Computational homogenization of mesoscale gradient viscoplasticity

Abstract

Variationally consistent selective homogenization is applied to a class of gradient-enhanced dissipative materials which is adopted for the mesoscale modeling. The adopted standard first order homogenization assumption results in the classical equilibrium equation for a local continuum on the macroscale, while the internal variables “live” on the mesoscale only. Among the issues considered in the paper, we note (i) the variationally consistent setting of homogenization of gradient theory on the mesoscale, (ii) the variational basis of the SVE-problem (SVE = Statistical Volume Element) in the time-incremental setting. The SVE-problem is formulated for the classical boundary conditions (Dirichlet and Neumann) pertinent to the standard momentum balance as well as the “micro-momentum” balance. The weak format of the SVE-problem constitutes the stationarity condition of an incremental SVE-potential, which represents an extension of the situation for a single-phase continuum model. The macroscale stress in a given time-increment is derivable from an incremental “macroscale pseudo-elastic strain energy”. Moreover, the saddle-point properties of the SVE-potential are shown to form the basis for establishing upper and lower bounds on the pseudo-elastic strain energy. Bingham viscoplasticity with gradient-enhanced hardening is chosen as the prototype model problem for the numerical evaluation. The computed stress–strain response relations confirm the theoretical predictions of the influence of different combinations of boundary conditions on the SVE.