D. Natroshvili and Sh. Zazashvili
abstract:
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.
Mathematics Subject Classification: 35J55, 74E30, 47G10
Key words and phrases: Elasticity theory, elastic mixtures, potential method, crack problems