Systematizing one-dimensional boundary problems of deformed solid mechanics

DOI: 10.34759/trd-2020-110-3

Аuthors

Korovaytseva E. A.

Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia

e-mail: katrell@mail.ru

Abstract

The presented work highlights four ways of problems description of deformed solid mechanics (DSM) existing at present, namely scalar, operator, tensor and vector-matrix. Features of each technique were analyzed from the viewpoint algorithm developing and subsequent programming. The article demonstrates that vector-matrix formalization ensures the most general problem setting, and allows developing most universal and effective solution algorithms. For performing one-dimensional DSM boundary problems systematization the author selected problems solving approach associated with the necessity of employing systems of differential equations.

Sufficiency of considering only one-dimensional problems is stipulated by the fact that multidimensional problems solution algorithms building creates no essential methodical problems compared with the case of one-dimensional problems. Independent of the type, structure, geometry, material, loading or fixing of the structures under study, a unified vector-matrix form is used for the given problem description. The engaged differential equations can be of two types, namely linear and nonlinear ones without any restrictions concerning the range of nonlinearities under study. Three forms are being considered in each type of boundary problem. They are:

1. the two-point boundary problem,

2. multipoint unbranched boundary problem,

3. multipoint branched boundary problem.

As a result, all six forms of boundary problems are assumed canonical. Characteristic feature of the introduced forms canonization is highlighting of the resolving vector of desired variables, being differentiated with respect to a coordinate, as well as vector function of initial values of the problem parameters.

Six more forms of boundary problems with additional algebraic relations were built as well, which are the most widely used when solving applied problems of structures deformation mechanics. These forms can be reduced to linear canonical ones by identity transformations (for linear problems), or to quasi-linear canonical forms when building solution algorithm based on the parameter differentiation method (for nonlinear problems).

Thus the suggested systematization of one-dimensional boundary problems of solids mechanics, based on vector-matrix formalization of the resolving relations, allows reducing the number of problems requiring solution and the number of methods and algorithms of their solution developed, at the same time rising generality of these algorithms.

The irreducible spectrum of one-dimensional problems of mechanics of solids solution algorithms is associated with the six canonical forms introduced in the presented work. When programming algorithms of all the considered boundary problems types solution, only three programs for linear problems solution are basic, corresponding to the first three canonical forms.

Keywords:

canonical form, two-point boundary problem, multipoint boundary problem, vector-matrix formalization

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