Homogenization of stress equation of motion in linear elastodynamics

Two methods of homogenization of elastic body with periodic heterogeneity based on a pure stress formulation of linear elastodynamics (cf [1-4]) are presented. The methods are closely related to the two representations of the displacement field in terms of the stress field: the first derived from the geometrical relations and Hooke's law and the second obtained by means of the equation of motion. Both methods lead to the same homogenized form of the stress equation of motion, and the resulting homogenized coefficients are identical to those of a displacement homogenization procedure. Also, a theorem is proved in which it is shown that a mixed initial-boundary value problem for a homogenized medium can be characterized by a mean stress field only.