A Comprehensive Approach via Global Relaxation to the Variational Modelling of Hierarchical Structured Deformations

TL;DR

This paper advances the mathematical modeling of hierarchical structured deformations by refining the variational framework and establishing equivalence of relaxation methods across multiple submacroscopic levels.

Contribution

It provides new mathematical tools and broadened analysis for the variational modeling of complex hierarchical deformations in materials.

Findings

Established equivalence of relaxation energies across hierarchical levels

Refined the mathematical framework for structured deformations

Extended the variational analysis to multiple submacroscopic scales

Abstract

The response of many materials to applied forces and boundary constraints depends upon internal geometric changes at multiple submacroscopic levels. Hierarchical structured deformations provide a mathematical setting for the description of such changes and for the variational determination of the corresponding energetic response. The research in this article provides substantial refinements and broadenings of the mathematical setting both for the underlying geometrical structure and for the variational analysis of energetic response. The mathematical tools employed in this research include the global method for relaxation and establish the equivalence of a relaxed energy obtained via relaxation under simultaneous geometrical changes at all levels and a relaxed energy obtained via iterated relaxations proceeding from the deepest submacroscopic level successively to the macroscopic level.