MIXED TYPE BOUNDARY VALUE PROBLEMS IN THE LINEAR THEORY OF ELASTIC MIXTURES FOR BODIES WITH INTERIOR CUTS

We consider two-dimensional mixed type boundary value problems for the equations of the linear theory of elastic mixtures. We assume that the elastic body under consideration contains interior cracks. On the exterior boundary of the body the mixed Dirichlet (displacement) and Neumann (traction) type conditions are given while on the crack sides the stress vector is prescribed. We apply generalized Kolosov Muskhelishvili type representation formulas and reduce the mixed boundary value problem to the system of singular integral equations with discontinuous coefficients. Fredholm properties of the corresponding integral operator are studied and the index is found explicitly. With the help of the results obtained we prove unique solvability of the original mixed boundary value problem.