Non-local models of stress field concentrations and effective thermoelastic properties of random structure composites

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions and subjected to inhomogeneous boundary conditions. We use the multiparticle effective field method (MEFM) based on the theory of functions of random variables and Green's functions; for references, see Buryachenko, VA. Applied Mechanics Reviews, 54, 1-47 (2001). Within this method, we derive a hierarchy of statistical moment equations for conditional averages of the stresses in the inclusions. The hierarchy is established by introducing the notion of an effective field. In this way the interaction of different inclusions is taken directly into account in the framework of the homogeneity hypothesis of the effective field. The non-local integral equation for the statistical average of stresses inside the inclusions is solved by three different methods: the quadrature method, the iteration method, and the Fourier transform method with subsequent comparative analysis. The standard scheme of iteration and Fourier transform methods permit us to obtain the explicit representations for the non-local integral and differential operators, respectively, of any order describing overall effective properties as well as the stress concentration factor in the components. It is shown that the integral operator reduces to the differential one for sufficiently smooth statistical average stress fields. With some additional assumptions, the proposed method is reduced to the perturbation method as well as to the "quasi-crystal line" approach. For some concrete numerical examples we can demonstrate the advantage of the quadrature method and the iteration method over the Fourier transform method.