Upscaling and spatial localization of non-local energies with persistent nonlinearities.

We describe submacroscopic geometrical changes via the multiscale geometry of structured deformations, and we introduce energy responses to changes at the microlevel near a given point $x$ in the body by means of a response function $\Psi$ that depends on weighted averages of the jump-discontinuities of deformation $u_n$ throughout a neighborhood of $x$ of a given size $r$. The deformation $u_n$ describes geometrical changes as viewed through a microscope with magnification power proportional to $\frac1n$, and the multiscale geometry provides that $u_{n}$ can be chosen so as to approach a given macroscopic deformation field $g$ and so that its gradient $\nabla u_n$ (away from sites of discontinuities) approaches a preassigned field $G$ as $n$ tends to infinity. We prove that the process of upscaling $n\to \infty$ results in a macroscale energy response that depends through the given non-linear response function $\Psi $ upon (1) the jumps $[g]$ of the macroscopic deformation $g$ and upon the \emph{disarrangement} field $\nabla g-G$, upon (2) the preassigned size $r$ of the neighborhood over which the jumps in $u_n$ were averaged, and upon (3) the weighting functions $\alpha _r$ employed in the microscale averaging. Consequently, we study here not only the process of upscaling to the macrolevel $n\to \infty $ but also the process of spatial localization $r\to 0$. We prove that the energetic response after the successive processes of upscaling and of spatial localization depends through the original function $\Psi$ upon the jumps of the macroscopic deformation $g$ and upon the disarrangement field $\nabla g-G$, but the upscaled and localized response no longer depends upon $r$ or upon the nature of the non-local averaging process. The resulting non-linear dependence of the energy upon the disarrangement field is significant for the description of yielding and hysteresis.