1 Introduction

In recent years, composition of several different materials is often used in structural components in order to optimize the thermal resistance and temperature distribution of structures subjected to thermal loading. For reducing the local stress concentration induced by abrupt transitions in material properties, the transition between different materials is made gradually. This idea used originally by Japanese researchers [1], leads to the concept of functionally graded materials (FGMs). These materials are expected to be used for thermal applications and high rate thermal loading. In the application of FGM cylindrical structures to aerospace, nuclear and automobile industries, analysis of transient heat transfer is of great importance. Analytical and computational studies of appointing thermal specifications and temperature distribution in cylindrical structures made of FGM have been carried out by some researchers.

Temperature and stress distribution were determined in a stress-relief-type plate of FGMs with steady state and transient temperature distributions by Awaji [2]. A multi-layered material model was employed to solve the transient temperature field in a FGM strip with continuous and piecewise differentiable material properties by Jin [3]. A general analysis of one dimensional steady state heat conduction and thermal stresses in a thick hollow cylinder under axisymmetric and non-axisymmetric loads were developed by Jabbari et al. [4, 5]. A long boundary integral equation method with the moving least squares approximation was applied to transient heat conduction analysis in functionally graded material by Sladek et al. [6]. Tarn et al. [7] have studied the end effects of steady state heat conduction in a hollow or solid cylinder of FGM under two-dimensional thermal loads with arbitrary end conditions. The sensitivity analysis of heat conduction for functionally graded materials and the steady state and transient problem treated with the direct method and the adjoint method were presented by Chen et al. [8]. Transient temperature field and associated thermal stresses in FGM have been determined by using a finite element-finite difference method by Wang et al. [9]. A finite element-finite difference method was developed also to solve the time dependent temperature field in non-homogeneous materials such as FGMs by Wang and Tian [10]. Hosseini et al. [11] studied the one dimensional transient heat conduction in an infinite functionally graded cylinder using an analytical method. They derived the temperature distribution by using Bessel functions.

In all of the mentioned cases the variation of volume fraction and properties of the FGMs are one-dimensional and properties vary continuously from one surface to the other with a prescribed function while in advanced machine elements temperature and load distribution may change in two or three directions. Therefore, if the FGM has two-dimensional dependent material properties, more effective material resistance can be obtained. Based on this fact a two-dimensional FGM whose material properties are bi-directionally dependent is introduced [12]. Recently a few authors have investigated 2D-FGM. Cho and Ha [13] characterized thermoelastic behavior of heat resisting FGMs, under given thermal loading and boundary conditions by the spatial distribution of volume fractions of constituent particles. They addressed a two-dimensional volume-fraction optimization procedure for relaxing the effective thermal stress distribution. Modeling the backing shell of the cemented acetabular cup with 2D-FGM and comparing its performance in the reduction of stress against 1-D FGM cup was done by Hedia et al. [14]. Reduction of thermal stresses in a plate using 2D-FGMs was investigated by Alla [15]. In his study a 2D-FGM rectangular plate under transient thermal loading is considered and the finite element method has been used to solve the governing equations. Employing the finite element method is a common approach for solving problems of this nature in literatures. But it should be noted that conventional finite element formulations use a single material property for each element so the property field is constant within an individual element, leading to significant discontinuity and inaccuracies for transient and dynamic problems [16]. Refinement of element size can reduce this effect to some extent but not completely. These inaccuracies will be more significant in 2-D FGM cases. Sentare et al. [16] showed that graded finite element can improve accuracy without increasing the number of degrees of freedom and decreasing the size of elements. Based on these facts the graded finite elements method is preferred for modeling the 2D-FGM problems.

Analysis of transient heat conduction of a thick hollow functionally graded cylinder with finite length can be rarely seen in literatures. And also the heat conduction in finite length cylinder is often investigated only in radial direction in these cases, while in real situations the heat conduction can be two dimensional in a finite length cylinder. On the other hand in all cases the material gradation of FGM is one dimensional. Therefore analysis of transient heat conduction of a thick hollow cylinder made of 2-D FGM can be of great importance. In the present study, we consider a thick hollow cylinder with finite length made of 2D-FGM for which its material properties are varied in the radial and axial directions with a power law function. The thermal behavior of the structure under thermal boundary conditions introduced suddenly is considered. Variation of temperature with time, two-dimensional distribution of temperature through the cylinder in different times and the effect of material composition profile are analyzed. The volume fraction distribution, thermal load and cylinder geometry are assumed to be axisymmetric but not uniform along the axial directions. The effects of two-dimensional material distribution on the time response of the structure and temperature distribution are studied. The finite element method with graded material properties within each element is used to model the material properties variations. Material properties are calculated by using linear rule of mixtures and prescribed volume fraction in each point. The Crank–Nicolson finite difference method is used to solve governing equations of heat transfer problem in the time domain.

2 Problem formulation

In this section the governing equations of heat transfer in axisymmetric cylindrical coordinates are obtained. Volume fraction distributions in the two radial and axial directions are introduced and graded finite element is used for modeling of the material non-homogeneity.

2.1 Governing equations

Consider a 2D-FGM thick hollow cylinder of internal radius r i , external radius r o , and finite length L. Because of axisymmetry of geometry and loading, coordinates r and z are used in the analysis.

2.1.1 Heat transfer equations

Without existence of heat sources, the equation of heat conduction in axisymmetric cylinder coordinates is obtained as

$$ {\frac{1}{r}}{\frac{\partial }{\partial r}}\left( {rk_{r} (r,z){\frac{\partial T(r,z,t)}{\partial r}}} \right) + {\frac{1}{r}}{\frac{\partial }{\partial z}}\left( {rk_{z} (r,z){\frac{\partial T(r,z,t)}{\partial z}}} \right) = \rho (r,z)c(r,z){\frac{\partial T(r,z,t)}{\partial t}} $$
(1)

where k r (r, z), k z (r, z), ρ(r, z) and c(r, z) are thermal conductivity in the radial and axial directions, mass density and heat capacity of functionally graded material, respectively, that vary in two directions. The thermal conductivities in the radial and axial directions are assumed to be the same.

The thermal boundary conditions are:

$$ {\text{temperature at inner radius}}\!:T(r_{i} ,z,t) = T_{c} (1 - e^{{c_{0} t}} ){ \sin }\left( {{\frac{\pi z}{L}}} \right), $$
(2)
$$ {\text{heat convection at the outer surface}}\!:k_{r} {\frac{{\partial T(r_{o} ,z)}}{\partial r}} + h_{o} (T - T_{\infty } ) = 0, $$
(3)
$$ {\text{heat convection at the lower surface}}\!: - k_{z} {\frac{\partial T(r,0)}{\partial z}} + h_{L} (T - T_{\infty } ) = 0 $$
(4)
$$ {\text{heat\;convection at the top surface}}\!:k_{z} {\frac{\partial T(r,L)}{\partial z}} + h_{T} (T - T_{\infty } ) = 0 $$
(5)

where h o , h L and h T are convection coefficients in outer side, lower and upper surfaces, respectively. T ∞ is the surrounding temperature and T c and c 0 are constant values. And the initial temperature of the cylinder is distributed uniformly.

2.2 Volume fraction and material distribution in 2D-FGM cylinder

Two-dimensional FGMs are usually made by continuous gradation of three or four distinct material phases in which one or two of them are ceramics and the others are metal alloy phases. The volume fractions of the constituents vary in a predetermined composition profile. Now consider the volume fractions of 2D-FGM at any arbitrary point in the 2D-FGM axisymmetric cylinder shown in Fig. 1. In the present cylinder the inner surface is made of two distinct ceramics and the outer surface from two metals.

Fig. 1
figure 1

Axisymmetric cylinder with two dimensional material distributions

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The volume fraction of the first ceramic material is changed from 100% at the lower surface to zero at the upper surface by a power law function. Also this volume fraction is changed continuously from inner surface to the outer surface. Volume fractions of the other materials are changed similar to the one mentioned, in two directions. The volume fraction distributions of each material can be explained as:

$$ V_{{c_{1} }} = \left[ {1 - \left( {{\frac{{r - r_{i} }}{{r_{0} - r_{i} }}}} \right)^{{n_{r} }} } \right]\left[ {1 - \left( {{\frac{z}{L}}} \right)^{{n_{z} }} } \right] $$
(6a)
$$ V_{{c_{2} }} = \left[ {1 - \left( {{\frac{{r - r_{i} }}{{r_{0} - r_{i} }}}} \right)^{{n_{r} }} } \right]\left[ {\left( {{\frac{z}{L}}} \right)^{{n_{z} }} } \right] $$
(6b)
$$ V_{{m_{1} }} = \left( {{\frac{{r - r_{i} }}{{r_{0} - r_{i} }}}} \right)^{{n_{r} }} \left[ {1 - \left( {{\frac{z}{L}}} \right)^{{n_{z} }} } \right] $$
(6c)
$$ V_{{m_{2} }} = \left( {{\frac{{r - r_{i} }}{{r_{0} - r_{i} }}}} \right)^{{n_{r} }} \left( {{\frac{z}{L}}} \right)^{{n_{z} }} $$
(6d)

where subscripts c 1, c 2, m 1, m 2 denote first ceramic, second ceramic, first metal and second metal respectively. Also n r , n z are non-zero parameters that represent the basic constituent distributions in r and z-directions. For instance, the volume fraction distribution of a basic material, c 1, for the typical values of n r  = 2 and n z  = 3 are shown in Fig. 2. In this case a thick hollow cylinder of inner radius r i  = 1 m, outer radius r o  = 1.5 m, and length L = 1 m is considered.

Fig. 2
figure 2

Volume fraction distribution of m 1 for n r  = 2 and n z  = 3

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Material properties at each point can be obtained by using the linear rule of mixtures, in which a material property P at any arbitrary point (r, z) in the 2D-FGM cylinder is determined by linear combination of volume fractions and material properties of the basic materials as shown in the following equation:

$$ P = P_{{c_{1} }} V_{{c_{1} }} + P_{{c_{2} }} V_{{c_{2} }} + P_{{m_{1} }} V_{{m_{1} }} + P_{{m_{2} }} V_{{m_{2} }} $$
(7)

The rule of mixtures is representative averaging estimate. This method is simple and convenient to apply for predicting the overall material properties and responses, however, owing to the assumed simplifications the validity is affected by the detailed microstructure and others. Because this estimate can not reflect the detailed constituent geometry, the dispersion structure and so on, its accuracy is well known to be highly questionable. But also it should be noted that most of the more precise methods are too complicated and many researchers considered them as the main problem for contribution [17, 18]. Also use of these models is usually developed for 1D-FGMs (two phase materials) and approximation models for 2D-FGMs are considered recently.

The basic constituents of the 2D-FGM cylinder are presented in Table 1:

Table 1 Basic constituents of the 2D-FGM cylinder
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The variation of a material property such as specific heat capacity based on the mentioned approach through the cylinder for the typical values of n r  = 2 and n z  = 3 is shown in Fig. 3.

Fig. 3
figure 3

Variation of specific heat capacity through the cylinder for n r  = 2 and n z  = 3

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2.3 Graded finite element modeling

In order to solve the governing equations the finite element modeling with graded element properties is used. To model a continuously non-homogeneous material, the material property function must be discretized according to the size of the element mesh. This approximation can provide significant discontinuities and these artificial discontinuities can cause enormous error in the results. Effects of these discontinuities will be more considerable in the 2D-FGMs because of its 2D non-homogeneity. Based on these facts the graded finite element is strongly preferable for modeling the present problem.

For modeling the heat transfer problem an axisymmetric ring element with triangular cross-section is used to discrete the domain. By taking the three nodal values of temperature as the degrees of freedom a linear model can be assumed as [19]:

$$ T^{e} (r,z,t) = [N(r,z)]\{ q\}^{e} , $$
(8)

where [N] is the matrix of linear interpolation functions and{q}e is the nodal temperature vector of the element as

$$ [N(r,z)] = [N_{i} (r,z) \; N_{j} (r,z) \; N_{k} (r,z)] $$
(9)
$$ \left\{ q \right\}^{e} = \left\{ {q_{i} \; q_{j} \; q_{k} } \right\} $$
(10)

Additional information is provided in the Appendix.

The variational form of the present problem leads to the following minimized integral

$$ I = 1/2\iiint\limits_{V} {\left[ {k_{r} r\left( {{\frac{\partial T}{\partial r}}} \right)^{2} + k_{z} r\left( {{\frac{\partial T}{\partial z}}} \right)^{2} + 2\rho cT{\frac{\partial T}{\partial t}}} \right]dv + 1/2\iint\limits_{S} {h(T^{2} - 2T_{\infty } T)ds}} $$
(11)

where V and S are the volume of the cylinder and the boundary on which the convective heat loss is specified.

The cylinder is now divided into the finite elements. The function I which is the sum of elemental quantities is minimized by using Eq. 8 subjected to the boundary and initial conditions. The graded finite element equations for each element are then derived as

$$ [k_{1} ]^{e} \{ q\}^{e} + [k_{2} ]^{e} \{ q\}^{e} + [k_{3} ]^{e} \{ \dot{q}\}^{e} = \{ P_{T} \}^{e} $$
(12)

where

$$ [k_{1} ]^{e} = 2\pi \iiint\limits_{V} {r[B]^{T} [D(r,z)][B]dV} $$
(13)
$$ [k_{2} ]^{e} = 2\pi \iint\limits_{S} {h\left\{ \begin{gathered} N_{i} \hfill \\ N_{j} \hfill \\ 0 \hfill \\ \end{gathered} \right\}}[\begin{array}{*{20}c} {N_{i} } & {N_{j} } & 0 \\ \end{array} ]rds $$
(14)
$$ [k_{3} ]^{e} = \iiint\limits_{V} {\rho (r,z)c(r,z)[N]^{T} [N]dV} $$
(15)
$$ \{ P_{T} \}^{e} = \iint\limits_{S} {hT_{\infty } [N]^{T} ds} $$
(16)
$$ [B] = \left[ {\begin{array}{*{20}c} {{\frac{{\partial N_{i} }}{\partial r}}} & {{\frac{{\partial N_{j} }}{\partial r}}} & {{\frac{{\partial N_{h} }}{\partial r}}} \\ {{\frac{{\partial N_{i} }}{\partial z}}} & {{\frac{{\partial N_{j} }}{\partial z}}} & {{\frac{{\partial N_{h} }}{\partial z}}} \\ \end{array} } \right] $$
(17)
$$ [D(r,z)] = \left[ {\begin{array}{*{20}c} {rk_{r} (r,z)} & 0 \\ 0 & {rk_{z} (r,z)} \\ \end{array} } \right] $$
(18)

For finding the characteristic matrices the integral must be taken over the elemental volume, considering the distribution function of material properties through each element.

Now by assembling the element matrices, the global matrix equation for the structure can be obtained as

$$ [K_{3} ] \left\{ {\mathop Q\limits^{.} } \right\} + [K]\{ Q\} = \{ P_{T} \} $$
(19)

where {Q} is the vector of nodal temperature of the cylinder and also

$$ [K_{3} ] = \sum\limits_{e = 1}^{M} {[K_{3} ]^{e} } $$
(20)
$$ [K] = \sum\limits_{e = 1}^{M} {[K_{1} ]^{e} + [K_{2} ]^{e} } $$
(21)
$$ \{ P_{T} \} = \sum\limits_{e = 1}^{M} {\{ P_{T} \}^{e} } $$
(22)

where M is the number of elements. Once the finite elements equations are established, the Crank–Nicolson finite difference method [20] with a suitable time step is used to solve the equations.

3 Implementation and validation

To verify the present work, since similar works are few, we simplify the present problem to be similar to the published literatures. At first a finite length 1D functionally graded cylinder under steady thermal loading solved semi-analytically in Ref. [21] using the series solution is considered. Then an infinite 1D-FGM cylinder under transient thermal loading solved analytically using Bessel functions in Ref. [11] is employed for verification and comparison of the results.

3.1 Finite length 1D-FGM cylinder under steady thermal loading

Consider a thick hollow cylinder in which the material distribution in the radial direction varies from ceramic (Mullite) at the inner surface to metal (Molybdenum) at the outer surface with a power law function. The geometrical parameters are as follows: length L = 5 m, inner radius r i  = 0.7 m and outer radius r o  = 1 m. The temperature distributions along surfaces are [21]:

$$ T(r_{i} ,z) = T_{o} \,{ \sin }\left( {{\frac{\pi z}{L}}} \right)\quad {\text{at}}\;r = r_{i} , $$
(23a)
$$ T(r_{o} ,z) = 0\quad {\text{at}}\;r = r_{o} , $$
(23b)
$$ T(r,0) = 0\quad {\text{at}}\;z = 0, $$
(23c)
$$ T(r,L) = 0\quad {\text{at}}\;z = L, $$
(23d)

where T o  = 200°C and all temperatures are in °C. The governing equations can be solved using a multi-layer approach [21]. For solving the mentioned problem by the graded finite element method developed here as a transient problem, we suppose that the thermal boundary conditions exerts as initial condition and the material properties vary in radial direction only. Therefore responses after a long time can be considered as the steady state result. Convergence of the time response is analyzed and the time step is chosen such that the response doesn’t depend on it. Also we suppose that the material properties vary in radial direction only and the volume fraction exponent and property coefficients are taken as: n z  = 0, n r  = 5, \( P_{{c_{1} }} = P_{{c_{2} }} = P_{c} \) and \( P_{{m_{1} }} = P_{{m_{2} }} = P_{m} \). where P c is the material property of the inner surface which is considered to be ceramic and P m is the material property of the metallic outer surface.

The variation of temperature with time in a specific point obtained using the present method and steady temperature of the same point obtained from the series solution [21] are shown in Fig. 4. The difference between the transient and the steady temperature is less than 10−5 after 6 × 104 s. So, comparison of the results shows good agreement between the two methods.

Fig. 4
figure 4

Variation of temperature with time at z/L = 0.5, r/r o  = 0.85

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3.2 Infinite 1D-FGM cylinder under transient thermal condition

In this case the distribution of mechanical properties are considered one-dimensional and across the thickness of the FG cylinder. The initial condition and boundary conditions on inner and outer radii are assumed to be the same as [11]:

$$ k_{r} {\frac{{\partial T(r_{i} ,z,t)}}{\partial r}} + h_{i} (T - T_{\infty } ) = 0\quad {\text{at}}\;r = r_{i} , $$
(24a)
$$ {\frac{{\partial T(r_{o} ,z,t)}}{\partial r}} = 0\quad {\text{at}}\;r = r_{o} , $$
(24b)
$$ T(r,z,0) = 200\,^{ \circ } {\text{C}}, $$
(24c)

Material properties are the same and r o /r i  = 1.5. In order to have radial heat conduction, thermal conductivity in the axial direction is eliminated. The cylinder is assumed to be too long and the temperature distribution along the radial direction is considered. The obtained results are then compared with the published data. The maximum difference for n.t = 0.5 and n.t = 2 are 0.82 and 0.74%, respectively. Figure 5 shows good agreement between the results at all points.

Fig. 5
figure 5

Time history of temperature’s radial distribution, n.t is the non-dimensional time

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4 Numerical results and discussion

A thick hollow cylinder of inner radius r i  = 1 m, outer radius r o  = 1.5 m, and length L = 1 m made of FGM with two–dimensional gradation of distribution profile is investigated. Constituent materials are two distinct ceramics and two distinct metals described in Table 1. Volume fraction of materials is distributed according to Eq. 6. Responses for some different powers of material composition profiles n r , n z will be compared.

Thermal boundary conditions are:

$$ T_{i} (z,t) = 1,000(1 - e^{{c_{0} t}} )\,{ \sin }(\pi z),\quad {\text{at}}\,r = r_{i} $$
(25)
$$ k_{r} {\frac{{\partial T(r_{o} ,z)}}{\partial r}} + h_{o} (T - T_{\infty } ) = 0,\quad {\text{at}}\,r = r_{o} $$
(26)
$$ - k_{z} {\frac{\partial T(r,0)}{\partial z}} + h_{L} (T - T_{\infty } ) = 0,\quad {\text{at}}\,z = 0 $$
(27)
$$ k_{z} {\frac{\partial T(r,L)}{\partial z}} + h_{T} (T - T_{\infty } ) = 0,\quad {\text{at}}\,z = 1 $$
(28)
$$ T(r,z) = 25^{ \circ } {\text{C}},\quad {\text{at}}\,t = 0 $$
(29)

where, h o  = h L  = h T  = 100(J/kg K), T ∞  = 25°C, c 0 = −2 and temperature unit is Centigrade.

In the following figures a cylinder with variation of volume fraction and material properties in two directions is considered. Figure 6 shows the temperature distribution through the cylinder at a specified time. The power law exponents of the material distribution profile in radial and axial directions are the same n r  = n z  = 2.

Fig. 6
figure 6

Temperature distribution through the cylinder after 2,000 s

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In order to have a more clear observation of effects of material distribution profile on transient responses, variation of temperature with time in two specified points near the upper and lower edges and two different points near the inner and outer surfaces of the cylinder are illustrated in Figs. 7 and 8 for different material distributions. In Fig. 7 the power law exponent in the radial direction is constant, but varies in the axial direction. Also the time response for a homogeneous cylinder (n r  = n z  = 0) is presented in this figure. In all of these cases the radial position of points are at \( \left( {{\frac{r}{{r_{o} }}} = 0.85} \right). \) It is clear that the time responses are strongly affected by material distribution tailoring. Increasing the radial power law exponent, the steady temperature of the specified point increases and also the temperature will become steady sooner. As an example increasing n z from 0.2 up to 5 in Fig. 7, increases the steady temperature by more than 25% at that specified point.

Fig. 7
figure 7

Transient temperature at points near the up and down edges for different material distributions

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Fig. 8
figure 8

Transient temperature at points near the inner and outer surfaces for different material distributions

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The same results are plotted for two different points near the inner and outer surfaces in Fig. 8. In these cases points are located on \( {\frac{z}{L}} = 05. \). Variation of temperature at the point located near the inner surface is less than the point near the outer surface of boundary condition at inner surface.

In order to have a more clear and comprehensive observation, distributions of temperature through the cylinder at different times are illustrated in Figs. 9, 10, 11 and 12 for various material compositions. The thermal initial and boundary conditions are as Eqs. 25–29 in these plots.

Fig. 9
figure 9

Temperature distribution (C) through the radial FG cylinder after 100 s

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Fig. 10
figure 10

Temperature distribution (C) through radial FG cylinder after 900 s

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In Figs. 9 and 10 material distributions are graded only in radial direction. It can be seen that in the same conditions the temperature distribution is different. The lower the value of n r , the lower the temperature is near the outer surface.

Volume fractions are graded in the two radial and axial directions in Figs. 11 and 12. Significantly different temperature distribution is obvious for the two specified times. For instance the temperature near the upper surface after 900 s is approximately 200°C for n r  = n z  = 0.2 while its value is more than 350°C for n r  = n z  = 5. Such different temperature distribution can be seen at the outer surface too.

Fig. 11
figure 11

Temperature distribution (C) through thme 2D-FGM cylinder for different material distributions after 100 s

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Fig. 12
figure 12

Temperature distribution (C) through the 2D-FGM cylinder for different material distributions after 900 s

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Also the temperature distribution through the 2D-FGM cylinder in which the inner and upper surfaces are subjected to the following boundary conditions is considered.

$$ T_{i} (z,t) = 100(1 - e^{{c_{0} t}} )z\,{ \sin }(\pi z),\quad {\text{at}}\,r = r_{i} $$
(30)
$$ k_{z} {\frac{\partial T(r,L)}{\partial z}} + h_{T} (T - T_{\infty } ) = 0,\quad {\text{at}}\,z = 1 $$
(31)

where T ∞ = 100°C. As the thermal boundary conditions aren’t symmetric with respect to the middle line in the axial direction in this case, two dimensional variations of materials distribution can be more effective to achieve a required temperature distribution. Figure 13 depicts the temperature distribution for two different values of n r while the values of n z are the same. Despite different temperature distributions a more effective change is expected. To this end the temperature distributions for two cases in which the volume fraction power exponents in radial and axial directions are different are illustrated in Fig. 14. Also two different values of power exponents are used in this case. So the temperature distribution is significantly affected and lower temperatures are achieved near the boundaries.

Fig. 13
figure 13

Temperature distribution (C) through the 2D-FGM cylinder for different material distributions after 500 s

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Fig. 14
figure 14

Temperature distribution (C) through the 2D-FGM cylinder for different material distributions after 500 s

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Radial distribution of temperature in the middle of the cylinder’s length at a specified time (t = 1,000 s) is shown in the Fig. 15 for different material distributions. That clearly shows the difference between temperature distributions especially the temperature at the outer surface.

Fig. 15
figure 15

Radial distribution of temperature for different material distributions on z = 2.5 after 1,000 s

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It is evident from these figures that both the magnitude and time delay of the responses are strongly affected by the material distribution powers n r and n z .

As it is seen in the results, temperature distribution and transient responses in two directions are strongly influenced by the material composition profile. In other words the time response, the maximum amplitude and the uniformity of temperature distribution through the cylinder can be modified to a required manner by selecting an appropriate material distribution profile. Variation of material distribution in two directions leads to a more flexible and desirable design which is very useful in optimization problems.

Although the manufacturing of multidimensional FGMs may seem to be costly or difficult, but it should be noted that while these technologies are relatively new, processes such as three-dimensional printing (3DP™) and Laser Engineering Net Shaping (LENS(R)) can currently produce FGMs with relatively arbitrary tree-dimensional grading [22]. With further refinements FGM manufacturing methods may provide the designers with more control over the composition profile of functionally graded components with reasonable cost.

5 Conclusion

A two-dimensional functionally graded cylinder with finite length under transient thermal loading is studied. Heat conduction equations are considered in two radial and axial directions. For modeling and simulation of governing equations a graded finite element method was used that has some advantages to the conventional finite element method. The transient responses of 2D-FGM cylinder are developed and variations of different parameters with volume fraction exponents are obtained. Effects of two-dimensional material distribution on the temperature distribution and time responses are considered. Based on the achieved results, 2D-FGMs have a powerful potential for designing and optimization of structures under multi-functional requirements.