Optimal Design of Fractured Media with Prescribed Macroscopic Strain

TL;DR

This paper develops a mathematical framework for the optimal design of fractured composite materials under prescribed macroscopic strain, accounting for microcracks and interfaces, with potential applications in plasticity modeling.

Contribution

It introduces a relaxed energy functional for fractured media using structured deformations, combining bulk and surface energies through a blow-up method.

Findings

Derived an integral representation for the relaxed energy functional.

Identified the interplay between interface optimization and microcrack diffusion.

Framework can incorporate plasticity in future models.

Abstract

In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media.