A. C. To, W. K. Liu and A. Kopacz
Computational Mechanics, 42, 531–541 (2008).
A finite temperature continuum theory of crystalline solid based on an approximate Helmholtz free energy expression is proposed. The free energy expression is specifically derived for simple implementation in atomistic-based continuum methods (i.e. quasicontinuum method) via the Cauchy-Born rule at finite temperature. It is obtained by the method of statistical moments via the quasi-harmonic approximation together with Taylor series expansion of a given interatomic potential. The phonons are assumed to follow the Bose-Einstein distribution so that the quantum effects at low temperature are accounted for. The resulting free energy is in terms of a given interatomic potential and a simple function of displacement that accounts for thermal expansion. It is employed to formulate two finite temperature continuum methods via Cauchy-Born rule and via the virtual atomic cluster (VAC). It is validated through comparison with experimental results of various thermodynamic quantities. In the case of fcc metals, the proposed free energy expression is shown to be valid for a wide range of temperatures above 50 K.