A singularity of stress at the crack tip, existing in a homogeneous medium for any shaped crack for both static and dynamic loading. The subject of dynamic fracture mechanics involves the analysis of structures with cracks when the load or the crack size changes rapidly. The determination of the Stress Intensity Factors (SIFs) and the T-stress is a fundamental task for elastoplastic fracture mechanics with functionally graded material in practical engineering.  It is well-known that the SIFs play a significant role in linear elastic fracture mechanics, as they characterise the crack-tip stress and the strain fields. However, the T-stress, which is the second constant in the asymptotic expression of the crack-tip stress fields, has to be considered in elastoplastic fracture mechanics. This project aimed to develop a series of the advanced numerical method, called Finite Block Methods, to investigate the fracture problems.