Computational Materials Science 96 (2015) 44–55

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

A multi-scale computational model using Generalized Methodof Cells (GMC) homogenization for multi-phase single crystal metals

http://dx.doi.org/10.1016/j.commatsci.2014.08.0450927-0256/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +1 315 268 4429.E-mail address: aachutha@clarkson.edu (A. Achuthan).

M. Ghorbani Moghaddam a, A. Achuthan a,⇑, B.A. Bednarcyk b, S.M. Arnold b, E.J. Pineda b

a Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, United Statesb NASA Glenn Research Center, OH 44135, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 March 2014Received in revised form 9 August 2014Accepted 29 August 2014

Keywords:Multi-scale computational modelMulti-phase metalsNi-based super alloysGeneralized Method of CellsHomogenizationCrystal plasticity constitutive model

A multi-scale computational model for determining the elastic–plastic behavior of a multi-phase metal isdeveloped on a finite element analysis (FEA) framework. A single crystal plasticity constitutive modelthat can capture the shear deformation and the associated stress field on the slip planes is employedat the microstructural (grain) length scale. The Generalized Method of Cells (GMC) micromechanicsmodel is used for homogenizing the local field quantities. At first, the ability of GMC for homogenizationis evaluated by analyzing simple problems using GMC as a stand-alone tool. A repeating unit cell (RUC) ofa two-phase CMSX-4 Ni-based superalloy with 72.9% volume fraction of c0 inclusion in the c matrix phaseis used for the evaluation. The evaluation is performed by comparing the results with those predicted by aFEA model incorporating the same crystal plasticity constitutive model. The average global stress–strainbehavior predicted by GMC demonstrated excellent agreement with FEA. The agreement between thelocal distribution of the field quantities predicted by GMC and FEA was satisfactory, especially when con-sidering the substantial savings in the computational cost due to homogenization. Finally, the capabilityof the developed multi-scale model, linking FEA and GMC, to solve real life sized structures is demon-strated by analyzing an engine disk component and determining the microstructural scale details ofthe field quantities of the two-phase CMSX-4 Ni-based superalloy.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Single crystal materials such as Nickel-based super alloys (e.g.,ME3) are widely used in engine blades and disks due to their excep-tional strength, fracture toughness, and resistance against creepand corrosion at elevated temperature. Nickel-based super alloysprimarily contain a two-phase microstructure; c0 inclusion phaseand c matrix phase. The c0-inclusion phase, a coherent, ordered Ni3-

Al inter-metallic of L12 crystal structure, can be considered asinclusions distributed in the c matrix phase, which is predomi-nantly an alloy of Ni, Cr and other alloying elements with a face-centered cubic (FCC) lattice structure [1,2]. The c0 inclusion, dueto the L12 microstructure, exhibits very low plastic behavior whencompared to the c matrix phase. As a result, the c0 inclusions act asobstacles to the motion of dislocations in the c matrix [1–3]. Conse-quently, the morphology and the distribution of c0 inclusions insidec matrix influence the deformation, resulting in excellent mechan-ical properties. Therefore, the volume fraction of inclusions and

their mean size and spacing have a major effect on the behaviorof these alloys under different loading and temperature conditions.

Constitutive models that can capture the interaction of variousmicrostructural features, such as different material phases, areneeded to determine the effect of these features on the plasticbehavior of multi-phase metals. The two key characteristics associ-ated with the plasticity are the permanent strain due to dislocationmotion under resolved shear stress and the strain hardening. Incrystal plasticity models, the dislocation motion is commonlyincorporated as a permanent shear strain in the various slip sys-tems (slip plane and slip direction) implemented with the aid ofa set of state variables [4,5]. The strain hardening is often imple-mented in terms of the strength of slip systems using another setof state variables. The nonlinear evolution laws are defined forthe permanent shear strains and the strength of slip systems, alongwith the equilibrium equations, in an incremental and iterativeapproach.

A major challenge in applying microstructure based plasticitymodels to real sized structural components is the computationalcost as this typically requires the solution of a system with anenormous number of degrees of freedom. A potential solution foraddressing this challenge is the careful implementation of an

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 45

appropriate and efficient homogenization technique to bridge themicroscale and the macroscale deformation without significantlycompromising on the accuracy of the solution, especially in thecritical regions of interest.

Various multi-scale methods were developed to address thischallenge [6,7]. These methods can be broadly classified into threemajor groups namely, sequential (or hierarchical), computationalhomogenization (so called FE2) and integrated.

In the sequential method, the macro model utilizes a number ofparameters which are obtained from a detailed microscale experi-mental analysis or are pre-computed [8–10]. As a result, thesemodels are macroscopic models in principle, but use internal statevariables for describing the material behavior at microscale. Hence,sequential methods are computationally very efficient, but lacks infidelity since detailed microstructural interactions, such as interac-tion between various phases, are not captured in these models. Thecomputational homogenization scheme, on the other hand, resultsin high fidelity models, accurately relating the complex materialbehavior at microscale to its macroscale response [11–14]. In thesemethods, the local governing behavior at the macroscale is deter-mined by solving the microscale boundary value problem on a rep-resentative volume element. This micro–macro transition can alsobe implemented by considering coupled boundary value problemsat the microscale and the macroscale, and solving them simulta-neously by means of finite element methods (FEA). The computa-tional homogenization scheme, though offering high fidelity, canstill be computationally expensive for many practical problems.The integrated method is based on applying a homogenizationtechnique to an embedded representative volume element in aFEA framework. The homogenization technique, most commonlybased on mean-field theories, serves as the transition step betweenthe microscale and macroscale [15–17]. The microstructure is dis-cretized and split into different phases with analytical relationsdescribing the interaction between the phases. While providingrelatively good accuracy, depending on the mean-field theory used,these methods are computationally very efficient. Recent studieshave demonstrated successful implementation of viscoplasticself-consistent schemes both under explicit [15,16] and implicitschemes [17].

Several studies on developing crystal plasticity based modelsfor the two-phase superalloys with constitutive models account-ing for shear flow and hardening rates have been reported inthe literature [18–36]. In many of these studies, constitutive mod-els were implemented for a homogeneous material volume usingeffective properties [18–25]. Such constitutive model, though easyto implement in FEA model, is limited in its ability to capture theeffect of the interaction of microstructural phases accurately. Inother studies, where different phases were explicitly consideredin order to capture the interaction between individual phases inmicroscale, the c matrix was considered to be elastoplastic, whilethe c0 inclusion is modeled as linear elastic [26–36]. The multi-scale models have also been developed for multi-phase materialslike Ni-based super alloys, adopting various approaches [30–36].This includes a Fourier series homogenization method [30,31],and a microstructure based method [32–34]. In some studies, con-stitutive models that account for strain-gradient effects in thevicinity of phase boundaries were implemented in a multi-scaleframework by deriving analytical expressions for response of anRUC [35,36].

The Generalized Method of Cells (GMC) [37–42] is a homogeni-zation method which easily links the macroscale to the microstruc-ture, in an integrated fashion, while considering the individualmicrostructural phases explicitly. GMC is a semi-analytical tech-nique which considers periodicity in the microstructure with arepeating unit cell (RUC). This method enables modeling ofcomplex unit-cell architectures, and is capable of studying the

elastic–plastic response of periodic heterogeneous materials undervarious loading. In GMC, an RUC is discretized into a number ofsub-volumes, referred to as subcells. Within each of these subcells,the displacement fields are approximated linearly. Traction anddisplacement continuity conditions are utilized to calculate astrain concentration matrix allowing localization of the applied,average, far-field strains to local subcell strains. GMC does notinclude coupling between shear and normal components of stress.Advanced methods such as High-Fidelity Generalized Method ofCells (HFGMC) [42] were proposed to account for the shear cou-pling through quadratic displacement approximations in the sub-cells. However, the increase in fidelity incurs a penalty in thecomputational efficiency.

In this article, GMC is evaluated as a potential method of homog-enization within a multi-scale model that can capture microscaleplastic deformation in multi-phase materials. To this end, a crystalplasticity model, in the form of a user material subroutine, is imple-mented at the microscale using GMC homogenization [43,44]. Atfirst, the performance of the stand-alone GMC, in terms of solutionaccuracy and computational time, was evaluated by comparingwith a FEA model that uses same material constitutive model sub-routine. The average stress–strain behavior exhibited an excellentagreement, while local distribution of field quantities predictedby GMC showed satisfactory agreement with FEA. It is importantto note that the objective of the homogenization is to obtain solu-tions within a desirable accuracy at a reasonable computationalcost. Following the evaluation of the stand-alone GMC, the multi-scale model was implemented within a FEA framework at the mac-roscale, with the material behavior at the integration points of theelements representing a microstructure homogenized by GMC. Themulti-scale model was evaluated by analyzing a realistic enginedisk component.

2. Theory and numerical Implementation

2.1. Single crystal plasticity model

The kinematics for the deformation mechanics of crystals fol-lows the pioneering work of Taylor [45], Hill [46], and Hill and Rice[47]. The theory is based on the assumption that any elastic–plasticdeformation can be split into two multiplicative operations, a plas-tic deformation where material is deformed through the rearrange-ment of lattices without any lattice stretching, followed by anelastic deformation associated with the stretching of lattices. Thetotal deformation gradient F is then given by,

F ¼ Fe � Fp ð1Þ

where Fp and Fe are the plastic and elastic deformation gradient,respectively.

Based on the deformation gradient definition, the total velocitygradient L is stated as

L ¼ _F � F�1 ¼ DþX ð2Þ

where the symmetric stretch rate D and the anti-symmetric spintensor X can be decomposed into lattice ( )⁄ and plastic parts ( )p,respectively.

D ¼ D� þ Dp; X ¼ X� þXp ð3Þ

The velocity gradient associated with the plastic deformation Lp

is then given in terms of Schmid’s law as,

Lp ¼ Fp � _Fp�1 ¼X

a

_� ama � na ð4Þ

where _� a is the rate of shear strain associated with the slipping of aslip system, m is the unit normal to the slip plane and n is the unit

46 M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55

vector parallel to slip direction. The incremental formulation ofplasticity theory is based on; (1) the evolution of Cauchy stress onthe corotational frame of reference that rotate with the crystal lat-tice, J⁄(r), (2) the slipping rate _� ðaÞ and (3) the strain hardening rate_gðaÞ as given below

J�ðrÞ þ rðI : D�Þ ¼ C : D�; ð5Þ

where r is the Cauchy stress, D⁄ is rate of stretching associated withelastic deformation and C is the tensor of elastic moduli. The gov-erning equation for slipping is defined as

_� ðaÞ ¼ _aðaÞsa

ga

� �n

ð6Þ

where sa is the resolved shear stress, ga is the strength, the _aðaÞ

refers to the slipping rate when resolved shear stress reaches thestrength. The governing equation for the strength can be modeledas

g_ ðaÞ ¼

Xb

hab�_ðbÞ ð7Þ

where hab are the slip hardening moduli. The sum ranges over allactivated slip systems. haa and hab are self and latent hardeningmoduli, respectively.

Hardening is defined based on the Pierce [48] and Asaro [49]model

haa ¼ hð� Þ ¼ h0sec2 h0�

ss � s0

�������� ðno sum on aÞ ð8Þ

where h0 is the initial hardening modulus, s0 is the yield stresswhich equals the initial value of current strength g(a)(0), ss is thestage-I stress (or the break-through stress where large plastic flowinitiates), and � is the total cumulative shear strain on all slip sys-tems, i.e.

� ¼X

a

Z t

0j _� ðaÞjdt ð9Þ

The latent hardening modulus is given by

hab ¼ qhð� Þ ða–bÞ ð10Þ

where q is a constant.For the numerical implementation of single crystal plasticity

model a user material subroutine in the form of UMAT for Abaqus[50] is used [4,5].

2.2. Generalized Method of Cells

GMC is a micromechanical formulation for predicting the over-all thermo-inelastic behavior of the multi-phase materials. Thereare four steps involved in this homogenization process [42]. Firstthe RUC is identified and discretized (Fig. 1a). A typical RUC is con-sisted of Na � Nb � Nc parallelepiped subcells (a ¼ 1; . . . ; Na; b ¼1; . . . ; Nb; c ¼ 1; . . . ; NcÞ in the x1, x2 and x3 directions, respectively,as shown in Fig. 1b. The individual subcells have the dimensions ofðda; hb; lcÞ and are related to the unit cell dimensions ðd; h; lÞ by,

d ¼XNa

a¼1

da; h ¼XNb

b¼1

db; l ¼XNc

c¼1

dc ð11Þ

As the second step, the relationships between the macroscopicaverage stresses and strains with the microscopic ones are stated.A linearly variable displacement uðabcÞ

i ði ¼ 1; 2; 3Þ is consideredinside each subcell. This displacement profile consists of pure dis-placement components in the center of each unit cell and themicro-variables representing the linear dependence of the dis-placement to the local coordinates.

The average strain in the RUC is defined as

�eij ¼1

dhl

XNa

a¼1

XNb

b¼1

XNc

c¼1

dahblc�eðabcÞij ð12Þ

which is based on the definition of strain tensor in each subcell �eðabcÞij

as:

�eðabcÞij ¼ 1

2@iu

ðabcÞj þ @ju

ðabcÞi

� �ð13Þ

Considering a general constitutive equation for the elastic-inelastic temperature-dependent material in each subcell (a, b, c),the average stress in that subcell is considered as,

�rðabcÞij ¼ CðabcÞ

ijkl�eðabcÞ

kl � �eIðabcÞkl � �eTðabcÞ

kl

� �ð14Þ

where the CðabcÞijkl is the subcell elastic tensor. The �eIðabcÞ

kl and �eTðabcÞkl are

the average inelastic and thermal strain tensors, respectively, ineach subcell.

Based on the volumetric summation of the average stresses inall subcells, the average stress in the entire unit-cell can be writtenas,

�rij ¼1

dhl

XNa

a¼1

XNb

b¼1

XNc

c¼1

dahblc �rðabcÞij ð15Þ

In the third step, the continuity of displacement and traction isapplied. Since RUCs are periodic, the interfaces of the subcells andthe boundaries of the RUCs follow the displacement and tractioncontinuity. Applying these conditions in an average sense resultsin elimination of the micro-variables in the displacement defini-tion and produces a set of continuity relationships. As a result, aset of NaðNb þ Nc þ 1Þ þ NbðNc þ 1Þ þ Nc equations stems fromthe continuity relations which can be declared as:

AGes ¼ J�e ð16Þ

with AG and J being matrices representing the geometrical details ofthe subcells and those related to the unit-cell itself, respectively.The �e is the average strain in the unit-cell, while es is defined as,

es ¼ �eð111Þ; . . . ; �eðNa ;Nb ;NcÞ� �

ð17Þ

The continuity of tractions, results in

AMðes � eIs � eT

s Þ ¼ 0 ð18Þ

which contains 6NaNbNc � ðNaNb þ NaNc þ NbNcÞ � ðNa þ Nb þ NcÞequations, in which AM includes the information regarding thematerial properties in each subcell. eI

s and eTs represent the inelastic

and thermal strains in each subcell, defined similarly to Eq. (17).The strain and traction continuity conditions (Eqs. (16) and

(17)) can be combined as

~Aes � ~DðeIs þ eT

s Þ ¼ K�e ð19Þ

where

~A ¼AM

AG

; ~D ¼

AM

0

; K ¼

0J

Rewriting this equation for each subcell as

�eðabcÞ ¼ AðabcÞ�eþ DðabcÞðeIs þ eT

s Þ ð20Þ

allows expressing the average strain in the subcell in terms of themacrostrain, and the inelastic and thermal strains by defining theconcentration matrices AðabcÞ and DðabcÞ.

In the fourth step, the overall macroscopic constitutive equa-tion of the material, based on the effective properties, isobtained. The average (homogenized) constitutive equation forthe RUC is

Fig. 1. (a) Repeating Unit Cells (RUCs) and (b) a sample RUC discretization consisted of Na = 3, Nb = 4 and Nc = 2 subcells.

Table 1Nominal composition of alloy (CMSX-4) (at.%) as given in [51].

Ni Al Ti Ta Cr Co W Mo Re Hf

CMSX-4 63.1 12.5 1.3 2.0 7.6 10.3 2.0 0.4 1.0 0.1Matrix (c) 54.9 – – – 20 18.6 2.7 0.8 3 –Inclusion (c0) 75 23 2 1 – – – – – –

Table 2Elastic properties for the CMSX-4 matrix and inclusion [51], and the crystal plasticitymodel parameters for the matrix [52].

E(GPa)

# G(GPa)

n _a h0

(MPa)s0

(MPa)ss

(MPa)

Matrix (c) 129 0.391 137 20 0.001 350 260 775Inclusion

(c0)114 0.405 130 – – – – –

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 47

�r ¼ B�ð�e� �eI � �eT Þ ð21Þ

in which the B⁄ represents the effective elastic tensor of the unitcell:

B� ¼ 1dhl

XNa

a¼1

XNb

b¼1

XNc

c¼1

dahblcCðabcÞAðabcÞ ð22Þ

The global inelastic strain tensor �eI and the average thermalstrain tensor �eT are also obtained in a similar fashion.

Fig. 2. Three different mesh refinements for studying the FEA solution convergence (Rinclusion). (a) One element each in matrix and inclusion in x and y directions (case (3x-etwo elements in matrix and 9 elements in inclusion (case (13x-element)).

�eI ¼ �B��1

dhl

XNa

a¼1

XNb

b¼1

XNc

c¼1

dahblcCðabcÞ DðabcÞeIs � �eIðabcÞ

� �ð23Þ

�eT ¼ �B��1

dhl

XNa

a¼1

XNb

b¼1

XNc

c¼1

dahblcCðabcÞ DðabcÞeTs � �eTðabcÞ

� �ð24Þ

Hence, given the local subcell thermal strains (known from thecurrent temperature and material coefficient of thermal expan-sion) and inelastic strains (known from the single crystal plasticitymodel), the homogenized macro (RUC-level) inelastic and thermalstrains are known from Eqs. (23) and (24). However, since the localsubcell inelastic strains depend on the local subcell stress state, theGMC solution is incremental and iterative. The process can be sum-marized as follows.

1. Given a converged solution at the previous loading incre-ment, apply an increment in the macro (RUC-level) stress/strain components. Any unknown (non-applied) macrostrain components are calculated from Eq. (21), where theprevious converged values of the macro inelastic strainsare assumed.

2. Localize to obtain the subcell strains and stresses from Eqs.(20) and (14), where the previous converged values of thesubcell inelastic strains are assumed.

3. Calculate increments in the local subcell strains using thesingle crystal plasticity model and add to previous con-verged values of the local subcell strains to obtain newvalues.

UC is cut open in the middle in z direction to show the element distribution inlement)), (b) two elements each in matrix and inclusion (case (6x-element)), and (c)

(a) (b)

(c)

0 0.002 0.004 0.006 0.008 0.010

200

400

600

800

1000

1200

Str

ess

(M

Pa)

Strain

3x-element (8noded)6x-element (8noded)13x-element (8noded)13x-element (20noded)

0 0.002 0.004 0.006 0.008 0.01 0.0120

200

400

600

800

1000

Str

ess

(M

Pa)

strain

3x-element (8noded)

6x-element (8noded)13x-element (8noded)

13x-element (20noded)

0 0.002 0.004 0.006 0.008 0.010

200

400

600

800

1000

1200

Str

ess

(M

Pa)

Strain

3x-element(8noded)6x-element(8noded)13x-element(8noded)13x-element(20nonded)

Fig. 3. Axial stress vs axial strain for the studied multi-phase single crystal for cases (3x-element), (6x-element), (13x-element) in (a) only matrix, (b) only inclusion, and (c)whole sample.

Fig. 4. Axial stress distribution on the cut plane for (a) (3x-element) case, (b) (6x-element) case, (c) (13x-element) case (8noded), (d) (13x-element) case (20noded) (lowerand upper bounds are set as 700 MPa and 1150 MPa, respectively, for all the three plots).

48 M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55

(a)

(b)

0 200 400 600 800 10000

200

400

600

800

1000

1200

von

Mis

es s

tres

s (M

Pa)

Applied axial stress (MPa)

FEA (Whole)

FEA (Matrix)

FEA (Inclusion)

GMC (Whole)

GMC (Matrix)

GMC (Inclusion)

0 0.005 0.01 0.0150

200

400

600

800

1000

1200S

tres

s (M

Pa)

Strain

FEA (Whole)

FEA (Inclusion)

FEA (Matrix)

GMC (Whole)

GMC (Inclusion)

GMC (Matrix)

Fig. 5. Comparison of GMC and FEA average axial stress–strain results for wholedomain and each individual inclusion and matrix phases, (a) axial stress vs axialstrain and (b) average von Mises stress vs applied axial stress.

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 49

4. Check for convergence of the local subcell strains comparedto the previous iteration. If converged, go to step 1.

5. Calculate new values of the macro inelastic strains from Eq.(23).

6. Based on the new macro inelastic strains, calculate updatedvalues of any unknown (non-applied) macro strain compo-nents from Eq. (21).

7. Based on the current local inelastic strain values, localize toobtain the subcell strains and stresses from Eqs. (20) and(14), and iterate by going to step 3.

2.3. Multi-scale model

The Micromechanics Analysis Code with Generalized Method ofCells (MAC/GMC), developed by NASA Glenn Research Center[35,36], which predicts the effective nonlinear response of hetero-geneous materials based on the known behavior of the constituentmaterials, is used in the present study. The MAC/GMC software,like most of the commercially available FEA packages, admitsuser-defined constitutive models. This capability was used toincorporate the single crystal plasticity model within the code suchthat the behavior of the individual constituent materials could berepresented with this model. GMC is called at the macroscaleFEA element level to represent the material response of that ele-

ment at its integration points using NASA’s FEAMAC program[43,44,50].

3. Results and discussion

In this study, CMSX-4 Nickel-based super alloy, which is exten-sively used in turbine blades, is considered. CMSX-4 is a two-phasesingle crystal, c0 inclusion and c matrix, with average inclusion sizeof 0.45 lm and volume fraction of roughly 70% (a value of 72.9% isused in the present study) [51,52]. CMSX-4 microstructure is char-acterized by a uniform distribution of well-shaped rectangularinclusions [52]. The composition of CMSX-4 for each phase isshown in Table 1 [51]. The Young’s modulus E, shear modulus G,and Poisson’s ratio # of the inclusion and the matrix phases [51],and the effective crystal plasticity model parameters of theCMSX-4 superalloy are shown in Table 2 [52]. Although, the effec-tive plastic properties are used in the present study due to lack ofexperimental data for the matrix, those are a good measure of thematrix behavior since inclusion exhibits very low plasticity com-pared to the matrix.

3.1. Validation of single crystal plasticity model using FEA

The two-phase single crystal (CMSX-4) is modeled first usingFEA. The inclusion was treated as linear elastic, while elastic–plas-tic behavior, in the framework of the crystal plasticity constitutivemodel described in Section 2.1, was defined for the matrix. A uni-axial loading of 1 GPa axial stress, at a loading rate of 1 GPa/s, wasapplied in the z direction. Periodic boundary conditions wereapplied in all three directions to simulate the periodicity of the unitcell in a bulk domain. The geometry of the modeled two-phase sin-gle crystal (CMSX-4) is illustrated in Fig. 2.

At first, the convergence of the FEA solution was studied. Threedifferent FE meshes with linear 8-node elements were consideredfor studying the convergence of the solution. Fig. 2a shows a meshcontaining one element each in the matrix and the inclusion alongx and y directions. This case is referred to as (3x-element). The sec-ond case, (6x-element), has two elements each in the matrix andthe inclusion, while the third case, (13x-element), has the numberof elements in the inclusion increased to nine as shown in Fig. 2band c, respectively. An additional case using 20-node higher orderelement with mesh similar to (13x-element) was also studied.

The average axial stress–strain curve of the sample is shown inFig. 3. All the studied cases demonstrated very good agreements forboth the matrix and the inclusion separately and as a result for theentire sample as well. The average values of axial stress and axialstrain for the matrix and inclusion were obtained by taking volumeaverage of these quantities for all the corresponding elementsobtained at their integration points. The axial stress distributionon an arbitrarily chosen cross-section is shown in Fig. 4. As themesh refinement increases the model captures the variation inthe individual phases. It should be noted that the stress valuesare maximum in the corners of the elastic inclusion region whichwere captured only in the (13x-element), 8-node and 20-node,cases. The results suggest that the solution is converged for (13x-element) case. Hence, only the case (13x-element) with 8-nodeFEA model was used for evaluating GMC stand-alone model as dis-cussed in the following section.

3.2. Evaluation of GMC as an integrated homogenization tool

The same RUC studied in the previous section is modeled usingGMC. Since in GMC, the refinement through subcells are onlyrequired for capturing various materiel phases and any furtherrefinement does not influence the solution significantly, the subcell

0 1 2 3 4 5x 10

-4

0

1

2

3

4

5 x 10-4

X

Y

Stress (S33)

700

750

800

850

900

950

1000

1050

1100

1150

(a)

0 1 2 3 4 5x 10

-4

0

1

2

3

4

5 x 10-4

X

Y

Stress (von-mises)

700

750

800

850

900

950

1000

1050

1100

1150

(b)

0 1 2 3 4 5

x 10-4

0

1

2

3

4

5 x 10-4

X

Y

Strain (E33)

0.009

0.0095

0.01

0.0105

0.011

(c)

Fig. 6. Results from comparison of GMC (left) and FEA (right) for (3x-element) mesh refinement case on the cut planes shown in Fig. 3, (a) axial stress distribution (lowerbound is 700 MPa, upper bound is 1150 MPa), (b) von Mises stress distribution (lower bound is 700 MPa, upper bound is 1150 MPa) and (c) axial strain (lower bound is 0.009and upper bound is 0.011).

50 M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55

structure same as (3x-element) FEA mesh was used. The GMCresults are then compared to the converged FEA solution obtainedusing (13x-element) 8-node case in order to evaluate the effective-ness of GMC for studying the two-phase single crystal materials.The comparison of the axial stress against the axial strain andthe von Mises stress against the applied axial stress demonstratedexcellent agreements for both the inclusion and the matrix phases,and hence for the entire domain as well (Fig. 5). The agreement inthe von Mises stress is an indication of the agreement in the overallshear deformation, not only in the axial stress–strain behavior.

The distributions of the axial stress, von Mises stress and axialstrain on an arbitrarily chosen cross-section were compared inFig. 6. Though the homogenized solution of the stress/strain distri-

bution on the cross-section did not agree as closely as in the case ofaverage global behavior (Fig. 5), it is still within a reasonable agree-ment with the FEA prediction. As expected, the homogenizationimpacts the accuracy of the results in terms of its ability to capturethe variation of field quantities in individual phases. Hence, GMCresults, or results from any homogenization scheme for that matter,may not be suitable for a simple, direct, deterministic type failureanalysis. However, from a statistical perspective (considering spa-tial variability in the material microstructure, variability in the localproperties, deviation of the idealistic conditions assumed in themodels compared to the real microstructural deformation), thedetermination of any additional microstructural deformation char-acteristics, representative of a statistically average behavior that is

(b)

0 0.2 0.4 0.6 0.8 1

x 10-3

0

0.2

0.4

0.6

0.8

1x 10

-3

700

750

800

850

900

950

1000

1050

1100

1150

(a)

Fig. 7. Two-phase microstructures having five inclusions in different sizes (a) the microstructure and (b) the comparison of von Mises stress distribution (MPa) predicted byGMC (left) and FEA (right) on a cut plane.

Fig. 8. RVEs representing complex microstructures of two-phase material (a) 2D microstructure and (b) 3D microstructure.

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 51

influenced by certain microstructural deformation phenomena ofinterest, provides opportunities for improved failure prediction. Itis also important to note that these reasonably accurate estimatesof that average values in individual phases can be used to obtaina good estimate of the stress and strain state at the interfacesbetween matrix and inclusion phases.

With regard to the computational cost comparison, the GMCdemonstrated 64, 156, 952 and 1425 times faster solution whencompared to the cases (3x-element), (6x-element), (13x-element)8-node and (13x-element) 20-node, respectively.

The comparison is extended to more realistic two-phase micro-structures with a larger number of inclusions and in different sizes,keeping the same inclusion volume fraction of 0.729. In the firstcase, an RUC having five inclusions is considered; four inclusions

each with a size of 0.45 � 0.45 � 0.45 lm3 at the bottom and onewith the size of 0.9 � 0.45 � 0.9 lm3 at the top (Fig. 7a). As inthe previous case shown in Fig. 5, an excellent agreement wasobtained for the average behavior of matrix, inclusions and wholedomain. The comparison of the distribution of von Mises alsoshowed very good agreement (Fig. 7b) validating the ability ofGMC homogenization to yield satisfactory predictions for the dis-tribution of field quantities. Again, it can be seen that GMC failsto capture the detailed variation in the individual phases asexpected.

Finally an RVE representing a more complex microstructurecomprising several inclusions with different sizes, as shown inFig. 8, is considered. Two configurations of the RVE, a 2D micro-structure without any variation in the thickness direction

Fig. 9. Average von Mises stress vs applied axial stress for the multi-phase single crystals having 1 inclusion, 5 inclusions and the more realistic RVE case in (a) matrix, (b)inclusion, and (c) whole domain.

(a)

0 1 2 3 4

x 10-3

0

0.5

1

1.5

2

2.5 x 10-3

X

Y

700

750

800

850

900

950

1000

1050

1100

(b)

Fig. 10. von Mises stress distribution (MPa) on a cut plane of the 2D RVE microstructure with inclusions in different sizes and shapes predicted by (a) GMC and (b) FEA.

52 M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55

(Fig. 8a) and a 3D microstructure by combining four randomly gen-erated 2D microstructure back-to-back (Fig. 8b), are studied. Both,the 2D and 3D microstructure samples have a cross-sectional sizeof 2.5 � 4 lm2, with an inclusion volume fraction of 0.729. A thick-ness of 1 lm for the 2D microstructure and a total thickness of2.5 lm (four layers of 0.6, 0.8, 0.4 and 0.7 lm) for the 3D micro-structure are considered. The average von Mises stress in matrix,inclusion and whole sample are shown for these two cases inFig. 9. In addition, results are also compared with the 1 inclusionand 5 inclusion cases. Overall, the GMC predictions of the behaviorof ME3 multiphase material using all these different representativesamples are in excellent agreement. The FEA results, though notshown in Fig. 9, were in excellent agreement with GMC resultsfor all these cases, except for the 3D RVE microstructure. For the

3D RVE microstructure, GMC prediction showed some deviationfrom the rest of the cases, while the FEA prediction was in agree-ment with them. This indicates that the GMC prediction is startingto deviate from the actual behavior as the microstructure becomesincreasingly complex. The excellent agreement for both the matrix(Fig. 9a) and the inclusion phases (Fig. 9b), and hence the wholedomain (Fig. 9c), indicate that all the different microstructural rep-resentations considered are capable to capture the average defor-mation of the two material accurately.

The von Mises stress distribution on an arbitrarily chosen cross-section predicted by the GMC and the FEA is compared for the 2DRVE microstructure in Fig. 10. Results are in excellent agreement.The prediction of von Mises stress distribution for the 3D RVEmicrostructure is illustrated in Fig. 11; the material index of the

(a) (b)

(d) (c)

0 1 2 3 4x 10

-3

0

0.5

1

1.5

2

2.5x 10

-3

X

Y

0.008

0.01

0.012

0.014

0.016

0.018

0 1 2 3 4x 10

-3

0

0.5

1

1.5

2

2.5x 10

-3

X

Y

0 1 2 3 4x 10

-3

0

0.5

1

1.5

2

2.5x 10

-3

X

Y

700

800

900

1000

1100

1200

Fig. 11. 3D RVE with two-phase microstructures (having inclusions in different sizes and shapes also in depth) (a) the microstructure of the studied cut plain, (b) axial straindistribution (MPa) predicted by GMC, (c) von Mises stress distribution (MPa) predicted by GMC and (d) von Mises stress distribution (MPa) predicted by FEA.

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 53

arbitrarily chosen cross-section is shown in Fig. 11a, while the dis-tribution of the axial strain is shown in Fig. 11b. By comparing thematerial index with the strain distribution it can be verified qual-itatively that GMC is capable of capturing the deformation reason-ably well. The GMC prediction of the von Mises stress distribution(Fig. 11c) on a given cross-section is compared with the corre-sponding FEA prediction (Fig. 11d). Though, the GMC and FEA pre-dictions are not in excellent agreement as in the previous simplecases, the GMC results are still reasonable, both in terms of theoverall general distribution over the cross-section, and also itsrange of 700–1200 MPa.

In summary, the obtained excellent agreement in average glo-bal stress–strain behavior, reasonably good agreement in the localdistribution of field quantities, and large savings in computationalcost demonstrate that GMC is a very promising homogenizationtool for modeling the multi-phase single crystal materials.

3.3. Multi-scale simulation

In order to implement a multi-scale analysis of a realistic, struc-tural-scale problem, a macroscale model was linked to the micro-scale material deformation through GMC homogenization. A FEAframework was used for the macroscale model. At the macroscaleelement level, FEA determines the stress field at the integrationpoints for a prescribed incremental displacement/strain and for agiven time step. In order to obtain the required material propertiesat the integration points, the prescribed strains at the integrationpoints of the macroscale model are applied as average quantities

on the microscale model that represents the two-phase Ni-basedsuperalloy microstructure at that integration point. At the micro-scale, the single crystal plasticity constitutive relation is appliedto the matrix phase to determine the strain concentration matricesunder GMC homogenization. The average values of stress compo-nents are then returned from the GMC analysis to the correspond-ing integration points in the macroscale model. State variables thatstore the total plastic strain, strain hardening, etc. at the microscaleare updated at each time step.

For the purpose of demonstrating this functionality, a segmentof a real-size turbine disk is analyzed. The disk is 254 mm long andis exposed to a centrifugal load (qx2) with the magnitude of 25.5kg rad2

=mm3 s2, corresponding to the squared angular speed of3 � 109 ðrad=sÞ2. Considering the symmetry of the geometry andthe load, 1/8 of the disk is analyzed applying appropriate symmet-ric boundary conditions. It is important to note that the symmetricassumption is valid strictly only at the macroscale. The two-phaseCMSX-4 superalloy was used as the material model for each ele-ment in the microscale domain. 20 node brick elements were usedfor the macroscale FEA model.

Fig. 12a shows the macroscale von Mises stress distribution inthe disk. To evaluate the multi-scale model, the von Mises stressdistribution was also obtained using the classical macroscale plas-ticity model as shown in Fig. 12b. In the classical plasticity model,the effective stress–strain curve obtained from the same two-phase CMSX-4 under periodic boundary condition was used asthe constitutive model. The stress distribution in the elements atthe macroscale shown in Fig. 12a and b follows the element

Fig. 12. von Mises stress distribution (�10�3 MPa) in the real-size disk example (a) multi-scale model using FEA/GMC and (b) standard FEA model.

0 1 2 3 4

x 10-3

0

0.5

1

1.5

2

2.5x 10

-3

X

Y

Stress (von-mises)

7

7.5

8

8.5

9

x 105

Fig. 13. von Mises stress distribution (�10�3 MPa) at the microscale shown for anarbitrarily chosen integration point in an arbitrarily chosen element (element k inFig. 12(a)).

54 M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55

interpolation. The von Mises stress distributions shown in Fig. 12aand b demonstrate that the multi-scale linking, by using GMChomogenization under the FEA framework, predicts expected dis-tribution at macroscale.

In the multi-scale model, in addition to the distribution of fieldquantities at the macroscale, GMC provides the local stress/straindistribution at the microscale consistent with the interpolated val-ues of the macroscale FEA. The von Mises stress distribution on achosen cross-section halfway through the depth of the RUC atmicroscale, corresponding to an arbitrarily selected integrationpoint of an arbitrarily chosen element (element k as shown inFig. 12a) is given in Fig. 13. The results show that the local fieldquantity distributions at the grain-level in a real-sized model canbe determined using the developed multi-scale model which isextremely difficult or nearly impossible with a standard FEA anal-ysis alone due to the exorbitant computational cost.

4. Conclusion

A multi-scale computational model with the ability to deter-mine the microstructural scale stress/strain field for real size

M. Ghorbani Moghaddam et al. / Computational Materials Science 96 (2015) 44–55 55

problems was developed employing FEA at macroscale and GMC atmicroscale. The crystal plasticity constitutive model was imple-mented in conjunction with GMC at the microscale. A two-phasesingle crystal material (CMSX-4 superalloy) was studied by explic-itly defining properties of individual phases. The ability of GMC asa homogenization tool for multi-scale modeling was evaluated bycomparing the GMC predicted results with FEA for samples withdifferent microstructures under periodic boundary conditions.The GMC analysis achieved high savings in computational cost ata minimal expense of accuracy in the local field quantities. GMCshowed excellent agreement with the FEA results in terms of theaverage stress–strain behavior and the average of field quantitiesin the individual phases. Finally, the multi-scale aspect of themodel (linking the microstructure consisting of two explicit phasesto the macroscale model), was demonstrated by implementingGMC as a homogenization technique within the FEA platform,and analyzing a real life turbine engine disk problem. The macro-scale results demonstrated similar stress distribution when com-pared to the standard FEA based analysis, thereby verifying themethod. The ability of the multi-scale model to obtain stress/strainfield at microstructure scale was demonstrated by extracting thevon Mises stress distribution at an arbitrarily selected integrationpoint of an arbitrarily selected element. The developed multi-scalemodel may allow engineers to tailor the microstructure for differ-ent structural applications.

Acknowledgments

Author A. Achuthan would like to thank NASA Glenn ResearchCenter and Ohio Aerospace Institute for the summer researchfellowship grant that supported part of this work.

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