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

Abstract

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.

Acknowledgments

The authors wish to thank Dr Torben Leffers (RISØ National Laboratory, Denmark) for his insights on the brass-type texture and for making his review paper on this subject available prior to publication, Dr Martin Idiart (University of Pennsylvania, USA) for his valuable comments on algorithmic aspects of field fluctuation calculation and Dr Olivier Castelnau (CNRS-Université Paris XIII, France) for discussions on the viscoplastic deformation of polycrystalline geomaterials. The work of PPC was supported by DOE grant DE-FG02-04ER46110.