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

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