The variational modeling of hierarchical structured deformations

TL;DR

This paper develops a variational framework for hierarchical structured deformations, providing theoretical foundations, an approximation theorem, and an iterative method to assign energies, thereby advancing the study of complex multi-level deformations.

Contribution

It introduces the first theoretical steps to define and analyze energies on hierarchical structured deformations using variational methods.

Findings

Established an approximation theorem for hierarchical structured deformations.

Developed an iterative procedure to assign energies to these deformations.

Validated the approach with an explicit example and outlined future research directions.

Abstract

Hierarchical (first-order) structured deformations are studied from the variational point of view. The main contributions of the present research are the first steps, at the theoretical level, to establish a variational framework to minimize mechanically relevant energies defined on hierarchical structured deformations. Two results are obtained here: (i) an approximation theorem and (ii) the assignment of an energy to a hierarchical structured deformation by means of an iterative procedure. This has the effect of validating the proposal made in [Deseri & Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J.~Elast., 135 (2019), 149--182] to study deformations admitting slips and separations at multiple submacroscopic levels. An explicit example is provided to illustrate the behavior of the proposed iterative procedure and relevant directions for future research are highlighted.