Upscaling and spatial localization of non-local energies with applications to crystal plasticity

We describe multiscale geometrical changes via structured deformations (g, G) and the non-local energetic response at a point x via a function Psi of the weighted averages of the jumps [u(n)](y) of microlevel deformations u(n) at points y within a distance r of x. The deformations u(n) are chosen so that lim(n -> infinity) del(un) = g and lim(n ->infinity) Delta u(n) = G. We provide conditions on Psi under which the upscaling "n -> infinity" results in a macroscale energy that depends through Psi on (I) the jumps [g] of g and the "disarrangement field" del(g) - G, (2) the "horizon" r, and (3) the weighting function a(r) for microlevel averaging of [u(n)](y). We also study the upscaling "n -> infinity" followed by spatial localization "r -> 0" and show that this succession of processes results in a purely local macroscale energy I(g, G) that depends through Psi upon the jumps [g] of g and the "disarrangement field" del g - G alone. In special settings, such macroscale energies I(g, G) have been shown to support the phenomena of yielding and hysteresis, and our results provide a broader setting for studying such yielding and hysteresis. As an illustration, we apply our results in the context of the plasticity of single crystals.