A finite temperature continuum theory based on interatomic potential in crystalline solids
A. C. To, W. K. Liu and A. Kopacz
Computational Mechanics, 42, 531-541 (2008).
ABSTRACT
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.