Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki (Dec 2020)
Analytical solution of elastostatic problems of a simply connected body loaded with nonconservative volume forces: theoretical and algorithmic support
Abstract
The possibility of constructing a full-parametric analytical solution of the stress-strain state problem for the body caused by the influence of volumetric forces is studied. In the general case of Cesaro, the displacements at each point of the body are determined through the volume forces by an integral expression with a singular nucleus. Therefore, with an arbitrary shape of the body, its elastic state can be constructed only numerically. A strict analytical solution is written in the classical version, corresponding to the potential forces. These forces are traditional objects of mechanics, but their list is quite limited. The current level of development of science and technology in the world requires the use of forces of an arbitrary nature, which can be generated both at the level of molecular interaction, and the interaction of electromagnetic fields inside the body. They certainly are not conservative. In addition, the use of perturbation methods in solving nonlinear elastostatic problems and thermoelasticity problems creates, at each iteration of the asymptotic approximation, artificially generated volume forces of a polynomial nature or forces fairly accurately approximated by polynomials. The ability to write out strict or highly accurate private decisions during the iteration provides an invaluable service to the calculator. New method of constructing a strict solution of the problem about the corresponding elastic state of the body for a very wide range of forces, approximated by polynomials from spatial coordinates or, even for a narrower class- polynomial forces, is formed. It is based on the isomorphism of Hilbert spaces of forces of this kind and their corresponding elastic states (sets of displacements, deformations, stresses).The existence theorem of isomorphic countable bases of these spaces is proved, and algorithms for their filling are constructed. The particular solution of the problem about the elastic field from polynomial forces is constructed by decomposition of a given load on an orthonormal basis, written simply in the final form, and in the analytical form. The correction from the particular solution is made to the boundary conditions of the homogeneous elasticity problem for the body, after which its solution is constructed. Computational approaches, oriented to computer algebra, provide analytical form of solution. A convenient variant of this approach is the method of boundary states (MBS), which has a number of advantages over widely used numerical (finite elements, boundary elements, finite differences, etc.) and one significant drawback: the MBS computational complex has not received a final completion. The advantages of MBS are briefly stated and its laconic description is given. The use of the MBS approach makes it possible to write out a full-parametric form of solutions for bodies of arbitrary geometric shape. MBS is used to construct a solution of the problem of linear-elastic flattened spheroid, loaded with a self-balanced system of volumetric forces. The solution was constructed for two variants of loading, namely potential, non-potential forces. The analytical version of the solution is given only for the displacement field (other characteristics of the elastic state are easily written out through the defining relations).Certain interest is the graphic illustration of stress fields, made at fixed values of parameters.
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