Abstract: A set of analytical approaches are outlined in the present paper for some critical problems in fracture mechanics. Chien Weizang's method, i. e.,a superposition technique of crack tip singularity terms on the conventional finite element displacement modes,is extended to the calculation of stress intensity factors for mode-I, mode I , as well as mixed mode linear elastic fracture problems. For the elastic-plastic crack problems , an approximate closed-form analysis for plane strain, mode-I situation is presented.