shear-band development in polycrystalline metal with strength-differential effect and plastic volume...

doi: 10.1098/rspa.2002.0971, 2243-2259458 2002 Proc. R. Soc. Lond. A

Mitsutoshi Kuroda and Toshihiko Kuwabara differential effect and plastic volume expansion

−band development in polycrystalline metal with strength−Shear

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10.1098/rspa.2002.0971

Shear-band development in polycrystallinemetal with strength-differential effect

and plastic volume expansion

By Mitsutoshi Kuroda1

and Toshihiko Kuwabara2

1Department of Mechanical Systems Engineering, Yamagata University,Yonezawa, Yamagata 992-8510, Japan

2Department of Mechanical Systems Engineering,Tokyo University of Agriculture and Technology,

Koganei, Tokyo 184-8588, Japan

Received 12 October 2001; accepted 18 February 2002; published online 15 July 2002

Experimental studies have shown that the flow stress of some metals is clearly influ-enced by superimposed hydrostatic pressure. Flow stress increases with the hydro-static pressure, and consequently it is often observed that the flow stress is clearlylarger in a uniaxial compression test than in a uniaxial tension test. This phenomenonis known as the strength-differential effect (SDE). In addition, metals undergoinguniaxial tension and compression show a permanent volume expansion (dilatancy)which is insensitive to the sign of the hydrostatic pressure. In this paper, shear-banddevelopment in polycrystalline metal with the SDE and dilatancy is studied, using arate-dependent crystal-plasticity model with a full three-dimensional, body-centred-cubic, slip-system structure. It is postulated that the appearances of the SDE anddilatancy are consequences of non-Schmid effects existing in each slip system. In thefinite-element analyses performed here, each Gaussian integration point in a finiteelement represents a polycrystal consisting of many of crystal grains having differ-ent orientations. As a boundary-value problem, a rectangular specimen subjected toplane-strain tension is considered. Although the finite-element geometry is chosen tobe two dimensional, the constitutive model incorporates the full three-dimensionalslip-system structure. As a polycrystal model to be embedded in each integrationpoint, an extended Taylor model is employed. Thus, macroscopic manifestations ofnon-Schmid effects are studied. The influences of hydrostatic stress, internal fric-tion, plastic volume expansion and strain-rate sensitivity on macroscopic shear-bandformation in polycrystals are investigated. Results are directly compared with pre-dictions from a classical Schmid-law-based crystal-plasticity theory.

Keywords: strain localization; crystal plasticity;finite-element method; hydrostatic stress

1. Introduction

Experimental studies have shown that the flow stress of some metals is clearly influ-enced by superimposed hydrostatic pressure, i.e. the flow stress increases with hydro-static pressure. Such a pressure dependency of the flow stress has been observed

Proc. R. Soc. Lond. A (2002) 458, 2243–22592243

c© 2002 The Royal Society



2244 M. Kuroda and T. Kuwabara

mainly for iron-based metals (Spitzig et al . 1975, 1976). As a consequence, it isobserved that the flow stress of iron-based metals is often clearly larger in uniaxialcompression than in uniaxial tension. This phenomenon is known as the strength-differential effect (SDE) (Leslie & Sober 1967; Kalish & Cohen 1969; Rauch &Leslie 1972; Chait 1972, 1973). The pressure dependency of flow stress has alsobeen observed for aluminium (Spitzig & Richmond 1984), but the SDE observed wasnegligibly small and could not be detected.

In addition, the iron-based high-strength metals undergoing uniaxial tension andcompression showed a permanent volume expansion (dilatancy) which is insensitiveto the sign of the hydrostatic stress. The observed magnitude of permanent volumeexpansion was much smaller than that predicted by a classical plasticity theory inwhich the volumetric plastic strain rate is associated with the hydrostatic pressureterm in the yield function (Spitzig et al . 1975, 1976; Spitzig & Richmond 1984). Thismeans that the associative flow rule is inadequate to explain the aforementionedphenomena, including the SDE.

Based on the concept of the ‘non-associative’ flow rule, Rudnicki & Rice (1975)proposed a simple generalization of the classical Prandtl–Reuss equation which canindependently account for the plastic dilatancy and the dependence of yielding onthe pressure. This proposition was originally motivated with reference to dilatant,frictional behaviour in rocks. Needleman & Rice (1978) used this constitutive modelto interpret the SDE reported by Spitzig et al . (1975). Based on the same concept,Brunig (1999, 2001) and Brunig et al . (2000) have studied numerically the effect ofhydrostatic stress sensitivity on shear-band development in iron-based high-strengthmetals, using generalized I1–J2 or I1–J2–J3 yield and flow potential functions. Brunigand co-workers concluded that the earlier onset of strain localization, as well as highlylocalized subsequent deformation, is caused by the deviations from the normality flowrule resulting from the non-associative structure of the constitutive models.

In the framework of a physically based crystal-plasticity model, a generalized ‘non-Schmid’ rule and its effects on localization of plastic deformation in single crystalwere first studied by Asaro & Rice (1977). Based on this approach, Dao & Asaro(1993) developed a rate-dependent constitutive model with non-Schmid effects. Inthis model, the hydrostatic pressure sensitivity can be incorporated by choosing aspecial set of non-Schmid parameter values. The influences of non-Schmid effects onstrain localization in a single crystal (Dao & Asaro 1996a, b) and on stress–strainbehaviour, texture development and localized deformation in polycrystals (Dao etal . 1996) were studied. In Dao & Asaro’s formulation, plastic volume expansion ordilatancy was not considered. A general constitutive framework for dilatant, frictionaland pressure sensitive materials was proposed by Nemat-Nasser (1983), who focusedon the behaviour in geomaterials. Very recently, this type of model has been improvedto account for the behaviour in granular materials (Nemat-Nasser 2000; Anand &Gu 2000). In these models, the dependency of yielding on internal friction caused bynormal stress on the slip plane and the plastic dilatancy can be treated independently.Following the generalized idea of Nemat-Nasser (1983), Brunig (1998) successfullyperformed numerical simulations of the experimentally observed SDE (Spitzig et al .1975), using a simplified two-dimensional double-slip crystal-plasticity model (Asaro1979) with rate-independent assumption.

In this paper, the shear-band development in polycrystals with the SDE and dila-tancy is studied, using a rate-dependent crystal-plasticity model with a full three-

Proc. R. Soc. Lond. A (2002)



Shear-band development in polycrystalline metal 2245

dimensional body-centred-cubic (BCC) slip-system structure. In the finite-elementanalyses performed here, each Gaussian integration point in a finite element repre-sents a polycrystal consisting of many of crystal grains having different orientations.As a boundary-value problem, a rectangular specimen subjected to plane-strain ten-sion is considered. It is important to note that although the finite-element geometryis chosen to be two dimensional, the constitutive model incorporates the full three-dimensional slip-system structure. As a polycrystal model is to be embedded in eachintegration point, an extended Taylor model (Asaro & Needleman 1985) is employed.Accounting for many of different grain orientations in each integration point, it isemphasized that this gives a reasonable approximation of a polycrystal metal, aslong as the grain size is much smaller than the size of the tensile specimen, but notfor grain sizes comparable with the specimen size. Thus, macroscopic manifestationsof non-Schmid effects incorporated in each slip system are studied. The influences ofhydrostatic stress, internal friction, plastic volume expansion and strain-rate sensi-tivity on macroscopic shear-band formation in polycrystals are investigated. Resultsare directly compared with predictions with a classical Schmid-law-based crystal-plasticity model (Kuroda & Tvergaard 2001b).

2. Constitutive model

We use notation commonly used in modern continuum mechanics. The tensors andvectors will be denoted by bold-face letters: e.g. a = aijei ⊗ ej , b = bijei ⊗ ej ,n = niei, where ⊗ denotes the tensor product and ei a Cartesian basis. The followingdefinitions for operation are used: ab = aikbkjei ⊗ ej , an = aijnjei, a : b = aijbij ,with proper extension to higher-order tensors.

(a) Relations for single crystal

It is assumed that the elastic strain is small, but the plastic strain may be large. Ina single crystal, it is assumed that the spatial velocity gradient L (= ∂vi/∂xjei ⊗ej ,where v is the velocity of a material particle and x is its current position) is additivelydecomposed into non-plastic and plastic parts (e.g. Asaro & Rice 1977; Nemat-Nasser1983; Anand & Gu 2000),

L = L∗ + Lp. (2.1)

The plastic contribution Lp is assumed to arise from slip on a finite number of slipsystems (Nemat-Nasser 1983),

Lp = Dp + W p, (2.2)

Dp =∑α

γ(α)N (α), W p =∑α

γ(α)w(α), (2.3)

where the superscript ‘(α)’ represents a quantity with respect to the αth slip system,γ(α) is the slip rate, the symmetric part Dp and the antisymmetric part W p are theplastic rate of deformation and the plastic spin, respectively, and

N (α) = p(α) + c sgn(γ(α))(m(α) ⊗ m(α)) + d sgn(γ(α))I, (2.4)

p(α) = 12(s(α) ⊗ m(α) + m(α) ⊗ s(α)), (2.5)

w(α) = 12(s(α) ⊗ m(α) − m(α) ⊗ s(α)). (2.6)





Here, m(α) is the unit vector normal to the slip plane, s(α) is the unit vector repre-senting the slip direction in the corresponding slip plane, c and d are material param-eters that characterize the plastic volume expansion, and I is the second-order unittensor. According to the flow rule of equations (2.3)–(2.5), the rate of plastic volumestrain is given by Dp

kk = I : Dp = (c + 3d)∑

α |γ(α)|. Thus, the volume of materialalways expands plastically for any plastic deformation, as long as the parameters cand d are chosen to be positive.

The contribution L∗ corresponds to the elastic lattice distortion and rotation. Theelastic constitutive relation for a crystal is assumed to be

◦σ∗ = σ − W ∗σ + σW ∗ = C : D∗, (2.7)

D∗ = 12(L∗ + L∗T), W ∗ = 1

2(L∗ − L∗T), (2.8)

where σ is the Cauchy stress, C represents the instantaneous crystal elasticity mod-uli, and the superscript ‘T’ denotes transpose. Denoting the Jaumann stress rate by�σ, which is based on the total continuum spin W = 1

2(L − LT), and the total rateof deformation D = 1

2(L + LT), it follows from equations (2.1)–(2.8) that�σ = σ − Wσ + σW = C : D − P , (2.9)

P =∑α

γ(α)(C : N (α) + w(α)σ − σw(α)). (2.10)

The evolution laws of s(α) and m(α) are simply assumed to be (Asaro & Rice 1977;Asaro 1979)

s(α) = W ∗s(α), m(α) = W ∗m(α). (2.11)With the resolved shear stress τ (α) = (s(α) ⊗ m(α)) : σ, the normal stress acting

on the slip plane σ(α) = (m(α) ⊗ m(α)) : σ, and the hydrostatic stress σkk/3, anextended viscoplastic power law can be written as

γ(α) = γ0 sgn(τ (α))[|τ (α)| + aσ(α) + bσkk

g(α)

]1/m

(2.12)

for |τ (α)| + aσ(α) + bσkk � 0 and |τ (α)| > 0,

γ(α) = 0 otherwise, (2.13)

where γ0 is a reference shear strain rate, m is a rate sensitivity parameter, a and b arenon-negative material parameters that characterize the dependence of ‘yielding’ onthe normal stress σ(α) and σkk, and g(α)(> 0) is the current hardness of the αth slipsystem. The sign of the slip rate γ(α) is assumed to follow that of τ (α). The absolutevalue is taken only for the resolved shear stress τ (α), because a particular value ofσ(α), as well as that of σkk, should have the same effect on yielding for the positiveand the negative directions of slip. It is obvious that when we set a = b = c = d = 0the present model reduces to the classical model, which is governed by the Schmidrule. Only for the cases a = c and b = d is the plastic volumetric strain associatedwith equation (2.12), which is regarded as the ‘dynamic yield function’ for each slipsystem. The evolution equation for g(α) is taken to be

g(α) =∑

β

hαβ|γ(β)|. (2.14)

A specific form of the slip hardening hαβ will be defined later.





In the present study, a BCC crystal with 24 slip systems of the types {110}〈111〉and {112}〈111〉 is considered.

Asaro & Rice (1977), Dao & Asaro (1993, 1996a, b) and Dao et al . (1996) con-sidered a more generalized non-Schmid rule, although they assumed no dilatancy.Indeed, equation (2.12) can be interpreted as a special case of their generalized con-siderations. However, they have stated that it is reasonable to interpret the findingsin Spitzig et al . (1975) as a consequence of hydrostatic stress sensitivity involved inslip systems. In this paper, the focus is mainly placed on the hydrostatic stress sen-sitivity which is governed by b in equation (2.12). For comparison purposes, we alsoconsider a friction type of effect caused by the normal stress σ(α), which is governedby the parameter a in equation (2.12).

(b) Polycrystal model

As a model for polycrystals, an extended Taylor model (Asaro & Needleman 1985)is adopted, in which the deformation in each grain is taken to be identical to themacroscopic deformation of the continuum, i.e. in the rate form, L(k) = L, whereL(k) and L denote the grain and macroscopic velocity gradients, respectively, withthe superscript ‘(k)’ denoting the grain indices (k = 1, . . . , N ; N represents the totalnumber of grains). Taking the volume fraction of each grain to be identical, themacroscopic stress, σ, is obtained from averaging the values over the total numberof grains, i.e.

σ =1N

N∑k=1

σ(k). (2.15)

Based on this approach, the macroscopic (average) constitutive relation becomes�σ = ˙σ − Wσ + σW = C : D − ˙P . (2.16)

The nominal stress rate Π(k) in the kth grain with respect to the current configura-tion is related to σ(k) as

Π(k) = σ(k) − Lσ(k) + (trL)σ(k). (2.17)

The volume average of equation (2.17) is given by

˙Π = ˙σ − Lσ + (trL)σ. (2.18)

Substituting equation (2.16) into equation (2.18) we have

˙Π = C : D − ˙P + Wσ − σW − Lσ + (trL)σ, (2.19)

which will be used directly for the finite-element formulation as shown below.

3. Numerical procedure and problem formulation

An updated Lagrangian finite-element formulation is employed. The finite-elementequation is derived on the basis of the rate-type principle of virtual work with neglectof the body-force effect (e.g. McMeeking & Rice 1975; Burke & Nix 1979),∫

V

˙ΠT : δLdV =∫

St

qδv dS, (3.1)





where V and S are the current volume and surface, respectively, δv and δL are thevirtual velocity and virtual velocity gradient, q is the nominal rate of traction perunit current area of surface, and St is the surface over which the traction rates areprescribed.

The finite-element equation is derived by substituting equation (2.19) into equation(3.1) and approximating the velocity field v in terms of finite-element shape func-tions. In the finite-element computations, the crystal-plasticity constitutive equationintroduced in the previous section is rewritten by use of a rate-tangent modulusmethod (Peirce et al . 1983). The term (trL)σ in equation (2.19) results in a smallnon-symmetry of the geometric stiffness matrix in the finite-element equation. But,this is not a disadvantage, at least in the present computations, because the consti-tutive equation with the rate-tangent modulus method (Peirce et al . 1983) itself isinherently non-symmetric.

Eight-node isoparametric plane-strain elements with four Gaussian integrationpoints (reduced integration) are used. The Taylor polycrystal model consisting ofN grains is used in each Gaussian integration point. Full details of the polycrystalmodel will be given later. It is important to note that although the element geometryis chosen to be two dimensional, the model incorporates the full three-dimensionalBCC slip-system structure.

A rectangular specimen is subjected to plane-strain tension, with x2 being the ten-sile axis and with unit length along the x3-direction (Kuroda & Tvergaard 2001b).The initial dimensions of the specimen are 2H0 along the x1-direction and 2L0 alongthe x2-direction. Considering symmetric conditions, only one quadrant of the speci-men is analysed. Thus, the boundary conditions for the one quadrant are

v1 = 0 and q2 = 0 on X1 = 0, (3.2)v2 = 0 and q1 = 0 on X2 = 0, (3.3)q1 = q2 = 0 on X1 = H0 + ∆H0, (3.4)

v2 = U and q1 = 0 on X2 = L0, (3.5)

where Xi denotes the position of a material particle in the initial configuration, U isthe prescribed rate of end-displacement, and ∆H0 represents an initial geometricalimperfection, which is taken to be of the form

∆H0 = H0[−ξ1 cos(πX2/L0)], (3.6)

with ξ1 being an imperfection amplitude. The end-displacement is obtained fromU =

∫U dt. The problem formulation and the finite-element mesh for one quadrant

of the specimen to be analysed are illustrated in figure 1. The initial aspect ratio ofthe specimen is taken to be L0/H0 = 3.

The finite-element discretization takes the form of 768 quadrilaterals as shownin figure 1. In each Gaussian integration point, a polycrystal model consisting of 48grains, whose orientations have been randomly chosen, is used. In figure 1, the stereo-graphic pole figure of {110} for the BCC polycrystal model is depicted. In a strictsense, this polycrystal model has exact symmetry neither about the x1–x3-plane norabout the x2–x3-plane, due to the finite number of grains. Hence, if we considered amodel of the full rectangular specimen, unsymmetric shear-band formations wouldbe predicted because of the small non-symmetries of the polycrystal. In the computa-tions performed in the next section, however, we are able to predict only symmetric





X2

L0

H0 + ∆H0 polycrystal model of BCC({110} pole figure for 48 grainswith random orientation)

X1O

Figure 1. Problem formulation of plane-strain tension for a polycrystal metal.

shear-band formation since the aforementioned model of one quadrant is used. Evenfor the one quadrant model, the computations are rather heavy, as understood fromthe fact that we consider in total 147 456 crystal grains (= 48 grains × 4 integrationpoints × 768 elements) in one batch of calculations. In order to check the effects of ele-ment discretization and number of grains, a few preliminary computations were done.A calculation with a coarser-mesh discretization (7 × 21 quadrilaterals) and a largernumber of grains (200 grains with random orientations for each integration point)was performed. With the increase in grain number, the assumption of initial isotropybecomes more realistic. The predicted shear-band development was very similar tothat obtained with the finer mesh and the 48 grains shown in figure 1. Furthermore,calculations with the mesh shown in figure 1 and a symmetrized arrangement of48 grain orientations were performed. The latter was constructed from a randomorientation distribution consisting of 12 grains, which was then symmetrized withrespect to the x1–x3- and x2–x3-planes. Such a symmetrized orientation distribu-tion is completely consistent with the assumed symmetric boundary condition of theone quadrant of the specimen shown in figure 1. For the symmetrized orientationdistribution of grains, the shear-band formation was slightly delayed, in comparisonwith the case of using the unsymmetrized orientation distribution shown in figure 1.However, it was confirmed that the element discretization, the grain number and thegrain-orientation distribution do not have any crucial influence on the main conclu-





sions in this study. All the computations reported in the next section were carriedout with the mesh and the grain-orientation distribution shown in figure 1.

4. Results

The components of the slip-hardening rate matrix, hαβ, in equation (2.14) are assum-ed to be

hαβ = h(γa) = h0

(h0γa

τ0n+ 1

)n−1

, γa =∫ ∑

α

|γ(α)| dt, (4.1)

where h0 is the initial slip-hardening modulus, γa is an accumulated magnitude ofslip, n is a strain-hardening exponent, τ0 is the initial value of g(α) (in equation(2.12)) for all the slip systems, and Taylor’s isotropic slip hardening is used in allthe computations here. The crystal elasticity moduli tensor C in equation (2.7) isassumed to be isotropic, determined only with Young’s modulus E and Poisson’sratio ν. The following values of the material constants are fixed throughout thepresent computations: E/τ0 = 1000, ν = 0.3, h0/τ0 = 30 and n = 0.1.

Before analysing the full boundary-value problem formulated in the previous sec-tion, we perform calculations for uniaxial tension/compression tests, by direct useof the extended Taylor model with the random 48 grain orientations the same asthe ones shown in figure 1. According to Spitzig et al . (1975, 1976), the observedmagnitude of plastic volume expansion was much smaller than that predicted by aplasticity theory in which the volumetric plastic strain rate is associated with thehydrostatic stress term in the yield function. Based on this observation, we assumehere a material that has no plastic dilatancy, but has an obvious sensitivity to thehydrostatic stress. In the case of the present crystal-plasticity model, this is accom-plished by setting c = d = 0 and a > 0 and/or b > 0.

Figure 2 shows computed curves of uniaxial true stress σ versus logarithmic strainε, where σ = σ22, ε =

∫D22 dt, D22 = γ0, m = 0.002, a = 0 and b = 0.015. The

amount of the SDE is defined by 2(|σC| − |σT|)/(|σC| + |σT|), where σC and σT are,respectively, the compressive and tensile uniaxial true stresses (Spitzig et al . 1975,1976). It is observed that the value of the SDE for b = 0.015 with a = 0 is realistic andalmost constant with increasing magnitude of uniaxial logarithmic strain. Almostthe same value of the SDE is predicted for a = 0.0225 with b = 0, although thecorresponding curves are not depicted here. In figure 2, curves for a = b = 0 (Schmidrule) are also shown. When adopting the Schmid rule, the magnitude of the truestress for tension is slightly larger than that for compression. This is attributed todifferent texture developments for tension and compression. It has been checked thatthis tendency is unchanged for the case of employing 400 grains.

Next, the plane-strain tensile problem shown in figure 1 is analysed. The rate ofthe end-displacement is prescribed as U = ˙ε(L0 + U), where the magnitude of theprescribed ‘average’ strain-rate, ˙ε, is taken to be equal to that of γ0. The initialimperfection amplitude ξ1 is taken to be 0.005.

We first consider a material with b = 0.015 and a = c = d = 0. The rate sensitivityparameter is taken to be m = 0.002. Figure 3a shows curves of the average nominalstress versus end-displacement. The average nominal stress is defined by Πave =P/H0, where P is the tensile load for one quadrant of the specimen, which is obtainedby summation of the nodal forces in the X2-direction at X2 = L0. In figure 3a, the





compression (b = 0.015)

tension (Schmid's rule)

compression (Schmid's rule)

tension (b = 0.015)

SDE (b = 0.015)

σ

7

6

5

4

3

2

1

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

SDE

| |ε

| |

/ 0

στ

Figure 2. Curves of uniaxial stress versus logarithmic strain for tension and compression; hydro-static stress sensitive (b = 0.015 and a = 0) and insensitive (Schmid rule) solids. No dilatancy(c = d = 0) is assumed.

result for the Schmid rule (a = b = c = d = 0) is also depicted. The curve forthe Schmid rule has been reproduced from Kuroda & Tvergaard (2001b). Figure 3bshows the same curves normalized by the maximum load values, Pmax. It is clearthat the curve for b = 0.015 drops earlier than the one for the Schmid rule.

Figure 4 illustrates deformed meshes at stages marked by � on the load–end-displacement curves in figure 3 (the �s in subsequent graphs of load versus end-displacement have the same meaning). The patterns of shear-band formation forboth the cases are the same, but the shear band in the solid with b = 0.015 hasformed earlier than that in the solid governed by the Schmid rule.

Results for a material with the same values of the parameters a–d and a largerviscosity, m = 0.02, are shown in figures 5 and 6. The amount of viscosity consideredhere is realistic for steels. Comparing the curves in figure 5 with those in figure 3,plastic instabilities are much delayed by the increase in viscosity, as is well known. It isnot clear from figure 6b whether an actual shear band has developed for the Schmid-rule material (reproduced from Kuroda & Tvergaard (2001b)), while a somewhatsharper strain localization is seen for the pressure-sensitive material with b = 0.015(figure 6a).

In the results shown in figures 3–6, no plastic volumetric strain (no dilatancy) wasassumed. Here, we consider the case of permitting plastic volume expansion. As anextreme case, it is assumed that the plastic volume expansion is associated with thehydrostatic stress term in ‘dynamic yield function’ of equation (2.12), i.e. b = d =0.015. The dilatant specimen exhibits more elongation than in the case when theSchmid rule is assumed, due to the plastic volume expansion, as seen in figure 7.But, eventually, a very clear shear-band formation which is similar to the case of nodilatancy (figure 4) is predicted, as seen in figure 8.





0

1

2

3

4

5

0.1 0.2 0.3U / L0

b = 0.015

Schmid's rule

ave

/ 0τ

0.1 0.2 0.3U / L0

0

0.4

0.8

1.2(a) (b)

P / P

max b = 0.015

Schmid's rule

Π

Figure 3. Curves of tensile load versus end-displacement for a hydrostatic stress-sensitive solid(b = 0.015 with a = 0 and c = d = 0) and for a Schmid-rule solid. (a) Average nomi-nal stress versus end-displacement. (b) Tensile load normalized by its maximum value versusend-displacement. The viscosity parameter is m = 0.002.

(a) (b)

Figure 4. Deformed meshes at the stages marked by � in figure 3: (a) hydrostatic stress-sensitivesolid (b = 0.015 with a = 0 and c = d = 0; U/L0 = 0.238); (b) Schmid-rule solid (U/L0 = 0.255).The viscosity parameter is m = 0.002.

Finally, we consider the influence of slip resistance produced by the normal stresson slip plane (friction type of effect). It is assumed that a = 0.0225 with b = 0, andno dilatancy, c = d = 0. In figure 9, an earlier gradual drop of the tensile load isobserved for the solid with friction type of effect (a = 0.0225). But, the rapid drop ofthe load at the final stage is much delayed, in comparison with the Schmid-rule solidand to the hydrostatic stress-sensitive solid with b = 0.015. A deformed mesh for thesolid with friction type of effect at the stage marked by � in figure 9 is illustrated infigure 10. Two shear bands are visible.





0

1

2

3

4

5

0.1 0.2 0.3U / L0

b = 0.015

Schmid's rule

ave

/ 0τ

0.1 0.2U / L0

0

0.4

0.8

1.2(a) (b)

P / P

max b = 0.015

Schmid's rule

0.4 0.3 0.4

Π

Figure 5. Curves of tensile load versus end-displacement for the hydrostatic stress-sensitivesolid (b = 0.015 with a = 0 and c = d = 0) and for the Schmid-rule solid. (a) Averagenominal stress versus end-displacement. (b) Tensile load normalized by its maximum value versusend-displacement. The viscosity parameter is m = 0.02.

(a) (b)

Figure 6. Deformed meshes at the stages marked by � in figure 5: (a) hydrostatic stress-sensitivesolid (b = 0.015 with a = 0 and c = d = 0; U/L0 = 0.376); (b) Schmid-rule solid (U/L0 = 0.393).The viscosity parameter is m = 0.02.

5. Discussion

It is obvious that single-crystal plasticity inherently involves the vertex type of effect.But also for polycrystal plasticity the vertex effect is involved as a well-known resultof the single-crystal behaviour. The vertex effect is very important in predictionsof plastic instabilities such as the shear-band development. The previous studies byTvergaard & Needleman (1993), Anand & Kalidindi (1994), Wu et al . (1997) and





0

1

2

3

4

5

0.1 0.2 0.3U / L0

b = d = 0.015

Schmid's ruleav

e /

0τ

0.1 0.2U / L0

0

1.2(a) (b)

P / P

max Schmid's rule

0.3

0.4

0.8

b = d = 0.015

Π

Figure 7. Curves of tensile load versus end-displacement for the hydrostatic stress-sensitive,dilatant solid (b = d = 0.015 with a = c = 0) and for the Schmid-rule solid. (a) Averagenominal stress versus end-displacement. (b) Tensile load normalized by its maximum value versusend-displacement. The viscosity parameter is m = 0.002.

Figure 8. Deformed mesh at the stage marked by � in figure 7. Hydrostatic stress-sensitive,dilatant solid (b = d = 0.015 with a = c = 0; U/L0 = 0.296). The viscosity parameter ism = 0.002.

Kuroda & Tvergaard (2001b) are typical examples of strain localization predictionsusing the crystal-plasticity model. In the present calculations, the successful shear-band predictions are themselves mainly attributed to the inherent vertex effect incrystal plasticity.

The experimentally observed magnitude of plastic volume expansion was muchsmaller than that predicted by a plasticity theory in which the volumetric plasticstrain rate is associated with the hydrostatic stress term in the yield function (Spitziget al . 1975). Thus, the assumption of no dilatancy is approximately valid, and this is





0

1

2

3

4

5

0.1 0.2 0.3U / L0

b = 0.015

Schmid's rule

ave

/ 0τ

0.1 0.2U / L0

0

1.2

(a) (b)

P / P

max

Schmid's rule

0.3

0.8

a = 0.0225

0.4

b = 0.015

a = 0.0225

Π

Figure 9. Curves of tensile load versus end-displacement for solid with frictional effect(a = 0.0225 with b = 0) and no dilatancy (c = d = 0). For comparison purposes, curves for theSchmid-rule solid and the hydrostatic stress-sensitive solid with no dilatancy (b = 0.015 witha = 0 and c = d = 0) are also depicted. (a) Average nominal stress versus end-displacement.(b) Tensile load normalized by its maximum value versus end-displacement. The viscosity param-eter is m = 0.002.

Figure 10. Deformed mesh at the stage marked by � in figure 9. Solid with frictional effect(a = 0.0225 and b = 0; U/L0 = 0.280). No dilatancy, c = d = 0, is assumed. The viscosityparameter is m = 0.002.

done by c = d = 0. In the present model, the material has a non-associative nature,as long as we assume a �= c and/or b �= d. The non-associative nature arising fromhydrostatic stress sensitivity, i.e. b > 0, a = b = c = 0, yields the earlier developmentof the shear band, as seen in figures 3–6.

It is interesting to compare the present results with the phenomenological resultswith hydrostatic stress-sensitive effect (Brunig 1999, 2001; Brunig et al . 2000). Theconstitutive models used by Brunig and his co-workers include a small non-normalityeffect resulting from the non-associative flow rule which is based on assumptions of





the hydrostatic stress sensitivity in the yield function and no (or very small) plasticdilatancy, but do not include the explicit vertex-type effects which are incorporatedin the theories proposed by Christoffersen & Hutchinson (1979), Simo (1987) andKuroda & Tvergaard (2001a). Using such simple non-associative models, we cannotpredict shear-band development in rather high-hardening materials. In fact, materi-als considered for shear-band predictions in Brunig (1999, 2001) and Brunig et al .(2000) were restricted to very-low-hardening materials. However, it has been at leastconfirmed that the non-associative nature in phenomenological models yields theearlier onset of strain localization. This is consistent with the present results for thephysically based polycrystal model with hydrostatic stress sensitivity.

Reliable phenomenological constitutive models will continue to be desired foractual engineering computations. In order to reproduce exactly the present polycrys-tal predictions by means of a phenomenological approach, we need a model whichinvolves both the vertex type of effect and the hydrostatic stress sensitivity. Thistype of approach was pioneered by Rudnicki & Rice (1975), mainly with referenceto the behaviour in rocks. Very recently, Kuroda & Tvergaard (2001a) proposed asimple phenomenological constitutive model with the vertex type of effect, whichaccurately represents observations in crystal plasticity based on the Schmid rule(Kuroda & Tvergaard 1999). Also, it has been confirmed that the predictions by thismodel are consistent with experimental observations (Kuwabara et al . 2000). There-fore, there are physically well-based grounds for using the material model proposedby Kuroda & Tvergaard (2001a). Incorporating the hydrostatic stress sensitivityinto this model with the vertex-type effect may be one possible way of building upa reliable phenomenological constitutive model.

For a comparison purpose, we have also considered the effect of slip resistanceproduced by the normal stress on the slip plane (friction type of effect). When wechoose appropriate values of the coefficients, both the assumptions of the internalfriction and the hydrostatic stress sensitivity have predicted the same amount of theSDE. However, using the same values of the coefficients, the much delayed shear-band formation is observed for the material with friction-type effect, as shown infigure 9. Delayed strain localization caused by the friction type of effect has alsobeen observed for a solid modelled by an idealized double-slip theory (Dao & Asaro1996b). This point should be the subject of further investigation.

The SDE has been discussed mainly for high-strength steels in the literature.However, according to the in-plane compression experiments for sheet metals carriedout by one of the present authors (Kuwabara et al . 1995, 2002), even Al-killed low-carbon steel sheets are found to exhibit the SDE, as well as the high-strength steelsheet with the relatively low tensile strength of ca. 370 MPa. Furthermore, hexagonalclose-packed materials, such as magnesium (Kelly & Hosford 1968) and titanium (Lee& Backofen 1966; Lowden & Hutchinson 1975; Kuwabara et al . 2001), are also knownto exhibit the SDE.

From a practical point of view, the SDE is expected to be an influential factorin some kinds of metal-forming processes, in addition to its influence on the predic-tions of ductility. In the press-stamping processes of sheet metal parts, for example,the geometry of the formed parts unexpectedly deviates from the design geome-try after removing the parts from the dies because of elastic recovery (springback).The springback phenomenon prohibits the production of precise sheet metal parts.The springback is caused by the residual bending moment, i.e. tension/compression





stresses mixed in the bent sheet. Therefore, in order to predict accurately the mag-nitude of springback in a numerical simulation, the difference of flow stresses intension and compression, i.e. the SDE, should be accounted for. Another example tobe noted is the influence of the SDE on earring behaviour in the cup-drawing of atitanium sheet (Kuwabara et al . 2001). It was found that the pure titanium sheet hasa significant SDE and, moreover, the magnitude of the SDE is quite different withthe strain levels and the testing directions. Since the variation of the circumferentialcompressive flow stresses in the flange of the drawn cup has a significant influence onthe earring behaviour in cup-drawing (Hill 1950), we cannot predict accurately theearring behaviour, if the constitutive law used in the simulation cannot reproduce theSDE appropriately. Thus, building up a reliable phenomenological constitutive modelaccounting for the SDE will be crucial also for precise metal-forming simulations, inaddition to the strain-localization predictions discussed here.

Financial support from Nippon Steel Corporation is gratefully acknowledged.

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