Derivation of macroscopic relations of the elasticity of complex crystal lattices taking into account the moment interactions at the microlevel

Abstract

A discrete mechanical model of a complex crystal lattice is proposed which contains particles possessing both translational and rotational degrees of freedom and which interact with one another by means of forces and moments. The transition to a continuum model of a crystal lattice is performed using the long-wave approximation, and, at the same time, it is shown that the dynamics of the continual model are described by the equations of the macroscopic moment theory of elasticity. Expressions are obtained for the macroscopic stiffness tensors which depend on the stiffness tensors of the interatomic bonds and the vectors determining the lattice geometry. A transition to the moment less theory of elasticity is made and it is shown that the macroscopic moduli of elasticity of the moment less theory depend both on the forces and the torque characteristics of the interatomic interaction. The stiffnesses of the interatomic bonds in a layer of graphite are calculated and it is shown that the transverse stiffness of an interatomic bond is comparable with the longitudinal stiffness, that is, a covalent bond is substantially non- central, which is only possible when there are torque interactions at the microlevel.