On separation of a layer from the half-plane: Elastic fixation conditions for a plate equivalent to the layer

The solution of the homogeneous problem on a half-infinite crack passing along the interface between a thin layer and an elastic half-plane made of materials with distinct properties is obtained and analyzed. Following [1–4], the two-sided Laplace transform is used to reduce the problem to a matrix Riemann problem. The class of combinations of elastic constants of the materials for which the matrix coefficient can be factorized by the method proposed in [1–4] is singled out. This factorization is used to generalize the problem studied in [1–4] to the case of distinct elastic constants of the layer and the half-plane (although they satisfy an additional condition). An asymptotic expression for the crack shore displacements far from the tip are obtained. It is shown that the leading terms of the asymptotics of the crack shore displacements correspond to the cantilever (plate) displacements under boundary conditions of the rigid fixation type, i.e., to the conditions that the displacements and the angle of rotation at the fixation point are proportional to the components of the resultant and the bending moment of the load. Some expressions for the entries of the matrix of elastic fixation coefficients are obtained.