Asymptotic solutions by the successive disordering method

Abstract

We develop the periodic componentmethod [1] and represent the solution of a stochastic boundary value elasticity problem for a random quasiperiodic structure with a given disordering degree of inclusions in the matrix via the deviations from the corresponding solution for a random structure with a smaller disordering degree. An example in which the tensor of elastic properties of a composite is calculated is used to illustrate the asymptotic and differential approaches of the successive disordering method. The asymptotic approach permits representing the quasiperiodic structure with a given chaos coefficient and the desired tensor of effective elastic properties as a result of small successive disordering of an originally ideally periodic structure of a composite with known tensor of elastic properties. In the differential approach, a differential equation is obtained for the tensor of effective elastic properties as a function of the chaos coefficient. Its solution coincides with the solution provided by the asymptotic approach. The solution is generalized to the case of piezoactive composites, and a numerical analysis of the effective properties is performed for a PVF (polyvinylidene fluoride) piezoelectric with various quasiperiodic structures on the basis of the cubic structure with spherical inclusions of a high-module elastic material.