Ricardo A. Lebensohn

Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations

We present a detailed description of the numerical implementation, within the widely used viscoplastic self-consistent (VPSC) code, of a rigorous second-order (SO) homogenization procedure for non-linear polycrystals. The method is based on a linearization scheme, making explicit use of the covariance of the fluctuations of the local fields in a certain linear comparison material, whose properties are, in turn, determined by means of a suitably designed variational principle. We discuss the differences between this second-order approach and several first-order self-consistent (SC) formulations (secant, tangent and affine approximations) by comparing their predictions with exact full-field solutions. We do so for crystals with different symmetries, as a function of anisotropy, number of independent slip systems and degree of non-linearity. In this comparison, the second-order estimates show the best overall agreement with the full-field solutions. Finally, the different SC approaches are applied to simulate texture evolution in two strongly heterogeneous systems and, in both cases, the SO formulation yields results in better agreement with experimental evidence than the first-order approximations. In the case of cold-rolling of low-SFE fcc polycrystals, the SO formulation predicts the formation of a texture with most of the characteristic features of a brass-type texture. In the case of polycrystalline ice, deforming in uniaxial compression to large strain, the SO predicts a substantial and persistent accommodation of deformation by basal slip, even when the basal poles become strongly aligned with the compression direction.

Modeling texture and microstructural evolution in the equal channel angular extrusion process

Abstract In this work, we develop a modeling framework for predicting the visco-plastic deformation, microstructural evolution (distributions of grain shape and size) and texture evolution in polycrystalline materials during the equal channel angular extrusion (ECAE) process, a discontinuous process of severe shear straining. The foundation of this framework is a visco-plastic self-consistent (VPSC) scheme. We consider a 90° die angle and simulate ECAE up to four passes for four processing routes, (A, C, B A and B C , as denoted in the literature) for an FCC polycrystalline material. We assume that the FCC single crystal has a constant critical resolved shear stress (CRSS), so that hardening by dislocation activity is suppressed, and the influence of grain shape distribution and texture as well as their interaction can be isolated. Many deformation microstructural features, such as grain size and shape distribution, texture, and geometric hardening–softening, were highly dependent on processing route. Using a grain subdivision criterion based on grain shape, route A was the most effective, then route B A and route B C and lastly route C, the least effective for grain size refinement, in agreement with redundant strain theory. For producing refined equiaxed grains, route B C was more effective than routes B A and A. We show that grain–grain interactions tend to weaken texture evolution and consequently geometric hardening and softening in all routes.

Viscoplastic modeling of texture development in polycrystalline ice with a self‐consistent approach: Comparison with bound estimates

Ice crystals deform easily by dislocation glide on basal planes, which provides only two independent easy slip systems. The necessary slip on other systems limits the strain rate of polycrystalline ice. The preferred c axis orientation of ice from polar ice sheets develops as a result of intracrystalline slip. An anisotropic viscoplastic self-consistent (VPSC) approach is used for predicting texture development and mechanical behavior of polycrystalline ice. Results are compared with lower and upper bound estimations. It is assumed that ice crystals deform by basal, prismatic, and pyramidal slip. The resistance of each slip system is determined from experimental data on monocrystals and isotropic polycrystals. The VPSC model can predict the behavior of isotropic polycrystalline ice on both the macroscopic and microscopic scale. This is not the case for the lower and upper bounds. Fabrics simulated in uniaxial extension and compression are qualitatively similar for all models. However, large differences in the rate of fabric development are found. This is explained by the different interaction stiffness between grain and matrix. Fabric concentration obtained with the VPSC model for uniaxial deformation is in close agreement with those observed in polar ices. In simple shear, the single maximum fabric found in situ cannot be reproduced without an extensive (and probably unrealistic) activity of nonbasal systems. The preferential growth of grains well oriented for basal glide associated with rotation recrystallization could be at the origin of the discrepancy between model results and natural simple shear fabrics. Distorted grain shape is found to slightly slow down fabric development.

Effects of grain size and boundary structure on the dynamic tensile response of copper

Plate impact experiments have been carried out to examine the influence of grain boundary characteristics on the dynamic tensile response of Cu samples with grain sizes of 30, 60, 100, and 200 μm. The peak compressive stress is ∼1.50 GPa for all experiments, low enough to cause an early stage of incipient spall damage that is correlated to the surrounding microstructure in metallographic analysis. The experimental configuration used in this work permits real-time measurements of the sample free surface velocity histories, soft-recovery, and postimpact examination of the damaged microstructure. The resulting tensile damage in the recovered samples is examined using optical and electron microscopy along with micro x-ray tomography. The free surface velocity measurements are used to calculate spall strength values and show no significant effect of the grain size. However, differences are observed in the free surface velocity behavior after the pull-back minima, when reacceleration occurs. The magnitude of th...

Macroscopic properties and field fluctuations in model power-law polycrystals: full-field solutions versus self-consistent estimates

This paper presents comparisons between full-field numerical results and homogenization estimates for the effective behaviour and statistical fluctuations of the stress and strain-rate fields in viscoplastic polycrystals. The full-field simulations make use of a recently introduced technique based on the fast Fourier transform (FFT) algorithm, while the homogenization results follow from the ‘second-order’ technique incorporating information about the averages and fluctuations of the fields in a suitably chosen ‘linear comparison polycrystal’, together with the standard self-consistent (SC) approximation for the linear comparison medium. An application is given for a model two-dimensional power-law polycrystal, for which exact estimates are available in the limit of linearly viscous behaviour. These exact results demonstrate the accuracy of the FFT method, even for relatively large values of the grain anisotropy parameter when the field fluctuations become significant. On the other hand, the ‘second-order’ SC estimates for both the effective behaviour and the statistical fluctuations of the stress and strain-rate fields in viscoplastic polycrystals are found to be in good agreement with the corresponding FFT results. This is the case even for strongly nonlinear systems with low strain-rate sensitivities, where the field fluctuations are found to be large, and where other, earlier versions of the SC approximation are shown to fail.

Comparison of finite element and fast Fourier transform crystal plasticity solvers for texture prediction

We compare two full-field formulations, i.e. a crystal plasticity fast Fourier transform-based (CPFFT) model and the crystal plasticity finite element model (CPFEM) in terms of the deformation textures predicted by both approaches. Plane-strain compression of a 1024-grain ensemble is simulated with CPFFT and CPFEM to assess the models in terms of their predictions of texture evolution for engineering applications. Different combinations of final textures and strain distributions are obtained with the CPFFT and CPFEM models for this 1024-grain polycrystal. To further understand these different predictions, the correlation between grain rotations and strain gradients is investigated through the simulation of plane-strain compression of bicrystals. Finally, a study of the influence of the initial crystal orientation and the crystallographic neighborhood on grain rotations and grain subdivisions is carried out by means of plane-strain compression simulations of a 64-grain cluster. (Some figures in this article are in colour only in the electronic version)

Dilatational viscoplasticity of polycrystalline solids with intergranular cavities

We propose constitutive models for polycrystalline aggregates with intergranular cavities and test them against full-field numerical simulations. Such conditions are prevalent in many engineering applications and failure of metallic components (e.g. HIPing and other forming processes, spallation under dynamic loading conditions, etc.), where the dilatational effects associated with the presence of cavities must be accounted for, and standard polycrystalline models for incompressible plasticity are not appropriate. On the other hand, it is not clear that the use of porous plasticity models with isotropic matrix behavior is relevant, particularly, when large deformations can lead to significant texture evolution and therefore to strong matrix anisotropy. Of course, finite strains can also lead to significant changes in the porosity and pore shape, resulting in additional anisotropy development. In this work, we make use of ‘variational linear-comparison’ homogenization methods to develop constitutive models simultaneously accounting for texture of the matrix, porosity and average pore shape and orientation. The predictions of the models are compared with full-field numerical simulations based on fast Fourier transforms to study the influence of different microstructural features (e.g. overall porosity, texture of the matrix phase, single-crystal anisotropy, etc.) and type of loading (triaxiality) on the dilatational viscoplastic behavior of voided polycrystals. The results are also compared with the predictions of isotropic-matrix porous plasticity models to assess the effect of the possible matrix anisotropy in textured samples.

Modeling the mechanical response of polycrystals deforming by climb and glide

This paper presents a crystallographically-based constitutive model of a single crystal deforming by climb and glide. The proposed constitutive law is an extension of the rate-sensitivity approach for single crystal plasticity by dislocation glide. Based on this description at single crystal level, a homogenization-based polycrystal model for aggregates deforming in a climb-controlled thermal creep regime is developed. To illustrate the capabilities of the proposed model, we present calculations of effective behavior of olivine and texture evolution of aluminum at warm temperature and low strain rate. In both cases, the addition of climb as a complementary single-crystal deformation mechanism improves the polycrystal model predictions.

A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids

In this work we consider the presence of ellipsoidal voids inside polycrystals subjected to large strain deformation. For this purpose, the originally incompressible viscoplastic selfconsistent (VPSC) formulation of Lebensohn and Tome (Acta Metall. Mater. 41 (1993) 2611) has been extended to deal with compressible polycrystals. In doing this, both the deviatoric and the spherical components of strain-rate and stress are accounted for. Such an extended model allows us to account for the void and for porosity evolution, while preserving the anisotropy and crystallographic capabilities of the VPSC model. The formulation can be adjusted to match the Gurson model, in the limit of rate-independent isotropic media and spherical voids. We present several applications of this extended VPSC model, which address the coupling between texture, plastic anisotropy, void shape, triaxiality, and porosity evolution.

On the Combined Effect of Pressure and Third Invariant on Yielding of Porous Solids With von Mises Matrix

In this paper, a new plastic potential for porous solids with von Mises perfectly-plastic matrix containing spherical cavities is derived using a rigorous limit analysis approach. For stress-triaxialities different from 0 and ±∞, the dilatational response depends on the signs of the mean stress and the third invariant of the stress deviator. The classic Gurson potential is an upper-bound of the new criterion. A full-field dilatational viscoplastic Fast Fourier Transform (FFT)-based approach is also used to generate numerical gauge surfaces for the porous material. The numerical calculations confirm the new features of the dilatational response, namely: a very specific dependence with the signs of the mean stress and the third invariant that results in a lack of symmetry of the yield surface.