Micro-Slip-Induced Multiplicative Plasticity: Existence of Energy Minimizers

Abstract

AbstractTo account for material slips at microscopic scale, we take deformation mappings asSBVfunctions$$\varphi $$φ, which are orientation-preserving outside a jump set taken to be two-dimensional and rectifiable. For their distributional derivative$$F=D\varphi $$F=Dφwe examine the common multiplicative decomposition$$F=F^{e}F^{p}$$F=FeFpinto so-called elastic and plastic factors, the latter a measure. Then, we consider a polyconvex energy with respect to$$F^{e}$$Fe, augmented by the measure$$|\textrm{curl}\,F^{p}|$$|curlFp|. For this type of energy we prove the existence of minimizers in the space ofSBVmaps. We avoid self-penetration of matter. Our analysis rests on a representation of the slip system in terms of currents (in the sense of geometric measure theory) with both$$\mathbb {Z}^{3}$$Z3and$$\mathbb {R}^{3}$$R3valued multiplicity. The two choices make sense at different spatial scales; they describe separate but not alternative modeling options. The first one is particularly significant for periodic crystalline materials at a lattice level. The latter covers a more general setting and requires to account for an energy extra term involving the slip boundary size. We include a generalized (and weak) tangency condition; the resulting setting embraces gliding and cross-slip mechanisms. The possible highly articulate structure of the jump set allows one to consider also the resulting setting even as an approximation of climbing mechanisms.