Fundamental-solution methods in stress-concentration problems for thin elastic shells

Abstract

This paper deals with fundamental-solution methods applied to stress-concentration problems for thin elastic shells. Publications concerned with the relevant division of the theory of plates and shells are reviewed. The theories behind the methods are described, and specific results for static and dynamic concentrated loads are presented. The capabilities of the methods are illustrated by fracture problems for orthotropic shells with notches and holes under mechanical loading and for isotropic shells with notches under thermal loading Introduction. The intensive use of thin-walled shells of different configurations in various branches of modern mechanical engineering (aeronautical engineering and space technology, chemical engineering, shipbuilding, etc.) raises new challenges associated with the theoretical strength analysis of shells with stress concentrators. This involves problems of stress concentration in shells in the presence of local mechanical and thermal fields, holes, notches, elastic or rigid punches, ribs, etc. There are solutions only to special cases of such problems for mainly spherical and cylindrical shells.Extending the research to shells with various elastic properties and geometries requires a mathematical apparatus that would allow us to quickly solve complicated boundary-value problems for shells. It is efficient to reduce such problems to integral or integro-differential equations. This approach involves construction of fundamental solutions to the equations of shell theory-solutions corresponding to single concentrated forces.Thus, developing methods to construct fundamental solutions for shells with arbitrary geometries and elastic properties and using these solutions to solve stress-concentration problems for shells under local mechanical and thermal loads are a crucial task of double interest. First, this is a mathematical apparatus for deriving boundary integral equations for mixed problems of shell theory. Second, this is a model for strength analysis of shells under concentrated (or distributed over small sites) mechanical and thermal loads.The present paper describes methods for stress-strain analysis of shells under various mechanical and thermal local loads (concentrated forces, moments, temperature, thermal sources, loads distributed over small areas, etc.). The results to be discussed have been obtained using a modern mathematical apparatus based on the theory of distributions, Fourier transforms, and the theory of fundamental solutions and special functions. 1. Review of Fundamental-Solution Methods. A great many domestic and foreign studies are concerned with the development of methods to construct fundamental solutions to the equations of the theory of thin elastic shells. Problem formulations, solution techniques, and some specific solutions can be found in publications by Ambartsumyan [3], Vlasov [16], Vorovich [18], Gatimov [19], Gol'denveizer [23-25], Grigolyuk and Tolkachev [30], Darevskii [32-36], Grigorenko [31], Zhigalko [47], Mossakovskii and Hudramovych [62], Novo...

Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

“…Thus, we will mostly omit them, and retain them only when it will be important for the signs -derivative from cosine give "-" and from sinus gives "+". Substitute (10) into axial equilibrium equation, it gives:…”

“…Yet these papers were aimed only at derivation of the maximal deflection of the point of force application and their relative accuracies can be explained by overwhelming contribution of terms at smaller number of in expansion as to circumferential coordinate. The detailed review of these works was given in work [10].…”

Stress state around a circular hole in a prestressed transversely isotropic spherical shell

“…The equilibrium equations for a prestressed transversely isotropic spherical shell were considered in [10,11] as a special case. Noteworthy is the review [16] devoted to stress-concentration problems for thin elastic shells solved by the method of fundamental solutions. The present paper addresses a stress problem for a shallow transversely isotropic spherical shell with a circular hole.…”