The Direct Boundary Integral Method for Elasticity

The direct boundary integral method for elasticity is based on Somigliana’s displacement formula, which gives the displacements at a point inside a region V, but not on its boundary S, in terms of the counterparts in elasticity for the single and double layer potentials in potential theory with density functions equal to (a) the tractions on S and (b) the displacements on the same boundary. For two dimensions, this formula can be discretized by dividing the boundary S into N straight line segments—boundary elements—joined end to end and assuming that both the tractions and the displacements are piecewise constant over the elements. For a well-posed boundary value problem either the displacements or the tractions or some combination of these will be known at the midpoint of each element, leading to a system of 2N simultaneous linear algebraic equations in 2N unknowns. Computational details are provided for both constant strength elements and straight line elements with piecewise linear approximations of the boundary parameters, and several examples are worked to demonstrate the accuracy of the approach. The chapter concludes with a discussion of curvilinear isoparametric elements with quadratic variation of the boundary parameters.