###### Abstract  
In this paper, the steady-state and transient heat transfer processes of functionally graded materials (FCMs) are analyzed using a coupled thermo-mechanical model in a GPU parallel multiphysics finite-discrete element software, namely MultiFracS. First, the coupled model to handle the heat transfer problem of heterogeneous materials is verified. Then, the advantages and disadvantages of FGMs and composite materials in response to thermal shock loads are compared and the results indicate that FGMs can overcome extreme environments better than composite materials. Finally, the influence of the geometric distribution characteristics of the double-edge cracks in the gradient material plate on the crack propagation is analyzed. The simulation results show that the interaction between the cracks affects the crack propagation path under the thermal load. The inclination angle and spacing of double-edge cracks greatly influence crack propagation. Specifically, a larger inclination angle and spacing can lead to a smaller crack propagation angle. The approach in this paper provides a new quantitative tool for investigating the thermal, elastic, and cracking of functionally graded materials. functionally graded materials; thermal cracking; thermal stress; composite materials; FDEM +
Footnote †: journal: Applied  
distribution in a functionally graded plate with a central crack under the action of two different directions of heat flow. [PERSON] et al. [3] studied the temperature distribution and the transient response of stress and displacement of functionally graded beams under one-dimensional thermal shock through Timoshenko beam theory and indicated that FGMs could effectively slow down the internal stress extremes. [PERSON] and [PERSON] [4] discussed the steady-state thermal stress in hollow cylindrical and hollow spherical FGMs shells, revealing the influence of radius on the stress and temperature field. [PERSON] et al. [5] studied the macro-response of an FGM plate under thermal load as a linear function and evaluated the magnitude of thermal stress by considering the change of material parameters as a linear function. [PERSON] [6] used first-order shear deformation theory to derive the equilibrium equation and stability equation of an FGM rectangular plate, assuming that the material parameters change as a power function with thickness. [PERSON] and [PERSON] [7] considered that the physical and mechanical parameters of FGM coatings vary exponentially with thickness and studied the influence of coating thickness on the internal temperature and stress of FGM coatings under thermal load. Similarly, [PERSON] [8] studied the effect of the gradient index, aspect ratio, and thickness-to-length ratio on FGM panels under the thermo-mechanical coupling based on the exponential change of the physical and mechanical parameters. [PERSON] and [PERSON] [9] assumed that the thermal expansion coefficient and elastic modulus of the FGM plate are distributed in the form of a power function volume fraction and further studied the influence of the loading type on the thermal buckling behavior in the sandwich FGM plate. [PERSON] et al. used ABAQUS software to study the load distribution in the threads of porous ZrO2/(ZrO2 + Ni) FGM bolt connection thermal protection system [10]. The above analytical methods are mainly based on simple geometrical shapes, such as plates and beams, and only consider the heat conduction in a single direction, or the material parameters that change in the form of simple functions (power functions, exponents, etc.) Additionally, the analytical solutions of steady-state and transient temperature fields and thermal stress distribution are derived. However, analytical methods are difficult to solve complex geometric shapes and boundary conditions and cannot fully reflect the spatial distribution of various parameters of FGMs. The numerical methods are widely used for the parameter analysis and material design of FGMs due to their low computation cost and strong applicability to boundary conditions.