S. P. Xiao and W. X. Yang
International Journal for Numerical Methods in Engineering, 69, 2099–2125 (2007).
A new homogenization technique, the temperature-related Cauchy-Born (TCB) rule, is proposed in this paper with the consideration of the free energy instead of the potential energy. Therefore, temperature effects at the nanoscale can be investigated using continuum approximation with the implementation of the TCB rule. The TCB rule is verified via stress analyses of several crystalline solids. Temperature-related material instability is also studied. In addition, a new hierarchical multiscale method is developed through implementing the TCB rule into meshfree particle methods. Quasicontinuum meshfree particle simulations are conducted to investigate bending of nanobeams, crack propagation in nanoplates and a three-dimensional nanoindentation problem.