Temperature-related Cauchy–Born rule for multiscale modeling of crystalline solids

In 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.