Boundary element method (BEM) is used to study propagation of crack under loading. First, all leading unknown coefficients in Williams series expansion and complete stress field for notched structures are calculated with BEM and former eigenanalysis for notch tip region. Then, with consideration of non-singular stress term and maximum circumferential stress criterion of brittle fracture, crack initiation extended direction from crack tip in a semicircular bending specimen is determined by BEM. Strategy for BEM tracking crack propagation is given. Numerical examples show that the method is correct and effective in simulating propagation of plane crack.

Based on asymptotic extension of displacement field at a composite notch tip, equilibrium equation for a notch subjected to anti-plane loading is transformed into a characteristic differential equation respects to notch singularity orders. A transformation is applied to convert the equation into a set of characteristic linear ordinary differential equations. Interpolate matrix method is introduced to solve the equations for getting notch singularity orders. A single material notch, a bi-material notch and a notch terminated at bimaterial interface are studied successively. Examples indicate that the method provides all stress singularity orders synchronously. Though singular stress state is not shown with regard to non-singular orders, non-singular stress orders are indispensable parameters as evaluating complete stress field at notch tip region.