Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

Cauchy–Born rule and stability of crystalline solids at finite temperature

Pages: 1461 – 1490



Tao Luo (School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, and Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China)

Yang Xiang (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

Jerry Zhijian Yang (School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China)

Cheng Yuan (School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, China)


We study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at finite temperature. We prove, under certain sharp stability conditions at zero temperature, that the solid is stable when temperature is low. This gives a criterion for the onset of instabilities of crystalline solids as temperature increases. Based on the stability conditions at both zero and finite temperature, we show that the finite temperature version of Cauchy–Born rule gives a correct nonlinear elasticity model in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with free energy functionals obtained from the Cauchy–Born rule at finite temperature. The convergence is proved for both simple and complex lattices.


Cauchy–Born rule, finite temperature, stability of crystalline solids, atomistic model

2010 Mathematics Subject Classification

74B20, 74E15, 74G65, 74N99

Received 27 February 2020

Accepted 14 January 2021

Published 2 August 2021