| Abstract: |
| We consider a nonlinear elastic composite with a periodic microstructure described by the nonconvex integral functional $$ \mathcal E_\varepsilon(u):=\int_\Omega W(\frac{x}{\varepsilon}, \nabla u(x)) -f(x)\cdot u(x)\,dx $$ As it is well-known, under suitable growth conditions, $\mathcal E_\varepsilon$ $\Gamma$-converges to a functional with a homogenized energy density $W_{\hom}$. 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 $W_{\hom}$ is in general given by an infinite-cell formula. Under appropriate assumptions on $W$ (e.g. frame indifference, minimality at identity, non-degeneracy) and on the microstructure (smooth but possibly touching inclusions), we show that in a neighbourhood of rotations $W_{\hom}$ is characterized by a single-cell homogenization formula. For this, we combine the construction of a matching convex lower bound and Lipschitz-estimates for sufficiently small solutions of nonlinear elliptic systems. Moreover, for small loads, we derive a quantitative two-scale expansion and establish existence and uniform Lipschitz estimates for minimizers of $\mathcal E_\varepsilon$. |
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