In this study, the nonlinear analytical analysis of a two-layer geometry made of functionally graded materials (FGMs) is examined. FGMs can be used in various engineering applications, such as building materials in civil engineering, due to the advantages of smoothly varying properties. The equations of stresses and displacements in the radial and circumferential directions (r, θ ) have been found by extracting the governing equations and defining them in the form of power-exponential functions. In the present paper, modulus of elasticity and heat conductivity coefficient (except for Poisson’s coefficient) are assumed to be expressed by power-exponential functions in radial and circumferential coordinates. The temperature distribution is also considered as a function of radius (r) and angle (θ). The analysis is implemented based on the theory of small elastic deformations and with the assumption of a very large length in plane strain mode. To analyze the governing equations, first, the heat transfer equations are obtained, and then the Navier’s equations are derived by combining the stress–strain, strain–displacement, and stress equilibrium equations. Then, the displacement equations and stress equations are obtained by solving the Navier’s equations. A direct method is presented to solve these equations analytically.