Archive for Rational Mechanics and Analysis

, Volume 183, Issue 2, pp 241–297

Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems



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.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA
  2. 2.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSSChinese Academy of SciencesBeijingChina

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