3-D steady temperature field in a laminated rectangular plate

According to the exact three-dimensional (3D) thermal theory, the steady temperature distribution in a laminated rectangular plate with zero temperature conditions on four lateral surfaces was studied. An analytical method was developed to solve the temperature field in the plate. Firstly, the general solution of the temperature field in a single-layer rectangular plate, which exactly satisfies the governing thermal differential equation, was derived out. Then, the temperature and heat flux relationships between the upper surface and the lower surface of the single-layer plate were obtained. Based on the continuity of the temperature and the heat flux on the interface of two adjacent layers, the temperature and the heat flux between the lowest layer and the top layer of the laminated plate were recursively obtained by using the transfer matrix method. The unknown coefficients in the solutions for every layer were uniquely determined by the use of the temperature conditions at the upper and lower surfaces of the plate. The temperature distribution in the laminated plate was given by substituting the unknown coefficients obtained back to the recurrent formulae and the solutions. The convergence of the solutions has been checked based on the number of series term. Comparing the results with those obtained from the finite element method, the accuracy and correctness of the present method were demonstrated. Finally, the effects of surface temperatures, thickness, layer number and material properties of the plate on the temperature distribution were discussed in detail.