Kinematics of elasto-plasticity: Validity and limits of applicability of F = F^eF^p for general three-dimensional deformations

Abstract

This article provides a multiscale justification of the multiplicative decomposition F=F^eF^p for three-dimensional elasto-plastic deformations, and sets its limits of applicability via a careful examination of the assumptions involved in the derivation. The analysis starts from the mesoscopic characterization of the kinematics at the level of discrete dislocations, where F_ϵ, F_ϵ^e and F_ϵ^p are uniquely defined, and the relationships F_ϵ≃F_ϵ^eF_ϵ^p and det F_ϵ^p≃1 are well-justified almost everywhere in the domain. The upscaling to the macroscale (i.e., F=F^eF^p and det F^p=1, with F, F^e and F^p defined as the limits of the analogous quantities at the mesoscale) is then rigorously derived on the basis of the following assumptions: sup_ϵ∥F_ϵ^e∥L^g(Ω)<∞ with 1 < g < ∞, sup_ϵ∥F_ϵ^p∥L∞(Ω)<∞,sup_ϵ|Curl F_ϵ^p|(Ω)<∞, and det F_ϵ^p=1. These may be interpreted, in suitable scenarios, as bounded local energy density and dissipation, finite density of dislocations and incompressibility of the plastic deformation, respectively. Although these assumptions are expected to hold in many single crystal elasto-plastic deformations, they may be violated in certain cases of physical relevance. Illustrative examples where each of the individual assumptions fails in turn are presented and their implications regarding finite multiplicative elasto-plasticity at the macroscale are examined in detail.

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