Wiley-VCH published the article “Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half-strip with stiffeners” in the journal Zeitschrift für Angewandte Mathematik und Mechanik (Web of Science Q3, Scopus Q2, Impact Factor JCR: 1.103). 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.
We have constructed an exact solution to the boundary value problem in the theory of elasticity for a half-strip with identical stiffeners along its long sides and a load acting at its end (even-symmetric deformation). The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from closed formulas. The solution includes two easily variable parameters that characterize the relative tensile-compressive and flexural rigidities of the stiffeners. The final formulas for the stresses and displacements in the plate as well as for the longitudinal and transverse forces and bending moment in the stiffeners are simple and can be easily used in engineering.
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).
The study of problems of contact interaction in continuum mechanics is an important task of science and technology, on the solution of which success in mechanical engineering, construction, electronics, seismic exploration and in other areas of human activity largely depends. In addition, great interest in problems of contact interaction is due not only to the importance of their technical applications, but also to the internal logic of the development of this modern branch of continuum mechanics, which, in turn, is a strong stimulus for the development of the corresponding fundamental branches of mathematics.
In this paper we construct an exact solution to the boundary value problem for a half-strip with identical stiffeners along its long sides. Their tensile-compressive and flexural rigidities are taken into account by two parameters that enter into the expressions for the characteristic equation of the boundary value problem and Papkovich–Fadle eigenfunctions. The basis for this solution is formed by the theory of expansions of functions into series in Papkovich–Fadle eigenfunctions developed by us previously for a smooth half-strip (without stiffeners).
We consider two classes of expanded functions: (a) generating (entire) functions and (b) general functions. In this paper, we assign to the latter class the piecewise continuous functions which are most commonly encountered in engineering applications (in particular, the delta function and its derivatives) and are equal to zero in the neighborhood of the half-strip corner points. The Lagrange coefficients for the functions from class (a) are found directly from the integral relations to determine the biorthogonal functions, while the Lagrange coefficients for the functions from class (b) are found by using the finite parts of the biorthogonal functions. If the expanded functions, i.e., the normal and tangential stresses specified at the half-strip end, belong to class (a), then the solution to the boundary value problem will always be regular. If, alternatively, they are from class (b), then the normal stresses must be self-balanced because otherwise the solution may turn out to be nonregular in the sense of Muskhelishvili’s definition. In this paper, we show how the Lagrange coefficients for the normal stresses need to be modified for the solution to the boundary value problem to be regular for non-self-balanced normal stresses too. From a formal mathematical viewpoint, regularizing the solution is equivalent to improving the convergence of the Lagrange series for the functions from class (b).
It is shown that, in the exact solution obtained, there is no singularity for the stresses at the corner points of the half-strip, which is known to be present in the solution to the analogous boundary value problem for an infinite wedge with a stiffener on its one side and stresses applied to the other side.
Source: Kovalenko M.D., Menshova I.V., Kerzhaev A.P., Yu G. Expansions in terms of Papkovich–Fadle eigenfunctions in the problem for a half-strip with stiffeners // Zeitschrift für Angewandte Mathematik und Mechanik, 2021. DOI: 10.1002/zamm.202000093