Theoretical Plasticity of Textured Aggregates Hill

7/22/2019 Theoretical Plasticity of Textured Aggregates Hill

1/13

Math. Proc. Camb. Phil. Soc. (1979), 85, 179 1 7 9Printed in Great Britain

Theoretical plasticity of textured aggregatesB Y R. HILL

Department of Applied Mathematics and Theoretical Physics,University of Cambridge(Received 26 June 1978)

Abstract. The plasticity of metal polycrystals with preferred orientations is con-sidered from a phenom enological stand po int. Some new general theore m s are proved,in particu lar th e existence of a work-equiva lent function of the tenso r strain -rate overany yield surface. The s tat us of the classical theory of plastic an isotropy is re-appraisedin the light of recent experiments, which are themselves critically reviewed. A newtyp e of yield function is proposed to a ccoun t for the so-called anom alous b ehav iour ofsome materials.1. Preliminary notions. Co nstitutive relations for the p lastic yielding and deforma-tion of anisotropic m etals a t a macroscopic level were prbposed long ago by the w riter

(1, 2). This theory was the simplest conceivable, but it has not been superseded andtechnologists c ontinue to find it useful. Howev er, inevitable lim itations on its range ofvalidity have eventually become apparent. My present aim is to suggest how thesemight be removed without sacrificing too much of the simplicity that was a mainattraction. Some necessary perspective is added by analysing some related aspects ofanisotropic plasticity from a general standpoint.At the ou tset it is imp erative to be precise abo ut the mean ing of' yie ld' in th e presentcontex t, since the word is ap t to be variously applied in the engineering and metallur-gical literatures. The principles underlying its intended sense are set out at length byHill (3), but a recapitulation may be found helpful. Consider a representative macro-element of the poly crystalline aggreg ate; it is initially at zero load tho ug h perhapsalready pre-strained (say by cold-rolling), and is afterwards subjected to macro-scopically uniform deformations. Suppose, to be definite, that on a primary path ofdeformation (e.g. uniaxial tension) the cartesian components of Cauchy macro-stressare kep t in fixed ratios p ti up to values cr^ = T/oi}-. The corresponding elastic domain inmacro-stress space is determinable by exploratory 'unloadings' from cri3-, namely byelastic pa ths to stat es where further plastic deformations becom e just observab le in afew grains. The do main bo un dary will be called the elastic limit associated with stageOy on the primary path; this boundary is of course directly dependent on theinhomogeneously distributed micro-stresses under o^-, more especially their localconc entrations, and it is therefore highly sensitive to th e te xt ur al micro-geometries innominally-equivalent specimens. The domain boundary is often alternatively calledth e yield surface associated w ith cr


2/13

180 R. H I L Lconcept suppose that, from a point i*p% on the current elastic limit, further plasticdeformation is produced by varying the components of macro-stress monotonically infixed ratios p*t (e.g. biaxial tension). Now imagine that further glide-hardening issuspended on every slip-system in every grain. The macro-stress on this secondary pathwould still increase as progressively more of the aggregate becomes plastic, but wouldtend asymptotically to an upper bound crt* = 7*p%, where T* > T*; also, this boundwould be closely approached within a strain of order 10~3 only, due to elastic constraint.According to standard theory, aft is the exact value at which deformation could begin ifthe elastic shearing moduli in every grain were infinite. Moreover, function T*(p*j)depends only on the critical stresses for glide on potential systems throughout theaggregate at stage cr,-3- on the primary path; in particular, unlike T*(p*j), it is inde-pendent of the micro-stress distribution under cr^. In reality, of course, glide-hardeningon the secondary path forces an increase of the macro-stress beyond o-*-; but, after amoderate deformation due to eri;- plus any original pre-strain, the needed rate ofincrease is two or more orders of magnitude less than a typical elastic modulus. There-fore, by a backward extrapolation of the experimental data from the portion of thesecondary path beyond a strain of 10~3 or so, the glide-hardening is removed and astress value is obtained which differs imperceptibly from


3/13

Theoretical plasticity of textured aggregates 181outward. At a vertex yieldpoint a set of limiting values of 8/dcri:j is defined by anyevanescent convex cap that would smooth the surface; such vfj- are normals to thesupporting hyperplanes and span a convex cone which is conjugate to the tangentcone.

(i) When (


4/13

182 R. HILLwhere e is any positive scalar and vi;- is given at a^ by (2-1) for some 0(o"y) representingthe particular yield surface. The corresponding work-rates (or work-increments) perunit current volume are expressible as


5/13

Theoretical plasticity of textured aggregates 183In the literature on plasticity, the search for a scalar measure of equivalence has

been a recurring theme, but in relation to strainhardening than to work expenditureperse. The question is mostly posed in the following way (2), pp. 23-33: can observedevolutions of yield surfaces with progressive deformation be described adequately interms of jus t one parameter; if so, is this either the work done or some functional of thestrain path ? The question is occasionally supplemented by another (2), p. 39; (5): whenare these alternatives not essentially different? In effect, an answer to the latter callsfor a consideration of work-equivalent deformations, but to my knowledge thisnotion has never been studied overtly and for its own sake.f By contrast, the presentexposition is free from the distraction of a supposed rule of strainhardening, and soallows work-equivalence to be analysed explicitly and with due generality. The mainoutcome is the universal existence theorem for function e(e^). Straightforward thoughthis may appear in retrospect, it is apparently not to be found in the literature. J

3. Quadratic yield functions. A certain quadratic invariant of Cauchy stress, ascribedto von Mises, is commonly used to approximate the yield surfaces of random aggre-gates of metal crystals. Its comparatively wide range of validity (covering severaltypes of simple lattice and even alloys) suggests that a non-invariant quadratic mightbe adequate for textured aggregates in some circumstances. One such function,proposed by the writer in 1948 (l), has provoked experimentation and debate eversince. Before adding retrospective comments, and coming to a new proposal, I believeit is illuminating to view the matter against the perspective of a general quadratic.

When represented by a homogeneous function of degree 1, the considered yieldsurface is W^^B^^l^c. (3-1)where the coefficients are apportioned so as to be symmetric under the interchangesi->j, k*-+l, and ij*-+kl. To begin with, suppose tensor flijkl to be positive-definite, sothat the surface is strictly convex. Then by (2-6), (2-8), and (2-9), a = c and

eii/e = vij = Pmi(Tkl/


6/13

184 R. HILLand this can be inserted either in (2-11) or (3-1) to give

e = K-wetj%)* (3-4)as the work-equivalent measure of strain-rate. It is reiterated that e need by no meansgovern the current rate of hardening.

Some obvious modifications are needed when plastic deformations are treated asvolume-preserving, as is well justified for metals. The surface is now merely convex, inthat expression (3-1) is insensitive to the hydrostatic part of cr^. This requires

Ai = 0 = / W (3-5)Then, in (3-2), eu = 0 automatically while


7/13

Theoretical plasticity of textured aggregates 185orientations, and (ii) un iaxial or biaxial loading , neither necessarily tensile nor in-planeb ut always co-directional with the orthotro py. According to the preceding theory , theresponse in (i) involves / :^ : h:n (the in-plane axes being numbered 1, 2) while in (ii) itinvolves / :^ : h only. For the simpler tex ture s (those th a t give rise to no more than fourears in a cupping operation), and for ma terials used in the sheet-forming industry (suchas alum inium, copper, zinc, killed steel, brass, titaniu m ) it is found that (3-8) with anoptim al choice of ratios pred icts well the orientation dependenc e of increm ental defor-mations in (i), both in the homogeneous and localized regimes (l, 2, 7, 8, 9, 10). Ofcourse, the incremental departures from the orthotropic ground-state must be smallenough not to alter its texture appreciably, yet large compared with elastic strains;then the optimal ratios may legitimately be associated with the ground-s^ate itself. Bycontrast, the implica tions of (3-8) in regard to (ii) have never been ade qu ately explored;the needed experiments are admittedly taxing, but not prohibitively so. To myknowledge, only the simplest confrontation has been attempted, namely uniaxialloadings along the principal directions; the respective theoretical ratios of the trans-verse strain-rates are h/g, f/h, g/f in cyclic order and so have product unity. This pre-diction is non -trivial for a sheet with in-plane anisotropy; it was precisely checked byBourne and Hill(8) for one texture in brass, while fair agreem ent can be seen in dataobtained by Lee and Backofen(l 1) for pure titanium and two alloys of tita niu m (theseauthors were not concerned with the specific issue, however).

There remains the problem of indep end ently testing (3-7) itself. In this connexion therele vant de finition of a yield surface is recalled (Section 1), from which it follows th a t, inprinciple, any surface is associated w ith one and only one path of plastic deformation.In technologically motivated investigations of textured materials, however, a severelylimited path-dependenc e is envisaged, namely such tha t the yield surface depends onone variable only. This is customarily taken to be either the total expenditure of workper unit volume or else some scalar functional of the strain (even jus t a nominal off-setcomponent). Experiments are thus conducted on the basis that a subsequent yieldsurface can be mapped by reaching different points on it via distinct paths of plasticdeformation. Correspondingly, in assessments of (3-7), a and the quadratic coefficientsare held to depend only on the one variable; often, indeed, the coefficient ratios arepresumed to be completely independent of path, implying geometric similarity of thefamily of surfaces. However, such limitations are not inherent in (3-7) and (3-8); thefunctional path-dependences of


8/13

186 R. H I L LWoodthorpe and Pearce (15) on commercial-purity aluminium pre-strained by coldrolling. In this case, the pre-strain yieldpoint under any type of loading can be as-signed with confidence by the backward-extrapolation construction, as explained inSection 1. Following each rolling reduction, Woodthorpe and Pearce reported uniaxialtension da ta (averaged over several in-plane orientations) and also equi-biaxial tensionda ta from a hydraulic bulge test on a circular diaphragm (which involves an indeter-minate weighting of in-plane anisotropy). Supposing the data to be neverthelesscomparable with theory for in-plane isotropy, we set/ = g and o~33 0 in (3-7) and (3-8),and reduce to ., . , , .,

/( 2, but not necessarily homogeneous,were early suggested for consideration (2). This was prompted by the associated theor-etical variation of the uniaxial yield stress with orientation in an orthotropic plane;specifically, its magnitude may have up to 4/t stationary values, each linked with co-axiality of the stress and strain-rate tensors, as in Section 2(i). Such coaxiality washeld to account for the inception of either an ear or hollow on the rim of a radially-drawn blank in a cupping operation; there is consequently a numerical correspondencebetween coaxiality and waviness. (The coincidences in location need not be exact


9/13

Theoretical plasticity of textured aggregates 187because of differential rotations of rim elements; this matter still awaits a thoroughanalysis.) It was inferred in (2) that, if 2/i ears along with the consequent 2/i hollowswere seen in an actual operation, the particular texture could only be represented by apolynomial with degree > /i, if at all. Orthotropic textures are in fact known where2, 4, 6, or 8 ears, respectively, are seen. The first construction of a polynomial with/t.> 2 is due to Bourne and Hill(8) who tried a homogeneous cubic for a 6-eared brassunder biaxial tension (note that the entire yield surface cannot be represented by asingle homogeneous polynomial of odd degree). Recently, Gotoh(7) has suggestedhomogeneous quartics for all textures, and has classified their coefficients according towhether 4, 6, or 8 ears are implied.

Polynomials will not be considered further here; while they may well find someapplication in practice, it is contended that more attractive alternatives are availablein general from the standpoints both of experimentalists and mathematicians. Thenon-orthodox yield functions that I have in mind are, in one distinctive respect,analogous to the non-orthodox strain-energy functions that I advocated for rubberelasticity in 1969 (and which have since become popular): namely, non-integer powersof stresses or their differences are as acceptable as non-integer powers of stretches ortheir reciprocals.

In this spirit, when the loading is coaxial with the orthotropy, we might consider

in principal components, where/, g, h are positive, m > 1, and


10/13

188 R. H I L LTo encompass the anomaly, we can extend (4-1) to/ k \

m\ \

m% ^ ^

jri\m + % i - o - j ^ + a^ a- ! - o -2-


11/13

Theoretical plasticity of textured aggregates 189We can suppose the pa ram eters chosen to agree with the measured values of 1, or 2 m - 1 < 1 +r when r < 1.The strain-rate ratio associated with (4-10) is most readily obtained by taking theflow-rule as , . ,, . . . ,, . nfa + e2) d{tTx +


12/13

190 R. H IL LIn the long run, too, it should be kept in mind that even for in-plane isotropy the

class (4-5) contains functions with still greater flexibility than (i), (ii), (iii) or (iv).Obv iously, considerable expe rimen tation will be needed to narrow th e choice w ithinth is class. For t h at reason it would be prem ature to propose functions with yet moredisposable parameters (albeit easy: for instance two different powers for the a,b,cand f,g,h term s respectively). In the mea ntime it is attr ac tive to calculate yieldfunctions theoretically for artificial tex ture s, say by th e homogen eous deformationmethod of Bishop and Hill (20). This has recently been applied by Bassani(i6) for face-cen tred aggregates with overall in-plane isotropy. Significantly, some of the computedloci display the anomaly, though often their shapes deviate appreciably from theovals (4-10). In th a t connexion, however, it has to be rememb ered th at the Bis hop -Hill method only delivers an outer bound to any yield surface, and that its degree ofappro xima tion is no t known with certainty (3, 4). Fu rthe r investigation of this aspectis plainly desirable.

I am muc h indebted to Professor P . B. Mellor, De partm ent of Mechanical Engineer-ing, Un iversity of Bradford, for unstinting information, over a period of several years,rega rding work in his own labo ratory and in the scattered literatur e on textu redmaterials. REFERENCES

(1) H I L L , R. A theory of the yielding and plastic flow of anisotropic metals. Proc. Boy. Soc.London. Ser. A 193 (1948), 281-297 .(2) H I L L , R . Mathematical Theory of Plasticity (Oxford: Clarendon Press, 1950).(3) H I L L , R. The essential struct ure of constitutive laws for m etal composites and p olycrystals.J. Mech. Phys. Solids 15 (1967), 7 9-95.(4) BISH O P, J. F . W. and H I L L , R. A theory of the plastic distortion of a polycrystalline aggre-gate under combined stresses. Philos. Mag. 42 (1951), 414-427.(5) BL A N D , D. R. The two m easures of work-hardening. Proc. 9th International Congress ofApplied Mechanics, Brussels (1956), 45-50.(6) H E L L A N , K. A note on the equivalence of the postulates of isotropic hardening. J. EngngMaterials Technology, Am . Soc. Mech. Eng. 96 (1974), 79-80.(7) GOTOH, M. A theory of plastic anisotropy based on a yield function of fourth order (planestress state ). Internal. J. Mech. Sci. 19 (1977), 505-520.

(8) B O U B N E , L. and H I L L , R. On the correlation of the directional properties of rolled sheet intension and cupping tests. Philos. Mag. 41 (1950), 671-681.(9) BBAMLEY, A. N. and MELLOR, P. B. Plastic flow in stabilized sheet steel. Internat. J. Mech.Sci. 8 (1966), 101-114.(10) BBAMLEY, A. N. and MELLOR, P . B. Plastic anisotropy of titani um and zinc sheet. I .Macroscopic approac h. Internat. J. Mech. Sci. 10 (1968), 211-219.(11) L E E , D. and BACKOFEN, W. A. An experimental determination of the yield locus for titan-ium and titanium alloy sheet. Trans. Metallurgical Soc, Am. Inst. Mech. Eng. 236 (1966),1077-1084.(12) P E A R C E , R. Some aspects of anisotropic plasticity in sheet metals. Internat. J. Mech. Sci.10 (1968), 995-1005.(13) TOZAWA, Y., NAKAMURA, M., and SHINKAI, I. Yield loci for pre-strained steel sheets. Proc.International Conference on the Science and Technology of Iron and Steel, Tokyo (1970),Trans. Iron Steel. Inst. Japan 11 (1971), 936-940.(14) TAGHVAIPOUR, M. and MELLOR, P. B. Plane strain compression of anisotropic sheet metal.Proc. Inst. Mech. Engrs. 185 (1970), 593-606.(15) WOODTHORPE, J. and PE A RCE , R. The anomalous behaviour of aluminium sheet underbalanced biaxial tension. Internat. J. Mech . Sci. 12 (1970), 341-347.


13/13

Theoretical plasticity of textured aggregates 191(16) BASSANI, J. L. Yield characterization of metals with transversely isotropic plastic proper-ties. Internat. J. Mech. Sci. 19 (1977), 651-660.(17) PABMAB, A. and MELLOE, P. B. Prediction of limit strains in sheet metal using a moregeneral yield criterion. Internat. J. Mech. Sci. 20 (1978), (to appear).(18) MELLOB, P. B. and PABMAB, A. Plasticity analysis of sheet metal forming. Proc. Symp.Mech. Metal Forming (1977) (Plenum Publ. Corporation: New York, 1979).(19) PABMAB, A. and MELLOB, P . B . Plasti c expansion of a circular hole in sheet metal subjectedto biaxial tensile stress. Internat. J. Mech. Sci. 20 (1978), (to appear).(20) BISHOP, J. F. W. and H I L L , R. A theorstical derivation of the plastic properties of a poly-crystalline face-centred metal. Philos. Mag. 42 (1951), 1298-1307.

Documents

Theoretical Plasticity of Textured Aggregates Hill