Integral representation for a relaxed optimal design problem for non-simple grade two materials

TL;DR

This paper develops a measure-based integral representation for a relaxed optimal design problem involving non-simple grade two materials, accounting for complex deformation and perimeter effects.

Contribution

It introduces a measure representation for the relaxed energy functional in optimal design problems with plastic deformations and non-simple materials, extending previous models.

Findings

Established a measure representation for the relaxed energy functional.

Extended the model to include non-simple grade two materials.

Provided a framework for analyzing optimal design with complex deformation energies.

Abstract

A measure representation result for a functional modelling optimal design problems for plastic deformations, under linear growth conditions, is obtained. Departing from an energy with a bulk term depending on the deformation gradient and its derivatives, as well as a perimeter term, the functional in question corresponds to the relaxation of this energy with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded hessian.