ALGORITHM OF SOLVING THE GENERAL THREE-DIMENSIONAL TASK OF ELASTICITY THEORY IN CYLINDRICAL SYSTEM OF COORDINATES FOR COMPUTER MATHEMATICS SYSTEMS
DOI:
https://doi.org/10.32782/KNTU2618-0340/2020.3.2-2.21Keywords:
a system computer mathematics (SCM), ossymmetrical task, cylindrical coordinates, symbolic solutionAbstract
This article is devoted to the problem of automation of analytical methods of the static theory of elasticity on electronic computers. Almost all bodies have to some extent the property of elasticity - the ability to return to their original shape during deformations caused by external forces. With elastic deformation, its value does not depend on the history and is completely determined by mechanical stresses, that is, it is an unambiguous function of stresses. For most engineering materials, this dependence can be considered with good accuracy as a direct proportionality, which is described by Hooke's law.
The main task of the static theory of elasticity is to determine the deformations of the body, their changes under given external forces. The system of equations for solving this problem is three equilibrium equations, which are closed by the equations of compatibility of deformations. A.I. Lurie and V.Z. Vlasov proposed one of the options for solving the system of equations - the analytical method of initial functions. V.V. Vlasov, F.A. Gochbaum improved the method of initial functions for a cylindrical coordinate system. However, due to the complexity of symbolic transformations, the method has not been used for a long time in mathematical modeling. This has now become possible with the development of computer mathematics systems. The proposed article shows the possibility of using the method of initial functions in mathematical modeling. The issues of constructing a common solution to the three-dimensional problem of the theory of elasticity by the methods of the initial functions of V.S. Vlasov, V.V. Vlasov are considered. The process of transition from Cartesian coordinates to cylindrical coordinates is described. An ossymmetrical task for body rotation is presented. The algorithm of building a symbolic solution in the form of differential operators in computer mathematics systems is proposed. The algorithm is programmed in the Maxima computer mathematics system. Entered the library of routines, written by the author to solve static problems of the theory of elasticity in two-dimensional and three-dimensional productions. Examples of work with the library in Maxima are given.
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