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 $\lim_{n\to \infty }u_{n}=g$ and $\lim_{n\to \infty }\nabla u_{n}=$ $G$. We provide conditions on $\Psi$ under which the upscaling "$n\to \infty$" results in a macroscale energy that depends through $\Psi$ on (1) the jumps $[g]$ of $g$ and the "disarrangment field" $\nabla g-G$, (2) the "horizon" $r$, and (3) the weighting function $\alpha _{r}$ for microlevel averaging of $[u_{n}](y)$. We also study the upscaling "$n\to \infty$" followed by spatial localization "$r\to 0$" 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 "disarrangment field" $\nabla g-G$, 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.