Micro-mechanical modeling of the inelastic behavior of ...

Mechanics of Materials 38 (2006) 203–217

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Micro-mechanical modeling of the inelastic behaviorof directionally solidified materials

K. Sai a, G. Cailletaud b,*, S. Forest b

a LGPMM, Ecole Nationale d’Ingenieurs de Sfax, BP W 3038, Tunisiab Centre des Materiaux/UMR 7633, Ecole des Mines de Paris/CNRS, BP 87, 91003 Evry Cedex, France

Received 20 April 2004; received in revised form 7 March 2005

Abstract

Micro-mechanically based constitutivemodels representing inelastic behavior of directionally solidified (DS)materialsare proposed. A finite element (FE) computation used to calibrate themodel is first presented. It is comparedwith amodelusing a static homogenization scheme (uniform stress throughout the material). These approaches lead to significantlydifferent results. A newmodel is then proposed with a suitable scale transition rule from the macroscopic level to the grainlevel. Satisfactory results are obtained for this second model, at the macroscopic level as well as for local responses.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Single crystal; Directionally solidified; b-Model; Inelastic behavior; Crystal plasticity

1. Introduction

Directionally solidified (DS) materials aresometimes a good compromise between too expen-sive single crystals and less resistant polycrystalsfor structures like turbine blades operating at hightemperatures. Nevertheless, despite the importanceof these materials, little has been done to developconstitutive models able to describe their overall

0167-6636/$ - see front matter � 2005 Elsevier Ltd. All rights reservdoi:10.1016/j.mechmat.2005.06.007

* Corresponding author. Tel.: +33 1 60 76 30 56; fax: +33 160 76 31 50.

E-mail address: [email protected] (G. Caille-taud).

mechanical properties. DS materials are knownfor their excellent tensile ductility and their highstrength at room temperature as well as at hightemperature. The morphology of DirectionallySolidified materials is such that the grains arepurely columnar (Fig. 1). The morphology exhibitsa perfect order in the direction of solidification(the x3 direction coincides with a h001i directionof each crystal), and a perfect random characterin the normal plane. This must be reflected bythe scale transition relation in the framework ofa micro–macro modeling. There is a growing inter-est for this type of application, but the state of theart of the modeling activity concerns only elastic

ed.

mailto:[email protected]

Fig. 1. Columnar morphology of DS grains.

204 K. Sai et al. / Mechanics of Materials 38 (2006) 203–217

properties (Yaguchi and Busso, 2005). The pur-pose of the present paper is to propose a strategyfor describing this scale transition, using finite ele-ment computation to build a reference for the DSaggregate. The local behavior in each grain is sup-posed to be properly represented by a phenomeno-logical constitutive equation involving nonlinearisotropic and kinematic hardenings. One of theopen question is to check if a mean field model isable to predict the global response of the material,even if the description of the intragranular hetero-geneities is missing. It should be mentioned thatsuch an attempt was successful in a previous studyon a cubic isotropic aggregate (Barbe et al., 2001).

The simplest and most widely used model topredict polycrystal behavior is the Taylor model(Taylor, 1938), where uniform plastic strain in eachgrain is assumed. Following the self-consistentplasticity model proposed by Hill (1965) manyauthors revisited the concept of self-consistency(Molinari, 1999) in order to improve polycrystalplasticity models. The aim of this work is to pro-vide a polycrystalline model with a suitable transi-tion rule able to describe DS behavior, having inview target materials like nickel-base directionallysolidified superalloys. The model is assessed bymeans of a finite element (FE) computation inwhich each grain is represented by a group of ele-ments with a columnar morphology. The overallbehavior of the polycrystalline model is comparedto the response of a FE model using volume aver-ages of stress and strain. The paper is organized

in the following manner: The main lines of consti-tutive equations for the single crystal model arerecalled briefly in the next section. In Section 3 apolycrystalline aggregate model is generated, andcomputed by the FE technique. The results of thissection will provide a database to assess an homo-genized polycrystalline model. In that section somestatistical analyses of the strain heterogeneity dis-tribution are shown in order to provide a deeperunderstanding of the local behavior of DS materi-als. Section 4 is devoted to the comparison betweenthe responses of (i) a ‘‘super single crystal’’ model,(ii) the FE model of the polycrystalline aggregate.In that section an analytical calculation of the yieldsurface of the ‘‘super single crystal’’ model is pre-sented. In Section 5, a polycrystalline model withan efficient so-called stress concentration proce-dure is presented. The parameters present in thetransition rule are identified from the macroscopicstress–strain curves of the FE model. To assess thereliability of the model, a comparison of local re-sponses is drawn between the polycrystalline modeland the FE model. For that purpose, stress–straincurves are compared grain per grain. After check-ing the response of the proposed model at macro-scopic and microscopic levels, a study of theinfluence of the grain position is performed. Anevaluation of the quality of the transition rule withrespect to intergranular heterogeneity is also made.

2. Constitutive equations for the single crystal

This section briefly presents the elastoviscoplas-tic single crystal model used in this work. It hasbeen developed in the framework of crystal plastic-ity theory (Meric et al., 1991), and widely used,especially for nickel-base superalloys. It is pre-sented here in its small perturbation version, whichuses an additive decomposition of the elastic andthe viscoplastic strain rates. The so-called resolvedshear stress ss acting on a particular slip system (s)is given by the relation:

ss ¼ rg : ms ð1Þwhere rg is the stress tensor in the crystal and ms isthe orientation tensor attributed to the slip system(s):

K. Sai et al. / Mechanics of Materials 38 (2006) 203–217 205

ms ¼ 1

2ðl s � ns þ ns � lsÞ ð2Þ

ns and l s being the ‘‘slip plane’’ normal vector andthe ‘‘slip direction’’ vector in this plane, respec-tively. The resolved shear stress ss can be relatedto corresponding shear rate _cs via a power lawexpression:

_cs ¼ jss xsj rs

k

� �n

signðss xsÞ ð3Þ

For each slip system, internal variables are intro-duced to describe the hardening of the material:isotropic hardening variables rs and kinematichardening variables xs. Viscoplastic flow reachesa rate-independent limit for large values of theparameters n or 1/k. The nonlinear evolution rulefor isotropic hardening involves an interaction ma-trix [H] which represents self-hardening (diagonalterms) and latent hardening (non-diagonal terms).In the present work, the components H rs are cho-sen equal to drs (Kronecker delta, self-hardeningonly), since this was shown to be a reasonableapproximation for nickel-base superalloys (Mericet al., 1991).

rs ¼ r0 þ QXr

H rsð1 ebvrÞ with _vr ¼ j _crj: ð4Þ

The following form of nonlinear kinematichardening is adopted:

_xs ¼ c _cs _vs dxs ð5Þwhere c and d are material parameters. For thecase of FCC materials, plastic deformation in thecrystal is the result of slip processes according to12 octahedral slip systems:

_epg ¼X12s¼1

ms _cs ð6Þ

Fig. 2. Mesh of the FE aggregate model.

Table 1Material parameters

Elasticity Isotropic hardening Kinematichardening

E (MPa) m r0 (MPa) Q (MPa) b c (MPa) d

196,000 0.3 111 35 7 1600 40

3. Finite element model

3.1. Computational tools and boundary conditions

The aim of this section is to provide an evalua-tion of the stress and strain fields in a DS aggre-gate. For that purpose, an aggregate model isgenerated and computed by FE technique (Caille-

taud et al., 2003). A cube formed by columnargrains is discretised as a regular triangle meshand the corresponding crystal orientation is attrib-uted to each Gauss point. To generate the meshcorresponding to the microstructure with arbitraryshapes, a 2D Voronoi polyhedra model (Gilbert,1962) has been used. The mesh includes 39,21015-nodes prisms with a triangular section and aquadratic nodal interpolation (Fig. 2). The FCCcrystal behavior of Section 2 is used with the mate-rial parameters shown in Table 1. Different tensileand shear simulations are performed with thefollowing boundary conditions:

• For shear tests, a displacement vector ui = eijrj isimposed on the contour (all the nodes of the sixfaces), where e is a symmetric tensor and~r is theposition of the considered nodes. The tensorcomponents are:


(e11 = 0, e22 = 0, e33 = 0, e12 = 1%, e23 = 0,e13 = 0) for shear tests in the plane 12,(e11 = 0, e22 = 0, e33 = 0, e12 = 0, e23 = 0,e13 = 1%) for the shear tests in the plane 13.

• For the tensile tests, four lateral faces are free,whereas an axial displacement is imposed onthe two remaining faces.

For the tensile tests in direction 11: u1 = 0 at the
nodes of the face (x1 = 0) and u1 = he11 at thenodes of the opposite face, such that e11 = 0.3%.For the tensile tests in direction 22: u2 = 0 at thenodes of the face (x2 = 0) and u2 = he22 at thenodes of the opposite face, such that e22 = 0.3%.For the tensile tests in direction 33: u3 = 0 atthe nodes of the bottom (x3 = 0) and u3 =he33 at the nodes of the top, such that e33 =0.5%.
Two strategies are tested to generate the setof grain orientations. The first one considers 64distinct orientations, each of them being gener-ated by rotation around the last Euler angle u.From now on, this model will be called FE-64-64. In this notation, the first number specifies thenumber of grains whereas the second number spec-ifies the number of distinct orientations. The sec-ond aggregate model considered by FE techniqueis called FE-64-8. In this aggregate, 8 differentorientations are considered. Each orientation isattributed to 8 column-like randomly distributedgrains.

3.2. Results of the FE computations

Fig. 3 shows typical contour plots for a tensiletest in direction 11 and for a shear test in the plane13. The von Mises contour plots show a rathersmall variation of stress field. For this reason, itis reasonable to begin with a static model in whicha uniform stress throughout the material is consid-ered. Three kinds of results can be deduced fromthe FE computations:

(i) The stress–strain curves produced by the glo-bal volume average. These results can beobtained by either FE-64-64 or FE-64-8simulations.

(ii) The volume average on each individual grainwhich can be obtained by FE-64-64 and FE-64-8 simulations.

(iii) The volume average in each phase (i.e. eachset of grains belonging to the same crystallo-graphic orientation class). These results areprovided by the FE-64-8 computations.

3.3. Statistical analysis

The FE analyses provide also local stress andstrain values inside the grains. These values areimportant for subsequent damage and lifetimeassessment. The stress and strain data generatedusing the FE simulations are analyzed using statis-tical techniques in order to study their distributioncharacteristics in a way similar to (Sarma andDawson, 1995). The FE aggregate of FE-64-8model is subjected to shear loading in the plane13 with four levels of average strain as summarizedin Table 2.

The deformation data generated using the finiteelement simulations where analysed using statisti-cal techniques in order to study their distributioncharacteristics. Fig. 4 shows the histogram of thee13 component in the volume. A Gaussian distribu-tion function is also plotted as a reference. Thisdemonstrates that the computed distribution isgenerally not Gaussian. A frequency histogram(Fig. 5) is also plotted for each average strain leveland for each last Euler angle u. The abscissa ofeach histogram represents the local strain level atGauss points, and the ordinate the frequency func-tion. These histograms have led to the followingremarks:

• As shown in Fig. 5a, most of the histograms arebell-shaped curves. Indeed they are almost sym-metric with respect to their mean value. How-ever the peak is always higher than the peakof the respective normal distribution. In suchcases, the kurtosis parameter is used to measurethe degree of peakedness of a distribution. Thekurtosis factor of a distribution is defined as:

k ¼ Eððx lÞ4Þr4

ð7Þ

Fig. 3. Contour plot in the FE aggregate: (a) tensile test 11, von Mises stress (MPa), (b) shear test 13, von Mises stress (MPa),(c) tensile test 11, cumulative inelastic strain (%), (d) shear test 13, cumulative inelastic strain (%).

Table 2Values of the average strain chosen for performing statisticalanalyses

he13i 0.195% 0.413% 0.588% 0.755%


where r is the standard deviation, l is the meanof x and E(t) denotes the expected value of t.Note that the kurtosis of a normal distribution

is 3. Fig. 5b gives for different loading levelskurtosis as function of the angle u. It appearsthat they are non-normal distributions.

• Each distribution is characterized by its meanvalue and its standard deviation. The meanvalue is almost equal to the corresponding aver-age strain. Fig. 5c shows that the spread (or thedispersion) increases with the loading level. In

0

0.1

0.2

0.3

0.4

0.5

0.24 0.45 0.67 0.89 1.10 1.32 1.54 1.75

ε13 (%)

Freq

uenc

y

Fig. 4. Histogram and comparison with Gaussian distribution.


the present case, the values of the standard devi-ations are respectively equal to 0.1335, 0.379,0.629 and 0.78.

0

0.1

0.2

0.3

0.4

0.203

0.351

0.499

0.647

0.794

0.942

1.090

1.238

ε13 (%)

Freq

uenc

y

FE-64-8<ε13> = 0.755 %ϕ = 0 ˚

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5Local shear strain 13 (%)

Freq

uenc

y

0.195%0.413%0.588%0.755%

<ε13>

FE-64-8ϕ = 30 ˚

(a)

(c)

Fig. 5. Statistical analysis for the strain controlled shear test 13. (a) Diof overall strain. (b) Kurtosis factors for different angles and differentwith a fixed angle. (d) Example of multimodal histograms for differen

• For some particular angles, the shape of the his-togram indicates a multimodal distribution forlow as well as for high average strain levels(Fig. 5d).

4. Super single crystal/static model

4.1. Constitutive equations

As a first approximation, it is considered herethat internal stresses remain low even during visco-plastic flow. This idea is also supported by the factthat all the grains have already a common axis.Using this simple static assumption and uniformelasticity, a uniform stress field is postulated inthe whole aggregate, and the stress concentration

2

4

6

8

10

12

14

0 20 40 60Angle ϕ (˚)

Kur

tosi

s fa

ctor

0.195%0.413%0.588%0.755%

<ε13>FE-64-8

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2Local shear strain 13 (%)

Freq

uenc

y

0.195%0.413%0.588%0.755%

<ε13>

FE-64-8ϕ = 70 ˚

(b)

(d)

stribution of local shear strain with fixed angle and fixed averageloading levels. (c) Spread evolution of the sample with loadingt loading levels with a fixed angle.

Table 3Transformation of stress tensors

r1 ¼r11cos

2u þ r22sin2u ðr11 r22Þ sinu cosu 0

ðr11 r22Þ sinu cosu r11sin2u þ r22cos

2u 00 0 0

0@

1A

r2 ¼r11cos

2u r11 sinu cosu 0r11 sinu cosu r11sin

2u 00 0 r33

0@

1A

r3 ¼0 0 r13 cosu0 0 r13 sinu

r13 cosu r13 sinu r33

0@

1A

r4 ¼r11cos

2ur12 sin2u 0:5r11 sin2uþr12 cos2u 00:5r11 sin2uþr12 cos2u r11sin

2uþr12 sin2u 00 0 0

0@

1A


procedure is reduced to the equality between themacroscopic stress and the stress in the grain:

rg ¼ r ð8ÞThe resulting aggregate is like a ‘‘super single crys-tal’’, or a single crystal having a collection of all theslip systems present in all the grains, since the samestress level is applied to each of them. Once the con-stitutive behavior of a single crystal is specified, onehas to find the overall behavior of an aggregateconsisting of columnar oriented grains. The plasticstrain of the representative volume element (RVE)is the volume average over all grains:

_ep ¼ _epg ð9ÞThe additive strain decomposition and Hooke�s

law give:

e ¼ ee þ ep; r ¼ C : ee ð10Þwhere C is the fourth-rank tensor of elasticmoduli.

4.2. Analytic calculation of the DS yield surface

Plotting the yield surface is a first solution tohave an estimation of the anisotropy of the DSaggregate. In this section, a perfect DS material isconsidered: All grains have a h001i axis parallelto direction x3. Let us call it [001]. A uniform distri-bution is then assumed for the crystal axis [100] ofthe grains. All the axes h001i are the same, they arein the x3 direction of the aggregate referential, sothat axes h100i for instance can take any directionin the x1–x2 plane. The resulting yield surfaces willlook like Tresca�s criterion in a subspace involvingx1 and x2 axes, but will keep a single crystal charac-ter if the axis x3 is involved. As an illustration, com-putations are made for a FCC material presentingoctahedral slip only. The initial resolved shearstress is 100 MPa. (R0) is the crystal frame [100][010] [001], and (R) is the coordinate frame associ-ated with the representative volume element. Thetransformation between (R0) and (R) is limited toa single rotation of angle u around the axis x3.

Four different stress tensors are considered, inorder to illustrate the yield surface for biaxial ten-sion and tension-shear for directions involving x3or not:

r1 ¼

r11 0 0

0 r22 0

0 0 0

0BB@

1CCA; r2 ¼

r11 0 0

0 0 0

0 0 r33

0BB@

1CCA;

r3 ¼

0 0 r13

0 0 0

r13 0 r33

0BB@

1CCA; r4 ¼

r11 r12 0

r12 0 0

0 0 0

0BB@

1CCA

The components of the four tensors in the crystalframe R0 are given in Table 3. Accordingly, the12 resolved stresses of the FCC slip systems are ex-pressed in Tables 4–7. The generic form of theseresolved stresses corresponds to a straight line:

sg ¼ AðuÞr1 þ BðuÞr2 ð11Þwhere r1 and r2 are the non-zero components ofthe considered stress tensor. The inner envelope(which characterizes the yield surface) of this fam-ily of straight lines can be obtained (Fig. 6) byintroducing a second family of straight lines ex-pressed in polar form:

r1 ¼ qðhÞ cos h and r2 ¼ qðhÞ sin h ð12ÞThe polar radius q is obtained by replacing r1

and r2 of Eq. (12) in Eq. (11):

qðh;uÞ ¼ sg

AðuÞ cos h þ BðuÞ sin hð13Þ

The yield surface is obtained when the polarradius is minimum. Accordingly, the partial deriv-ative of q(h,u) with respect to utreating h asconstant must be equal to zero:

Fig. 6. Methodology used to calculate analytically yieldsurfaces of the static model.

Table 4Resolved shear stresses for the slip systems in the space r11r22ffiffiffi6

ps1 ¼ ðcos2u sinu cosuÞr11 ðsin2u sinu cosuÞr11ffiffiffi6

ps2 ¼ ðsin2u sinu cosuÞr11 ðcos2u sinu cosuÞr22ffiffiffi6

ps3 ¼ cos 2uðr22 r22Þffiffiffi6

ps4 ¼ ðcos2u þ sinu cosuÞr11 ðsin2u þ sinu cosuÞr22ffiffiffi6

ps5 ¼ ðsin2u þ sinu cosuÞr11 ðcos2u þ sinu cosuÞr22ffiffiffi6


ps7 ¼ ðsin2u þ sinu cosuÞr11 ðcos2u þ sinu cosuÞr22ffiffiffi6

ps8 ¼ cos 2u ðr22 r11Þffiffiffi6

ps9 ¼ ðcos2u þ sinu cosuÞr11 ðsin2u þ sinu cosuÞr22ffiffiffi6


ps11 ¼ ðcos2u þ sinu cosuÞr11 þ ðsin2u sinu cosuÞr22ffiffiffi6

ps12 ¼ ðsin2u þ sinu cosuÞr11 þ ðcos2u sinu cosuÞr22


ps1 ¼ ðcos2u sinu cosuÞr11 þ r33ffiffiffi6

ps2 ¼ ðsin2u sinu cosuÞr11 þ r33ffiffiffi6

ps3 ¼ r11 cos 2uffiffiffi6

ps4 ¼ ðcos2u þ sinu cosuÞr11 þ r33ffiffiffi6

ps5 ¼ ðsin2u þ sinu cosuÞr11 þ r33ffiffiffi6


ps7 ¼ ðsin2u þ sinu cosuÞr11 þ r33ffiffiffi6


ps9 ¼ ðcos2u þ sinu cosuÞr11 þ r33ffiffiffi6


ps11 ¼ ðcos2u þ sinu cosuÞr11 r33ffiffiffi6

ps12 ¼ ðsin2u þ sinu cosuÞr11 r33


ps1 ¼ r13 sinu þ r33ffiffiffi6

ps2 ¼ r13 cosu þ r33ffiffiffi6

ps3 ¼

ffiffiffi2

pr13 sinðu p=4Þffiffiffi

6p

s4 ¼ r13 sinu þ r33ffiffiffi6

ps5 ¼ r13 cosu þ r33ffiffiffi6

ps6 ¼

ffiffiffi2

pr13 cosðu p=4Þffiffiffi

6p

s7 ¼ r13 cosu þ r33ffiffiffi6

ps3 ¼

ffiffiffi2

pr13 cosðu p=4Þffiffiffi

6p

s9 ¼ r13 sinu þ r33ffiffiffi6

ps10 ¼

ffiffiffi2

pr13 cosðu þ p=4Þffiffiffi

6p

s11 ¼ r13 sinu r33ffiffiffi6

ps12 ¼ r13 cosu r33


ps1 ¼ ðcos2u sin 2u=2Þr11 þ ðsin 2u þ cos 2uÞr12ffiffiffi6

ps2 ¼ ðsin2u sin 2u=2Þr11 ðsin 2u þ cos 2uÞr12ffiffiffi6

ps3 ¼ r11 cos 2u þ 2r12 sin 2uffiffiffi6

ps4 ¼ ðcos2u þ sin 2u=2Þr11 þ ðsin 2u þ cos 2uÞr12ffiffiffi6

ps5 ¼ ðsin2u þ sin 2u=2Þr11 ðsin 2u cos 2uÞr12ffiffiffi6

ps6 ¼ r11 cos 2u 2r12 sin 2uffiffiffi6

ps7 ¼ ðsin2u þ sin 2u=2Þr11 þ ðcos 2u sin 2uÞr12ffiffiffi6


ps9 ¼ ðcos2u þ sin 2u=2Þr11 þ ðsin 2u þ cos 2uÞr12ffiffiffi6


ps11 ¼ ðcos2u þ sin 2u=2Þr11 ðsin 2u cos 2uÞr12ffiffiffi6

ps12 ¼ ðsin2u þ sin 2u=2Þr11 þ ðsin 2u þ cos 2uÞr12


oqou

¼ 0 hence A0ðuÞ cos h þ B0ðuÞ sin h ¼ 0 ð14Þ

This last equation is then substituted into Eq. (13),and different forms of yield surface functions arethen obtained (smooth for r4 and presentingcorners for the other tensors). All the curves

(Fig. 7a–d) are plotted with a resolved shear stresssc = 100 MPa.

For the stress tensor r1, the yield surface isdelimited by the six straight lines:

ðffiffiffi2

pþ 1Þr11 ð

ffiffiffi2

p 1Þr22 ¼ �2

ffiffiffi6

psc ð15Þ

ðffiffiffi2

p 1Þr11 ð

ffiffiffi2

pþ 1Þr22 ¼ �2

ffiffiffi6

psc ð16Þ

r11 r22 ¼ �ffiffiffi6

psc ð17Þ


ðffiffiffi2

pþ 1Þr11 þ 2r33 ¼ �2

ffiffiffi6

psc ð18Þ

ðffiffiffi2

p 1Þr11 þ 2r33 ¼ �2

ffiffiffi6

psc ð19Þ

r11 ¼ �ffiffiffi6

psc ð20Þ

-300

-100

100

300

-300 -100 100 300

σ11 (MPa)

σ 22

(MPa

)

(2,12)(1,4,5,7)

(1,4,5,7)

(2,12)

(3,6,8,10)

(3,6,8,10)

-300

-200

-100

0

100

200

300

-300 -100 100 300

σ13 (MPa)

σ 33

(MPa

)

(3,6,8,10) (3,6,8,10)

(1,2,4,5,7,9,11,12)

-300

-200

-100

0

100

200

300

-300 -100 100 300

σ11 (MPa)

σ 33

(MPa

)

(1,4,9,11)

(1,4,9,11)(2,5,7,12)

(2,5,7,12)(3,6,8,10)

(3,6,8,10)

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250

σ11 (MPa)

σ 12

(MPa

)(1,2,4,5,7,9,11,12)(3,6,8,10)

(a) (b)

(c) (d)

Fig. 7. Analytical yield surface for the static model: (a) space r11r22, (b) space r11r33, (c) space r13r33, (d) space r12r11.



r33 þ r13 ¼ �ffiffiffi6

psc ð21Þ

r33 r13 ¼ �ffiffiffi6

psc ð22Þ

r13 ¼ �ffiffiffi3

psc ð23Þ

From Fig. 7c, one can easily show that the poly-gonal area inside the yield surface in space r13r33

is almost equal to 10:97s2c . For the stress tensorr4, the yield surface is delimited by the curves:

q ¼ �2ffiffiffi6

psc

cos h þffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3sin2h

p ð24Þ

q ¼ �ffiffiffi6

pscffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 3sin2hp ð25Þ

with

r11 ¼ q cos h and r12 ¼ q sin h ð26ÞIt can be shown that the intersection of the two

last curves is obtained for an angle hi ¼Arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4

ffiffiffi2

p 5Þ=ð19 12

ffiffiffi2

pÞ

qwhich is approxi-

matively equal to 35�. Note that this value is inde-pendent of the critical resolved shear stress.Accordingly, the value of the area inside the yieldsurface in space r11r12 can be calculated by inte-gration of two domains from 0 to hi and from hito p

2. This area is almost equal to 8:15s2c . Hence,

the area inside the yield surface in space anr11r12 is smaller then the area inside the yield sur-face in space r13r33. Indeed, the equivalent stressin space r11r12 is always perpendicular to theh001i axes. As a result, the possibility of findingan active system is larger than in the space r13r33.


4.3. Comparison with FE model

The aim of this section is to assess the quality ofthe static model. The polycrystalline model usingthe simplest transition rule of Section 4.1 (uniformstress assumption, Eq. (8)) is considered. Thiscorresponds to a representative volume element(RVE) computation in which the aggregate is se-lected so that the orientation distribution is thesame as in FE-64-64. The same tensile and sheartests are performed with the static model and thencompared to the FE-64-64 responses. Fig. 8ashows no difference for tensile test in 33 direction,whereas small differences are observed for 11 and22 directions. However, a significant deviationis observed for shear test (Fig. 8b), especially for12 test, meanwhile either 12 and 13 shear levelsare underestimated by the static model. This dis-crepancy is just a remainder that a model mustbe checked for all types of loadings, and demon-strates that using a super single crystal approachis not a valid assumption for a general purposeDS material model. It is then necessary to intro-duce an appropriate transition rule in an improvedmodel. This is the subject of the following section.

5. The b-model

In the previous section, it was shown that thestatic assumption may fail to describe correctlythe DS behavior. This is mainly due to the pres-

0

100

200

300

0 0.1 0.2 0.3 0.4 0.5Axial Strain (%)

Axi

al S

tres

s (M

Pa)

FE-64-64, direction 33

Static Model, direction 33

FE-64-64, directions 11 and 22

Static Model, directions 11 and 22

(a)

Fig. 8. Fitting on uniaxial strain controlled test. Comparison betweentests, (b) shear tests.

ence of non-uniform stresses in the different grainsin the case of shear test and tensile tests in direc-tions 11 and 22. The material presenting a randomcharacter in the plane normal to x3, the self-consis-tent scheme is a good candidate to represent theintergranular interaction. Instead of using the fullself-consistent formalism, what is presented here isan explicit transition rule, which requires no inte-gral differential equation to be solved for charac-terizing the scale transition. The model involves aset of parameters (only one for isotropic behavior)that can be calibrated before use by means of FEcomputations on representative cells.

5.1. Constitutive equations

The initial idea of the model is to start fromKroner�s approach of the elastic accommodationproblem which uses Eshelby�s solution (Eshelby,1957) for an ellipsoidal inclusion in an infinitemedium. The corrective term proposed by Kronerto compute the local stresses is the product of afourth order tensor by the difference between aver-age strain and local strain (Eq. (27)). The fourthorder tensor involves the elastic moduli tensor C,the fourth order unit tensor I and Eshelby�s tensorS (Sanchez-Palencia and Zaoui, 1987). In the pres-ent model (Eq. (28)), these tensors are still used,but the local strain is replaced by a phenomenolog-ical variable bg, that presents a nonlinear evolutionwith respect to plastic strain (Eq. (29)). Replacingtotal strain by b

assumes that the main source of

0

100

200

300

0 0.2 0.4 0.6 0.8 1

Shear Strain (%)

Shea

r St

ress

(M

Pa)





(b)

the static model and the aggregate FE-64-64 model: (a) tensile


heterogeneity is plasticity more than elasticity.This new variable was shown able to correctly cap-ture the plastic accommodation which comes fromthe self-consistent formalism (Cailletaud, 1987;Cailletaud and Pilvin, 1994; Pilvin, 1996).

rg ¼ r þ C : ½ðI SÞ : ðe egÞ� ð27Þrg ¼ r þ C : ½ðI SÞ : ðb bgÞ� ð28Þ_bg ¼ _epg D : bg

Xs2g

j _csj ð29Þ

Since the grains are elongated in one direction, thechosen S tensor is Eshelby�s tensor for a cylindricalinclusion, obtained from the elliptic solution byassuming that two dimensions are equal (a1 = a2)and that the third one tends to infinity (a3 ! 1)(Mura, 1987).

In the material axes, the tensor D has the fol-lowing form, with six independent materialparameters:

D ¼

D11 D12 D23 0 0 0

D12 D11 D23 0 0 0

D23 D23 D23 0 0 0

0 0 0 D44 0 0

0 0 0 0 D55 0

0 0 0 0 0 D55

0BBBBBBBB@

1CCCCCCCCA

ð30Þ

Using the deviatoric property of the accommoda-tion variable trace _b

g ¼ 0 the following additionalcondition is enforced:

D33 ¼ D11 þ D12 D23 ð31Þ

0

100

200

300

0 0.1 0.2 0.3 0.4 0.5

Axial Strain (%)

Axi

al S

tres

s (M

Pa)


FE-64-64, directions 11 and 22

β-Model, directions 11 and 22

β-Model, direction 33

(a)

Fig. 9. Fitting on uniaxial strain controlled test. Comparison between(b) shear tests.

On the other hand, the form of S is thefollowing:

S ¼

S1111 S1122 S1133 0 0 0

S2211 S2222 S2233 0 0 0

0 0 0 0 0 0

0 0 0 S1212 0 0

0 0 0 0 S1313 0

0 0 0 0 0 S2323

0BBBBBBBBBB@

1CCCCCCCCCCA

ð32Þ

This Eshelby tensor S corresponds to the planestrain hypothesis. This must be complemented bythe following condition which is enforced for eachcrystal orientation:

eg33 ¼ e33 ð33Þ

After some algebraic manipulations, a conditionbetween the components of the local and globalstress tensors is derived:

rg33 ¼ r33 þ mðrg

11 r11Þ þ ðrg22 r22Þ ð34Þ

5.2. b-model–FE identification

The FE-64-64 computation is taken as a refer-ence to calibrate the scale transition rule of theb-model. The material parameters describing crys-tal behavior are the same in both computations

0

100

200

300

0 0.2 0.4 0.6 0.8 1

Shear Strain (%)

Shea

r St

ress

(M

Pa)





(b)

the b-model and the aggregate FE-64-64 model: (a) tensile tests,

0

100

200

300

0 0.1 0.2 0.3Strain (%)

Stre

ss (

MPa

)


FE-64-64, direction 11β-Model, direction 11


-350

-250

-150

-50

50

150

250

350

-1 -0.5 0 0.5 1

ε13 (%)

σ13 (MPa)

FE-64-64

β Model

(a)

(b)

Fig. 10. Validation of the identification procedure: comparisonbetween the b-model and the aggregate FE-64-64 model: (a)tensile biaxial test, (b) shear strain controlled cyclic loading.


(FE and b-model). The only parameters to deter-mine are those introduced in the matrix D definedin Eq. (30). The global response of the RVE de-scribed by FE-64-64 is computed by averagingthe stress and strain component on the wholemesh. The database generated with FE computa-tions includes tensile tests (Fig. 9a) and shear tests(Fig. 9b). The scale transition parameters in matrixD are identified by comparing the loading curvescoming from FE and b-model integration. Asshown by the figures, the responses obtained withthe two approaches are in good agreement with thechosen set of coefficients (D11 = 1000, D12 = 75,D23 = 100, D44 = 10, D55 = 100).

5.3. Model validation

Fig. 8a shows a comparison between the b-model and the FE-64-64 aggregate model for abiaxial test not included in the database of theidentification procedure. The two models are ingood agreement. On the other hand, the b-modelcan be used in principle for non-monotonic load-ing conditions. It is then necessary to compare itsresponse for a cyclic loading and that of the FEaggregate model. Fig. 8b shows the comparisonfor a test in which a symmetric shear strain isimposed. This test was not included in the identi-fication procedure and is thus a validation of themodel. This remarkable agreement shows the kine-matic character of intergranular stresses (Fig. 10).

After having validated the b-model on a macro-scale, it is necessary to assess its reliability at thegrain level (level 2). Note that a different strategywould consist in solving the inverse problem madeof the global response and the local responsestaken together to identify the material parameters.The present approach is more simple, and isproved to provide an acceptable solution. Two ori-entations are selected for that purpose u = 20� andu = 50�. Fig. 11a–c show that the b-model is ingood agreement with FE calculations for tensiletests as well as for shear tests.

An other mean to make the assessment at themicroscopic level, for a given macroscopic strainlevel, is to compare the mean stress per orientationpredicted by the b-model and the FE calculation.In Fig. 11d the X-coordinate of a given point

(corresponding to a given orientation) is thelocal stress obtained by the b-model whereas theY-coordinate is the local stress given by the FEsimulation. It can be seen that the group of pointsis close to the first bissectrice.

5.4. Intergranular heterogeneity

Fig. 12a shows for two selected orientation,u = 20� and u = 60�, the influence of the locationof the grain. It can be seen that these variations areweak. So that global and local responses of FE-64-64 simulations and FE-64-8 simulations are almostidentical.

0

50

100

150

200

250

300

0 0.1 0.2 0.3

ε11 (%)

σ 11

(MPa

)σ 1

3 (M

Pa)

x FE-64-64, ϕ=20˚- δ Model, ϕ=50˚o FE-64-64, ϕ=20˚- δ Model, ϕ=50˚

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1

ε13 (%)

x FE-64-64, ϕ=20˚- β Model, ϕ=50˚o FE-64-64, ϕ=20˚- β Model, ϕ=50˚

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1

ε12 (%)

σ 12

(MPa

)

x FE-64-64, ϕ=20˚- β Model, ϕ=50˚o FE-64-64, ϕ=20˚- β Model, ϕ=50˚

Average strain : <ε13>=0.8%

200

220

240

260

280

300

200 220 240 260 280 300

σ13 (MPa), β-Model

σ 13

(MPa

), F

E-6

4-64

(a) (b)

(c) (d)

Fig. 11. Comparison between the b-model and the FE aggregate model, local stress–strain curves in uniaxial strain controlled tests:(a) tensile test 11, (b) shear test 12, (c) shear test 13, (d) local shear stress in uniaxial strain controlled shear test 13. b-model vs FE-64-64model.


After checking the response of the b-model onmacroscopic model on both macroscopic andmicroscopic stress–strain curves, it is also impor-tant to evaluate the quality of the transition rulewith respect to intergranular heterogeneity. Thisis made by comparing the average response ofthe elements having the same orientation in theFE computation with the corresponding localresponse for the same crystallographic phase inthe b-model. Note that due to the additionalcondition (34) the intergranular strain heteroge-neities vanish for the 33 direction. The heteroge-neities are more significant for shear tests thanfor tensile tests (Fig. 12b–d). Furthermore, inthe yield surfaces presented in Section 4.2 it wasshown that when the stress components arecontained in a plane perpendicular to the h001iaxes, there is a large amount of relative orienta-

tions between the crystals and the loading frame.This is why the 12 shear test exhibits a more sig-nificant intergranular heterogeneity than 13 sheartest.

6. Conclusions

The purpose of the paper was to test the capa-bilities of a micro-mechanical mean field modelto represent mechanical behaviour of DS materi-als. A FE model in which grains are representedby a group of elements with columnar morphologyis used to generate a material database. A numberof loading cases are considered to account for theorthotropic character of the resulting unit cell. Theresults of the FE computations provide informa-tion at three levels:

0

100

200

0 0.2 0.4 0.6 0.8 1

ε12 (%)

σ 12

(MPa

)

grains with angle ϕ=20˚grains with angle ϕ=60˚

0

100

200

300

0 0.5 1 1.5 2

ε12 (%)

σ 12

(MPa

)

0

100

200

300

0 0.1 0.2 0.3 0.4

ε11 (%)

σ 11

(MPa

)

0

100

200

300

400

0 0.5 1 1.5

ε13 (%)

σ 13

(MPa

)

(a) (b)

(c) (d)

Fig. 12. (a) FE study of the influence of the location of the grain (FE-64-8). Uniaxial strain controlled shear test 12. Intergranularheterogeneity in b-model. (b) Strain controlled tensile test 11. (c) Strain controlled shear test 12. (d) Strain controlled shear test 13.


• Macroscopic level, using the volume averageover all Gauss points.

• Phase scale, using the volume average over theGauss points of the elements having the sameorientation.

• Intragranular level, which concerns the individ-ual Gauss points.

An attempt is first made with a static model, butit is demonstrated that such an approach gives sat-isfactory results only for a tension along the growthaxis, and fails to represent any type of shear load-ing. A transition rule is then necessary to accountfor residual stresses on the phase level. A b-rule isused for that purpose. It is successfully calibratedon the whole numerical data base, using the macro-scopic responses. A detailed analysis is then shown,in order to evaluate the scatter to be expected onthe other scales. The b-rule correctly represents

the stress and strain range on the level of the phase.Further developments have to be made to capturean evaluation of the intragranular scatter.

It is worth noting that this good predictioncapability can be obtained without capturing theintragranular heterogeneities. This has to be re-lated to the fact that the global response is the re-sult of a series of averages, which are acceptablewith the present microstructure (texture and slipsystem). It does not demonstrate that the resultis general, nevertheless, it has already be validatedin similar conditions on a FCC aggregate (Barbeet al., 2001). The b-model is implemented in theFE element code Zebulon (Besson et al., 1998).In the present state, it is possible to use the b-modelas the constitutive equations in the FE method toanalyse inelastic behaviours of machine compo-nents made of DS material submitted to multi-axial thermomechanical loadings.


References

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Pilvin, P., 1996. The contribution of micromechanicalapproaches to the modelling of inelastic behaviour ofpolycrystals. In: Andre, P., Georges, C., Trevor, L. (Eds.),Fourth Int. Conf. on Biaxial/Multiaxial Fatigue andDesign, ESIS, 21. Mechanical Engineering Publications,London.

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