Calculus of Variations and Geometric Measure Theory
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M. Friedrich - M. Perugini - F. Solombrino

$Γ$-convergence for free-discontinuity problems in linear elasticity: Homogenization and relaxation

created by solombrin on 14 Oct 2020
modified by solombrino on 12 Jan 2022


Accepted Paper

Inserted: 14 oct 2020
Last Updated: 12 jan 2022

Journal: Indiana University Mathematics Journal
Year: 2022

ArXiv: 2010.05461 PDF


We analyze the $\Gamma$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $\Gamma$-convergence and represent the $\Gamma$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on $GSBD^p$, and cover the classical case of periodic homogenization.

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