On the boundary integral formulation of the plane theory of elasticity (computational aspects)

Abstract

A numerical scheme is proposed for the solution of the system of field equations and boundary conditions of the static, linear plane strain problem of the Theory of Elasticity in stresses. The work is based on a previous analytical approach by the authors within the Boundary Integral Method for a homogeneous and isotropic elastic material occupying a simply connected region. The differential and integral operators appearing in the resulting system of integro-differential equations for the determination of the boundary values of four unknown harmonic functions are discretized and the problem at this stage is reduced to finding the solution of a linear system of algebraic equations. Formulae are provided for the determination of the displacements and stresses at the boundary. As an illustration, the proposed numerical scheme is applied to find the solution of the following problems: (i) a nearly-circular boundary subject to a uniform pressure or to a uniform normal displacement; (ii) a square subject to a uniform pressure or to different prescribed normal displacements. This choice of the problems has allowed to assess the efficiency of the method through comparison with known analytical solutions and to test the accuracy of the results under different smoothness properties of the boundary and boundary conditions. Whenever new, the obtained results were thoroughly discussed. Future work in this field will be devoted to the extension of the method to include multiply connected regions and mixed boundary conditions, and to investigate the prediction and isolation of the singularities in the solution, caused by boundary irregularities.