Integro-differential approach to solving problems of linear elasticity theory

Abstract

In order to construct efficient algorithms for solving problems of linear elasticity theory, the approach is used based on the introduction of integral relations between components of the stress and strain tensors. In the framework of the model proposed, the integro-differential boundary value problem is reduced to a variational problem to which well-developed methods of numerical analysis are applicable [1, 2]. To demonstrate the potentialities of the proposed approach, we employ the numerical-analytical method of finding approximations for desired stress functions and displacement functions. 1. We consider an elastic body that occupies a certain region Ω with boundaries γ . We assume that the displacement and stresses are given at the parts γ u and γ σ of the boundary, respectively, ( γ u  γ σ = 0, γ u ← γ σ = γ ). The stress‐strain state of the body is described by the set of differential equations of the linear elasticity theory: