Study On Heat Conduction Of FGM Structure Based On Variable Separation Method

Functionally graded material (FGM) is a new type of nonhomogeneous composite material, its composition and physical properties of the internal structure show continuous gradient change. Because of its excellent resistance to high temperature and high pressure, corrosion-resistant and wear-resistant and effective relaxation on stress mutation and other excellent properties, FGMs are widely used in aerospace, nuclear reactor, pressure vessel and new type building and so on research fields.According to the existing research situation on the mechanics problems in FGM structure, the paper puts forward the application of the variables separation method for solving the analytical solutions of the steady and transient heat conduction in the one-dimensional and two-dimensional FGM plates and the steady heat conduction of three-dimensional FGM structure.Firstly, as a result of the mathematical complexity of the analytical method and restricted to the initial boundary value problem, so the thermal conductivity in the FGM plate was assumed to be exponential function in order to simplify the process and increase the possibility of solving the problem. At the same time, specific heat capacity was aslo assumed the same type of function as thermal conductivity when the transient problem was considered. On the basis, the variables separation method and the trigonometric function orthogonality were applied to solve the general solution of the heat conduction equations in FGM structure.Then, structural boundary conditons were given, the undertermined parameters of the equations were derived, the only form of the analytical solution of the FGM plate heat conduction of corresponding to different boundary function was determined. In the paper, the general solutions of steady and transient heat conduction analytical function in the FGM plate under the variable temperature in the upper and lower boundary (upper boundary temperature function is the constant function, linear function and trigonometric function) and the constant temperature in the left and right boundary were derived. In order to exame the method, the finite element method based on weighted residuals was applied to solve the same condtion problem, the result showed that two methods have good consistency. Finally, the effects of material gradient parameters, geometric composition and different boundary temperature functions on temperature field distribution of FGM plate were studied. The results from one-dimensional and two-dimensional FGM were compared, the results showed that the internal temperature distributions of two kinds of FGM plates were obviously different, the study has important significance.In a word, to simulate the real engineering environment better, the different functions are applied to FGM sturcture in the paper, especially in two-dimensional FGM plate. The results can be used to instruction for the FGM design and evalute the other numerical methods. Also, the method can be extended to derive the analytical solution of steady-transient heat conduction problem of two-dimensional FGM axisymetric hollow cylindrical and the transient heat conduction of the three-dimensional FGM structure.