Mathematical modeling of static deformation of a layered construction with incompressible layers

This paper presents a mathematical model of static deformation of a layered medium with alternating layers of different stiffness. The proposed regularization of the problem is implemented using the parameter inverse to the volume compressibility modulus. The obtained numerical results are compared with analytical solution to verify a numerical algorithm convergence. The problem of stress-strain state calculation in a thick-walled pipe under internal pressure is considered as a test problem. Test problem solution is obtained applying the proposed methodology using regularization parameters corresponding to the Poisson ratio of 0.35, 0.45, and 0.49. The problem with compressible medium is also solved at the Poisson ratio of 0.49 and 0.499. The grid convergence of the problem solution is analyzed and the relative error is calculated. When considering the calculations obtained for compressible media at the Poisson ratio equal to 0.49, the relative error as regard to analytical solution exceeds an acceptable level, and when the ratio is of 0.499 for coarse meshes, the calculated results are different from analytical data. When using the proposed approach with regularization parameters corresponding to a range of Poisson ratio from 0.35 up to 0.45, the numerical solution is stable and the error obtained for calculated displacements, stresses, and strains for any kind of grid is less than 0.5%. Thus, in this paper, the rational value of regularization parameter is determined and the special approach that can be used to solve real technical problems is proposed.

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