Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method

Abstract

In this paper, the boundary element method (BEM) based on elasticity theory is developed for two-dimensional (2-D) thin structures with the thickness to length ratio in the micro (10−6) or nano (10−9) scales. An efficient analytical method is developed to deal with the nearly-singular integrals in the boundary integral equation (BIE) for 2-D thin structures. The nearly-singular integrals, which are line integrals for 2-D problems and arise when two boundary curves are close to each other, are transformed into function evaluations at the two end points of the element of integration. In addition, a new nonlinear coordinate transformation is developed for nearly weakly-singular integrals to further increase the numerical accuracy. For the test problems studied, very promising results are obtained when the thickness to length ratio is in the orders of 10−6 to 10−9, which is sufficient for modeling most thin structures in the micro- or nano-scales. The developed method can be applied readily to model layered coatings, thin films or other layered structures to analyze contact stresses, interfacial cracks, thermal effects and nonlinear deformations.