Expansions in Papkovich–Fadle functions in the problem for a strip with a crack

Source abstract

The paper discusses a method for solving boundary value problems of the theory of elasticity in an infinite isotropic strip with an internal vertical crack (plane stress state). On the sides of the strip, homogeneous boundary conditions are prescribed. In general, the solution is represented as series of Papkovich — Fadle eigenfunctions, the coefficients of which are determined explicitly. The problem can be even- or odd-symmetric with respect to any of the central coordinate axes. On the vertical cut, horizontal or vertical displacements of the cut edges, as well as normal or shear boundary stresses, can be prescribed. In the first case, when the discontinuity profile is known, the solutions are obtained using the Papkovich orthogonality relation. In the second case, when stresses are prescribed on the cut edges, the solution becomes more complicated since it is first necessary to find on the cut the displacements corresponding to the prescribed stresses, and then to use the Papkovich orthogonality relation for determining the unknown coefficients of the expansions into series of Papkovich — Fadle eigenfunctions. The solution scheme is illustrated with an example of an even-symmetric problem for a free strip with a central vertical crack. The problem is solved in two formulations: 1) the discontinuity profile of horizontal displacements is known on the cut; 2) normal stresses are applied to the cut edges. In the second case, to determine the corresponding displacements, we introduce such a function constructed from Papkovich — Fadle eigenfunctions that it admits an analytical continuation to the entire strip, except for the cut. To construct the analytical continuation, the Borel transform is used. The obtained solution for a strip with a crack is compared with the corresponding solution for an unbounded plane, as well as with the numerical solution obtained using the finite element method.