Upper bounds for the homogenization problem in nonlinear elasticity: the incompressible case

Abstract

We consider periodic homogenization of hyperelastic models incorporating incompressible behavior via the constraint $\det (\nabla u)=1$. We show that the 'usual' homogenized integral functional $\int W_{\mathrm{hom}}(\nabla u)dx$, where $W_{\mathrm{hom}}$ is the standard multicell-formula of non-convex homogenization restricted to volume preserving deformations, yields an upper bound for the $\hat{I}“$-limit as the scale of periodicity tends to zero.

17 pages