$Γ$-convergence and homogenisation for free discontinuity functionals with linear growth in the space of functions with bounded deformation

TL;DR

The paper establishes a comprehensive Γ-convergence framework for free discontinuity functionals with linear growth in BD, deriving compactness, partial and full integral representations, and homogenisation results. It introduces BD-specific minimisation constructs on small cubes to identify bulk and surface integrands $f$ and $g$, and extends the representation to the Cantor part under an $x$-independence assumption. A stability subclass $\mathfrak{F}^{\alpha,\infty}$ is developed to ensure Γ-closure and full integral representation, with precise conditions linking the Cantor density to recession data $f^{\infty}$. The work then treats oscillatory and stochastic homogenisation, proving almost-sure Γ-convergence to homogenised functionals $F^{f_{lim},g_{lim}}$, with $g_{lim}$ tied to $f_{lim}^{\infty}$, and establishing deterministic results in periodic settings. Overall, the results provide a robust BD-based analogue of BV homogenisation, enabling both deterministic and stochastic analyses for a broad class of cohesive fracture-type energies.

Abstract

We study the $Γ$-convergence of sequences of free discontinuity functionals with linear growth defined in the space ${\rm BD}$ of functions with bounded deformation. We prove a compactness result with respect to $Γ$-convergence and outline the main properties of the $Γ$-limits, which lead to an integral representation result. The corresponding integrands are obtained by taking limits of suitable minimisation problems on small cubes. These results are then used to study the deterministic and stochastic homogenisation problem for a large class of free discontinuity functionals defined in ${\rm BD}$.

Theorems & Definitions (84)

• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Remark 3.6
• Remark 3.7
• Proposition 3.8
• proof
• Theorem 4.1