Geometric indexes of atomic configurations and microscopic expressions for elastic constants of solids

Elastic constants, are usually obtained experimentally, since it has some problems to predict elastic constants of materials analytically from atomistic viewpoint. In the previous papers, the authors proposed a method for expressing internal forces from motions of atoms, and the conservation laws for solids are introduced microscopically. In the present paper, constitutive equations and elastic constants not only for stresses but also for higher-order stresses are derived by dividing the kinematical quantities of atoms into the macroscopic deformation and thermal motion. In the process of derivation, a concept of a hierarchical Reynolds decomposition is introduced. It is an expansion in power series which can be divided into an average value and a deviation in each hierarchy. The hierarchical deviation terms are expressed by characteristic tensors which can be called P-tensors. The P-tensors are the indexes of atomic configurations and are used effectively for the expression of the elastic constants.