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Cauchy-Born rule and the stability of crystalline solids: Static problems. (English) Zbl 1106.74019
Summary: We study the connection between atomistic and continuum models for elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model can be given by classical Cauchy-Born rule in the sense that elastically deformed states of atomistic model are closely approximated by solutions of continuum model with stored energy functionals obtained from 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 in crystalline solids.

MSC:
74E15 Crystalline structure
74A25 Molecular, statistical, and kinetic theories in solid mechanics
82D25 Statistical mechanical studies of crystals
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