Calculus of Variations and Geometric Measure Theory

M. Amar - J. Matias - M. Morandotti - E. Zappale

Periodic homogenization in the context of structured deformations

created by morandott on 25 Mar 2022
modified on 21 Jul 2022


Published Paper

Inserted: 25 mar 2022
Last Updated: 21 jul 2022

Journal: ZAMP
Volume: 73
Pages: Art. n. 173
Year: 2022
Doi: 10.1007/s00033-022-01817-6

ArXiv: 2203.12769 PDF


An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process proper of the theory of homogenization.

Keywords: relaxation, Homogenization, Structured deformations, multiscale geometry