Difference schemes for the solution of some static problems in the theory of elasticity

Abstract

IN this paper we investigate difference schemes for the solution of basic plane problems in the theory of elasticity in a rectangle, which are described by a set of equilibrium equations in displacements and boundary conditions of one of the three following types: 1. 1) the second boundary value problem (stresses are given on the boundary); 2. 2) the problem of hard contact or the third boundary value problem (the normal component of the displacement vector and the tangential component of the stress vector are given on the boundary); 3. 3) a mixed problem (conditions of various types are given at different parts of the boundary). The existence and uniqueness of the solution of the difference problems is proved and an apriori estimation of the solutions in the network norm [1] is obtained, which is homogeneous in h, and from which the convergence of the constructed difference scheme with speed O ( h 1 2 + h 2 2) follows for a sufficiently smooth solution, where h 1 and h 2 are network steps in the direction of x 1 and x 2. Difference schemes for the solution of a number of problems in the theory of elasticity are discussed in [1–7]. In [1] a decomposition scheme was developed for the solution of a two-dimensional dynamical problem in a rectangle with given displacements on the boundary and its convergence with speed O ( τ 2) was proved for ƒ h i = const . In [2] an iteration process was developed for the solution of a two-dimensional static problem in a rectangle with the same boundary conditions as in [1] and an estimate of the speed of convergence was obtained. In [3] both dynamic and static problems of the theory of elasticity in a p-dimensional parallelopiped are considered. Displacements on the boundary are also given. For the solution of the dynamic problem difference schemes are developed which are absolutely stable and converge with speed O ( τ + h 2) and O ( τ 2 + h 2). For the solution of the static problem an iteration process is developed and an estimate of the rate of convergence obtained. In [4] a problem is considered which converges to a plane problem of the theory of elasticity in a rectangle with boundary conditions which contain derivatives of the displacements. For this by a variational method a difference scheme of the second order of approximation is developed and an estimate obtained of the accuracy of O ( h) in the integral metric of W 2 (1) (the solution of the difference problem was used semilinearly). Some problems in the theory of elasticity in a rectangle (in particular a mixed one) are considered in [5,6]. In [7] a difference scheme is developed for the second boundary value problem of the theory of elasticity in a parallelopiped by a variational method; the boundary conditions are approximated with order O ( h). For its solution an iteration process is developed.