Temperature-related Cauchy-Born rule for multiscale modeling of crystalline solids
S. P. Xiao and W. X. Yang
Computational Materials Science, 37, 374-379 (2006).
ABSTRACT
n this study, we develop a temperature-related Cauchy-Born (TCB) rule for
multiscale modeling of crystalline solids based on the assumptions that
deformation is locally homogeneous and atoms have the same local vibration mode.
When employing the TCB rule in the nanoscale continuum approximation, the first
Piola-Kirchhoff stress can be explicitly computed as the first derivative of the
Helmholtz free energy density to the deformation gradient. Since the Helmholtz
free energy is temperature-dependent, multiscale methods consisting of the TCB
rule embedded continuum model can be used to elucidate temperature-related
physical phenomena at the nanoscale. Stress analyses of canonical ensembles
verify the continuum approximation with the TCB rule by comparing the calculated
Cauchy stresses with the outcomes of molecular dynamics simulations. As an
application of the TCB rule in multiscale modeling, the nanoscale meshfree
particle method with the TCB rule demonstrates the same crack propagation
phenomenon in a nanoplate as molecular dynamics. This example shows that the
temperature effects are significant on the crack propagation speed when the
temperature is in a particular range.