Periodic homogenization in the context of structured deformations

TL;DR

This paper develops a method to derive an effective energy model for structured deformations in periodic homogenization, combining relaxation, cell formulas, and blow-up techniques to represent bulk and interfacial energies.

Contribution

It provides an integral representation of the relaxed energy for structured deformations using homogenization and blow-up methods, advancing the theoretical understanding of energy relaxation.

Findings

Derived integral representation of the energy densities

Established asymptotic cell formulas for energy averaging

Combined blow-up and homogenization techniques successfully

Abstract

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.