Effective energy integral functionals for thin films on curl-free vector fields in the Orlicz–Sobolev space setting

We consider an elastic thin film $\omega\subset \mathbb{R}^2$ with three dimensional bending moment. The effective energy functional defined on the Orlicz–Sobolev space over $\omega$ is obtained by $\Gamma$-convergence and $3D$-$2D$ dimension reduction techniques in the case when the energy density function is cross-quasiconvex. In the case when the energy density function is not cross-quasiconvex we obtained both upper and lower bounds for the $\Gamma$-limit. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function $M$. Here $M, M^*$ are assumed to be non-power-growth-type and to satisfy the condition $\Delta_{2}^{\text{glob}}$ (that imply the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by $M$), and $M^*$ denotes the complementary (conjugate) Orlicz $N$-function of $M$.