Homogenization of second order energies on periodic thin structures

Abstract

We study the homogenization of integral functionals depending on the Hessian matrix over periodic low-dimensional structures in \(\mathbb{R}^n\). To that aim, we follow the same approach as in [6], where the case of first order energies was analyzed. Precisely, we identify the thin periodic structure under consideration with a positive measure μ, and we associate with μ an integral functional initially defined just for smooth functions. We prove that, under a suitable connectedness assumption on μ, the homogenized energy is an integral functional of the same kind, now with respect to the Lebesgue measure, whose effective density is obtained by solving an infimum problem on the periodicity cell. Such a problem presents basic differences from the first order case, as it involves both the microscopic displacement and the microscopic bending (Cosserat field). This feature is a consequence of the relaxation result for second order energies on thin structures proved in [7]. In the case when the initial energy density is quadratic and isotropic, we apply the main homogenization theorem to obtain some bounds on the eigenvalues of the homogenized tensor and to compute explicitly the effective density for several examples of geometries in the plane.