Derivations of Integral Equations of Elasticity

Abstract

In this paper, we derive the integral equations of elasticity, which may be considered to be a very general formulation for solutions of (cracked and uncracked) elasticity problems. The formulation is general enough to be a starting point for an analytical study or for a numerical treatment. The theory can be developed either by utilizing Betti's law or the weighted residual method, or directly resorting to physical meaning, as in the potential theory. To show that the results of the derivations are consistent with one another, we also prove four lemmas of the properties of the kernel functions. The derivations are continued by applying two commutative operations, traction and trace, leading naturally to the concept of Hadamard principal value. Consequently, singularity, often present in problems involving geometry degeneracy, causes no particular difficulties. Finally, we employ two examples to demonstrate the usefulness of the resulting dual boundary integral equations in both analytical and numerical solution procedures.