On a General Theory of Composite Materials and Micro-Inhomogeneous Elastic Media

A method is developed to average over the volume the differential equations of equilibrium describing inhomogeneous elastic composite media with markedly different elastic moduli. A chain of macro-equilibrium equations is obtained involving macro tensors of stress couples and other stresses, all of increasing rank. These tensors are in general unisymmetric due to their definition of average quantities with respect to the oriented surface elements. The system of equations reduces to a single equation involving a series of derivatives of stresses of increasing order, averaged over the volume and residual term which is a derivative of stress averaged over the surface. By truncation of the series for an assumed accuracy a differential equation is obtained which is sufficient in the case when a single kinematic quantity is a vector of macro displacement, A structure of equation is derived typical for a gradient in the nonlocal theory of elasticity. Numerical calculations were carried out for a polycrystal provided that averaging over the volume is equivalent to averaging over the number of possible realizations.