Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions.The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

Uniqueness and existence theorems of solutions of the boundary value problems of the theory of consolidation with double porosity

In this paper we consider the Aifantis' theory of consolidation with double porosity and we prove the uniqueness and existence theorems of solutions of basic boundary value problems (BVPs) of statics for the two-dimensional finite and infinite domains. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.     

Application of Analogues of General Kolosov–Muskhelishvili Representations in the Theory of Elastic Mixtures

The existence and uniqueness of a solution of the first, the second and the third plane boundary value problem are considered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov–Muskhelishvili representations from [1], these problems can be split and reduced to the first and the second boundary value problem for an elliptic equation which structurally coincides with the equation of statics of an isotropic elastic body.