Speaker
Description
I will present a recent quantitative result concerning the stochastic homogenization of the
so-called Griffith type model arising in fracture mechanics : the energy $E_\varepsilon(u)$ for $u\in\mathrm{SBV}$ takes the form of
$$
E_\varepsilon(u) = \int_{\Omega\setminus S_u} F\left(\frac{\cdot}{\varepsilon},\nabla u\right)-f\cdot u+\int_{S_u} g\left(\frac{\cdot}{\varepsilon}\right)\,\mathrm{d}\mathcal{H}^{d-1},
$$
where $F$ denotes the stored elastic energy, f the external forces, 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 $\varepsilon \downarrow 0$ in the equation above;
and in particular the two main constitutive properties of the system have been derived: the
homogenized elastic energy and the homogenized 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.
