Crystal Plasticity and Evolution of Polycrystalline Microstructure

Abstract

The continuum slip theory of crystals provides a canonical framework of finite plasticity with physically well‐defined roots in the dislocation mechanics of metals. Reliable algorithmic settings of crystal plasticity are not only needed for structural analyses of single crystals but also provide a cornerstone for multiscale computations of evolving anisotropic microstructures in metallic polycrystals. This article summarizes a current status of computational crystal plasticity and its application to the texture analysis of polycrystals. It outlines a basic constitutive structure of crystal plasticity, alternative deformation‐driven stress update algorithms and a computational scheme for the nonlinear homogenization of polycrystalline aggregates. Particular emphasis is put on a canonical incremental variational formulation for Schmid‐type crystal plasticity with potential hardening where an incremental stress potential is obtained from a local minimization problem with respect to the internal variables. The existence of this stress potential allows the formulation of IBVPs for elastic–plastic crystals as a sequence of incremental minimization problems. Furthermore, the nonlinear homogenization of polycrystalline microstructures may be recast into a principle of minimum averaged incremental energy. Within this variational setting of homogenization, a quasi‐hyperelastic incremental macro‐stress potential is obtained from a global minimization problem with respect to the fine‐scale displacement fluctuation field. The performance of the formulations is demonstrated for benchmarks of single crystal plasticity and texture developments in polycrystals.

A numerical study of crack tip constraint in ductile single crystals

“…The single crystal plasticity theory, discussed in detail by Asaro (1983) and Miehe and Schotte (2004), is used in this work. The deformation gradient F is assumed to decompose multiplicatively, as,…”

“…The plastic deformation map F p defines the cumulative effect of dislocation motion. The evolution equation for the plastic deformation map is written as (Miehe and Schotte, 2004)…”

“…Assuming small elastic strains, a quadratic form of macroscopic free energy function is postulated as (Miehe and Schotte, 2004) …”

Fast texture updates in fcc polycrystal plasticity based on a linear active-set-estimate of the lattice spin

“…To this end, we compare the new purely geometric formulation of rigid crystal plasticity with results obtained by elasto-plastic crystals formulations with active set search outlined in Miehe et al (2002) and Miehe and Schotte (2004) and by results obtained for the standard power-type setting of rigid crystal plasticity. It is shown that the new geometric approach yields very accurate estimates for the reorientation of single crystals and provides an extremely attractive fast estimate for texture developments in polycrystals.…”

“…We refer to the standard setting of crystal plasticity with power laws by Peirce et al (1982) and the advanced computational settings based on active set definitions by Cuitin˜o and Ortiz (1992), Anand and Kothari (1996), Miehe et al (2002), and Ortiz and Stainier (1999). The recent review article Miehe and Schotte (2004) provides an overview. Micro-macro linking hypotheses drawn for the general theoretical framework of homogenization are outlined in Miehe et al (1999) and Miehe et al (2002).…”