# An exact solution of an inhomogeneous problem for an elastic half-strip

Sage Publications published the article “An inhomogeneous problem for an elastic half-strip: An exact solution” in the journal Mathematics and Mechanics of Solids (Web of Science Q2, Scopus Q2, Impact Factor JCR: 2.040). Among the authors are employees of the Institute of Earthquake Prediction Theory and Mathematical Geophysics of the Russian Academy of Sciences (IEPT RAS): Leading Researcher, D. Sc. (Phys.-Math.) M.D. Kovalenko, Senior Researcher, Cand. Sc. (Phys.-Math.) I.V. Menshova and Senior Researcher, Cand. Sc. (Phys.-Math.) A.P. Kerzhaev.

In the article, in the form of series in Papkovich–Fadle eigenfunctions, exact solutions to two inhomogeneous boundary value problems of the theory of elasticity for a half-strip with free long sides are constructed: (1) the half-strip end is free, (2) the half-strip end is rigidly clamped. First, a solution to the inhomogeneous problem for an infinite strip is constructed. Then, the corresponding solutions for a half-strip are added to this solution, with the help of which the boundary conditions at the end are satisfied. To solve the inhomogeneous problem in a strip, the Papkovich orthogonality relation is used.

The research was carried out with the support of the Russian Science Foundation (project No. 19-71-00094) and the Russian Foundation for Basic Research in cooperation with the National Natural Science Foundation of China (project No. 20-51-53021).

Many fewer publications are devoted to inhomogeneous boundary value problems in the theory of elasticity (equilibrium equations with the right-hand side) than to homogeneous ones. To a certain extent, this is because the process of their solution is very laborious and the formulas describing the solution are quite complex. Furthermore, considerable mathematical skills are required for the construction of solutions.

In the proposed article, exact solutions for a half-strip with free long sides are obtained. An external surface load acts inside the half-strip along its axis (even-symmetric deformation). The half-strip end is free (the stresses are zero) or rigidly clamped (the displacements are zero). Fig. 1. Scheme of the inhomogeneous problem for a half-strip with a free end. Fig. 2. Scheme of the inhomogeneous problem for a half-strip with a rigidly clamped end.

First, a solution to the inhomogeneous problem for an infinite strip is constructed. Then, the corresponding solutions for a half-strip are added to this solution, with the help of which the boundary conditions at the end are satisfied. The solution to the inhomogeneous problem in the strip is constructed by using the Papkovich orthogonality relation, which quickly leads to achieving the goal. This solution is represented in the form of series in Papkovich–Fadle eigenfunctions. The Papkovich orthogonality relation also remains valid for other types of homogeneous boundary conditions, in particular, when its sides are rigidly clamped. Therefore, this method can be used to find simple exact solutions for a wide range of inhomogeneous boundary value problems in a strip with various homogeneous boundary conditions on its long sides. These solutions will also be represented by series in Papkovich–Fadle eigenfunctions.

The equivalence of solving problems with discontinuities of longitudinal normal stresses and inhomogeneous problems is shown by an example of solving the inhomogeneous problem of the theory of elasticity for a half-strip with free long sides.

It is shown that in the case of a clamped end, the stresses have no singularities at the corner points of the half-strip, in contrast to the corresponding solution for an infinite rectangular wedge. The reason is that in the solution for a wedge the type of boundary conditions changes when moving along one coordinate line (this becomes obvious for a straight wedge).

The solutions to all the problems are exact. They are represented by series in Papkovich–Fadle eigenfunctions, the theory of expansions in which is based on a fundamentally new mathematical apparatus based on the Borel transform in the class of quasi-entire functions of exponential type (Kerzhaev A.P., Kovalenko M.D., Menshova I.V.: Borel transform in the class W of quasi-entire functions. Complex Anal. Oper. Theory 12(3), 571–587 (2018). DOI: 10.1007/s11785-017-0643-y).

Source: Kovalenko M.D., Menshova I.V., Kerzhaev A.P., Yu G. An inhomogeneous problem for an elastic half-strip: An exact solution // Mathematics and Mechanics of Solids, 2021. DOI: 10.1177/1081286521996418.