Metric Description of Singular Defects in Isotropic Materials

Raz Kupferman*, Michael Moshe, Jake P. Solomon

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

Abstract

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.

Original languageAmerican English
Pages (from-to)1009-1047
Number of pages39
JournalArchive for Rational Mechanics and Analysis
Volume216
Issue number3
DOIs
StatePublished - 1 Jun 2015

Bibliographical note

Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

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