ALEX 2018 Workshop: Abstracts

Quantitative homogenization in nonlinear elasticity

Stefan Neukamm and Mathias Schäffner

Technische Universität Dresden, Faculty of Mathematics (Germany)

We consider a nonlinear elastic composite with a periodic microstructure described by the nonconvex energy functional

$${\int }_{\mathrm{\Omega }}W(\frac{x}{\epsilon },\nabla u(x))-f(x)\cdot u(x)dx.$$

It is well-known that under suitable growth conditions the energy $\mathrm{\Gamma }$-converges to a homogenized functional with a homogenized energy density $W_{\mathrm{hom}}$. One of the main problems in homogenization of nonlinear elasticity is that long-wavelength buckling prevents the possibility of homogenization by averaging over a single period cell, and thus $W_{\mathrm{hom}}$ is in general given by an infinite-cell formula. Under appropriate assumptions on $W$ (frame indifference, minimality at identity, non-degeneracy) and on the microstructure (e.g., a piecewise constant composite with smooth inclusions that might touch), we show that in a neighbourhood of rotations $W_{\mathrm{hom}}$ is characterized by a single-cell homogenization formula. In particular, we prove that correctors are available — a property that we exploit to derive a quantitative two-scale expansion and uniform Lipschitz estimates for minimizers. The presentation is based on [1] and work in progress.

Acknowledgments: The study was funded by the Deutsche Forschungsgemeinschaft in the context of TU Dresden’s Institutional Strategy “The Synergetic University” .