Dimension reduction in the context of structured deformations

TL;DR

This paper investigates the effects of applying dimension reduction and structured deformations sequentially and simultaneously on a three-dimensional continuum, revealing conditions under which energies coincide or differ.

Contribution

It introduces a modified relaxation approach for combined dimension reduction and structured deformations, providing explicit energy density calculations and comparisons.

Findings

Relaxed energies are identical when processes are applied in any order for surface-only initial energy.

Simultaneous relaxation can yield lower energy densities than sequential procedures.

Explicit formulas for energy densities are derived for specific initial conditions.

Abstract

In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.