Integral representations in elastostatics and their application to an alternative boundary element method

We obtain a new representation for derivatives and anti-derivatives of any order of the displacement and stress fields for elastostatics problems when the boundary data is given in terms of polynomials of arbitrary degree. The result includes, as a special case, Somigliana’s theorem. Based on this identity, we propose an alternative algorithm for the boundary element method that uses polynomial approximations of arbitrary order. The method provides accurate results for two-dimensional elastostatics boundary value problems and has the advantage that it is easy to implement. The formula can also be used to accurately compute stresses and strains. For domains bounded by polygons, we provide closedform analytical expressions for the terms that appear in the stiffness matrix and the load vector for polynomials of arbitrary degree, thus avoiding numerical integration. We analyse the accuracy of the numerical solution as a function of the degree of the polynomial approximation by solving a representative boundary value problem