Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity

: In this study, a closed-form analytical solution is derived to compute the stress formulations for a thick cylindrical pressure vessel made of functionally graded material (FGM) with varying parameters, which are mechanical and thermal boundary conditions. The assumed mechanical boundary condition is the time-dependent pressure acting on the internal surface of the cylinder, while the assumed thermal boundary condition is the transient temperature distribution over the cylinder thickness. The material properties are considered to be graded exponentially in the radial direction, except Poisson ’ s ratio which is assumed to be constant. The stress and displacement formulations are evaluated using Mathematica software for the uncoupled thermo-mechanical analysis. The results of radial, hoop, and axial stress are plotted at various times for two FGM cylinders, the SS304-Alumina FGM cylinder and the TZM-SIC FGM cylinder, to study the impact of using di ﬀ erent materials for the same boundary conditions on the results. The results obtained in this study are bene ﬁ cial as these contribute to the design and modeling of cylinders that are exposed to time-dependent internal pressure and transient temperature pro ﬁ les.


Introduction
The use of functionally graded materials (FGMs) is increasing in versatile engineering applications [1][2][3][4][5].FGMs introduce good material properties that enable them to be used under harsh working conditions such as high-temperature environments in addition to the presence of mechanical loadings.FGMs are materials designed and tailored as per engineering requirements, which makes them unique.Usually, these materials are made from a mixture of two materials with different properties to present new material and have the combined properties of the two consistent materials.The most famous two materials used in producing FGMs are ceramic and metal [6][7][8][9][10][11][12][13][14].
Cylinders can be used as hollow cylinders, thin-walled shells, pressure vessels and pipes, cylindrical panels, etc. Cylinders are being used in many applications in which they shall be subjected to mechanical loads or thermal loads or both at the same time.Hence, cylinders are of great interest to be investigated as FGMs under different loading conditions.Some researchers considered the cylinder material properties to change according to power functions of radius r; for example, Abrinia et al. [15] and Jabbari et al. [16] developed a one-dimensional steady-state thermal stress analysis in a hollow thick cylinder made of FGM.The general thermal boundary condition is assumed to be in the form of variations of the temperature along the radial direction; as a result, the temperature of the internal surface is different from that of the outer surface and mechanical boundary conditions as a result of different values of pressure on both the inner and outer surfaces.The direct method is used to solve the heat conduction and Navier equations.Wang et al. [17] presented an analytical solution of the thermo-mechanical stresses for the FGM cylinder with properties changing through the thickness according to a power law function subjected to internal pressure and a thermal load.A numerical model of the FGM cylinder was developed using the finite element (FE) method to validate the analytical solution.Xie et al. [18] obtained an exact solution of stresses in the FG cylindrical/spherical pressure vessel.Three different boundary conditions are considered during the study of the FG shells.The analytical solutions of the FG hollow cylinder/spherical shell were obtained and compared with the existing classical theoretical and numerical solutions.The results showed that the gradient parameter and geometric size have a considerable effect on the mechanical response of the FG hollow cylinder/spherical shell under different boundary conditions.Das et al. [19] studied the thermo-mechanical behavior of FGM hollow cylinders subjected to both mechanical loading and steady thermal stresses.The analytic solution of stresses and displacement was obtained, and numerical models were built using the FE method.The effect of inhomogeneity in the FGM thick cylinder was investigated.It is concluded that by altering the β value, the properties of the FGM shall be altered accordingly to achieve the lowest stress values.Many other studies, such as those of Nejad et al. [20] and Chen [21], dealt with the stresses in cylinders of FGM with materials varying according to power functions.
Sondhi et al. [22] investigated the limit elastic yield pressure of an internally and externally pressurized thick hollow cylinder made of FGM for different gradient indices.The mechanical properties of the material are assumed to follow exponential functions.An approximate solution is found, and the results are reported for deformation and stress distributions along the radial direction.Additionally, the variation of limit elastic yield pressure in normalized form for various values of the gradient index is plotted and hence, the limit pressure can be predicted for a cylinder with given grading parameters and geometry parameter.
Other researchers considered the material properties to change according to exponential properties via the thickness; for example, Celebi et al. [23] presented a numerical solution for the calculation of axisymmetric thermal and mechanical stresses in a thick hollow cylinder made of FGM.The temperature distribution was assumed to be a function of the cylinder radius with thermal and mechanical boundary conditions on the internal and external cylinder surfaces of the cylinder.A complementary function method is employed in the analysis and is presented.Habib et al. [24] presented an analytical solution for a pressure vessel made of FGM and subjected to both thermal and mechanical loadings with material properties varying according to the exponential function.The resulting solution is validated by the FE modeling method.Tutuncu [25], Ghannad and Gharooni [26], and Manthena et al. [27], and others studied the axisymmetric stresses in FGM cylinders with varying properties that follow the exponential function.
Later, researchers focused on varying loading conditions.Most researchers who dealt with the varying parameters focused on either the transient/non-uniform thermal loads or varying non-uniform mechanical loads, although in many applications such as gun barrels, boilers, pipes of nuclear reactors, and FGMs, both loading conditions coexist.Hence, this research focused on studying the stresses resulting in cylinders under varying loading conditions, both mechanical and thermal.
Khoshgoftar et al. [28] investigated the thermoelastic analysis of functionally graded cylinders with variable thickness under non-uniform pressure.The article aims to show the capability of matched asymptotic solutions for different non-homogeneous cylinders with different shapes and different non-uniform pressures.Nejad et al. [20] investigated an FG rotating thick hollow cylinder with variable thickness semi-analytically considering an arbitrarily non-uniform pressure acting on the inner surface.The FG cylinder material properties are assumed to vary according to a power law function in the axial direction of the cylinder.The problem is solved using the first-order shear deformation theory considering that the nonhomogeneous cylinder is divided into n homogenous disks.Then, the results are compared by using an FE method solution.
Others investigated the stresses in cylinders as a result of non-uniform or varying thermal loads such as Rani et al. [29] studied the thermal stresses of an FG hollow thick cylinder with non-uniform internal heat generation.Exact analytical solutions found for the temperature distribution and the thermal stresses resulted in the FG hollow cylinder due to the non-uniform internal heat generation based on the theory of uncoupled thermoelasticity.
The novelty of this study is investigating the impact of transient thermal boundary condition in the presence of time-dependent mechanical boundary condition for the FGM pressure vessel analysis as most literature surveys focused on studying either the impact of transient thermal boundary condition or the impact of variant or time-dependent mechanical boundary condition.In the present study, stress and displacement analysis is presented for a cylinder subjected to varying mechanical load in the form of internal pressure that varies with time and transient thermal load through the cylinder wall thickness.The material properties such as Young's modulus, coefficient of thermal expansion, and thermal conductivity are considered to follow an exponential function of the cylinder's radius.By using equilibrium equation and constitutive equations, a second-order nonhomogeneous differential equation is introduced and solved to get the displacement.Accordingly, stress formulations are found.By considering two materials for the FGM cylinder, the stresses' values with respect to time are graphically represented.

Analytical solution
In this section, an analytical solution is presented by following some key steps to solve the problem at hand and get the required stress and strain formulations.First, the problem is identified by presenting the cylinder's geometrical parameters and material properties.Second, the constitutive and equilibrium equations are specified.Third, the transient temperature distribution equation is evaluated.Last, the thermoelastic analysis is applied by finding the governing equations and applying the boundary conditions.

Problem identification
As shown in Figure 1, the under-study problem is assumed to be an FGM pressure vessel subjected to a varying internal pressure P with respect to time t and a transient temperature T acting in the radial direction.The cylinder's inner radius is R i and the outer radius is R o .A thermal stress component σ T and a mechanical stress component σ M result from the thermal loading mechanical loading, respectively.The cylinder is considered to be under a linearelastic uncoupled thermo-mechanical conduction condition, for which the combined stresses shall be the sum of mechanical and thermal stresses for the radial, hoop, and axial stresses.
The material properties are considered to follow an exponential function, Poisson's ratio is assumed to be a constant, while the Young's modulus, the coefficient of thermal expansion, and the thermal conductivity of the FGM are defined as follows [24]: where r, E, α, and λ are the radial radius, Young's modulus, coefficient of thermal expansion, and thermal conductivity as a function in r.The symbols R o , E o , α o , and λ o represent the radial radius, Young's modulus, coefficient of thermal expansion, and thermal conductivity of the outside wall of the cylinder, respectively.And β, ϕ, and γ are the graded factors of the FGM pressure vessel, and Poisson's ratio is assumed to be a constant.

Constitutive and equilibrium equations
The constitutive equations of the pressure vessel subjected to both mechanical and thermal loads are given as According to the infinitesimal strain theory, the equilibrium equation and the strain displacement relations for an axisymmetric cylinder subjected to axisymmetric load in only the radial direction [30] and dependent on time can be written, respectively, as where σ r , σ θ , and σ z are the radial, hoop, and axial stresses, respectively, as a function of both radius r and time t.ε r and ε θ are the radial and hoop strain, respectively, as a function of both r and t, while ε z is the axial strain and is assumed to be a function of only t as it is considered to be constant with respect to r, considering the large length/ radius ratio.u is the radial displacement as a function of r and t and w is the axial displacement as a function of length and t. υ is Poisson's ratio, and the body forces are ignored. ( )σ r 1,0 is the first derivative of σ r with respect to r.

Transient temperature distribution evaluation
The transient temperature heat conduction through the wall thickness is considered one-dimensional heat conduction.The transient temperature through the cylinder wall for initial temperature T i, which is a function of r acting on the inner surface and with boundary conditions of prescribed surface temperatures T a and T b on both inner and outer surfaces, respectively, can be expressed as follows [30]: where And s n are eigen values of the eigen function The boundary conditions for this problem are

Thermo-elastic analysis
The model is considered to be under a linear-elastic uncoupled thermo-mechanical conduction condition, for which the combined stresses will be the summation of both mechanical and thermal stresses for radial, hoop, and axial stresses.The mechanical and thermal stresses will be derived separately by substituting Eqs. ( 1)-(3) and Eq.(5) into Eq.( 4) and then solving Eq. ( 4) to find the governing equations in the cylindrical part.The resulting governing equation is in the form of a second-order linear nonhomogeneous differential equation with variable coefficients which when solved will give the general equation for the displacement u as a function of r and t Eq. ( 6).The resulting equation has some complex integrals that can be solved using the Taylor series approximation: Eq. ( 6) is the general form of equation of the displacement, where f 1 , f 2 , and f 3 are functions of r and t, while a 1 and a 2 are constants of integration and are only dependent on t.To find the integration constants a 1 and a 2 , along with the axial strain ϵ z , the following boundary conditions are applied: where Eqs. ( 7) and ( 8) are the mechanical and thermal loading boundary conditions, respectively.In Eqs. ( 7) and ( 8), σ r M , σ r T , σ z M , and σ z T are the mechanical radial stress component, the thermal radial stress component, the mechanical axial stress component, and the thermal axial stress component, respectively.P is the internal pressure applied and is a function of t.By substituting the integration constants a 1 and a 2 , along with the axial strain ϵ z in Eqs. ( 2) and ( 4), then the radial stress σ r , hoop stress σ θ , axial stress σ z, and the axial strain equations for both the mechanical and thermal solutions separately can be found.
Hence, the mechanical stresses and axial strain equations can be written as follows (Eq.(9a-c)): where )

T β T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T
1 e e 2 Ei ,

Results and numerical examples
In the previous analysis, a thermo-elastic closed-form solution to compute the stress formulations for a thick cylindrical pressure vessel made of FGM with varying parameters is derived.This section is dedicated to numerical examples to show and present different parameters that affect or enhance the results.A cylinder with an internal radius of 330 mm and an outer radius of 400 mm is considered for the analysis.The cylinder material is assumed to be 94% alumina for ceramic at the inner surface and stainless 304 at the outer surface with and is graphically represented in Figure 3. Figure 4 shows the temperature distribution over the cylinder thickness with respect to radius r at different values of time t.
The temperature distribution is calculated according to Eq. ( 5) considering T a = 100°C, T b = 40°C, and T i as follows: where T i = 70°C is the temperature of the inner surface at the steady state phase and T o = 25°C is the temperature of the outer surface.
By substituting with the aforementioned SS304-Alumina material properties and loading conditions in Eqs.(9a-c) and (10a-c), stress values can be determined.Figures 5-7 show the total radial stress σ r , hoop stress σ Θ , and axial stress σ z  acting on the cylinder wall with respect to time t at different cylinder's radii, respectively.Figure 8 shows the von Mises equivalent stress a with respect to time t at different cylinder radii, while Figure 9 shows the axial strain with respect to time t as it is constant throughout the thickness.

Effect of using different materials
Considering two other materials with better properties subjected to the same boundary conditions to compare and seek better results choosing stainless steel (SS304) and alumina (Al 2 O 3 ) materials under the mentioned boundary conditions resulted in relatively high values of strain and stresses as presented in Figures 7 and 8.The materials are silicon carbide (SiC) for the inner surface and molybdenum alloy TZM for the outer surface and are selected to enhance        11) and using Eq. ( 5), the new temperature distribution over the cylinder thickness with respect to radius r at different values of time are presented in Figure 9.
By substituting with the aforementioned TZM-SiC material properties and loading conditions in Eqs.(9a-c) and (10a-c), stress values can be determined.Figures 10-12 show the total radial stress σ r , hoop stress σ Θ , and axial stress σ z acting on the cylinder wall with respect to time t at different cylinder's radii, respectively.Figure 13 shows the von Mises equivalent stress a with respect to time t at different cylinder radii, while Figure 14 shows the axial strain with respect to time t as it is constant through the thickness.Note that the trend lines in the above figures are different than those of the SS304-alumina case, which is denoted as a result of different property indices values of materials [19].

Effect of using different material indices
To show the effect of material indices on the results, the total stresses are calculated and plotted as shown in Figure 15 for the TZM-SiC FGM cylinder considering different material indices β = 0.763185, φ = 0.087838, and γ = −2.114225.Comparing the results of Figure 15 with those of Figures 10-13, it is evident that there is a significant difference in the values of the results which proves that the material indices have a great impact on the results.

Domination of thermal boundary conditions
Both the mechanical and the thermal stresses for the TZM-SiC FGM cylinder are presented in Figure 16.Comparing these results of Figure 16 with the total stresses presented in Figures 10-13, it is evident that the thermal boundary conditions have a major effect over the total results and this is confirmed with the results introduced by Das et al. [19].

Conclusion
In this study, stresses and strains are analyzed in an inhomogeneous hollow cylinder under a time-dependent internal pressure and transient temperature profile.The material properties are assumed to be graded according to an exponential function in the radial direction.The stress formulations are derived for both mechanical and thermal loads and the total results of the radial, hoop, axial stress are plotted for two FGM cylinders: SS304-alumina FGM cylinder and TZM-SiC FGM cylinder.
In conclusion, the results demonstrate that the FGM must be selected based on the application for which the vessels are intended to be used and the loading conditions to which they are subjected.As mentioned in the results, the mechanical and thermal properties affect directly the distribution of the resulting stress.The type of the cylinder material must be carefully chosen, as shown in the results of Figures 4-8 vs Figures 10-14, respectively; the materials with better mechanical and thermal properties (molybdenum-SiC FGM) offer very good results compared to those of the material with lower properties (SS304-Al 2 O 3 ).Additionally, the study emphasizes that the material inhomogeneity indices are a very helpful design parameter of the FGM cylinder and regulate stress distributions with the application for which the material shall be chosen as explained and proven in the results.The same results are concluded by Houari et al. [19,[31][32][33].Finally, this research highlights the major effect of thermal boundary conditions on the stress results.As shown earlier, the results show that the thermal boundary condition has a major effect on the total results and this agrees with the results obtained by Das et al. [19] and Yıldırım et al. [34].
The results obtained in this work are useful as these results contribute to designing and modeling cylinders subject to time-dependent internal pressure and transient temperature profile.

Figure 1 :
Figure 1: Configuration of model loads and boundary conditions.

Figure 2 :
Figure 2: Variant pressure P (MPa) acting on the internal surface of the SS304-Al 2 O 3 FGM cylinder with respect to time t (s).

Figure 3 :
Figure 3: Transient temperature T (°C) through the wall thickness of the SS304-Al 2 O 3 FGM cylinder at different times with respect to radius r (mm).

Figure 4 :
Figure 4: Radial stress σ r (MPa) acting on the SS304-Al 2 O 3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 5 :
Figure 5: Hoop stress σ Θ (MPa) acting on the SS304-Al 2 O 3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 7 :
Figure 7: von Mises stress σ v (MPa) acting on the SS304-Al 2 O 3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 8 :
Figure 8: Axial strain ∈ z (mm) as a result of mechanical and thermal loads acting on the SS304-Al 2 O 3 FGM cylinder with respect to time t (s).

Figure 9 :
Figure 9: Transient temperature T (°C) through the wall thickness of the TZM-SiC FGM cylinder at different times with respect to radius r (mm).

Figure 6 :
Figure 6: Axial stress σ z (MPa) acting on the SS304-Al 2 O 3 FGM cylinder wall with respect to time t (s) at different cylinder radii.
the results: E o = 325 GPa, and the considered grade factors are β = 1.52637, φ = 0.1756769, and γ = −4.22845.Molybdenum has better mechanical properties than that of SS304 and SIC has better thermal properties than that of Al 2 O 3 .Considering the new materials, properties' indices, pressure variation Eq. (

Figure 10 :
Figure 10: Radial stress σ r (MPa) acting on the TZM-SIC FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 12 :
Figure 12: Axial stress σ z (MPa) acting on the TZM-SiC FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 13 :
Figure 13: von Mises stress σ v (MPa) acting on the TZM-SIC FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 14 :
Figure 14: Axial strain ∈ z (mm) as a result of mechanical and thermal loads acting on the TZM-SiC FGM cylinder with respect to time t (s).

Figure 11 :
Figure 11: Hoop stress σ Θ (MPa) acting on the TZM-SIC FGM cylinder wall with respect to time t (s) at different cylinder radii.