Home Technology Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity
Article Open Access

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

  • El-Sayed Habib , Araby I. Mahdy , Gamal Ali , Abla El-Megharbel EMAIL logo and Eman El-Shrief
Published/Copyright: August 22, 2023
Download Article (PDF)
Curved and Layered Structures
From the journal Curved and Layered Structures Volume 10 Issue 1

Abstract

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 different materials for the same boundary conditions on the results. The results obtained in this study are beneficial as these contribute to the design and modeling of cylinders that are exposed to time-dependent internal pressure and transient temperature profiles.

Keywords: FGM cylinder; time-dependent pressure; transient temperature; closed-form

1 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.

2 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.

2.1 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 linear-elastic 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.

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

Configuration of model loads and boundary conditions.

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]:

(1) E = ⅇ β 1 − r R o E o , α = ⅇ φ 1 − r R o α o , λ = ⅇ γ 1 − r R o λ o ,

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.

2.2 Constitutive and equilibrium equations

The constitutive equations of the pressure vessel subjected to both mechanical and thermal loads are given as

σ r = E 1 + ν ϵ r + ν 1 − 2 ν ( ϵ r + ϵ θ + ϵ z ) − α E T 1 − 2 ν ,

σ θ = E 1 + ν ϵ θ + ν 1 − 2 ν ( ϵ r + ϵ θ + ϵ z ) − α E T 1 − 2 ν ,

(2) σ z = E 1 + ν ϵ z + ν 1 − 2 ν ( ϵ r + ϵ θ + ϵ z ) − α E T 1 − 2 ν .

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

(3) σ r ( 1 , 0 ) + σ r − σ θ r = 0 ,

(4) ϵ r = ∂ u ∂ r , ϵ θ = u r , ϵ z = ∂ w ∂ z ,

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.

2.3 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]:

(5) T = T a + ( T b − T a ) ln r / R i ln R o / R i − π ∑ n = 1 ∞ T a J 0 ( s n R o ) − T b J 0 ( s n R i ) J 0 2 ( s n R o ) − J 0 2 ( s n R i ) J 0 ( s n R o ) f ( s n , r ) ⅇ − κ s n 2 t − π 2 2 ∑ n = 1 ∞ s n 2 J 0 2 ( s n R o ) f ( s n , r ) J 0 2 ( s n R o ) − J 0 2 ( s n R i ) × ∫ R i R o T i ( η ) f ( s n , η ) η d η ⅇ − κ s n 2 t ,

where

f ( s n , r ) = Y 0 ( s n R i ) J 0 ( s n r ) − J 0 ( s n R i ) Y 0 ( s n r ) .

And s n are eigen values of the eigen function f ( s n , R o ) = 0 .

The boundary conditions for this problem are

T = T a on r = R i ,

T = T b on r = R o .

2.4 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:

(6) u = f 1 a 1 + f 2 ϵ z + f 3 a 2 ,

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:

σ r M = − P when r = R i ,

σ r M = 0 when r = R o ,

(7) ∫ R i R o σ z M 2 π r d r = P π R i 2 ,

σ r T = 0 when r = R i ,

σ r T = 0 when r = R o ,

(8) ∫ R i R o σ z T 2 π r d r = 0 ,

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)):

(9a) σ Mr = − ⅇ β − r β R o E o r ( 1 + ν ) ( − 1 + 2 ν ) ( a M 1 ( ν f M 1 − r ( − 1 + ν ) f M 1 ( 1 , 0 ) ) + ϵ Mz ( ν ( r + f M 2 ) − r ( − 1 + ν ) f M 2 ( 1 , 0 ) ) + a M 2 ( ν f M 3 − r ( − 1 + ν ) f M 3 ( 1 , 0 ) ) ) ,

(9b) σ M θ = − ⅇ β − r β R o E o r ( 1 + ν ) ( − 1 + 2 ν ) ( a M 1 ( − ( − 1 + ν ) f M 1 + r ν f M 1 ( 1 , 0 ) ) + ϵ Mz ( − ( − 1 + ν ) f M 2 + r ν ( 1 + f M 2 ( 1 , 0 ) ) ) + a M 2 ( − ( − 1 + ν ) f M 3 + r ν f M 3 ( 1 , 0 ) ) ) ,

(9c) σ Mz = − ⅇ β − r β R o E o r ( 1 + ν ) ( − 1 + 2 ν ) ( ν a M 1 ( f M 1 + r f M 1 ( 1 , 0 ) ) + ϵ Mz ( r − r ν + ν f M 2 + r ν f M 2 ( 1 , 0 ) ) + ν a M 2 ( f M 3 + r f M 3 ( 1 , 0 ) ) ) ,

where

a M 1 = − ( ( − Z M 3 Z M 5 + Z M 2 Z M 6 ) ( − π P R i 2 Z M 2 − P ( − f M 5 [ R i ] + f M 5 [ R o ] ) ) + P Z M 5 ( − Z M 3 ( − f M 5 [ R i ] + f M 5 [ R o ] ) + Z M 2 ( − f M 6 [ R i ] + f M 6 [ R o ] ) ) ) / ( ( − Z M 3 Z M 5 + Z M 2 Z M 6 ) ( Z M 2 ( − f M 4 [ R i ] + f M 4 [ R o ] ) − Z M 1 ( − f M 5 [ R i ] + f M 5 [ R o ] ) ) − ( Z M 2 Z M 4 − Z M 1 Z M 5 ) ( − Z M 3 ( − f M 5 [ R i ] + f M 5 [ R o ] ) + Z M 2 ( − f M 6 [ R i ] + f M 6 [ R o ] ) ) ) ,

a M 2 = − ( π P R i 2 Z M 2 Z M 4 − π P R i 2 Z M 1 Z M 5 + P Z M 5 f M 4 [ R i ] − P Z M 5 f M 4 [ R o ] − P Z M 4 f M 5 [ R i ] + P Z M 4 f M 5 [ R o ] ) / ( Z M 3 Z M 5 f M 4 [ R i ] − Z M 2 Z M 6 f M 4 [ R i ] − Z M 3 Z M 5 f M 4 [ R o ] + Z M 2 Z M 6 f M 4 [ R o ] − Z M 3 Z M 4 f M 5 [ R i ] + Z M 1 Z M 6 f M 5 [ R i ] + Z M 3 Z M 4 f M 5 [ R o ] − Z M 1 Z M 6 f M 5 [ R o ] + Z M 2 Z M 4 f M 6 [ R i ] − Z M 1 Z M 5 f M 6 [ R i ] − Z M 2 Z M 4 f M 6 [ R o ] + Z M 1 Z M 5 f M 6 [ R o ] ) ,

ϵ Mz = − ( π P R i 2 Z M 3 Z M 4 − π P R i 2 Z M 1 Z M 6 + P Z M 6 f M 4 [ R i ] − P Z M 6 f M 4 [ R o ] − P Z M 4 f M 6 [ R i ] + P Z M 4 f M 6 [ R o ] ) / ( − Z M 3 Z M 5 f M 4 [ R i ] + Z M 2 Z M 6 f M 4 [ R i ] + Z M 3 Z M 5 f M 4 [ R o ] − Z M 2 Z M 6 f M 4 [ R o ] + Z M 3 Z M 4 f M 5 [ R i ] − Z M 1 Z M 6 f M 5 [ R i ] − Z M 3 Z M 4 f M 5 [ R o ] + Z M 1 Z M 6 f M 5 [ R o ] − Z M 2 Z M 4 f M 6 [ R i ] + Z M 1 Z M 5 f M 6 [ R i ] + Z M 2 Z M 4 f M 6 [ R o ] − Z M 1 Z M 5 f M 6 [ R o ] ) ,

f M 1 = ⅇ − r ( − 2 + t ) β 2 R o r ,

f M 2 = 1 ( − 1 + t ) 3 β 2 ( − 1 + ν ) ν R o 4 ( − 1 + t ) 3 β ( − 2 + t ) t + ⅇ 1 2 t 1 − 1 + t + r β R o ( − 1 + t ) β Ei 1 2 t 1 1 − t − r β R o + 1 r ( − 2 + t ) 2 − 8 + 14 t − 6 t 2 + ⅇ − ( − 2 + t ) ( r ( − 1 + t ) β + R o ) 2 ( − 1 + t ) R o ( − 2 + t ) 2 Ei 1 2 ( − 2 + t ) 1 − 1 + t + r β R o R o ,

f M 3 = ⅇ rt β 2 R o ,

f M 4 = − 2 ⅇ β − rt β 2 R o π ( − 2 + t ) ν E o t ( − 1 + ν + 2 ν 2 ) ,

f M 5 = 2 ⅇ − r β R o π − ⅇ r β R o β 2 ν ∫ ⅇ β − r β R o f M 2 ⅆ r + ∫ ⅇ β − r β R o r f M 2 ( 1 , 0 ) ⅆ r − ⅇ β ( − 1 + ν ) R o ( r β + R o ) E o β 2 ( − 1 + ν + 2 ν 2 ) ,

f M 6 = − 2 ⅇ β + r ( − 2 + t ) β 2 R o π ν ( r ( − 2 + t ) t β − 4 R o ) E o ( − 2 + t ) 2 β ( 1 + ν ) ( − 1 + 2 ν ) ,

Z M 1 = − ⅇ β − β R i R o E o ( ν f M 1 [ R i ] − ( − 1 + ν ) R i f M 1 ( 1 , 0 ) [ R i ] ) ( 1 + ν ) ( − 1 + 2 ν ) R i ,

Z M 2 = − ⅇ β − β R i R o E o ( ν ( R i + f M 2 [ R i ] ) − ( − 1 + ν ) R i f M 2 ( 1 , 0 ) [ R i ] ) ( 1 + ν ) ( − 1 + 2 ν ) R i ,

Z M 3 = − ⅇ β − β R i R o E o ( ν f M 3 [ R i ] − ( − 1 + ν ) R i f M 3 ( 1 , 0 ) [ R i ] ) ( 1 + ν ) ( − 1 + 2 ν ) R i ,

Z M 4 = − E o ( ν f M 1 [ R o ] − ( − 1 + ν ) R o f M 1 ( 1 , 0 ) [ R o ] ) ( 1 + ν ) ( − 1 + 2 ν ) R o ,

Z M 5 = − E o ( ν ( R o + f M 2 [ R o ] ) − ( − 1 + ν ) R o f M 2 ( 1 , 0 ) [ R o ] ) ( 1 + ν ) ( − 1 + 2 ν ) R o ,

Z M 6 = − E o ( ν f M 3 [ R o ] − ( − 1 + ν ) R o f M 3 ( 1 , 0 ) [ R o ] ) ( 1 + ν ) ( − 1 + 2 ν ) R o .

The thermal stresses and axial strain equations can be written as follows (Eq. (10a–c)):

(10a) σ r T = 1 r R o 2 ( 1 + ν 2 ) ⅇ β − r β R o a T 1 R o 2 E o ν 1 ( 1 + ν 2 ) 2 + ⅇ β a T 2 R o E o ν 1 ( 1 + ν 2 ) ( r β + ( r β + R o ) ν 2 ) − r R o 2 ( 1 + ν 2 ) f 1 + ⅇ β − r β R o R o ( 1 + ν 2 ) f 2 − ⅇ β ( r β + ( r β + R o ) ν 2 ) f 3 + ⅇ β Ei − r β R o ( a T 1 R o E o ν 1 ( 1 + ν 2 ) ( r β + ( r β + R o ) ν 2 ) + ( r β + ( r β + R o ) ν 2 ) f 2 + r ( r Z T 1 ( r β ( 1 + ν 2 ) + R o ( 2 + 3 ν 2 ) ) ϵ Tz + R o ( 1 + ν 2 ) f 2 ( 1 , 0 ) ) ) − ⅇ β r R o f 3 ( 1 , 0 ) − ⅇ β r R o ν 2 f 3 ( 1 , 0 ) + ⅇ β − r β R o ϵ Tz r R o 2 E o ν 1 ν 2 ( 1 + ν 2 ) − ⅇ r β R o r β Z T 1 ( 1 + ν 2 ) f 4 + R o Z T 1 r r − ⅇ r β R o f 4 ( 1 , 0 ) + ν 2 r 2 − ⅇ r β R o f 4 − ⅇ r β R o r f 4 ( 1 , 0 ) ,

(10b) σ ϴ T = 1 r R o 2 ( 1 + ν 2 ) ⅇ β − r β R o a T 1 R o 2 E o ν 1 ν 2 ( 1 + ν 2 ) + ⅇ β a T 2 R o E o ν 1 ( 1 + ν 2 ) ( r β ν 2 + R o ( 1 + ν 2 ) ) − r R o 2 ( 1 + ν 2 ) f 1 + ⅇ β − r β R o R o ν 2 f 2 − ⅇ β R o f 3 − ⅇ β r β ν 2 f 3 − ⅇ β R o ν 2 f 3 + ⅇ β Ei − r β R o ( a T 1 R o E o ν 1 ( 1 + ν 2 ) ( r β ν 2 + R o ( 1 + ν 2 ) ) + r β ν 2 ( f 2 + r 2 Z T 1 ϵ Tz ) + R o ( ( 1 + ν 2 ) f 2 + r ( r Z T 1 ( 1 + 3 ν 2 ) ϵ Tz + ν 2 f 2 ( 1 , 0 ) ) ) ) − ⅇ β r R o ν 2 f 3 ( 1 , 0 ) + ⅇ β ϵ Tz ⅇ − r β R o r R o 2 E o ν 1 ν 2 ( 1 + ν 2 ) − r β Z T 1 ν 2 f 4 + R o Z T 1 − f 4 + ν 2 ⅇ − r β R o r 2 − f 4 − r f 4 ( 1 , 0 ) ,

(10c) σ z T = 1 r R o 2 ( 1 + ν 2 ) ⅇ β − r β R o a T 1 R o 2 E o ν 1 ν 2 ( 1 + ν 2 ) + ⅇ β a T 2 R o ( r β + R o ) E o ν 1 ν 2 ( 1 + ν 2 ) − r R o 2 ( 1 + ν 2 ) f 1 + ⅇ β − r β R o R o ν 2 f 2 − ⅇ β r β ν 2 f 3 − ⅇ β R o ν 2 f 3 + ⅇ β Ei − r β R o ν 2 ( a T 1 R o ( r β + R o ) E o ν 1 ( 1 + ν 2 ) + ( r β + R o ) f 2 + r ( r ( r β + 3 R o ) Z T 1 ϵ Tz + R o f 2 ( 1 , 0 ) ) ) − ⅇ β r R o ν 2 f 3 ( 1 , 0 ) + ⅇ β − r β R o ϵ Tz r R o 2 E o ν 1 ( 1 + ν 2 ) 2 − ⅇ r β R o r β Z T 1 ν 2 f 4 + R o Z T 1 ν 2 − ⅇ r β R o f 4 + r r − ⅇ r β R o f 4 ( 1 , 0 ) ,

where

ϵ Tz = 2 β 2 R i 2 Z T 3 Z T 5 ν 2 + β Z T 2 ( − 2 ( R i − R o ) ( β R i + ( 2 + β ) R o ) Z T 6 ν 2 + Z T 9 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) ) + β R i ( 2 R o Z T 6 Z T 8 E o ν 1 ν 2 ( 1 + ν 2 ) 2 − Z T 5 ( − 4 R o Z T 3 ν 2 + β Z T 9 ( 1 + ν 2 ) 2 ) ) + R o ν 2 ( R o ( − 2 β ( 2 + β ) Z T 3 Z T 5 + 2 Z T 6 Z T 8 E o ν 1 ν 2 ( 1 + ν 2 ) ) − ( 1 + ν 2 ) ( β Z T 5 Z T 9 + 2 Z T 3 Z T 8 E o ν 1 ( β + ( 1 + β ) ν 2 ) ) ) ) / ( − β Z T 10 ( 1 + ν 2 ) ( Z T 2 ( β + ( 1 + β ) ν 2 ) − Z T 5 ( R o ν 2 + β R i ( 1 + ν 2 ) ) ) + 2 ν 2 ( β 2 R i 2 ( − Z T 4 Z T 5 + Z T 2 Z T 7 ) − β R i R o ( 2 Z T 4 Z T 5 + Z T 7 ( − 2 Z T 2 + Z T 8 E o ν 1 ( 1 + ν 2 ) 2 ) ) + R o ( Z T 4 Z T 8 E o ν 1 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) + R o ( β ( 2 + β ) Z T 4 Z T 5 − Z T 7 ( β ( 2 + β ) Z T 2 + Z T 8 E o ν 1 ν 2 ( 1 + ν 2 ) ) ) ) ) ,

a T 1 = β ( − ( Z T 4 Z T 6 − Z T 3 Z T 7 ) ( 2 ( R i − R o ) ( β R i + ( 2 + β ) R o ) Z T 7 ν 2 − Z T 10 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) ) + ( Z T 10 Z T 6 − Z T 7 Z T 9 ) ( 1 + ν 2 ) ( − Z T 4 ( β + ( 1 + β ) ν 2 ) + Z T 7 ( R o ν 2 + β R i ( 1 + ν 2 ) ) ) ) ) / ( β ( Z T 4 Z T 5 − Z T 2 Z T 7 ) ( 2 ( R i − R o ) ( β R i + ( 2 + β ) R o ) Z T 7 ν 2 − Z T 10 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) ) + ( 1 + ν 2 ) ( β Z T 10 Z T 5 − 2 R o Z T 7 Z T 8 E o ν 1 ν 2 ) ( Z T 4 ( β + ( 1 + β ) ν 2 ) − Z T 7 ( R o ν 2 + β R i ( 1 + ν 2 ) ) ) ,

a T 2 = ⅇ − β ( Z T 10 ( − β Z T 3 Z T 5 + β Z T 2 Z T 6 ) + Z T 7 ( − β Z T 2 Z T 9 + 2 R o Z T 3 Z T 8 E o ν 1 ν 2 ) + Z T 4 ( β Z T 5 Z T 9 − 2 R o Z T 6 Z T 8 E o ν 1 ν 2 ) ) ) / ( R o E o ν 1 ( β Z T 10 ( 1 + ν 2 ) ( Z T 2 ( β + ( 1 + β ) ν 2 ) − Z T 5 ( R o ν 2 + β R i ( 1 + ν 2 ) ) ) − 2 ν 2 ( β 2 R i 2 ( − Z T 4 Z T 5 + Z T 2 Z T 7 ) − β R i R o ( 2 Z T 4 Z T 5 + Z T 7 ( − 2 Z T 2 + Z T 8 E o ν 1 ( 1 + ν 2 ) 2 ) ) + R o ( Z T 4 Z T 8 E o ν 1 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) + R o ( β ( 2 + β ) Z T 4 Z T 5 − Z T 7 ( β ( 2 + β ) Z T 2 + Z T 8 E o ν 1 ν 2 ( 1 + ν 2 ) ) ) ) ) ) ,

ν 1 = 1 1 + ν , ν 2 = ν 1 − 2 ν ,

f 1 = α E T 1 − 2 ν , f 2 = ∫ ⅇ β ( − 1 + r R o ) r R o f 1 ( 1 , 0 ) d r ,

f 3 = ∫ ⅇ β ( − 1 + r R o ) r Ei [ − r β R o ] R o f 1 ( 1 , 0 ) d r , f 4 = r 2 Ei − r β R o + ⅇ − r β R o R o ( r β + R o ) β 2 ,

f 5 = ∫ r f 3 [ r , t ] ⅆ r , f 6 = ∫ r Ei − r β R o f 2 ⅆ r ,

f 7 = ∫ ⅇ β − r β R o f 2 [ r , t ] ⅆ r , f 8 = ∫ Ei − r β R o f 2 ⅆ r ,

f 9 = ∫ f 3 [ r , t ] ⅆ r , f 10 = ∫ r Ei − r β R o f 2 ( 1 , 0 ) ⅆ r ,

f 11 = ∫ r f 3 ( 1 , 0 ) ⅆ r , f 12 = ∫ r f 4 ⅆ r ,

f 13 = ∫ f 4 ⅆ r , f 14 = ∫ r f 4 ( 1 , 0 ) ⅆ r ,

f 15 = ∫ r f 1 ⅆ r , Z T 1 = 1 2 β E o ν 1 ν 2 ,

Z T 2 = ⅇ β R o E o ν 1 ( 1 + ν 2 ) − ⅇ − β R i R o R o ( 1 + ν 2 ) − Ei − β R i R o ( R o ν 2 + β R i ( 1 + ν 2 ) ) ,

Z T 3 = ⅇ β R o − ⅇ − β R i R o 1 + ν 2 + ⅇ β R i R o Ei − β R i R o ν 2 f 2 [ R i ] + ν 2 f 3 [ R i ] + R i ( 1 + ν 2 ) R o 2 f 1 [ R i ] − ⅇ β β Ei − β R i R o f 2 [ R i ] − f 3 [ R i ] − ⅇ β R o Ei − β R i R o f 2 ( 1 , 0 ) [ R i ] − f 3 ( 1 , 0 ) [ R i ] ,

Z T 4 = − ⅇ β Ei − β R i R o R i 2 Z T 1 ( β R i ( 1 + ν 2 ) + R o ( 2 + 3 ν 2 ) ) − ⅇ β − β R i R o R i 2 R o Z T 1 ( 1 + ν 2 ) − ⅇ β R i R o R o Z T 1 ν 2 f 4 [ R i ] + R i ( 1 + ν 2 ) R o 2 E o ν 1 ν 2 − ⅇ β R i R 0 β Z T 1 f 4 [ R i ] − ⅇ β R i R o R o Z T 1 f 4 ( 1 , 0 ) [ R i ] ,

Z T 5 = − R 0 E o ν 1 ( 1 + ν 2 ) 2 − ⅇ β Ei [ − β ] R o E o ν 1 ( 1 + ν 2 ) ( β + ( 1 + β ) ν 2 ) ,

Z T 6 = R o 2 ( 1 + ν 2 ) f 1 [ R o ] − ( 1 + ν 2 ) f 2 [ R o ] − ⅇ β Ei [ − β ] ( β + ( 1 + β ) ν 2 ) f 2 [ R o ] + ⅇ β ( β + ( 1 + β ) ν 2 ) f 3 [ R o ] − ⅇ β Ei [ − β ] R o ( 1 + ν 2 ) f 2 ( 1 , 0 ) [ R o ] + ⅇ β R o f 3 ( 1 , 0 ) [ R o ] + ⅇ β R o ν 2 f 3 ( 1 , 0 ) [ R o ] ,

Z T 7 = − ⅇ β Ei [ − β ] R o 2 Z T 1 ( 2 + β + ( 3 + β ) ν 2 ) − R o 2 ( 1 + ν 2 ) ( Z T 1 + E o ν 1 ν 2 ) + ⅇ β Z T 1 ( β + ( 1 + β ) ν 2 ) f 4 [ R o ] + ⅇ β R o Z T 1 ( 1 + ν 2 ) f 4 ( 1 , 0 ) [ R o ] ,

Z T 8 = − ⅇ β β 2 Ei − β R i R o R i 2 + ⅇ β β − ⅇ − β R i R o − 2 Ei − β R i R o R i R o + 1 − ⅇ β − β R i R o + β + ⅇ β β ( 2 + β ) Ei [ − β ] R o 2 ,

Z T 9 = 1 1 + ν 2 4 ( R o 2 f 15 [ R i ] + ν 2 ( ⅇ β β f 5 [ R i ] − ⅇ β β f 6 [ R i ] + R o ( − f 7 [ R i ] − ⅇ β f 8 [ R i ] + ⅇ β f 9 [ R i ] − ⅇ β f 10 [ R i ] + ⅇ β f 11 [ R i ] + R o f 15 [ R i ] ) ) − R o 2 f 15 [ R o ] − ν 2 ( ⅇ β β f 5 [ R o ] − ⅇ β β f 6 [ R o , t ] + R o ( − f 7 [ R o ] − ⅇ β f 8 [ R o ] + ⅇ β f 9 [ R o ] − ⅇ β f 10 [ R o ] + ⅇ β f 11 [ R o ] + R o f 15 [ R o ] ) ) ) ,

Z T 10 = 1 β 3 ( 1 + ν 2 ) R o 4 ( ( 6 + 6 β + 3 β 2 + β 3 + ⅇ β β 3 ( 4 + β ) Ei [ − β ] ) Z T 1 ν 2 − 4 β ( 1 + β ) E o ν 1 ( 1 + ν 2 ) 2 ) − 4 ⅇ β β 4 Z T 1 ν 2 f 12 [ R o ] − ⅇ β − β R i R o ⅇ β R i R o β 4 Ei − β R i R o R i 4 Z T 1 ν 2 + β 3 1 + 4 ⅇ β R i R o Ei − β R i R o R i 3 R o Z T 1 ν 2 + 3 β 2 R i 2 R o 2 Z T 1 ν 2 − 2 β R i R o 3 ( − 3 Z T 1 ν 2 + 2 β E o ν 1 ( 1 + ν 2 ) 2 ) − 2 R o 4 ( − 3 Z T 1 ν 2 + 2 β E o ν 1 ( 1 + ν 2 ) 2 ) + 2 ⅇ β R i R o β 4 Z T 1 ν 2 f 12 [ R i ] + 2 ⅇ β R i R o β 3 R o Z T 1 ν 2 ( f 13 [ R i ] + f 14 [ R i ] ) − 4 ⅇ β β 3 R o Z T 1 ν 2 ( f 13 [ R o ] + f 14 [ R o ] ) .

f j ( 1 , 0 ) denotes the first partial derivative relative to r, f j [ R i ] denotes f j at r = R i, f j [ R o ] denotes f j at r = R o, f j ( 1 , 0 ) [ R i ] denotes f j ( 1 , 0 ) at r = R i, and f j ( 1 , 0 ) [ R o ] denotes f j ( 1 , 0 ) at r = R o.

3 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 E o = 200 GPa, ν = 0.29, and the considered grade factors are β = 2.316943, φ = − 4 . 336242 , and γ = 0.60206 . An arbitrary pressure function acting on the internal surface in the following form (Figure 2):

(11) P = P 0 + P 1 t + P 2 t 2 + P 3 t 3 + P 4 t 4 ,

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.

Figure 2 Variant pressure P (MPa) acting on the internal surface of the SS304–Al2O3 FGM cylinder with respect to time t (s).
Figure 2

Variant pressure P (MPa) acting on the internal surface of the SS304–Al2O3 FGM cylinder with respect to time t (s).

Figure 3 Transient temperature T (°C) through the wall thickness of the SS304–Al2O3 FGM cylinder at different times with respect to radius r (mm).
Figure 3

Transient temperature T (°C) through the wall thickness of the SS304–Al2O3 FGM cylinder at different times with respect to radius r (mm).

Figure 4 Radial stress σ r (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.
Figure 4

Radial stress σ r (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

The temperature distribution is calculated according to Eq. (5) considering T a = 100°C, T b = 40°C, and T i as follows:

T i = K 2 + ⅇ − γ Ei r γ R o K 1 λ o ,

where

K 1 = − ⅇ γ ( T i − T o ) λ o Ei [ γ ] − Ei γ R i R o , K 2 = − − Ei [ γ ] T i + Ei γ R i R o T o Ei [ γ ] − Ei γ R i R o ,

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.

Figure 5 Hoop stress σ Θ (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.
Figure 5

Hoop stress σ Θ (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 6 Axial stress σ z (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.
Figure 6

Axial stress σ z (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 7 von Mises stress σ v (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.
Figure 7

von Mises stress σ v (MPa) acting on the SS304–Al2O3 FGM cylinder wall with respect to time t (s) at different cylinder radii.

Figure 8 Axial strain ∈ z (mm) as a result of mechanical and thermal loads acting on the SS304–Al2O3 FGM cylinder with respect to time t (s).
Figure 8

Axial strain ∈ z (mm) as a result of mechanical and thermal loads acting on the SS304–Al2O3 FGM cylinder with respect to time t (s).

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 9

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

3.1 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 (Al2O3) 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 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 Al2O3. Considering the new materials, properties’ indices, pressure variation Eq. (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].

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

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 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 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 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 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 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).

3.2 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.

Figure 15 (a), (b), (c), and (d) represent the total radial, hoop, axial, and von Mises stresses respect to time “t” (s) at different cylinder’s radii for the TZM-SiC FGM cylinder considering material indices’ values of β = 0.763185, φ = 0.087838, and γ = ‒2.114225.
Figure 15

(a), (b), (c), and (d) represent the total radial, hoop, axial, and von Mises stresses respect to time “t” (s) at different cylinder’s radii for the TZM-SiC FGM cylinder considering material indices’ values of β = 0.763185, φ = 0.087838, and γ = ‒2.114225.

3.3 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].

Figure 16 (a), (c), (e), and (g) represent the mechanical radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM cylinder, while (b), (d), (f), and (h) represent the thermal radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM with respect to time t (s) at different cylinder radii.
Figure 16 (a), (c), (e), and (g) represent the mechanical radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM cylinder, while (b), (d), (f), and (h) represent the thermal radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM with respect to time t (s) at different cylinder radii.
Figure 16

(a), (c), (e), and (g) represent the mechanical radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM cylinder, while (b), (d), (f), and (h) represent the thermal radial, hoop, axial, and von Mises stresses for the TZM-SiC FGM with respect to time t (s) at different cylinder radii.

4 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–Al2O3). 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.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: Authors state no conflict of interest.

References

[1] Petit C, Montanaro L, Palmero P. Functionally graded ceramics for biomedical application: Concept, manufacturing, and properties. Int J Appl Ceram Technol. 2018;15(4):820–40.10.1111/ijac.12878Search in Google Scholar

[2] Tornabene F, Reddy JN. FGM and Laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: A GDQ solution for static analysis with a posteriori stress and strain recovery. J Indian Inst Sci. 2013;93(4):635–88.Search in Google Scholar

[3] El-Galy IM, Saleh BI, Ahmed MH. Functionally graded materials classifications and development trends from industrial point of view. SN Appl Sci. 2019;1:1378.10.1007/s42452-019-1413-4Search in Google Scholar

[4] Saleh B, Jiang J, Fathi R, Tareq AH, Xu Q, Wang L, et al. 30 Years of functionally graded materials: An overview of manufacturing methods. Appl Future Chall Compos Part B: Eng. 2020;201:108376.10.1016/j.compositesb.2020.108376Search in Google Scholar

[5] Taghizadeh M, Babaei M, Dimitri R, Tornabene F. Assessment of critical buckling load of bi-directional functionally graded truncated conical micro-shells using modified couple stress theory and Ritz method. Mech Based Des Struct Mach. 2023;1–32.10.1080/15397734.2023.2202230Search in Google Scholar

[6] Suresh S, Mortensen A. Fundamentals of functionally graded materials: Processing and thermomechanical behaviour of graded metals and metal-ceramic composites. Mater Today. 1998;1(4):18.10.1016/S1369-7021(98)80023-0Search in Google Scholar

[7] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stresses. 1998;21(6):593–626.10.1080/01495739808956165Search in Google Scholar

[8] Sobczak J, Drenchev L. Metal based functionally graded materials. Engineering and Modeling Saif Zone. Sharjah, United Arab Emirates: Bentham Science Publishers; 2009.Search in Google Scholar

[9] EL-Wazery MS, EL-Desouky AR. A review on functionally graded ceramic-metal materials. Mater. Environ. Sci. 2015;6(5):1369–76.Search in Google Scholar

[10] Gupta A, Talha M. Recent development in modeling and analysis of functionally graded materials and structures. Prog Aerosp Sci. 2015;79:1–14.10.1016/j.paerosci.2015.07.001Search in Google Scholar

[11] Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Compos Struct. 2013;96:833–49.10.1016/j.compstruct.2012.09.001Search in Google Scholar

[12] Loh GH, Pei E, Harrison D, Monzón MD. An overview of functionally graded additive manufacturing. Addit Manuf. 2018;23:34–44.10.1016/j.addma.2018.06.023Search in Google Scholar

[13] Li Y, Feng Z, Hao L, Huang L, Xin C, Wang Y, et al. A review on functionally graded materials and structures via additive manufacturing: From multi-scale design to versatile functional properties. Adv Mater Technol. 2020;5(6):1900981.10.1002/admt.201900981Search in Google Scholar

[14] Merdaci S, Adda HM, Hakima B, Tornabene DR, Higher-Order F. Free vibration analysis of porous functionally graded plates. J Compos Sci. 2021;5:11.10.3390/jcs5110305Search in Google Scholar

[15] Abrinia K, Naee H, Sadeghi F, Djavanroodi F. New analysis for the FGM thick cylinders under combined pressure and temperature loading. Am J Appl Sci. 2008;5(7):852–9.10.3844/ajassp.2008.852.859Search in Google Scholar

[16] Jabbari M, Sohrabpour S, Eslami MR. Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int J Press Vessel Pip. 2002;79:493–7.10.1016/S0308-0161(02)00043-1Search in Google Scholar

[17] Wang ZW, Zhang Q, Xia LZ, Wu JT, Liu PQ. Stress analysis and parameter optimization of an FGM pressure vessel subjected to thermo-mechanical loadings. Procedia Eng. 2015;130:374–89.10.1016/j.proeng.2015.12.230Search in Google Scholar

[18] Xie J, Hao S, Wang W, Shi P. Analytical solution of stress in functionally graded cylindrical/spherical pressure vessel. Archive Appl Mech. 2021;91:3341–63.10.1007/s00419-021-01970-wSearch in Google Scholar

[19] Das P, Islam MA, Somadder S, Hasib MA. Analytical and numerical analysis of functionally graded (FGM) axisymmetric cylinders under thermo-mechanical loadings. Mater Today: Commun. 2022;33:104405.10.1016/j.mtcomm.2022.104405Search in Google Scholar

[20] Nejad MZ, Jabbari M, Ghannad M. Elastic analysis of axially functionally graded rotating thick cylinder with variable thickness under non-uniform arbitrarily pressure loading. Int J Eng Sci. 2015;89:86–99.10.1016/j.ijengsci.2014.12.004Search in Google Scholar

[21] Chen YZ. A novel numerical solution for a functionally graded hollow cylinder with arbitrary elastic property along the radial direction. Int J Press Vessel Pip. 2021;191:104301.10.1016/j.ijpvp.2021.104301Search in Google Scholar

[22] Sondhi L, Kumar R, Bhowmick S. Limit elastic yield pressure of internally and externally pressurized functionally graded thick cylinder. Mater Today: Proc. 2019;18:5507–14.10.1016/j.matpr.2019.07.582Search in Google Scholar

[23] Celebi K, Yarımpabuc D, Keles I. A novel approach to thermal and mechanical stresses in a fgm cylinder with exponentially-varying properties. J Theor Appl Mech. 2017;55(1):343–51.10.15632/jtam-pl.55.1.343Search in Google Scholar

[24] Habib ES, El-Hadek MA, El-Megharbel A. Stress analysis for cylinder made of FGM and subjected to thermo-mechanical loadings. Metals. 2018;9(1):4.10.3390/met9010004Search in Google Scholar

[25] Tutuncu N. Stresses in thick-walled FGM cylinders with exponentially-varying properties. Eng Struct. 2007;29:2032–5.10.1016/j.engstruct.2006.12.003Search in Google Scholar

[26] Ghannad M, Gharooni H. Elastic analysis of pressurized thick FGM cylinders with exponential variation of material properties using TSDT. Lat Am J Solids Struct. 2015;12:1024–41.10.1590/1679-78251491Search in Google Scholar

[27] Manthena VR, Lamba N, Kedar G. Thermal stress analysis in a functionally graded hollow elliptic-cylinder subjected to uniform temperature distribution. Appl Appl Mathematics. 2017;12(1):612–32.Search in Google Scholar

[28] Khoshgoftar MJ, Mirzaali MJ, Rahimi GH. Thermoelastic analysis of non-uniform pressurized functionally graded cylinder with variable thickness using first order shear deformation theory(FSDT) and perturbation method. Chin J Mech Eng. 2015;28(6):1149–56.10.3901/CJME.2015.0429.048Search in Google Scholar

[29] Singh K, RaniP, Muwal R. Thermal stresses due to non-uniform internal heat generation in functionally graded hollow cylinder. Int J Appl Mech Eng. 2021;26(2):186–200.10.2478/ijame-2021-0027Search in Google Scholar

[30] Eslami MR, Hetnarski RB, Ignaczak J, Noda N, Sumi N, Tanigawa Y. Theory of Elasticity and Thermal Stresses. Dordrecht: Springer Science+Business Media; 2013.10.1007/978-94-007-6356-2_14Search in Google Scholar

[31] Houari A, Bouchikhi AS, Mokhtari MM, Madani K, Bouchikhi HA, Mokhtari AS, et al. Numerical analysis of the elastic-plastic behavior of a tubular structure in FGM under pressure and defect presence. Frat ed Integrita Strutt. 2022;59:212–31.10.3221/IGF-ESIS.59.16Search in Google Scholar

[32] Benchallal R, Benslimane A, Bidgoli O, Hammiche D. Analytical solution for rotating cylindrical FGM vessel subjected to thermomechanical loadings. Mater Today: Proc. 2022;53(2):24–30.10.1016/j.matpr.2021.12.212Search in Google Scholar

[33] Tornabene F. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng. 2009;198:2911–35.10.1016/j.cma.2009.04.011Search in Google Scholar

[34] Yıldırım A, Yarımpabuç D, Arikan V, Eker M, Celebi K. Nonlinear thermal stress analysis of functionally graded spherical pressure vessels. Int J Press Vessel Pip. 2022;200:104830.10.1016/j.ijpvp.2022.104830Search in Google Scholar
Received: 2023-03-15
Revised: 2023-06-11
Accepted: 2023-06-28
Published Online: 2023-08-22

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. Investigation of differential shrinkage stresses in a revolution shell structure due to the evolving parameters of concrete
  3. Multiphysics analysis for fluid–structure interaction of blood biological flow inside three-dimensional artery
  4. MD-based study on the deformation process of engineered Ni–Al core–shell nanowires: Toward an understanding underlying deformation mechanisms
  5. Experimental measurement and numerical predictions of thickness variation and transverse stresses in a concrete ring
  6. Studying the effect of embedded length strength of concrete and diameter of anchor on shear performance between old and new concrete
  7. Evaluation of static responses for layered composite arches
  8. Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam
  9. Numerical study of the FRP-concrete bond behavior under thermal variations
  10. Parametric study of retrofitted reinforced concrete columns with steel cages and predicting load distribution and compressive stress in columns using machine learning algorithms
  11. Application of soft computing in estimating primary crack spacing of reinforced concrete structures
  12. Identification of crack location in metallic biomaterial cantilever beam subjected to moving load base on central difference approximation
  13. Numerical investigations of two vibrating cylinders in uniform flow using overset mesh
  14. Performance analysis on the structure of the bracket mounting for hybrid converter kit: Finite-element approach
  15. A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation
  16. Strength analysis of marine biaxial warp-knitted glass fabrics as composite laminations for ship material
  17. Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity
  18. Structural function analysis of shear walls in sustainable assembled buildings under finite element model
  19. In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation
  20. Optimization of structural parameters and numerical simulation of stress field of composite crucible based on the indirect coupling method
  21. Numerical study on crushing damage and energy absorption of multi-cell glass fibre-reinforced composite panel: Application to the crash absorber design of tsunami lifeboat
  22. Stripped and layered fabrication of minimal surface tectonics using parametric algorithms
  23. A methodological approach for detecting multiple faults in wind turbine blades based on vibration signals and machine learning
  24. Influence of the selection of different construction materials on the stress–strain state of the track
  25. A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells
  26. Comparative study of shell element formulations as NLFE parameters to forecast structural crashworthiness
  27. A size-dependent 3D solution of functionally graded shallow nanoshells
  28. Special Issue: The 2nd Thematic Symposium - Integrity of Mechanical Structure and Material - Part I
  29. Correlation between lamina directions and the mechanical characteristics of laminated bamboo composite for ship structure
  30. Reliability-based assessment of ship hull girder ultimate strength
  31. Finite element method on topology optimization applied to laminate composite of fuselage structure
  32. Dynamic response of high-speed craft bottom panels subjected to slamming loadings
  33. Effect of pitting corrosion position to the strength of ship bottom plate in grounding incident
  34. Antiballistic material, testing, and procedures of curved-layered objects: A systematic review and current milestone
  35. Thin-walled cylindrical shells in engineering designs and critical infrastructures: A systematic review based on the loading response
  36. Laminar Rayleigh–Benard convection in a closed square field with meshless radial basis function method
  37. Determination of cryogenic temperature loads for finite-element model of LNG bunkering ship under LNG release accident
  38. Roundness and slenderness effects on the dynamic characteristics of spar-type floating offshore wind turbine
https://doi.org/10.1515/cls-2022-0207
Keywords for this article
FGM cylinder; time-dependent pressure; transient temperature; closed-form
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Investigation of differential shrinkage stresses in a revolution shell structure due to the evolving parameters of concrete
  3. Multiphysics analysis for fluid–structure interaction of blood biological flow inside three-dimensional artery
  4. MD-based study on the deformation process of engineered Ni–Al core–shell nanowires: Toward an understanding underlying deformation mechanisms
  5. Experimental measurement and numerical predictions of thickness variation and transverse stresses in a concrete ring
  6. Studying the effect of embedded length strength of concrete and diameter of anchor on shear performance between old and new concrete
  7. Evaluation of static responses for layered composite arches
  8. Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam
  9. Numerical study of the FRP-concrete bond behavior under thermal variations
  10. Parametric study of retrofitted reinforced concrete columns with steel cages and predicting load distribution and compressive stress in columns using machine learning algorithms
  11. Application of soft computing in estimating primary crack spacing of reinforced concrete structures
  12. Identification of crack location in metallic biomaterial cantilever beam subjected to moving load base on central difference approximation
  13. Numerical investigations of two vibrating cylinders in uniform flow using overset mesh
  14. Performance analysis on the structure of the bracket mounting for hybrid converter kit: Finite-element approach
  15. A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation
  16. Strength analysis of marine biaxial warp-knitted glass fabrics as composite laminations for ship material
  17. Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity
  18. Structural function analysis of shear walls in sustainable assembled buildings under finite element model
  19. In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation
  20. Optimization of structural parameters and numerical simulation of stress field of composite crucible based on the indirect coupling method
  21. Numerical study on crushing damage and energy absorption of multi-cell glass fibre-reinforced composite panel: Application to the crash absorber design of tsunami lifeboat
  22. Stripped and layered fabrication of minimal surface tectonics using parametric algorithms
  23. A methodological approach for detecting multiple faults in wind turbine blades based on vibration signals and machine learning
  24. Influence of the selection of different construction materials on the stress–strain state of the track
  25. A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells
  26. Comparative study of shell element formulations as NLFE parameters to forecast structural crashworthiness
  27. A size-dependent 3D solution of functionally graded shallow nanoshells
  28. Special Issue: The 2nd Thematic Symposium - Integrity of Mechanical Structure and Material - Part I
  29. Correlation between lamina directions and the mechanical characteristics of laminated bamboo composite for ship structure
  30. Reliability-based assessment of ship hull girder ultimate strength
  31. Finite element method on topology optimization applied to laminate composite of fuselage structure
  32. Dynamic response of high-speed craft bottom panels subjected to slamming loadings
  33. Effect of pitting corrosion position to the strength of ship bottom plate in grounding incident
  34. Antiballistic material, testing, and procedures of curved-layered objects: A systematic review and current milestone
  35. Thin-walled cylindrical shells in engineering designs and critical infrastructures: A systematic review based on the loading response
  36. Laminar Rayleigh–Benard convection in a closed square field with meshless radial basis function method
  37. Determination of cryogenic temperature loads for finite-element model of LNG bunkering ship under LNG release accident
  38. Roundness and slenderness effects on the dynamic characteristics of spar-type floating offshore wind turbine
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 20.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/cls-2022-0207/html
Scroll to top button