Second-order structured deformations: relaxation, integral representation and applications

TL;DR

This paper extends the theory of structured deformations to second order, deriving an integral representation for relaxed energies that account for bending and curving effects in continua, with applications to inhomogeneous materials.

Contribution

It introduces a second-order relaxation framework for structured deformations, providing explicit formulas and integral representations for energies including inhomogeneous densities.

Findings

Derived an integral representation for second-order relaxed energies.

Provided explicit formulas for bulk relaxed energies.

Outlined potential applications in material modeling.

Abstract

Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.