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Analytical analysis for non-homogeneous two-layer functionally graded material

  • Farhad Belalpour Dastjerdi EMAIL logo and Mohsen Jabbari
Published/Copyright: November 4, 2022
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 11 Issue 1

Abstract

In this study, the nonlinear analytical analysis of a two-layer geometry made of functionally graded materials (FGMs) is examined. FGMs can be used in various engineering applications, such as building materials in civil engineering, due to the advantages of smoothly varying properties. The equations of stresses and displacements in the radial and circumferential directions (r, θ ) have been found by extracting the governing equations and defining them in the form of power-exponential functions. In the present paper, modulus of elasticity and heat conductivity coefficient (except for Poisson’s coefficient) are assumed to be expressed by power-exponential functions in radial and circumferential coordinates. The temperature distribution is also considered as a function of radius (r) and angle (θ). The analysis is implemented based on the theory of small elastic deformations and with the assumption of a very large length in plane strain mode. To analyze the governing equations, first, the heat transfer equations are obtained, and then the Navier’s equations are derived by combining the stress–strain, strain–displacement, and stress equilibrium equations. Then, the displacement equations and stress equations are obtained by solving the Navier’s equations. A direct method is presented to solve these equations analytically.

Keywords: two-layer; functionally graded material; analytical analysis

1 Introduction

Since traditional materials do not meet the current needs of industry, the production of a new generation of materials, such as functionally graded materials (FGMs), is expected to meet the recent needs of the industry. FGMs have a heterogeneous but isotropic microstructure and their mechanical properties change gradually and continuously. FGM is preferred in terms of mechanical behavior compared to materials with a fibrous structure. The gradual change in properties with respect to their dimensions has led to the growth of the application of these materials [1]. It is anticipated that global industry would benefit significantly from research into FGM materials. Therefore, the investigation of these materials is crucial for the design of structures. Recently, materials having functionally graded characteristics have been widely studied. Miyamoto et al. [2] published the first book on FGM. Various studies have been conducted on FGM [3,4,5,6]. Elkafrawy et al. [3] investigated the linear eigenvalue buckling of uniaxially loaded FGM plates. Their study determined how the critical buckling load is impacted by the size and geometry of the openings in the FGM plate. A free vibration of 2D FGM beams with continuous material property variation was analyzed by Aminbaghai et al. [5]. Nonlinear strain–displacement relations of a sheet with FGM was studied by Praveen and Reddy [6].

Some research studies have presented a mathematical solution [7,8,9,10,11,12,13,14,15,16,17]. A mathematical model was provided by Njim et al. [7] to assess the buckling stress of FGM rectangular plates. Cheng and Batra [8] studied the buckling and vibrations of a FGM sheet analytically. The critical load and the natural frequencies of the sheet were examined in their study. Wang [9] presents an analytical solution for the piezoelectric cylindrical construction and scaled function. This article examines the electro-elastic behavior with a changing Young’s modulus along the radius. An analytical solution using functionally graded plates was presented by Bouazza et al. [10]. A comprehensive review of FGM behavior was described by Swaminathan et al. [11] and both analytical and numerical approaches were considered in their study. A comprehensive review of multi-layered plates in terms of wave transmission features was presented by Zarastvand et al. [12]. They also focused on the fundamental equations of composite shell structure. They proposed an analytical solution for the first time to determine the vibration response of composite shell structure with a double curvature [13]. In another work, Zarastvand et al. [14] provided a solution strategy for acoustic equations which were solved along the stiffened shell equations.

Some researchers studied FMG in cylindrical forms [9,15,16,17,18,19]. Wang [9] obtained an analytical solution for functionally graded piezoelectric cylindrical structures. Their results illustrated that non-homogeneity of material has main effect on the electro-elastic field. Sharma and Kaur [16] presented a numerical study by using finite element method. Tokovyy and Ma [18] conducted an analytical approach for radially inhomogeneous geometry. Their method was based on the direct integration approach. The stresses and strains produced by a cylinder of FGM were determined by Fukui and Yamanaka [20]. Wang and Shao [21] conducted a three-dimensional analysis of FGM with a limited length and subjected to non-uniform loads. The steady-state thermo-elasticity of a functionally graded panel has been theoretically investigated by Ootao and Tanigawa [22]. They presented a precise solution. Oral and Anlas [23] have investigated the stress distribution in a heterogeneous anisotropic cylinder.

Most research studies analytical analysis of FGM material only with one-layer in only one direction. In this work, the analytical analysis of a two-layer geometry made of FGM is studied. Derivation of governing equation in two-layer FGM is more challenging than one-layer FGM. The novelty of this work is to propose an analytical solution for two-layer FGM graded materials in two dimensions. Since there is a lack of study in two-layer FGM with the material properties graded along (r, θ) coordinates in both r and θ directions, this work fulfils the gap in the existing literature. Navier’s equations primarily help us to better describe the mechanics of the engineering phenomena. Navier’s equations are derived by combining the stress–strain, strain–displacement and stress equilibrium equations and a direct method is presented to solve these equations analytically.

2 Equation derivations

In most research, the potential function is a commonly used approach for investigating the corresponding stresses in FGM in two dimensions and three dimensions. It should be mentioned that this method has some limitations in choosing boundary conditions. Therefore, it is important to apply a method that does not have these limitations. The direct method is a common method for solving equations which does not have these limitations. At present, the displacement equation and stress equations are obtained by solving the Navier’s equations using the direct method.

2.1 Mathematical modeling

The governing equations for the mechanical behavior of FGM are different from the governing equations for isotropic materials, so it is necessary to determine their equations according to the type of function which is appropriate for their properties. In this study, the investigated problem is a two-layer cylinder made of FGM. An analytical solution is presented for axially asymmetric thermal stresses. The analysis is performed based on the theory of small deformations and the assumption of a very large length in the plane strain mode. In this work, combined power-exponential models are used to express the changes in the properties of FGM. First, the heat transfer equations are obtained and then the equilibrium equations in terms of displacement, known as Navier’s equations, are derived by combining the stress–strain, strain–displacement and stress equilibrium equations. Second, displacement and stress equations are obtained by solving the Navier’s equations. The direct method is applied to solve the Navier’s equations.

The functional relationship of the material properties must be known in order to develop the equations. The mathematical models that can be used to express the changes in the mechanical properties of FGM materials include the exponential model, the power model, and the power–exponential model. In this work, a combined power-exponential model is used to express the changes in the mechanical properties of FGM. The material properties are asymmetric and are assumed in Eq. (1) [24,25] to be graded along (r, θ) coordinates. The modulus of elasticity and the heat conductivity coefficient are assumed to be described with a power-exponential model.

(1) p ( r , θ ) = p 0 r a e b θ ,

where a and b are power-exponential model indices of the material that can be positive or negative, p(x) and p 0 are material properties and material constant, respectively.

2.2 Heat conduction problem in two-layer FGM

Two-dimensional strain–displacement relationships in the radial and circumferential directions are defined. The radial, circumferential strain, and angular strain are obtained from Eq. (2) [21]:

(2) ε r r = ∂ u ∂ r , ε θ θ = 1 r ∂ v ∂ θ + u r ε r θ = 1 2 1 r ∂ u ∂ θ + ∂ v ∂ r − v r .

The strain components are obtained in the radial and circumferential directions. The equation of stress and strain are calculated in the (r, θ, Z) coordinate system. The equilibrium equations with consideration of unit thickness are obtained as follows:

(3) ∂ σ r r ∂ r + 1 r ∂ σ r θ ∂ θ + 1 r ( σ r r − σ θ θ ) = 0 ∂ σ r θ ∂ r + 1 r ∂ σ θ θ ∂ θ + 2 r ( σ r θ ) = 0 .

According to the law of conservation of energy, we have

(4) r ∂ q r ∂ r + q r + ∂ q θ ∂ θ = 0 .

Derivation of heat transfer equations in two dimensions:

By taking the derivative of the law of the conservation of energy [21], and assuming k = k(r) we have

(5) ∂ 2 T ∂ r 2 + k ′ k + 1 r ∂ T ∂ r + 1 r 2 ∂ 2 T ∂ θ 2 = 0 .

Using the theory of linear elasticity [21]

(6) ε i j = 1 2 ( u i , j + u j , i ) ,

(7) ε i j = 1 2 μ σ i j − λ 3 λ + 2 μ σ k k δ i j + α ( T ) δ i j .

Using the elasticity constants E = μ ( 3 λ + 2 μ ) λ + μ and ν = λ 2 ( λ + μ ) , λ and μ are the Lame´ coefficients related to E, and by substituting σ kk , which is σ kk = (3λ + 2μ)(∈ kk − 3α(T)), in Eq. (7) we have

(8) σ i j = 2 μ ε i j + λ ∈ k k δ i j − ( 3 λ + 2 μ ) α ( T ) δ i j .

By using the constants, δ ii = δ jj = δ rr = δ θθ = 1, δ ij = δ rθ = 0, ∈ kk = (ε rr + ε θθ ) and also ε i j ( T ) = α ( T ) δ i j , the stress-strain relations for the FGM in the plane strain are calculated as follows:

(9) σ r r = ( λ + 2 μ ) ε r r + λ ε θ θ − ( 3 λ + 2 μ ) α T ( r , θ ) σ θ θ = ( λ + 2 μ ) ε θ θ + λ ε r r − ( 3 λ + 2 μ ) α T ( r , θ ) σ r θ = 2 μ ε r θ .

The equations of heat transfer are derived in first and second layer FGM. The geometry of the problem is a cylinder with an inner radius r 1, an outer radius of the first layer r 2 and the outer radius of the second layer r 3.

In general, in this problem we have

(10) 1 r ( k r T , r ) , r + 1 r 2 ( k T , θ ) , θ = 0 r 1 ≤ r ≤ r 3 , 0 ≤ θ ≤ 2 π ,

where k is the heat conductivity coefficient, C ij is a constant parameter, f 1(θ) and f 2(θ) are the known functions of the problem. With the boundary conditions as Eq. (11)

(11) C 11 T ( r 1 , θ ) + C 12 T , r ( r 1 , θ ) = f 1 ( θ ) C 21 T ( r 3 , θ ) + C 22 T , r ( r 3 , θ ) = f 2 ( θ ) .

2.2.1 Equation for the first layer

For the first layer k ( r , θ ) = k 0 r m 3 e m 6 θ , where k 0, m 3 and m 6 are material parameters. Based on the Eqs. (10) and (11), we obtain the following:

(12) T , r r + ( m 3 + 1 ) 1 r T , r + 1 r 2 ( m 6 T , θ + T , θ θ ) = 0 .

By considering T(r, θ) in the form of a complex Fourier series, T ( r , θ ) = ∑ − ∞ + ∞ T n ( r ) e in θ & T n ( r ) = 1 2 π ∫ 0 2 π T ( r , θ ) e − in θ d θ , and substituting in the heat transfer equation, Euler’s equation is determined

(13) T n ″ ( r ) + ( m 3 + 1 ) 1 r T n ' ( r ) + 1 r 2 ( i m 6 n − n 2 ) T n ( r ) = 0 .

To solve Eq. (13), T n (r) = A n r β and β 2 + m 3 β + (im 6 n−n 2) = 0, the roots of the equation are determined as β n 1 , 2 = − m 3 2 ∓ [ m 3 2 4 − ( i m 6 n − n 2 ) ] 1 / 2 . Therefore, we find the following:

(14) T n ( r ) = A n 1 r β n 1 + A n 2 r β n 2 T ( r , θ ) = ∑ − ∞ + ∞ ( A n 1 r β n 1 + A n 2 r β n 2 ) e i n θ .

The constants of the problem calculated by Cramer’s method:

(15) A n 1 = 1 2 π ∫ 0 2 π [ ( C 21 b β n 2 + C 22 β n 2 b β n 2 − 1 ) f 1 ( θ ) − ( C 11 a β n 2 + C 12 β n 2 a β n 2 − 1 ) f 2 ( θ ) ] e − in θ d θ / ( C 1 ^ − C 2 ^ )

A n 2 = 1 2 π ∫ 0 2 π [ ( C 11 a β n 1 + C 12 β n 1 a β n 1 − 1 ) f 2 ( θ ) − ( C 21 b β n 1 + C 22 β n 1 b β n 1 − 1 ) f 1 ( θ ) ] e − in θ d θ / ( C 1 ^ − C 2 ^ ) ,

where, C 1 ^ = ( C 11 a β n 1 + C 12 β n 1 a β n 1 − 1 ) ( C 21 b β n 2 + C 22 β n 2 b β n 2 − 1 ) and C 2 ^ = ( C 11 a β n 2 + C 12 β n 2 a β n 2 − 1 ) ( C 21 b β n 1 + C 22 β n 1 b β n 1 − 1 ) .

2.2.2 Equation for the second layer

For the second layer k ( r , θ ) = k 0 r m ¯ 3 e m ¯ 6 θ , where k 0, m ¯ 3 and m ¯ 6 are material parameters. Finally, the heat transfer equation is obtained

(16) T , r r + ( m ¯ 3 + 1 ) 1 r T , r + 1 r 2 ( m ¯ 6 T , θ + T , θ θ ) = 0 .

Just as in the first layer, by considering T(r,θ) in the form of a complex Fourier series, Euler’s equation obtained:

(17) T n ″ ( r ) + ( m ¯ 3 + 1 ) 1 r T n ' ( r ) + 1 r 2 ( i m ¯ 6 n − n 2 ) T n ( r ) = 0 .

To solve the above equation, T n ( r ) = A n r β ¯ and β ¯ 2 + m ¯ 3 β ¯ + ( i m ¯ 6 n − n 2 ) = 0 the roots of the equation are β ¯ n 1 , 2 = − m ¯ 3 2 ∓ [ m ¯ 3 2 4 − ( i m ¯ 6 n − n 2 ) ] 1 / 2 . Therefore, we obtain the following:

(18) T n ( r ) = A n 3 r β ¯ n 1 + A n 4 r β ¯ n 2 T ( r , θ ) = ∑ − ∞ + ∞ ( A n 3 r β ¯ n 1 + A n 4 r β ¯ n 2 ) e in θ .

The constants of the problem are calculated by Cramer’s method:

(19) A n 3 = 1 2 π ∫ 0 2 π [ ( C 21 b β ¯ n 2 + C 22 β ¯ n 2 b β ¯ n 2 − 1 ) f 1 ( θ ) − ( C 11 a β ¯ n 2 + C 12 β ¯ n 2 a β ¯ n 2 − 1 ) f 2 ( θ ) ] e − i n θ d θ / ( C 1 ^ − C 2 ^ ) A n 4 = 1 2 π ∫ 0 2 π [ ( C 11 a β ¯ n 1 + C 12 β ¯ n 1 a β ¯ n 1 − 1 ) f 2 ( θ ) − ( C 21 b β ¯ n 1 + C 22 β ¯ n 1 b β ¯ n 1 − 1 ) f 1 ( θ ) ] e − i n θ d θ / ( C 1 ^ − C 2 ^ ) ,

where C 1 ^ = ( C 11 a β ¯ n 1 + C 12 β ¯ n 1 a β ¯ n 1 − 1 ) ( C 21 b β ¯ n 2 + C 22 β ¯ n 2 b β ¯ n 2 − 1 ) and C 2 ^ = ( C 11 a β ¯ n 2 + C 12 β ¯ n 2 a β ¯ n 2 − 1 ) ( C 21 b β ¯ n 1 + C 22 β ¯ n 1 b β ¯ n 1 − 1 ) .

In the equations for T(r, θ), Eqs. (14) and (18), we have four unknowns (A n1, A n2, A n3, A n4) which are obtained by applying the boundary conditions and continuity between layers.

(20) T f ( r 1 , θ ) = ∑ n = − ∞ ∞ ( A n 1 r 1 β n 1 + A n 2 r 1 β n 2 ) e i n q = f 1 ( θ ) T f ( r 2 , θ ) first layer  = T f ( r 2 , θ ) second layer T f ( r 3 , θ ) = ∑ n = − ∞ ∞ ( A n 3 r 3 β ¯ n 1 + A n 4 r 3 β ¯ n 2 ) e i n θ = f 2 ( θ ) T f ' ( r 2 , θ ) first layer = T f ' ( r 2 , θ ) second layer .

We have four equations (Eq. (20)) and four unknowns.

(21) f 1 ( θ ) = ∑ n = − ∞ ∞ f 1 0 e i n θ f 2 ( θ ) = ∑ n = − ∞ ∞ f 2 0 e i n q f 1 0 = 1 2 π ∫ 0 2 π f 1 ( θ ) e − i n θ d θ f 2 0 = 1 2 π ∫ 0 2 π f 2 ( θ ) e − i n θ d θ .

By substituting the equations and simplifying them we have:

(22) A n 1 r 1 β n 1 + A n 2 r 1 β n 2 = f 1 0 A n 3 r 3 β ¯ n 1 + A n 4 r 3 β ¯ n 2 = f 2 0 A n 1 r 2 β n 1 + A n 2 r 2 β n 2 = A n 3 r 2 β ¯ n 1 + A n 4 r 2 β ¯ n 2 A n 1 β n 1 r 2 β n 1 − 1 + A n 2 β n 2 r 2 β n 2 − 1 = A n 3 β ¯ n 1 r 2 β ¯ n 1 − 1 + A n 4 β ¯ n 2 r 2 β ¯ n 2 − 1 .

2.3 Navier’s equations

By combining the equations of stress, strain, displacement, and equilibrium in terms of stress, the equations of equilibrium in terms of displacement, known as Navier’s equations, are obtained.

For the first layer, the modulus of elasticity and heat expansion coefficient with the power-exponential model are assumed as E ( r , θ ) = E 0 r m 1 e m 4 θ and α ( r , θ ) = α 0 r m 2 e m 5 θ where m 1, m 2, m 4, m 5 are the power-exponential model indices of the material and E 0 and α 0 are material constants. Variables A and B are defined as A = ν ( 1 + ν ) ( 1 − 2 ν ) and B = 1 2 ( 1 + ν ) . By simplifying the stress equations and substituting into the first equilibrium equation we have

(23) u , r r + ( m 1 + 1 ) 1 r u , r + [ A A + 2 B m 1 − 1 ] 1 r 2 u + B A + 2 B m 4 1 r 2 u , θ + B A + 2 B 1 r 2 u , θ θ − B A + 2 B m 4 1 r 2 v + B A + 2 B m 4 1 r v , r + A + B A + 2 B 1 r v , r θ + [ A A + 2 B ( m 1 − 1 ) − 3 B A + 2 B ] 1 r 2 v , θ = 3 A + 2 B A + 2 B α 0 e m 5 θ [ ( m 1 + m 2 ) r m 2 − 1 T + r m 2 T , r ] .

The Navier’s equation is obtained as follows:

(24) u , r r + ( m 1 + 1 ) 1 r u , r + [ ν 1 − ν m 1 − 1 ] 1 r 2 u + 1 − 2 ν 2 − 2 ν m 4 1 r 2 u , θ + 1 − 2 ν 2 − 2 ν 1 r 2 u , θ θ − 1 − 2 ν 2 − 2 ν m 4 1 r 2 v + 1 − 2 ν 2 − 2 ν m 4 1 r v , r + 1 2 − 2 ν 1 r v , r θ + [ ν 1 − ν ( m 1 − 1 ) − 3 − 6 ν 2 − 2 ν ] 1 r 2 v , θ = 1 + ν 1 − ν α 0 e m 5 θ [ ( m 1 + m 2 ) r m 2 − 1 T + r m 2 T , r ] .

In the following, by substituting the relevant terms in the second equilibrium equation, the second Navier’s equation is obtained as follows:

(25) v , r r + ( m 1 + 1 ) 1 r v , r − ( m 1 + 1 ) 1 r 2 v + 2 − 2 ν 1 − 2 ν m 4 1 r 2 v , θ + 2 − 2 ν 1 − 2 ν 1 r 2 v , θ θ + 2 − 2 ν 1 − 2 ν m 4 1 r 2 u + 2 ν 1 − 2 ν m 4 1 r u , r + 1 1 − 2 ν 1 r u , r θ + [ ( m 1 + 3 ) + 2 ν 1 − 2 ν ] 1 r 2 u , θ = 2 + 2 ν 1 − 2 ν α 0 r m 2 − 1 e m 5 θ [ ( m 4 + m 5 ) T + T , θ ] .

For the second layer, the modulus of elasticity and heat expansion coefficient with the power-exponential model are assumed as E ¯ ( r , θ ) = E ¯ 0 r m ¯ 1 e m ¯ 4 θ and α ¯ ( r , θ ) = α ¯ 0 r m ¯ 2 e m ¯ 5 θ , and we also have A ¯ = ν ¯ ( 1 + ν ¯ ) ( 1 − 2 ν ¯ )   and B ¯ = 1 2 ( 1 + ν ¯ ) . By applying the same method as the first layer, the equilibrium is determined as:

(26) u , r r + ( m ¯ 1 + 1 ) 1 r u , r + [ A ¯ A ¯ + 2 B ¯ m ¯ 1 − 1 ] 1 r 2 u + B ¯ A ¯ + 2 B ¯ m ¯ 4 1 r 2 u , θ + B ¯ A ¯ + 2 B ¯ 1 r 2 u , θ θ

− B ¯ A ¯ + 2 B ¯ m ¯ 4 1 r 2 v + B ¯ A ¯ + 2 B ¯ m ¯ 4 1 r v , r + A ¯ + B ¯ A ¯ + 2 B ¯ 1 r v , r θ + [ A ¯ A ¯ + 2 B ¯ ( m ¯ 1 − 1 ) − 3 B ¯ A ¯ + 2 B ¯ ] 1 r 2 v , θ = 3 A ¯ + 2 B ¯ A ¯ + 2 B ¯ α ¯ 0 e m ¯ 5 θ [ ( m ¯ 1 + m ¯ 2 ) r m ¯ 2 − 1 T + r m ¯ 2 T , r ] .

The Navier’s equation is obtained as Eq. (27)

(27) u , r r + ( m ¯ 1 + 1 ) 1 r u , r + ν ¯ 1 − ν ¯ m ¯ 1 − 1 1 r 2 u + 1 − 2 ν ¯ 2 − 2 ν ¯ m ¯ 4 1 r 2 u , θ + 1 − 2 ν ¯ 2 − 2 ν ¯ 1 r 2 u , θ θ − 1 − 2 ν ¯ 2 − 2 ν ¯ m ¯ 4 1 r 2 v + 1 − 2 ν ¯ 2 − 2 ν ¯ m ¯ 4 1 r v , r + 1 2 − 2 ν ¯ 1 r v , r θ + ν ¯ 1 − ν ¯ ( m ¯ 1 − 1 ) − 3 − 6 ν ¯ 2 − 2 ν ¯ 1 r 2 v , θ = 1 + ν ¯ 1 − ν ¯ α ¯ 0 e m ¯ 5 θ [ ( m ¯ 1 + m ¯ 2 ) r m ¯ 2 − 1 T + r m ¯ 2 T , r ] .

By substituting the relevant terms in the second equilibrium equation, the second Navier’s equation is obtained as follows:

(28) v , r r + ( m ¯ 1 + 1 ) 1 r v , r − ( m ¯ 1 + 1 ) 1 r 2 v + 2 − 2 ν ¯ 1 − 2 ν ¯ m ¯ 4 1 r 2 v , θ + 2 − 2 ν ¯ 1 − 2 ν ¯ 1 r 2 v , θ θ + 2 − 2 ν ¯ 1 − 2 ν ¯ m ¯ 4 1 r 2 u + 2 ν ¯ 1 − 2 ν ¯ m ¯ 4 1 r u , r + 1 1 − 2 ν ¯ 1 r u , r θ + [ ( m ¯ 1 + 3 ) + 2 ν ¯ 1 − 2 ν ¯ ] 1 r 2 u , θ = 2 + 2 ν ¯ 1 − 2 ν ¯ α ¯ 0 r m ¯ 2 − 1 e m 5 θ [ ( m ¯ 4 + m ¯ 5 ) T + T , θ ] .

3 Results

The displacement equations and stress equations are presented by solving the Navier’s equations.

3.1 Solutions of Navier’s equations in two-layer FGM

To solve the Navier’s equations, we expand the displacement components in the form of a Fourier series, u ( r , θ ) = ∑ n = − ∞ + ∞ u n ( r ) e ( i n + m 5 ) θ , v ( r , θ ) = ∑ n = − ∞ + ∞ v n ( r ) e ( i n + m 5 ) θ , u n ( r ) = 1 2 π ∫ 0 2 π u ( r , θ ) e − ( i n + m 5 ) θ d θ , and v n ( r ) = 1 2 π ∫ 0 2 π v ( r , θ ) e − ( in + m 5 ) θ d θ , and substitute in Eqs. (24) and (27):

(29) u n ″ + ( m 1 + 1 ) 1 r u n ′ + ν m 1 1 − ν − 1 + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) m 4 + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) 2 1 r 2 u n + ( 1 − 2 ν ) m 4 2 − 2 ν + ( i n + m 5 ) 2 − 2 ν 1 r v n ′ + ν ( m 1 − 1 ) 1 − ν − 3 − 6 ν 2 − 2 ν ( i n + m 5 ) − ( 1 − 2 ν ) m 4 2 − 2 ν 1 r 2 v n = 1 + ν 1 − ν α 0 [ ( m 1 + m 2 + β n 1 ) A n 1 r β n 1 + m 2 − 1 + ( m 1 + m 2 + β n 2 ) A n 2 r β n 2 + m 2 − 1 ] ,

(30) v n ″ + ( m 1 + 1 ) 1 r v n ′ + 2 − 2 ν 1 − 2 ν ( in + m 5 ) m 4 − ( m 1 + 1 ) + 2 − 2 ν 1 − 2 ν ( in + m 5 ) 2 1 r 2 v n + 2 ν m 4 1 − 2 ν + ( in + m 5 ) 1 − 2 ν 1 r u n ′ + ( m 1 + 3 ) + 2 ν 1 − 2 ν ( in + m 5 ) + ( 2 − 2 ν ) m 4 1 − 2 ν 1 r 2 u n = ( 2 + 2 ν ) α 0 1 − 2 ν [ ( m 4 + m 5 + in ) ( A n 1 r β n 1 + m 2 − 1 + A n 2 r β n 2 + m 2 − 1 ) ] .

Eqs. (29) and (30) are a system of ordinary differential equations with general and particular solutions. To solve these equations, for general solutions of the problem, we have u n g (r) = Br η and v n g (r) = Cr η , therefore

(31) η ( η − 1 ) + ( m 1 + 1 ) η + ν m 1 1 − ν + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) m 4 + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) 2 − 1 B + ( 1 − 2 ν ) m 4 2 − 2 ν + ( i n + m 5 ) 2 − 2 ν η + ν ( m 1 − 1 ) 1 − ν − 3 − 6 ν 2 − 2 ν ( i n + m 5 ) − ( 1 − 2 ν ) m 4 2 − 2 ν C = 0 2 ν m 4 1 − 2 ν + ( i n + m 5 ) 1 − 2 ν η + ( m 1 + 3 ) + 2 ν 1 − 2 ν ( i n + m 5 ) + ( 2 − 2 ν ) m 4 1 − 2 ν + η ( η − 1 ) + ( m 1 + 1 ) η + 2 − 2 ν 1 − 2 ν ( i n + m 5 ) m 4 − ( m 1 + 1 ) + 2 − 2 ν 1 − 2 ν ( i n + m 5 ) 2 C = 0 .

For the non-trivial solution of Eq. (31)

(32) η ( η − 1 ) + ( m 1 + 1 ) η + ν m 1 1 − ν + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) m 4 + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) 2 − 1 η ( η − 1 ) + ( m 1 + 1 ) η + 2 − 2 ν 1 − 2 ν ( i n + m 5 ) m 4 − ( m 1 + 1 ) + 2 − 2 ν 1 − 2 ν ( i n + m 5 ) 2 − 2 ν m 4 1 − 2 ν + ( i n + m 5 ) 1 − 2 ν η + ( m 1 + 3 ) + 2 ν 1 − 2 ν ( i n + m 5 ) + ( 2 − 2 ν ) m 4 1 − 2 ν ( 1 − 2 ν ) m 4 2 − 2 ν + ( i n + m 5 ) 2 − 2 ν η + ν ( m 1 − 1 ) 1 − ν − 3 − 6 ν 2 − 2 ν ( i n + m 5 ) − ( 1 − 2 ν ) m 4 2 − 2 ν = 0 .

For the general solution of Eq. (32), u n g ( r ) = ∑ j = 1 4 B n j r η n j , v n g ( r ) = ∑ j = 1 4 N n j B n j r η n j and the relation between B nj and C nj constants is defined as N nj , where, j = 1, 2, 3, 4.

N n j = − η n j ( η n j − 1 ) + ( m 1 + 1 ) η n j + ν m 1 1 − ν + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) m 4 + 1 − 2 ν 2 − 2 ν ( i n + m 5 ) 2 − 1 ( 1 − 2 ν ) m 4 2 − 2 ν + ( i n + m 5 ) 2 − 2 ν η n j + ν ( m 1 − 1 ) 1 − ν − 3 − 6 ν 2 − 2 ν ( i n + m 5 ) − ( 1 − 2 ν ) m 4 2 − 2 ν

For the particular solutions of Eq. (32), u n p ( r ) = D n 1 r β n 1 + m 2 + 1 + D n 2 r β n 2 + m 2 + 1 and v n p ( r ) = D n 3 r β n 1 + m 2 + 1 + D n 4 r β n 2 + m 2 + 1 . By placing them in Eqs. (29) and (30) we have

(33) d 1 D n 1 r β n 1 + m 2 − 1 + d 2 D n 2 r β n 2 + m 2 − 1 + d 3 D n 3 r β n 1 + m 2 − 1 + d 4 D n 4 r β n 2 + m 2 − 1 = d 5 r β n 1 + m 2 − 1 + d 6 r β n 2 + m 2 − 1 d 7 D n 3 r β n 1 + m 2 − 1 + d 8 D n 4 r β n 2 + m 2 − 1 + d 9 D n 1 r β n 1 + m 2 − 1 + d 10 D n 2 r β n 2 + m 2 − 1 = d 11 r β n 1 + m 2 − 1 + d 12 r β n 2 + m 2 − 1 ,

where d 1 to d 12 are the defined constants. Equating the power coefficients and using Cramer’s method:

(34) D n 1 = d 5 d 7 − d 3 d 11 d 1 d 7 − d 3 d 9 D n 2 = d 6 d 8 − d 4 d 12 d 2 d 8 − d 4 d 10 D n 3 = d 1 d 11 − d 5 d 9 d 1 d 7 − d 3 d 9 D n 4 = d 2 d 12 − d 6 d 10 d 2 d 8 − d 4 d 10 .

The complete solutions are obtained as:

(35) u ( r , θ ) = ∑ n = − ∞ + ∞ ∑ j = 1 4 B n j r η n j + D n 1 r β n 1 + m 2 + 1 + D n 2 r β n 2 + m 2 + 1 e ( i n + m 5 ) θ v ( r , θ ) = ∑ n = − ∞ + ∞ ∑ j = 1 4 N n j B n j r η n j + D n 3 r β n 1 + m 2 + 1 + D n 4 r β n 2 + m 2 + 1 e ( i n + m 5 ) θ .

Therefore, by using Eq. (35) we calculated ε rr , ε θθ , ε rθ , σ rr , σ θθ and σ rθ .

(36) ε r r = ∑ n = − ∞ + ∞ ∑ j = 1 4 η n j B n j r η n j − 1 + D n 1 ( β n 1 + m 2 + 1 ) r β n 1 + m 2 + D n 2 ( β n 2 + m 2 + 1 ) r β n 2 + m 2 e ( i n + m 5 ) θ ε θ θ = ∑ n = − ∞ + ∞ ∑ j = 1 4 ( ( i n + m 5 ) N n j + 1 ) B n j r η n j − 1 + ( D n 1 + ( i n + m 5 ) D n 3 ) r β n 1 + m 2 + ( D n 2 + ( i n + m 5 ) D n 4 ) r β n 2 + m 2 e ( i n + m 5 ) θ ε r θ = 1 2 ∑ n = − ∞ + ∞ ∑ j = 1 4 ( ( i n + m 5 ) + ( η n j − 1 ) N n j ) B n j r η n j − 1 + ( ( i n + m 5 ) D n 1 + ( β n 1 + m 2 ) D n 3 ) r β n 1 + m 2 + ( ( i n + m 5 ) D n 2 + ( β n 2 + m 2 ) D n 4 ) r β n 2 + m 2 e ( i n + m 5 ) θ ,

and,

(37) σ r r = E 0 ( 1 + ν ) ( 1 − 2 ν ) ∑ n = − ∞ + ∞ ∑ j = 1 4 ( ( 1 − ν ) η n j + ν ( ( i n + m 5 ) N n j + 1 ) ) B n j r ( η n j + m 1 − 1 ) + ( ( 1 − ν ) D n 1 ( β n 1 + m 2 + 1 ) + ν ( D n 1 + ( i n + m 5 ) D n 3 ) − ( 1 + ν ) α 0 A n 1 ) r ( β n 1 + m 1 + m 2 ) + ( ( 1 − ν ) D n 2 ( β n 2 + m 2 + 1 ) + ν ( D n 2 + ( i n + m 5 ) D n 4 ) − ( 1 + ν ) α 0 A n 2 ) r ( β n 2 + m 1 + m 2 ) e ( i n + m 4 + m 5 ) θ σ θ θ = E 0 ( 1 + ν ) ( 1 − 2 ν ) ∑ n = − ∞ + ∞ ∑ j = 1 4 ( ( 1 − ν ) ( ( i n + m 5 ) N n j + 1 ) + ν η n j ) B n j r ( η n j + m 1 − 1 ) + ( ν ( β n 1 + m 2 + 1 ) D n 1 + ( 1 − ν ) ( D n 1 + ( i n + m 5 ) D n 3 ) − ( 1 + ν ) α 0 A n 1 ) r ( β n 1 + m 1 + m 2 ) + ( ν ( β n 2 + m 2 + 1 ) D n 2 + ( 1 − ν ) ( D n 2 + ( i n + m 5 ) D n 4 ) − ( 1 + ν ) α 0 A n 2 ) r ( β n 2 + m 1 + m 2 ) e ( i n + m 4 + m 5 ) θ σ r θ = E 0 ( 1 + ν ) ∑ n = − ∞ + ∞ ∑ j = 1 4 ( ( i n + m 5 ) + ( η n j − 1 ) N n j ) B n j r ( η n j + m 1 − 1 ) + ( ( i n + m 5 ) D n 1 + ( β n 1 + m 2 ) D n 3 ) r ( β n 1 + m 1 + m 2 ) + ( ( i n + m 5 ) D n 2 + ( β n 2 + m 2 ) D n 4 ) r ( β n 2 + m 1 + m 2 ) e ( i n + m 4 + m 5 ) θ .

Now for the second layer, we repeat the same above calculation.

(38) d ¯ 1 D ¯ n 1 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 2 D ¯ n 2 r β ¯ n 2 + m ¯ 2 − 1 + d ¯ 3 D ¯ n 3 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 4 D ¯ n 4 r β ¯ n 2 + m ¯ 2 − 1 = d ¯ 5 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 6 r β ¯ n 2 + m ¯ 2 − 1 d ¯ 7 D ¯ n 3 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 8 D ¯ n 4 r β ¯ n 2 + m ¯ 2 − 1 + d ¯ 9 D ¯ n 1 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 10 D ¯ n 2 r β ¯ n 2 + m ¯ 2 − 1 = d ¯ 11 r β ¯ n 1 + m ¯ 2 − 1 + d ¯ 12 r β ¯ n 2 + m ¯ 2 − 1 .

Based on Cramer’s method, the complete solutions are calculated:

(39) u ( r , θ ) = ∑ n = − ∞ + ∞ ∑ j = 1 4 B ¯ n j r η ¯ n j + D ¯ n 1 r β ¯ n 1 + m ¯ 2 + 1 + D ¯ n 2 r β ¯ n 2 + m ¯ 2 + 1 e ( i n + m ¯ 5 ) θ

v ( r , θ ) = ∑ n = − ∞ + ∞ ∑ j = 1 4 N ¯ n j B ¯ n j r η ¯ n j + D ¯ n 3 r β ¯ n 1 + m ¯ 2 + 1 + D ¯ n 4 r β ¯ n 2 + m ¯ 2 + 1 e ( i n + m ¯ 5 ) θ .

Accordingly, ε rr , ε θθ , ε rθ , σ rr , σ θθ , and σ rθ are calculated by using Eq. (39). It should be noted that parameters B nj and B ¯ n j are determined by considering the boundary conditions for displacements and stresses u(r 1, θ) = g 1(θ), u(r 3, θ) = g 2(θ), v(r 1, θ) = g 3(θ), v(r 3, θ) = g 4(θ), σ rr (r 1, θ) = g 5(θ), σ rr (r 3, θ) = g 6(θ), σ rθ (r 1, θ) = g 7(θ), and σ rθ (r 3, θ) = g 8(θ), as well as four continuity conditions σ rr (r 2, θ)1 = σ rr (r 2, θ)2, σ rθ (r 2, θ)1 = σ rθ (r 2, θ)2, u(r 2, θ)1 = u(r 2, θ)2, and v(r 2, θ)1 = v(r 2, θ)2 ).

Therefore, we have eight unknowns (B n1, B n2, B n3, B n4, B ¯ n 1 , B ¯ n 2 , B ¯ n 3 , B ¯ n 4 ) and eight equations. A complex Fourier series ( g j ( θ ) = ∑ n = − ∞ + ∞ G j ( n ) e ( i n θ ) , j = 1 , … , 4 ) is implemented for solving the equations, where m 5 = m ¯ 5 and m 4 = m ¯ 4 .

3.2 Example

We assumed the following inputs: inner radius r 1 = 1m and outer radius r 3 = 1.4m and the radius of the middle layer r 2 = 1.2 m. Assume that Poisson’s ratio, modulus of elasticity, and coefficient of heat expansion of the first layer are ν = 0.3, E 0 = 200 GPa, and α 0 = 1.2 × 10−6 1/°C, and for the second layer, ν = 0.2, E 0 = 70 GPa and α 0 = 1.5 × 10−6 1/°C, respectively.

The stress distributions in the functionally graded two-layer cylinder under the heat boundary conditions are presented to evaluate the solution of the applied method under the defined boundary conditions. There are six independent power-exponential parameters in the equations, which indicates that the parametric solution of the problem has not been obtained for the specific case. Since the obtained equations are general, different parameters can be applied in any desired range for the material. Due to the existence of six independent parameters such as m 1, m 2, m 3, etc., choosing their value without observing a specific basis for the ratio between the parameters enables many possible situations.

For example, consider a stress-free inner boundary with temperature distribution of T(r 1, θ) = 60 sinθ. The temperature of the outer boundary is assumed to be zero. Therefore, by using four boundary conditions and four continuity conditions for stress and displacement, it can be concluded σ rr (r 1, θ) = 0, σ rθ (r 1, θ) = 0, u(r 3, θ) = 0, v(r 3, θ) = 0, σ rr (r 2, θ)1 = σ rr (r 2, θ)2, σ rθ (r 2, θ)1 = σ rθ (r 2, θ)2, u(r 2, θ)1 = u(r 2, θ)2, v(r 2, θ)1 = v(r 2, θ)2, T f (r 1, θ) = f 1(θ), T f (r 3, θ) = f 2(θ), T f (r 2, θ)firstlayer = T f (r 2, θ)secondlayer, and T f '(r 2,θ)firstlayer = T f '(r 2, θ)secondlayer. Heat distribution coefficients are obtained by placing the boundary conditions and continuity of stress and displacement and using the heat boundary conditions.

The power-exponential indices of material properties at θ = π 2 and r = [1, 1.4] ; for the first layer m 1 = m 2 = m 3 = m, m = [−1, 0.0001, 1], m 4 = m 5 = m 6 = 0, and also for the second layer m ¯ 1 = m ¯ 2 = m ¯ 3 = m , m ¯ 4 = m ¯ 5 = m ¯ 6 = 0 . The above conditions are assumed for the material properties change along the radial direction without considering the distribution of the material properties in the circumferential direction.

T values are shown in Table 1 in different radii for three different m values.

Table 1

T values in different radii (r/r 1)

r/r 1 T(°C)
m = 1 m = 0.0001 m = −1
1.00 60.00 60.00 60.00
1.04 51.47 52.79 54.01
1.08 43.65 45.93 48.10
1.12 36.44 39.37 42.26
1.16 29.77 33.10 36.47
1.20 23.58 27.08 30.73
1.24 17.91 21.29 24.92
1.28 12.79 15.70 18.96
1.32 8.13 10.30 12.82
1.36 3.88 5.07 6.50
1.40 0.00 0.00 0.00

Heat distribution is also shown in Figure 1. Figure 1 shows that the amount of heat distribution decreases with the increase in the parameter m.

Figure 1 Heat distribution in radial direction.
Figure 1

Heat distribution in radial direction.

Displacement components are shown in Figure 2. According to this figure by increasing m values, the value of V decreases and the value of U increases.

Figure 2 Displacement components in radial direction, (a) circumferential displacement and (b) radial displacement.
Figure 2

Displacement components in radial direction, (a) circumferential displacement and (b) radial displacement.

Thermal stress component distributions are shown in Table 2 and Figure 3.

Table 2

Stress components (σ θθ , σ rr , and σ rθ ) in different radii (r/r 1)

r/r 1 σ θθ × 108 (Pa) σ rr × 108 (Pa) σ rθ × 108 (Pa)
m = 1 m = 0.0001 m = −1 m = 1 m = 0.0001 m = −1 m = 1 m = 0.0001 m = −1
1.00 −6.40 −6.35 −6.33 0.00 0.00 0.00 0.00 0.00 0.00
1.04 −5.92 −5.78 −5.64 −1.23 −1.20 −1.17 −1.97 −1.92 −1.86
1.08 −5.38 −5.18 −4.97 −1.82 −1.75 −1.70 −2.91 −2.80 −2.72
1.12 −4.85 −4.58 −4.32 −2.19 −2.10 −2.01 −3.50 −3.36 −3.22
1.16 −4.37 −4.06 −3.76 −2.42 −2.30 −2.20 −3.86 −3.68 −3.52
1.18* −3.90 −3.56 −3.25 −2.46 −2.35 −2.25 −3.94 −3.76 −3.60
1.20 −2.24 −1.92 −1.63 −2.49 −2.39 −2.29 −3.98 −3.82 −3.66
1.24 −1.81 −1.49 −1.21 −2.47 −2.37 −2.27 −3.95 −3.79 −3.63
1.28 −1.35 −1.07 −0.83 −2.43 −2.33 −2.23 −3.89 −3.73 −3.57
1.32 −0.91 −0.69 −0.50 −2.34 −2.25 −2.16 −3.75 −3.59 −3.45
1.36 −0.41 −0.29 −0.19 −2.23 −2.13 −2.05 −3.57 −3.41 −3.27
1.40 0.00 0.00 0.00 −2.10 −1.99 −1.90 −3.35 −3.18 −3.03

*This value is 1.19 for σ θθ .

Figure 3 Stress component distributions in radial direction, (a) circumferential stress (σ θθ ), (b) radial stress (σ rr ), and (c) shear stress (σ rθ ).
Figure 3

Stress component distributions in radial direction, (a) circumferential stress (σ θθ ), (b) radial stress (σ rr ), and (c) shear stress (σ rθ ).

Our findings indicate that according to Figure 3, by increasing m values, the value of the stress components (σ θθ , σ rr , and σ rθ ) increases.

4 Conclusion

Due to the benefits of gradually varying properties, FGM can be employed in a variety of engineering applications, such as building materials in civil engineering. This study presents an analytical solution in a two-layered geometry made of FGM. In this research, the direct method and power-exponential model are implemented to determine the solution of the governing equations instead of using other approaches such as the potential function method. The advantage of the direct method in comparison with the potential function method is its generality and mathematical ability to apply different boundary conditions. The approach implemented helps us to better describe the mechanics of the engineering phenomena. In addition, direct method gives us precise answers. However, for more complex equations it might be better to implement other methods such as iteration method.

Since increasing the values of the parameter m, the values of modulus of elasticity increases, it enables us to grade the geometry from a low modulus of elasticity to a high modulus of elasticity. According to our results by increasing m values, the value of circumferential displacement (V) decreases and the value of radial displacement (U ) increases. Moreover, by increasing m values, the value of the stress components (σ θθ , σ rr , and σ rθ ) increases. For further work, it would also be beneficial to apply other methods to solve the governing equations. This enables us to compare results of different approaches to achieve a better understanding of the problem.

Acknowledgments

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: 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.

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Received: 2022-07-25
Revised: 2022-09-10
Accepted: 2022-10-03
Published Online: 2022-11-04

© 2022 Farhad Belalpour Dastjerdi and Mohsen Jabbari, published by De Gruyter

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

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https://doi.org/10.1515/nleng-2022-0258
Keywords for this article
two-layer; functionally graded material; analytical analysis
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  23. Synchronizing a synchronverter to an unbalanced power grid using sequence component decomposition
  24. Distinguishability criteria of conformable hybrid linear systems
  25. A new computational investigation to the new exact solutions of (3 + 1)-dimensional WKdV equations via two novel procedures arising in shallow water magnetohydrodynamics
  26. A passive verses active exposure of mathematical smoking model: A role for optimal and dynamical control
  27. A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1 + 2)-physical models
  28. Exploration of peristaltic pumping of Casson fluid flow through a porous peripheral layer in a channel
  29. Investigation of optimized ELM using Invasive Weed-optimization and Cuckoo-Search optimization
  30. Analytical analysis for non-homogeneous two-layer functionally graded material
  31. Investigation of critical load of structures using modified energy method in nonlinear-geometry solid mechanics problems
  32. Thermal and multi-boiling analysis of a rectangular porous fin: A spectral approach
  33. The path planning of collision avoidance for an unmanned ship navigating in waterways based on an artificial neural network
  34. Shear bond and compressive strength of clay stabilised with lime/cement jet grouting and deep mixing: A case of Norvik, Nynäshamn
  35. Communication
  36. Results for the heat transfer of a fin with exponential-law temperature-dependent thermal conductivity and power-law temperature-dependent heat transfer coefficients
  37. Special Issue: Recent trends and emergence of technology in nonlinear engineering and its applications - Part I
  38. Research on fault detection and identification methods of nonlinear dynamic process based on ICA
  39. Multi-objective optimization design of steel structure building energy consumption simulation based on genetic algorithm
  40. Study on modal parameter identification of engineering structures based on nonlinear characteristics
  41. On-line monitoring of steel ball stamping by mechatronics cold heading equipment based on PVDF polymer sensing material
  42. Vibration signal acquisition and computer simulation detection of mechanical equipment failure
  43. Development of a CPU-GPU heterogeneous platform based on a nonlinear parallel algorithm
  44. A GA-BP neural network for nonlinear time-series forecasting and its application in cigarette sales forecast
  45. Analysis of radiation effects of semiconductor devices based on numerical simulation Fermi–Dirac
  46. Design of motion-assisted training control system based on nonlinear mechanics
  47. Nonlinear discrete system model of tobacco supply chain information
  48. Performance degradation detection method of aeroengine fuel metering device
  49. Research on contour feature extraction method of multiple sports images based on nonlinear mechanics
  50. Design and implementation of Internet-of-Things software monitoring and early warning system based on nonlinear technology
  51. Application of nonlinear adaptive technology in GPS positioning trajectory of ship navigation
  52. Real-time control of laboratory information system based on nonlinear programming
  53. Software engineering defect detection and classification system based on artificial intelligence
  54. Vibration signal collection and analysis of mechanical equipment failure based on computer simulation detection
  55. Fractal analysis of retinal vasculature in relation with retinal diseases – an machine learning approach
  56. Application of programmable logic control in the nonlinear machine automation control using numerical control technology
  57. Application of nonlinear recursion equation in network security risk detection
  58. Study on mechanical maintenance method of ballasted track of high-speed railway based on nonlinear discrete element theory
  59. Optimal control and nonlinear numerical simulation analysis of tunnel rock deformation parameters
  60. Nonlinear reliability of urban rail transit network connectivity based on computer aided design and topology
  61. Optimization of target acquisition and sorting for object-finding multi-manipulator based on open MV vision
  62. Nonlinear numerical simulation of dynamic response of pile site and pile foundation under earthquake
  63. Research on stability of hydraulic system based on nonlinear PID control
  64. Design and simulation of vehicle vibration test based on virtual reality technology
  65. Nonlinear parameter optimization method for high-resolution monitoring of marine environment
  66. Mobile app for COVID-19 patient education – Development process using the analysis, design, development, implementation, and evaluation models
  67. Internet of Things-based smart vehicles design of bio-inspired algorithms using artificial intelligence charging system
  68. Construction vibration risk assessment of engineering projects based on nonlinear feature algorithm
  69. Application of third-order nonlinear optical materials in complex crystalline chemical reactions of borates
  70. Evaluation of LoRa nodes for long-range communication
  71. Secret information security system in computer network based on Bayesian classification and nonlinear algorithm
  72. Experimental and simulation research on the difference in motion technology levels based on nonlinear characteristics
  73. Research on computer 3D image encryption processing based on the nonlinear algorithm
  74. Outage probability for a multiuser NOMA-based network using energy harvesting relays
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