TY - JOUR
T1 - Cauchy-Born rule and the stability of crystalline solids
T2 - Static problems
AU - Weinan, E.
AU - Ming, Pingbing
N1 - Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.
PY - 2007/2
Y1 - 2007/2
N2 - 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.
AB - 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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U2 - 10.1007/s00205-006-0031-7
DO - 10.1007/s00205-006-0031-7
M3 - Review article
AN - SCOPUS:33846861564
VL - 183
SP - 241
EP - 297
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
IS - 2
ER -