Homogenization of second order energies on periodic thin structures (Q1890018)

The authors study the homogenization of functionals depending on the Hessian matrix over periodic low-dimensional structures in \(\mathbb{R}^n\). The thin periodic structure is identified by a positive measure \(\mu\), and \(\mu\) is associated with an integral functional, defined initially for smooth functions. Under a suitable connectedness assumption on \(\mu\), the homogenized energy converges to an integral functional of the same kind, with respect to the Lebesgue measure, whose effective density is obtained by solving an infimum problem on the periodicity cell. The limit problem presents differences from the first order case, since it involves both the microscopic displacement and the microscopic bending (Cosserat field), as a consequence of the relaxation result for second order energies on the thin structures. If the initial energy density is quadratic and isotropic, by homogenization the authors obtain some bounds on the eigenvalues of the homogenized tensor and they compute explicitly the effective density for several examples of geometries in the plane.