No abstract

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.

No abstract

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 [Basheleishvili, Georgian Math. J. 4: 223–242, 1997], these problems can be splitted 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.

The basic mixed plane boundary value problem of equations of statics of the elastic mixture theory is considered in a simply connected domain when the displacement vector is given on one part of the boundary and the stress vector on the remaining part. The problem is investigated using the general displacement vector and stress vector representations obtained in [Basheleishvili, Georgian Math. J. 4: 223–242, 1997]. These representations enable us to reduce the considered problem to a system of singular integral equations with discontinuous coefficients of special kind. The solvability of this system in a certain class is proved, which implies that the basic plane boundary value problem has a solution and this solution is unique.