Mathematical models and methods of the mechanics of stochastic composites

Abstract

The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it