Loss of polyconvexity by homogenization
The study of the effective properties of composite or cellular elastic materials requires the description of the asymptotic behavior as \(n\) tends to \(+ \infty\) of variational integrals of the form \({\mathcal F}_ n(u) = \int_ \Omega W(nx, Du(x))dx\), where \(Du\) denotes the gradient matrix of the displacement \(u\), and \(W\) is an energy density characterizing the microscopic properties of the elastic body. In this paper we show with an explicit example that the variational limit of a sequence of oscillating functionals of the form \({\mathcal F}_ n\) as above, with \(W(x,\cdot)\) polyconvex for all \(x\), is in general a variational integral \({\mathcal F}_{\text{hom}} (u) = \int_ \Omega W_{\text{hom}}(Du(x)) dx\), with \(W_{\text{hom}}\) quasiconvex but not polyconvex.
scientific article; zbMATH DE number 687669