Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions

We derive thermo-mechanical continuum equations from Molecular Dynamics (MD) equations using the Generalized Mathematical Homogenization (GMH) theory developed by the authors for 0 K applications. GMH constructs an array of atomistic unit cell problems coupled with a thermo-mechanical continuum problem. The unit cell problem derived is a molecular dynamics problem defined for the perturbation from the average atomistic displacements subjected to the deformation gradient and temperature extracted from the continuum problem. The coarse scale problem derived is a constitutive law-free continuum thermo-mechanical equation. Attention is restricted to heat transfer by lattice vibration (phonons). The method is verified on several model problems against the reference molecular dynamics solution.