Conference on Calculus of Variations in Lille - 4th edition - June 24-28 2024

Quantitative stochastic homogenization of variational models arising in fracture mechanics

Jun 26, 2024, 11:00 AM

30m

Polytech Lille, Chappe auditorium, Cité Scientifique (Laboratoire Paul Painlevé)

Speaker

Nicolas Clozeau (IST Austria)

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 $E_{\epsilon }(u)$ for $u\in \mathrm{SBV}$ takes the form of
$$E_{\epsilon }(u)={\int }_{\mathrm{\Omega }\setminus S_u}F(\frac{\cdot }{\epsilon },\mathrm{\nabla }u)-f\cdot u+{\int }_{S_u}g\left(\frac{\cdot }{\epsilon }\right)\mathrm{d}{\mathcal{H}}^{d-1},$$ where $F$ denotes the stored elastic energy, f the external forces, g the toughness and $\epsilon \ll 1$ 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 $\mathrm{\Gamma }$-limit as $\epsilon \downarrow 0$ 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.