CauchyBorn Rule and the Stability of Crystalline Solids: Static Problems

by: Weinan, Pingbing Ming

Archive for Rational Mechanics and Analysis (February 2007), pp. 241-297.

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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.

@article{citeulike:2702462, 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.}, author = {Weinan and Ming, Pingbing }, citeulike-article-id = {2702462}, doi = {10.1007/s00205-006-0031-7}, issn = {0003-9527}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {elasticity}, month = {February}, pages = {241--297}, posted-at = {2008-04-22 16:35:49}, priority = {2}, publisher = {Springer}, title = {CauchyBorn Rule and the Stability of Crystalline Solids: Static Problems}, url = {http://dx.doi.org/10.1007/s00205-006-0031-7}, year = {2007} }

RIS record

TY - JOUR ID - citeulike:2702462 TI - CauchyBorn Rule and the Stability of Crystalline Solids: Static Problems JF - Archive for Rational Mechanics and Analysis SP - 241 EP - 297 PB - Springer SN - 0003-9527 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. KW - elasticity AU - Weinan AU - Ming, Pingbing PY - 2007/02// UR - http://dx.doi.org/10.1007/s00205-006-0031-7 ER -

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