## 2 The Governing Equation and General Solution
### The Analytical Solution of the FGM-Coated Circular Plate
Additionally, the stresses along  
Furthermore, let us examine how the thickness of the coating influences mechanical behavior. In this analysis, we consider the ratios \(R_{1}/R=0.1,\ 0.2,\ 0.5,\ 1\). The variations of stresses and moments along the radial direction for these ratios are presented in Figures 5 and 6. The results indicate the following observations: First, smaller values of the ratio \(R_{1}/R\), corresponding to thicker FGM layers, result in lower stresses. Second, in the case of a homogeneous circular plate (\(R_{1}/R=1\)), the maximum stress or moment is located at the center of the circular plate. However, when the circular plate is coated with a thicker FGM layer (smaller \(R_{1}/R\)), the maximum stress or moment shifts towards the inner part of the FGM-coated circular plate. Third, the stress and bending moment curves exhibit similar patterns, indicating a proportional relationship for certain values of \(n\), \(m\), and \(P\). As mentioned in Section 3.2, the thermal stresses and moments exhibit the relations \(M_{j}=h^{3}\sigma_{j}/(12z)\), \(j=r,\theta\) for the special case when \(P=0\). These findings shed light on the impact of the coating thickness on the mechanical behavior of the circular plate and provide valuable insights for practical applications. Figure 4: The distributions of stresses \(\sigma_{r}\) and \(\sigma_{\theta}\) of the radially FGM-coated circular plate (\(n=2\)) subjected to different radially thermal loads (\(m=0,\ 1,\ 2\)). Figure 5: The distributions of stresses \(\sigma_{r}\) and \(\sigma_{\theta}\) of the radially FGM-coated circular plate (\(n=2\)) subjected to thermal load (\(P=0\), \(m=2\)) while the coated thickness changes with \(R_{1}/R=0.1,\ 0.2,\ 0.5,\ 1\). To further analyze the effect of radially applied thermal load on the mechanical behavior of the radially FGM-coated circular plate, let us consider the thermal parameters \(P=-0.5\) and \(m=0.5,\ 1,\ 2,\ 3\). It is worth noting that \(P=-0.5\) signifies the absence of transverse thermal loads. Figures 7 and 8 present the stresses and bending moments of the FGM-coated circular plate under radially applied thermal load. Observations from Figures 7 and 8 are as follows: Firstly, the stresses of the homogeneous portion of the circular plate are functions of \(r^{m}\), contributed by the radially applied thermal load. On the other hand, the stresses of the FGM layer follow a polynomial \(r^{m+n}\), influenced by both the radially applied thermal load and material gradation. Consequently, the stresses \(\sigma_{r}\) and \(\sigma_{\theta}\) in Figure 7 of the radially FGM-coated circular plate subjected only to the radially thermal load (\(P=-0.5\)) exhibit continuity with noticeable inflection points in slope at the interfaces (\(R_{1}=2R/3\)). Secondly, for \(P=-0.5\), the thermal stresses and moments do not exhibit a proportional relationship. Figure 6: The distributions of bending moments \(M_{r}\) and \(M_{\theta}\) of the radially FGM-coated circular plate (\(n=2\)) subjected to thermal load (\(P=0\), \(m=2\)) when the coated thickness changes with \(R_{1}/R=0.1,\ 0.2,\ 0.5,\ 1\). Figure 7: The distributions of the stresses \(\sigma_{r}\) and \(\sigma_{\theta}\) of the radially FGM-coated circular plate (\(n=2\)) subjected to the radially thermal load only (\(P=-0.5,m=0.5,\ 1,\ 2,\ 3\)). Notably, the proposed analytical solution can be applied to a homogeneous circular plate by setting the material index \(n=0\) in the FGM-coated circular plate. When the FGM-coated circular plate degenerates into a homogeneous circular plate (\(n=0\)), the stress and moment behaviors under transverse thermal loads (\(P=0,\ 2\)) and radial thermal loads (\(m=1,\ 2,\ 3\)) are depicted in Figure 9. These results exhibit a dependence on the function \(r^{m}\) for the homogeneous circular plate. Moreover, it is noteworthy that the effects of radial thermal loads are more pronounced compared to transverse thermal loads when the value of \(P\) is small.