零場積分方程求解含圓形邊界之應力集中問題 Null-field Integral Equation Approach for Solving Stress Concentration Problems with Circular Boundaries Po-Yuan Chen and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University. Keelung, Taiwan Abstract In the paper, boundary value problems with circular boundaries are formulated in a unified manner by using null-field integral equation in conjunction with degenerate kernels and Fourier series expansions. Laplace problems of circular holes as well as Helmholtz problem of SH-wave impinging on circular cavities and/or inclusions were studied. The fundamental solution is expanded to degenerate form by separating the source point and field point in the polar coordinate. The main gain of using degenerate kernels for interior and exterior expansions is free of calculating the principal values. In order to fully describe the circular boundary, the present method employs the Fourier series to approximate the boundary potential. By collocating the null-field points on the real boundary with the same number of Fourier coefficients, the unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Five advantages, well-posed model, principal value free, elimination of boundary-layer effect, mesh-free approach and exponential convergence, are achieved. Finally, several examples involving bending, and infinite domain with cavity and half-plane with alluvial valleys and inclusions problem were given to demonstrate the validity of the proposed method. Also, the numerical results agree well after comparing with available solutions in the literature. A general-purpose program for multiple circular cavities and/or inclusions of various radii and arbitrary positions was developed. Degenerate kernelFourier series Collocation point and matching B.C. Adaptive observer system Linear algebraic equation Fourier coefficients Potential of domain point Stress field Vector decomposition Numerical Analytical Flowchart of present method Degenerate kernels for the fundamental solution Fourier series expansion for the boundary density Vector decomposition Vector decomposition for the potential gradient in the hypersingular equation Numerical examples Bending problem Stress concentration factor of cavities problem subject to the incident SH-wave Half-plane problems with inclusions subject to the incident SH-wave Stress concentration versus b for a=0.12, R=1.0 and three different values of Contour plot for b=0.4, a=0.12, R=1.0 and three different values of Shear stress around the cavity of a full-plane problem subject to the horizontally incident SH wave. A full-plane problem with a cavity subject to SH-wave. Surface displacements as a function of x/a and η for the vertical incidence η=0.1 η=0.25 Surface amplitudes of two-inclusions problem 國立台灣海洋大學河海工程學系 力學聲響振動實驗室 ~ Copyright 中華民國第八屆結構工程研討會 The 8th National Conference on Structure Engineering Sun Moon Lake, Taiwan, R. O. C., 1-3 Sept Paper No. F-006