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

To account for material slips at microscopic scale, we take deformation mappings as SBV functions ϕ, which are orientation-preserving outside a jump set taken to be two-dimensional and rectifiable. For their distributional derivative F = Dϕ we examine the common multiplicative decomposition F = FeF p into so-called elastic and plastic factors, the latter a measure. Then, we consider a polyconvex energy with respect to Fe, augmented by the measure |curl F p|. For this type of energy we prove the existence of minimizers in the space of SBV maps. 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 Z3 and R3 valued 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.