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The generalized method of self-consistency for composite materials with random elastic properties of phases of compound or hollow inclusions

Pan'kov A.A.

#### Abstract:

We develop the generalized method of self-consistency for problems of prediction of tensor of effective elastic properties in order to analyze the statistic characteristics of fields of deformation of the elements of composite materials’ structures with random elastic properties of the phases of compound and hollow inclusions. The generalized method of self-consistency allows us to reduce the solution of these problems to the solution of a system of sequences or chains of more simple averaged boundary value problems for the single inclusions with corresponding intermediate layers and boundary conditions in the medium with the effective properties sought for. The elastic properties of the f-th phase of a single inclusion for some g-th averaged problem in the corresponding f-th chain can be calculated via the solution of the g-th and the followed (g+1)-th averaged problems; at that, the elastic properties of the other phases of the single inclusion do not depend on g and can be derived from the other chains of averaged problems. The elastic properties of the corresponding g-th intermediate layers take into account the peculiarities of random relative position and the variation in dimensions and elastic properties of the composite material’s inclusions via the special averaged indicator functions. In order to facilitate the numerical applications of the generalized method of self-consistency presented, we develop an approach based on approximation of the deformations in the phases of composite material’s inclusions by the finite sums of power series with respect to the random parameters of dispersion of the elastic properties of corresponding phases of composite material’s inclusions. The coefficients of these expansions and the tensor of effective elastic properties of composite material sought for should be estimated by simultaneous solution of the corresponding boundary value problems. We present the results of calculation of the isotropic tensor of effective elastic properties of a spheroplastic material with random elastic properties of hollow spherical inclusions derived using the generalized method of self-consistency and compare the results obtained with the known solution.

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