arch. comput. meth. engng. vol. 11, 1,

94
Arch. Comput. Meth. Engng. Vol. 11, 1, 3-96 (2004) Archives of Computational Methods in Engineering State of the art reviews Modelling the Plastic Anisotropy of Metals A.M. Habraken Senior Research Associate FNRS epartement de M´ ecanique des Mat´ eriaux et Structures Universit´ e de Li` ege 1, Chemin des Chevreuils, Bˆat B52/3, 4000 Li` ege, Belgique e-mail: [email protected] Summary This work is an overview of available constitutive laws used in finite element codes to model elastoplastic metal anisotropy behaviour at a macroscopic level. It focuses on models with strong links with the phenom- ena occurring at microscopic level. Starting from macroscopic well-known models such as Hill or Barlat’s laws, the limits of these macroscopic phenomenological yield loci are defined, which helps to understand the current trends to develop micro-macro laws. The characteristics of micro-macro laws, where physical behaviour at the level of grains and crystals are taken into account to provide an average macroscopic answer are described. Some basic knowledge about crystal plasticity models is given for non-specialists, so every one can understand the microscopic models used to reach macroscopic values. The assumptions defining the transition between the microscopic and macroscopic scales are summarized: full constraint or relaxed Taylor’s model, self-consistent approach, homogenisation technique. Then, the two generic families of micro- macro models are presented: macroscopic laws without yield locus where computations on discrete set of crystals provide the macroscopic material behaviour and macroscopic laws with macroscopic yield locus defined by microscopic computations. The models proposed by Anand, Dawson, Miehe, Geers, Kalidindi or Nakamachi belong to the first family when proposals by Montheillet, Lequeu, Darrieulat, Arminjon, Van Houtte, Habraken enter the second family. The characteristics of all these models are presented and commented. This paper enhances interests of each model and suggests possible future developments. 1 INTRODUCTION Micro-macro modelling is a typical multi-disciplinarily field as it requires mechanical, met- allurgical and computational knowledge. Here “micro” means at the crystal level, not at the atomistic scale. This state-of-the-art review mainly comes out of “Aggregation in Higher Education Thesis”, Habraken (2001). It aims to gather the necessary information to un- derstand the various proposals found in the literature in the field of micro-macro models dedicated to the anisotropic plastic behaviour of metals. However the first question is probably why micro-macro models appear around 1985 (Lequeu et al. 1987a and b, Arminjon, 1988, Van Houtte 1988, Mathur & Dawson 1989) and are currently more and more developed (Miehe et al. 1999, Geers et al. 2000, Feyel & Chaboche 2000, Hoferlin 2001). What are the interests of industrial groups like ARCELOR, CHRYSLER, ALCOA, ... in such models? In practice, different metal forming processes such as deep drawing or folding processes are required to manufacture automotive parts, beverage or food cans, steel sheet panels used in aeronautics or civil engineering applications. Computer models try to replace the expensive and time-consuming trial-and-error methods used in conventional design. The Finite Element Method (FEM) is quite successful to simulate metal forming processes, but accuracy relies on the used constitutive law and the material parameter identification. Following paragraphs emphasize the necessity of efficient constitutive models to predict final shape, springback, rupture, fatigue behaviour of the forming products. For instance, final shape of the product is strongly linked to the plastic material flow. Plastic anisotropy explains the ondulated rims called ears and present in a cup produced by cylindrical tools applied on a circular blank. Figure 1 describes the typical case of a c 2004 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: December 2002

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Page 1: Arch. Comput. Meth. Engng. Vol. 11, 1,

Arch. Comput. Meth. Engng.Vol. 11, 1, 3-96 (2004) Archives of Computational

Methods in EngineeringState of the art reviews

Modelling the Plastic Anisotropy of Metals

A.M. HabrakenSenior Research Associate FNRSDepartement de Mecanique des Materiaux et StructuresUniversite de Liege1, Chemin des Chevreuils, Bat B52/3, 4000 Liege, Belgiquee-mail: [email protected]

Summary

This work is an overview of available constitutive laws used in finite element codes to model elastoplasticmetal anisotropy behaviour at a macroscopic level. It focuses on models with strong links with the phenom-ena occurring at microscopic level. Starting from macroscopic well-known models such as Hill or Barlat’slaws, the limits of these macroscopic phenomenological yield loci are defined, which helps to understandthe current trends to develop micro-macro laws. The characteristics of micro-macro laws, where physicalbehaviour at the level of grains and crystals are taken into account to provide an average macroscopic answerare described. Some basic knowledge about crystal plasticity models is given for non-specialists, so everyone can understand the microscopic models used to reach macroscopic values. The assumptions definingthe transition between the microscopic and macroscopic scales are summarized: full constraint or relaxedTaylor’s model, self-consistent approach, homogenisation technique. Then, the two generic families of micro-macro models are presented: macroscopic laws without yield locus where computations on discrete set ofcrystals provide the macroscopic material behaviour and macroscopic laws with macroscopic yield locusdefined by microscopic computations. The models proposed by Anand, Dawson, Miehe, Geers, Kalidindior Nakamachi belong to the first family when proposals by Montheillet, Lequeu, Darrieulat, Arminjon,Van Houtte, Habraken enter the second family. The characteristics of all these models are presented andcommented. This paper enhances interests of each model and suggests possible future developments.

1 INTRODUCTION

Micro-macro modelling is a typical multi-disciplinarily field as it requires mechanical, met-allurgical and computational knowledge. Here “micro” means at the crystal level, not at theatomistic scale. This state-of-the-art review mainly comes out of “Aggregation in HigherEducation Thesis”, Habraken (2001). It aims to gather the necessary information to un-derstand the various proposals found in the literature in the field of micro-macro modelsdedicated to the anisotropic plastic behaviour of metals.

However the first question is probably why micro-macro models appear around 1985(Lequeu et al. 1987a and b, Arminjon, 1988, Van Houtte 1988, Mathur & Dawson 1989)and are currently more and more developed (Miehe et al. 1999, Geers et al. 2000, Feyel &Chaboche 2000, Hoferlin 2001). What are the interests of industrial groups like ARCELOR,CHRYSLER, ALCOA, ... in such models? In practice, different metal forming processessuch as deep drawing or folding processes are required to manufacture automotive parts,beverage or food cans, steel sheet panels used in aeronautics or civil engineering applications.Computer models try to replace the expensive and time-consuming trial-and-error methodsused in conventional design. The Finite Element Method (FEM) is quite successful tosimulate metal forming processes, but accuracy relies on the used constitutive law andthe material parameter identification. Following paragraphs emphasize the necessity ofefficient constitutive models to predict final shape, springback, rupture, fatigue behaviourof the forming products.

For instance, final shape of the product is strongly linked to the plastic material flow.Plastic anisotropy explains the ondulated rims called ears and present in a cup producedby cylindrical tools applied on a circular blank. Figure 1 describes the typical case of a

c©2004 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: December 2002

Page 2: Arch. Comput. Meth. Engng. Vol. 11, 1,

4 A.M. Habraken

mild steel supplied by SOLLAC and compares experimental and simulated earing profiles.The classical isotropic elasto-plastic von Mises’law predicts no rim. In Figure 1, the simplequadratic anisotropic elasto-plastic Hill’s law simulates a wrong earing profile when Ferron-Tourki’s proposal (Tourki et al. 1996) is closer to experimental points. Such ears have tobe removed in order to produce a usable can. So material suppliers are required by canmanufactures to minimize the earing tendency of their material and model suppliers arerequired to provide accurate earing predictions.

MILD STELL Ferron - TourkiHill 1948Experimental points

50.00

48.00

46.00

44.00

42.00

40.00

38.000.00 60.00 120.00 180.00 240.00 300.00 360.00

Angle from Rolling Direction (degrees)

Cu

p h

eigh

t (m

m)

RD

Cupheight

angle

Figure 1. Deep drawing of a cylindrical cup: comparison between experimentaland calculated earing profiles (from Tourki et al. 1996)

Springback (Figure 2a) is the shape distortion of sheet metal when removed from thedie after forming. It is a major problem in stamping high-strength steel and even more sowith aluminium sheet used to manufacture lightweight vehicles. Accurate analysis of thisphenomenon requires that at the end of the forming process before removal from the dieand trimming, the stress state and material state (isotropic and kinematic hardening) arecorrectly computed. If the yield locus size and shape and the stress field are well known,then the elastic return inducing springback during the die removal can be simulated. Fig-ure 2c presents ADINA results for the two-dimensional draw bending benchmark problemproposed by the organizers of NUMISHEET 93 Conference.

During the forming operation, rupture events can appear. Research has already broughtsome solution. For instance, Forming Limit Diagram (FLD) helps to define a safe process. Itdivides the plane of major principal strain versus minor principal strain into a safe zone anda failure zone (see Figure 3 adapted from Barata da Rocha 1985). Again, this approachrequires accurate models to predict strain field during the forming operation. The FLDprediction itself provides a very sensitive way to check behaviour model coupled or notwith FEM. The following references give only a small idea of the important effort dedicatedto theoretical FLD prediction: Narasimhan & Wagoner 1991, Boudeau et al. 1996, Hora etal. 1996, Cayssials 1998 and 1999, Hoferlin et al. 1998, Vegter et al. 1999a. This proves theinterest in FLD from the automotive or can deep drawing industries. Note that the modelproposed by Cayssials 1999 is very effective for classical steels. Coupling plastic instabilitytheory with a damage approach, Cayssials’s model accurately predicts FLD of the metalsheet with data measured by tensile tests.

When the piece manufacturing has been successful, the piece is used. Here, it is a keyfactor to provide fatigue life prediction under periodical or random multi-axial stress state.

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Modelling the Plastic Anisotropy of Metals 5

Figure 2. Draw bending test. (a) Shape of a metal sheet: initial, after stamping, afterspring back; (b) measurement technique; (c) comparison between experimentaldata (circles and squares) and simulated results (dashed lines), (from Kawka& Bathe 2001)

As checked by Weber et al. 1999, the forming process can strongly modify the materialstate and by consequence, its fatigue characteristics. So life estimation techniques rely onthe final plastic state of the material (accurate size and shape of its yield locus).

Previous comments clearly show why accurate behaviour models are required in FEMto simulate forming operations. A complete answer to the question why developing micro-macro approaches, must however explain the interests of such laws by comparison withclassical phenomenological laws. Advantages and limitations of phenomenological constitu-tive laws are detailed in Section 2. As such models roughly consist of the fitting of functionson experimental results, it is not surprising that they provide only average behaviour whentoo simple functions are chosen and that they are quite difficult to identify when complexfunctions covering large fields of behaviour are applied. When models are based on the realphysical phenomena, their internal parameters are usually less numerous and their identi-fication of reasonable difficulty. As crystal plasticity has been studied for more than 100years (Ewing & Rosenhain 1900), material scientists know quite accurately the behaviourof monocrystals. Average field theory and homogenisation theory, steps to polycrystallinebehaviour were already proposed by Sachs 1928, Taylor 1938 in metal. Such knowledgeas well as FEM and computer developments explain that about 20 years ago, micro-macro

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6 A.M. Habraken

β=-1

α=-1

α=0

β=-0.5

α=0.5

β=0

ε

ε

1

2

restrai

nedzone stretchedzonefailure zone

α =1

β =1

Symmetrical

biaxialtension

plane

strain

state

uniaxial

tension

pureshear

safe zone

minor principal strain

Figure 3. Forming Limit Diagram for an isotropic material, whith α stress ratio(α = σ2/σ1) and β strain increment ratio (β = dε2/dε1), adapted fromBarata da Rocha 1985

models have appeared.This paper summarizes the required notions about crystal plasticity in Section 3, then

the different possibilities to go from the crystal to the polycrystal are presented in Section 4.More details are provided on FEM micro-macro models without or with microscopic yieldlocus in Sections 5 and 6 respectively. Finally, Section 7 concludes with comments on theinterests of the different model families and the perspectives of the micro-macro models inmetal forming simulations.

2 ANISOTROPIC PHENOMENOLOGICAL YIELD LOCI

The yield locus is the boundary between the elastic and plastic domains. It is a continuoussurface in stress space, Fp(σij) = 0, corresponding to all stress states σij that cause yield-ing. During plastic deformation, the updated yield locus will expand or contract, translateand distort. This phenomenon is described by a hardening rule. In this section, the ini-tial shape of the yield locus associated with the first yielding is analysed. Experimentalevidence and theoretical considerations concerning the plastic behaviour of materials haveled to some restrictions on the mathematical representation of a yield surface. FollowingBridgman’s experimental observations (Bridgman 1923, 1952), hydrostatic pressure (25000bars) does not induce plasticity in metals. More recently, Barlat et al. 1991 have measureda relative density change due to plastic deformation of the order of 10−3. Extrapolatingan approximate limit analysis of a porous medium, Gurson’s model 1977 proposes a plasticbehaviour law affected by pressure. His goal is fracture prediction but far from rupture, thepressure effect is limited. In a hot sintering process, the initial porosity can be very highand the pressure effect is important. However, in general cases of sound classical metalscharacterized by low porosity, yield surfaces are taken to be pressure independent. Drucker1951 showed that, based on a stability postulate, the yield surface must be convex. Therelationship between the components of the strain-rate tensor and the stress tensor is calledthe flow rule.

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Modelling the Plastic Anisotropy of Metals 7

Another consequence of Drucker’s 1951 postulate is that the flow rule for stable materialsis directly associated to the yield criterion: the strain rate vector is orthogonal to the yieldsurface. The associated flow rule is:

εpij = λ

∂Fp

∂σij(1)

Hecker 1976 or Phillips 1979, who reviewed numerous critical experiments to assess theyield surface shape, found that the normality rule was never violated. This is also confirmedby the work of Hayakawa & Murakami 1998 in damage mechanics. Indeed, all the yieldsurfaces listed hereafter respect these 3 characteristics. Table 1 summarizes the evolutionof the well-known Hill’s 1948 equation, then Table 2 introduces the models of Barlat andKarafellis. Finally, Vegter’s models are shortly presented since they seem to be well vali-dated and adapted to FEM computations. In addition to these tables, some comments helpto understand the origin of the presented models, their parameters identification methodas well as their experimental validation. Let us note that except for Karafillis’model andVegter’s model, all the others are restricted to anisotropic materials exhibiting orthotropicsymmetry, i.e. to materials which possess 3 mutually orthogonal planes of symmetry atevery point. Actually, most scientists agree that this restriction is not a very significantconstraint because most mechanically processed materials are orthotropic in their initialstate. Typical examples are rolled sheets or plates.

Reviews of phenomenological anisotropic models can be found in Vial et al. 1983,Kobayashiet al. 1985, Barlat 1987, Arminjon et al. 1994, Mahmudi 1995, Kuwabara & VanBael 1999, Banabic 2000, Habraken 2001, Barlat et al. 2002. Note that sheet anisotropyis usually characterized by the Lankford’s coefficient. This coefficient is determined byuni-axial tensile tests on sheet specimens in the form of a strip. This anisotropy coefficientis defined by:

r =ε2

ε3(2)

where ε2, ε3 are the strains in the width and thickness directions, respectively. This ratiois strongly correlated with the deep drawability of the sheet.

2.1 Hill’s Approach

Table 1 is dedicated to Hill’s research. The classical quadratic yield criterion, Hill 1948 (seeTable 1) is well suited to specific metals and textures, but lacks flexibility. This criteriongives a better correlation with metals having an average Lankford’s coefficient r greaterthan 1 (steel) but is less acceptable when r is less than 1 (aluminium). This average valueis computed by:

r =14(r0 + 2r45 + r90) (3)

It can be seen that if r < 1, the yield locus predicted by Hill 1948 criterion is locatedinside the one given by von Mises; if r > 1, the Hill 1948 yield locus is outside the von Misesyield locus. The angle α defines a direction, the reference is always the Rolling Direction(RD). For instance, the notation σα means the current yield stress in uni-axial tension alonga direction at an angle α with the Rolling Direction.

Only 3 tensile tests at 0, 45, 90 are required to determine the material parameters ofHill 1948 in plane stress state but a biaxial test is necessary for the more recent versions.Figure 4 compares experimental data and analytical yield loci for experiments performedon brass. Cases of a steel sheet and an aluminium one are treated by Figure 5.

Paulo
Highlight
Paulo
Highlight
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8 A.M. Habraken

Table 1. Hill’s models

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Modelling the Plastic Anisotropy of Metals 9

Table 2. Barlat’s and Karafillis’ models

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10 A.M. Habraken

1.0

Brass

M - Mo S2T - Teflon

Hill 1948

Hill 1979

Hosford 1979

1.00.50

0.5

σ / σx F

σ /

σy

F

MT

MT

Figure 4. Brass yield loci normalized by uniaxial tension at strain of 0.1, biaxialexperimental points, horizontal and vertical tangent defined by planestrain tests (from Vial et al. 1983)

σ x

σy

200

150

100

50

0

0 50 100 150 200

Experiment

Hill 1993

Hill 1948

von Misesσy

σ x

Experiment

Hill 1993

Hill 1948

von Mises

16012080400

160

140

120

100

80

60

40

20

0

a) ST 1405 b) Al Mg Si 1

Figure 5. Biaxial tensile points compared with analytical yield loci for a steel andan aluminium alloy (from Banabic et al. 1999)

2.2 Karafellis-Barlat-Banabic Family

The presentation of Karafillis et al. 1993 is reproduced as it provides a nice general frameto explain his proposal as well as Barlat’s 1989, 1991, 2002. In fact, the first idea is tofind a general form of isotropic yield surfaces. Then, this form is extended to anisotropicbehaviour with the help of tensor s, a projection of the stress tensor σ .

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Modelling the Plastic Anisotropy of Metals 11

First looking at isotropic yield surfaces, Karafillis recalls that Mendelson 1968 has de-rived from symmetry and convexity considerations, bounds in an isotropic yield surface.The lower bound coincides with the maximum shear stress yield surface as described byTresca’s 1864 criterion, whereas the upper bound was proposed by Hosford 1972. In the de-viatoric plane section of the yield surface, von Mises’ yield circle lies between upper boundand lower one (Figure 6). This deviatoric plane, also called Π plane, is perpendicular tothe line representing hydrostatic pressure state and contains the stress origin point. Therelations to compute the projection of a stress state onto the Π plane can be found in VanBael 1994 or Khan & Huang 1995.

1σ3σ

π - plane

von Mises

Tresca

upper bound

Figure 6. The upper bound, the lower bound and the von Mises’ yield surfacesin the Π-plane (from Karafillis et al. 1993)

Isotropic yield surfaces lying between the bounds defined by von Mises’ yield surfaceand Tresca’s yield surface can be mathematically described by Hosford’s 1972 criterion. Hisproposal is a modification of the von Mises’ mathematical description of a yield surface,where an exponent other than 2 is used:

(σ1 − σ2)2a + (σ2 − σ3)2a + (σ3 − σ1)2a = 2σ2aF (4)

with a integer > 1, σ1 σ2 σ3 the principal values of σ the deviatoric stress tensor and σF

the yield stress under uni-axial tension. This relation is equivalent to von Mises’ equationwhen a = 1 and to Tresca’s equation when a = ∞. Intermediate values of a describe allthe yield surfaces lying between the two proposals (Figure 7). Equation (4) is identical torelation “Karafillis 1993 (a)” in Table 2 if σ is substituted by s.

In fact, the isotropic surfaces lying between von Mises’ yield surface and the upper bound(Figure 8) are described by relation “Karafillis 1993 (b)” in Table 2 if s is substituted by σ .When a = 1 , the yield surface corresponds to von Mises’ yield surface whereas when a = ∞,the upper bound yield surface is recovered. So to describe a generic isotropic yield surfacelying between the lower bound and the upper bound, a generalized mathematical relationmixing Karafillis 1993 relations (a) and (b) in Table 2 is proposed (with s substitutedby σ ). It is the goal of relation (c) where the constant C (Figure 9) belongs to [0,1]. Byvarying the value of the mixing factor C, a family of yield surfaces is created which spansthe space between the 2 bounds, set by the choice of a (Figure 9). In this approach, thereare two parameters C and a that may be used to “adjust” the shape of this isotropic yieldlocus whereas the other criteria use only one parameter (exponent a for Hosford 1972) forthis purpose.

Page 10: Arch. Comput. Meth. Engng. Vol. 11, 1,

12 A.M. Habraken

a = 1 (von Mises)a = 3a = 5a = 8a = 16a = + (Tresca)∝

σF

σF

σ 1 /

σ2σ 1 -

σ 2

/

section

Φ1

- 0.50 0.00 0.50 1.00 1.50- 0.75

- 0.50

- 0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Figure 7. Influence of parameter a in φ1 function (from Karafillis et al. 1993)

a = 1 (von Mises)a = 3a = 5a = 8a = 15a = + (Upper bound)

σFσ 1 /

σF

σ 2

/

σ2σ 1 -

section

Φ2

- 0.50 0.00 0.50 1.00 1.50- 0.75

- 0.50

- 0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Figure 8. Influence of parameter a in φ2 function (from Karafillis et al. 1993)

The extension of the previous isotropic approach to anisotropic cases depends on thechoice of s tensor. Different proposals by Barlat, Karafellis, Banabic ... are known. A shortreview is proposed hereafter.

Barlat et al. 1991 use a linear transformation of the 6 components of a stress state(Barlat 1991 (b), Table 2). The obtained transformed stress state is then used in Hosford’s1972 yield criterion (4). Different identification procedures are proposed in Banabic 2000.Practically, a is larger or equal 6, depending on the anisotropy induced by texture and on

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Modelling the Plastic Anisotropy of Metals 13

c = 1 c = 0.99c = 0.95c = 0.75c = 0.00

σFσ 1 /

σF

σ 2

/

- 0.50 0.00 0.50 1.00 1.50- 0.75

- 0.50

- 0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

a = 15

Figure 9. Influence of C parameter in global φ function (from Karafillis et al. 1993)

the crystal type. In fact, a can be tuned to optimise the prediction of the yield locus shape.Barlat’s 1989 work already contains the same type of approach but is limited to plane

stress state (see Table 2). This formulation has the advantage of clearly showing the effectof the shear component, which modifies the sections of the yield locus. Figure 10 presentsyield locus section of a material containing 50 % brass texture and 50 % of randomlydistributed grains. The 4 parameters A, h, p, a of Barlat’s 1989 model are determined asexplained by Barlat & Lian 1989 or Berg et al. 1998:

a = 6 for b.c.c. metals; a = 8 for f.c.c. metals (from polycrystalline model comparisons)

A = 2 − 2√

r0

1 + r0

r90

1 + r90h =

√r0

1 + r0

r90

1 + r90(5)

p =σF

τ

(2

2A + 2a(2 − A)

) 1a

(6)

where σF is the yield stress in uni-axial tension, τ is the yield stress for pure shear. Inpractice, p varies since the ratio σF/τ is not a constant function of the equivalent strain.This affects the shape of the yield surface as demonstrated by Berg et al. 1998.

For anisotropic material, Karafillis et al. 1993 consider a tensor s , resulting of a lineartransformation L of the actual stress tensor σ . In this proposal, the transformed tensor sis introduced in the general isotropic yield criterion defined by relation “Karafellis 1993 (c)”in Table 2. The fourth order tensor L chosen by Karafillis is more flexible than the simplechoice made by Barlat and can be associated with material symmetries which range fromthe lowest level of triclinic symmetry to the highest level of full symmetry. This providesa potential for the representation of any anisotropic state of the material. More details onsymmetry properties and constraints applied to L can be found in Karafillis et al. 1993.Finally, for orthotropic materials, 3 uni-axial tensile tests of specimens cut at 0, 45, 90with respect to Rolling Direction allow to completely determine C and L. Remark that, inthis orthotropic case, Karafillis’ transformation tensor L reduces to Barlat’s 1991 (Table 2).Figure 11 shows Karafillis’results compared with other models and experiments.

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14 A.M. Habraken

σFσ x /

σ Fσ

y/

1.50

1.00

0.50

0.00

- 0.50

- 1.00- 1.00 - 0.50 0.00 0.50 1.00 1.50

a = 14

r = 1.0 90

r = 5.0 45

r = 0.7 0

s = 0.432s = 0.4

s = 0.3s = 0.2

s = 0.0

Hill 1948

Figure 10. Sections of Barlat & Lian 1989 yield surface, with s = σxy/σF (adapted fromBarlat & Lian 1989)

ANGLE FROM ROLLINGS DIRECTION [DEG]

2008 - T4

ExperimentKarafillis 1993

Barlat 1991T B H

0.0 15.0 30.0 45.0 60.0 75.0 90.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

B = 0

r

Figure 11. Curves of Lankford coefficient versus angle with the rolling direction for 2008-T4 aluminum alloy , TBR =Taylor 1938-Bishop & Hill 1951 polycrystal model(from Karafillis et al. 1993)

Hill 1948 (Table 1), Barlat 1989 and Karafillis 1993 (Table 2) were implemented inthe commercial code LS-DYNA3D. Andersson et al. 1999 use it to simulate deep drawingtests of aluminum sheets and compare their FEM results to experiments. Their results showthat Karafillis’model yields the best agreement with the experimental results and that CPUtimes are in the same range: Hill 1948 = 1, Barlat 1989 = 1.5 and Karafillis 1993 = 1.1.

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Modelling the Plastic Anisotropy of Metals 15

Barlat’s 1997 proposal (Table 2) generalizes his 1991 model. For the plane stress case,only 6 independent coefficients characterize anisotropy. They can be deduced from 4 tests:uni-axial tension at 0, 45, 90 to the Rolling Direction and equal biaxial bulge test.

Let us note that Banabic et al. 2000 propose a new yield criterion BBC2000 for or-thotropic sheet metals under plane stress conditions. It is derived from Barlat 1989 criterion.Two additional coefficients have been introduced in order to allow a better representation:

[b(cΓ + dΨ)2a + b(cΓ − dΨ)2a + (1 − b)(2dΨ)2a]1/2a = σF (7)

where a, b, c, d are material parameters, while Γ and Ψ are functions of the second andthird invariants of the fictitious stress tensor s defined by Karafellis (relation “Karafellis1993 (d)” in Table 2).

Drucker’s 1949 isotropic criterion:

J32 − cJ2

3 = k2 (8)

where J2 = 1/3trσ 2 and J3 = 1/3trσ 3 are the second and third invariants of the stressdeviator has been extended by Cacazu & Barlat 2001 to anisotropy. They replace the devi-atoric stress invariants J2 and J3 by homogeneous functions of degree 2 and 3 respectively.These functions are pressure insensitive, invariant to any transformation belonging to thesymmetry group of the material and reduce to J2 and J3 in isotropic conditions.

Barlat et al. 2002 propose to use two different linear transformations L′ and L′′ of thestress tensor σ to define 2 tensors s′ and s′′ (see relation “Barlat 1991 (b)” Table 2). Thenthey introduce them in isotropic functions:

Φ′(s) = |s1 − s2|a Φ′′(s) = |2s2 + s1|a + |2s1 + s2|a (9)

leading to the resulting anisotropic yield function:

Φ′(s′) + Φ′′(s′′) = 2σaF (10)

This approach is only developed in plane stress and uses 10 anisotropy coefficients, 2are zero and the other 8 coefficients can be determined using stress and r values in tensionalong three directions, the balanced biaxial flow stress σb and rb. The latter coefficient rb isdefined by analogy to Lankford’s coefficient but tensile test is replaced by a disk compressiontest (see Barlat et al. 2002 for further details).

Banabic’s et al. 2000 (BBC2000), Cacazu & Barlat’s 2001 (CB2001) and Barlat et al.2002 (Yld2000-2d) models have been identified for AA3103-0 aluminum alloy (Barlat et al.2002). Figures 12 and 13 present comparison between experimental and predicted valuesof the distributions of the uni-axial yield stress and Lankford’s coefficient with respect tothe angle with the Rolling Direction.

2.3 Vegter’s Approach

Vegter’s approach (Vegter et al. 1999a,b) proposes a yield criterion directly based on ex-perimental data of the pure shear test, the uni-axial test, the plane strain test and thebiaxial test. Figure 14 clearly summarizes this criterion for a planar isotropic materialbehavior. A complete yield surface is described by 12 Bezier interpolation functions. Whenσ1 and σ1 are defined in such a way that σ1 ≥ σ2, only the part of the surface beneath theline σ1 = σ2, is required. As the material is assumed to behave identically in tension andcompression, the part of the surface beneath the line σ1 = −σ2 can be derived from thepart above that line. As a result, the complete material yielding can be described using

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16 A.M. Habraken

0.95

1.00

1.05

1.10

1.15

0.5

0.55

0.6

0.65

0 20 40 60 80

Nor

mal

ized

str

ess

•r valu

e ∇

Angle from rolling

3103-O

r value

stress

Yld2000-2dCB2001BBC2000

Figure 12. Curves of uniaxial yield stress and Lankford coefficient versus angle with therolling direction for AA31303-0 aluminum alloy (from Banabic et al. 2002)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00 0.25 0.50 0.75 1.00 1.25 1.50

σxx

/ σ

σ yy/

σ

3103-O

Yld2000-2dCB2001BBC2000

Figure 13. Sections of analytical yield surfaces and experimental points (circles)for AA31303-0 aluminum alloy (from Banabic et al. 2002)

the measurements for a quarter of the yield surface. Vegter’s yield function is restricted toplane stress state. It is defined as:

σveg(σx, σy , σxy) − σyield = 0 (11)

where σveg is a kind of equivalent stress depending on the stress components in orthotropicaxes σx, σy, σxy and σyield is the yield stress.

In case of planar isotropic material behavior, σyield is the uni-axial yield stress σF . Thesecond order Bezier interpolations allow to describe the normalized yield surface. For a

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Modelling the Plastic Anisotropy of Metals 17

σ2

σ1

H

H

H

Biaxial

Plane strain

Uniaxial

Pure shear

Figure 14. Vegter’s yield locus in principal stress space, black circles= experimentalpoints, white circles = hinge points, adapted from Carleer et al. 1997

σ1

σ2

reference point 2 : p ref 2

reference point 1 : p ref 1

hinge point : H

Figure 15. Bezier curve in Vegter’s yield locus, adapted from Carleer et al. 1997

part going from reference point 1 to reference point 2 with hinge point H (Figure 15), onehas: σ1

σveg= (1 − β)2pref1

1 + 2β(1 − β)phinge1 + β2pref2

1 (12)

σ2

σveg= (1 − β)2pref1

2 + 2β(1 − β)phinge2 + β2pref2

2 (13)

where σ1, σ2 are principal stresses, β is a curvilinear coordinate increasing from 0 to 1 andrepresenting the location on the curve between the reference points, prefj

i and phingei are

respectively the components i of the reference point j and the hinge point. Both equations(12) and (13) give an expression for σveg. Equating these expressions allows to find βas a solution of a second order equation. This solution is chosen to satisfy the condition0 ≤ β ≤ 1. The expression of σveg is then found by substituting β into (11).

For planar anisotropy, the 4 tests (pure shear, uni-axial, plane strain and biaxial tests)are performed for directions of 0, 45, 90 from the Rolling Direction and an interpolationfunction pref(α) is used to find interpolated reference points from the measured points pref

α :

pref(α) =pref

0 + 2pref45 + pref

90

4+

pref0 − pref

90

2cos 2α +

pref0 − 2pref

45 + pref90

4cos 4α (14)

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18 A.M. Habraken

Here α is the angle of the principal stresses to the Rolling Direction. The function pref(α)can be regarded as a Fourier series with only 3 terms; it can be extended if orthotropicproperties are not respected. So, with the new interpolated reference points, Vegter’s yieldcriterion can be constructed.

Figure 16 presents the benchmark of the Numisheet 96 conference. Vegter’s modelpredicts minor strains along a symmetry axis of a Limiting Dome Height test that are veryclosed from experiments for the studied draw quality mild IF steel. Figure 17 shows thequality of FLD predictions with Vegter’s model.

203.3

105.7R 6.35

R 50.8

132.6

DIE

Punch

0.02 experimentHill 1948Vegter

- 0.080 60original x-coordinate (mm)

min

or s

train

Figure 16. Predicted and measured minor strains along a symmetry axis of a Lim-iting Dome Height test, draw quality mild IF steel (Pijlman et al. 1998)

VegterHill 1990Hill 1993Barlat 1991Experiment

0.60.40.20-0.2-0.40

0.2

0.4

ε2

ε1

0.6

0.8

Figure 17. FLD predictions applied on bi-axial pre-strained specimens of an IFsteel (from Vegter et al. 1999a)

However the identification of Vegter’s model is quite expensive: pure shear, uni-axial,plane strain and biaxial tests in one direction in case of planar isotropy or in 3 directions forplanar anisotropy. As pointed by Vegter et al. 1999b, it is important to reach high experi-mental accuracy and to correct experimental results if non homogeneity or non isothermalconditions happen (Piljman et al. 1999).

An important characteristic of Vegter’s model is its flexibility. For instance, Figure 18presents a variant called Vegter-flat, in which the reference point corresponding to planestrain is very close to the hinge point located between plane strain and biaxial states. The

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Modelling the Plastic Anisotropy of Metals 19

σF

σ 2

/

σFσ 1 /

0

0.5

0 0.5 1

1

A

Hill 1990

Hill 1993

Vegter

σF

σ 2

/

σFσ 1 /

0.70.7

B

Barlat 1991

Vegter - flat

Vegter

x

0.8 0.9 1 1.1

0.8

0.9

1

1.1

x x x xxxxxxxxxx

x

xxxxxxxx

Figure 18. Comparison of yield loci descriptions for Al-6000, A global yield locus, B zoomaround biaxial state (from Vegter et al. 1999a)

Figure 19. FLD predictions and measurements for Al-6000 (90 to RD) (from Veg-ter et al. 1999a)

agreement with the measured Forming Limit Diagram shown on Figure 19 is significantlyimproved. This last effect shows the extreme sensitivity of the FLD prediction to the shapeof the yield locus.

2.4 Summary of Described Phenomenological Models

To summarize this section devoted to phenomenological yield loci, scientists have found newformulations of anisotropic yield loci that seem closer to experimental evidence. However,the number of parameters used to determine these new criteria increases with their flexibil-ity. Table 3 summarizes the experimental values necessary to identify the yield loci listedabove in plane stress state, in their initial shape, without hardening description. Usually,the exponent a is fitted according to polycrystalline yield locus, common value are a = 6for b.c.c. metals and a = 8 for f.c.c. metals. This table also tells if the model is limited toplane strain or developed for three dimensional state and if it is able to simulate anomaliesA1, A2 defined in Table 1.

Note that in some cases, the user can choose which data he uses to identify his model. Forinstance, the parameters of Hill 1948 can be identified from the yield stresses or the Lank-ford’s coefficients given by a uni-axial tensile tests. Choosing stress values or Lankford’s

Page 18: Arch. Comput. Meth. Engng. Vol. 11, 1,

20 A.M. Habraken

Name 3D σ0 σ45 σ90 σb τ r0 r45 r90 Addit. A1 A2param.

Hill 1948 X X X X XHill 1979 X X X X XHill 1990 X X X X X X XHill 1993 X X X X X XBarlat 1989 X X X X XBarlat 1991 X X X X X XKarafellis 1993 X X X X X X X X XBarlat 1997 X X X X X X X X XBanabic 2000 X X X X X X X X XCacazu 2001 X X X X X X X X σ30 σ75 X X

r30 r75

Barlat 2002 X X X X X X X rb X XVegter 1999 X X X X in X in plane strain X X

3 dir. 3 dir. in 3 dir.

Table 3. Identification of phenomenological models

coefficients is not equivalent and provides two sets of parameters. One set of parametersrecovers stress information and the other strain information. This apparent difficulty comesfrom the approximations of this phenomenological model, unable to reproduce experimentswith perfect accuracy.

2.5 Discussion on Phenomenological Yield Criteria

In practice, the specific strain path or loading history undergone by the material inducesmicro-structural and textural evolutions. The initial phenomenological yield criteria de-scribed in the preceding sections must be associated with hardening rules to model plasticdeformation beyond plasticity initiation. Hardening models can be isotropic or anisotropic.The former corresponds to an expansion of the yield surface without distortion. Any otherform of hardening, such as kinematic hardening, which is defined by the translation of theyield surface, is anisotropic. This type of hardening leads to different properties in differentdirections after deformation, even if the material is initially isotropic. The hardening mod-els depend on internal variables and allow to compute the new size, shape and position ofthe yield locus. The updated yield locus connect points representing stress states charac-terized by identical values of internal variables. For simple macroscopic hardening models,the plastic work is often taken as the only internal state variable, which validates the use ofthe work contours to measure updated yield loci, even though this surface connects stresspoints related to different material textures. However, for hardening approaches with strongmicroscopic roots, the equality of internal hardening state variables effectively implies thatall the points connected by the yield locus are related to the same material state.

Note that for sheet forming simulations, Barlat et al. 2002 propose a kind of alternativeapproach to accurate hardening models. They use stresses (called flow stresses) and rvalues defined at a given amount of plastic work to characterize the average behaviour ofthe material over a finite deformation range. In this case, it would be more appropriateto talk about flow function and flow surface instead of yield function and yield surfacealthough, mathematically, yield or flow functions are identical.

Classical arguments for the use of phenomenological yield loci are:

1. Easy to understand for mechanical engineers;

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Modelling the Plastic Anisotropy of Metals 21

2. Available in commercial codes (Hill 1948 is already in most of them);

3. Low CPU time and storage memory requirement compared to micro-macro approach;

4. Easy to implement in a Finite Element code;

5. Accurate enough;

6. Low number of parameters, easy to identify.

If the two first points are clearly true, the other ones are a source of controversy:

3. With hardware improvement, parallel computation and efficient formulation of micro-macro models, the gap concerning CPU time decreases, even if it is still a stronglimitation for some of them.

4. As reported by Banabic 2000, all models are not straightforward to implement, somehave a quite unfriendly form of the yield function and require mathematical abilities.

5. The list of models should stop growing if scientists were happy with their accuracy.According Barlat et al. 2002 because mechanical data are used as input, phenomeno-logical models can be more accurate than polycrystalline models when the strainamount is moderate, which is generally the case for sheet forming. However, forlarger strains or for abrupt strain path changes, microstructure evolution is an issueand is the subject of much research at the present time (Teodosiu and Hu 1998, Lopeset al. 2002). Because polycrystalline models can track the lattice rotation of eachindividual grain, the material anisotropy is naturally evolutional, which makes thisapproach very attractive.

6. To increase accuracy, authors add new parameters that require more and more me-chanical tests for their identification. Such tests must be performed with care andcost time and money. Is it really less expensive to do biaxial and bulk tests than tomeasure a texture?

This discussion explains why micro-macro models were worth to be developed in additionto phenomenological models. Another argument is that micro-macro models help to deeplyunderstand the material behaviour, they allow to numerically verify the microscopic modelsand to select the driving mechanisms of plastic deformation in polycrystals.

3 BASIC KNOWLEDGE ABOUT MICROSCOPIC MODELS

3.1 Description of Crystal Events During Plastic Deformation

Ewing & Rosenhain 1900 have established that the plastic deformation of a crystal occurs byslipping on some crystallographic planes (slip or glide planes) and in some crystallographicdirections (slip or glide direction) or by twinning, another crystallographic phenomenon.However, the elementary theoretical estimation of the shear strength of a perfect crystalgiven by Frenckel 1926 yields values of the order of 10−1 G (G = shear modulus of thecrystal). Knowing that the observed shear strength is of the order of 10−6 ≈ 10−4 G,scientists have proposed the concept of crystal defects, called dislocations, that reduce thestrength of the crystal. Figures 20 and 21 present two basic types of dislocations, the edgeand screw dislocations, and show the dislocation line AB. The motion of dislocations resultsin crystal shearing.

Burgers’ vector b is introduced to describe the slip direction of a dislocation. It is de-termined by Burgers’circuit, a closed path involving two lattice directions and surroundingthe dislocation line (Figure 22).

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22 A.M. Habraken

A

B

Figure 20. Edge dislocation (from Magnee, 1994)

D

B

A

C

Figure 21. Screw dislocation (from Khan & Huang, 1995)

F

S b

F S

b

A

( a ) ( b )

Figure 22. Burgers’ circuit for edge (a) and screw (b) dislocation defining Burgers’vector (from Khan & Huang, 1995)

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Modelling the Plastic Anisotropy of Metals 23

It should be pointed out that, for screw dislocation, Burgers’ vector is parallel to itsdislocation line, while for edge dislocation, Burgers’ vector is perpendicular to the dislo-cation line. A slip or glide system is characterized by the combination of a slip plane anda slip direction. Experimental observations show that, in most metals, the slip planes areusually those planes with the closest atomic packing, while the slip directions are alwaysthe closest packed directions on the slip planes. Once reference axes are defined, a slipsystem is described by its geometrical matrix Ks

ij also called Schmid’s tensor:

Ksij = bs

insj (15)

nsj unit vector normal to the slip plane for the slip system s, bs

j unit vector in the slipdirection for the slip system s, As

ij and Zsij are respectively the symmetric and the anti-

symmetric parts of Ksij .

Another approach to characterize one slip system is to use Miller’s indices to representthe slip planes and directions. For instance, for b.c.c metals, 24 110 < 111 > and112 < 111 > slip systems are available and, for f.c.c metals, 12 111 < 110 > aregenerally assumed (Van Houtte 1995).

The critical resolved shear stress CRSS τ sc is the shear stress to apply in order to sus-

tain the long range propagation of a dislocation according to a slip system s, under thesuperposed effects of all coexisting structural features which represent obstacles of differ-ent strengths. In practice, the resolved shear stress τ s acting on a slip system s can bederived by projecting the microscopic stress σ micro on the slip plane with the help of thecorresponding As

ij matrix. This determination can be checked, if one computes the energydissipated by the strain rate tensor ε associated to a single shear of one active slip system:

γsτ s = εpij σ

microij = As

ij γsσmicro

ij

τ s = Asijσ

microij (16)

The slip s occurs according to Schmid’s law if the shear stress τ s reaches the CRSS τ sc .

This defines the yield locus of a single crystal:

−τ sc ≤ τ s ≤ τ s

c (17)

Equals signs hold for plastic deformation, while inequalities apply to the elastic domain.In this elastoplastic formulation, the detection of active slip systems can be done either byBishop-Hill’s approach or by Taylor’s approach. These methods are dual ones, shortlysummarized hereafter, more details can be find in Van Houtte 1988.

3.2 Taylor’s Single Crystal Plasticity Model

At the crystal level, the plastic microscopic velocity gradient generated by a particular slipsystem s is given by:

Lsij = Ks

ij γs (18)

where Schmid’s tensor Ksij is defined by relation (15) and γs is the slip rate acting on

this slip system s. In practice, multiple slips occur together. Neglecting the elastic strainsas in Asaro & Needleman 1985, Becker 1990, Neale 1993 or Dawson & Kumar 1997, themicroscopic velocity gradient applied on a crystal Lmicro is given by:

Lmicro =∑

s

Ls + Ω L (19)

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24 A.M. Habraken

(a) pΩ =∑

s

ss

Z γ& = plastic spin or slip induced spin

(b) LΩ = rate of crystal lattice rotation as a rigid body, used to update texture

Figure 23. Representation of the 2 terms of the crystal microscopic spin

where Ω L is the rate of crystal lattice rotation. The establishment of this well-knownrelation is recalled in Van Houtte 1995. Its link with the classical formalism of ContinuumMechanics is summarized in Section 3.5. This microscopic velocity gradient can be splitinto a microscopic plastic deviatoric strain rate ε p micro and a microscopic spin Ω micro:

ε p micro = sym(Lmicro) =∑

s

Asγs (20)

Ω micro = skw(Lmicro) =∑

s

Zsγs + Ω L (21)

The 2 components of the microscopic spin Ω micro are symbolically represented on Fi-gure 23. The first term of (21),

∑s

Zsγs, is called plastic spin Ω p.

Several different combinations of slip rates may achieve the prescribed strain rate. Ac-cording to Taylor 1938, the one which minimizes the power dissipation is chosen:

W p =∑

s

τ sc |γs| = min (22)

Taylor roughly assumes that all slip systems have a common value τc of their CriticalResolved Shear Stress τ s

c (CRSS). This common value τc is only reasonable for annealedcondition and crystals like b.c.c. ones for instance, where no strong variations exists betweenslip systems. Computing different CRSS is a classical improvement for scientists using thisapproach.

The prescribed strain rate ε p micro can be split into a scalar magnitude εpeq computed

by the classical von Mises’ formula and a strain rate mode Uεp :

Uε p =ε p micro

εp microeq

(23)

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Modelling the Plastic Anisotropy of Metals 25

Introducing also the slip rate γsscaled per unit equivalent strain rate:

γsscaled =

γs

εp microeq

(24)

advantage can be taken of the assumed strain-rate insensitivity to simplify the formulation.Indeed, dividing relation (22) by the strain rate magnitude and the reference CRSS τc leadsto:

W p

εp microeq τc

=∑

s

|γsscaled| = minimum (25)

with, according to equation (20), the minimization constraint:

Uεp =∑s

Asγsscaled (26)

These two equations are called Taylor’s equations and may be efficiently solved by meansof linear programming (Simplex Method) as explained by Van Houtte 1988. The solutionγs

scaled relies only on the prescribed strain mode Uεp , it depends neither on the magnitudeεeq nor on the value of the reference CRSS τc.

In the Simplex Method, relation (25) is called the cost function, to be minimized underthe constraint (26). This primal problem in the space of slip rates is transformed into adual problem where the stresses are now the independent variables. In the stress space, thevalue W p/εp micro

eq τc represents a maximum:

W p

εp microeq τc

= Uεp :σ micro

τc= maximum (27)

This retrieves the approach proposed by Bishop-Hill, which assumes that the admissiblestress states satisfy the yield locus constraints expressed by relation (17) and that the realstress state maximizes the external plastic work. Taylor’s and Bishop-Hill’s methods arestrictly equivalent, that is why they are very often referred to as the Taylor-Bishop-Hill(TBH) crystal model.

The Simplex Method produces both slip rates and microscopic stresses at the crystallevel. In practice, using the same CRSS τc for all slip systems leads to multiple solutions.Different selections of activated slip systems achieve the prescribed strain rate (20) withthe same minimum power dissipation (22). Van Houtte 1988 proposes different methods tochoose one particular solution. From an energy point of view, all the solutions are equiv-alent. However, they produce different slip rates, microscopic stresses and hence differentcrystal lattice rotations. Once the set of active slip systems and the corresponding slip ratesare known by the resolution of equations (25), (26), the crystal rotation Ω L produced byone imposed velocity gradient can be reached using relation (21). It is expressed by:

Ω L = Ω micro −∑s

Zsγs (28)

Let us define M (g,Uεp) the Taylor’s factor, which is an important scalar in all themicroscopic models. This factor is associated with a crystal of orientation g for a givenstrain mode Uεp . It is conventionally derived from the plastic power dissipation per unitvolume W p by the following relation:

M (g,Uεp) =W p

εp microeq τc

=∑

s

|γsscaled| =

1τc

σmicroij (Uεp)ij (29)

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26 A.M. Habraken

Physically, it represents a certain amount of dislocation glide rate associated to thecrystal orientation and to the applied strain rate.

3.3 Strain Rate Sensitivity Approach for Single Crystal Plasticity Model

Another single crystal plasticity model is issued from the theory of thermally activateddislocation glide. It consists in adopting a viscoplastic flow rule:

γs = γ0

∣∣∣∣τ s

τ sc

∣∣∣∣1n

sign(τs) (30)

where γs is the slip rate on the slip system s, γ0 is a reference slip rate defined so that|γs| = γ0 when |τ s| = τ s

c and the parameter n characterizes the material rate-sensitivity.Under isothermal conditions and in a narrow range of strain rates, the above simple power-law equation has proven its validity. For cold deformation, the rate sensitivity is rather lowand the stress dependence on the slip rate can be reasonably approximated with n close to 0.The slip system shear rate γs does not vanish as long as the resolved shear stress τ s on thecorresponding slip system s is not identically zero. It keeps however a low value if τ s is notclose to τ s

c . This equation applies a posteriori the activation condition defined by Schmid’slaw by filtering out the inactive systems. For low value of n, this approach provides aninteresting alternative to TBH model as it overcomes the problem of non uniqueness inselecting a set of active slip systems by the TBH approach. It is a popular approach usedby many scientists (Asaro & Needleman 1985, Anand et al. 1997, Dawson et al. 1997)its equivalence to TBH approach has been studied and verified by Neale 1993, Anand &Kothari 1996.

3.4 Evolution Rule for CRSS Value

Whatever the solution is chosen to model single crystal behaviour (TBH model or vis-coplastic flow rule), the CRSS τ s

c appears and evolves in a different way for each individualslip system s. This implies the knowledge of the initial value of τ s

c as well as its evolutionequation. Slip on any slip system generally induces hardening for all slip systems. This istaken into account by adopting an evolution equation of the CRSS of the form:

τ sc =

∑u

hsuγu (31)

where hsu is the so-called hardening matrix. Diagonal components of this matrix corre-spond to the self-hardening effect, while off-diagonal components describe cross-hardeningeffects. Franciosi 1988 proposes an evolution rule for the components of this matrix. Thisrule depends on the pair of slip systems, their shearing rates and their temperature. Thisapproach could seem sophisticated. However, it is still not accurate enough. The fact thathardening defined by one fixed structure of dislocations is affected by structure modifica-tion is not taken into account. The choice for hsu constitutes a distinction between themicroscopic models used in micro-macro approaches implemented in FEM codes. Khan &Cheng 1996 and Teodosiu 1997 review some different proposals.

3.5 Mechanical Frame for Single Crystal Plasticity

In the frame of Continuum Mechanics, researchers as Asaro 1983, Anand & Kothari 1996 usean isoclinic configuration, which is an intermediate conceptual local configuration definedby Figure 24, to decompose the deformation gradient tensor:

F =∂x∂X

=∂x∂x∗

∂x∗

∂X= F∗Fp (32)

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Modelling the Plastic Anisotropy of Metals 27

Figure 24. Reference, deformed and isoclinic configurations, multiplicative decom-position of the deformation gradient tensor F = F ∗F p

X coordinates in the initial configuration expressed in axes Xi, x coordinates in the de-formed configuration expressed in axes xi, x∗ coordinates in the isoclinic configurationexpressed in axes Xi, F∗ sum of an overall “elastic” distortion of the lattice and the rigidrotation of the lattice, Fp “plastic” simple shears that do not disturb the geometry of thelattice.

The lattice in the isoclinic relaxed configuration has the same orientation as the latticein the reference configuration. The incremental deformation of a crystal is taken as theresult of the contributions from two independent atomic mechanisms:

• the sum of an overall “elastic” distortion of the lattice and a rigid rotation of thelattice (F∗);

• “plastic” simple shears that do not disturb the lattice geometry (Fp).

The microscopic velocity gradient is linked to the deformation gradient tensor:

Lmicro =∂x∂x

=∂x∂X

∂X∂x

= FF−1 (33)

Lmicro = F∗F∗−1+ F∗FpFp−1F∗−1

(34)

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28 A.M. Habraken

The plastic shear rate is expressed through a certain number of slip systems s (the activeones in the TBH model and all slip systems in the rate sensitive approach):

Lmicro = F∗F∗−1+∑s

γsK∗s (35)

Schmid’s tensor K∗s is expressed in deformed configuration, but as the crystal latticeis not affected by Fp, one has:

K∗s = F∗KsF∗−1(36)

So in the micro-mechanical frame, one gets (35) which should be equivalent to relation(19) in Taylor’s model. Since in (19), the elastic part of F∗ has been neglected, the strainrates deduced from (35) and (19) are different:

(35) → ε = sym(Lmicro) = ε elastic + ε p micro = ε elastic +∑s

A∗sγs (37)

(19) → ε = sym(Lmicro) = ε p micro =∑

s

A∗sγs (38)

The skew-symmetric parts of both velocity gradients are identical and are called thespin:

Ω micro = Ω L + Ω p = Ω L +∑s

Z∗sγs (39)

where Ω L is the crystal lattice rotation due to both the global rigid body rotation of themacroscopic body and the particular crystal rotation due to texture updating.

4 MICRO-MACRO APPROACHES

On a macroscopic point of view, one works with polycrystals that contain a lot of crystallites.In consequence, models defined for single crystals cannot directly be applied. A micro-macrotransition has to be developed. Before reviewing the different micro-macro proposals inSection 4.2, let us look at the decomposition of a polycrystal into its crystallite orientations.

4.1 Crystallographic Texture

In a polycrystalline sample, each crystal is characterized by its volume fraction dV and byits orientation, symbolically designated by g. More specifically, the set of Euler’s angles(ϕ1, φ, ϕ2), describing the orientation of the crystal reference system with respect to thesample (external) reference system, is most often used (Bunge, 1982).

Figure 25 shows the physical meaning of ϕ1, φ, ϕ2. With dV (g)/V as the volumefraction of crystals having their orientation within dg around g, the statistical crystalliteorientation distribution function (ODF) f(g) is then defined as:

dV (g)V

= f(g) dg with dg =1

8π2sin φ dϕ1 dφ dϕ2 (40)

It provides a quantitative description of the crystallographic texture of the polycrystal;high values indicate preferred orientations, and f(g) ≡ 1 a completely random texture.

Because of the crystal symmetry, several symmetrically equivalent choices exist for acrystal reference system. The classical ranges for the three Euler’s angles (0 ≤ ϕ1 ≤ 2π,0 ≤ φ ≤ π and 0 ≤ ϕ2 ≤ 2π) may therefore be reduced. Additionally, symmetries in theforming process may lead to an initial texture with similar statistical symmetries. This

Page 27: Arch. Comput. Meth. Engng. Vol. 11, 1,

Modelling the Plastic Anisotropy of Metals 29

φ

φ

φ

ϕ1

ϕ1

ϕ1

ϕ2

ϕ2

ϕ2

z

y

x

z’y’

x’

φ

φφ

ϕ1

ϕ1

ϕ1

z

y

x

z’y’

x’

ϕ1

ϕ1

ϕ1

z

y

x

z’

y’

x’

z

y

x

z’

y’

x’K = CrystalB

K = SpecimenA

( a ) ( b )

( d )( c )

Figure 25. Definition of Euler’s angles (from Van Houtte 1995)

so-called sample symmetry allows again for a reduction of the part of Euler’s space tobe considered. For example, in the case of cubic crystals without sample symmetry, theranges [0, 2π]x[0, π/2]x[0, π/2] are to be used; with additional orthorombic sample symme-try, [0, π/2]x[0, π/2]x[0, π/2] will be sufficient (Van Bael 1994).

Using the harmonic method proposed by Bunge 1982, the ODF f(g) can be representedby a series expansion:

f(g) ∼=lmax∑l=0

µmax(l)∑µ=1

νmax∑ν=1

Cµνl

˙Tµν

l (g) (41)

with lmax the maximum degree of the series expansion, ˙Tµν

l (g) harmonic functions of Euler’sangles and Cµν

l their Fourier’s coefficients describing the texture. For instance, truncatingat lmax = 22, an ODF for cubic crystals and orthorombic sample symmetry is representedby a set of 185 Cµν

l - coefficients (355 in the absence of sample symmetry). These coefficientscan for example be obtained, after appropriate mathematical processing, from a set of X-ray diffraction pole figures, measured on a sample with the help of a texture goniometer(Bunge, 1982).

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30 A.M. Habraken

4.2 Polycrystalline Plasticity Models

Starting from the state of individual crystallites, different assumptions have been proposedto deduce the state of the corresponding polycrystal characterized by the macroscopic strainrate tensor ε macro and the stress tensor σ macro. The most logical approach is a volumeaverage weighted by the ODF function f(g) defined by relation (40):

ε macro =∮

ε micro(g)f(g) dg (42)

σ macro =∮

σ micro(g)f(g) dg (43)

Although the assumption of homogeneous microscopic stress tensor σ micro and strainrate tensor ε micro in each single crystal is already a simplification, most macroscopic modelsare bound by stronger assumptions:

• Sachs 1928 applies the loading state (uni-axial tension) at each individual crystalliteas if it were a free-standing single crystal and he assumes that each crystal has onlyone active slip system (Van Houtte 1995). This assumption leads to more or lesssevere violations of geometric compatibility at the grain boundaries and, in general,quite unsatisfactory results (Sevillano et al. 1980). In particular, the predictions oftexture evolutions due to deformation do generally not compare favourably with theexperimental results.

• Full Constraint Taylor 1938 (FC) assumes an homogeneous distribution of plasticvelocity gradient, that induces an homogeneous plastic strain rate distribution:

Lmacro = Lmicro =⇒ ε p macro = ε p micro (44)

This assumption leads to equilibrium violations at grain boundaries. For cubic metals,as reported by Van Houtte et al. 2002, the predictions of this model are first-orderapproximations of deformation textures for all possible strain histories and all possibleinitial textures, though they do not perform well for low stacking-fault energy f.c.c.(brass, silver, etc.). “First-order approximation” means that predictions are goodfrom a qualitative point of view but not from a quantitative point of view, sincethe final locations of the final orientations may be up to 10 off, and intensities andvolume fractions may be up to a factor 2 wrong. However this model often computes,for many practical applications, more acceptable results than the previous one. Theassumption (44) provides an upper bound solution when models derived from Sachsapproach computes lower bounds. It is generally the Full Constraint Taylor hypothesislinked with Taylor-Bishop-Hill single crystal plasticity model that is applied whenliterature provides “polycrystalline TBH yield locus” (Figure 26).

• The Full Constraint Taylor’s model enforces rigorous geometric compatibility at theexpense of stress equilibrium. The so-called Relaxed Constraint Taylor’s (RC) modeldrops this strict assumption by relaxing the compatibility of well chosen componentsof the velocity gradient tensor. In the lath model, component L(RD)(ND) is not pre-scribed and the pancake model does not enforce the component L(TD)(ND) either,where RD, TD and ND are the rolling, transverse and normal directions. The pan-cake model is advocated for materials with flat, elongated grains produced by rolling.Figure 27 shows for instance L(RD)(ND) relaxation for such a flat grain. L(TD)(ND)

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Modelling the Plastic Anisotropy of Metals 31

σ = - 1σ xy /

σσ Fx /

σF

σ y

/1.5

1.5

1.0

1.0

0.0

0.0

0.5

0.5

- 0.5

- 0.5

- 1.0

- 1.0- 1.5

- 1.5

Experiments

T B H

Barlat 1991Barlat 1997

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

Crystal plasticity

6th-order plasticpotential

σ11/τ0

σ22/τ0

(a) (b)

Figure 26. TBH yield loci compared with other predicted yield loci, a) Al-2.5%Mg(from Barlat et al. 1997 b) IF steel (from Li et al. in press)

ND

TD

RD

L(RD)(ND)

Figure 27. Relaxation of L(RD)(ND) in a flat elongated grain of the pancake model(from Van Houtte et al. 2002)

and L(RD)(ND) represent shears that would not cause any rotation of the top or bot-tom surfaces of the flat elongated grains. It is believed by pancake model’s advocatesthat the relaxation of such shears would cause fewer problems of strain misfits withneighbouring grains that other kind of relaxations would. These components are cal-culated by methods assuming that the associated shear stresses are set to zero. Thesecomponents take different values in each crystallite, depending on the lattice orien-tation. Such approach does not take the interaction with other grains into account.As shown by Van Houtte 1988, these kinds of relaxations can be implemented in anelegant way using pseudo-slip system.

Compared to FC models, RC models sometimes produce better and more detailedresults, especially for rolling texture predictions, although their justification is stillunder debate. The problem of accommodating the strain misfits inherent to RCmodels leads scientists like Wagner, Lucke, Arminjon, Imbault, Van Houtte to propose

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32 A.M. Habraken

several more advanced models which try to take these misfits into account. Thesummary presented in Van Houtte 1996 confirms that such models lie between TBHand self-consistent approaches and yield improved results compared to classical FCor RC models.

In general, Taylor type models are quite successful for f.c.c and b.c.c metals wherea large number of slip systems ensures that individual crystals can accommodate anarbitrary deformation. However, these types of models are not valid for materialswhose crystals have an insufficient number of deformation modes to sustain an arbi-trary strain. Such kinematically rank deficient materials are not rare: semi-crystallinepolymers, minerals and other geological materials, superconducting ceramics, metalsof hexagonal close-packed crystal structure such as zinc, zirconium and titanium. Twoproposals adapted to such cases are described in Prantil, Dawson and Chastel 1995.

• Self-consistent models (Berveiller & Zaoui 1979, Canova & Lebensohn 1995, Molinari1997, Masson & Zaoui 1999) consider a grain as a solid inhomogeneity embedded ina homogeneous infinite matrix subjected to macroscopic loading. All the grains aretreated that way one after the other, the matrix behaviour results from the weightedaverage of the individual contributions of all the grains. Both strain rate and stressheterogeneities are allowed, but they are linked by an interaction formula based onEshelby’s 1957 work. For elastic cases, Kroner 1961 uses the interaction formula atthe grain level:

(σ micro − σ macro) = −L∗: (ε micro − ε macro) (45)

where L∗ is a 4th rank tensor called the “interaction tensor”. The interested readercan find the method to identify L∗ in Van Houtte 1995.

This approach is not straightforward to extend to nonlinear cases. For elastoplasticcases, an incremental linearization proposed by Hill 1965 seems to be adapted:

σ micro − σ macro = −LH : (ε micro − ε macro) (46)

where LH is called the Hill’s constraint tensor. It depends on the elastoplastic mod-ulus, on the shape and orientation of the crystals. Masson & Zaoui 1999 summarizesthe scientific controversies on this topic and demonstrates that Hill’s conception couldbe adopted even for elastoviscoplasticity.

Ponte Castaneda’s variational procedure (Ponte Castaneda 1991) developed for nonlinear composites has recently been applied by Gilormini et al. 2002 to self-consistentapproach describing titanium polycrystals behaviour. The variational procedure in-troduces a linear comparison polycrystal, (identified by an optimisation problem),which is then used to give a self-consistent estimate of the response of the non linearpolycrystal.

Whatever self-consistent approach is used, a rather long iterative solution procedureis required. This is mainly due to the fact that, at the end of the iterative process, themacroscopic values must coincide with the average of the grain responses. However,unlike the Taylor’s models, the self-consistent models allow to take into account botheffects of texture and grain morphology on the mechanical response of the material.They provide an answer that respects in average both equilibrium and compatibilitybetween grains. Self-consistent models coupled with FEM are clearly one way toincrease accuracy of micro-macro models, but more efficient numerical approachesare still required to solve non academic problems.

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Modelling the Plastic Anisotropy of Metals 33

Figure 28. Illustration of the three types of relaxation considered by the Lamelmodel. Type I corresponds to 12-shear; Type II to 23-shear; Type IIIto 12-shear

• N-point models propose to look at n grains simultaneously to caputre some specificeffects due to the interaction of neighbouring grains. For instance, Van Houtte etal. 1999 and 2002 propose different versions of the Lamel model, that studies twograins at the same time. As the pancake model, it has been developped for rolling andassumes that grains tend to become flattened and elongated. Each of the two grainstaken separately undergoes a homogeneous strain, which however does not need to beequal to the macroscopic average strain. In its first version, the FC Taylor conditionwas satified for the two grains taken together, when the second version of this modelrelaxes this requirement. All the relaxations that maintain the geometry integrity ofthe boundary between the two grains and keep the boundary plane parallel with therolling plane are possible. Figure 28 presents the three types of relaxation considered.The local velocity of two interacting grains are computed by minimising the plasticwork under the following conditions:

– Plastic deformation proceeds by dislocation slip.

– The two grains coopeatively accommodate the macroscopic deformation (exceptL12 in the second version of Lamel).

– No grain boundary sliding occurs along the planar boundary separating the grain.

– Minimisation of the rate of plastic work in each grains.

This model still closed to RC Taylor model allows however to take into account thegrain interaction.

• Homogenization technique, used by Smit et al. 1998, Miehe et al. 1999 or Geers et al.2000, is directly based on a mathematical procedure already applied to find micro-macro links in composite materials. Two levels of finite element models are used:a mesh of the entire structure and a mesh of the Representative Volume Element(RVE). At each interpolation point of the macroscopic mesh, the finite element modelof the RVE is called to provide the stress-strain behavior of the material.

Sections 5 and 6 present some finite element applications of the above micro-macrohypotheses.

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34 A.M. Habraken

4.3 Texture Updating

If a texture is described by a set of representative crystals orientations (Toth & Van Houtte1992), the rotation of each representative crystal leads to the up-dated texture. Taylor’sassumption about an homogeneous velocity gradient:

Lmacro = Lmicro (47)

provides the rotation of each crystal, if Taylor’s single crystal plasticity model is appliedfor each crystal. In a first step, relation (20) identifies the active slip systems and thenrelation (21) defines the rate of crystal lattice rotation. This approach will be extensivelyused in the micro-macro approaches presented in Sections 5 and 6 to update textures, it isof common use in scientific works dedicated to micro-macro approaches (Van Houtte et al.1989, Hirsch 1991, Winther et al. 1997, Aukrust et al. 1997).

4.4 Average Taylor’s Factor

The TBH theory described in Section 3.2 for one crystal can directly be applied to poly-crystals with Taylor’s assumption (44). So, plastic strain rate tensors ε p are supposed tobe homogeneous throughout the polycrystal. In addition, Taylor assumes that the commonreference CRSS τc (common reference value for all the slip systems in one crystal) is thesame for all crystallites in the polycrystal in one representative volume element. It is calledthe average common reference CRSS τc:

τc = τc (48)

Again, this assumption seems acceptable for materials in their annealed state, thoughit becomes questionable after accumulation of a certain deformation (Bunge et al. 1985).With relations (29), the average plastic power dissipation in a polycrystal is easily computedfrom its expression for a single crystal (22) and the ODF (40):

Wp(ε p macro) =

∮W p(ε p micro, g)f(g) dg = εp macro

eq τcM(U ˙εp) = σmacro

eq εp macroeq (49)

with the average Taylor’s factor:

M(U ˙εp) =

∮M (U ˙ε

p , g)f(g) dg (50)

Hereafter any over-lined variable indicates a value which is assumed to be an averagefor all crystallites belonging to the same elementary volume considered in the macroscopicapproach. Note that the average Taylor’s factor only depends on the texture of the poly-crystalline material f(g) and on the given strain rate mode U ˙ε

p but not on the strain ratemagnitude εp macro

eq .Relation (49) yields the important micro-macro link:

σmacroeq = τcM(U ˙ε

p) (51)

Both geometric (textural) hardening related to M and material (strain) hardening re-lated to τc (linked to dislocation density) coexist and appear in a micro-macro approach.Experimental works also demonstrate that the anisotropic behaviour cannot be attributedto only one of these factors. For instance, Jensen & Hansen 1987 measured the 0.2 %tensile yield stress on specimens cut at different angles α to the Rolling Direction. Theirmaterial was a sheet of commercially pure aluminum, which was previously cold-rolled at an

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Modelling the Plastic Anisotropy of Metals 35

ε = 2.0F

LOW

S

TR

ES

S

(0.2

%)/

M

(MN

/m )2

ANGLE TO ROLLING DIRECTION

20 40 60 80

25

30

40

45

ε = 0.2

Figure 29. Results of tensile tests performed in different directions: yield stress dividedby M in a pure aluminium submitted to cold rolling of 20 or 200%

equivalent true strain of up to 200 %. Textures of the rolled sheet were determined and theaverage Taylor’s factors M were computed for two different levels of deformation. Figure 29shows the evolution of the ratio σmacro

eq /M versus the angle α to the Rolling Direction, thisratio corresponds to τc according to (51). At very large strains, τc is almost constant, indi-cating that the observed plastic anisotropy can mainly be attributed to the crystallographictexture. On the contrary, after moderately large monotonic strains, τc increases with α,showing a strong influence of the intragranular microstructure on the plastic anisotropy.According to Teodosiu’s 1997 review, this conclusion seems true for polycrystalline f.c.c:

• for moderately cold-rolled sheet, plastic anisotropy seems due to the orientation ofthe dislocation structures (sheets of high dislocation densities more or less parallel tothe 111 slip planes);

• for heavily cold-rolled sheets, plastic anisotropy mainly seems due to the crystal-lographic texture, because dislocations are arranged in thick-walled, equiaxed cells,providing an almost isotropic hardening.

Let us note that the further work of Winther, Jensen and Hansen 1997 studies the com-bined effect of texture and microstructure on the flow stress anisotropy of metals containingdislocation boundaries with a macroscopic orientation with respect to the sample axes. As-suming that dense dislocation walls and micro-bands resist like ordinary grain boundaries,a value of the CRSS depending on a Petch-Hall equation is adopted. In their approach,

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36 A.M. Habraken

Taylor’s or Sachs’ polycrystal assumption is used on a set of 1152 crystals, representative ofthe texture. It has been proved that the presence of dislocation boundaries has a significanteffect on anisotropic yield.

4.5 Link Between the Evolution of the Reference CRSS and Macroscopic StrainHardening

Taylor 1938 proposes that the common reference CRSS τc evolves as a function of the totalslip Γ in each crystal:

Γ(g) =∫ t

0Γ(g) dt (52)

where g defines the crystal orientation and the total slip rate Γ is:

Γ(g, ε p micro) =∑s

|γs| = εp microeq M (g,U ˙ε

p) (53)

where ε p is a strain rate tensor represented by its mode U ˙εp and magnitude εp micro

eq .Using Taylor’s assumptions ( ˙ε = ε p micro = ε p macro and τc = τc) the total polycrys-

talline slip rate ¯Γ is defined by:

¯Γ( ˙ε p) =∮

Γ(g, ˙ε p)f(g) dg = ˙ε eqM (g,U ˙εp) (54)

The total polycrystalline slip Γ is then computed by integration of the slip rate and animportant micro-macro link appears:

dΓ = dεpeqM (55)

If one works with a uni-axial test where εp macroeq = ε and σmacro

eq = σ, the macroscopicwork hardening can easily be deduced from relations (51) and (55):

dε= M 2 dτc

dΓ+ τc

dM

dε(56)

The first term (material hardening) on the right side of equation (56) indicates anisotropic hardening at the polycrystal level, since an average is done whatever slip systemis activated and for all crystals. The second term (geometrical hardening) is due to theevolution of texture resulting from plastic deformation. It is often neglected in macroscopicmodels or even in simple micro-macro models (Schmitz 1995, Winters 1996). Relation (56)is well known and comments can be found in Aernoudt et al. 1987.

5 FEM MICRO-MACRO MODELS WITHOUT MACROSCOPIC YIELDLOCUS

5.1 Introduction

The Finite Element Method is currently used to simulate the behaviour of the materialsat the microscopic level (Teodosiu et al., Kalidindi et al. 1992, Anand & Kothari 1996,Bertram et al. 1997, Barbe et al. 1999, Acharya & Beaudoin 2000, Van Houtte et al. 2002...). Finite element simulations allow to treat a crystalline aggregate as a continuum, bysimply requiring the equilibrium of the stress tensors and the continuity of the displacementsacross the grain boundaries. In such a model, each crystallite consists of at least one elementof the FE mesh. Figure 30 shows the mesh used by Kalidindi to model an IF-steel that hasbeen industrially cold-rolled to a reduction of 70 % (Van Houtte et al. 2002). In this case,

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Modelling the Plastic Anisotropy of Metals 37

Figure 30. Initial (a) and final mesh (b) of an IF steel 70% cold-rolling simulation(from Van Houtte et al. 2002)

each element represents one crystal, which is simply assumed cubic. In other cases, exactboundaries of observed grains are represented as in Figure 31, where the eleven grainsdetected on the front face of a pure copper tensile sample have been meshed (Teodosiuet al. 1992). Another possibility is to use simple cubic elements to mesh grains. Forinstance, Acharya & Beaudoin 2000 discretizes one grain by 12 to 96 elements accordingto coarse or fine discretizations (Figure 32). In all these simulations, the lattice orientationcharacterizes each grain. The constitutive equations of the microscopic model (crystalplasticity model) are implemented as a material model. Such simulations aim to validatethe chosen microscopic model (Acharya & Beaudoin 2000), to study the effect of grainsinteractions on texture prediction (Van Houtte et al. 2002) or to allow parametric studies,difficult to carry out experimentally.

Regarding microscopic strain heterogeneity, microscopic FEM models are in principlesuperior to methods such as the FC or RC Taylor’s models or Lamel model. However, therequired calculation time is at least one order of magnitude larger, which makes the methodunsuited for large scale FE simulations of a forming process of a fine-grained material.

This article is focused on another use of the FEM coupled with microscopic models.It concerns accurate macroscopic simulations, where the material behaviour is taken intoaccount at a microscopic level. This time, each finite element has a macroscopic size, itrepresents numerous crystallites. Clearly the FEM discretization is on a macroscopic scale.In such a micro-macro approach, a lot of averaging methods to extract the macroscopicbehaviour from microscopic analyses are possible. Some examples are listed in Sections5.2. to 5.3. They concern approaches where no macroscopic yield locus is computed, theypresent one generic family of micro-macro models.

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38 A.M. Habraken

( a )

( b )

8

1

7

10

5

6

2 311

4

9

Figure 31. (a) Observed 11 crystallites on the front face of a copper tensile sample; (b)single layer of pentahedral finite elements (from Teodosiu et al. 1992)

Figure 32. Von Mises equivalent stress (Mpa) computed by a FEM mesh for atensile strain of 20% of nickel polycrystal aggregate (grain size 32µm),from Acharya & Beaudoin 2000

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Modelling the Plastic Anisotropy of Metals 39

The most simple micro-macro FEM model of this type is proposed by Nakamachi, it is adirect extension of the FEM analysis applied at the microscopic level. At each macroscopicintegration point, the behaviour law of a single crystallite characterized by its orientationand the hardening state of each of its slip system is adopted. So no average procedure of thepolycrystalline behaviour is applied. Nakamachi & Dong 1997 have applied their approachto the Limiting Dome Height test (NUMISHEET’96 benchmark test, Figure 16). Thequarter of the 180× 100 mm blank is simulated by 1125 eight-node SRI (Selected ReducedIntegration) solid elements. A dynamic explicit finite element code is used with an elas-tic/crystalline viscoplastic constitutive law. The strain heterogeneity between integrationpoints directly follows from the nodal displacements. The equilibrium is assumed but notchecked since an explicit finite element scheme has been used. Such a simple micro-macroapproach can gives interesting results at least close to experimental ones (Nakamachi et al.1997,1999a, 1999b).

Section 6 is dedicated to the macroscopic approximation of yield loci based on micro-scopic assumptions (3G model, Aifantis’models) or microscopic computations (see describedmodels of Montheillet, Darrieulat, Arminjon, Bacroix, Imbault, Van Houtte). These modelsbelongs to a second generic family of micro-macro models.

5.2 Macroscopic FEM Simulations Relying on Discrete Set of Crystals

In this type of macroscopic FEM simulations, the response of each integration point de-pends on the response of a multitude of single grains, representative of this material point.Each crystallite is described by a microscopic model. This can be done only thanks tomassive parallel computations. Simulations like hydroforming process, performed by Daw-son already 10 years ago (Dawson et al. 1992), show that, with powerful computers, thisapproach can be applied to real problems of limited size. Clearly, in addition to the usualchoices in a FEM approach (lagrangian, eulerian formulation, explicit, implicit scheme, ...),the scientist must determine further assumptions:

• Step 1: the micro-macro link (Full Constraint, Relaxed Constraint Taylor’s modelor one of their variants, self-consistent model, homogenisation technique). It allowsto go from a macroscopic velocity gradient to microscopic values and to provide,after computations at microscopic level, the macroscopic stress as well as the stiffnessmatrix for implicit scheme.

• Step 2: the set of representative crystals. The number of crystals must be deter-mined as well as their orientation, shape, size, slip system and associated CRSS. Theinitialisation of these crystal data is essential for the accuracy of the results.

• Step 3: the behaviour model of each crystal. In this type of models, elasticity can beneglected or taken into account. The crystal plasticity model defines the link betweenthe resolved shear stress and the slip system rate. The CRSS hardening rule is alsoa key function that induces differences between the models. As shown hereafter, twodifferent proposals exist: viscoplasticity or rate independent plasticity. Finally, themicroscopic stress related to the microscopic velocity gradient is computed.

Of course, all these options interact and have strong effects on the accuracy and com-putation time. Well-known scientists have proposed various models in order to describepolycrystalline models. Some of these models are presented hereafter, together with theiradvantages and drawbacks.

5.2.1 TBH single crystal model + FC Taylor’s polycrystalline model

Such polycrystalline models are described for instance in Asro & Needleman 1985, Mathur& Dawson 1989, Becker 1990, Neale 1993, Beaudoin et al. 1994, Kalidindi & Anand 1994,

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40 A.M. Habraken

Anand et al. 1997. The three general choices: micro-macro link, set of representativecrystals and behaviour model of each crystal are successively identified hereafter and someapplications are presented to illustrate this type of models.

a) Micro-Macro link

Full Constraint Taylor’s polycrystalline model is used. This implies that the local de-formation gradient in each grain is set homogeneous and identical with the macroscopicdeformation gradient.

The elasticity is generally neglected in the work performed by Dawson’s team (Mathur& Dawson 1989, Beaudoin et al. 1994):

Fmacro = Fmicro = R∗Fp (57)

R∗ being a rotation matrix. But elasticity is taken into consideration by scientists workingwith Anand (Kalidindi & Anand 1994, Anand et al. 1997):

Fmacro = Fmicro = F∗Fp (58)

with F∗ defined by Figure 24.If the N representative grains or crystals have equal volume, a simple average is used

to link micro and macro stress tensors (Kalidindi & Anand 1994).

σ macro =1N

N∑k=1

σ micro(k) (59)

Otherwise, a weighted average based on the volume fraction Wk of each crystal orien-tation is used (Beaudoin et al. 1994).

σ macro =N∑

k=1

W kσ micro(k) (60)

Clearly, the average used to reach the macroscopic stress tensor depends on the choiceof the representative set of crystals.

b) Representative set of crystals

Number of crystals: N varies depending on simulations and authors. For instance, 180 or200 crystals per integration point are used for compression tests (Kalidindi & Anand 1994),32 for cup-drawing (Anand et al. 1997), 256 for hydroforming process (Beaudoin et al.1994), 200 for titanium rolling (Dawson & Kumar, 1997) ...Orientation of crystals: experimental data from X-ray diffraction measurements of crystal-lographic texture give discrete intensities of diffracted energy as a function of goniometricangle position. This can be transformed into a “Crystal Orientation Distribution Function”ODF (see Section 4.1) and can be used to generate a set of “weighted Euler angles”.

According to “popLA package” (Kallend et al. 1991), two approaches are proposedto approximate a texture defined by a continuous density distribution through a set ofdiscrete orientations. In the first approach, all orientations have the same weight. Since thecrystals are assumed to have the same volume, they are located in the orientation space sothat higher density regions are more densely populated. The second approach consists inrandomly populating the orientation space with discrete orientations and assigning to eachof them an initial weight that minimizes the effect of the density fluctuations arising from theEuler’s space distortion. The actual density in the associated volume of orientation spacethen multiplies the weight of each discrete orientation. This provides a weighted orientation

Page 39: Arch. Comput. Meth. Engng. Vol. 11, 1,

Modelling the Plastic Anisotropy of Metals 41

approach generally used by Dawson and his co-workers (see relation (60)), while Anand’steam generally uses the first approach with identical weights (see relation (59)).

This initial set of orientations is updated during the computations. Section 4.3 explainshow the computation of the crystal lattice rotation (Ω L for a single crystal is directlyapplicable, thanks to Taylor’s assumption of identical deformations at macroscopic andmicroscopic levels (see relations (19) - (21)).Crystal slip systems: the slip systems are well known for f.c.c materials like aluminium,copper or b.c.c materials like steel, tantalum, h.c.p. materials like zinc, zircalloy ...CRSS: as explained in Section 3.4, the hardening of each CRSS associated to a slip systemshould be taken into account as well as its initial value. Very often, the values of theinitial CRSS of all slip systems are assumed equal or close to each other (Van Bael 1994).In an annealed state for f.c.c or b.c.c materials, this seems reasonable. From numeroussimulations of homogeneous deformations of f.c.c materials (Kalidindi et al. 1992), it hasbeen observed that, after large deformations, the values of the CRSS for the various slipsystems in an aggregate are quite close to each other. This common value is estimated froma macroscopic simple compression test.Crystal shape: this cannot be taken into account in a classical polycrystalline Taylor’smodel.

c) Behaviour model of each crystal

For instance, Mathur & Dawson 1989 and Beaudoin et al. 1994 neglect elasticity (57) anduse a rate dependent plasticity model described for each slip system by relation (30). Theyalso adopt a common average value for all CRSS in one crystal:

τ sc = τc (61)

and choose a simple evolution law of Voce’s type:

τc = H0τsat − τc

τsat − τc0

and τsat = f(Γ) (62)

where Γ = total shear slip on all slip systems of the crystal; τsat = saturation value of thecommon reference CRSS; τc0 = initial value of the common reference CRSS; H0 = materialparameter.

In Beaudoin et al. 1994, a hydroforming process (Figure 33) is chosen to validate thenumerical FEM model. This choice seems very well adapted to check the model of theblank behaviour as i) applied pressure assures stability, ii) contact and friction modelsdo not introduce inaccuracy; punch and blank are in sticking conditions and blank-flangecontact is assumed frictionless.

Consequently the final deformed shape depends on the material anisotropy and is mea-sured by the percentage of earring. As shown in Figure 34, numerical results are close toexperimental measurements. For this case, it was checked that the texture evolution duringthe process and its effect on the shape of the yield locus are minimal.

Another example is provided by Anand & Kothari 1996 who compare two different mod-els at the microscopic level: a viscoplastic flow rule (relation (30)) and a rate-independentcrystal plasticity model. A robust calculation scheme determines a unique set of active slipsystems and the corresponding shear increments in the rate-independent theory. In bothmodels, the evolution rule (31) of the CRSS associated to each slip system τ s

c is applied.This relation comes from Asro & Needleman 1985 and the chosen hardening matrix hsu for

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42 A.M. Habraken

CONTROLLED PRESSURE

STROKE

OIL

RUBBER

Figure 33. Axisymmetrical hydroforming process of an aluminium sheet (adaptedfrom Beaudoin et al. 1994)

00

10 20 30 40 50 60 70 80 90

Radial distance from RD ( degrees )

5.0

5.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

% E

arin

g

ExperimentSimulation

Figure 34. Earing measurement and prediction at the final stage of a hydroformingprocess (from Beaudoin et al. 1994)

the 12 slip systems of f.c.c crystals is:

hsu = hu

A qA qA qA

qA A qA qA

qA qA A qA

qA qA qA A

with A =

1 1 11 1 11 1 1

(63)

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Modelling the Plastic Anisotropy of Metals 43

where systems 1, 2, 3 are coplanar, as systems 4, 5, 6 and 7, 8, 9 and 10, 11, 12 are. Forcoplanar systems, the ratio of the latent hardening rate to the self hardening rate is equalto unity. For non coplanar systems, it is evaluated by means of factor q = 1.4. The functionhu is defined by:

hu = h0

(1 − τu

c

τsat

)a

(64)

where h0, a and τsat are slip system hardening parameters which are taken identical for allslip systems. In fact, τsat should be an increasing function of strain rate but this can beneglected at low temperature.

0.2 0.4 0.6 0.80.0

ε

0

100

200

300

400

σ(

MP

a )

Strain Rate Jump Test

0.5 1.0 1.50.0ε

0

100

200

300

400

σ(

MP

a )

Constant True Strain Rate Test

a b

Figure 35. Experimental and numerical compression tests. (a) Strain rate jumptest; (b) Constant true strain rate test (from Kalidindi & Anand 1994)

These parameters are determined by curve-fitting between the results of simple com-pression tests and numerical simulations using “Taylor simulation”. Figures 35 illustratethe identification of copper parameters. By “Taylor simulation”, one means: the TBHmodel applied at the crystal level with Taylor’s assumption of equality between micro-scopic and macroscopic strain rates at the polycrystalline level. Here, this model uses (31)and (64) to follow the evolution of the CRSS of each slip system in each crystal. The ini-tial copper isotropic texture is represented by 200 crystals. One can check that the initialvalue of CRSS τc0 influences the macroscopic initial yield, τsat the final saturated value ofmacroscopic stress, h0 affects the initial hardening rate and a modifies the shape of thepolycrystalline stress-strain curve between the initial yield and saturation. If the TBHmodel is replaced by a rate dependent visco-plastic law (30), two additional parameters:γ0, n appear. For copper, according to Kalidindi & Anand 1994, γ0 is assumed to have aconstant value of 0.012 and n is chosen equal to the corresponding macroscopic parame-ter. This latter is identified thanks to a strain rate jump experiment on a polycrystallinespecimen in compression state at room temperature.

Once all the microscopic parameters τc0 , τsat, h0, a and γ0, n are identified, Anandand co-workers (Kalidindi & Anand 1994, Anand & Kothari 1996, Anand et al. 1997)can apply a macroscopic FEM model (ABAQUS) where each interpolation point averagesthe microscopic behaviour of 180 crystallites. In particular, compression tests of cylindri-cal specimens, plane strain compression experiments, forging experiments (Figure 36), cupdrawing experiments (Figure 37) are performed. Experimental and numerical loads, geo-metric shapes and texture evolutions (Figure 38) are compared as well. Such a model hasbeen successfully applied to high deformation rate of tantalum (Anand et al. 1997).

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44 A.M. Habraken

Figure 36. (a) Plane strain forging experiment performed on an isotropic copper;(b) coarse FEM mesh with each integration point relying on 180 singlecrystals (from Kalidindi & Anand 1994)

d) Conclusion

In short, the previous examples using Taylor’s model, coupled with finite elements, showthat such approach is feasible and has been validated for b.c.c and f.c.c materials. If a highnumber of representative crystals at the microscopic level is used, this can provide accu-rate predictions of texture, geometry and stress history during any cold forming process.The texture evolution is directly implemented in this approach. This means that the “ge-ometrical hardening”, responsible for the shape modification of the yield locus is directlytaken into account. This characteristic constitutes an advantage of this type of modelswith respect to others, such as the phenomenological yield loci coupled with kinematicand isotropic hardening. These ones cannot easily represent texture evolution effect. Asexplained above, this micro-macro method does not use a global yield locus but an averageanswer computed from the plastic behaviour of a set of representative crystals.

Taylor’s model coupled with finite elements requires a high CPU time and memorystorage, directly proportional to the number N of crystals associated to one integrationpoint. It is quite surprising that this number N has not been more investigated. Veryoften, measured pole figures are qualitatively compared to the ones computed from thediscrete set of orientations (Anand & Kothary 1996). No further study from a mechanicalpoint of view, such as stress response computed for the same strain and different N values,

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Modelling the Plastic Anisotropy of Metals 45

50

40

30

20

10

0

Cup

Hei

ght (

mm

)

0 90 180 270 360

Angle from Rolling Direction ( degrees )

Experiment

Polycrystal model

Figure 37. Experimental and predicted earing profiles resulting of the deep draw-ing of a cylindrical cup (from Anand et al. 1997)

is performed. For each of the N representative crystals associated with one integrationpoint, one must store:

• the crystal orientation, typically defined by 3 variables;

• hardening variable(s): either one reference CRSS τc, as in relation (61) if a commonreference value for all slip systems is adopted, or one CRSS τ s

c for each slip systemss as in equation (31).

5.2.2 Self-consistent polycrystalline models

As explained in Section 4.2, the self-consistent approach respects, on the average, bothcompatibility and equilibrium between grains. It is intensively used by scientists aiming tounderstand and predict the macroscopic material behaviour thanks to micro-macro models(Canova & Lebensohn 1995, Molinari 1997, Nikolov & Doghri 2000). However, this greatadvantage is shaded by the increase of computation time. This fact explains why it is seldomused for coupling with FEM models. It is not surprising that self-consistent approachapplications appear for hexagonal materials like Zircalloy (Chastel et al. 1998) becausethe low number of deformation modes in such crystals yields inaccurate predictions withTaylor’s model.

Chastel et al. 1998 present a viscoplastic self-consistent polycrystalline model (Leben-sohn & Tome 1993) coupled to a 3D eulerian finite element code LAM3 (Hacquin et al.1995), applied to the hot extrusion of Zircalloy. In practice, the finite element calcula-tion starts with an isotropic rheology, which provides a first deformation gradient at eachintegration point. Then, the polycrystalline model is locally activated and provides ananisotropic response of the material which induces subsequent calculation in LAM3. Forthis case, the final flow patterns reached by the macroscopic approach and by the micro-macro computation are very close to each other as they are mainly fixed by the kinematicboundary computations. Discrepancies between predicted and measured texture evolutions

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46 A.M. Habraken

Figure 38. Experiment and predicted 111 and 110 pole figures in the deformed specimenat the point indicated in the deformed mesh (from Kalidindi & Anand 1994)

are attributed to the occurrence of recrystallization and/or recovery phenomena, which arenot taken into account in the model.

To summarize, such a method allows to consider directly “textural hardening” as Tay-lor’s model coupled with finite elements does. It is more satisfactory from a scientific pointof view since, at the microscopic level, both compatibility and equilibrium are approached.The grain shape can be taken into account at the microscopic level. The case of crystalswith a low number of deformation modes seems to require this type of model for accuratepredictions. The cost in memory is identical with Taylor’s approach. However, the CPUtime, which is already a problem with Taylor’s model, is even worse here, as an itera-tive process is required to find the heterogeneous strain repartition between crystals. This

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Modelling the Plastic Anisotropy of Metals 47

explains why so few macroscopic FEM models are linked to self-consistent microscopic mod-els. Progresses in this direction are on their way. It must be pointed out that with “objectprogramming”, it is not a real problem to replace a Taylor’s model by a self-consistent one.

5.2.3 Homogenisation polycrystalline models

The three general choices: micro-macro link, set of representative crystals and behaviourmodel of each crystal are successively identified hereafter and some applications are pre-sented to illustrate this type of models.

a) Micro-macro link

Proposals by Smit et al. 1998, Miehe et al. 1999, Geers et al. 2000 or Feyel & Chaboche2000 are directly based on mathematical procedures already applied in composite materials.Two levels of finite element models are used: at every interpolation point of the macroscopicFEM mesh, another microscopic FEM model, simulating a Representative Volume Element(RVE), is called to provide the stress-strain behaviour of the material. In other words, theconstitutive law at a macroscopic point results from the global response given by the FEManalysis of a set of representative crystals (RVE) described by a microscopic behaviourmodel. These 2 levels of computation are represented in Figure 39.

macro micro

X micro

Macro - Variables Micro - Variables

X macro

Macro - Continuum Micro - Structure

Figure 39. Macroscopic and microscopic levels (adapted from Miehe et al. 1999)

The homogenisation approach provides the mathematical background to go from themicroscopic level to the macroscopic level. Figure 40 presents the mathematical descriptionof the same deformation at macroscopic and microscopic levels. In Figures 39, 40, 41the superscript macro relates to the macroscopic mesh and the superscript micro identifiesvariables attached to the RVE considered at the microscopic simulation level. The followingnotations are introduced:

Γmacro or micro initial configuration,γmacro or micro deformed configuration associated to the initial one,Xmacro or micro coordinate tensor in the initial configuration,xmacro or micro coordinate tensor in the deformed configuration,Fmacro = ∂xmacro

∂Xmacro macroscopic deformation gradient tensorFmicro = ∂xmicro

∂Xmicro microscopic deformation gradient tensor,nΓ outward normal of the initial configuration of the RVE,nγ outward normal of the deformed configuration of the RVE.

The following averaging relations define the macroscopic gradient tensor Fmacro and themacroscopic first Piola Kirchhoff stress τ macro from their values at a microscopic scale:

Fmacro =1V

∫Γmicro

Fmicro dV (65)

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48 A.M. Habraken

macro macro

micromicro

x microX micro

x micro micro= F macro w+ ~X

F micro = F macro + ~F

a. Deformation of Macro - Continuum

b. Deformation of Micro - Structure

x macro

F macro

x macroX macro

Figure 40. Definition of the deformation at macroscopic and microscopic level(adapted from Miehe et al. 1999)

F micro = F macro +~F

x micro micromacro + ~X

Figure 41. (a) Details of the deformation at microscopic level; (b) (c) (d) initialFEM meshes of the RVE where crystals are identified by different greycolours (adapted from Miehe et al. 1999)

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Modelling the Plastic Anisotropy of Metals 49

τ macro =1V

∫Γmicro

τ micro dV (66)

The application of Gauss’ theorem leads to:

Fmacro =1V

∫Γmicro

Fmicro dV =1V

∫∂Γmicro

xmicro ⊗ NΓ dA (67)

τ macro =1V

∫Γmicro

τ micro dV =1V

∫∂Γmicro

tmicro ⊗ Xmicro dA (68)

where the tension on the boundary ∂Γmicro is defined by:

tmicro = τ micronΓ (69)

Figure 41a. illustrates the above theoretical considerations. Figures 41b., c., d. presentdifferent microscopic finite element meshes: one element per crystal grain (b), regular squareelements mesh (c), mesh of triangles applied on the crystals (d).

The deformation assumption in the RVE is related to macroscopic values by:

xmicro = FmacroXmacro + w (70)

Fmicro = Fmacro + F (71)

The assumption (71) coupled with the previous average equation (67) leads to:

1V

∫Γmicro

FdV =1V

∫∂Γmicro

w ⊗ nΓ−dA +

1V

∫∂Γmicro

w ⊗ nΓ+dA = 0 (72)

with NΓ+= NΓ−

at 2 associated points of the contour (Figure 41).This general mathematical frame shows that 3 alternative possibilities directly satisfy

relation (72):

• w = 0 everywhere in Γmicro: Taylor’s assumption Fmicro = Fmacro is recovered, nosuperimposed deformation field at the micro scale;

• w = 0 on the contour ∂Γmicro: zero fluctuation of the superimposed deformation fieldw on the boundary but non zero fluctuation inside the RVE;

• w+ = w− on the contour ∂Γmicro: periodic fluctuation of the superimposed deforma-tion field w on the boundary.

b) Set of representative crystals

All the examples presented in Miehe et al. 1999 start from an isotropic texture easilyrepresented by random crystal orientations. According to the size of the macroscopic mesh,the number of finite elements in the RVE varies.

For instance, in a validation test simulating a simple shear loading, one single macro-scopic element is used, coupled with 100 finite elements in the RVE. Each of them has oneintegration point associated with one crystal orientation. The texture predictions resultingfrom the 3 alternative choices for the w field are quite close to each other.

Another example of punch indentation was computed using 100 macroscopic finite ele-ments, each of them linked with 400 crystal grains in the RVE.

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50 A.M. Habraken

c) Behaviour model of each crystal

Miehe et al. 1999 provides one of the most complete micro-macro models: it considersthe anisotropic elasticity in each crystal coupled with a thermoplastic or thermoviscoplasticanisotropic behaviour and temperature effect. The rate dependent plastic behaviour is quiteclassical and the rate independent plastic approach proposes an extension of the work ofKothari & Anand 1996.

d) Conclusion

The advantages of this approach are that texture and grain shape or size effects are directlytaken into consideration. Better than in the self-consistent approach, both equilibrium andcompatibility between crystals are accurately reached in the RVE. The effects of 3 differentassumptions at the microscopic scale (Taylor’s hypothesis, no fluctuation or periodic fluc-tuation on the boundary) are easily considered. One significant drawback is the amountof computations and memory requirements, which are even worse than with self-consistentand Taylor’s models coupled with macroscopic FEM simulations. For instance, all the vari-ables describing the microscopic FEM simulation (nodal positions, state variables at everymicroscopic integration point: crystal orientation, reference CRSS or CRSS for each slipsystems ...) associated to every macroscopic integration point must be stored. Feyel &Chaboche 2000 proposes however an industrial application.

5.3 FEM Analysis Applied on Both ODF Evolution and Mechanical Fields

The discrete set of representative crystals used in FEM simulations as described in Section5.2.1 b) suffers from shortcomings because such a characterization comes with few analy-tical tools. No direct means are available, for instance, to develop quantitative measuresto differentiate between textures associated with distinct discrete aggregates. As a result,considerations on differences between textures are often qualitative, or obtained throughprojections onto alternate representations. In dealing with spatially inhomogeneous tex-tures, there is often a need to interpolate or to project across textures. This requirementappears when initialising from experiment, or computing spatial gradients of texture asmeasures of the inhomogeneity degree. Such measures are important when the consideredmaterial is initially inhomogeneous and when substantial inhomogeneities develop over thecourse of the process. In rolling, for instance, highly localized regions of inhomogeneitydevelop through the thickness of the sheet due to roll induced shearing.

Kumar & Dawson 1995a, 1995b and 1996 and Dawson & Kumar 1997 propose a quitecomplex approach that uses directly the ODF without relying on discrete sets of representa-tive crystals. This approach offers the advantage of an easier and more accurate possibilityto compare and interpolate textures. 2 FEM simulations applied in different spaces areconnected to each other:

• in the crystal orientation space, a finite element mesh describes the ODF representingthe material; a “metallurgical-orientation” finite element analysis is used to solve theODF conservation equation (Clement 1982);

• in the classical geometrical space, a spatial steady state simulation using eulerian me-chanical finite elements is applied to an industrial process such as rolling for instance.

It is fundamentally different from the approach proposed in Section 5.2.3 which consists of2 coupled FEM simulations in the classical geometrical space but performed on differentscales: microscopic and macroscopic.

Of course, in Kumar and Dawson’s work, both FEM simulations are coupled:

• the microscopic FEM requires the strain evolution to compute texture evolution;

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Modelling the Plastic Anisotropy of Metals 51

• the macroscopic FEM uses the up-dated texture as well as crystal plasticity withTaylor’s or Sachs’ assumption to get the macroscopic constitutive behaviour.

5.3.1 Crystal orientation representation

Kumar and Dawson have not chosen Euler’s angles but Rodrigues’parameters to representthe crystal orientations. The crystal orientation is defined as the rotation R required toalign the crystal lattice frame with a fixed sample reference frame. Three independentparameters are sufficient to describe such a rotation. An alternative class of representationuses an axis of rotation n and an angle of rotation φ. In this case:

R(n, φ) = n ⊗ n + (I− n ⊗ n) cos φ + I ∧ n sin φ (73)

where I is the second order unit tensor. A particular representation of the crystal orientationis defined by Rodrigues by vector r:

r = n tg φ (74)

The parameters r1, r2, r3 define the so-called Rodrigues’ space.

The advantages of this choice are the following ones:• A simple fundamental region can be computed in the orientation parameters space.

Rodrigues’ proposal defines all the possible crystal orientations without redundancydue to crystal symmetry. For instance, the case of f.c.c crystal leads to the funda-mental region represented on Figure 42.

• The space distortion is limited and there is no singularity in this space. This can bededuced from the form of the invariant volume element:

dv = cos2(

φ

4

)dr1 dr2 dr3 (75)

Note that Euler’s space distortion and singularity can be identified from relation (40).

D

E C

F

F

CA

B

E C

D

F

E

B

B

AA

D

D

A

C

F

E

B

r3

r2r1

Figure 42. F.c.c fundamental region for Rodrigues’space (from Kumar & Dawson 1995)

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52 A.M. Habraken

With this parametrization, the Orientation Distribution Function (ODF) is defined bythe probability density f(r). The volume fraction V ∗ of crystals whose orientations belongto a subset Ω∗ of the orientation space is given by:

V ∗ =∫Ω∗

f(r) dΩ (76)

The whole domain Ω of orientation space depends on symmetries exhibited by thecrystal. f(r) is scaled in such a way that:

l =∫Ω

f(r) dΩ (77)

5.3.2 Micro-macro link

The macroscopic behaviour is the average of the behaviour of the individual crystallitesof the aggregate. If xmicro is an arbitrary crystal quantity, its macroscopic value xmacro isgiven by:

xmacro =∫Ω

xmicro(r)f(r) dΩ (78)

Here is a difference with respect to approaches using a discrete sample of orientationsand simply computing a weighted average of the crystal quantities (see relations (59) and(60)). Dawson and Kumar apply a viscoplastic constitutive law at the crystal level asproposed in Section 3.3, with identical rate sensitivity and CRSS for all slip systems andfor all the crystallites related to one interpolation point. Either Taylor’s assumption ofequality between micro and macro plastic strain rates or the assumption of micro andmacro stress equality (Prantil et al. 1995) is applied.

5.3.3 Evolution rule of the orientation distribution function

Restricting his attention to the lattice reorientation caused by crystallographic slips, Clement1982 proposes to model texture evolution by integrating an equation for the rate of changeof the probability density f(r, t). This equation results from requiring that the materialderivative of relation (78) vanishes.

∂f

∂t+ v grad f + f div v = 0 (79)

where v is the reorientation velocity. Note that Arminjon 1988 provides a demonstration ofthis type of equation using physical hypotheses. The above formulation is an eulerian rep-resentation in which f is associated to particular locations in the orientation space, ratherthan to particular crystals whose orientations change. Alternatively, a reference texture f0

can be specified, for instance f at time t0. Then, this reference position remains fixed forall times and is used to define initial lagrangian coordinates for crystals. Relation (79) canbe written either with respect to current eulerian coordinates f(r, t) or with respect to thelagrangian coordinates f(r0, t). Euler’s choice (79) leads to difficulties in the FEM formu-lation because of the convective contribution associated with the term v grad f . This stillremains a problem, even if some solutions are proposed in computational fluid mechanics.The alternative is to use the lagrangian representation. However, this adds the cost of anexplicit computation of crystal trajectories in the orientation space. Both approaches aredescribed in Kumar & Dawson 1996.

Another difficulty is the extreme behaviour of the ODF. The ODF can evolve expo-nentially, sometimes tending asymptotically to Dirac’s function. Clearly, inaccuracies are

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Modelling the Plastic Anisotropy of Metals 53

inevitable as the finite element size cannot tend to 0. An effective strategy applied by Ku-mar and Dawson is to moderate the evolution of the ODF by the following transformation:

p = ln f (80)

Relation (79) to compute the new ODF uses the reorientation velocity field v, which islinked to crystal plasticity and to the relationship between the crystal velocity gradient andits macroscopic counterpart. Here again the relation (21) is applied and the relationship be-tween the rate of crystal lattice rotation Ω L and v depends on the specific parametrizationemployed for rotations. For Rodrigues’ parameters, Kumar & Dawson 1995a or Dawson &Kumar 1997 have established the adapted formula.

5.3.4 Applications

a) Texture prediction under monotonic deformations

The following example shows a FEM analysis applied to solve the texture conservationrelation under the assumption of plane strain compression of f.c.c polycrystals. In Figure 43,the 3D FEM mesh of 28672 - 4 nodes tetrahedral elements is showed.

r3

r2r1

Figure 43. f.c.c fundamental region for Rodrigues’space (from Kumar & Dawson 1995)

As Taylor’s hypothesis is applied, the reorientation velocity, developed under a mono-tonic deformation, is invariant with strain. So crystal computations are done only once.The developed texture is adequately represented by the ODF on the boundaries of thefundamental region; consequently only outside views of the ODF are represented. Theideal components of f.c.c plane strain compression texture are compared with the com-puted results on Figure 44. Texture development is dominated by two fibers: an α fiberconnecting the ideal Goss and Brass’ orientations and an β fiber connecting Brass andTaylor’s orientations.

b) Application to aluminum rolling

The flat rolling of a 1100 aluminum (Figure 45) being a steady state process, a macroscopiceulerian FEM approach is applied to model the macroscopic mechanical problem. In fact,the FEM analysis of texture evolution is coupled with the macroscopic FEM computation.Two FEM scales are present and the details on their parallel implementation can be found

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54 A.M. Habraken

T

B G B

T

= 0.2

( a ) ( b )

r3

r2r1

ε = 0.2

Figure 44. Ideal (a) and computed (b) components of plane strain compressiontexture of a f.c.c polycrystal (from Kumar & Dawson 1995b)

Workpiece

Tractionfree

Freesurface Rigid

rolls

Tractionfree

Slidingfriction

Constant angularroll velocity

Figure 45. Schematic diagram of flat rolling (from Dawson & Kumar 1997)

in Kumar & Dawson 1995b. Figure 46 shows the equivalent plastic deformation rate, thereference CRSS and the scalar measure of the spatial gradients of the ODF defined by:

∇A =∫

Ω|gradf(r,x)|dΩ (81)

where x identifies a material point of the workpiece.As expected, the microstructure hardens primarily within the deformation zone under

the roll. Hardening and texture gradients appear through the workpiece thickness. It isobserved that the texture gradients are rather important.

5.3.5 Conclusion

Kumar and Dawson’s approach is interesting from a scientific point of view. It showsanother proposal for the micro-macro coupling, not limited to the mechanical point ofview. Here the FEM formulation is directly applied for texture prediction and, hence,provides metallurgical information. However, the amount of computations seems to beeven greater than in Section 5.2 since the mesh discretization of the texture problem isalready of large size. Another problem, more “sociological” than scientific is the fact that

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Modelling the Plastic Anisotropy of Metals 55

20.0 65.0 110.0

ε peq

28.0 41.5

τ c

55.0 MPa

0.9 2.8 8.5 x 10 6Z

Y

X

Α

Figure 46. Equivalent plastic deformation rate εpeq , reference CRSS τc, scalar measure of the

spatial gradients of the ODF ∆A (from Kumar & Dawson 1995).

the ODF description is neither pole figures nor section in the Euler’s space well known bypeople with a metallurgical background.

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56 A.M. Habraken

5.4 Discussion about Micro-Macro Approaches without Yield Locus

Clearly, with parallel computers, the above selection of scientific works demonstrates thatmicro-macro approaches attract a great interest and become closer to practical problems.However, for each type of applications, it appears that more efforts should be devoted esti-mating the size of the set of representative crystals or the RVE or the texture discretization.This is directly linked to the accuracy of the results and to the CPU time.

From this review, it appears that a visco-plastic formulation is very often used at thecrystal level with the advantage of avoiding the choice between the multiple solutions ofTaylor’s model. However, at room temperature, the strain rate sensitivity coefficient is verylow, which leads to numerical difficulties and explains the further research dedicated to thestrain rate independent approach (Anand & Kothari 1996, Miehe et al. 1999). Finally bothmodels can be chosen according to their availability.

The use of a common CRSS evolution rule for all the slip systems in a crystal or evenfor all crystallites in a polycrystal at an integration point, seems a logical simplification tolimit the number of state variables. However, a clear information on the accuracy benefitof choosing a distinct CRSS for each slip system seems unavailable.

The effect of the micro-macro links which have been used (FC or RC or modified Taylor’smodel, self-consistent model, homogenisation method) has been the most studied problem.The choice clearly depends on the number of available slip systems, the desired accuracyand the CPU requirement. For b.c.c or f.c.c materials, Taylor’s model already seems togive interesting results from a mechanical point of view (stress, strain) but it also givesqualitative texture prediction with the lowest CPU time.

6 FEM MICRO-MACRO MODELS WITH MACROSCOPIC YIELD LOCUS

6.1 Introduction

In Section 5, the drawback of high CPU time to compute the state of representative crys-tals, then to reach, by an averaging operation, the macroscopic behaviour has often beenmentioned. Scientists have investigated other micro-macro approaches that are less greedyfrom a CPU time point of view.

A first option is to develop new macroscopic elasto-plastic or elasto-visco-plastic modelswith general features imbued from plasticity models in single crystals. Such approaches aredescribed in Section 6.2. (3G model, Aifantis’model).

Another option is presented in Section 6.3. In this case, outside any FEM code, modelsat crystal level and micro-macro links are applied to estimate an accurate expression of theyield locus in polycrystal materials. Then, this accurate yield locus function is used duringmacroscopic FEM computations. The evolution of size and position of this yield locusduring the process is defined by macroscopic isotropic and kinematic hardening rules. Suchhardening models can be macroscopic but with strong links to microscopic phenomena.Even if their accuracy can be quite high (see Teodosiu & Hu 1998 and Miller & Mc Dowel1996), such models generally neglect the “texture hardening”, i.e. the fact that due totexture evolution, the yield locus shape should be updated. In some cases, this phenomenonis really negligible and using an accurate description of the initial yield locus conjugatedwith elaborate hardening models yields a very good accuracy at low CPU cost. The timereduction of such approaches as compared to FEM codes directly coupled with microscopiccalculations is difficult to estimate. However we do not speak of a ratio of 2 or 3 but rather10, 100 and even greater, if no parallel computation is applied.

About the memory requirement, such approaches spare all the variables required atevery integration point to store the orientation and the average CRSS (or even the CRSSassociated to each slip system) of each crystal of the representative set. This explains

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Modelling the Plastic Anisotropy of Metals 57

why, even if some yield locus description and their hardening behaviour need 100 or moreconstants and 50 internal variables, they are still much more economical than the micro-macro approaches presented in Section 5.

Of course these FEM models predict only the mechanical behaviour. If the textureevolution is an interesting result, the strain history during the process must be stored andused in post-processor modules to predict the final texture. Such an approach is proposedby Winters 1996, who simulates a cup drawing using the yield locus described in Section6.3.4. in stress space. Then he uses the strain history to predict the texture evolution andcompares it to texture measurements. His simulation results are quite accurate. Anotherexample is the work by Schoenfeld & Asaro 1996. They study the texture gradients throughthickness in rolled polycrystalline alloys by means of FEM rolling simulations using a phe-nomenological constitutive law. Once the displacement time history of the roll gap hasbeen calculated, Taylor’s model is applied at locations of interest through the thickness ofthe workpiece to predict the final texture and material anisotropy.

6.2 Macroscopic Models Imbued from Single Crystal Plasticity

Such models are no micro-macro models but propose an interesting alternative to phe-nomenological models described in Section 2. For instance, Khan and Cheng (1996, 1998)propose to extend crystal plasticity model to polycrystalline model just by using a largernumber of slip systems. Two other examples are shortly summarized here: 3G and Aifantis’models. Such proposals directly extend the knowledge of crystal plasticity to macroscopicplasticity models. Each one makes some assumptions to provide “simple” useful macro-scopic constitutive laws. Their applications verify that the actual behaviour under complexloading paths can be predicted. It is however clear that each specific path (tension +torsion, pre-straining ...) may need some adjustments of the models.

Such models are neither straightforward to implement in FEM code nor to identify. Soit is not surprising that they generally seem to be used only by the teams that have devel-oped them. They provide more accurate results than simple phenomenological approaches.Let us note that Aifantis’ model also gives a theoretical justification for more advancedphenomenological models such as Karafillis & Boyce 1993.

6.2.1 3G model

Hage Chehade 1990, Monfort et al. 1991, Montfort & Defourny 1993, Montfort & Defourny1994 have developed the so-called “3G model”, which is an anisotropic non associated visco-plastic model. To understand this approach, first consider the plane stress state in a planarisotropic material as described on Figure 47.

The strain tensor associated to this loading state is defined by (82) where G12 is associ-ated to the shear stress τ12 applied on the planes oriented at 45 from principal directions1 and 2.

ε p =

G12 0 00 −G12 00 0 0

ref axes 1,2,3

ε p =

0 G12 0G12 0 00 0 0

ref axes a,b,3

(82)

The extension of this simple state to general cases assumes that plastic strains happenby plastic slips on planes oriented at 45 from principal axes and along directions whereshear stresses are maximal. As in b.c.c metals, 24 families of slip planes exist, one canassume that there is always a crystallographic plane oriented nearly along a direction ofmaximum shear stress. This generalisation is represented by Figure 48 and by relation (83)where axes λ, µ, ν are not orthogonal axes.

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58 A.M. Habraken

Figure 47. Pure shear state in axes a, b, rotated by 45 from the principal stressaxes 1,2 (adapted from Hage Chehade 1990)

12

3

λ

µν

Figure 48. General view of assumed slip planes in general cases (adapted fromHage Chehade 1990)

The plastic strain tensor is expressed by:

ε p =

0 G12 G13

G12 0 G32

G13 G32 0

ref axes λ,µ,ν

(83)

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Modelling the Plastic Anisotropy of Metals 59

where the 3G shear strains, responsible for the model name, appear. For simplicity, relation(83) is written in principal axes 1,2,3:

ε p =

ε11 0 00 ε22 00 0 ε33

=

G12 − G13 0 00 G23 − G12 00 0 G31 − G23

ref axes 1,2,3

(84)

Shear stresses acting on octahedral planes are associated to the 3G shear strains:

τ12 → G12, τ23 → G23, τ13 → G13 (85)

With the previous concepts in mind, the isotropic planar version of the 3G model isdeveloped and applied to a thin sheet where the principal stress σ3 in the thickness directionis low. It assumes that any plastic strain results from the superposition of plastic shearsoriented at 45 from principal stresses directions as represented on Figure 49.

σ2

σ3

σ3

σ2σ1

σ1

Figure 49. Sheet strains decomposed into plastic slips in planes oriented at 45

from principal stresses (from Hage Cheade, 1990)

If the material presents a planar anisotropy, more crystals are oriented in specific di-rections for which a macroscopic deformation is easier or more difficult. In other words,the resistance to deformation varies with the direction in the sheet plane. Extending theisotropic planar approach, macroscopic strain occurs in families of planes presenting themost favourable ratio between the applied shear stress and the intrinsic resistance to defor-mation. In general, such planes are deviated from the 45 directions to principal stresses.The shift angle strongly depends on the Lankford coefficient. It can be mathematicallydetermined as well as its incidence on the total deformation.

As far as the hardening rule is concerned, the 3G model applies a generalization ofBergstrom’s model based on dislocation balance (Bergstrom 1969). Using the 3G as strain

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60 A.M. Habraken

measures allows to extend Bergstrom’s work on a more physical basis for any stress andstrain state. It is demonstrated (Monfort & Defourny, 1994) that the plastic shear stressused in Tresca’s theory can be related to the 3G strain measures:

2τ = A + B[1 − exp(−ar45+β(|G12| + r90+β|G23| + rβ|G31|)] 12 (86)

where A, B and a are material constants, rα is Lankford’s coefficient for a direction at anangle α from the Rolling Direction, β is the angle between the first principal stress and theRolling Direction and τ is the stress limit in Tresca’s criterion.

As the 3G model is based on stresses acting on the maximum shear planes, a Tresca-likecriterion is used, which is consistent with the mechanism taken into account. However theexact shape of the plasticity criterion is of secondary importance, since it only specifiesthe limiting stress level of plasticity initiation. The subsequent plastic behaviour, in termsof stresses and strains is obviously more important with this non associated visco-plasticmodel. It is described by a flow rule developed according physical considerations. Return-ing to the physics, the evolution of the shear strain dGij is due to the creation and thepropagation of mobile dislocations. It is proportional to:

• the time increment dt;

• the rate of mobile dislocation creation∗ itself proportional to the total amount ofimmobile dislocations and to a thermal activation factor;

• the path free of mobile dislocations, which is assumed constant and equal to theaverage dimension of the dislocation cell as in Bergstrom’s model;

• the inverse of Lankford’s coefficient;†

• the probability that a dislocation moves along a given shear plane, which is a functionof the shear stresses.

With its parameters related to fundamental physics, the 3G model potentially has theability to take into account the main macroscopic features of deformation under complexstrain paths. For instance, pre-strain effect, or change in dislocation cell shapes and sizescan be handled with simple modifications. This advantage appears for any model basedon Bergstrom’s extension. The interesting feature of the 3G model is to take directly intoconsideration changes in Lankford’s coefficients; however as their evolution is not providedin the current version of the model, it neglects texture evolution effect.

The planar anisotropic behaviour of the 3G model is validated by theoretical and ex-perimental considerations on the fracture of cylindrical cups (Montfort & Defourny 1994).In conclusion, along simple strain paths, the 3G model gives slightly more accurate resultsthan the classical theory but its main advantage is to give a metallurgical background to themathematical formulation of the material behaviour. It has the basic ingredients to modelthe events happening in complex loading paths but specific improvements are needed topredict texture evolution or pre-strain effects. Its identification is not too heavy and itsFEM implementation is not straightforward but tractable.

∗The rate of mobile dislocation creation is related to a potential function of stresses (Mecking & Lucke1970).

†The texture effect responsible for Lankford’s coefficient value directly modifies the flow rule.

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Modelling the Plastic Anisotropy of Metals 61

6.2.2 Aifantis’ models

Aifantis 1987 is a typical isotropic macroscopic model imbued from single crystal plasticity.It is based on a scale invariance concept, which assumes that the structure of the resultingequations is preserved during a transition from the micro-scale to the macro-scale.

The single crystal model proposed by Aifantis considers a single family of dislocationsmoving along a slip. The vector n is the normal unit to the chosen slip plane and b a unitvector in the slip direction. The microscopic state is represented by two sets of equations:i) the mass and momentum balance of dislocations; ii) constitutive equations linking stressstate and dislocation state. Such a microscopic model helps to predict heterogeneity ofplastic flow such as shear bands or Portevin-Le Chatelier bands (Aifantis 1987).

The microscopic stress quantities σ micro, σ micro − α , α (dislocation stress) are as-sumed to preserve their character and interrelationship during the transition to macro scale.They are respectively identified with the macroscopic total stress σ macro, the macroscopiceffective stress σ effective and the macroscopic back stress α macro. As macroscopic plas-ticity smoothes out plastic micro-heterogeneities, it is assumed that, at macroscopic level,the divergence terms of the microscopic model have no influence and can be dropped. Themacroscopic plasticity theory neglecting volume changes, the climb process of dislocationsis also neglected and just the glide components are kept. The stress tnn⊗n, which accountsfor presence of dislocation dipole and decomposition, can be kept but is often neglected asin Prager’s kinematic hardening rule. Consequently, the macroscopic relations chosen byAifantis are:

ρ = c(ρ, jb, τeffective) (87)

a1b − a2bτeffective − a3bjb = 0 (88)

σ effective = σ macro − α macro (89)

α macro = tAA + tnn ⊗ n (90)

with ρ the dislocation density, jb the glide dislocation flux in the slip direction b , A is anorientation tensor “equivalent” to the symmetric part of the Schmid’s tensor (see relation(15)):

ε p macro = γpA (91)

τeffective is the CRSS transported in a macroscopic scale: projection of σ effective on Atensor, see relation (16). c represents the generation, immobilisation or annihilation ofdislocations. a1b, a2b, a3b are assumed functions of ρ. The coefficient a1b measures thelattice-dislocation interactions and is responsible for yielding. The coefficients a2b, measuresthe effect of Peach-Koehler’s force, while a3b measures the drag associated with dislocationmotion and is responsible for internal damping and viscoplastic flow. The coefficient tAmeasures the interaction forces between dislocations and is responsible for the developmentof back stresses. The macroscopic yield condition is directly deduced from (88).

In Aifantis 1987, the macroscopic A tensor results from the macroscopic principle ofmaximum dissipation for maximum entropy production. This is related to the power asso-ciated with a dislocation motion along its slip plane:

τ effectivejb > 0 ⇐⇒ (σ macro − α macro): ε p macro > 0 (92)

This leads to a maximization problem:

maximum of (σ effectiveij Aij)

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62 A.M. Habraken

with applied constraints:

trA = 0; trA2 =12

(93)

by analogy to the microscopic analysis where A is the symmetric part of Schmid’s tensor.The solution found by Aifantis is:

A =σ effective

2√

12(σ effective)2

(94)

where σ effective is the deviatoric part of stress tensor σ effective. This solution allows toretrieve Prandtl-Reuss’ flow rule and von Mises’ yield criterion:

ε p macro =γp

2√

J2σ effective

√J2 =

a1b

a2b= σF (95)

with J2 = (1/2)σ effectiveij σ effective

ij .Using standard kinematic arguments in conjunction with relation (91), Aifantis obtains,

at the macroscopic level, the plastic spin formulation. His result is equivalent to the ex-pression adopted in usual theories of crystal plasticity for the plastic spin if only one slipsystem is considered (see Figure 23):

Ω p = γpZ (96)

where Z is the non-symmetric part of Schmid’s tensor. By analogy to the microscopic state,the practical macroscopic relation for the plastic spin is derived by eliminating tensor Zfrom (96) thanks to the use of expressions (90) and (91):

Ω p = −t−1n (α macroε p − ε pα macro) (97)

The above relations define Aifantis’ macroscopic model, when a random texture resultsin an isotropic macroscopic behaviour. Its interest is for instance to be able to predictthe development of axial strain due to torsion in free-end cylindrical specimens or thedevelopment of axial stress due to torsion in fixed-end cylindrical specimens (Swift effect).

In cases where the effect of grain orientations cannot be neglected, Ning & Aifantis 1996propose a micro-macro model relying on discrete set of crystals. This approach is closedto the ones presented in Section 5.2., but has the advantage of providing final analyticalmacroscopic relations. The simplified relations (87)-(90), (94), (95), (97) are now assumedto be the constitutive relations for a single crystal. Aifantis’ anisotropic model appliesTaylor’s assumption of equality of the velocity gradient at microscopic and macroscopiclevels. Aifantis introduces an additional “texture spin” Ω t not mentioned in Section 3.2.There the material rotation was only subdivided into a plastic spin Ω p and a crystal latticerotation Ω L. Ning & Aifantis’ texture spin Ω t results from grain boundary constraintsto maintain deformation compatibility. The crystal lattice rotation Ω L is derived as thedifference between the global spin (Ω micro = Ω macro), related to the macroscopic velocitygradient, and two types of plastic spins, the usual plastic spin Ω p due to crystal slip givenby relation (96) and the texture spin Ω t due to grain boundary effects:

Ω L = Ω macro −Ω p − Ω t (98)

Ning & Aifantis define a crystal orientation by a unit vector a oriented in the crystaldirection, expressed as a function of the slip system identifiers n, b and of the angle between

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Modelling the Plastic Anisotropy of Metals 63

a and b. Then, they use the classical concept of Orientation Distribution Function f , whichrepresents the probability for a grain to be oriented along a at time t. The evolution ofthe ODF is prescribed by a conservation rule as already presented in Section 5.3.3, whichcan be solved analytically in the case of an initial random texture and simple deformationgradient tensor.

Ning & Aifantis’ micro-macro link differs from the classical volume average procedureexpressed in its integrated form (78), or in its discrete form (59) or (60). Such a classicaverage procedure does not consider the effect of morphological texture. For instance, thefact that the orientation of large grains may have a more pronounced effect than that ofsmall grains. The following improved average relation is proposed:

xmacro =∮

K(a):xmicrof(a, t) da (99)

where xmicro is either the deviatoric Cauchy stress or the back stress at the crystal level andxmacro is the associated macroscopic value. The fourth order tensor K, called “texture”tensor, is a function of a. Due to the stress symmetry, it is a transversely isotropic tensorfinally defined by only three independent material parameters. The overall plastic flowrule and yield condition are then obtained by combining the flow rule and yield conditionassumed at the crystal level (95) and the average stress relation (99). This results in themacroscopic flow rule:

ε p =γp

2σF (γp)〈K〉−1: σ effective (100)

and the yield condition:√12σ effectif〈K〉−1〈K〉−1σ effectif = σF (γp) (101)

where the overall texture tensor 〈K〉, is defined by the average formula:

〈K〉 =∮

Kf(a, t) da (102)

Relation (100) is similar to the previous phenomenological relations describing yieldbehaviour of metallic materials by the introduction of “modified” stress tensors as, forinstance, in Karafillis & Boyce 1993, presented in Section 2.2.

The evolution of the macroscopic back stress and the plastic spin Ω p are deducedfrom their microscopic value, Taylor’s assumption and the same averaging procedure. Themacroscopic counterpart of the additional texture spin Ω t describes the overall averagegrain rotation due to grain boundary constraints. Thus, it cannot be expressed in termsof the average slip processes alone. The average procedure (99) is slightly modified andapplied now to a vector associated with the skew-symmetric tensor Ω t. The fourth ordertensor K is reduced to a second order tensor, isotropic function of vector a, taking intoaccount 2 material parameters related to plastic strain history and temperature as well asto grain size and shape.

Like Aifantis’isotropic macroscopic model, Aifantis’anisotropic macroscopic model hasbeen applied to the prediction of Swift’s effect. For this simple torsion loading state, itis possible to find an analytical form of the ODF function. Analytical and experimentalresults are very close as shown in Figure 50.

The yield behaviour in tension coupled to torsion has been further studied for a the-oretical material. For instance, Figures 51 and 52 show the evolution of the initial yieldlocus. They respectively correspond to a tension dominated deformation state and a shear

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64 A.M. Habraken

0

-5

-10

-15

-20

-250.0 0.5 1.0 1.5 2.0 2.5 3.0

Shear strain

Axi

al s

tres

s (

MP

a )

Experiment

Theory

Figure 50. Predicted and experimental values for the non-monotonic evolution of axial stressin a copper bar under fixed-end torsion (from Ning & Aifantis 1996)

-2

-1

0

-4 -2 0

Normalized Axial Stress

Nor

mal

ized

She

ar S

tres

s

2 4

1

2

3

4

0.0

0.1

0.5

Figure 51. Evolution of yield surface in tension-torsion for different tensile strains: 0, 0.1,0.5, and a ratio shear/axial strain of 1 (from Ning & Aifantis 1996)

dominated deformation mode. The results show that the texture development causes bothrotation and distortion of the yield surface, the shear mode having a more pronounced ef-fect than the tension mode. A comparison of model predictions with available experimentaldata in small deformations (effect of plastic spin neglected) is presented in Figure 53 for304 stainless steel in tension-torsion.

In short, these macroscopic models imbued from single crystal plasticity can predictmechanical behaviour quite accurately. Ning & Aifantis 1996 analytically compute the ODFevolution in some simple cases. However, the extension of this model to a general velocitygradient seems heavy from both computational and theoretical points of view. The lattermodel has the advantage of giving microscopic fundamental bases for phenomenologicalmodels, which propose the same form for the yield locus function.

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Modelling the Plastic Anisotropy of Metals 65

-2

-1

0

-5 -3 0

Normalized Axial Stress

Nor

mal

ized

She

ar S

tres

s

2 5

1

2

4

0.0

0.1

0.5

431-1-2-4

3

5

6

Figure 52. Evolution of yield surface in tension-torsion for different tensile strains: 0; 0.1; 0.5and a ratio shear/axial strain of 5 (from Ning & Aifantis 1996)

-300

-200

-100

0

Axial Stress (MPa)

Shea

r St

ress

(MPa

)

200

0

100

300

400-200-400

200

400

500

Experim ent ( = 0.0)Experim ent ( = 127 E-6)Experim ent ( = 713 E-6)Theory

γpγpγp

Figure 53. Comparison between theoretical predictions and experimental data forthe yield surface of 304 stainless steel in tension-torsion, γp means hereequivalent plastic strain (from Ning & Aifantis 1996)

6.3 Analytical Yield Loci Computed from Texture Data

Section 6.3.1 presents the classical way to obtain a macroscopic yield locus thanks to clas-sical single crystal plasticity combined with texture description. As this approach does notgive an analytical formulation easy to implement in FEM codes, different ways to derive ananalytic function for the flow surfaces have been proposed. Some are described in Sections6.3.2 to 6.3.4.

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66 A.M. Habraken

6.3.1 Polycrystal yield locus obtained by Taylor’s approach

The average Taylor’s factor (see Section 4.4 relation (50)) appears as a material parameterable to express the response of a textured material submitted to a given strain rate. At thelevel of a polycrystal, the average plastic work dissipation per unit is (see relation (49)):

εp macroeq τcM(U ˙ε

p) = σmacroeq εp macro

eq = σ macro: ε p macro (103)

One can see that, if the macroscopic strain rate ε p macro is known, M(U ˙εp) can be

computed and all stress tensors σ macro that satisfy (103) constitute hyperplanes in stressvectors space. The strain rate vectors ε p macro are perpendicular to these hyperplanes.In practice, as already explained, only the deviatoric stress tensors are considered in theplastic state, so the stress space is a 5 dimensional space. The deviatoric stress σ macro

corresponding to ε p macro is one point of the hyperplane. According to the normality rule,the yield locus must be tangent to the hyperplane at the location of σ macro (see Figure 54).Since the yield locus must be convex, it is the inner envelope of the hyperplanes associatedwith all possible strain modes. The practical way to build 2-dimensional projections of theyield surface is described in Canova et al. 1985, Lequeu et al. 1987a, Van Houtte 1992.Some details on Van Houtte’s approach will be given in Section 6.3.4.

Figure 54. Example of projection of the yield locus onto the subspace σ11−σ332

, σ13

(from Canova et al 1985)

6.3.2 Proposals applying Montheillet’s concepts

Lequeu et al. 1987b follow the approach of Montheillet et al. 1985, where the principal axesof anisotropy are chosen to coincide with the axes of the studied texture component, ratherthan with the symmetry axes of the workpiece. In summary, this approach adjusts ananalytical function on Taylor-Bischop-Hill’s single crystal locus; then, by rotation, adaptsit according to macroscopic axes and uses a Sachs’ approach when the material has morethan one texture component.

Darrieulat & Piot 1996 apply the same type of approach to f.c.c materials characterisedby 12 slip systems. However, they consider a more accurate representation of the micro-scopic behaviour and take into account the ODF function to represent the texture effect.It is described hereafter.

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Modelling the Plastic Anisotropy of Metals 67

Beginning with the plastic behaviour at the single crystal level, Darrieulat & Piot 1996express Schmid’s law by an analytical form obtained through a classical mathematicalproperty of power averages already proposed by Arminjon 1988:

Fp(σ ) =

(12∑

s=1

∣∣∣∣σ :As

τ sc

∣∣∣∣n) 1

n

= 1 (104)

where As is the symmetric part of Schmid’s tensor (15) associated with the slip system s.Function (104) is differentiable, strictly convex and arbitrarily close to the inner envelopeof the hyperplanes of equations:∣∣∣∣ σ :As

τ sc

∣∣∣∣ = 1 s = 1, . . . , 12 (105)

Applying the normality rule to the yield surface, the plastic strain rate is computed by:

ε p = λF1n−1

p

12∑s=1

sgn(σ :As)∣∣∣∣σ :As

∣∣∣∣n−1

As (106)

It is interesting to note that this form is close to the macroscopic flow rule derived whenthe rate sensitive approach is applied at the crystal level (see relation (30)). Figure 55compares yield loci based on expression (104) for different values of n with results fromLequeu et al. 1987a.

Figure 55. For cube orientation, yield stress versus the angle to the Rolling Direc-tion, values predicted by Darrieulat & Piot and by Lequeu (Darrieulat& Piot 1996)

When an orthotropic material is known through its N crystallographic components, re-lation (104) provides a differentiable representation of the mechanical behaviour that can beattributed to each component. The macroscopic behaviour is some average between them.A simple physical assumption close to Khan & Cheng’s 1996 proposal is that polycrystalsbehave like crystals possessing not 12 slip systems but 12×N slip systems, the orientations

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68 A.M. Habraken

of which are given by the texture data. Each of the N sets contributes proportionally toits volume fraction.

Let the texture be defined by an ODF f(g) function of Euler’s angles ϕ1, φ, ϕ2. If Euler’sspace is covered by a step ∆, one value fi of the ODF represents the average intensity of theorientations in the solid angle [ϕ1−∆/2, ϕ1+∆/2]x[φ−∆/2, φ+∆/2]x[ϕ3−∆/2, ϕ3+∆/2].It is also the volume fraction of crystals belonging to this orientation space sector. Thenumber of different crystallographic orientations taken into account N directly depends onthe size of ∆.

Using the same power average as in relation (104), the plastic behaviour of the poly-crystalline material is given by:

Fp =

(N∑

i=1

fi

12∑s=1

∣∣∣∣σ microi :As

i

∣∣∣∣n) 1

n

= τc (107)

where Taylor’s assumption of a common value of CRSS is adopted for all the slip systemsand all the crystals. The subscript i identifies the texture component. This tensorialexpression being the average of strictly convex and differentiable terms, it is also strictlyconvex and differentiable. It can be easily computed no matter what the chosen axes are.

If n = 2, relation (107) is a quadratic criterion similar to Hill’s 1948 yield locus. Dar-rieulat & Piot 1996 give the 6 coefficients of Hill’s 1948 criterion as functions of fi and As.

The macroscopic mechanical behaviour (σ macro, ε p macro) of a polycrystalline materialscan be predicted from each of its component (σ micro

i , ε p microi ) if:

• either a uniform stress (σ microi = σ macro) or a uniform strain rate (ε p micro

i =ε macro) assumption is accepted. In Darrieulat & Piot (1996), the stress uniformity ischosen;

• a condition of homogenisation is applied:

N∑i=1

fiσ i: ε i = σ macro: ε macro (108)

The details are given in Darrieulat & Piot’s 1996 paper. Their conclusions are that pro-posal (107) gives good results for low values of n (up to 10) but assigns a too large influenceto crystallographic texture when n increases. Their validations consist in predictions ofuniaxial yield stresses and Lankford’s coefficients as functions of the angle with the rollingdirection. This is done for a single component texture or real materials such as aluminium3004 or 5182. Note that Darrieulat and Montheillet (2003) just proposed a new version oftheir model restricted to Hill type quadratic assumption.

6.3.3 Arminjon, Bacroix, Imbault ... ’s potential formulation

Arminjon 1988, Arminjon & Bacroix 1991, Arminjon et al. 1994 propose an identificationof the plastic work rates to derive a yield criterion applied to polycrystalline materials.When a quadratic form is assumed, the macroscopic anisotropy parameters become explicitfunctions of the texture coefficients. Their approach is summarized hereafter.

In visco-plastic models, the existence of two dual convex potentials E and Ec is usuallyassumed. They define the relationship giving the stress tensor as function of the plasticstrain rate tensor as well as the inverse relation:

σ =∂E

∂ε p ε p =∂Ec

∂σ(109a,b)

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Modelling the Plastic Anisotropy of Metals 69

where functions E and Ec exchange by Legendre’s transformation. In case of rate-independentstandard plasticity, function E is simply the rate of plastic work W p. The complementaryfunction Ec cannot be defined, since the strain rate ε p is only determined by the stresstensor σ up to the plastic multiplier λ(ε p). The yield locus function and the energy po-tential are equal in associated plasticity model. The flow rule described by relation (1) isused:

ε p = λ(ε p)∂Fp

∂σ(110)

In this section, the yield locus is expressed as:

Fp(σ ) = τc (111)

Following Hill 1987, Fp can be formulated as an homogeneous function of order one withrespect to positive multipliers. This property implies :

Fp(σ ) = σ :∂Fp

∂σ(112)

Relations (110) - (112) allow to give the following expression of the rate of plastic work:

W p = σ : ε p = Fp(σ )λ(ε p) = τcλ(ε p) (113)

Using (109a) and (113), Arminjon writes:

σ =∂W p

∂ε p =∂(τcλ(ε p))

∂ε p = Fp(σ )∂λ(ε p)

∂ε p (114)

Comparing the macroscopic expression of the plastic power dissipation (113) with themicro-macro one (49), presented as an average value, one can find physical interpretationsfor each term. τc is the average CRSS in the polycrystal. λ is naturally expressed from thetexture alone and is characterized by the average Taylor’s factor (relation (50)). The aboverelations assume the equality between the macroscopic strain rate and the microscopic one.So, all the further analytical formulation is based on the so-called Taylor’s hypothesis.

Arminjon & Bacroix 1991 define four requirements that functions λ(ε p) in strain ratespace, or Fp in stress space, have to fulfil:

• convexity,

• respect of orthotropic material symmetry,

• homogeneity,

• use of deviatoric tensors only.

If one chooses a polynomial function, the convexity must be checked. For instance,Arminjon and Bacroix propose a 4th order homogeneous function F of plastic strain rate,which respects the three other characteristics:

F =22∑

k=1

αkXk

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70 A.M. Habraken

X1 = (εp11)

4 X2 = (εp22)

4 X3 = (εp23)

4 X4 = (εp13)

4

X5 = (εp12)

4 X6 = (εp11)

3εp22 X7 = (εp

22)3εp

11 X8 = (εp11)

2(εp22)

2

X9 = (εp11)

2(εp23)

2 X10 = (εp11)

2(εp13)

2 X11 = (εp11)

2(εp12)

2 X12 = (εp22)

2(εp23)

2

X13 = (εp22)

2(εp13)

2 X14 = (εp22)

2(εp12)

2 X15 = (εp23)

2(εp13)

2 X16 = (εp23)

2(εp12)

2

X17 = (εp13)

2(εp12)

2 X18 = εp11ε

p22(ε

p23)

2 X19 = εp11ε

p22(ε

p13)

2 X20 = εp11ε

p22(ε

p12)

2

X21 = εp11ε

p23ε

p13ε

p12 X22 = εp

22εp23ε

p13ε

p12

(115)where symmetry εij = εji is assumed and only i ≤ j terms are present. A scaled functionis used in practice as function λ:

λ(ε p) =W p

τc= F (ε p)/(εp

eq)3 =

22∑k=1

αkΨk(ε p) (116)

with Ψk(ε p) = Xk(ε p)/(εpeq)3.

Note that symmetry σij = σji is not assumed in (114) but taken into account in (116).So the stresses are computed by:

σij = τc∂λ

∂εpij

if i = j

σij =12τc

∂λ

∂εpij

if i < j (117)

Let us explain how the choice (116) leads to represent the coefficients αk by a linearfunction of Cµν

l coefficients of the ODF. Here for simplicity, a single index notation Ci isadopted to identify these Cµν

l coefficients (relation 41):

f(g) ∼=lmax∑l=0

µmax(l)∑µ=1

νmax∑ν=1

Cµνl

˙Tµν

l (g) =⇒ f(g)I∑

i=1

CiTi(g) (118)

where I is the maximum index of Fourier’s coefficients identified by a single index notation.Looking at the above relations (116), (118), (49) and (50), one can express the function λ,after algebraic manipulations, by a linear function of coefficients Ci:

λ(ε p) =W p

τc=

W p

τc=

I∑i=1

CiM∗i (ε p) (119)

where function M ∗i is computed from Taylor’s factor and the harmonic function Ti. The

identification of the material parameters αk, introduced in relation (116), directly resultsfrom the 2 expressions (119) and (116) of function λ(ε p):

αk =I∑

i=1

βikCi k = 1, . . . , 22 (120)

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Modelling the Plastic Anisotropy of Metals 71

Arminjon et al. 1994 have found that, for a given i, the best approximation of coefficientsβi

k is based on a set of 16200 values of ε p and the comparison between:

M ∗∗i (ε p) =

22∑k=1

βikΨk(ε p) and M ∗

i (ε p) (121)

Such computation is done only once with a Taylor’s model and it has been checkedthat for steel texture, I = 12 is enough. This fact means that the 22 coefficients αk arenot independent because relation (119) uses 13 coefficients to describe plastic dissipation(12 coefficients Ci and the value of τc). The above process of parameters identification isindependent of the Ci values and hence independent of texture. This is why it can be doneonly once.

The above explanations give the procedure to start from the choice of a scaled fourthorder polynomial function λ(ε p) and to finally provide an expression of this function pa-rameters as linear relations of coefficients Ci. The same approach can be used if one choosesλ as a second order polynomial function or Hill’s yield locus. The application of this methodwill give a direct computation of 2nd order series or Hill’s coefficient from Ci coefficients.

Figure 56 compare sections of the yield loci computed by different methods for 2 in-dustrial steels I1 (Al killed) and I5 (IF, Ti). The following sections are represented:σ12/τc = 0, 0.5, 1 and 1.5 and, at the center, the value of σ12/τc for σ11 = σ22 = 0 isgiven. The above approach computes an analytical yield locus from the initial texture.Figure 57 for steel I1 (Al killed) shows Lankford’s coefficients and yield loci deduced fromtexture measurements in the initial state and after biaxial tension stopped at two differentlevels of deformation. In this Figure, one observes that the effect of texture evolution can-not always be neglected. Imbault & Arminjon 1993 propose a semi-analytical method totake it into account in their analytical expression of the yield locus. In fact, the principleis simple. They assume that a linear operator, which is adjusted by comparison with apolycrystalline model, can express the evolution with strain of the Cµν

l coefficients. Ofcourse, this linear operator depends on strain rate, which means that numerous strain ratetensors (1800) must be used to establish it. Once the Cµν

l coefficients are known, the aboveidentification process of the analytical expression for the yield locus can be activated.

6.3.4 Van Houtte’s potential formulation

As Arminjon, Van Houtte 1994 uses the method of the dual plastic potentials to deriveconvenient formulae for calculating yield loci of rate insensitive anisotropic materials. Inpractice, the implementation in FEM code of such a yield locus in strain space has beenperformed by one of his Ph.D. student, Van Bael 1994, in collaboration with the universityof Birmingham (Wang et al. 1992). The yield locus in stress space has been implementedin another FEM code by Winters 1996, another Ph.D. student in collaboration with M&Steam (Munhoven et al. 1995a and b). Further developments such as the coupling withTeodosiu’s hardening model (Hiwatashi et al. 1997) and the formulation in strain ratespace in the LAGAMINE code (Hoferlin et al. 1998, Hoferlin et al. 1999b) is described indetails in Hoferlin’s Ph.D. thesis (2001). The hereafter description summarizes Van Houtteand co-workers’ approach.

For rate-independent standard plasticity and formulation in strain rate space, the rela-tion (109a) computes the stress tensor where the potential E is directly given by the rateof plastic work W p macro. In practice, as already suggested by Lequeu et al. 1987a, VanHoutte’s team works in a five dimensional space. The plastic strain rate is classically as-sumed to be deviatoric and only the deviatoric stress tensor matters with regard to plasticdeformation (Van Houtte et al. 1989). So, the tensors have only 5 independent componentsand are replaced by 5-dimensional vectors. This transformation, “tensor Vij - vector vp”,

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72 A.M. Habraken

Hill 1948

Hill 1948

- Taylor- Bishop- Hill

Serie 4Arminjon

22 c

22 c

11 c

- Taylor- Bishop- Hill

Serie 4Arminjon

11 c

Figure 56. Yield locus sections computed by classical Hill’s model, crystallographicTaylor-Bishop-Hill’s model or Arminjon’s 4th order series, for 2 indus-trial steels I1 (Al killed) and I5 (IF, Ti) (from Arminjon et al. 1994)

can be defined in different ways. The version chosen by Winters 1996 assumes, as VanHoutte 1988:

v1 =1√2(V11 − V22) v2 =

√32(V11 + V22) = −

√32V33

v3 =√

2V23 v4 =√

2V31 v5 =√

2V12 (122)

This vector representation has the following property:

V:W = VijWij = vw = vpwp (123)

The vector forms of plastic strain rate and deviatoric stress tensors are respectivelynoted e and s and their components ep and sp. Using the potential relation (109a) Taylor’shypothesis and micro-macro relations (49), (50), the deviatoric stress is computed by:

σ =∂

∂ε p

(∮W p micro(ε p micro, g)f(g) dg

)=

∂ε p (εp macroeq τcM(U ˙ε

p)) (124)

σ = τc∂

∂ε p (εp macroeq M(U ˙ε

p)) (125)

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Modelling the Plastic Anisotropy of Metals 73

2.2

2.1

2

1.8

1.6

1.4

1.2

1.0

0.8

rL

ankf

ord

coef

ficie

nt

0.0 1.5 3.0 4.5 6.0 7.5 9.0

Alpha angle

initialbiaxial = 0.2εbiaxial = 0.52ε

- 4

-3

-2

-1

0.0

1

2

3

4

- 4 - 3 - 2 - 1 0.0 1 2 3 4

biaxial

tension = 0.2ε

1.56

- 4

-3

-2

-1

0.0

1

2

3

4

- 4 - 3 - 2 - 1 0.0 1 2 3 4

1.57

initial

- 4

-3

-2

-1

0.0

1

2

3

4

- 4 - 3 - 2 - 1 0.0 1 2 3 4

biaxial

tension = 0.52ε

1.54

11 c

11 c11 c

22c

22

c

22

c

Figure 57. Lankford’s coefficients and yield loci computed from texture measure-ments for a steel in its initial state and after biaxial tests performed upto 2 different levels (from Imbault & Arminjon 1993)

where U ˙εp = ε p macro/εp macro

eq is a strain mode as defined in relation (23). This is the strainrate space formulation. The relation (125) assumes that the average CRSS is independentof further applied strain rate modes. It helps to understand how Van Houtte and co-workerscompletely dissociate the size (τc) and the shape of the yield locus ∂/∂ε p(εp macro

eq M(U ˙εp)).

Winters 1996 proposes to use an isotropic hardening model. He updates τc via a simpleSwift’s law applied at the macroscopic stress strain level. The micro-macro link used toidentify τc(Γ) is based on the first term of the relation (56) where Γ is the total polycrystalslip. The stress σ and the strain ε are those of a uniaxial test. Neglecting the secondterm of (56) means that the texture evolution is dropped. This will be assumed in furtherdevelopments, except when clearly specified.

Hiwatashi et al. 1997 applies a kinematic hardening assumption and slightly modifies

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74 A.M. Habraken

the relation (125) which becomes:

σ − α = τc∂

∂ε p (εp macroeq M (U ˙ε

p)) (126)

where α is the back-stress and defines the updated center of the yield locus. The evolutionsof α and τc follow the model proposed by Teodosiu & Hu 1998.

The size and position of the yield locus being defined by the choice of the hardeningmodel, what about its shape? Relation (125) clearly shows the role of the texture viathe average Taylor’s factor M (U ˙ε

p). u or up are the vector forms of the strain mode.The average Taylor’s factor can be approximated by an analytical function Q(up) of thecomponents up:

M (U ˙εp) = M (u) ≈ Q(u) = Fp1p2p3...pN

up1up2up3 . . . upN

N = order of series expansion pi = 1, . . . , 5 i = 1, . . . ,N (127)

For instance, if N is reduced to 2, there are 15 coefficients:

Q(u) = F11u1u1 + F12u1u2 + F13u1u3 + F14u1u4 + F15u1u5 + F22u2u2 + . . . F55u5u5 (128)

Van Bael 1994 has extensively described the symmetry properties of such an analyticalexpression, the main drawback of which is its lack of convexity. He explains that an oddorder choice gives a non centro-symmetrical yield locus, which allows to model stress differ-ential effects. It has been checked by Van Bael et al. 1996 and Munhoven et al. 1997 thatthe 6th order is required to reproduce, with such an analytical description, the accuracy ofpolycrystal approaches (Figure 58).

6

4

2

2.5

2.0

1.5

1

0 3 0 6 0 9 0

Lan

kfo

rdco

effi

cien

t

M-

serie

so

rder

Angle to RD (°)α

Taylor - Bishop - Hill

Figure 58. Lankford’s coefficient of a classical interstitial steel computed by a poly-crystal model or by means of 2nd, 4th, 6th order series in strain ratespace (from Munhoven et al. 1997)

This choice of N = 6 leads to 210 coefficients Fp1...p6 . To identify them, it is interesting tonote that any strain mode tensor can be represented by means of 4 independent parametersand not 5 because U ˙ε

p :U ˙εp = 3/2. Van Houtte 1994 demonstrates:

U ˙εp =

U11 U12 U13

U22 U23

SYM U33

U ˙ε

p = RTUX˙εpR (129)

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Modelling the Plastic Anisotropy of Metals 75

with:

R = R(β1, β2, β3) and (β1, β2, β3) ∈ (0, 2π) × (0, π/2) × (0, π) (130)

UX˙εp =

UX11

UX22

UX33

but tr (UX

˙εp) = 0 and UX

˙εp :UX

˙εp =

32

(131)

UX11 = cos

(β4 − π

3

), UX

22 = cos(

β4 +π

3

), UX

33 = − cos(β4) (132)

The 3 Euler’s angles β1, β2, β3 define the orientation of the strain rate mode principaldirections with respect to the sample reference system. The fourth angle gives the deviationof the current strain rate mode with respect to an axisymmetric compression along thethird principal axis. Finally discrete variations of βi, for instance ∆β1 = ∆β2 = ∆β3 = 10and ∆β4 = 7.5 define a discrete set of strain rate modes (around 70300). The averageTaylor factor is computed by the texture and crystalline approach (full Constrained Taylor-Bischop-Hill’s model) for each of these modes. Then, the coefficients of the series expansionare provided by a least square fit of Q function (127).

Relation (125) calculating the stress from the dissipation, is modified to take into ac-count both the vector formulation and the analytical expression of the average Taylor’sfactor:

sp = τc∂(εp

eqQ(u))∂ep

(133)

In FEM, one usually needs the yield locus point corresponding to a given stress direc-tion s∗. So, Legendre’s transformation must be applied. Let ss∗ describe one stress vectorbelonging to the yield locus: the scalar factor is the vector norm √

spsp of the stress point,it is called stress radius, and s∗ is its unit vector direction. Assuming that u is the strainrate mode associated with this stress direction s∗ and using the average Taylor’s factordefinition, one gets:

M (u) = min u′ss∗

τcu′ = minimum in strain rate space (134)

where u′ represents all the possible strain modes of the strain rate space. As u is actuallyunknown, one has to minimize the ratio:

s

τc= min u′

M(u′)u′s∗

(135)

Or with the approximation (127):

s

τc= min u′

Q(u′)u′s∗

(136)

The scaled factor τc clearly shows that the shape does not depend on the size of theyield locus. If a formulation in strain rate space is adopted, Q(u) is known and (136) is usedto perform the minimization. Figure 59 explains this procedure graphically. As reportedabove, the field of strain rate modes can be covered by 4 independent parameters, whichlimits the operation duration. Additionally, Hoferlin et al. 1999a propose a way to speed up

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76 A.M. Habraken

M ( u )1

M ( u )2

s*

u2

u1

σ

/τ22

cσ / τ11

c

Figure 59. Graphical view of the minimization procedure to find a point on theyield locus (adapted from Hoferlin et al. 1999a)

significantly the minimization: with a BFGS type method instead of a classical Newton’smethod and by splitting the minimization process.

As the convexity of the function Q(u) in strain rate space is not perfect, gathering allstress points computed by (136) does not produce a convex yield locus in stress space. Somefishtails appear as demonstrated by Figure 60. This drawing applies to an almost singlef.c.c. crystal texture generated around the Goss’ orientation (gaussian distribution with11 spread around (011)[100]). Left Figure 60 shows the (π-section of one yield locus instrain rate space. For a pure single crystal, linear segments compose this locus. The seriesexpansion reproduces linear segments by oscillating around them. The application of theminimization (136) provides a yield locus in stress space with fishtails: see right Figure 60.

Real materials generally present a less sharp texture; so their yield loci are smootherand the 4th or 6th order series expansions should be convex descriptions. However as nonconvex loci fail to bring convergence in FEM simulations, some secure approach should beimplemented. Hoferlin’s proposal consists in repeating the minimization (136) with differentstarting guesses for u′ and choosing the smallest result s, as the one giving the point on theyield locus. This procedure cuts off the fishtails and prevents convergence problems dueto lack of convexity. Recent publication Van Bael & Van Houtte 2002 proposes anotheranalytical expression of the average Taylor coefficient (127) with the advantage that theconvexity can be strictly imposed.

Another choice can be to implement an analytical yield locus in stress space, computedfrom texture and crystal plasticity approaches. The above presentation explains how tofind points belonging to the yield locus. For a set of directions s∗ in stress space, (135)or (136) provide the stress radius s. These two ways to obtain sets of yield points are notexactly equivalent:

majesus
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Modelling the Plastic Anisotropy of Metals 77

+ +

N = 4

N = 6

22p11

p

33p

1 c

3 c

2 c

p

11ε

P

22ε

P

33ε

τσ /ˆ11

τσ /ˆ22

τσ /ˆ33

Figure 60. π-plane sections of the yield loci expressed in strain rate space (left)and in deviatoric stress space (right) for a f.c.c polycrystal with Goss’texture component (from Hoferlin et al. 1999a)

3

21

451

2

3

1

2

3

ε22p

ε33p

ε11p

3

21

451

2

3

1

2

3

σ τ / 22 c

σ τ / 33 c

σ τ / 22 c

Figure 61. π-plane sections of the yield loci expressed in strain rate space (left)and in deviatoric stress space (right), relative to a polycrystal with cubetexture component and a Gaussian spreading of 16.5 (adapted fromWinters 1996)

• The first way uses relation (135), where M (u′) has no analytical expression and iscomputed for a certain number of strain rate modes. It is similar to the polycrystallinemethod to get yield locus sections (see Section 6.3.1.). By analogy, this method iscalled “geometrical approach”. These yield points generally describe a nearly convexyield locus.

• The second way uses relation (136) and gives a set of points defining an approximateyield locus, not necessarily convex as shown by Figure 61.

In both cases, one has not yet reached an analytical yield formulation of the stress yieldlocus Fp. To reach this goal, one must fit an analytical function on these points. By analogy

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78 A.M. Habraken

to the work in strain rate space, a series expansion is applied. Relations (137) and (138)present two possible choices:

s

τc= Q(s∗) → Fp =

s

Q(s∗)− τc = 0 → u∗

p = λ∂(s/Q(s∗))

∂s∗p(137a,b, c)

s

τc=

1Q(s∗)

→ Fp = sQ(s∗) − τc = 0 → u∗p = λ

∂(sQ(s∗))∂s∗p

(138a,b, c)

where u∗ is chosen as a unit vector. Its relation with the previous identification of the strainmodes in vector form is u∗ = u/

√uu. It defines the direction of the strain rate mode u.

The third relations (137c), (138c) express a weak form of the normality rule in whichthe vector norms are not defined since both s∗ and u∗ are unit vectors. The scalar λ iscomputed to keep the norm of the strain rate mode tensor equal to unity.

In practice, Winters 1996 chooses (138) in order to use all the routines already developedto compute the formulation (133) in strain rate space. His function Q(s∗) is a 6th orderseries expansion, the coefficients of which are fitted to the inverse of the stress radii:

Q(s∗) = Gp1p2...pNs∗p1

s∗p2. . . s∗pN

=τc

s(139)

This set of points τc/s is provided by the first way presented above. So, the yield locuswill exhibit no fishtail. However as illustrated by Figure 61, it must be observed that thisyield locus in stress space is not completely convex. The locus on the left in strain ratespace results from the approximation (127) in strain rate space while the one in stress spaceon the right is computed by (138b) and the first way to get the set of stress points.

Figure 62 summarizes all the required steps to reach the yield locus shape expressed instress or strain rate spaces. Figure 63 explains that if the FEM approach is coupled with,for instance, a FC Taylor model that computes texture updating at each integration pointwith the FEM computed velocity gradient, then the yield locus shape can be updated andtexture evolution can be taken into account during the FEM computation.

Finally, is it better to use a formulation in strain rate or in stress space? The answerdepends on your primary interest: accuracy, low CPU time, necessity of texture updating.Further investigations have computed Lankford’s coefficient: from calculations in stressspace, from calculations in strain rate space and from Taylor-Bishop Hill’s model. Bothworks from Winters 1996 and Van Bael et al. 1996 reach the same conclusion that theyield locus in stress space is less accurate than the formulation in strain rate space. Table 4summarizes the advantages and drawbacks of each approach.

One point is completely missing in this description: the numerical way to identifythe position of material axes. The solution proposed by Munhoven et al. 1995a can beapplied: each step is characterized by a constant local velocity gradient, which determinesthe evolution of material axes. Another explanation of the same mathematical approachis proposed by Hoferlin et al. 1999b, using Ponthot’s 1995 constant co-rotational strainrate tensor. Another possibility is to determine the material axes position by the Mandelspin (Peeters et al. 2001), which is the average of the spins of all the crystal lattices of thepolycrystal. An efficient way to compute Mandel spins for all possible strain modes hasbeen proposed by Van Houtte 2001.

6.3.5 Discussion about micro-macro approaches with yield locus

Sections 6.3.2 to 6.3.4 propose models able to describe the behaviour of real materials. Theycan be implemented in FEM and each one presents some advantages and drawbacks. Dar-rieulat’s micro-macro approach seems more accurate than Lequeu’s one. However, for real

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Modelling the Plastic Anisotropy of Metals 79

Coefficients describing the shape of the

yield locus in stress space

Initial texture measurements

Identification of µνl

C coefficients of the Orientation Distribution

Computation of average Taylor’s factor )( PUεM for a set of strain rates modes

Minimization of Taylor’s factor to

find the stress radii *)(ss for a set of

stress directions *s

Fitting of a 6th

order series on the

inverse of the stress radii

Fitting of a 6th order series on the

average Taylor’s factor

Coefficients describing the shape of the yield locus in strain rate space

Figure 62. Flowchart to reach the shape description of the yield locus in stress orstrain rate space

Stress space sτc

= 1Q(s∗) Strain-rate space W p = τcε

peqQ(u)

Speed of FEM Faster slower 4-dim. MinimizationTexture Slower Fasterevolution texture → M → texture → M → Q

4-dim. minimization → QSharp textures one stress direction one stress direction

→ one stress point → fishtails (extra cost)Accuracy Lower Higher

Table 4. Comparison between stress and strain rate formulations for analytical yield lociimplemented in FEM codes (adapted from Hoferlin et al. 1999a)

materials where important number of texture components are present, the FEM computa-tions must be quite lengthy as their approach finally considers the yield locus associated toeach texture component. Uniform stress approaches are not straightforward to implementin usual FEM integration scheme. The proposals by Arminjon, Van Houtte and co-authorsseem quite interesting. The identification of yield locus coefficients from texture coefficientscan be optimized outside the FEM code. Taylor’s assumption leads to a direct macroscopicstress-strain formulation. Arminjon’s semi-analytical method to take texture updating into

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80 A.M. Habraken

ODF of initial texture

Discretization technique

SlipsystemsCRSS…

Discrete set of orientationswith associated volume fraction

updating

Texture prediction

Macroscopievelocity gradient

Foreach

InterpolationPoint

Micro - Macrotransition

Yield locusshape in stress

space

F E Mcode

GeometryForces

Hardening…

data

Figure 63. Complete flowchart for coupling yield locus and crystallographic texture

account is probably difficult to use for arbitrary velocity gradient. The approach proposedby Hoferlin (yield locus in strain rate space) seems more adapted in a FEM context toupdate the material state with the texture prediction during FEM simulations (Li et al.2003). However, the CPU time is still important, as working in the strain rate space isslower than in stress space.

7 A FEM MICRO-MACRO MODEL WITH LOCAL DESCRIPTION OF MA-CROSCOPIC YIELD LOCUS

7.1 Model Description

This section summarizes the method developed by Habraken and co-workers. It aims totake into account texture updating during the FEM computation in an efficient way. Thismethod relies on a set of representative crystal orientations defined at each integrationpoint of the FE mesh but most of the time no microscopic computations are performed,an approximated local zone of the yield locus is used. So this approach states between themodels described in Sections 5.2 and 6.3.

This local description of the yield locus is assumed sufficient as the stress state generally

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Modelling the Plastic Anisotropy of Metals 81

remains in a local zone of the yield locus for few increments of a FEM computation. Inthe deviatoric stress space, five or six points in the interesting part of the yield locus arecomputed by the full constraint Taylor’s model applied on a set of representative crystals.This is done at each integration point of the FE mesh. No global yield locus is defined.Just a direct interpolation between these points is achieved. These microscopic points mustbe computed in two cases:

• when, at the current integration point, the plastic strains significantly deform thematerial and induce changes in the crystallographic orientations (texture updating);

• when, during a loading increment, the new stress state at the current integrationpoint leaves the described part of the yield locus so that an updated local zone ofthe yield locus is required. In most cases, the definition of this updated local zoneonly requires the calculation of one new point in the vicinity of the points alreadyavailable.

Compared to micro-macro models without macroscopic yield locus, which systematicallycall the microscopic model, the present approach proposes an important computation timedecrease as five or six calls to the microscopic model are only required from time to time.Compared to micro-macro models with macroscopic yield locus, these five or six calls replacethe thousands calls required to update the full yield locus description. Hoferlin’s thesis2001 and Van Houtte 2001 define the exact amount of runs of the Taylor’s program intheir approach. As verified by Imbault & Arminjon 1993, important effects on yield locusanisotropy are induced by texture updating, so the interest of the proposed method is clear.

All the numerical details of this approach can be found in Duchene 2000 or in Habraken& Duchene (in press). In fact two local descriptions have been tested. The most simpleone “Hyperplane approach” was a set of five pieces of hyperplane, each one containing onecentral point and four from the five points describing the local zone of the yield locus. Thislocal description of the yield locus suffers of strong discontinuities as shown on Figure 64 andintroduces convergence problems in FEM simulations. The second approach “Interpolationapproach” extends the interpolation concept of isoparametric triangle finite element to thefive-dimensions deviatoric stress space and provides a more continuous and stable approach.For a SPXI steel sheet, Figure 64 shows π-sections of the yield locus computed by fivedifferent methods:

• 6th order = global yield locus in stress space described by a 6th order series from VanHoutte (see Section 6.3.4).

• Hyp. 1 or 20 = Hyperplane approach with an angle of 1 or 20 between the strainrate directions used as domain limit vectors.

• S-s I 1 or 20 = Stress-strain rate Interpolation approach with an angle of 1 or 20between the strain rate directions used as domain limit vectors.

For the local approaches with large domains (Hyp. 20 and S-s I 20), the associatedyield locus normals, which represent the deviatoric plastic strain rates, are also plotted.Normals from the hyperplane approach are clearly discontinuous. When small local domainsare used (1), their associated yield loci are superposed and very close to the yield locuscomputed by the 6th order series. The latter locus is validated and close to Taylor-Bishop-Hill locus. For larger domain size (20), the results of the hyperplane and interpolationmethods strongly differ: while the interpolation approach is continuous and close to theyield loci computed by the 6th order series, the hyperplane result is discontinuous anddiverges from the 6th order series result.

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82 A.M. Habraken

Norm. (s-s I 20°)

Norm. (hyp. 20°)

S-s I 1°

Hyp. 1°

S-s I 20°

Hyp. 20°

6th orderS3

S1 S2

Figure 64. π-sections of the yield locus and normals to the yield loci computed bystress-strain rate interpolation and hyperplane method with a size of20 (from Duchene et al. 1999c)

7.2 Deep-Drawing Simulations

In order to show up the influence of the texture evolution during a forming process, thedeep drawing of a cylindrical cup with a hemispherical punch has been simulated. Itis one of the benchmarks proposed by NUMISHEET 1999. The geometry is defined inFigure 65. The drawing ratio is 2.0; the blankholder force is 80 kN; the simulation isachieved up to a drawing depth of 85 mm. The proposed material “DDQ” (mild steel) hasbeen chosen. A Coulomb law is used to model the friction with a coefficient of 0.15. All thematerial parameters such has hardening behaviour and Lankford coefficients were measuredby simple tensile tests in different directions and available for benchmark’s participants (seeNumisheet 1999). As we focus on the texture, the shape of the yield locus is deduced fromits Orientation Distribution Function (ODF), which has been measured by X-ray diffractionby Nakamachi, who provided us the C coefficients describing the ODF of this mild steel.LAGAMINE FEM code developed by M & S department has been used with two layers ofsolid 3D-mixed type finite elements BLZ3D (Zhu & Cescotto 1996) and contact elementsCFI3D (Habraken & Cescotto 1994).The simulation results computed with the stress-straininterpolation method with and without texture updating are compared to a classical Hill1948 constitutive law and to 2 experimental results (4 experimental results were availablebut for Figure readability we kept only 2 defining the experimental range). Figure 66 definesdraw-in, curvilinear abscissa and sections where results are compared.

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Modelling the Plastic Anisotropy of Metals 83

Figure 65. Geometry of the chosen benchmark of Numisheet 1999

Draw-in:Distance betweenblank boundary line andpart boundary line in a section 1)

Rolling DirectionRD

45°

45°B: Draw-in at 45°

A: Draw-in at 90°

C: Draw-in at 0°

Cup

Blank

B

C

A

Figure 66. Definitions of sections A, B, C, curvilinear abscissa s and draw-in (fromNUMISHEET 1999)

Figure 67 shows the maximum principal strain distribution on a section along the trans-verse direction for Hill and texture based laws and for experimental results. It should benoticed that the strain is relatively small at the top of the cup (near the pole; for s smallerthan 50 mm) while large displacements take place. Indeed, in that part of the cup, the steelsheet is applied against the punch and follows the punch travel without large deformation.Then, at the flange of the cup, the maximum principal strain suddenly increases and ismaximum for s being equal to 90 mm. This maximum is located at the vertical part of theflange where the cup is free of contact with the punch and the matrix. After the maximum,the principal strain ε1 decreases as the contact with the matrix reduces the tension of thesheet. The resulting maximum principal strain obtained with Hill law is a little bit too highwhile texture based laws are in agreement with experimental results.

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84 A.M. Habraken

Strain Distribution ε1 along Transverse Direction

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140s: distance from pole (mm)

ε1

Constant TextureEvolving TextureHillExperimentalExperimental

Figure 67. Maximum principal strain distribution ε1 along section A (transversesection)

Strain Distribution ε2 along Transverse Direction

-0.55

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0 20 40 60 80 100 120 140

s: distance from pole (mm)

ε2

Constant Texture

Evolving Texture

Hill

Experimental

Experimental

Figure 68. Second principal strain distribution ε2 along section A (transverse section)

Figure 68 shows the second (smallest) principal strain along the same section. Hereagain, the same evolution can be noticed: low constant value near the pole, followed bya minimum (or a maximum in absolute value) in the flange of the cup and then lowerdeformations under the blankholder. Looking ε1 and ε2 together, we can see that fourtypical regions can be identified:

• Near the pole (s < 40 mm), an equi-biaxial tension state or stretched zone is present(ε1 ≈ ε2),

• In the flange (40 < s < 90), increasing tensile and compression strains can be noticed;a restrained zone is determined (ε1 > 0, ε2 < 0 and ‖ε1‖ > ‖ε2‖),

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Modelling the Plastic Anisotropy of Metals 85

Strain Distribution ε3 along Transverse Direction

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 20 40 60 80 100 120 140

s: distance from pole (mm)

ε3

Constant TextureEvolving TextureHillExperimentalExperimental

Figure 69. Thickness strain distribution ε3 along section A (transverse section)

• Near the matrix curvature (90 < s < 110), the restrained zone presents a decrease inthe absolute value of the strains (ε1 > 0, ε2 < 0 and ‖ε1‖ > ‖ε2‖),

• Under the blankholder (110 < s < 135), the compression strain ε2 becomes largerthan the tensile strain ε1(ε1 > 0, ε2 < 0 and ‖ε1‖ < ‖ε2‖). This strain state wouldgive rise to instability and wrinkling without the action of the blankholder.

Then Figure 69 shows the third principal strain distribution along section A (transversedirection). It corresponds to the thickness strains. Near the pole, in the equi-biaxial tensionstrain zone, ε3 is negative: corresponding to a thickness reduction. Then progressively, ε3

grows and becomes positive (the thickness is increasing during the process) in the fourthregion described here above.

The punch force as a function of the punch travel is presented on Figure 70. Thesecurves are not linked to the anisotropy of the steel sheet but to the global stiffness of thematerial and then to the hardening behaviour. It can be noticed that the curve with theevolving texture is very close to the experimental curve; the constant texture is a little bittoo low at the end of the process and Hill is too high.

Finally, Figure 71 is directly linked to the anisotropy of the steel sheet. For finiteelement simulations, this anisotropy is introduced in the constitutive law (either the Hillcoefficients or the texture data). The Flange Draw-In is defined as the length of the flangethat is swallowed under the blankholder (see Figure 66). It is the opposite of the earingprofile. The experimental results exhibit a maximum draw-in along section B (45) andlower values along sections A and C. The Hill based constitutive law shows a good draw-inprofile (maximum at section B) but the amplitude is too high (variation between the threesections). The draw-in profiles obtained by the texture based laws are very similar to theexperimental one. The anisotropy is then well represented by this constitutive law but ashift is observed along both directions: the draw-in is too low with these texture basedlaws. The behaviour of the material is less ductile numerically than experimentally. Theamplitude of the draw-in profile is a little bit higher when the texture is updated duringthe process.

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86 A.M. Habraken

Punch force as a function of punch travel

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80Punch travel (mm)

Pun

ch fo

rce

(kN

)

Constant TextureEvolving TextureHillExperimentalExperimental

Figure 70. Punch force as function of punch displacement

Flange Draw-In at sections A, B and C

0

5

10

15

20

25

30

35

40

A : 90° B : 45° C : 0°Section : angle from Rolling Direction

Dra

w-I

n (m

m)

Constant Texture

Evolving Texture

Hill

Experimental

Experimental

Figure 71. Flange draw-in at sections A,B,C

8 CONCLUSION

This overview work provides an idea of the main models implemented today in FEM codesto represent macroscopic plastic anisotropic material behaviour. Concerning micro-macroapproach with yield loci, it is focused on the initial yield locus shape and on geometric ortextural hardening. The latter describes the yield locus updating in shape, position and sizedue to the effect of crystallites rotation. Clearly no review of the material hardening, relatedto dislocation density is provided. As dislocations constitute obstacles to the productionand motion of further dislocations, this phenomenon also induces shape, position and size ofthe yield locus (Bouvier et al. 2002, Tedodosiu, Peeters 2002, Lopes et al. 2003). Completestate of the art review of material hardening is another story, let us just underline that

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Modelling the Plastic Anisotropy of Metals 87

proposals based on microscopic events like the ones from Bergstrom-van Liempt-Vegter(Vegter’s et al. 1999a), Follansbee & Kocks 1988, Teodosiu & Hu 1998 are interestingalternatives to classical isotropic and kinematic hardening models. Another key point inFEM simulations not addressed here is the choice of orthotropic material axes evolution inthe FEM description. Peeters et al. 2001 summarizes clearly various choices.

Some care has been brought to provide not only the theoretical models, but their iden-tification methods as well. Whenever possible, links between models are presented, such as,for instance, Aifantis’s proposal that gives physical basis to von Mises and Karafillis’ laws.

In the future, hardware development and parallel computation will reduce problemsof CPU time. However, each time that this happens, engineers increase the sizes of theproblems that they want to solve by finite elements ... So, simple phenomenological lawsdescribed in Section 2 that allow escaping to microscopic computations in the macroscopicFEM simulations, retain their interest. This also forces researchers to identify the importantfeatures necessary to capture material behaviour. For instance, it is clear that textureevolution effects on yield locus are not necessary in all deep-drawing simulations.

The actual question is: what is really useful to take into account? The answer is not thesame according to the goal of the simulations: shape prediction after spring back, textureprediction, residual stress field, wrinkling and necking prediction. It is clear that criteriaexist to predict necking and wrinkling, but they rely on accurate stress and strain fieldcomputations. The spring back prediction is quite hard if your model neglects elasticity.The final shape and size of the yield locus after forming processes are important if your goalis to apply accurate fatigue models to predict the life of the pieces. The model descriptionsprovided in this review should help to choose the adapted model to fit one’s requirements.

One direction not investigated in this overview is the formulation of the Finite Elementitself. Going from a simple displacement formulation to a mixed or hybrid formulation canalready provide a better convergence and a smoother stress answer, even if a low number ofcrystals is used per integration point (Beaudoin et al. 1995). Another possibility is to applya simple macroscopic analysis coupled with a micro-macro analysis only where some event,such as strain localization, appears and requires a finer scale. For instance, Garikipati &Hughes (2000) propose such a so-called variational multiscale approach.

ACKNOWLEDGEMENT

A. M. Habraken is mandated by the National Fund for Scientific Research (Belgium). Shealso thanks the Belgian Federal Science Policy Office (Contract P5/08) for its financialsupport.

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