Mixed boundary value problems in the theory of elasticity in an infinite strip

We propose a method of solving mixed boundary value problems in the theory of elasticity in an infinite horizontal strip that has two points of change in the type of boundary conditions located on the upper and lower sides of the strip and lying at the ends of a vertical segment—the line of the joint between the left and right half-strips. The solutions to the right and the left of the joint line are represented as series in eigenfunctions that correspond to particular boundary conditions on the horizontal sides of the right and left half-strips. Solution continuity conditions or, on the contrary, discontinuities of the displacements or stresses can be specified at the joint between the half-strips. We illustrate the main idea of the method initially using a mixed boundary value problem for the harmonic equation as an example. Here, biorthogonal functions are constructed, their properties and the properties of the coefficients of expansions are investigated, and examples are given to illustrate the correctness of the obtained solutions. Subsequently we solve a mixed boundary value problem in which two trigonometric systems of functions with different eigenvalues are involved at the joint between the half-strips. We consider examples where (1) displacement and stress continuity conditions, (2) a discontinuity of the longitudinal displacements, and (3) a discontinuity of the transverse displacements are specified at the joint between the half-strips.