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

Abstract

We describe multiscale geometrical changes via structured deformations $(g,G)$ and the non-local energetic response at a point x via a function $\mathrm{\Psi }$ 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 $\mathrm{\Psi }$ under which the upscaling “ $n\to \infty$ ” results in a macroscale energy that depends through $\mathrm{\Psi }$ on (1) the jumps $[g]$ of g and the “disarrangement 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 $\mathrm{\Psi }$ upon the jumps $[g]$ of g and the “disarrangement 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.