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 9 of the weighted averages of the jumps [un](y) of microlevel deformations un at points y within a distance r of x. The deformations un are chosen so that limn→∞ un = g and limn→∞ ∇un = G. We provide conditions on 9 under which the upscaling “n → ∞” results in a macroscale energy that depends through 9 on (1) the jumps [g] of g and the “disarrangement field” ∇g − G, (2) the “horizon” r, and (3) the weighting function αr for microlevel averaging of [un](y). We also study the upscaling “n → ∞” 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 9 upon the jumps [g] of g and the “disarrangement field” ∇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.