Development of the Galerkin Finite Element Method for Stress-Based Elasticity Problems

This paper proposes a finite element approach to the numerical solution of boundary value problems of linear elasticity theory formulated directly in terms of the stress tensor components. In contrast to the classical finite element formulation, in which displacements are the primary unknowns, the approach considered here treats the stress components as the sought quantities. Two forms of the boundary value problem are investigated. The first is based on the joint use of the equilibrium equations and the Beltrami–Michell equations, while the second is a transformed system of Poisson-type equations for the stress tensor components. Variational relations based on the Galerkin method are obtained for both formulations. Using linear basis functions on triangular finite elements, local matrices are constructed, and global systems of algebraic equations are formed, whose unknowns are the nodal values of the stresses. The numerical implementation of the developed schemes is carried out in an in-house C++ program and in the FreeFEM++ software environment. To verify the reliability of the proposed mathematical and numerical models, the classical Kirsch problem of a stretched elastic plate with a circular hole—characterized by a pronounced stress concentration near the edge of the hole—is solved. The numerical values of the stress components are compared with the analytical solution obtained using the Airy stress function method, as well as with the results of calculations performed with the FreeFEM++ software package. The comparison shows good agreement between the obtained solutions and confirms the possibility of determining stresses directly, without first computing the displacement field. A mesh-refinement study further showed a systematic reduction in the numerical error: on the finest mesh considered, the relative error decreased to 3.189% for formulation A and to 7.415% for formulation B. The proposed approach extends the applicability of the finite element method to boundary value problems of elasticity theory formulated in terms of stresses and can be used to study problems with complex geometry and local stress concentration.