Equilibrium configurations of defective crystals
type: Item
instance of: scholarly article
title: Equilibrium configurations of defective crystals
zbMATH Open document ID: 0785.73007
DOI: 10.1007/BF00375027
author: Irene Fonseca
author: Gareth P. Parry
published in: Archive for Rational Mechanics and Analysis
publication date: 1992-12-14T00:00:00Z
review text: Davini's model [\textit{C. Davini}, Arch. Ration. Mech. Anal. 96, 295-317 (1986; Zbl 0623.73002)] for a slightly defective crystal uses a tensor field to represent locally a triad of `average' lattice vectors, a scalar field to represent density variations due to vacancies or extra atoms and a set of functionals which, in some sense, measure the defects and remain unchanged when the crystal is deformed elastically. Defect-preserving (neutral) deformations may involve also atomic rearrangements representing the slip mechanism of plasticity theory. Pursuing earlier work on this model by \textit{C. Davini} and the second author [Proc. R. Soc. Lond., Ser. A 439, No. 1906, 247-256 (1992; Zbl 0766.73057)], the authors, in this very interesting paper, study equilibria of defective crystals within a variational framework; they take the view that equilibria correspond to minimizers of the elastic energy functional \(E\) when the class \(A\) of admissible variations includes all neutral changes of state. The issue is not the existence of reasonably regular minimizers (actually they may be measure-valued), rather it is the study of minimizing sequences leading to values of energy and stresses. The conclusion is startling: the Cauchy stress must be pressure; this apparent paradox reminds one of the d'Alembert paradox for perfect fluids. If one will be able to eschew the paradox through the artifice of adding somehow friction (as the authors suggest) or if one need invent rather some missing deeper concept (as was the boundary layer concept in the case of fluids), only further study will tell.
Mathematics Subject Classification ID: 74E10
Mathematics Subject Classification ID: 74A60
Mathematics Subject Classification ID: 74M25
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74P10
zbMATH DE Number: 78054
zbMATH Keywords: variational formulation
zbMATH Keywords: average lattice vectors
zbMATH Keywords: defect-preserving deformations
zbMATH Keywords: density variations
zbMATH Keywords: elastic energy functional
zbMATH Keywords: existence
zbMATH Keywords: regular minimizers
zbMATH Keywords: minimizing sequences
zbMATH Keywords: Cauchy stress
reviewed by: Gianfranco Capriz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00375027
OpenAlex ID: W1980118021
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description: scientific article; zbMATH DE number 78054