On the Definitions of Effective Stress and Deformation Gradient for Use in MD: Hill’s Macro-Homogeneity and the Virial Theorem
As engineering applications have moved to the submicron- and nano-scales, classically-trained engineers started using tools from statistical mechanics and solid state physics like molecu- lar dynamics. As it turns out, there were no theoretical results available to tie together the averaging notions from homogenization theory of composite materials with the averaging pro- cesses carried out in molecular dynamics simulations. The central contribution of this paper is the construction of a mathematically rigorous bridge between molecular dynamics, as an intrinsically discrete approach to the estimation of material properties, and continuum mechanics. In particular, the authors have established a rigorous limit process for the matching of the classical Cauchy stress and the so-called virial stress.
Costanzo, F., G. L. Gray, and P. C. Andia (2005) “On the Definitions of Effective Stress and Deformation Gradient for Use in MD: Hill’s Macro-Homogeneity and the Virial Theorem,” International Journal of Engineering Science, 43(7), pp. 533–555.