Homogenization of second order energies on periodic thin structures

• Published in: Calculus of variations and partial differential equations 2004-06, Vol.20 (2), p.175-211
• Main Authors: Bouchitt, Guy, Fragal, Ilaria, Rajesh, M.
• Format: Article
• Language: English
• Subjects: Calculus of variations; Energy; Homogenization; Mathematics; Partial differential equations; Thin films
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-003-0232-y

We study the homogenization of integral functionals depending on the Hessian matrix over periodic low-dimensional structures in Rn. 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 mu, and we associate with mu an integral functional initially defined just for smooth functions. We prove that, under a suitable connectedness assumption on mu, 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. [PUBLICATION ABSTRACT]