Boundary element method for three-dimensional couple stress elastostatic analysis

Abstract

A boundary element formulation for three-dimensional size-dependent couple stress elastostatic analysis is developed for the first time in the present work. The resulting computational method can play an important role in evaluating the mechanical response of a wide variety of components and systems at the micro- and nano-scale within a continuum framework. Initially, the infinite space fundamental solution is obtained by following the systematic Kupradze method and the remaining kernel functions due to point forces and point couples are derived. Via the reciprocal theorem, the boundary integral representation is then developed, and details of the numerical implementation are provided. In this process, regularization techniques are introduced, along with a novel five-node hybrid displacement-rotation boundary element, to eliminate the need for Cauchy principal value and Hadamard finite part integrals despite the deeply singular nature of the couple stress kernels. Several prototype computational examples are studied to explore the convergence of this new boundary element method and to elucidate some interesting behavior of couple stress theory, including the importance of three-dimensional analysis.