The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity
DOI:
https://doi.org/10.32347/2410-2547.2019.103.208-218Keywords:
cylindrical cavity in a layer, Lame's equation, generalized Fourier method, infinite systems of linear algebraic equationsAbstract
An analytic - numerical algorithm for solving a substantially spatial problem of the theory of elasticity for a layer with a longitudinal cylindrical cavity and the conditions of the first principal problem, given on the boundary surfaces, is developed.
The solution is based on the generalized Fourier method applied to the Lame equation system. The cavity is considered in the cylindrical coordinate system, the layer - in Cartesian. By satisfying the boundary conditions and using special formulas for the transition between coordinate systems for basic solutions, we create infinite systems of linear algebraic equations, which are solved by the method of cutting. The numerical study of the determinant gives reason to claim that this system of equations has a single solution. As a result, stresses were obtained at different points in the elastic body.
Cutting parameters were chosen so that the accuracy of the boundary conditions reaches 10-2. As the cutoff parameter increases, the accuracy of boundary conditions increases, but the duration of the calculation increases.
The numerical solution of the problem is performed for a layer with a cylindrical cavity and a normal balanced load at the boundary of the layer. The analysis of the stress state gives grounds to state:
1. The solution of the algebraic system of equations can be found with any degree of accuracy by the method of reduction, which is confirmed by the high accuracy of boundary conditions.
2. The presence of a cavity gives rise to a redistribution of stresses, in which maximum values occur on the surface of the cavity, and tensile stresses occur at the top of the layer (which is compressed without a cavity).
This method can be used in the calculation of structures, the calculation scheme of which is a layer with a cylindrical cavity, under given boundary conditions in the form of balanced stresses. Stress analysis enables the selection of geometric characteristics at the initial design stage.
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