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

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2025年11月11日

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Matthias Ruf,
Mathias Schäffner

摘要

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