# Crystal plasticity-based finite element analysis of...

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J O U R N A L O F M A T E R I A L S S C I E N C E 3 8 (2 0 0 3 ) 307 – 322

Crystal plasticity-based finite element analysis

of deformation and fracture of polycrystalline

lamellar γ-Tial + α2-Ti3al alloys

M. GRUJICIC, G. CAO, S. BATCHUDepartment of Mechanical Engineering, Program in Materials Science and Engineering,Clemson University, Clemson SC 29634, USAE-mail: [email protected]

Deformation behavior of fully-lamellar polycrystalline γ -TiAl + α2-Ti3Al alloys has beenanalyzed using a finite element method. A three-dimensional rate-dependent, finite-strain,crystal-plasticity based materials constitutive model is used to represent the deformationbehavior of the bulk material. The constitutive behavior of γ -TiAl/γ -TiAl lamellar interfacesand lamellae-colony boundaries, on the other hand, are modeled using a cohesive-zoneformulation. The interface/boundary potentials used in this formulation are determinedthrough the use of atomistic simulations of the interface/boundary decohesion. Theconstitutive relations for both the γ -TiAl + α2-Ti3Al bulk material and the lamellar interfacesand colony boundaries are implemented in the commercial finite element programAbaqus/Standard within which the material state is integrated using an Euler-backwardimplicit formulation. The results obtained show that plastic flow localizes into deformationbands even at an overall strain level of only 0.5% and that incompatibilities in plastic flowbetween the adjacent colonies can give rise to high levels of the hydrostatic stress and, inturn, to intercolony fracture. Furthermore, it is found that when lamellar interfaces areadmitted into colonies, fracture is delayed and the materials fail in a more gradual manner.C© 2003 Kluwer Academic Publishers

1. IntroductionTwo-phase γ -TiAl + α2-Ti3Al alloys with micron-scale lamellar microstructures generally exhibit a goodcombination of high-temperature properties such ascreep resistance, microstructural stability, oxidation re-sistance, etc. Consequently, there has been much inter-est in developing these alloys as viable materials forhigh-temperature structural applications. However, alack of tensile ductility and fracture toughness in thesealloys at the ambient temperature is one of the majorshortcomings which hampers their wide use. There areseveral comprehensive reviews [e.g., 1–3], that sum-marize the major advances in development of these al-loys. It should be pointed out, however, that the mainimprovements in alloy properties have been realizedlargely in polysynthetically-twinned single-crystallineform of these materials, and that it has been quite diffi-cult to achieve similar successes in the polycrystallinematerials of this type. The latter typically fail at ten-sile strains less than 3% and generally have a fracturetoughness level KIC below 30 MPa

√m. While single

crystalline materials of this type possess quite attractiveproperties, their use is cost prohibitive. Thus, achievinga superior combination of properties in conventionallyprocessed polycrystalline γ -TiAl + α2-Ti3Al alloys re-mains an important, though formidable, engineeringchallenge.

During solidification, Ti-(48–50) at.% Al alloys,which are considered in the present work, first forma disordered hexagonal-close-packed (h.c.p.) α-phasewhich during cooling orders into an α2-Ti3Al-typephase with the DO19 crystal structure and then trans-forms to (or near) completion into an ordered face-centered-tetragonal (f.c.t.) γ -TiAl phase with the L10crystal structure. The final microstructure typically con-sists of colonies each containing micron-thick parallelγ -TiAl and α2-Ti3Al lamellae with a standard f.c.c.-h.c.p. type orientation relationship between the twophases: {111}γ ‖ (0001)α2 and 〈1–10]γ ‖ 〈11–20〉α2. Inaddition, the γ /α2 and γ /γ lamellar boundaries havethe following crystallographic orientations: {111}γ ‖(0001)α2 and {111}γ ‖{111}γ , respectively. Under uni-directional solidification condition, the microstructureof Ti-(48–50) at.% Al can be obtained which consists ofa single colony of γ -TiAl + α2-Ti3Al lamellae [e.g., 4].Such alloys are referred to as having polysynthetically-twinned single crystalline microstructure.

The microstructure of polycrystalline γ -TiAl + α2-Ti3Al alloys can be modified using post-solidificationheat treatments at different temperatures, holding timesand cooling rates [e.g., 5]. The resulting microstructuresare generally classified as: near-gamma, duplex, nearlylamellar, and fully lamellar. For Ti-(48–50) at.% Al al-loys, fully-lamellar and duplex microstructure impart

0022–2461 C© 2003 Kluwer Academic Publishers 307

the best combinations of mechanical properties. Thefully-lamellar microstructure consists of colonies ofγ -TiAl + α2-Ti3Al lamellae where the colony size iscontrolled by the holding time at the heat-treating tem-perature while the rate of cooling to room temper-ature governs the thickness of lamellae. The duplexmicrostructure contains in addition to the γ -TiAl andα2-Ti3Al lamellae, small γ -TiAl phase particles typi-cally located at the colony boundaries. In this case, theheat-treating temperature and the holding time affectsthe volume fraction of the phases, while the cooling ratedetermines the lamellar thickness. In the present work,only fully-lamellar polycrystalline γ -TiAl + α2-Ti3Alalloys will be analyzed.

The mechanical response of γ -TiAl + α2-Ti3Alpolysynthetically twinned single-crystalline alloys ishighly anisotropic at the macroscopic, microscopic andcrystal structure length scales [e.g., 4]. At the macro-scopic length scale, properties such as flow stress,fracture stress, fracture strain, crack growth rate andothers exhibit strong orientation dependence. In theγ -TiAl + α2-Ti3Al lamellar microstructure, γ -TiAl isthe softer phase and its flow properties are highlyanisotropic due to the lamellar geometry of its crys-tals. Shear deformation parallel to the lamellar inter-faces is considerably easier (the soft mode) than thatnormal to them (the hard mode). In the latter case,slip in the γ -TiAl phase is constrained by the harderα2-Ti3Al phase [4]. Since the soft-to-hard mode sliplength ratio is typically on the order of 100, the Hall-Petch effect is significant. In addition, the Hall-Petchslopes are also anisotropic, with typical values 0.273and 0.440 MPa/

√m for the soft and the hard modes,

respectively [6]. Hence, at the microscopic length-scale, materials anisotropy is caused primarily by largedifferences in the soft and hard mode deformation re-sistances within each phase and differences in the de-formation resistances of the two phases. At the crystal-structure length scale, materials anisotropy is derivedfrom differences in deformation resistance of differ-ent slip systems within the same phase. For example,〈a〉-slip systems associated with 〈1120〉 slip directionsare generally substantially softer than any of the pos-sible (c + a)-slip systems in α2-Ti3Al. Since the lattersystems are needed to achieve a general state of strain,the α2-Ti3Al phase can be considered as kinematicallyconstrained in the c-direction [e.g., 7].

The anisotropic behavior of polysynthetically-twinned γ -TiAl + α2-Ti3Al single crystals describedabove is, to some extent, retained in the polycrystallineform of these materials. However, additional changes inthe mechanical behavior of lamellar γ -TiAl + α2-Ti3Alpolycrystals arises from variations in the microstruc-ture such as: the shape, size, and volume fraction of theconstituent phases. For example, fully lamellar and du-plex alloys exhibit an inverse tensile elongation/fracturetoughness relationship [5]. That is, while the flow be-havior (yield stress, rate of strain hardening, etc.) isquite similar in the two microstructures, fully lamellaralloys have low ductility (∼<3%) and relatively highfracture toughness (KIC ≈ 22–32 MPa

√m), whereas

the duplex alloys have high ductilities (∼5–10%) andlow fracture toughness (KIC ≈ 10–12 MPa

√m) [5].

While tensile fracture strains on the order of30% have been found in polysynthetically-twinnedγ -TiAl + α2-Ti3Al single crystals [10], polycrystallineforms of these materials show a drastically lower ten-sile ductility (<3%) [7]. Such disparities in mechani-cal behavior of single and polycrystalline alloys havebeen widely reported, however the origin of this phe-nomenon is still not well understood. Often, it is pos-tulated to arise from the constraints to plastic flow ina colony or grain imposed by the surrounding coloniesor grains, the phenomenon, which is present only inpolycrystalline materials.

The objective of the present work is to provide a suit-able rationale for the aforementioned limited ductilityand fracture toughness in polycrystalline γ -TiAl + α2-Ti3Al alloys and to help establish relationships betweenthe microstructure (particularly the effect of γ -TiAl/γ -TiAl lamellar interfaces) and properties in these ma-terials. Toward that end, a rate-dependent, finite strain,crystal-plasticity material constitutive model is devel-oped, implemented into the commercial finite elementpackage Abaqus/Standard [8] and employed to pre-dict the global response of polycrystalline material.The approach used provides an opportunity to studythe evolution of strain localization and to help re-veal the contribution strain localization makes to theglobal mechanical response of the material. For exam-ple, due to the soft-vs. hard-mode flow-stress disparity,non-uniform deformation is likely to occur in poly-crystalline γ -TiAl + α2-Ti3Al alloys containing ran-domly oriented lamellar colonies. In fact, plastic flownon-uniformity leading to strain localization near grainboundaries and three grain junctions are observed atsmall plastic strains even in f.c.c. metallic materials inwhich due to a large number of crystallographicallyequivalent slip systems all of the identical (initial) de-formation resistance, strain localization is more diffi-cult to develop [9, 10]. Since γ -TiAl and α2-Ti3Al pos-sess slip systems of different crystallographic naturewhose deformation resistances can differ substantiallyso that these materials can be considered kinematicallyconstrained with respect to certain deformation modes,strain localization is more likely to occur and shouldoccur at lower plastic strains. Strain localizations inthe regions near the colony/grain boundaries and threecolony/grain junctions are generally associated withlarge local stress concentrations and since these stressesmight be precursors to material failure, understandingtheir evolution during deformation, their relationshipsto the local microstructure and their contribution to theglobal materials response is highly important and willbe studied in the present work. A comprehensive finiteelement analysis of the deformation and fracture be-havior of the two phase γ -TiAl + α2-Ti3Al alloys wasconducted in a series of papers by Asaro and coworkers[7, 11, 12]. However, while the analysis of Asaro andcoworkers [7, 11, 12] provided important insight intothe effect of colony to colony misorientation on the localincompatibility of plastic flow and the resulting stressconcentration, no provision was made for the materi-als fracture. To overcome this limitation, fracture alongγ -TiAl/γ -TiAl lamellar interfaces and colony bound-aries is allowed in the present work through the use of

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a cohesive zone formulation. Also, a three-dimensionalcrystal-plasticity materials constitutive model recentlyproposed by Grujicic and Batchu [13] which incorpo-rates kinematic constraints arising from the retentionof the crystallographic nature of γ -TiAl/α2-Ti3Al in-terlamellar boundaries and the retention of the γ -TiAlvs. α2-Ti3Al orientation relationship is used.

The organization of the paper is as follows: A briefoverview of the derivation of the constituent responseof lamellar interfaces and colony boundaries and thederivation of the stiffness matrix for the correspondinginterfacial elements suitable for implementation into afinite element analysis are provided in Section 2.1. Thecrystal plasticity model for polysynthetically-twinnedγ -TiAl + α2-Ti3Al single crystals and its implemen-tation into the commercial finite element packageAbaqus/Standard are reviewed in Section 2.2. An out-line of the boundary value problem analyzed in thepresent work through the use of finite element methodis presented in Section 2.3. The main computational re-sults are shown and discussed in Section 3. A summaryof the key conclusions resulted from the present workis presented in Section 4.

2. Computational procedure2.1. γ -TiAl/γ -TiAl lamellar-interface and

colony-boundary constitutive relationsTo comply with the experimental observations [e.g., 5],fracture is allowed to occur along the γ -TiAl/γ -TiAllamellar interfaces and colony boundaries. Towardthis end, the mechanical constitutive response of theγ -TiAl/γ -TiAl lamellar interfaces and the boundariesbetween γ -TiAl + α2-Ti3Al lamellar colonies is mod-eled using the cohesive zone framework originally in-troduced by Needleman [14]. The cohesive zone is as-sumed to have a negligible thickness when comparedto other characteristic lengths of the problem, such asthe lamellae thickness, typical lengths associated withthe gradient of the fields, etc. The mechanical behav-ior of the cohesive zone is characterized by a traction-displacement relation which is introduced through thedefinition of an interface potential, �. Stable equilib-rium for an interface boundary corresponds to a per-fectly bonded configuration, where the potential has aminimum and all tractions vanish. For any other con-figuration, the value of the potential is taken to dependonly on the displacement jump across the interface. Ingeneral, an interface displacement jump is expressedin terms of its normal component, Un , and two tan-gential components, Ut and Ub. Differentiation of theinterface potential function � with respect to Un , Ut

and Ub yields, respectively, a normal and two tangen-tial components of the traction per unit interface areain the deformed configuration. Thus the interface con-stitutive relations are fully defined by specifying theform for the interface potential function. In the presentwork, the approach based on the use of MolecularStatics method recently proposed by Grujicic and Lai[15] is used to determine the interface potential for theγ -TiAl/γ -TiAl lamellar interfaces and colony bound-aries. A brief overview of this approach is givenbelow.

Figure 1 Schematic representation of the γ -TiAl/γ -TiAl (γ 1/γ 2)bicrystal used for determination of the (111)γ 1/(111)γ 2 lamellar interfa-cial potential.

To determine the potential function for a par-ticular γ -TiAl/γ -TiAl lamellar interface or colonyboundary, a γ -TiAl/γ -TiAl atomistic bicrystal is firstconstructed from two appropriately orientated singlecrystals and with the appropriate orientation of the in-terface/boundary separating the single crystals. An ex-ample of the γ -TiAl/γ -TiAl atomistic bicrystal is givenin Fig. 1. The single crystals edge lengths are expressedin terms of the number of interplanar spacings d(uvw)of the corresponding (uvw) planes, where (uvw) are theplane Miller indices. The atomic interactions are ac-counted for through the use of the Embedded AtomMethod (EAM) interatomic potential [16]. Interatomicpotentials, which account for the atomic interactions ina crystal, are the required input for any atomistic simu-lation study and therefore their reliability and accuracyare very critical for the atomistic simulations to faith-fully represent the materials behavior. In contrast to thetraditional pair potentials, the EAM interatomic poten-tials take into account, in an implicit way, the many-body effects and have therefore been proven more re-liable in representing the atomic interactions in metals[e.g., 17]. As mentioned earlier, γ -TiAl has an L10-typeordered crystal structure, and hence the interatomic po-tential functions must account for the Ti-Ti, Al-Al andTi-Al interactions and for the fact that the two atomicspecies (Ti and Al) reside on separate crystal sublat-tices. Farkas [18] recently developed the EAM-type in-teratomic potentials for γ -TiAl and showed that a rea-sonable agreement can be obtained between the modelpredictions and the experimentally measured proper-ties of this phase. In addition, this potential has beenextensively tested and validated in our previous work,Grujicic and Dang [19, 20], Grujicic and Lai [21] andGrujicic and Zhang [22, 23], and hence is used in thepresent study.

Prior to constructing a simulation bicrystal, the equi-librium lattice parameters of γ -TiAl at 0 K are de-termined by carrying out the Molecular-statics type

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Figure 2 Equilibrium configuration of the [111]γ 1//[111]γ 2 lamellarinterface. Atomic positions projected onto: (a) the [011]γ 1//[011]γ 2

plane; and (b) the [111]γ 1//[111]γ 2 interface plane. Interfacial disloca-tions are marked as D1 while the magnitude of their Burger’s vector isdenoted as b.

potential energy minimization through the use of theConjugate Gradient Method [24]. The following latticeparameters are obtained for γ -TiAl: aγ = 3.944 A andcγ = 4.010 A, resulting in the cγ /aγ ratio of 1.05 whichis in fair agreement with its experimental counterpartcγ /aγ = 1.03 [25].

Before the appropriate γ -TiAl/γ−TiAl interface po-tentials can be evaluated, the reference stress-free equi-librium structure of the interface must be determined.The equilibrium interface structure at 0 K is determinedin each case by minimizing the potential energy of thecorresponding bicrystals using the Conjugate GradientMethod under the flexible periodic boundary conditionsin two mutually orthogonal directions parallel with theplane of interface and the free surface boundary condi-tions in the direction normal to the interface. The use ofthe free surface boundary conditions allows the spacingof the planes parallel to and near the interface to adjustto differences in the atomic environment. An exam-ple of the equilibrium atomistic configuration for theγ -TiAl/γ -TiAl lamellar interface is shown in Fig. 2aand b. Two views of the interface structure are given inFig. 2a and b. In Fig. 2a, atoms of the two crystals areprojected on a plane normal to the interface. In Fig. 2b,on the other hand, the atoms are projected on the planeof the interface. As indicated in Fig. 2a and b, the in-terface structure is dominated by an array of parallelscrew dislocations marked D1. The characteristics ofthe array D1 are given in Table I.

The interface potential functions for each of thelamellar interfaces and colony boundaries analyzedare determined by first rigidly displacing the two sin-

TABLE I Parameters characterizing dislocation structure in theγ -TiAl/γ -TiAl lamellar interface shown in Fig. 2a and b

Parameter Value

Line direction[011]γ1

Burger’s vector1/2 aγ cγ /

√a2

γ + c2γ [011]γb

Dislocation spacing25aγ

√(a2

γ + 2c2γ

)/(a2

γ + c2γ

)λ

gle crystals from the equilibrium configuration of thebicrystal such as the one shown in Fig. 2a and b, in thenormal n and tangential t and b directions by differentamounts. These three displacements are respectivelydesignated as: Un , Ut and Ub. The potential energy ofthe perturbed bicrystals is next minimized under theconstraint that the average displacements of the atomsin the interface planes in each of the two single crys-tals remain equal to the rigid displacements Un, Ut andUb. The difference between the energies of the bicrys-tal in the displaced configuration and in the stress-freeequilibrium configuration expressed per unit area of theinterface is then defined as the value of the interface po-tential � at the given values of Un , Ut and Ub.

The calculation results of the interface potential �

for the γ -TiAl/γ -TiAl lamellar interface, whose equi-librium atomic configuration is shown in Fig. 2a and b,are shown in Fig. 3a and b. The variation of � with thenormal interface displacement Un at various values ofthe tangential displacement Ut and Ub = 0 is given inFig. 3a. The variation of � with the tangential displace-ment Ut at various values of the tangential displacementUb and the zero value of normal displacement Un isshown in Fig. 3b. The tangential direction b is cho-sen to coincide with the line vector ([011]γ 1‖[011]γ 2)of the interface dislocations while the tangential vectort([211]γ 1‖[211]γ 2) is taken to be normal to it. Based onthe analysis of over 50 γ -TiAl/γ -TiAl interfaces (theresults for only one given here for brevity), the overallbehavior of the computed-interface potentials with thevariation in the three displacements can be summarizedas following:

(a) For the pure normal decohesion process whereUt = Ub = 0, the interface potential increases monoton-ically with the normal displacement Un and asymptot-ically approaches a value defined as the work of deco-hesion, �dec.

(b) At non-zero values of the tangential displace-ments, (for example for Ut = 10 nm and Ub = 0 as inFig. 3a), the interface potential first decreases and thenincreases with the normal displacement Un .

(c) The shear behavior of the interface is periodic withthe periodic length governed by the spacing of the inter-facial dislocations. For example, the periodic length for� in Fig. 3b is equal to the spacing l of the interfacialdislocations D1 in Fig. 2a and b.

(d) The pure shear behavior is unstable in some cases(not shown for brevity) since perturbations in the nor-mal Un or in the orthogonal shear displacements cangive rise to a reduction in the potential.

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Figure 3 Variation of the interface potential for the (111)γ 1//(111)γ 2

lamellar interface with the normal displacement Un in the[111]γ 1//[111]γ 2 direction and tangential displacements Ut and Ub inthe [011]γ 1//[011]γ 2 and [211]γ 1//[211]γ 2 directions, respectively.

(e) In some cases (not shown for brevity) the shearbehavior is not symmetric which has been shown to berelated to the non-planar step + ledge character of theinterface [15].

Since every γ -TiAl/γ -TiAl lamellar interfaces andcolony-boundary appearing in the finite element rep-resentation of the microstructure in polycrystallineγ -TiAl + α2-Ti3Al materials will be analyzed using thecohesive zone approach, it is useful to express the inter-face/boundary potential for all of them using the samefunctional form. The fact that pure shear is unstablein some cases, rules out the use of the interface po-tential function proposed by both Needleman [14] andSocrate [26]. Furthermore, due to the nonsymmetricalcharacter of the interface shear, the universal functionfor ideal shear proposed by Bozzolo et al. [27] couldalso not be used. After analyzing a number of combi-nations of the aforementioned interface potential func-tions, the following function is found to best mimic

the overall relationship between � and Un , Ut and Ub

for all γ -TiAl/γ -TiAl lamellar interfaces and colonyboundaries analyzed:

� = σmaxδn

[e −

(Un

δn+ 1

)exp[(Un/δn)+1]

]

+ 1

2α0

(6Un

δn+ 1

)exp(6Un/δn)

×{α1 +

[α2 sin

4π (Us − 2)

λs+ cos

(2πUs

λs+ α3

)

+ α4

](α5 + cos

2πUb

λb

)}(1)

Table II gives a summary of the parameters appearingin Equation 1 for the lamellar interfaces whose atom-istic structure is displayed in Fig. 2a and b and forone colony boundary (atomistic structure not shownfor brevity).

It should be noted that several parameters appearingin Equation 1 have a particular physical meaning:

– σs max is the normal interface strength associ-ated with the pure normal decohesion processes(Ub = Ut = 0) and corresponds to the normal trac-tion at the normal interfacial displacement Un = δn .

– λb and λt are the periodic lengths for pure shearin the two shear directions b and t , respectivelyand are related to the spacing of the correspondinginterfacial dislocations.

– The shear interface strengths in the two direc-tions, τmax, b and τmax,t , correspond to two pureshear modes of interface decohesion and oc-cur at Ub = 0.25λb, Un = Ub = 0 and Ut = 0.25λt ,Un = Ub = 0, respectively.

The difference between the interface potential inthe limit Un → ∞, and its equilibrium magnitude atUn = Ub = Ut = 0, is generally termed the work of de-cohesion and is taken to be in the range of 1–10 J/m2.

TABLE I I Decohesion potential parameters for one γ -TiAl/γ -TiAllamellar interface and one colony boundary

Interface type

(111)γ 1/(111)γ 2 (001)γ 1/(011)γ 2

Parameter Lamellar boundary Colony boundary

n-Direction [111]γ 1/[111]γ 2 (001)γ 1/(011)γ 2

t-Direction [111]γ 1/[111]γ 2 (110)γ 1/(111)γ 2

b-Direction [111]γ 1/[111]γ 2 (110)γ 1/(211)γ 2

δn , nm 0.05 0.05λt , nm 12.35 3.67λb , nm 0.578 1.07α1 −0.039 −0.145α2 −1 0.42α3 0 −0.31α4 0 −3.42α5 0 1.17α0 0 −5.02σmax (GPa) 2.01 1.21τmax,t (GPa) 0.160 12.7τmax,b (GPa) 1.65 0.73�(Un → ∞) (J/m2) 0.561 0.374

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The results obtained such as the one given in Table IIare generally consistent with this range. The normal in-terface strength σmax is generally assumed to be 0.01to 0.02 times the material Young’s modulus, whichyields 1.6–3.2 GPa for the γ -TiAl phase. The normalinterface strength data given in Table II are consis-tent with this range of values. As for the shear in-terface strength, it is generally assumed to be a rela-tively small fraction of σmax. An examination of theresults given in Table II, shows that this is true forτmax,t for the (111)γ 1/(111)γ 2 lamellar boundary butnot for the (001)γ 1/(011)γ 2 colony boundary. In addi-tion, the shear resistances in the b and t directions arequite different from each other and, as in the case of the(001)γ 1/(011)γ 2 colony boundary, the shear resistanceτmax,t can be significantly larger than σmax. This findingis a direct consequence of the non-planar interface mor-phology, which consists of steps and ledges aligned inthe b direction. Similarly, the presence of a single arrayof dislocations in the case of γ -TiAl/γ -TiAl lamellarinterface causes the two shear resistances, τmax,t andτmax,b to differ by more than an order of magnitude.

Once the γ -TiAl/γ -TiAl interface/boundaries poten-tials are derived, they are used to construct (continuum)interfacial elements, which are suitable for a finite ele-ment analysis. A brief description of the derivation ofthe stiffness matrix of continuum interfacial elementsis given in Appendix.

2.2. Materials constitutive model forpolysynthetically-twinned γ -TiAl + α2Ti3Al single crystals

The deformation behavior of polysynthetically-twinned γ -TiAl + α2-Ti3Al single crystals has been re-cently modeled by Grujicic and Batchu [13] using arate-dependent, isothermal, elastic-viscoplastic, finite-strain, crystal-plasticity formulation. The continuummechanics foundation for this model can be traced tothe work of Teodosiu [28], Hill and Rice [29], Mandel[30], Teodosiu and Sidoroff [9], Asaro and Rice [10]and Asaro [31]. The work of Kocks et al. [32], Frostand Ashby [33], and Argon [34], on the other hand, pro-vides a more materials science viewpoint of the subjectmatter.

It is well established [e.g., 4] that plastic deformationin polysynthetically-twinned γ -TiAl + α2-Ti3Al singlecrystals parallel to the interlamellar γ -TiAl/α2-Ti3Alinterfaces is substantially easier (the soft mode) thandeformation normal to these interfaces (the hard mode).Therefore, the two modes of deformation are consid-ered separately.

In γ -TiAl lamellae, plastic deformation on planesparallel to the (111)γ ‖(0001)α2 interfaces is controlledboth by the 〈110](111) and the 〈101](111) slip systems.Grujicic and Batchu [13] found that slip resistances andstrain hardening characteristics differ in these two slipsystems in γ -TiAl single crystals. In polysynthetically-twinned γ -TiAl + α2-Ti3Al, each γ -TiAl lamella con-sists of multiple crystallographically-equivalent do-mains of this phase. The domains differ from each otherwith respect to along which of the six 〈1120〉 directionsof the α2-Ti3Al phase is 〈101] direction of the γ -TiAl

aligned in the (111)γ ‖(0001)α2 lamellar interface. Sincethe probability for formation of each domain is thesame, the effective slip properties of γ -TiAl lamella areassumed to be essentially equal in all close-packed di-rections in the (111) plane parallel to the lamellar inter-face. Since there are twice as many 〈101] directions than〈110] directions in a (111) plane of the γ -TiAl phase,and the probability for occurrence of each domain ina γ -TiAl lamella is essentially the same, Grujicic andBatchu [13] defined the effective properties for the soft-mode deformation of the γ -TiAl lamellae as a weightedaverage of those associated with 〈110](111) (weightingfactor = 1/3) and 〈101](111) (weighting factor = 2/3)slip systems in γ -TiAl single crystals.

For the soft-mode plastic deformation of γ -TiAllamellae, which involves slip in the close-packed di-rections parallel to the interlamellar boundaries buton planes inclined with respect to these boundaries.Grujicic and Batchu [13] found that its deformation re-sistance is somewhat higher than that associated withslip on planes parallel to the lamellar interfaces.

The contribution of plastic deformation of theα2-Ti3Al lamellae in a direction parallel to the(111)γ ‖(0001)α2 lamellar boundaries is neglected byGrujicic and Batchu [13] for the following reasons: (a)The soft-mode plastic deformation is expected to bedominated by the softer (γ -TiAl) phase; and (b) sincethe volume fraction of the α2-Ti3Al lamellae is typi-cally an order of magnitude smaller than that of theγ -TiAl lamellae, the role of α2-Ti3Al in the soft-modedeformation is expected to be minimal.

The situation is quite different for the hard-mode ofplastic deformation, i.e. for plastic deformation, whichincludes a component normal to the (111)γ ‖(0001)α2lamellar interfaces. In this case slip is controlled bythe phase which exerts more resistance toward thismode of plastic deformation, the α2-Ti3Al phase. Asestablished experimentally [35], the only slip systemin α2-Ti3Al which gives rise to this type of plas-tic deformation is the {1121}〈1126〉-〈c + a〉 slip sys-tem. Hence, this slip system is expected to control thehard-mode of plastic deformation in polysynthetically-twinned γ -TiAl + α2-Ti3Al single crystals. However,since the volume fraction of α2-Ti3Al is typically lessthan one tenth of that of γ -TiAl, and γ -TiAl appears asmultiple lamellae, limited hard-mode plastic deforma-tion in γ -TiAl is allowed. This possibility is incorpo-rated implicitly by Grujicic and Batchu [13] by allowingthe {1121}〈1126〉 slip systems in α2-Ti3Al lamellae ofpolysynthetically-twinned γ -TiAl + α2-Ti3Al to have alower deformation resistance relative to that in α2-Ti3Alsingle crystals.

To summarize, the soft-mode plastic deformationbehavior of polysynthetically-twinned γ -TiAl + α2-Ti3Al single crystals is assumed to be controlled bythe {111}〈110〉 slip systems of γ -TiAl with the slipdirection parallel with the lamellar interfaces, whilethe hard mode plastic deformation is taken to be con-trolled by the {1121}〈1126〉 slip of α2-Ti3Al. A detailedanalysis of deformation and strain-hardening behaviorof polysynthetically-twinned γ -TiAl + α2-Ti3Al singlecrystals is given in reference [13].

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The materials constitutive model developed byGrujicic and Batchu [13] assumes that each materialpoint contains a single material which is obtained byproper homogenization of the two phases: γ -TiAl andα2-Ti3Al. The (initial) reference configuration in thismaterial is taken to consist of a perfect stress-free crys-tal lattice and the embedded material. The position ofeach material point in the reference configuration isgiven by its position vector X . In the current configu-ration, each material point is described by its positionvector, x , and hence, mapping of the reference con-figuration into the current configuration is describedby the deformation gradient, F = ∂x/∂ X . In order toreach the current configuration, the reference configu-ration must be deformed both elastically and plasticallyand, hence, the total deformation gradient can be mul-tiplicatively decomposed into its elastic, Fe, and plas-tic, F P, parts as F = Fe F p. In other words, the defor-mation of a single-crystal material point is consideredto be the result of two independent atomic-scale pro-cesses: (i) an elastic distortion of the crystal lattice cor-responding to the stretching of atomic bonds and; (ii) aplastic deformation which is associated with atomicplane slippage which leaves the crystal lattice undis-turbed. While twinning is a potential inelastic deforma-tion mechanism at lower temperatures and higher strainrates, only slip on well defined crystallographic planesin low-index crystallographic directions is considered.

The constitutive model of Grujicic and Batchu [13]is based on the following governing variables: (i) TheCauchy stress, T ; (ii) The deformation gradient, F ;(iii) Crystal slip systems, labeled by integers α. Eachslip system is specified by a unit slip-plane normal nα

0 ,and a unit vector mα

0 aligned in the slip direction, bothdefined in the reference configuration; (iv) The plasticdeformation gradient, F p, with det F p = 1 (plastic de-formation by slip does not give rise to a volume change)and; (v) The slip system deformation resistance sα > 0which has the units of stress.

Based on the aforementioned multiplicative decom-position of the deformation gradient, the elastic defor-mation gradient Fe which describes the elastic distor-tions and rigid-body rotations of the crystal lattice, canbe defined by:

Fe ≡ F F p−1, det Fe > 0. (2)

The plastic deformation gradient, F p, on the otherhand, accounts for the cumulative effect of shearing onall slip systems in the crystal.

Since elastic stretches in intermetallic materials suchas γ -TiAl and α2-Ti3Al are generally small, the consti-tutive equation for stress under isothermal conditionscan be defined by the linear relation:

T ∗ = C[Ee] (3)

where C is a fourth-order anisotropic elasticity ten-sor, and Ee and T ∗ are respectively the Green elasticstrain measure and the second Piola-Kirchhoff stressmeasure relative to the isoclinic configuration obtainedafter plastic shearing of the lattice as described by F p.

Ee and T ∗ are respectively defined as:

Ee ≡ (1/2){FeT Fe − I T ∗ ≡ (det Fe)Fe−1T Fe−T

(4)

where I is the second order identity tensor.The elasticity tensor C (expressed as a 6 × 6 square

matrix with respect to the appropriate crystal-structurebased Cartesian coordinate system), for the tetragonalγ -TiAl phase contains six independent non-zero com-ponents: C11 = C22, C33, C12, C13 = C23, C44 = C55,and C66. The hexagonal α2-Ti3Al phase exhibits trans-verse elastic isotropy and, hence, its elasticity tensorcontains six different (five independent) non-zero com-ponents: C11 = C22, C33, C12, C13 = C23, C44 = C55,C66 = 0.5(C11 − C12). The effective elastic constantsfor the polysynthetically-twinned γ -TiAl + α2-Ti3Alsingle crystals are obtained as a proper weightedaverage.

The evolution equation for the plastic deformationgradient is defined by the flow rule:

F p F p−1 =∑

β

γ α Sα0 , Sα

0 ≡ mα0 ⊗ nα

0 , (5)

where Sα0 is the Schmid tensor and ⊗ denotes the ten-

sorial product of the two vectors.The components of a unit slip plane normal nα

0 anda unit slip direction mα

0 in the reference configurationsof the two phases are defined relative to the respectiveorthonormal bases in the two phases.

The plastic shearing rate γ α on a slip system α is de-scribed using the following simple power-law relation:

γ α = ˙γ|τα||sα|

1/m

sign(τα) (6)

where ˙γ is a reference plastic shearing rate, τα andsα are the resolved shear stress and the deformationresistance on slip system α, respectively and m is thematerial rate-sensitivity parameter.

Since elastic stretches in intermetallic materials isgenerally small, the resolved shear stress on slip systemα can be defined as:

τα = T ∗ · Sα0 (7)

where the raised dot denotes the scalar product betweentwo second order tensors.

Finally, the slip system resistance is taken to evolveas:

sα =∑

β

hαβ |γ β | (8)

where hαβ describes the rate of strain hardening onthe slip system α due to the shearing on the coplanar(self-hardening) and non-coplanar (latent-hardening)slip systems β and is given the following simple form:

hαβ = qαβhβ (9)

where, hβ denotes the self-hardening rate while qαβ isa matrix describing the latent hardening behavior for

313

which the following simple form is adopted:

qαβ = 1ql

{if α and β are coplanar slip systems,otherwise

(10)

The polysynthetically-twinned γ -TiAl + α2-Ti3Alsingle crystal as modeled by Grujicic and Batchu[13] has one γ -TiAl based set of coplanar slip sys-tems defined as following: (111)[110], (111)[101], and(111)[011].

The self-hardening rate hβ is defined as:

hβ = hβ

0

∣∣∣∣∣1 − sβ

sβs

∣∣∣∣∣τ

sign

(1 − sβ

sβs

). (11)

where hβ

0 is the initial hardening rate and sβs the satura-

tion slip deformation resistance and r power exponent.Finite element analysis of deformation and fracture

behavior of polysynthetically-twinned polycrystallineγ -TiAl + α2-Ti3Al materials requires the integration ofthe material state represented by Equations 9, 10 and 12along the loading path. A detailed description of thisprocedure and its implementation into a User MaterialSubroutine (UMAT) of Abaqus/Standard is given in ourprevious work [23].

2.3. Polycrystalline finite element methodTo analyze the deformation and fracture behavior ofpolycrystalline γ -TiAl + α2-Ti3Al materials, a finite el-

Figure 4 (a) The finite element mesh consisting of 971 quadrilateral and 364 triangular elements; (b) The polycrystalline reference configurationconsisting of 27 lamellar colonies; (c) The finite element mesh consisting of 612 quadrilateral and 36 triangular elements, and (d) The polycrystallinereference configuration consisting of three lamellar colonies, each containing two γ -TiAl/γ -TiAl interlamellar boundaries.

ement mesh shown in Fig. 4a consisting of 971 quadri-lateral and 364 triangular elements is used in the presentwork. The mesh is partitioned into 27 equiaxed (hexag-onal) lamellar colonies, Fig. 4b. The materials con-stituent response of each of the colonies is described us-ing the crystal-plasticity model of Grujicic and Batchu[13] for polysynthetically twinned γ -TiAl + α2-Ti3Alsingle crystals which was reviewed in the previous sec-tion. No provisions are made for the presence of in-terlamellar boundaries within the colonies. However,colony boundaries are modeled using the cohesive zoneapproach, which allows boundary decohesion. The ini-tial orientation of colonies 14, 19 and 20 are chosen insuch a way that in each of these colonies, the lamellarinterfaces are normal to the x–y plane of the finite ele-ment mesh, Figure 4(b), and their traces make an angleof 0◦, 60◦ and 120◦ with the x-axis, respectively. Theinitial orientation of the remaining colonies, as definedby the appropriate set of three Euler angles, is assignedat random. The initial (reference) configuration is as-sumed to be stress free and not to contain any latticeperturbations. The polycrystalline aggregates definedin Fig. 4b is loaded in the y direction while constrain-ing the four straight edges of the aggregates to remainstraight and parallel to their original orientation withno strain being allowed in z direction. While the aggre-gate does not possess any special symmetry and, hence,is not expected to deform in the enforced orthotropicmanner, the aggregate is considered here as a “mate-rial point” and according to the Taylor assumption, the

314

deformation gradient of each material point is assumedto be equal to the global deformation gradient.

Based on the results obtained for the polycrystallineaggregate, the triangular region marked using dashedlines in Fig. 4a is selected for a further, fine-scale fi-nite element analysis. This region is divided into 612quadrilateral and 36 triangular elements, Fig. 4c, andpartitioned into three colonies, Fig. 4d. The crystallo-graphic orientation of the material in the three coloniesis set to be identical to that of colonies 14, 19 and 20 inFig. 4b. Each colony boundary is again modeled usingthe cohesive zone approach. In addition, each colonyis allowed to have two γ -TiAl/γ -TiAl lamellar inter-faces. The position of the lamellar interfaces within thecolonies is chosen at random. The loading on the three-colony aggregate shown in Fig. 4c is applied by settingthe displacement history of the boundary nodes of thisaggregate to be identical to the deformation history ofthe corresponding triangular region marked in Fig. 4b.

3. Results and discussion3.1. Aggregate consisting of 27

γ -TiAl + α2-Ti3Al lamellar coloniesIn this section, the deformation fields are shown and an-alyzed for the 27-colony γ -TiAl + α2-Ti3Al aggregate,Fig. 4b, loaded in plane-strain tension. Specifically, thecontour plots are shown for: (a) the equivalent plas-tic strain; (b) the angle of the lattice rotation; (c) thehydrostatic stress and; (d) the equivalent stress.

Figure 5 Contour plots in the 27-colony aggregate at the overall noraml strain of 1.5% in the vertical direction (the darker the shade of gray the higherthe value of the corresponding quantity): (a) the equivalent plastic strain (0.05–2.44%); (b) the lattice rotation (−1.4–1.8◦); (c) the hydrostatic stress(−83–460 MPa) ; and (d) the equivalent stress (52–409 MPa).

The distributions of the equivalent plastic strain, thelattice-rotation angle, the hydrostatic stress and theequivalent stress for the 27-colony aggregate at the totalnormal strain in the y-direction of 1.5% are shown inFig. 5a–d, respectively.

The equivalent plastic strain contour plot shown inFig. 5a clearly indicates a nonuniform distribution ofthe equivalent plastic strain. The values of the equiva-lent plastic strain in Fig. 5a range between 0.05% (thelightest shade of gray) and 2.44% (the darkest shade ofgray). Furthermore, localization of the strain into defor-mation bands inclined relative to the x-axis at an angleof ≈±40◦, as marked by arrows in Fig. 5a, is apparent.The same deformation bands could be first observed atthe total normal strain in the y-direction of ≈0.5% (thecontour plot not shown for brevity) and as deformationproceeded, they become more pronounced.

In addition to the nonuniformity in the distributionof the equivalent plastic strain on a length-scale of the27-colony aggregate, the distribution of the equivalentplastic strain is quite nonuniform within each colonyas well. For example, within colonies 8, 13, 14, 19,etc., at the total normal strain in the y direction of1.5%, Fig. 5a, the equivalent plastic strain ranges be-tween 0.2 and 2.44%. Furthermore, some colonies con-tain regions of high gradients of the equivalent plasticstrain. Such regions are often located either near thecolony boundaries (e.g. colony boundaries associatedwith colonies 7 and 12 and colonies 10 and 15) or near

315

the three-colony junctions (e.g. 7-12-13 and 10-15-16colony junctions).

The distribution of the magnitude of the lattice ro-tation in the 27-colony aggregate at the overall plas-tic strain of 1.5% is shown in Fig. 5b. Counterclock-wise lattice rotations are described as positive while theclockwise lattice rotations are denoted as negative. Thevalues of the lattice rotations in Fig. 5b range between−1.4 (the lightest shade of gray) and +1.8 degrees(the darkest shade of gray). A careful examination ofFig. 5b reveals that the largest rotations take place nearthe colony boundaries (e.g. the boundaries betweencolonies 12 and 13, colonies 9 and 10, etc.) and near thethree-colony junctions (e.g. 3-4-9, 12-13-18, 19-20-25,etc. three-colony junctions). Furthermore, a compari-son of the corresponding results shown in Fig. 5a and bsuggests that the largest rotations take place within thecolonies located in the deformation bands (e.g. colonies5,13,19 and 20 in one deformation band, and colonies9 and 22 in another deformation band). This is not sur-prising since the lattice rotations are the natural conse-quence of the effect of the orthogonal kinematic con-straints imposed by the boundary conditions (e.g. thefour straight edges of the 27-colony aggregate are re-quired to remain straight and parallel to their originalorientation) and the constraints associated with the sur-rounding colonies on the crystallographic shear withineach colony. Hence, the regions characterized by largestlevels of the equivalent plastic strain are generally ex-pected to experience largest magnitudes in the latticerotation.

The distribution of the hydrostatic stress in the27-colony aggregate at the overall plastic strain of1.5% is shown in Fig. 5c. The values of the hydro-static stress shown in Fig. 5c range between −83(the lightest shade of gray) and 460 MPa (the dark-est shade of gray). This figure clearly shows that thehydrostatic stress is distributed quite nonuniformly notonly among the colonies but also within each colony.Some colonies (e.g. 7, 12 and 23, etc.) are subject topositive (tensile) hydrostatic stresses, while the others(e.g. 14, 19, 20, etc.) experience both positive (tensile)and negative (compressive) hydrostatic stresses. High-magnitude hydrostatic stresses are generally concen-trated near the colony boundaries or near the three-colony junctions. In particular, the three-colony junc-tions associated with colonies 7, 12 and 13 and colonies14, 19, and 20 are characterized by a high gradient anda large magnitude of the positive hydrostatic stress. Acomparison of the results shown in Fig. 5c with the re-sults shown in Fig. 5a and b suggests that the most prob-able cause for the observed high magnitude and highgradient of the hydrostatic stress at the 7-12-13 three-colony junction is the incompatibility of the plastic de-formation in the three colonies which is manifested bya large variation in the equivalent plastic strain and inthe lattice rotations across the 7-13 and 12-13 colonyboundaries. On the other hand, since neither the equiv-alent plastic strain nor the lattice rotations vary greatlyin the region surrounding the 14-19-20 three-colonyjunction, the concentration of the hydrostatic stressesin this case, appears to be associated with the elastic

anisotropy and pronounced misorientation of the crys-tal lattice in the three joining colonies.

The distribution of the equivalent stress in the 27-colony aggregate at the overall plastic strain of 1.5%is shown in Fig. 5d. The values of the equivalent stressshown in Fig. 5d range between 52 (the lightest shadeof gray) and 409 MPa (the darkest shade of gray). Thisfigure clearly shows that the distribution of the equiv-alent stress is quite nonuniform both on a length-scaleof the 27-colony aggregate as well as on the length-scale of a single colony. By comparing the results inFig. 5d with the results shown in Fig. 5a and b, no clearcorrelation between the distributions of the equivalentplastic strain and the lattice rotations on one hand andthe equivalent stress on the other can be established.Similar comparison of the results shown in Fig. 5d withthe results shown in Fig. 5c, however, suggests that theregions of a high concentration of the hydrostatic stress(e.g. 7-12-13 and 14-19-20 three-colony junctions) aregenerally associated with high gradients in the equiva-lent stress. Furthermore, in many cases, several closedequivalent-stress contours could be seen within a sin-gle colony (e.g. colonies 8, 18, 19, etc.). The presenceof such contours is a clear indication of the kinematicconstraints imposed by the surrounding colonies on thegiven colony.

The equivalent plastic-strain, the lattice rotation an-gle, the hydrostatic stress and the equivalent stress con-tour plots in the 27-colony aggregate at the total nor-mal strain in the y direction of 3.0% are shown inFig. 6a–d, respectively. Nonuniform distribution of theabove-mentioned quantities, which was observed andanalyzed at the total normal strain in the y-direction of1.5%, is retained and even somewhat more pronounced.For instance, the shear bands seen in Fig. 6a are sharperthan the corresponding ones observed in Fig. 5a. Theequivalent plastic strain shown in Fig. 6a ranges be-tween 0.06 and 5.57% in comparison to the 0.05 to2.44% equivalent plastic strain range associated withFig. 5a. Similar observations can be made by compar-ing the contour plots in Fig. 6b–d with the correspond-ing ones displayed in Fig. 5b–d. The range of the re-spective quantities (the angle of the lattice rotation, thehydrostatic stress, and the equivalent stress) are givenrespectively in the captions of Fig. 6b–d.

At this point it is worth noting that the observed dis-tributions in hydrostatic and equivalent stresses, equiv-alent plastic strains and lattice rotations at the overallplastic strains of 1.5% and 3.0%, Figs 5a–d and 6a–d,are quite similar to their counterparts reported by Asaroand co-workers [7, 11]. This finding is reasonable sinceat these levels of the overall plastic strain, the extent ofdecohesion along colony boundaries (modeled in thepresent work but not in the work of Asaro and co-workers [7, 11]) is quite limited.

At the overall strain of 3.19%, pronounced de-cohesion occurred along the 14-19 and 14-20colony-boundaries causing the execution of theAbaqus/Standard job to terminate. With the exceptionof the hydrostatic stress, the equivalent plastic strain,the lattice rotation and the equivalent stress contourplots at the total normal strain in the y-direction of

316

Figure 6 Contour plots in the 27-colony aggregate at the overall noraml strain of 3.0% in the vertical direction (the darker the shade of gray the higherthe value of the corresponding quantity): (a) the quivalent plastic strain (0.06–5.58%); (b) the lattice rotation (−3.3–4.3◦); (c) the hydrostatic stress(94–522 MPa); and (d) the equivalent stress (19–439 MPa).

Figure 7 The hydrostatic stress contour plots in colonies 14, 19 and 20 of the 27-colony aggregate at the overall noraml strains in the vertical directionof: (a) 3.0% and (b) 3.19%. To assist comapisonof, the contour lines in (a) and (b) are are placed at the same levels of the hydrostaic stress.

3.19% are very similar to the ones at 3% of the sametype of strain, Fig. 6a and b and d. Hence these con-tour plots are not shown. As far as the hydrostaticstress is concerned, it undergoes a significant changein the range of the total normal strain in the y direc-tion between 3.0 and 3.19 only in the vicinity of the14-19-20 three-colony junction. Hence, in Fig. 7a–b,contour plots for the hydrostatic stress are shown onlyfor the region comprising colonies 14, 19 and 20. Toassist comparison of the results shown in Fig. 7a and b,the contour lines are placed at the same levels of the hy-drostatic stress in the two figures. As evident from theresults shown in Fig. 7a and b, the hydrostatic stressdecreases as loading is proceeded between the total

normal strain in the y direction of 3.0% and 3.19%.As mentioned earlier, this is caused by decohesionwhich takes place along the 14-19 and 14-20 colonyboundaries. The decohesion process along the 14-19and 14-20 colony-boundaries and its relationship withthe presence of lamellae in the three colonies is exam-ined in more details in the next section.

3.2. Aggregate consisting of threeγ -TiAl + α2-Ti3Al lamellar colonies

In order to better understand the conditions leading tomaterial fracture by decohesion along colony bound-aries and the effect of lamellar interfaces within thecolonies on the strain to fracture, a finite element

317

Figure 8 Contour plots for the equivalent plastic strain, (a)–(c), and the hydrostatic stress, (d)–(f) in a region surrounding the three-colony junctionof the 3-colony aggregate without lamellar interfaces at the overall noraml strain: (a) and (d) 2.5%; (b) and (e) 3.0% and (c) and (f) 3.19%. The twoquantitis vary between 0.05% and 500 MPa (the lightest shade of gray) and 5.0% and 1300 MPa (the darkest shade of gray), respectively.

analysis of the evolution and the distributions of thedeformation fields discussed in the previous section, ata smaller length scale, the length scale of the triangularregion surrounding the 14-19-20 three-colony junctionis carried out in this section. As discussed in Section 2.3,external loading of this region is achieved by prescrib-ing the displacements along its external boundaries. Theevolution of these displacements during loading is setto be identical to that for the corresponding triangularregion in the 27-colony aggregate, Fig. 4b.

The distributions of the equivalent plastic strain andthe hydrostatic stress in the triangular 3-colony ag-gregate without lamellar interfaces at the total normalstrain in the y-direction of 2.5, 3.0 and 3.19% are shownin Fig. 8a–f, respectively. For improved clarity only theinnermost rectangular region surrounding the 14-19-20three-colony junction is shown in these figures. The re-sults shown in Fig. 8a–f can be summarized as follows:

– In accordance with the equivalent plastic strain dis-tribution results at the length-scale of the 27-colonyaggregate, Figs 5a and 6a, the level of the equiv-

alent plastic strain in colony 19 is considerablyhigher than those in colonies 14 and 20 whenviewed at the length-scale of the 3-colony aggre-gate, Fig. 8a–c. This finding suggests that the load-ing applied to the 3-colony aggregate is consistentwith the loading experienced by the correspond-ing triangular region in the 27-colony aggregate,Fig. 4b.

– While some of the equivalent plastic strain con-tours located within colony 19, Fig. 8a–c, are gen-erally aligned with the +40◦ deformation band,Figs 5a and 6a, the equivalent plastic strain dis-tribution near the three-colony junction is domi-nated by incompatibilities in plastic deformationof the adjoining colonies and colony-boundariesdecohesion.

– Even at an overall normal strain in the y direc-tion of 2.5%, prominent 14-19 and 14-20 colony-boundary decohesion is observed. However, at thislevel of the overall normal strain in the y direc-tion, decohesion is localized to a small region sur-rounding the 14-19-20 three-colony junction. As

318

Figure 9 Contour plots for the equivalent plastic strain, (a)–(c), and the hydrostatic stress, (d)–(f) in a region surrounding the three-colony junction ofthe 3-colony aggregate with two lamellar interfaces per colony at the overall normal strain: (a) and (d) 2.5%; (b) and (e) 3.0% and (c) and (f) 3.19%.The two quantities vary between 0.05% and 500 MPa (the lightest shade of gray) and 5.0% and 1300 MPa (the darkest shade of gray), respectively.

loading is continued, on the other hand, decohe-sion extends along the two colony-boundaries. Itshould be noted that in order to improve the clar-ity of the results, displacements in Fig. 8a–f aremagnified by a factor of 50.

– The progression of decohesion is closely related tothe distribution of the hydrostatic stress, Fig. 8d–f.That is, as loading is continued, the hydrostaticstress increases, causing an increase in the nor-mal displacement jump across colony boundaries.However, once displacement jumps exceed a crit-ical value, (which is dependent on the magnitudesof the corresponding colony-boundary shear dis-placement jumps) the boundary’s ability to supportthe load decreases and local (decohesion-induced)unloading takes place. Some evidence of such un-loading can be seen by comparing the hydrostaticstress contours in Fig. 8e and f.

– The distributions of the equivalent plastic strainand the hydrostatic stress in the triangular 3-colonyaggregate in which each colony contains twolamellar interfaces at the total normal strain in

the y-direction of 2.5, 3.0 and 3.19% are shownin Fig. 9a–f, respectively. To enable a compar-ison with the results shown in Fig. 8a–f, thesame rectangular region surrounding the 14-19-20three-colony junction is shown in Fig. 9a–f. Inaddition, same levels of the equivalent plasticstrain/hydrostatic stress are represented in the twosets of figures using the same shades of gray. Theresults shown in Fig. 9a–f can be summarized asfollows:

– As in the case of the 3-colony aggregate with-out interlamellar boundaries, Fig. 8a–c, the equiv-alent plastic strain is also localized near the14-19-20 three-colony junction in the present case,Fig. 9a–c. However, significant localization of theequivalent plastic strain near intersections of thecolony boundaries and lamellar interfaces can alsobe observed.

– At the identical levels of the overall normal strainin the y direction, the extent of decohesion alongcolony boundaries in the material containing in-terlamellar boundaries, Fig. 9a–f, is significantly

319

Figure 10 Normal stress versus normal strain in the y direction in a3-colony aggregate without interlamellar boundaries (“No Lamellae”)and in two 3-colony aggregates with interlamellar boundaries (“WithLamellae #1” and “With Lamelae #2”).

lower than in the material without lamellar inter-faces, Fig. 8a–f. However, in the former case, con-siderable decohesion takes place also along inter-lamellar boundaries.

– In comparison with the results shown in Fig. 8d–f,the hydrostatic-stress contour plots for the mate-rial with interlamellar boundaries, Fig. 9d–f, showthat hydrostatic stress is lower in magnitude andmore uniformly distributed. This finding suggeststhat the presence of lamellar interfaces facilitatesaccommodation of the incompatibilities in plasticflow of the adjacent colonies. In addition, the re-sults shown in Fig. 9d–f indicate that relatively highhydrostatic stresses are situated not only near the14-19-20 three-colony junction but also near thecolony-boundary/lamellar-interface intersections.

The average normal stress versus normal strain (in they direction) curves for the 3-colony aggregates withoutand with interlamellar boundaries are shown in Fig. 10.The curve denoted as “No Lamellae” corresponds tothe 3-colony aggregate without lamellar interfaces andshows that in this case the material undergoes an abruptfracture at the overall normal strain of 3.19%. Contrary,in the 3-colony aggregates containing lamellar inter-faces (Curves labeled “With Lamellae #1” and “WithLamellae #2”, fracture is somewhat delayed (fracturestrain 3.8–3.9%) and occurs in a more gradual man-ner. The curve labeled “With Lamellae #1” correspondsto the 3-colony aggregate for which the contour plotsare displayed in Fig. 9a–f. The curve labeled “WithLamellae #2”, on the other hand, corresponds to an-other 3-colony aggregate (not shown for brevity) witha different location of lamellar interfaces. The second3-colony aggregate with lamellar interfaces is used toconfirm that the observed effects of increased strainto fracture and a more gradual nature of the fractureprocess are primary due to the presence of lamellar in-terfaces and not due to their fortuitous location in thefirst 3-colony aggregate.

4. ConclusionsBased on the results obtained in the present study thefollowing main conclusions can be drawn:

– The incompatibilities in plastic flow betweenthe adjacent lamellar colonies in polycrystallineγ -TiAl + α2-Ti3Al materials resulting from a pro-nounced plastic anisotropy in these materials cangive rise to a large build-up of the tensile hydro-static stress in regions surrounding certain three-colony junctions and, in turn, can lead to materialfracture via colony-boundary decohesion.

– The fracture behavior of the material following theonset of colony-boundary decohesion appears to begreatly affected by the absence/presence of inter-lamellar boundaries. In the absence of interlamellarboundaries, decohesion along colony-boundariescontinues to take place quite rapidly leading to im-minent fracture of the material. In sharp contrast,when interlamellar boundaries are present, theyalso undergo some decohesion. Consequently, in-stead of decohesion taking place rapidly and solelyalong colony boundaries, it also spreads out to ad-jacent lamellar interfaces. As a result, fracture oc-curs in a more gradual manner and the materialbecomes more damage resistant.

– The tensile fracture strains (3–4%), predicted in thepresent work are very comparable to experimen-tal counterparts [e.g., 19] suggesting that colony-boundary decohesion plays a critical role in frac-ture of the materials at hand. This observation isconsistent with the scanning electron microscopyresults of Kad et al. [7], which showed that whilethe fracture surface reveals a combination of theintercolony (along colony boundaries) and trans-colony (along interlamellar boundaries) fracturemodes, the region underneath the fracture sur-face contains numerous colony-boundary cracks.In other words, the fracture appears to be initiatedand, most likely controlled, by colony boundarydecohesion.

– The presence of interlamellar boundaries is foundto improve materials tensile ductility, which is fullyconsistent with the experimental observations ofGrujicic and Dang [19] that lamellar γ -TiAl-baseintermetallics posses higher strain to fracture thantheir single-phase (non-lamellar) counterparts.

Appendix A: Derivation of the interfaceelements stiffness matrixThe γ -TiAl/γ -TiAl lamellar interface and colonyboundary potentials developed in Section 2.1 are incor-porated into the UEL subroutine of Abaqus/Standardto compute the stiffness matrix of the correspondinginterfacial elements. The UEL subroutine allows theuser to define the contribution of interfacial elementsto the global finite element model. In other words, forthe given nodal displacements of the interface elementsprovided to UEL by Abaqus/Standard, the contributionof the interfacial elements to the global vector of resid-ual forces and to the global Jacobian (element stiff-ness matrix) is determined in the UEL subroutine and

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Figure A1 Definition of a linear, four-node interface element. Nodes 1 and 4 and nodes 2 and 3 coincide in the reference configuration. Internal nodesA and B located at the midpoints of segments connecting corresponding nodes on the opposite sides of the interface, two integration points markedas + and a local s-n coordinate system are also indicated.

passed to Abaqus/Standard. The implementation of theγ -TiAl/γ -TiAl interface/boundary potentials (derivedin Section 2.1) in the UEL subroutine is discussedbelow.

Since only two-dimensional analysis is carried outin the present work, each interface element is definedas a four-node isoparametric element on the γ -TiAl/γ -TiAl lamellar interface/colony boundary S, as shownschematically in Fig. A1. In the undeformed configura-tion (not shown for brevity), nodes 1 and 4, and nodes2 and 3 coincide, respectively. A local coordinate sys-tem, consistent with the directions, which are tangent(s) and normal (n) to the interface, is next assigned tothe each element. Direction s is colinear with directionst and b used in Section 2.1 so that Ut = Uscos α andUb = Ussin α where Ut , Ub and Us are displacementsalong the three tangential directions and α is the anglebetween s and t shear directions. Next, two ‘internalnodes’, A and B, located at the midpoints of the lines1-2 and 3-4, connecting the corresponding interface/boundary nodes of the two lamellae/colonies are intro-duced. The interface displacements at the internal nodesA and B are expressed in terms of the displacements ofthe element nodes 1–4 as:

U An = (

U 4y − U 1

y

)cos θ − (

U 4x − U 1

x

)sin θ (A1)

U As = (

U 4y − U 1

y

)cos θ − (

U 4x − U 1

x

)sin θ (A2)

U Bn = (

U 3y − U 2

y

)cos θ − (

U 3x − U 2

x

)sin θ (A3)

U As = (

U 4y − U 1

y

)cos θ − (

U 4x − U 1

x

)sin θ (A4)

An isoparametric coordinate η is next intro-duced along the tangent direction with η(A) = −1

and η(B) = 1 and two linear Langrangian interpola-tion functions are defined as NA(η) = (1 − η)/2 andNB(η) = (1 + η)/2.

The interpolation functions given above allow thenormal and the tangential components of the interfacedisplacements to be expressed in the form of their valuesat the internal nodes A and B as:

Us(η) = NA(η)U As + NB(η)U B

s (A5)

Un(η) = NA(η)U An + NB(η)U B

n (A6)

The tangential and normal components of the forcesat nodes A and B, i.e. F A

s , F Bs , F A

n and F Bn , which

are work conjugates of the corresponding nodal dis-placements U A

s , U Bs , U A

n and U Bn are next determined

through the application of the virtual work to the inter-facial element as:

∫ 1

−1δ�(η)Lπr (η) dη =

∑I=n.s

∑N=A.B

F NI δU N

I (A7)

where L is the A-B element length. The perturbationof interface potential is expressed in terms of the per-turbations of the interface displacements at the internalnodes A and B, U A

s , U Bs , U A

n and U Bn as:

∂� = ∂�[Us(η), Un(η)]

∂Un

[NA(η)δU A

n + NB(η)δU Bn

]+ ∂φ[Us(η), Un(η)]

∂Us

[NA(η)δU A

s + NB(η)δU Bs

](A8)

321

By substituting Equation A8 into A7 and by choosingone of the δU N

I (N = A, B; l = s, n) perturbations at atime to be unity and the remaining perturbations to bezero, the corresponding F N

I component of the nodalforce can be expressed as:

F NI =

∫ 1

−1

∂�[Us(η), Un(η)]

∂UINN (η)Lπr (η) dη (A9)

Using a straightforward geometrical procedure andimposing the equilibrium condition, the correspondingresidual nodal forces Ri

x and Riy (i = 1–4) in the global

x-y coordinate system, are defined as:

R1x = −R4

x = F As cos θ − F A

n sin θ

R1y = −R4

y = F As sin θ − F A

n cos θ

R2x = −R3

x = F Bs cos θ − F B

n sin θ

R2y = −R3

y = F Bs sin θ − F B

n cos θ (A10)

The components of the element Jacobian are nextdefined as:

∂ Rij

∂U kI

=∑I=n·s

∑N=A·B

∑j=n·s

∑M=A·B

∂ Rij

∂ F NI

∂ F NI

∂U MJ

∂U MJ

∂U kI

(A11)

where the components of the internal Jacobian ∂ F NI

∂U MJ(i , j = n, t ; N , M = A, B) are calculated by differenti-

ation of Equation A9. The residual nodal forces givenby Equation A10 and the element Jacobian given byEquation A11 are computed in the UEL subroutine, andpassed to Abaqus/Standard for accurate assessment ofkinematics in its global Newton scheme.

AcknowledgementsThe material presented here is based on work supportedby the National Science Foundation, Grant NumbersDMR-9906268 and CMS-9531930 and by the U.S.Army Grant Number DAAH04-96-1-0197. The authorsare indebted to Dr. K. L. Murty of NSF and Dr. David M.Stepp of ARO for the continuing interest in the presentwork. The authors also acknowledge the support of theOffice of High Performance Computing Facilities atClemson University.

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Received 4 December 2001and accepted 18 September 2002

322