,

where rl is 1 or 0 according as (P-l m,) ii is positive or negative, andP=SM=Ms=(L* +L)-l

as in (7).(ii) Suppose, now, that the macroscopic constitutive law of an elastoplasticpolycrys~ is known and can be approximated piecewise linearly :

where each t is 1 or 0 according as its associated m i is locally positive or negative.Consider the corresponding auxiliary problem where the crystal has the constitutivelaw (21). I f the stress-rate at all paints in the outer phase induces one and thesame branch of &I , the material behaviour is quasi-elastic and the previous analysisremains rigorously valid for the particular overall loading. I t would be possiblein principle to determine the loading ranges for which this happened, is respectof any given branch and any lattice orientation.

A more practicable course is to take as the representative outer phase a trulyelastic material whose properties coincide with the actual branch of &I associatedwith whatever overall stress-rate may be in question. Naturally, for certain loadings,the resulting solution of the auxiliary problem will sometimes violate the original(27) locally in the outer phase, but we can reasonably hope that the net error inthe overall. constraint will be negl~gib$e.

Expressed formally, our hypothesis means that (27) is modified to the extent


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Contour micro-mechanicsof el8stogh3ticpolyer@als 99that each q is taken as 1or 0 according as its associated m G is positive or negative.The analysis in (i) above is then rigorously valid within each region of the space ofoverall stress-rate given by this dissection. In the formulae it is only necessaryto insert the appropriate branch of L and 32 from the modified (27), together withthe associated L * , M *, S, T, P and Q.

(iii) KR~NER (1961) has suggested a different scheme. In BUDIANSKYand Wuselaboration of this (1962), adopted also by HUTCHINSON 1964), the representativeouter phase is given the constitutive law

where J is the elastic compliance (assumed isotropic) of the aggregate and of theindividual crystals. This is an artificial relation, patently not quasi-elastic; never-theless Eshelbys analysis is still presumed to apply. The writer has not understoodthe subsequent line of argument, but the eventual formulae imply that

CC -_2= - A* (iC - G)

and so, in particular, that the aggregate deforms homogeneously when the crystalsare rigid/plastic (&, .4* = 0). Thus, in effect if not by expressed intention,KR~NER et al. assign to the outer phase the overall isotropic constraint that theaggregate would in fact exert if its incremental deformation were always purelyelastic. This disregards the pronounced directional weaknesses in the constraintof an already yielded aggregate.

I f used with the present formulae in (i) above, Krijners proposal would entailreplacing L * , M *, S, T, P and Q by their elastic counterparts. Neither Budianskyand Wu nor Hutchinson obtain such formulae, but calculate instead the rate ofhardening of typical crystals in a random aggregate loaded either by uniaxialtension or pure shear. At the same time the overall hardening of the aggregateitself is computed by the self-consistent method.

6. SELF-CONSISTENTMODEL WITH PLASTIC FLOWIt is proposed to combine the hypothesis of 8 5 (ii) with the self-consistent

method of averaging. The object of calculation is now the macroscopic constitu-tive law itself. The unknown matrix M will certainly have the structure (27),since not only are its piecewise linearity and diagonal symmetry intrinsic to thistheory but they are also clearly compatible with the law (21) assumed for theindividua1 crystals. Of course the branches of M will not normally be foreseeableat the outset. They are determinable in principle with the help of the requirementthat the overall stress-rate should always depend continuously on strain-rate.

Fortunately, the elastic branch at least can be located already, It is associatedwith the pyramidal region defined by the inequalities

m,(.M* +yKE)-l(cly* fS4V)S (0 ,......,for all crystals and modes. _R* is the reciprocal constraint tensor associated withthe overall elastic compliance M satisfying

(,K* + .4)-r = ((A* + .dej--l)


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100 R. HILLES in (18). The hyperplanes bounding this elastic region in overall stress-ratespace correspond of course to the tangent planes at the currently given stress-pointon the yield surface of the aggregate. The vertex is more or less pronouncedaccording to whether the directions

(I + -K,9Lp*)- m,, . . . . . )span a large or small solid angle.

Without resort to computation, the analysis can be carried farther only inparticular cases. Suppose, for example, that the crystals are elastically isotropicand that the current internal stress is such as to activate just one mode m, whosedirection in strain-rate space is moreover the same for all grains. The typicalcrystal compliance can then be written as

where only the hardening parameter varies from grain to grain. Then, in thefamily of auxiliary problems for the various orientations, 7 is simultaneously1 or 0 according as m (M* + &)-I (&f* + M) t is positive or negative; inparticular, for the elastic branch A this criterion expression becomes m f . Bysolving (B,} = I for M , when the concentration factor B, is the indicated speciali-zation of (24), one finds that

regardless of what M * might be, where[h + m (M* + .4)-l m l -l = ([h e + m (M* + -4 -l m l -l >.

Finally, by virtue of the a posteriori reduction(28)

m (M* + A)-1 (M* + Al) G = m I + q (M* + d )- y I qm (M* + A)-- m

h Im , + ,

it is confirmed that the criterion expression is essentially the same for both branches(the factor in square brackets being positive). The conclusion is, therefore, that theaggregate also deforms by the one plastic mode, for the overall stress, and that itsoverall rate of hardening is implicitly given by (28). Since M* depends on M,which is itself a function of h, this equation is quite complicated. Leaving it andreturning to (24), we obtain the following expression for the inhomogeneity of theinternal fields :

$ - ic = L * (EC qm G (29)h, + m (M* + 4-l m (Ail* + A)-- m .

The inhomogeneity in strain-rate is seen to be of the same type in all grains, thoughdeflected from direction m .


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Continuum micro-mechanics of eiastopbratic polycrystals 101The latter effect is purely elastic and would vanish if the crystals and aggregate

were rigid/plastic, i.e. if & = 0. In that event the strain-rate would necessarilyalways be homogeneous in the auxiliary problem, and so M* = 0 while m ic = h,and m $ = h. Thus h = {h,) is the limiting form of (~8). To determine the stress-rate field uniquely, the auxiliary problem must naturally be re-set in the mannerappropriate to rigid/plastic solids with a (piecewise) linear constitutive law (HILL1956). When the direction of the only activated mode is not the same in all grains,combining (B,> = I with (25) or {A,} = 1 with (26) leads to the implicit equation

P= mc m,rlhc+m,L*m, > (39)for L or M. The accompanying conditions on 7 define a pyramidal dissection ofthe space of the overall stress-rate by hyperplanes with normals S-l m,, for thevarious branches of S.Further consequences are left for future investigation.

ACKNOWLEDGMENTThis work is part of a programme of research on mechanics of materials which is supported

by a grant from the Department of Scientific and Industrial Research.

BI~R~AVA, R. D. and~DHAKRIkiNA, H. C. 196%1968b

1964BUDIANSKY, B., HUHIN, Z.

and SANDERS, J. L. 1960

BUDIANSKY, B.and WV, T. T.

ESEELBY, J. D.1962195119571961

HERSHEY, A. V. 1954HILL, R. 1956

19621963

HUTCHINSON, J. W. 1964KNEER, G. 1963KR~NER, E. 1958

1961WILLIS, J. R. 1964

REFERENCES

Proc. Camb. Phil. Sot. 59, 811.I bid 59, 821.J. Phys. Sot. Japan 19, 396.Proc. 2nd Symp. Naval Stwctural Mechani cs, 239

(Pergamon Press, Oxford).Pmt. 4th U.S. Nat. Gong. Appl . Mech. 1175.Phi l. Trans. Roy. Sot. 244, 87.Proc. Roy. Sot. A241, 376.Progress in Solid Mechanics (Edited by I. N. SNEDDON

and R. HILL), Vol. 2, Chap. III (North-HollandPub. Co., Amsterdam).J. Appl . Mech. 21, 286, 241.

J. Mech. Phys. Soli ds 4, 247.I bid. 10, 1.I bid. 11, 357.J. Mech. Phys. Soli ds 12, 11 and 25.Physica Status Soli di 3, K331.2. Physik 151, 504.Acta M et. 9, 155.Quart. J. Mech. Appl. Math. 17, 157.

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