Non-convex potentials and microstructures in finite-strain plasticity

A mathematical model for a finite-strain elastoplastic evolution problem is proposed in which one time-step of an implicit time-discretization leads to generally non-convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip-system and von Mises plasticity illustrate that finite-strain elastoplasticity generates reduced problems with non-quasiconvex energy densities and so allows for non-attainment of energy minimizers and microstructures.