Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ana Cristina Barroso, José Matias, Marco Morandotti, David R. Owen, Elvira Zappale

TL;DR

This work advances variational modelling of materials with multi-scale internal geometry by formulating a global relaxation framework for hierarchical structured deformations with $L$ levels. It proves that the fully simultaneous relaxation energy $I(g,G_1,\dots,G_L)$ is equivalent to an iterated relaxation, and it extends the Approximation Theorem to $(g,G_1,\dots,G_L)$ under Carathéodory energy densities $W$ and $\\psi$. The paper provides integral representations for the relaxed energies: for the 2-level case via densities $H_{1,p}$ and $h_{1,p}$, and for the 3-level case via densities $f_{2,p}$ and $\Phi_{2,p}$, with $\Phi_{2,p}(x_0,\lambda,\nu)=h_{1,p}(x_0,\lambda,\nu)$ and independence properties of the derivatives. These results generalize prior 2-level frameworks (CF1997, BMMOZ, BMZ2024) to hierarchical settings, enabling rigorous multi-scale energy accounting in heterogeneous materials and informing continuum models of submacroscopic dissipation across levels.

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.

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

TL;DR

This work advances variational modelling of materials with multi-scale internal geometry by formulating a global relaxation framework for hierarchical structured deformations with levels. It proves that the fully simultaneous relaxation energy is equivalent to an iterated relaxation, and it extends the Approximation Theorem to under Carathéodory energy densities and . The paper provides integral representations for the relaxed energies: for the 2-level case via densities and , and for the 3-level case via densities and , with and independence properties of the derivatives. These results generalize prior 2-level frameworks (CF1997, BMMOZ, BMZ2024) to hierarchical settings, enabling rigorous multi-scale energy accounting in heterogeneous materials and informing continuum models of submacroscopic dissipation across levels.

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.
Paper Structure (7 sections, 14 theorems, 96 equations)

This paper contains 7 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and main definitions
  4. Approximation theorem and auxiliary results
  5. A generalization of the relaxation theorem for $2$-level structured deformations
  6. The relaxation theorem for $3$-level structured deformations
  7. The case $L=3$

Key Result

Theorem 2.1

Let $f \in L^1(\Omega; \mathbb{R}^{d{\times} N})$. Then there exist $u \in SBV(\Omega; \mathbb{R}^{d})$, a Borel function $\beta\colon\Omega\to\mathbb{R}^{d{\times} N}$, and a constant $C>0$ depending only on $N$, such that

Theorems & Definitions (33)

  • Theorem 2.1: AL
  • Lemma 2.2: CF1997
  • Definition 2.3
  • Definition 2.4: book
  • Remark 2.5
  • Theorem 2.6: Approximation Theorem for $(L+1)$-level structured deformations
  • Corollary 2.7
  • proof
  • Remark 2.8
  • Theorem 2.9
  • ...and 23 more