Micromechanical modelling of polycrystalline materials under non-proportional loading paths

8/14/2019 Micromechanical modelling of polycrystalline materials under non-proportional loading paths

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Volume 52 2007 Issue 2

K. KOWALCZYK-GAJEWSKA , Z. MRZ, R. B. PCHERSKI,

MICROMECHANICAL MODELLING OF POLYCRYSTALLINE MATERIALS UNDER NON-PROPORTIONAL

DEFORMATION PATHS

MODELOWANIE MIKROMECHANICZNE MATERIAW POLIKRYSTALICZNYCHDLA NIEPROPORCJONALNYCH CIEEK DEFORMACJI

The rigid-plastic crystal plasticity model with single yield surface of 2n-degree is applied to simulate the polycrystallinebehaviour and the crystallographic texture development under non-proportional deformation paths. The role of two controllingparameters: the amplitude and frequency for the processes of tension or compression assisted by cyclic torsion of thin-walledtubes made of copper is analysed. The effect of micro-shear bands on the reduction of global hardening rate is described bymeans of the contribution function of shear banding in the rate of plastic deformation. The conclusions drawn from the study

can find also application in the extension of the analysis for high strength and hard deformable materials.

Przeprowadzono symulacje zachowania si polikrysztau oraz rozwoju tekstury krystalograficznej dla nieproporcjonalnychcieek deformacji wykorzystujc sztywno-plastyczny model plastycznoci krysztau z powierzchni plastycznoci stopnia 2 n.Przeanalizowano wpyw dwch parametrw kontrolnych: amplitudy i czstoci dla procesw rozcigania lub ciskania przyudziale cyklicznego skrcania cienkociennych rurek wykonanych z miedzi. Efekt dziaania mikropasm cinania w postaci re-dukcji globalnego moduu wzmocnienia odksztaceniowego zosta opisany poprzez funkcj udziau pasm cinania w przyrociedeformacji plastycznej. Wnioski wysnute z powyszego studium mog znale docelowo zastosowanie w rozszerzeniu analizydla materiaw o podwyszonej wytrzymaoci oraz materiaow trudno odksztacalnych.

1. Motivation

Metal forming processes such as extrusion assistedby cyclic torsion have demonstrated essential advantageswith respect to classical forming processes. The signif-icant reduction of required load for forming, growth ofductility, possible reduction of dissipated energy of form-ing and finer grain structure are the main beneficial fac-tors, cf. e.g. [5, 10, 11]. One of possible explanationsof these advantages is the appearance of a multi-scalehierarchy of shear localization modes, so-called shearbanding, that replaces the crystallographic multiple slipor twinning and reduces strain hardening. In the paperthe constitutive model developed in [13] will be applied

to study macroscopic behaviour of polycrystalline mate-rials that are subjected to non-proportional deformationpaths as so called KOBO-type processes related withcyclic torsion [10]. Plastic anisotropy evolution connect-ed with crystallographic texture development will be alsostudied.

In the next section the concise description of theconstitutive model is given. More information can be

found in [12] and [13]. In section 3 the results of nu-merical simulation are presented and discussed. Finally,in the last section the conclusions are outlined.

2. The constitutive modelling

2.1. The single grain model with the yield surfaceof 2n-degree

The rigid-plastic model is considered. In order tostudy the texture development as well as the polycrys-

talline behaviour for the advanced plastic deformationprocess the large strain formulation of the crystal plastic-ity theory has to be applied [3]. We use the multiplicativedecomposition of the total deformation gradient into theelastic part and the plastic part Fp. The assumption ismade that the elastic stretches are negligible comparing

INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH PAS, 00-049 WARSZAWA, 21 WITOKRZYSKA STR., POLAND

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to the analysed plastic strains so that the elastic part isrestricted to the rigid rotation R. It is easy to showthat such form of the deformation gradient results in theadditive decomposition of the velocity gradient L intothe elastic spin , the plastic spin p and the rate ofplastic deformation tensor Dp (the symmetric part of theplastic part of the velocity gradient)

L = +p + Dp. (2.1)

It is generally accepted that during the elastic regimecrystallographic lattice and the material undergo thesame deformation while plastic deformation does not in-duce lattice motion which is described only by the elasticspin. Plastic deformation occurs by slip in the specifiedcrystallographic direction m on the specified crystallo-graphic plane with the unit normal n. The number M andtype of slip systems {n, m} depend on the lattice type. Inthis paper the numerical simulations are performed forthe f.c.c. crystals with 12 slip systems {111} < 110 >.The plastic part of the velocity gradient is then describedas follows

Lp = p + Dp =M

r=1

rmr nr (2.2)

In the classical S c h m i d multi-surface plasticitythe slip on the considered slip system is initiated if theresolved shear stress r = mr nr reaches the criticalvalue rc being the material parameter. Instead of thisformulation the regularized S c h m i d law (the crys-tal plasticity with single yield surface) is used [12]. Inview of this model the crystallographic slip mechanismis activated when the following yield condition for the

C a u c h y stress is fulfilled

f() =1

2n

Mr=1

r

rc

2n m = 0, (2.3)

where n is the positive exponent and m is the materialparameter independent of the lattice orientation [7]. Therate of plastic deformation tensor is associated with thisyield condition

Dp = Mr=1

1rc

r

rc

2n1Pr (2.4)

and for the plastic spin p the following specification isused

p =

Mr=1

1rc

r

rc

2n1Wr, (2.5)

where Pr and Wr are symmetric and skewsymmetricparts ofmrnr diad, respectively. Comparing the above

flow rules with the kinematic description (2.2) one mayeasily identify that

r = 1

rc

r

rc

2n1. (2.6)

In its mathematical structure this model is similar to therate-dependent formulation [4], however, the referencevelocity, which is the material parameter in the latterapproach, is here replaced by the plastic multiplier

obtained from the consistency condition for the yieldsurface (2.3). The explicit formula for may be foundin [7]. In the case of deformation controlled process itmay be calculated from the formula

=Dp

Mr=1 1rc

r

rc

2n1Pr

,

where A =

A A = Ai jAi j.Slip occurs by motion of dislocations that have to

overcome both short-range and long-range obstacles. Ac-

cumulation of dislocations makes its further movementmore and more difficult. This qualitative description ofphysical nature of slip is captured in the framework ofcrystal plasticity by the hardening rule for the critical re-solved shear stress rc. In this study we use the formulaproposed in [3]

rc =

Mq=1

hrq |q|, (2.7)

where hardening matrix hrq enables to account for thelatent and self-hardening by setting the parameter q 1

in its specification

hrq = hq(q + (1 q)rq ).

For the function hq responsible for the self-hardening weassume the following equation [2]

hq = ho

1

qc

sat

(2.8)

with the material parameters {ho, qc0, sat, } that have tobe identified in the experiments. Here r

c0are the initial

values ofqc .Most of the results discussed in this paper is ob-

tained applying the presented model. The material pa-rameters have been identified with use of the experimen-tal results for polycrystalline copper reported in [9], [1].They have been obtained by fitting the stress-strain curvefor the simple compression. More details concerning theidentification procedure may be found in [14].


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2.2. Simplified description of the reductionof the hardening rate due to micro shear bands

General conclusion that follows from experiments[5, 10, 11] is that by persistent changing of the defor-mation path it is possible to reduce significantly theglobal strain hardening rate. The inventors of KOBOmethod explained their pivotal idea as follows. Expe-rience gained from the conducted tests allow them to ar-gue that the change of mode of plastic flow destabilizes

the material substructure that has been developed andleads to the initiation of the second mode of deforma-tion. The homogeneous crystallographic slip is graduallyreplaced by the localized heterogeneous deformation bymicro-shear bands.

In order to account for the reduction of the hardeningrate due to appearance of micro-shear bands we followthe concept introduced in [18]. The plastic velocity gra-dient Lp is decomposed into two parts: one connectedwith the crystallographic slip Lp

slipand one connected

with micro-shear bands LpMS

. For the crystallographic

slip the rate-independent single grain model presentedin the previous subsection is used. The constitutive ap-proach for the part connected with the micro-shear bandswas thoroughly described in the series of papers [13,1821].

In the present analysis the simplified model describ-ing the effect of micro-shear bands is applied. The yieldcondition for initiation of micro-shear bands is the sameas the yield condition for slip mechanism (2.3). In [13]two flow rules for micro-shear bands were considered.In this paper we focus on the model in which the flowrule for the part of the rate of plastic deformation Dp

MS

connected with micro-shear bands is associated with theyield condition, so both tensors DpMS

and Dpslip

have thesame directions. Using the following notation:

Dp = d, Dpslip

= dslip,

DpMS

= dMS fMS =dMS

d

the total rate of plastic deformation reads

Dp =

1

fMS

M

r=1

1r

c

r

rc

2n1Pr, fMS < 0, 1) (2.9)

and the constitutive function fMS describes the contribu-tion of shear banding in the rate of plastic deformation.

Since the hardening rate depends on the incrementof crystallographic slip (Eqs (2.6) and (2.7)), the highercontribution of micro-shear bands the higher is the re-duction of the global strain hardening rate. According tothe invoked experimental results the reduction of strain

hardening is more pronounced for plane strain com-pression than for the simple compression [9]. Followingthis observation the fMS function of logistic dependenceon accumulated plastic strain proposed and identified in[19], [16], was extended in [13] to incorporate the effectof the type of strain path so that the following formulawas proposed

fMS () =f

MS

1 + exp(a

b),

where

=

32

peq(1 | cos3|), 0 1 (2.10)

and

peq =

23

Dp Dp, | cos3| = 3

6| det(Dp )|,

Dp =Dp

Dp

.

The coefficient is an additional material parameter thatdescribes the following ratio

= 1 cos3=1cos3=0

for the same value of the accumulated plastic strain fortwo extreme types of strain paths. In the next section thenon-proportional strain paths will be analysed, therefore,contrary to the analysis in [13] the distinction has to bemade between magnitude of plastic strain Ep and theaccumulated plastic strain peq. For the purpose of thisanalysis the second one must be used.

For the plastic spin connected with micro-shearbands we use the rule expressed in [6] that relates theplastic spin to the non-coaxiality factor between corre-sponding the plastic strain rate tensor and the C a u c h ystress

p

MS= fMS

DpMS

DpMS

DpMS

DpMS

, (2.11)

where

p

= ,

p

MS= MS

fMS =

MS

.

For the second constitutive function fMS connectedwith micro-shear bands two possibilities are considered,namely

fMS = 0, and fMS = fMS . (2.12)

In the first case there is no plastic spin due to micro-shearbanding and in the second case the contribution of


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micro-shear bands mechanism to the total plastic spinis the same as the contribution to the rate of plasticdeformation tensor.

The material parameters {a, b, fMS , } connectedwith the incorporation of shear banding have been iden-tified by comparison of the experimental results for sim-ple compression and plane strain compression reportedin [9], [1].

3. Simulations

3.1. Program of simulations

In the simulations non-proportional deformationpaths are studied. The analysis is performed at the ma-terial point on the level of a polycrystalline sample. Asthe representative volume element (RVE) for this macro-scopic material point, the grain aggregate composed of500 grains with initially random distribution of latticeorientations is assumed. As a micro-macro transition rulethe Taylor hypothesis is applied assuming that the veloc-ity gradient Lg in every grain is the same and equal tothe macroscopic velocity gradient L. The macroscopicstress and plastic dissipation are calculated as follows

=

Ng=1

gg, Wp =

Ng=1

gg Dp Wp =t

0Wpdt,

(3.13)

where g is the volume fraction of grains with the ori-entation g in RVE.

Consider a thin walled tube of initial outer radiusRo, length Lo and wall thickness to subjected to axialtension or compression, Fig. 1. Assume the axial strainrate x to be specified and constant. The alternating tor-sion is imposed in order to reduce the axial stress andthe applied axial force in order to execute the process.Such processes of tension or compression assisted bycyclic torsion were studied in [15] in the framework ofsmall strain formalism and rigid-plastic or elastic-plasticmaterial model with the H u b e r-Vo n M i s e s yieldcondition.

Here these processes will be studied in the frame-

work of large strain formalism and crystal plasticity. Inorder to do that the macroscopic velocity gradient of theform

Li j = x

1.0 0 0(t) 0.5 0

0 0 0.5

in {ek} (3.14)

is assumed where x is the constant rate of extensionwhile (t) x is the rate of shear in cyclic torsion. Simi-larly to [15] piecewise linear and harmonic programs ofcyclic torsion will be considered that is

(t) =

0sign[cos(2t/T)]

2 o cos(2t/T)

(3.15)

and

E = xT,

where o controls the maximum rate of shear. The re-sulting stress-strain curves will allow us to compare theinfluence of the process parameters {o, m} on the axialstress reduction and the plastic dissipation. The textureevolution will allow to study the microstructure changes.The simulations have been performed for the set of con-trolling parameters collected in Table 1. The calculationshave also been done for o = 0. The analysed set of pa-

rameters roughly corresponds to the program of experi-ments performed by G r o s m a n and P a w l i c k i [8]on cylinders made of copper. For the upper row of ovalues the maximum shear angle m is close to 2o andfor the lower row it is close to 8o.

TABLE 1Set of controlling parameters for the simulation program

Strain period E = x T

0.085 0.031 0.013

o 1.64 4.56 10.54

6.56 18.2 42.2

The deformation history may be derived using theknown relation

F = LF, Fi j(t) = Lik(t)Fk j(t)

which actually consists of 9 differential equations. Theycan be integrated to obtain

Fi j =

exp(x) 0 0

f(x)exp12 x

exp

12 x

0

0 0 exp12 x

in {ek},(3.16)

where x = xt and in the case of harmonic torsionoscillations


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f(x) =oE

162 + 9E2exp

3

2x

3Ecos(2

x

E) 4 sin(2 x

E) 3E

.

The initial conditions of the form F(0) = I, Fi j(0) = i jhave been assumed. In the case of piecewise linear os-cillations it is obtained for the regimes of constant

f(x) = F012 exp(12

0) 23

exp(

32

x) exp(32

0)

,

where F12(0) = F012 and x = 0 indicates the momentof last reversing of torsion deformation.

Using the specified deformation gradient (3.16) onemay derive stretch ratios along axis of a tube and alongthe tube radius (see Fig. 1)

1 = exp(x)

1 + f2(x)exp(3x), 3 = exp1

2x

as well as sinus of an angle of shear between initiallyperpendicular directions e1 and e2 (see Fig. 2)

sin =f(x)exp(32 x)

1 + f2(x)exp(

3x)

.

Finally, the rate of equivalent plastic strain peq in theanalysed case (compare (2.1)) reads

pep = x

1 +

13

2

so the results presented in the next subsection must beinterpreted with care because axial plastic strain is forthe considered process not equal to the equivalent plasticstrain.

Fig. 1. Geometry of a cylindrical tube (up) and stretch ratios for tension assisted by cyclic torsion ( = 6.56, E = 0.085) (down)


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Fig. 2. Variation of component F21 of deformation gradient (solid grey line), L21 component od velocity gradient (dashed grey line) and

shear angle (solid black line) with x ( = 6.56, E = 0.085)

3.2. Results

In this subsection first the results of simulations con-cerning stress-strain curves will be presented and nextthe crystallographic texture development will be dis-cussed.

3.2.1. Strain-stress curves

In Fig. 3 the variation of axial component of themacroscopic Cauchy stress 11 with F11 component ofdeformation gradient are presented for the compressionassisted by cyclic torsion. The material model discussed

in subsection 2.1 has been applied. The following valuesof the material parameters have been identified in [14]with use of the experimental results for polycrystallinecopper [1, 9]

n = 20, c0 = 14MPa, h0 = 232.75MPa,

sat = 138.6MPa, q = 1.4, = 2.5.

Gradual reduction of the value of 11 is observed withthe increasing value of the parameter o. Also, for largershear strain amplitude m the reduction of axial stress ismore pronounced.


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Fig. 3. Variation of axial component of the macroscopic Cauchy stress with F11 for compression assisted by harmonic torsion oscillations

(upper row) and piecewise linear torsion oscillations (lower row), eta = o in the labels of curves

These results have been recalculated to obtain vari-ation of compressive axial force F with cylindrical tubeshortening u. The initial cross-section A0 and initiallength Lo of a tube was assumed as those used in Gros-man experiments on cylinders [8]. The results are pre-sented in Fig. 4. For 0 = 0 quantitative agreement withthose experiments were achieved while in other cases

the results may be compared only qualitatively due to adifferent geometry of the analysed specimens (a cylin-drical tube versus a cylinder)1). The observed influenceof controlling parameters on the axial force reduction insimulations corresponds to the one observed in experi-ments.

Fig. 4. Variation of compressive force with the cylindrical tube shortening for compression assisted by harmonic torsion oscillations (left)

and piecewise linear torsion oscillations (right), eta = o in the labels of curves

1) Large number of considered grains makes the calculations time-consuming so that it is possible to perform analysis only at the materialpoint or for a body that undergoes uniform deformation


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Fig. 5. Variation of axial component of the macroscopic Cauchy stress with F11 for tension assisted by harmonic torsion oscillations (upper

row) and piecewise linear torsion oscillations (lower row), eta = o in the labels of curves

Fig. 6. Variation of axial component of the macroscopic Cauchy stress with the accumulated plastic strain peq for tension (left) or

compression (right) assisted by harmonic torsion oscillation (upper row) and piecewise linear torsion oscillation (lower row), eta = o in

the labels of curves the effect of incorporation of micro-shear banding


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The same program of simulations has been also con-ducted for tension assisted by cyclic torsion. Due tothe applied large strain formalism and the role of theplastic anisotropy induced by the texture developmentthe results concerning the stress-strain curves for ten-sion and compression differs contrary to what has beenobtained in the case of small strain and the classicalH u b e r-V o n M i s e s plasticity analysed in [15] forthe same type of processes. In Fig. 5 the variation ofmacroscopic axial stress with F11 is presented. General

tendency concerning the role of process parameters aswell as the curve character is similar but some differ-ences may be noticed in the shape of curves comparingto the compression process.

Now, let us look at the effect of incorporation ofmicro-shear banding mechanism of plastic deformationinto the constitutive modelling on the axial stress reduc-tion. The following material parameters for the functionfMS were identified with use of experimental data pre-sented in [1, 9]

= 1.0, fMS = 0.85, a = 5.0, b = 9.62.

In Fig. 6 the variation of macroscopic axial stresswith the equivalent plastic strain is presented. One mayobserve that in the process of tension or compressionassisted by cyclic torsion the accumulated plastic strainis much larger than in the traditional process representedby the curve = 0. Extension of the constitutive model

accounting for the shear banding enables to obtain fur-ther reduction of axial stress comparing to the modelwithout introduction of fMS function.

We conclude this subsection with the presentation ofthe results concerning the plastic dissipation (Fig. 7). Asit can be expected, similarly to the results in [15], withthe increasing parameter o the plastic dissipation is alsoincreasing. Some reduction of plastic dissipation due tothe reduction of the strain hardening rate is obtainedwhen applying the model with shear banding (Fig. 8).

Fig. 7. Increase of plastic work during the process of compression (left) and tension (right) assisted by harmonic torsion oscillation,eta = o in the labels of curves

Fig. 8. Increase of plastic work during the process of compression (left) and tension (right) assisted by harmonic torsion oscillation the

effect of incorporation of shear banding, eta = o in the labels of curves


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3.2.2. Crystallographic texture development

In this subsection we will present the results con-cerning the crystallographic texture evolution in the anal-ysed processes. In order to clarify the effects observedfor the grain aggregate, first, we look at the behaviourof single, arbitrary selected orientation, Fig. 9.

Fig. 9. Comparison of reorientation of arbitrary selected grain during

the simple compression and compression assisted by cyclic torsion standard pole figure {111}, eta = o in the labels of points

One may observe that the orientation path is differ-ent for the process assisted by cyclic torsion than for thesimple compression. In the first case it is composed of

reversible rotation connected with torsion and reorienta-tion due to the compression. Furthermore, the final ori-entation of a grain differs from that achieved in the con-ventional process. It is due to the fact that in the analysedprocess the direction of strain rate tensor is completelydifferent than in the simple compression/tension processso the plastic flow is executed by the different set ofslip systems. Consequently, crystallographic texture de-veloped in the analysed processes with non-proportionalstrain paths predicted by the crystal plasticity models

differs significantly (Figs 10 and 11) from the widely re-ported textures of simple compression or extrusion (seefor example [1]). Note that the applied model fairly wellpredicts these textures (compare Figs 10a and 11a).

When the effect of micro-shear banding is added,depending on the prescribed assumption (2.12), with in-creasing contribution of micro-shear banding the evolu-tion of crystallographic texture is impeded or the finaldeveloped texture is changed with respect to the mod-el without micro-shear banding, respectively. In Fig. 12the second case is presented. The observation that in thepresence of micro-shear bands the developed crystallo-graphic texture is different [17, 22] may be explainedby the localized character of plastic deformation. Out-side the zones of micro-shear banding the rate of plasticdeformation is much smaller than the averaged rate ofplastic deformation in the whole volume element.

Fig. 10. Final standard pole figures {111} for compression assisted by harmonic torsion oscillations: a) o = 0, b) and c) m 2o, o = 1.64and o = 10.54, d) and e) m 8o, o = 6.56 and o = 42.2


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Fig. 11. Final standard pole figures {111} for tension assisted by harmonic torsion oscillations: a) o = 0, b) and c) m 2o

, o = 1.64 ando = 10.54, d) and e) m 8o, o = 6.56 and o = 18.2

Fig. 12. Final standard pole figures {111} for tension (a) and compression (b) assisted by harmonic torsion oscillations, m 2o, o = 10.54 effect of micro-shear banding ( fMS = fMS )

4. Conclusions

The polycrystalline behaviour and texture develop-ment have been studied for the advanced plastic deforma-tion processes with non-proportional strain paths, name-ly tension or compression assisted by torsional oscilla-tions. The crystal plasticity model with one yield surfaceexpressed by the homogeneous yield function of degree2n has been applied within large strain formalism. Inorder to account for the effect of micro-shear bandingmechanism on the reduction of the strain rate hardeningthe simplified model developed in [13] has also beenapplied.

The applied micromechanical model enables also tosimulate the crystallographic texture development that is

a result of lattice rotation within the grain aggregate sub-jected to large plastic deformations. The theoretically ob-tained crystallographic texture evolution in the analysedprocesses is significantly different than that predicted forthe simple compression or extension. The reason is thefact that in the crystal plasticity framework the textureevolution strongly depends on the set of deformationmechanisms constituting the deformation path (compareEqs. (2.1), (2.5)). It must be stressed that the successin the texture prediction lies in the proper reproductionof the deformation path in the analysed process togetherwith the knowledge of the potential set of deformationmechanisms for the considered material. The experimen-tal measurements of texture development in the process-es with non-proportional deformation paths should serve


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as a tool for verification of this input information and theapplied micromechanical model.

Two controlling parameters for the analysed defor-mation programs, m and o, affect the process as fol-lows: the higher m and o the more significant is theaxial stress reduction. In the same time the plastic workis always increased as compared to the processes with-out torsional oscillations. These conclusions correspondto the results of theoretical analysis performed in [15] inthe case of phenomenological plasticity theory. It should

be stressed that the results of simulations presented inthis paper are not directly comparable to the situationmet in the KOBO process of extrusion assisted by cyclictorsion. In the simulated process the whole sample un-dergoes plastic deformation while in the KOBO processthe zone of plastic yielding is strongly localized.

Acknowledgements

Most of the results reported in the paper was obtained in the

framework of the research project No. PBZ-KBN-102/T08/2003 sup-

ported by the State Committee for Scientific Research of Poland.

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Received: 10 March 2007.

Micromechanical modelling of polycrystalline materials under non-proportional loading paths

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