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

Technical Summary: $\Gamma$-Convergence and Homogenisation for Free Discontinuity Functionals in BD

Problem Statement
The paper addresses the $\Gamma$-convergence of sequences of free discontinuity functionals with linear growth defined on the space of functions with bounded deformation, $BD(U)$. This space consists of vector fields $u \in L^1(U; \mathbb{R}^d)$ whose symmetric distributional gradient $Eu$ is a bounded Radon measure. The study is motivated by applications in elasto-plasticity and the variational modeling of fracture mechanics (specifically the Francfort-Marigo model), where energy functionals typically take the form:
$$F(u, U) = \int_U f(x, Eu) \, dx + \int_U f^\infty\left(x, \frac{dE^c u}{d|E^c u|}\right) \, d|E^c u| + \int_{J_u \cap U} g(x, [u], \nu_u) \, d\mathcal{H}^{d-1}$$
Here, the energy comprises a bulk term, a Cantor part (singular with respect to the Lebesgue measure but vanishing on sets of finite $(d-1)$-dimensional Hausdorff measure), and a surface term associated with the jump set $J_u$. The authors investigate the asymptotic behavior of such functionals as they undergo homogenisation, including both deterministic and stochastic settings.

Methodology
The authors employ the classical localisation method for $\Gamma$-convergence, adapted to the specific challenges of the $BD$ setting. The core methodology involves:

1. Compactness and Abstract Representation: The authors first establish a compactness theorem for sequences of abstract functionals satisfying specific growth, locality, and continuity estimates. They prove that any $\Gamma$-limit functional admits a decomposition into absolutely continuous, Cantor, and jump parts.
2. Integral Representation via Minimisation Problems: The integrands of the limit functional are characterised as limits of minimum values of auxiliary minimisation problems on small cubes. Specifically, the bulk integrand $f$ and surface integrand $g$ are derived from limits of $m_F(\ell_A, Q(x, \rho))/\rho^d$ and $m_F(u_{x,\zeta,\nu}, Q_\nu(x, \rho))/\rho^{d-1}$, respectively.
3. Handling the Cantor Part: A significant technical hurdle in $BD$ (compared to $BV$) is the treatment of the Cantor part $E^c u$. Previous results in $BV$ relied on uniform continuity with respect to translations or sophisticated truncation arguments. The authors note that such truncation arguments are not available in $BD$. Instead, they utilise a characterisation of the Radon-Nikodým derivative of the Cantor part derived by Caroccia, Focardi, and Van Goethem [17], combined with specific growth conditions (defined in Definition 6.4) that control the rate at which the integrands approach their recession functions. This allows them to prove a full integral representation without assuming translation invariance of the limit functional.
4. Homogenisation via Rescaling: To study homogenisation, the authors consider functionals with rapidly oscillating coefficients $f(x/\varepsilon_n, \cdot)$ and $g(x/\varepsilon_n, \cdot)$. They utilise a change of variables to relate the minimum values of the oscillating functionals on small cubes to the minimum values of the original functional on large cubes.
5. Stochastic Homogenisation: For the stochastic case, the authors apply the Subadditive Ergodic Theorem of Akcoglu and Krengel to prove the almost sure existence of the limits defining the homogenised integrands.

Key Contributions and Results

• Compactness Theorem (Theorem 4.1): The authors prove that any sequence of functionals in a specific class $\mathcal{F}$ (satisfying linear growth and continuity estimates) admits a $\Gamma$-convergent subsequence in the $L^1_{loc}$ topology. The limit functional belongs to the same class and satisfies a measure property.
• Partial Integral Representation (Section 5): They establish that the absolutely continuous and jump parts of the limit functional can be represented by integral functionals with integrands defined via cell problems (Lemma 5.3).
• Full Integral Representation (Theorem 7.1): Under additional growth conditions (Definition 6.4) and assuming the bulk integrand limit is independent of the spatial variable $x$, the authors provide a complete integral representation for the limit functional, including the Cantor part. The limit functional is shown to be:
  $$F(u, B) = \int_B f(Eu) \, dx + \int_B f^\infty\left(\frac{dE^c u}{d|E^c u|}\right) \, d|E^c u| + \int_{J_u \cap B} g(x, [u], \nu_u) \, d\mathcal{H}^{d-1}$$
• Characterisation of $\Gamma$-limits (Theorem 8.3): The paper provides sufficient conditions for $\Gamma$-convergence based on the existence of limits of minimum values on small cubes. If these limits exist and are independent of $x$ (for the bulk term), the sequence $\Gamma$-converges to the functional defined by these limits.
• Deterministic and Stochastic Homogenisation (Theorems 9.7 and 10.6):
  • The authors prove that for a large class of functionals with oscillating coefficients, the $\Gamma$-limit exists and is given by a functional with constant (homogenised) integrands.
  • In the stochastic setting, they prove the almost sure existence of the homogenised integrands using the Subadditive Ergodic Theorem.
  • A specific scaling is chosen for the surface term in the oscillating functional (Eq. 1.11) to avoid the need for vertical truncation arguments, which are not feasible in $BD$.
• Relation between Bulk and Surface Limits: The paper establishes that the homogenised surface integrand $g_{lim}$ is the recession function of the homogenised bulk integrand $f_{lim}$ evaluated at the symmetrised tensor product of the jump and normal vectors (Eq. 9.27 and 10.3).

Significance and Claims
The paper claims to extend the theory of $\Gamma$-convergence and homogenisation for free discontinuity problems from the space of functions of bounded variation ($BV$) to the space of functions of bounded deformation ($BD$). The authors highlight that this extension is non-trivial due to the lack of certain approximation properties and the inability to use truncation arguments that are standard in $BV$.

Specifically, the authors note that while previous works in $BV$ (e.g., [16, 26, 30]) could handle surface terms that are not strictly 1-homogeneous by using truncation, their approach in $BD$ requires the surface term to be positively 1-homogeneous in the jump variable $\zeta$ (or a specific scaling as in Eq. 1.11) to circumvent these difficulties. The results provide a rigorous variational framework for the homogenisation of elasto-plastic materials and cohesive fracture models where the deformation is described by $BD$ functions, covering both periodic and random microstructures. The work relies on and extends recent results by Caroccia, Focardi, and Van Goethem [17], adapting them to a setting where translation invariance is not assumed a priori.