Analytical Solution Of Axisymmetric Steady Heat Conduction In Two-dimensional Functionally Graded Cylinder

This paper studied the steady temperature field distribution of two-dimensionalFGM cylinder in the first thermal boundary conditions. We assumed that the thermalconductivity of materials distribute in the form of power function. According to the lawof conservation of energy and the Fourier, we established the heat-conductiondifferential equations of the FGM cylinder. Combining with the orthogonality oftrigonometric functions, we obtained the analytical solution of steady temperature fieldof the2D-FGM cylinder in the corresponding boundary conditions by the use ofseparation of variables. On this basis, through the transformation of thermalconductivity, we obtained the analytical solution of the steady temperature field in the1D-FGM under the corresponding conditions.Using the finite element method, we also obtained the numerical solution of the2D-FGM cylinder under the corresponding condition. Comparing these two solutions,we found that the results of two methods are basically consistent，and the error is in theallowable range. Therefore，the correctness of the analytical solutions was examined.In addition, using specific examples, we studied the steady temperature fielddistribution of the2D-FGM cylinder under the following temperature boundaryconditions: constant function, linear function and sinusoidal function, and analyzed theaffecting factors of the temperature field distribution. The main considering aspectswere as follows:(1) The influence of different boundary temperature functions on the steadytemperature field of the1D-FGM cylinder and2D-FGM cylinder.(2) The influence of different geometrical size on the steady temperature field ofthe1D-FGM cylinder and2D-FGM cylinder.(3) The influence of different gradient parameters on the steady temperature fieldof the2D-FGM cylinder.The main analysis results are as follows:(1) On condition that other factors are same, under different thermal boundaryfunctions, the curve and change trend of the steady temperature field distribution ofFGM cylinder are markedly different.(2) The thermal boundary conditions and gradient parameters c are the same, when the radial dimension r of the FGM cylinder is fixed, with the increasing of axial size z,the change of temperature field distribution curve is more obvious. when the axial sizez of the cylinder is fixed, with the increasing of radial size, the change of temperaturefield distribution curve is smaller．When the radial size is larger, the change oftemperature field distribution curve is smaller.(3) The boundary condition and the geometrical size are the same, the c=0isbounded, when the c>0, c is large, the thermal conductivity of the materials is larger,the change of temperature field distribution curve is more obvious, when the c <0, cis greater, the thermal conductivity of the material is smaller, the change of temperaturefield distribution curve is smaller.This paper mainly studied the analytical solution of steady heat conductionproblem in the two-dimensional FGM cylinder, the models of similar to actual situationand the boundary temperature functions were considered. The obtained results canprovide analysis of suitable gradient parameters and geometry to meet the need fordesign, application and heat stress.