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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 analysisof deformation and fracture of polycrystallinelamellar -Tial + 2-Ti3al alloys

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

    Deformation behavior of fully-lamellar polycrystalline -TiAl + 2-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 -TiAl/ -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 -TiAl + 2-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 -TiAl + 2-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., 13], 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 -TiAl + 2-Ti3Al alloys re-mains an important, though formidable, engineeringchallenge.

    During solidification, Ti-(4850) at.% Al alloys,which are considered in the present work, first forma disordered hexagonal-close-packed (h.c.p.) -phasewhich during cooling orders into an 2-Ti3Al-typephase with the DO19 crystal structure and then trans-forms to (or near) completion into an ordered face-centered-tetragonal (f.c.t.) -TiAl phase with the L10crystal structure. The final microstructure typically con-sists of colonies each containing micron-thick parallel -TiAl and 2-Ti3Al lamellae with a standard f.c.c.-h.c.p. type orientation relationship between the twophases: {111} (0001)2 and 110] 11202. Inaddition, the /2 and / lamellar boundaries havethe following crystallographic orientations: {111} (0001)2 and {111} {111} , respectively. Under uni-directional solidification condition, the microstructureof Ti-(4850) at.% Al can be obtained which consists ofa single colony of -TiAl + 2-Ti3Al lamellae [e.g., 4].Such alloys are referred to as having polysynthetically-twinned single crystalline microstructure.

    The microstructure of polycrystalline -TiAl + 2-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-(4850) at.% Al al-loys, fully-lamellar and duplex microstructure impart

    00222461 C 2003 Kluwer Academic Publishers 307

  • the best combinations of mechanical properties. Thefully-lamellar microstructure consists of colonies of -TiAl + 2-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 -TiAl and2-Ti3Al lamellae, small -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 -TiAl + 2-Ti3Alalloys will be analyzed.

    The mechanical response of -TiAl + 2-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 the -TiAl + 2-Ti3Al lamellar microstructure, -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 -TiAl phase is constrained by the harder2-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 2-Ti3Al. Since the lattersystems are needed to achieve a general state of strain,the 2-Ti3Al phase can be considered as kinematicallyconstrained in the c-direction [e.g., 7].

    The anisotropic behavior of polysynthetically-twinned -TiAl + 2-Ti3Al single crystals describedabove is, to some extent, retained in the polycrystallineform of these materials. However, additional changes inthe mechanical behavior of lamellar -TiAl + 2-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 (

  • 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 -TiAl/2-Ti3Al in-terlamellar boundaries and the retention of the -TiAlvs. 2-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-twinned -TiAl + 2-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. -TiAl/ -TiAl lamellar-interface and

    colony-boundary constitutive relationsTo comply with the experimental observations [e.g., 5],fracture is allowed to occur along the -TiAl/ -TiAllamellar interfaces and colony boundaries. Towardthis end, the mechanical constitutive response of the -TiAl/ -TiAl lamellar interfaces and the boundariesbetween -TiAl + 2-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