Cauchy-Born rule and the stability of crystalline solids: Static problems

E. Weinan, Pingbing Ming

Research output: Contribution to journalReview article

116 Scopus citations

Abstract

We study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model is given by the classical Cauchy-Born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the Cauchy-Born rule. The analysis is carried out for both simple and complex lattices, and for this purpose, we develop the necessary tools for performing asymptotic analysis on such lattices. Our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids.

Original languageEnglish (US)
Pages (from-to)241-297
Number of pages57
JournalArchive for Rational Mechanics and Analysis
Volume183
Issue number2
DOIs
StatePublished - Feb 1 2007

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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