The aim of this talk is to present some recent developments in the theory of quantitative stochastic homogenization of non-linear problems arising in mechanics. In a first part, I will discuss a quantitative homogenization theory for non-linear elliptic PDEs of p-Laplacian, that can be seen as the Euler-Lagrange equation of the energy of an elastic body with random microstructure (in the case of small deformations). In particular, I will present two results : optimal rates for the convergence towards the homogenized problem; and a work in progress concerning a large-scale regularity theory for such non-linear elliptic PDEs. The is based on joint works with Antoine Gloria and Mathias Schffner.

In a second part, I will move forward variational problems arising in fracture mechanics that are described by Griffith-type energies. Based on a work by Dal Maso, Scardia and Zeppieri that identifies qualitatively the effective fracture toughness by means of a cell-formula; I will present a recent result that establishes rates for the convergence of the cell-formula towards the effective quantity. This is based on a joint work with Antonio Agresti and Julian Fischer.