Homogeneous deformation of an ordered crystalline solid at finite temperature can cause non-affine transformation of atomic trajectories. In such a case, continuum measures based on affine transformation of trajectories are insufficient to ensure energetic equivalence between the atomic and continuum scales. We use molecular dynamics simulation of fcc aluminum in NVT ensembles to demonstrate that the second moments of atomic positions about equilibrium show increasing deviation from affine behavior with strain and temperature. While the Cauchy-Born rule enforces affine deformation of the crystal in the static sense, second moments have been used to quantify non-affinity in thermal vibrations due to atomic trajectory transformation under macroscopic deformation. The evolution of second moments with applied strain becomes highly non-linear at high temperatures in the tensile regime as a result of the anharmonic potential energy surface of the material. The vibrational entropy is computed using the time-averaged second moments of position from the molecular dynamics simulation. Stress computed using the resulting free energy is significantly lower than the virial stress in the high tensile strain – high temperature regimes.
Ranganathan Parthasarathy, "Thermal vibration contribution to continuum stress in the elastic regime", Mechanics Research Communications, Volume 92, 2018, Pages 101-106, ISSN 0093-6413, https://doi.org/10.1016/j.mechrescom.2018.08.009.