Abstarct: In recent years functional materials due to mechanical and thermal properties desirable and manageable for engineers and researchers have been ever-increasing attention. These materials benefit high toughness high mechanical strength of mextals, heat resistance, corrosion resistance and wear ceramic material simultaneousl. So checking the behavior of the failure of these materials considering their application fields has particular importance. In this thesies, the fracture behavior of functionally graded materials under thermo-mechanical shocks in a thick-walled cylinder is investigated. For this purpose, classical coupled thermaoelastic equations are used in calculations.First, these equations are discretized with extendedfinite element method in the space domain and then they are solved by the Newmark method in the time domain. In this thesis, micromechanical models for conventional composites with assist of generalized isoparametric element formulation, which continuously models the material properties across the elements, are used. Also, the most general form of interaction integral is extracted for axially symmetric circumferential crack in a cylinder under thermal and mechanical loads in functional graded materials and is used to calculate dynamic stress intensity factors. All stages of problem solution from mesh generation to obtaining results has implemented in MATLAB programming environment.The accuracy of written program is verified using several solved numerical examples and compared with existing analytical and numerical results in literature. Finally, effects of length of crack, change of properties, rotational speed and change internal and external pressure in thermal loading condition are investigated on stress intensity factors. Also, variation of stress intensity factor against time in dynamic load scenario is depicted. Ti-6Al-4V/Zro2 is functional graded material used in this thesis.