Optimal flux densities for linear mappings and the multiscale geometry of structured deformations
type: Item
instance of: scholarly article
title: Optimal flux densities for linear mappings and the multiscale geometry of structured deformations
zbMATH Open document ID: 1330.49049
author: D. R. J. Owen
author: Roberto Paroni
published in: Archive for Rational Mechanics and Analysis
publication date: 2015-11-04T00:00:00Z
review text: The authors study expressions for interfacial terms in the relaxed energy associated to structured deformations of continua. Structured deformations were defined in [\textit{G. Del Piero} and \textit{D. R. Owen}, Arch. Ration. Mech. Anal. 124, No. 2, 99--155 (1993; Zbl 0795.73005)] for the study of deformations at different length scales particularly in models of plasticity and fracture. For a continuum \(U\), a structured deformation consists of a macroscopic deformation \(g\), a microscopic deformation \(G\), and a rectifiable surface in \(U\) indicating the site of disarrangement. The least energy needed to deform \(U\) by \(g\) to affect a deformation by \(G\) on small pieces is realized as the infimum of limit inferiors of energies defined on sequences \(u_n\) with \(u_n \to g\) and \(\nabla u_n \to G\). Representation formulae in [\textit{R. Choksi} and \textit{I. Fonseca}, Arch. Ration. Mech. Anal. 138, No. 1, 37--103 (1997; Zbl 0891.73078)] show that it suffices to compute the infimum of a related energy, which for \(x_0 \in U\) involves maps on the unit cube having the average gradient \(G(x_0)\) and the boundary value \(\nabla g(x_0)\). The authors study a particular form of this energy measuring switches, separations and interpenetration along the disarrangement site. The computation amounts to the evaluation of \[ \text{inf}_V \frac{1}{|V|} \int_{\partial V} |M x \cdot \nu |d\mathcal{H}^{n-1}(x) \] for \(M\) in a dense set of matrices. Here the infimum is taken over sets \(V \subset U\) with finite perimeter. The authors show that the infimum equals \(|\text{tr} M|\). Since \(|\text{tr} M|\) is a lower bound for the integral, equality comes by modifying \(V\) along the flow \(x \mapsto Mx\) to create regions in \(\partial V\) where the integrand is minimized. This provides an interesting relation between minimal flux of a linear vector field \(x \mapsto Mx\) and the minimal amount of switching, separating and interpenetrating needed to form the structure…
reviewed by: Christopher Policastro
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 74M25
Mathematics Subject Classification ID: 74A60
Mathematics Subject Classification ID: 74A99
zbMATH DE Number: 6504072
zbMATH Keywords: structured deformation
zbMATH Keywords: multiscale geometry
zbMATH Keywords: optimal flux densities
zbMATH Keywords: relaxation
zbMATH Keywords: integral representation formula
MaRDI profile type: Publication
full work available at URL: s00205-015-0890-x
OpenAlex ID: W377204975
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DOI: 10.1007/S00205-015-0890-X
description: scientific article; zbMATH DE number 6504072