Numerical Solution of the Two-Dimensional Elasticity Problem in Strains

Abstract

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.