Published Paper

Inserted: 18 jun 2021
Last Updated: 18 apr 2023

Journal: Arch Rational Mech Anal
Year: 2021
Doi: https://doi.org/10.1007/s00205-023-01867-8
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Abstract:

To account for material slips at microscopic scale, we take deformations as SBV functions $\varphi$, which are orientation-preserving outside a jump set taken to be two-dimensional and rectifiable. For their distributional derivative $F=D\varphi$ we admit the common multiplicative decomposition $F=F^{e}F^{p}$ into so-called elastic and plastic factors. Then, we consider a polyconvex energy with respect to $F^{e}$, augmented by the measure $\vert curl F^{p}\vert$. For this type of energy we prove existence of minimizers in the space of SBV maps with appropriate constraints such as the one avoiding interpenetration of matter. Our analysis rests on a representation of the slip system in terms of currents with both $\mathbb{Z}^{3}$ and $\mathbb{R}^{3}$ valued multiplicity. The first choice is particularly significant in periodic crystalline materials at a lattice level, while the latter covers a more general setting and requires to account for an energy extra term involving the slip boundary size.

Keywords: calculus of variations, Geometric measure theory, dislocations, plasticity, energy minimization

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