Work conjugate strain of virial stress

Certain stress tensor and strain tensor form a conjugate pair if there exists a scalar valued strain energy function such that the stress tensor is equal to the derivative of strain energy function with respect to the strain tensor. Virial stress is widely accepted as the stress measurement in molecular dynamics (MD). However, its conjugate strain is not yet identified. An atomic logarithmic strain is proposed and numerically verified as the conjugate strain of virial stress at temperature. The strain energy is calculated by virial stress and the proposed atomic logarithmic strain equals to the interatomic potential energy density. This conclusion is numerically verified with (1) Coulomb-Buckingham potential, Lenard-Jones potential, or arbitrary nonlinear pair potential and (2) randomly generated atomic configurations and deformation gradients. Examples are given in determining the stress-strain relation for magnesium oxide with MD simulation. The result shows that the atomic logarithmic strain is identical to engineering strain when deformation is small.