R.A. Lebensohn

N-SITE MODELING OF A 3D VISCOPLASTIC POLYCRYSTAL USING FAST FOURIER TRANSFORM

We present a formulation to compute the local response of elastic and viscoplastic anisotropic 3D polycrystals based on the Fast Fourier Transform (FFT) algorithm. This formulation is conceived for periodic heterogeneous microstructures and also for materials with random spatial distribution of heterogeneities. The approach is of the n-site kind, provides an exact solution of the equilibrium equation and has better numerical performance than small-scale FEM. The viscoplastic FFT formulation combined with an ad-hoc microstructure updating scheme is used to predict local states, morphology and texture evolution of ideal f.c.c. polycrystals. The model predicts strain localization, intragranular misorientation and subgrain formation and overall textures which are smoother than those obtained with classical 1-site schemes, in better agreement with experiments.

MECHANICAL RESPONSE OF ZIRCONIUM—I. DERIVATION OF A POLYCRYSTAL CONSTITUTIVE LAW AND FINITE ELEMENT ANALYSIS

Simulating the forming of anisotropic polycrystals, such as zirconium, requires a description of the anisotropy of the aggregate and the single crystal, and also of their evolution with deformation (texture development and hardening). Introducing the anisotropy of the single crystal requires the use of polycrystal models that account for inhomogeneous deformation depending on grain orientation. In particular, visco- plastic self-consistent models have been successfully used for describing strongly anisotropic aggregates. As a consequence, using a polycrystal constitutive law inside finite element (FE) codes represents a considerable improvement over using empirical constitutive laws, since the former provides a physically based description of anisotropy and its evolution. In this work we develop a polycrystal constitutive description for pure Zr deforming under quasi-static conditions at room and liquid nitrogen temperatures. We use tensile and compressive experimental data obtained from a clock-rolled Zr sheet to adjust the constitutive parameters of the polycrystal model. Twinning is accounted for in the description. The polycrystal model is implemented into an explicit FE code, assuming a full polycrystal at the position of each integration point. The orientation and hardening of the individual grains associated with each element is updated as deformation proceeds. We report preliminary results of this methodology applied to simulate the three-dimensional deformation of zirconium bars deforming under four-point bend conditions to maximum strains of about 20%. A critical comparison between experiments and predictions is done in a second paper (Kaschner et al., Acta mater. 2001, 49(15), 3097-3107). Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.

A self-consistent approach for modelling texture development of two-phase polycrystals application to titanium alloys

A large strain self-consistent viscoplastic model is proposed, developed and applied to a two-phase polycrystal. This model accounts for crystallographic textures and grain morphologies, as well as for the phase correlation, both in space and orientation. The basic formulation is shown and the case of lamellar (α + β) Ti alloys in rolling is studied. In these alloys, the two phases exhibit specific morphologic and crystallographic correlations. The present study shows that the model leads to better texture predictions when all these correlations are accounted for.

Modelling deformation and recrystallization textures in calcite

A self-consistent polycrystal plasticity theory is applied to model the deformation and recrystallization behavior of rhombohedral calcite. In this mineral deformation occurs by slip and twinning and in experimentally deformed aggregates a strong crystallographic texture develops. Whereas the Taylor theory has failed to model axial compression, the self-consistent theory automatically introduces heterogeneous deformation by curling and simulates the correct texture. A deformation-based recrystallization model, that balances growth and nucleation, predicts a so far enigmatic high temperature texture. Aspects of mechanical twinning are also discussed. Many of the conclusions are directly applicable to other low-symmetry materials, especially hexagonal metals.

A convection model to explain anisotropy of the inner core

Seismic evidence suggests that the solid inner core of the Earth may be anisotropic. Several models have been proposed to explain this anisotropy as the result of preferred orientation of crystals. They range from a large annealed single crystal, growth at the melt interface, to deformation-induced texture. In this study texture development by deformation during inner core convection is explored for e-iron (hcp) and γ-iron (fcc). Convection patterns for harmonic degree two were investigated in detail. In the model it is assumed that traces of potassium are uniformly dispersed in the inner core and act as a heat source. Both for fee and hep iron, crystal rotations associated with intracrystalline slip during deformation can plausibly explain a 1–3% anisotropy in P waves with faster velocities along the N-S axis and slower ones in the equatorial plane. The effect of single crystal elastic constants is explored.

Heterogeneous deformation and texture development in halite polycrystals: comparison of different modeling approaches and experimental data

Modeling the plastic deformation and texture evolution in halite is challenging due to its high plastic anisotropy at the single crystal level and to the influence this exerts on the heterogeneity of deformation over halite polycrystals. Three different assumptions for averaging the single crystal responses over the polycrystal were used: a Taylor hypothesis, a self-consistent viscoplastic model, and a finite element methodology. The three modeling approaches employ the same single crystal relations, but construct the polycrystal response differently. The results are compared with experimental data for extension at two temperatures: 20 and 100 degreesC. These comparisons provide new insights of how the interplay of compatibility and local equilibrium affects the overall plastic behavior and the texture development in highly anisotropic polycrystalline materials. Neither formulation is able to completely simulate the texture development of halite polycrystals while, at the same time, giving sound predictions of microstructural evolution. Results obtained using the finite element methodology are promising, although they point to the need for greater resolution of the individual crystals to capture the full impact of deformation heterogeneities

Development of preferred orientation and microstructure in sheared quartzite: comparison of natural data and simulated results

Abstract c-axis fabric and microstructures in a quartzite sample, sheared and extensively recrystallized under greenschist facies conditions, have been analyzed and compared with theoretical predictions using a viscoplastic self-consistent model modified to incorporate the effects of dynamic recrystallization. An asymmetric small-circle c-axis fabric about the finite shortening z-axis with a small half opening angle (35°) is present in the sample; it consists of four orientation components which are represented by host grain c-axis orientations (referred to as A, B, C and D): A and B are at high angles to the foliation plane, displaced against and with the sense of shear, respectively; C is in an intermediate direction between the Y- and Z-axis of finite strain, and D forms a subsidiary concentration around the intermediate strain (Y-) axis. B- and C-grains are favorably oriented for basal (0001) and pyramid {10 1 1}〈a〉 slip, respectively, and strongly deformed, while A- and D-grains are unfavorably oriented for the slip systems and little or moderately deformed. Some of A-grains are even fractured. The degree of dynamic recrystallization increases with increasing strain undergone by differently oriented grains (in the sequence of A-, D-, C- and B-grains). Microstructural evidence and theoretical predictions indicate that harder A-, C- and D-grains were significantly consumed by the grain boundary migration of the softer recrystallized B-component (although the consumption of A-grains was not really documented in the quartzite sample). The conclusion is supported by the fact that the B-component is much more dominant in the recrystallized than in the host c-axis fabric. Hence, the c-axis maximum nearly perpendicular to the shear plane and apparently displaced with the sense of shear commonly found in naturally sheared quartzites (correlated with the B-component) is presumably developed by the growth of soft orientations for basal (0001) slip by grain boundary migration at large strains.

Stacking fault energy and microstructure effects on torsion texture evolution

A series of experiments and simulations that vary the texture and microstructure simultaneously are used to establish the role of the microstructure in texture formation in FCC metals. The stacking fault energy (SFE) of the metal, which is known to have a strong impact on texture and microstructure, is the vital parameter used to make these variations. It was determined that the wide variety of textures and microstructures observed as a function of SFE and temperature was developed by slip processes alone; twinning was not necessary, as previously thought. The different textures are caused by (i) variations in local slip patterns within a single grain, as revealed by grain subdivision into differently deforming cell blocks; and (ii) more subtly by the cell–block shape. The local selection of slip systems creating the lattice rotations within a cell block is altered by the planarity of slip. Slip planarity is controlled by the SFE and temperature. It is hypothesized that the new texture components, that are distinct from the generally accepted ideal components, are created by the different slip processes occurring as a result of low SFE and low temperature. A more subtle effect of grain subdivision is related to the cell–block shapes that develop as a function of SFE and temperature and correspond to the different textures observed. The shape of the cell block is related to the level of constraint required by the deformation. The slip pattern changes and cell–block shapes correlate with the presence or absence of certain ideal texture components whose evolution is not simulated. Materials and conditions with similar deformation microstructures developed similar textures in the experiments.

Plastic flow of γ-TiAl-based polysynthetically twinned crystals: micromechanical modeling and experimental validation

Abstract A micromechanical model for the calculation of the plastic behavior of a lamellar structure is presented. This model is based on a rate-sensitive approach to describe the plasticity at the single crystal (lamella) level and on the relaxed constraints theory to account for the influence of the lamellar morphology on the overall plastic response of the structure. The equations for the cases of 2- and N -lamellae structures undergoing states of applied stress or strain rate are presented. The model is applied to a lamellar matrix–twin pair which is a simplified representation of a γ -TiAl polysynthetically twinned (PST) crystal. For this case, a morphology-based classification of the critical stresses of the γ -TiAl deformation systems is also presented. This model for PST plasticity is successfully validated by comparison with available experimental data.

Calculation of intergranular stresses based on a large-strain viscoplastic self-consistent polycrystal model

We present here an extension of the viscoplastic self-consistent (VPSC) polycrystal model for the calculation of the intergranular Cauchy stresses in an aggregate. This method, which is based on the self-consistent treatment of incompressible aggregates proposed in 1987 by Molinari et al, is formulated using the inclusion formalism and full anisotropy is incorporated into it. The complete stress state in the grains is obtained by computing the deviatoric and the hydrostatic local deviations with respect to the overall corresponding magnitudes applied to the polycrystal. The extended VPSC model, followed by an elastic self-consistent unloading, is used to obtain the intergranular residual strains in the aggregate after large plastic deformation. The texture evolution and the hardening of the material are explicitly taken into account in the model. As an application, the model is used to predict intergranular residual states in Incoloy-800 plate after uniaxial deformation.