Orateur
Description
I will present a recent quantitative result concerning the stochastic homogenization of the so-called Griffith type model arising in fracture mechanics: given a body $\Omega\subset \mathbb{R}^3$, the energy $E_\varepsilon(u)$ for deformation $u\in \mathrm{SBV}(\Omega)$ takes the form of
$$E_\varepsilon(u):=\int_{\Omega\backslash S_u} F(\tfrac{\cdot}{\varepsilon},\nabla u)+\int_{S_u}g(\tfrac{\cdot}{\varepsilon})\dd\mathcal{H}^{d-1},$$ where $F$ denotes the stored elastic energy, $g$ the toughness and $\varepsilon\ll 1$ the scale of the microstructure. Since the work of Cagnetti, Dal Maso, Scardia and Zeppieri, the homogenized model has been identified qualitatively by taking the $\Gamma$-limit as $\varepsilon\downarrow 0$ of \eqref{Energy}; and in particular the two main constitutive properties of the system have been derived: the effective stored elastic energy and the effective fracture toughness, both given explicitly by means of cell-formulas. I will explain in this talk how we can derive quantitative estimates for the convergence of the cell-formula for the effective toughness. This is based on a joint work with Julian Fischer and Antonio Agresti.