Calculus of Variations and Geometric Measure Theory

A. C. Barroso - J. Matias - M. Morandotti - D. R. Owen - E. Zappale

The variational modeling of hierarchical structured deformations

created by morandott on 26 Aug 2022
modified on 28 Nov 2022


Accepted Paper

Inserted: 26 aug 2022
Last Updated: 28 nov 2022

Journal: J. Elast.
Year: 2022

ArXiv: 2208.11785 PDF


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.

Keywords: relaxation, Integral representation, energy minimization, Structured deformations, hierarchies