Calculus of Variations and Geometric Measure Theory

J. Matias - M. Morandotti - D. R. Owen - E. Zappale

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

created by morandott on 05 Jul 2019
modified on 25 Oct 2020


Accepted Paper

Inserted: 5 jul 2019
Last Updated: 25 oct 2020

Journal: Mathematics and Mechanics of Solids
Pages: 30
Year: 2020

ArXiv: 1907.02955 PDF


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.

Keywords: Non-local energies, Structured deformations, crystal plasticity, upscaling