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

TL;DR

This paper establishes that the standard homogenized integral functional provides an upper bound for the limit of hyperelastic models with incompressibility constraints in periodic homogenization, advancing understanding in nonlinear elasticity.

Contribution

It demonstrates that the multicell-formula for non-convex homogenization restricted to volume-preserving deformations bounds the $ ext{Gamma}$-limit from above in the incompressible hyperelastic setting.

Findings

The homogenized integral functional bounds the $ ext{Gamma}$-limit from above.

The multicell-formula applies to incompressible hyperelastic models.

The result advances theoretical understanding of homogenization in nonlinear elasticity.

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 $\int W_{\rm hom}(\nabla u)\,dx$, where $W_{\rm hom}$ is the standard multicell-formula of non-convex homogenization restricted to volume preserving deformations, yields an upper bound for the $\Gamma$-limit as the scale of periodicity tends to zero.