Investigation of the First and the Second Exterior Plane Boundary Value Problems of Steady State Oscillations in the Linear Theory of Elastic Mixtures

Displacement vectors are represented by combinations of special potentials; singular integral equations of the normal type with zero index are obtained for the first and the second boundary value problem of steady oscillations in the theory of elastic mixtures. It is proved that in the case of positive frequencies the corresponding homogeneous singular integral equations have only a trivial solution.

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 [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.

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.

Asymptotic Behavior of Eigenfunctions and Eigenfrequencies of Oscillation Boundary Value Problems of the Linear Theory of Elastic Mixtures

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.