Abstract: From the Hamiltonian governing equations of an elastic sectorial domain, the variable separation and eigenfunction expansion techniques are employed to formulate two circular singular hyper-analytical -elements. The two hyper-analytical-elements give a precise description of mode Ⅰ and mode Ⅱ Dugdale crack tip fields respectively. The new analytical elements can be implemented into FEM program systems to compute the plastic zone sizes and crack tip opening or sliding displacement based on mode Ⅰ or mode Ⅱ Dugdale model for plane crack problems with arbitrary shapes and loads. Numerical results for typical problems show that the method is simple and effective.