Three-Dimensional Piezothermoelastic Stress of a Finite Functionally Graded Cylindrical Shell with Piezoelectric Layer

Abstract

Three-dimensional piezothermoelastic solutions for a finite functionally graded cylindrical shell with piezoelectric layer are carried out in this paper. The cylindrical shell is simply supported at four end edges and is subjected to axisymmetric thermomechanical loads. The piezoelectric layers are polarized along radial direction as a sensor. The material properties are assumed to be temperature independent and radially dependent but are assumed to be homogeneous in each layer; the variables are expanded in Fourier series to satisfy the boundary conditions and multilayer approach is used. Numerical results of mullite/molybdenum functionally graded cylindrical shell are presented; the temperature change, stresses, electric potential, and electric displacement distributions are given and briefly discussed.

1. Introduction

Functionally graded materials (FGMs) are new kind of inhomogeneous composite materials with continuously varying properties. The spatial grading in composition and microstructure of FGMs can be adjusted in order to obtain specific properties or to reduce the magnitude of mechanical and thermal stresses. Piezoelectric materials are probably the most popular active materials which act as sensors and actuators and have been used widely in structural health monitoring, vibration, and noise control as well as many other areas. Among these, piezoelectric bimorph and multimorph are commonly employed. Usually, these structures are made of two or more layers of piezoelectric sheets and are jointed by bonding agents. However, such laminated piezoelectric structures suffer from high stress concentration near the interface due to the abrupt changes in both material composition and thermoelectroelastic properties, which can cause severe deterioration of the bonding layer strength and reduce the lifetime of the structures. To overcome the drawbacks and meet some particular requirements for performance and reliability, the functionally graded materials (FGMs) and piezoelectric material are used together, which called functionally graded piezoelectric materials (FGPMs). A new type of structure was developed.

The fabrication and property investigation of FGPMs have attracted great attention from the research community. Among the early investigators, PintoCarreia et al. [1] derived and solved the coupled displacement and electrical field equations for a piezoelectric cylindrical shell based on the third-order shear deformation theory and the finite element method. Using classical laminate theory, the piezothermoelastic behavior of a piezoelectric composite actuator with functionally graded microstructure was analyzed [2]. Ootao and Tanigawa [3–7] exactly analyzed the transient piezothermoelastic problem of a functionally graded hollow sphere, a simply supported cylindrical composite panel which composed of crossply or angleply and piezoelectric layer under the state of generalized plane deformation, a rectangular composite plate composed of cross-ply laminate and piezoelectric material, and a functionally graded rectangular plate bonded to a piezoelectric plate due to partial heat supply. Using the power series expansion method, Chen and his coworkers [8, 9] successfully derived two-dimensional exact elasticity solution and studied both the direct and inverse piezoelectric effects of the piezothermoelastic behavior of circular laminated piezoelectric cylindrical shells and functionally graded piezoelectric cylindrical shells subjected to axisymmetric loading. Kapuria and Achary [10] presented a new coupled consistent third-order theory for static thermoelectromechanical response of hybrid piezoelectric laminated plates. Shao [11] presented solutions for temperature, displacement and stress fields in an FG hollow cylinder using a multilayered approach based on the laminated composite theory. Huang et al. [12] derived the analytical solutions for FGP beams under both mechanical and electrical loads from the two-dimensional equations of piezoelectricity, in which the elastic, piezoelectric and dielectric coefficients of the piezoelectric beams were assumed to vary along the beam thickness direction only. Yang and Xiang [13] used the Timoshenko beam theory to investigate the static bending and dynamic response of FGP actuators under combined thermalelectromechanical loading. Ying and Wang [14] employed the separation of variable technique to carry out two-dimensional elastic-dynamic analysis of simply supported hollow cylinders of finite length excited by nonuniform thermal shock. Reference [15] proposed improved layered shell finite element solution for the coupled thermoelectromechanical responses of smart fiber reinforced composite shell panels under piezothermoelastic loading and showed that piezoelectric effect has a significant influence on the response of such shells. The bending behavior of a circularly curved FGP cantilever actuator under an applied electrical load and heat conduction was investigated [16]. Infinitesimal axisymmetric deformation of a functionally graded shell with piezoelectric layers perfectly bonded to its inner and outer surfaces subjected to thermo-electro-mechanical loads was studied by Alibeigloo [17]. Alashti and Khorsand [18] carried out three-dimensional static analyses of FG cylindrical shells with piezoelectric layers under the effect of thermoelectromechanical loads, using the differential quadrature method.

In this paper, a functionally graded cylindrical shell with piezoelectric layer is investigated. The thermal and mechanical loads are applied on the cylinder asymmetrically. In order to obtain analytical solutions for temperature, displacements, and stresses for the three-dimensional piezothermoelastic problem, it is assumed that the piezoelectric functionally graded cylindrical shell is composed of 𝑁 fictitious layers and one piezoelectric layer, and each layer of the functionally graded material is homogeneous and isotropic.

2. Basic Equations

A piezoelectric functionally graded cylindrical shell with finite length 𝑙, internal radius 𝑟_𝑎, external radius 𝑟_𝑏, and circumferential angle 𝜃₀ is considered (as shown in Figure 1), and it is assumed that the piezoelectric layer is perfectly bonded, the inner radius of piezoelectric layer is 𝑟_𝑐, and the radius of the interface between the 𝑖th and (𝑖+1)th layer is 𝑟_𝑖. Cylindrical coordinates 𝑟, 𝜃, and 𝑧 are used in analysis. This functionally graded cylindrical shell is simply supported at its four end edges and subjected to nonuniform steady-state thermal loads 𝑇_𝑎(𝜃,𝑧) and 𝑇_𝑏(𝜃,𝑧) on the inner and outer surface. The temperatures at end edges keep zero. Nonuniform pressure loadings 𝑞_𝑎(𝜃,𝑧) and 𝑞_𝑏(𝜃,𝑧) are applied to the inner and outer surfaces of the panel, respectively. Details of piezoelectric functionally graded cylindrical shell are shown in Figure 1.

Figure 1 

A piezoelectric functionally graded cylindrical shell.

It is assumed that the piezoelectric functionally graded cylindrical shell is composed of 𝑁 fictitious layers and one piezoelectric layer, and each layer of the functionally graded material is homogeneous and isotropic. The law of mixtures proposed by Kerner, which was derived based on the assumption that there is a granular phase embedded in a matrix phase, is applied to describe the variations of material properties in the radial direction of the functionally graded cylinder shell. Young’s modulus 𝐸, Poisson’s ratio 𝜇, thermal conductivity coefficient 𝜆, and thermal expansion coefficient 𝛼 can be expressed as 𝐸(𝑟)=𝑉_𝑚𝐸_𝑚+1−𝑉_𝑚𝐸_𝐶,𝜇(𝑟)=𝑉_𝑚𝜇_𝑚+1−𝑉_𝑚𝜇_𝐶,𝜆(𝑟)=𝑉_𝑚𝜆_𝑚+1−𝑉_𝑚𝜆_𝐶,𝛼(𝑟)=𝑉_𝑚𝛼_𝑚+1−𝑉_𝑚𝛼_𝐶, (1) where the subscripts 𝑚 and 𝑐 represent the metal and ceramic, respectively. 𝑉_𝑚 is the volume fraction in the functionally graded material: 𝑉_𝑚(𝑟)=𝑟−𝑟_𝑎𝑟_𝑏−𝑟_𝑎^𝑝. (2) Here, 𝑝 is a constant material. The effects of the constant 𝑝 on the temperature, displacements, and thermal stresses are fully discussed in [6, 19]. For the sake of brevity, discussion of these effects is omitted here.

The basic piezothermoelastic equations for the 𝑖th layer can be expressed as follows.

2.1. Conduction Equations and Equilibrium Equations

The heat conduction equation can be expressed as 𝜕²𝜕𝑟²+1𝑟𝜕+1𝜕𝑟𝑟²𝜕²𝜕𝜃²+𝜕²𝜕𝑧²𝑇_𝑖=0, (3) where 𝑇_𝑖 is the temperature in the 𝑖th layer. While the equilibrium equation of the piezoelectric functionally graded cylindrical shell is 𝜕𝜎_𝑟𝑟𝑖+1𝜕𝑟𝑟𝜕𝜎_𝑟𝜃𝑖+𝜕𝜃𝜕𝜎_𝑟𝑧𝑖+𝜎𝜕𝑧^𝑟𝑟𝑖+𝜎_𝜃𝜃𝑖𝑟=0,𝜕𝜎_𝑟𝜃𝑖+1𝜕𝑟𝑟𝜕𝜎_𝜃𝜃𝑖+𝜕𝜃𝜕𝜎_𝜃𝑧𝑖+𝜕𝑧2𝜎_𝑟𝜃𝑖𝑟=0,𝜕𝜎_𝑟𝑧𝑖+1𝜕𝑟𝑟𝜕𝜎_𝜃𝑧𝑖+𝜕𝜃𝜕𝜎_𝑧𝑧𝑖+𝜎𝜕𝑧𝑟^𝑟𝑧𝑖=0, (4)

the electrostatics equation can be written as 𝜕𝐷_𝑟+𝐷𝜕𝑟𝑟^𝑟+1𝑟𝜕𝐷_𝜃+𝜕𝜃𝜕𝐷_𝑧𝜕𝑧=0. (5)

The Constitutive equation can be 𝜎_𝑟𝑟𝑖𝜎_𝜃𝜃𝑖𝜎_𝑧𝑧𝑖𝜎_𝜃𝑧𝑖𝜎_𝑟𝑧𝑖𝜎_𝑟𝜃𝑖=𝐶_11𝑖𝐶_12𝑖𝐶_13𝑖𝐶000_21𝑖𝐶_22𝑖𝐶_23𝑖𝐶000_13𝑖𝐶_23𝑖𝐶_33𝑖000000𝐶_44𝑖000000𝐶_55𝑖000000𝐶_66𝑖𝜀_𝑟𝑟𝑖𝜀_𝜃𝜃𝑖𝜀_𝑧𝑧𝑖𝜀_𝜃𝑧𝑖𝜀_𝑟𝑧𝑖𝜀_𝑟𝜃𝑖−𝛽_𝑟𝑖𝑇_𝑖𝛽_𝜃𝑖𝑇_𝑖𝛽_𝑧𝑖𝑇_𝑖000−𝑒₁𝑒00₂𝑒00₃0000000𝑒₅0𝑒₆0𝐸_𝑟𝐸_𝜃𝐸_𝑧, (6) where 𝛽_𝑟𝑖=𝐶_11𝑖+𝐶_12𝑖+𝐶_13𝑖𝛼_𝑖𝛽_𝜃𝑖=𝐶_12𝑖+𝐶_22𝑖+𝐶_23𝑖𝛼_𝑖𝛽_𝑧𝑖=𝐶_13𝑖+𝐶_23𝑖+𝐶_33𝑖𝛼_𝑖𝑖=1,2,3,…,𝑁+1, (7)

In the case of the stress-strain relations of functionally graded cylindrical shell (𝑖=1 to 𝑁), the terms 𝑒_𝑘 and 𝐸_𝑘 on the right side in (5) vanished. And 𝐶_𝑘𝑙𝑖, 𝛽 are elastic constants and thermoelastic coefficients.

The constitutive equations for the electric field are 𝐷_𝑟=𝑒₁𝜀_𝑟𝑟+𝑒₂𝜀_𝜃𝜃+𝑒₃𝜀_𝑧𝑧+𝜂₁𝐸_𝑟+𝑃₁𝑇₁,𝐷_𝜃=𝑒₆𝜀_𝑟𝜃+𝜂₂𝐸_𝜃,𝐷_𝑧=𝑒₅𝜀_𝑟𝑧+𝜂₃𝐸_𝑧,𝜀_𝑟𝑟𝑖=𝜕𝑢_𝑖𝜕𝑟;𝜀_𝜃𝜃𝑖=1𝑟𝜕𝑣_𝑖𝜕𝜃+𝑢_𝑖;𝜀_𝑧𝑧𝑖=𝜕𝑤_𝑖𝜕𝑧;𝜀_𝑟𝜃𝑖=1𝑟𝜕𝑢_𝑖+𝜕𝜃𝜕𝑣_𝑖−𝜕𝑟𝜕𝜈_𝑖𝑟;𝜀_𝜃𝑧𝑖=𝜕𝑣_𝑖+1𝜕𝑧𝑟𝜕𝑤_𝑖𝜕𝜃;𝜀_𝑧𝑟𝑖=𝜕𝑢_𝑖+𝜕𝑧𝜕𝑤_𝑖;𝐸𝜕𝑟_𝑟=−𝜕𝜙𝜕𝑟;𝐸_𝜃1=−𝑟𝜕𝜙𝜕𝜃;𝑖=1,2,3,…,𝑁+1. (8)(9)

2.2. Boundary and Continuity Conditions

Consider the following:𝜃=0,𝜃₀∶𝑇_𝑖=0𝑙𝑧=0,𝑟₀∶𝑇_𝑖=0𝑟𝑟=¹𝑟₀∶𝑇₁=𝑇_𝑎(𝜃,𝑧)𝑟𝑟=^𝑁+2𝑟₀∶𝑇_𝑁+1=𝑇_𝑏(𝜃,𝑧)𝑟𝑟=^𝑖+1𝑟₀∶𝑇_𝑖=𝑇_𝑖+1;𝜆_𝑖𝜕𝑇_𝑖𝜕𝑟=𝜆_𝑖+1𝜕𝑇_𝑖+1𝜕𝑟,𝑖=1,2,3,…,𝑁, (10a)(10b)(10c)(10d)(10e)where 𝜆_𝑖 and 𝜆_𝑖+1 are the thermal dimensionless conductivity parameter in 𝑖th and 𝑖+1th layer, respectively. 𝑇_𝑎(𝜃,𝑧) and 𝑇_𝑏(𝜃,𝑧) are temperatures in the inner and outer surfaces.

For the simply supported panel, we assumed that the edges of the piezoelectric layer are electrically grounded, so the boundary conditions and continuous conditions between layers of displacements and stresses can be expressed as𝜃=0,𝜃₀∶𝑢_𝑖=𝑤_𝑖=0;𝜎_𝜃𝜃𝑖=𝜎_𝑟𝜃𝑖=𝜎_𝜃𝑧𝑖𝑙=0;𝜙=0,𝑖=1,2,3,…,𝑁+1,𝑧=0,𝑟₀∶𝑢_𝑖=𝑣_𝑖=0;𝜎_𝑧𝑧𝑖=𝜎_𝑟𝑧𝑖=𝜎_𝜃𝑧𝑖𝑟=0;𝜙=0,𝑖=1,2,3,…,𝑁+1,𝑟=¹𝑟₀∶𝜎_𝑟,1=𝑞_𝑎(𝜃,𝑧);𝜎_𝑟𝑧,1=𝜎_𝑟𝜃,1𝑟=0𝑟=^𝑁+1𝑟₀𝑟∶𝜙=0,𝑟=^𝑁+2𝑟₀∶𝐷_𝑟=0;𝜎_{𝑟𝜃,𝑁}=𝜎_{𝑟𝑧,𝑁}=0;𝜎_{𝑟𝑟,𝑁+1}=𝑞_𝑏()𝑟𝜃,𝑧𝑟=^𝑖𝑟₀∶𝜎_𝑟𝑟𝑖=𝜎_{𝑟𝑟,𝑖+1};𝜎_𝑟𝜃𝑖=𝜎_{𝑟𝜃,𝑖+1};𝜎_𝑟𝑧𝑖=𝜎_{𝑟𝑧,𝑖+1};𝑢_𝑖=𝑢_𝑖+1;𝑣_𝑖=𝑣_𝑖+1;𝑤_𝑖=𝑤_𝑖+1,𝑖=1,2,3,…,𝑁+1+1. (11a)(11b)(11c)(11d)(11e)(11f)

To simplify the solving process of the 3D piezothermomechanical problem, the following dimensionless variables are introduced: 𝑟𝑟=𝑟₀,𝑧𝑧=𝑟₀,𝑙𝐿=𝑟₀,𝛼_𝑘=𝛼_𝑘𝛼₀,𝜆_𝑘=𝜆_𝑘𝜆₀,𝑘_𝑘=𝑘_𝑘𝑘₀,𝑇_𝑘=𝑇𝑇₀,𝑇_𝑎=𝑇_𝑎𝑇₀,𝑇_𝑏=𝑇_𝑏𝑇₀,𝑞_𝑎=𝑞_𝑎𝛼₀𝑌₀𝑇₀,𝑞_𝑏=𝑞_𝑏𝛼₀𝑌₀𝑇₀,𝜎_𝑘𝑙𝑖=𝜎_𝑘𝑙𝑖𝛼₀𝑌₀𝑇₀,𝜀_𝑘𝑙𝑖=𝜀_𝑘𝑙𝑖𝛼₀𝑇₀,𝑢_𝑖,𝑣_𝑖,𝑤_𝑖=𝑢_𝑖,𝑣_𝑖,𝑤_𝑖𝛼₀𝑟₀𝑇₀,𝛼_𝑘𝑖=𝛼_𝑘𝑖𝛼₀,𝐶_𝑘𝑙𝑖=𝐶_𝑘𝑙𝑖𝑌₀,𝐸_𝑘=𝐸_𝑘𝑑₁𝛼₀𝑇₀;𝑑_𝑘=𝐷_𝑘𝛼₀𝑇₀𝑌₀𝑑₁,𝜙_𝑘=𝜙𝑑₁𝛼₀𝑇₀𝑟₀,𝑒_𝑘=𝑒_𝑘𝑌₀𝑑₁,𝜂_𝑘=𝜂_𝑘𝑌₀𝑑₁²,𝑃₁=𝑃₁𝛼₀𝑌₀𝑑₁, (12) where 𝑟₀ is reference value of radius and 𝑌₀, 𝛼₀, 𝑘₀, 𝜆₀ and 𝑇₀ are reference values of Young’s modulus, thermal expansion coefficient, thermal diffusion coefficient, thermal conductivity coefficient, and temperature, respectively, where 𝜎_{𝑘𝑙,𝑖} are the stress components, 𝜀_{𝑘𝑙,𝑖} are the strain components, (𝑢_𝑖, 𝑣_𝑖, 𝑤_𝑖) are the displacement components, 𝐶_{𝑘𝑙,𝑖} are the elastic stiffness constants, 𝐷_𝑘 are the electric displacement components, 𝑒_𝑘 are the piezoelectric coefficients, 𝜂_𝑘 are the dielectric constants, 𝑃₁ is the pyroelectric constant, and 𝑑₁ is the piezoelectric modulus.

The dimensionless displacement-expressed equilibrium equations can be expressed as𝐶_11𝑖𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟−𝐶_66𝑖𝜕²𝑟²+𝜕𝜃𝐶_55𝑖𝜕²𝜕𝑧²−𝐶_22𝑖𝑟²𝑢_𝑖+1𝑟𝐶_12𝑖+𝐶_66𝑖𝜕²𝑣_𝑖𝜕−1𝑟𝜕𝜃𝑟²𝐶_22𝑖+𝐶_66𝑖𝜕𝑣_𝑖+𝜕𝜃𝐶_13𝑖+𝐶_55𝑖𝜕²𝑤_𝑖𝜕𝑟𝜕𝑧+𝐶_13𝑖−𝐶_23𝑖1𝑟𝜕𝑤_𝑖𝜕𝑧−𝛽_𝑟𝑖𝜕𝑇_𝑖𝜕𝑟−𝛽_𝑟𝑖−𝛽_𝜃𝑖𝑇_𝑖𝑟+𝑒₁𝜕²𝜙𝜕𝑟²+𝑒₆𝜕²𝜙𝑟²𝜕𝜃²+𝑒₅𝜕²𝜙𝜕𝑧²+𝑒₁−𝑒₂1𝑟𝜕𝜙𝜕𝑟=0,𝐶_12𝑖+𝐶_66𝑖1𝑟𝜕²𝑢_𝑖𝜕+𝑟𝜕𝜃𝐶_22𝑖+𝐶_66𝑖1𝑟²𝜕𝑢_𝑖+𝜕𝜃𝐶_66𝑖𝜕²𝑣_𝑖𝜕𝑟²+1𝑟𝜕𝑣_𝑖𝜕𝑟−𝑣_𝑖𝑟²+𝐶_22𝑖1𝑟²𝜕²𝑣_𝑖𝜕𝜃²+𝐶_44𝑖𝜕²𝑣_𝑖𝜕𝑧²+𝐶_23𝑖+𝐶_44𝑖1𝑟𝜕²𝑤_𝑖𝜕𝜃𝜕𝑧−𝛽_𝜃𝑖1𝑟𝜕𝑇_𝑖+𝜕𝜃𝑒₂+𝑒₆1𝑟𝜕²𝜙𝜕𝑟𝜕𝜃=0,𝐶_13𝑖+𝐶_55𝑖𝜕²𝑢_𝑖𝜕𝑟𝜕𝑧+𝐶_23𝑖+𝐶_55𝑖1𝑟𝜕𝑢_𝑖𝜕𝑧+𝐶_23𝑖+𝐶_44𝑖1𝑟𝜕²𝑣_𝑖𝜕𝜃𝜕𝑧+𝐶_55𝑖𝜕²𝑤_𝑖𝜕𝑟²+𝐶_55𝑖1𝑟𝜕𝑤_𝑖𝜕𝑟+𝐶_44𝑖𝜕²𝑤_𝑖𝑟²𝜕𝜃²+𝐶_33𝑖𝜕²𝑤_𝑖𝜕𝑧²−𝛽_𝑧𝑖𝜕𝑇_𝑖𝜕𝑧+𝑒₃+𝑒₆𝜕²𝜙𝜕𝑟𝜕𝑧+𝑒₅1𝑟𝜕𝜙𝜕𝑧=0. (13a)(13b)(13c)

The dimensionless displacement-expressed electrostatic equation can be expressed as 𝑒₁𝜕²𝑢_𝑖𝜕𝑟²+𝑒₁+𝑒₂1𝑟𝜕𝑢_𝑖𝜕𝑟+𝑒₆1𝑟²𝜕²𝑢_𝑖𝜕𝜃²+𝑒₅𝜕²𝑢_𝑖𝜕𝑧²+𝑒₂+𝑒₆1𝑟𝜕²𝑣_𝑖−𝜕𝑟𝜕𝜃𝑒₆1𝑟²𝜕𝑣_𝑖+𝜕𝜃𝑒₃+𝑒₅𝜕²𝑤_𝑖𝜕𝑟𝜕𝑧+𝑒₃1𝑟𝜕𝑤_𝑖𝜕𝑧−𝜂₁𝜕²𝜙𝜕𝑟²+1𝑟𝜕𝜙𝜕𝑟−𝜂₂1𝑟²𝜕²𝜙𝜕𝜃²−𝜂₃𝜕²𝜙𝜕𝑧²+𝑝₁𝜕𝑇_𝑖𝜕𝑟+1𝑟𝑇_𝑖=0. (14) Using Navier trigonometric series, solutions of (3), (13a) to (13c) and (14) which satisfy the ends displacements boundary conditions (11a) to (11b), (14) can be written as𝑇_𝑖(𝑟,𝜃⋅𝑧)=^∞∞_𝑛=1𝑚=1𝑇_𝑛𝑚𝑖(𝑟)sin(𝑎𝜃)sin(𝑏𝑧)𝑖=1,2,3,…,𝑁+1,𝑢_𝑖(𝑟,𝜃⋅𝑧)=^∞∞_𝑛=1𝑚=1𝑢_𝑛𝑚𝑖(𝑟)sin(𝑎𝜃)sin(𝑏𝑧)𝑖=1,2,3,…,𝑁+1,𝑣_𝑖(𝑟,𝜃⋅𝑧)=^∞∞_𝑛=1𝑚=1𝑣_𝑛𝑚𝑖(𝑟)cos(𝑎𝜃)sin(𝑏𝑧)𝑖=1,2,3,…,𝑁+1,𝑤_𝑖(𝑟,𝜃⋅𝑧)=^∞∞_𝑛=1𝑚=1𝑤_𝑛𝑚𝑖(𝑟)sin(𝑎𝜃)cos(𝑏𝑧)𝑖=1,2,3,…,𝑁+1,𝜙_𝑖(𝑟,𝜃⋅𝑧)=^∞∞_𝑛=1𝑚=1𝜙_𝑛𝑚𝑖(𝑟)sin(𝑎𝜃)sin(𝑏𝑧)𝑖=𝑁+1, (15a)(15b)(15c)(15d)(15e)where 𝑎=𝑚𝜋/𝜃⁰, 𝑏=𝑛𝜋/𝐿; 𝑇_𝑛𝑚𝑖, 𝑢_𝑛𝑚𝑖, 𝑣_𝑛𝑚𝑖, 𝑤_𝑛𝑚𝑖, and 𝜙_𝑛𝑚𝑖 are unknown functions, Substituting trigonometric series (15a) to (15e) into (3) and (13a) to (13c) and (14), we can obtain 𝑑²𝑇_𝑛𝑚𝑖𝑑𝑟²+1𝑟𝑑𝑇_𝑛𝑚𝑖𝑑𝑟−𝑎²𝑟²𝑇_𝑛𝑚𝑖−𝑏²𝑇_𝑛𝑚𝑖=0,𝑖=1,2,3,…,𝑁+1, (16)𝐶_11𝑖𝑑²𝑑𝑟²+1𝑟𝑑𝑑𝑟−𝑎²𝐶_66𝑖1𝑟²−𝑏²𝐶_55𝑖−𝐶_22𝑖1𝑟²𝑢_𝑛𝑚𝑖−𝑎𝐶_12𝑖+𝐶_66𝑖1𝑟𝑑𝑣_𝑛𝑚𝑖𝑑𝑟+𝑎𝐶_22𝑖+𝐶_66𝑖1𝑟²𝑣_𝑛𝑚𝑖−𝑏𝐶_13𝑖+𝐶_55𝑖𝑑𝑑𝑟+𝐶_13𝑖−𝐶_23𝑖1𝑟𝑤_𝑛𝑚𝑖−𝛽_𝑟𝑖𝑑𝑇_𝑛𝑚𝑖𝑑𝑟−𝛽_𝑟𝑖−𝛽_𝜃𝑖𝑇_𝑛𝑚𝑖𝑟+𝑒₁𝑑²𝜙_𝑛𝑚𝑖𝑑𝑟²−𝑎²𝑒₆1𝑟²𝜙_𝑛𝑚𝑖−𝑏²𝑒₅𝜙_𝑛𝑚𝑖+𝑒₁−𝑒₂1𝑟𝑑𝜙_𝑛𝑚𝑖𝑑𝑟𝑎=0,𝐶_12𝑖+𝐶_66𝑖1𝑟𝑑𝑑𝑟+𝐶_22𝑖+𝐶_66𝑖1𝑟²𝑢_𝑛𝑚𝑖+𝐶_66𝑖𝑑²𝑑𝑟²+1𝑟𝑑𝑑𝑟−1𝑟²𝑣_𝑛𝑚𝑖−𝑎²𝐶_22𝑖1𝑟²𝑣_𝑛𝑚𝑖−𝑏²𝐶_44𝑖𝑣_𝑛𝑚𝑖−𝑎𝑏𝐶_23𝑖+𝐶_44𝑖1𝑟𝑤_𝑛𝑚𝑖−𝑎𝛽_𝜃𝑖1𝑟𝑇_𝑛𝑚𝑖+𝑎𝑒₂+𝑒₆1𝑟𝜙_𝑛𝑚𝑖𝑏=0,𝐶_13𝑖+𝐶_55𝑖𝑑𝑑𝑟+𝐶_23𝑖+𝐶_55𝑖1𝑟𝑢_𝑛𝑚𝑖−𝑎𝑏𝐶_23𝑖+𝐶_44𝑖1𝑟𝑣_𝑛𝑚𝑖+𝐶_55𝑖𝑑²𝑤_𝑛𝑚𝑖𝑑𝑟²+𝐶_55𝑖1𝑟𝑑𝑑𝑟−𝛼²𝐶_44𝑖1𝑟²−𝑏²𝐶_33𝑖𝑤_𝑛𝑚𝑖−𝑏𝛽_𝑧𝑖𝑇_𝑛𝑚𝑖+𝑏𝑒₃+𝑒₆𝑑𝑑𝑟+𝑒₅1𝑟𝜙_𝑛𝑚𝑖=0, (17a)(17b)(17c)𝑒₁𝑑²𝑑𝑟²+𝑒₁+𝑒₂1𝑟𝑑𝑑𝑟−𝑎²𝑒₆1𝑟²−𝑏²𝑒₅𝑢_𝑛𝑚𝑖−𝑎𝑒₂+𝑒₆1𝑟𝑑𝑑𝑟−𝑒₆1𝑟²𝑣_𝑛𝑚𝑖−𝑏𝑒₃1𝑟𝑤_𝑛𝑚𝑖−𝑏𝑒₃+𝑒₅𝑑𝑤_𝑛𝑚𝑖𝑑𝑟−𝜂₁𝑑²𝑑𝑟²+1𝑟𝑑𝑑𝑟−𝑎²𝜂₂1𝑟²−𝑏²𝜂₃𝜙_𝑛𝑚𝑖+𝑝₁𝑑𝑑𝑟+1𝑟𝑇_𝑛𝑚𝑖=0. (18)

Furthermore, substituting the trigonometric series (15a) to (15e) into boundary and continuous conditions (10c) to (10e) and (11c) to (11f), we can obtain𝑟𝑟=¹𝑟₀∶𝑇_𝑛𝑚1=𝑇_𝑛𝑚𝑎,𝑟𝑟=^𝑁+2𝑟₀∶𝑇_{𝑛𝑚,𝑁+1}=𝑇_𝑛𝑚𝑏,𝑟𝑟=^𝑖+1𝑟₀∶𝑇_𝑛𝑚𝑖=𝑇_{𝑛𝑚,𝑖+1};𝜆_𝑖𝑑𝑇_𝑛𝑚𝑖𝑑𝑟=𝜆_𝑖+1𝜕𝑇_{𝑛𝑚,𝑖+1}𝜕𝑟 (19a)(19b)(19c)𝑟𝑟=¹𝑟₀∶𝐶₁₁₁𝑑𝑢_𝑛𝑚1𝑑𝑟+𝐶₁₂₁𝑢_𝑛𝑚1𝑟−𝑎𝐶₁₂₁𝑣_𝑛𝑚1𝑟−𝑏𝐶₁₃₁𝑤_𝑛𝑚1−𝛽_𝑟1𝑇_𝑛𝑚1=𝑞_𝑛𝑚𝑎𝑏𝑢_𝑛𝑚1+𝑑𝑤_𝑛𝑚1𝑑𝑟𝑎=0,𝑢_𝑛𝑚1𝑟+𝑑𝑣_𝑛𝑚1𝑑𝑟−𝑣_𝑛𝑚1𝑟=0𝑟𝑟=^𝑁+2𝑟₀∶𝜙_{𝑛𝑚,𝑁+1}=0,𝑒₁𝑑𝑢_𝑛𝑚𝑖𝑑𝑟+𝑒₂𝑟𝑢_𝑛𝑚𝑖−𝑎𝑣_𝑛𝑚𝑖−𝑏𝑒₃𝑤_𝑛𝑚𝑖−𝜂₁𝑑𝜙_𝑛𝑚𝑖𝑑𝑟+𝑝₁𝑇_𝑛𝑚𝑖=0𝐶_11𝑖𝑑𝑢_𝑛𝑚𝑖𝑑𝑟+𝐶_12𝑖−𝑎𝑟𝑣_𝑛𝑚𝑖+𝑢_𝑛𝑚𝑖−𝑏𝐶_13𝑖𝑤_𝑛𝑚𝑖−𝛽_𝑟𝑖𝑇_𝑛𝑚𝑖+𝑒₁𝑑𝜙_𝑛𝑚𝑖𝑑𝑟=𝑞_𝑛𝑚𝑏𝐶_66𝑖𝑎𝑟𝑢_𝑛𝑚𝑖+𝑑𝑣_𝑛𝑚𝑖𝑑𝑟−𝑣_𝑛𝑚𝑖𝑟+𝑎𝑒₆𝑟𝜙_𝑛𝑚𝑖=0,𝐶_55𝑖𝑑𝑤_𝑛𝑚𝑖𝑑𝑟+𝑏𝑢_𝑛𝑚𝑖+𝑏𝑒₅𝜙_𝑛𝑚𝑖=0𝑟𝑟=^𝑖+1𝑟₀∶𝑢_𝑛𝑚𝑖=𝑢_{𝑛𝑚,𝑖+1},𝑣_𝑛𝑚𝑖=𝑣_{𝑛𝑚,𝑖+1},𝑤_𝑛𝑚𝑖=𝑤_{𝑛𝑚,𝑖+1}𝐶_11𝑖𝑑𝑢_𝑛𝑚𝑖𝑑𝑟+𝐶_12𝑖−𝑎𝑟𝑣_𝑛𝑚𝑖+𝑢_𝑛𝑚𝑖−𝑏𝐶_13𝑖𝑤_𝑛𝑚𝑖−𝛽_𝑟𝑖𝑇_𝑛𝑚𝑖=𝐶_11,𝑖+1𝑑𝑢_{𝑛𝑚,𝑖+1}𝑑𝑟+𝐶_12,𝑖+1−𝑎𝑟𝑣_{𝑛𝑚,𝑖+1}+𝑢_{𝑛𝑚,𝑖+1}−𝑏𝐶_13,𝑖+1𝑤_{𝑛𝑚,𝑖+1}−𝛽_𝑟,𝑖+1𝑇_{𝑛𝑚,𝑖+1}𝐶_66𝑖𝑎𝑟𝑢_𝑛𝑚𝑖+𝑑𝑣_𝑛𝑚𝑖𝑑𝑟−𝑣_𝑛𝑚𝑖𝑟=𝐶_66,𝑖+1𝑎𝑟𝑢_{𝑛𝑚,𝑖+1}+𝑑𝑣_{𝑛𝑚,𝑖+1}𝑑𝑟−𝑣_{𝑛𝑚,𝑖+1}𝑟𝐶_55𝑖𝑑𝑤_𝑛𝑚𝑖𝑑𝑟+𝑏𝑢_𝑛𝑚𝑖=𝐶_55,𝑖+1𝑑𝑤_{𝑛𝑚,𝑖+1}𝑑𝑟+𝑏𝑢_{𝑛𝑚,𝑖+1}. (20a)(20b)(20c)

Using the orthogonality of trigonometric functions, we can obtain 𝑇_𝑛𝑚𝑎𝑇_𝑛𝑚𝑏𝑞_𝑛𝑚𝑎𝑞_𝑛𝑚𝑏=4𝑟₀𝜃₀𝑙^𝜃0₀^𝑙/𝑟0₀𝑇_𝑎(𝜃,𝑧)𝑇_𝑏(𝜃,𝑧)𝑞_𝑎(𝜃,𝑧)𝑞_𝑏(𝜃,𝑧)sin(𝑎𝜃)sin(𝑏𝑧)𝑑𝜃𝑑𝑧. (21)

3. Theoretical Analysis

Based on the series solving method of ordinary differential equations, if the coefficient items of the ordinary differential equations were analytical at point (𝑟−1) and could be expressed as Taylor’s series of (𝑟−1), the solutions of (17a)–(17c) and (18) can also be expressed as the following Taylor’s series [17]: 𝑇_𝑛𝑚𝑖(𝑟)=^∞_𝑘=1𝐴_𝑘𝑖(𝑠)(𝑟−1)^𝑘𝑖=1,𝑁+1𝑢_𝑛𝑚𝑖(𝑟)=^∞_𝑘=1𝐵_𝑘𝑖(𝑠)(𝑟−1)^𝑘𝑖=1,𝑁+1𝑣_𝑛𝑚𝑖(𝑟)=^∞_𝑘=1𝐶_𝑘𝑖(𝑠)(𝑟−1)^𝑘𝑖=1,𝑁+1𝑤_𝑛𝑚𝑖(𝑟)=^∞_𝑘=1𝐷_𝑘𝑖(𝑠)(𝑟−1)^𝑘𝑖=1,𝑁+1𝜙_𝑛𝑚𝑖(𝑟)=^∞_𝑘=1𝐹_𝑘𝑖(𝑠)(𝑟−1)^𝑘𝑖=𝑁+1. (22)

Substituting series (22) into (16) to (18), comparing the coefficient of (𝑟−1)^𝑘, we can obtain the following recurrence equation: −(𝑘+2)(𝑘+1)𝐴_𝑘+2,𝑖=(2𝑘+1)(𝑘+1)𝐴_𝑘+1,𝑖+𝑘²𝐴_𝑘𝑖−𝑎²+𝑏²𝐴_𝑘𝑖−2𝑏²𝑘_𝑖𝜆_𝑖𝐴_{𝑘−1,𝑖}−𝑏²𝑘_𝑖𝜆_𝑖𝐴_{𝑘−2,𝑖} (23)−(𝑘+2)(𝑘+1)𝐵_𝑘+2,𝑖=(2𝑘+1)(𝑘+1)𝐵_𝑘+1,𝑖+𝑘²−𝐶_22𝑖+𝑏²𝐶_55𝑖+𝑎²𝐶_66𝑖𝐶_11𝑖𝐵_𝑘𝑖−2𝑏²𝐶_55𝑖𝐶_11𝑖𝐵_{𝑘−1,𝑖}−𝑏²𝐶_55𝑖𝐶_11𝑖𝐵_{𝑘−2,𝑖}−𝑎𝐶_12𝑖+𝐶_66𝑖𝐶_11𝑖(𝑘+1)𝐶_𝑘+1,𝑖+𝑎𝐶_22𝑖+𝐶_66𝑖𝐶_11𝑖𝐶_𝑘𝑖−𝑎𝑘𝐶_12𝑖+𝐶_66𝑖𝐶_11𝑖𝐶_𝑘𝑖−𝑏𝐶_13𝑖+𝐶_55𝑖𝐶_11𝑖(𝑘+1)𝐷_𝑘+1,𝑖𝑘−2𝑏𝐶_13𝑖+𝐶_55𝑖𝐶_11𝑖𝐷_𝑘,𝑖−𝑏𝐶_13𝑖−𝐶_23𝑖𝐶_11𝑖𝐷_𝑘,𝑖−𝛽_𝑟𝑖𝐶_11𝑖(𝑘+1)𝐴_𝑘+1,𝑖()−𝑏𝑘−1𝐶_13𝑖+𝐶_55𝑖𝐶_11𝑖+𝐶_13𝑖−𝐶_23𝑖𝐶_11𝑖𝐷_{𝑘−1,𝑖}−(2𝑘+1)𝛽_𝑟𝑖−𝛽_𝜃𝑖𝐶_11𝑖𝐴_𝑘,𝑖−𝑘𝛽_𝑟𝑖−𝛽_𝜃𝑖𝐶_11𝑖𝐴_{𝑘−1,𝑖}+(𝑘+2)(𝑘+1)𝑒₁𝐶_11𝑖𝐹_𝑘+2,𝑖+(𝑘+1)(2𝑘+1)𝑒₁−𝑒₂𝐶_11𝑖𝐹_𝑘+1,𝑖+1𝐶_11𝑖𝑘²𝑒₁−𝑘𝑒₂−𝑎²𝑒₆𝐹_𝑘,𝑖−𝑏²𝑒₅𝐶_11𝑖𝐹_𝑘,𝑖−𝑏²𝑒₅𝐶_11𝑖2𝐹_{𝑘−1,𝑖}+𝐹_{𝑘−2,𝑖}−(𝑘+2)(𝑘+1)𝐶_𝑘+2,𝑖=(2𝑘+1)(𝑘+1)𝐶_𝑘+1,𝑖+𝑘²−𝑎²𝐶_22𝑖+𝑏²𝐶_44𝑖𝐶_66𝑖−1𝐶_𝑘,𝑖𝑏−2²𝐶_44𝑖𝐶_66𝑖𝐶_{𝑘−1,𝑖}−𝑏²𝐶_44𝑖𝐶_66𝑖𝐶_{𝑘−2,𝑖}+𝑎𝐶_12𝑖+𝐶_66𝑖𝐶_66𝑖(𝑘+1)𝐵_𝑘+1,𝑖+𝑎𝑘𝐶_12𝑖+𝐶_66𝑖𝐶_66𝑖𝐵_𝑘,𝑖+𝑎𝐶_22𝑖+𝐶_66𝑖𝐶_66𝑖𝐵_𝑘,𝑖−𝑎𝑏𝐶_23𝑖+𝐶_44𝑖𝐶_66𝑖𝐷_𝑘𝑖+𝐷_{𝑘−1,𝑖}−𝑎𝛽_𝜃𝑖𝐶_66𝑖𝐴_𝑘𝑖+𝐴_{𝑘−1,𝑖}−𝑎𝑒₂+𝑒₆𝐶_66𝑖(𝑘+1)𝐹_𝑘+1,𝑖+𝑘𝐹_𝑘𝑖−(𝑘+2)(𝑘+1)𝐷_𝑘+2,𝑖=(2𝑘+1)(𝑘+1)𝐷_𝑘+1,𝑖+𝑘²−𝑎²𝐶_44𝑖+𝑏²𝐶_33𝑖𝐶_55𝑖𝐷_𝑘𝑖−2𝑏²𝐶_33𝑖𝐶_55𝑖𝐷_{𝑘−1,𝑖}−𝑏²𝐶_33𝑖𝐶_55𝑖𝐷_{𝑘−2,𝑖}+𝑏𝐶_13𝑖+𝐶_55𝑖𝐶_55𝑖(𝑘+1)𝐵_𝑘+1,𝑖+𝑏𝐶_55𝑖2𝑘𝐶_13𝑖+𝐶_55𝑖+𝐶_23𝑖+𝐶_55𝑖𝐵_𝑘,𝑖−𝑎𝑏𝐶_23𝑖+𝐶_44𝑖𝐶_55𝑖𝐶_𝑘,𝑖+𝐶_{𝑘−1,𝑖}−𝑏𝛽_𝑧𝑖𝐶_55𝑖𝐴_𝑘,𝑖+2𝐴_{𝑘−1,𝑖}+𝐴_{𝑘−2,𝑖}+(2𝑘+1)𝑏𝑒₅𝐶_55𝑖𝐹_𝑘,𝑖+2𝑏𝑘𝑒₃𝐶_55𝑖𝐹_𝑘,𝑖+𝑏𝐶_55𝑖(𝑘−1)𝑒₃+𝑘𝑒₅𝐹_{𝑘−1,𝑖}(𝑘+2)(𝑘+1)𝐹_𝑘+2,𝑖=−(2𝑘+1)(𝑘+1)𝐹_𝑘+1,𝑖−𝑘²+𝑎²𝜂₂𝜂₁+𝑏²𝜂₃𝜂₁𝐹_𝑘𝑖+𝑏²𝜂₃𝜂₁2𝐹_{𝑘−1,𝑖}+𝐹_{𝑘−2,𝑖}+𝑝₁𝜂₁(𝑘+1)𝐴_𝑘+1,𝑖+(2𝑘+1)𝐴_𝑘,𝑖+𝑘𝐴_{𝑘−1,𝑖}+𝑒₁𝜂₁(𝑘+2)(𝑘+1)𝐵_𝑘+2,𝑖+𝑒₁𝜂₁(2𝑘+1)(𝑘+1)𝐵_𝑘+1,𝑖+𝑒₂𝜂₁(𝑘+1)𝐵_𝑘+1,𝑖+1𝜂₁𝑒₁𝑘²+𝑒₂𝑘−𝑎²𝑒₆−𝑏²𝑒₅𝐵_𝑘,𝑖𝑏−2²𝑒₅𝜂₁𝐵_{𝑘−1,𝑖}−𝑎𝑒₂+𝑒₆𝜂₁(𝑘+1)𝐶_𝑘+1,𝑖−𝑏²𝑒₅𝜂₁𝐵_{𝑘−2,𝑖}−𝑎𝑒₂𝑘+𝑎𝑒₆(𝑘−1)𝜂₁𝐶_𝑘,𝑖−𝑏𝑒₃+𝑒₅𝜂₁(𝑘+1)𝐷_𝑘+1,𝑖−1𝜂₁𝑏𝑒₃(2𝑘+1)𝐷_𝑘,𝑖−2𝑏𝑒₅𝑘𝜂₁𝐷_𝑘,𝑖−𝑏𝑘𝑒₃𝜂₁𝐷_{𝑘−1,𝑖}−𝑏(𝑘−1)𝑒₅𝜂₁𝐷_{𝑘−1,𝑖}, (24a)(24b)(24c)(24d)where 𝐴_−1,𝑖=𝐴_−2,𝑖=0. From the recurrence equations (23) and (24a) to (24d), we can see that all coefficients 𝐴_𝑘𝑖, 𝐵_𝑘𝑖, 𝐶_𝑘𝑖, 𝐷_𝑘𝑖, and 𝐹_𝑘𝑖 in Taylor’s series can be briefly expressed by 𝐴_𝑜𝑖, 𝐴_1𝑖, 𝐵_𝑜𝑖, 𝐵_1𝑖; 𝐶_𝑜𝑖, 𝐶_1𝑖; 𝐷_𝑜𝑖, 𝐷_1𝑖; 𝐹_𝑜𝑖, and 𝐹_1𝑖: 𝐴_𝑘𝑖=𝜉^𝑘_1𝑖𝐴_0𝑖+𝜉^𝑘_2𝑖𝐴_1𝑖;𝐵_𝑘𝑖=𝜁^𝑘_1𝑖𝐵_0𝑖+𝜁^𝑘_2𝑖𝐵_1𝑖+𝜁^𝑘_3𝑖𝐶_0𝑖+𝜁^𝑘_4𝑖𝐶_1𝑖+𝜁^𝑘_5𝑖𝐷_0𝑖+𝜁^𝑘_6𝑖𝐷_1𝑖+𝜁^𝑘_7𝑖𝐹_0𝑖+𝜁^𝑘_8𝑖𝐹_1𝑖+𝜁^𝑘_9𝑖𝐴_0𝑖+𝜁^𝑘_10𝑖𝐴_1𝑖,𝐶_𝑘𝑖=𝐿^𝑘_1𝑖𝐵_0𝑖+𝐿^𝑘_2𝑖𝐵_1𝑖+𝐿^𝑘_3𝑖𝐶_0𝑖+𝐿^𝑘_4𝑖𝐶_1𝑖+𝐿^𝑘_5𝑖𝐷_0𝑖+𝐿^𝑘_6𝑖𝐷_1𝑖+𝐿^𝑘_7𝑖𝐹_0𝑖+𝐿^𝑘_8𝑖𝐹_1𝑖+𝐿^𝑘_9𝑖𝐴_0𝑖+𝐿^𝑘_10𝑖𝐴_1𝑖,𝐷_𝑘𝑖=𝜌^𝑘_1𝑖𝐵_0𝑖+𝜌^𝑘_2𝑖𝐵_1𝑖+𝜌^𝑘_3𝑖𝐶_0𝑖+𝜌^𝑘_4𝑖𝐶_1𝑖+𝜌^𝑘_5𝑖𝐷_0𝑖+𝜌^𝑘_6𝑖𝐷_1𝑖+𝜌^𝑘_7𝑖𝐹_0𝑖+𝜌^𝑘_8𝑖𝐹_1𝑖+𝜌^𝑘_9𝑖𝐴_0𝑖+𝜌^𝑘_10𝑖𝐴_1𝑖,𝐹_𝑘𝑖=𝛾^𝑘_1𝑖𝐵_0𝑖+𝛾^𝑘_2𝑖𝐵_1𝑖+𝛾^𝑘_3𝑖𝐶_0𝑖+𝛾^𝑘_4𝑖𝐶_1𝑖+𝛾^𝑘_5𝑖𝐷_0𝑖+𝛾^𝑘_6𝑖𝐷_1𝑖+𝛾^𝑘_7𝑖𝐹_0𝑖+𝛾^𝑘_8𝑖𝐹_1𝑖+𝛾^𝑘_9𝑖𝐴_0𝑖+𝛾^𝑘_10𝑖𝐴_1𝑖, (25a)(25b)(25c)(25d)(25e)where the coefficients 𝜉^𝑘_1𝑖, 𝜉^𝑘_2𝑖; 𝜁^𝑘_1𝑖∼𝜁^𝑘_10𝑖; 𝐿^𝑘_1𝑖∼𝐿^𝑘_10𝑖; 𝜌^𝑘_1𝑖∼𝜌^𝑘_10𝑖 and 𝛾^𝑘_1𝑖∼𝛾^𝑘_10𝑖 can be derived from the recurrence equations (23) and (24a) to (24d). Substituting the solved series coefficients (25a) to (25d) into Taylor series (22), we can obtain 𝑇_𝑛𝑚𝑖(𝑟), 𝑢_𝑛𝑚𝑖(𝑟), 𝑣_𝑛𝑚𝑖(𝑟), 𝑤_𝑛𝑚𝑖(𝑟) and 𝜙_𝑛𝑚𝑖(𝑟).

Where 𝐴_0𝑖, 𝐴_1𝑖; 𝐵_0𝑖, 𝐵_1𝑖; 𝐶_0𝑖, 𝐶_1𝑖; 𝐷_0𝑖, 𝐷_1𝑖; 𝐹_0𝑖 and 𝐹_1𝑖 are unknown constants and they can be determined by substituting solutions (24a)–(24d) into the boundary and continuous conditions (19a) to (19c) and (20a) to (20c).

Substituting solutions 𝑇_𝑛𝑚𝑖(𝑟), 𝑢_𝑛𝑚𝑖(𝑟), 𝑣_𝑛𝑚𝑖(𝑟), 𝑤_𝑛𝑚𝑖(𝑟), and 𝜙_𝑛𝑚𝑖(𝑟) into trigonometric series (15a)–(15e), we can obtain the steady-state piezothermomechanical solution for the finite cylindrical composite panel. Furthermore, using geometric equation and constitutive equation, the analytical solutions of thermo-mechanical stresses for the laminated cylinder can be obtained.

4. Numerical Results and Discussion

To illustrate the above analysis, we investigate the piezothermoelastic stresses in a functionally graded cylindrical shell with a piezoelectric layer on outside. The dimensions 𝐿=6, 𝜃₀=𝜋/3, 𝑅_𝑎=0.95, and 𝑅_𝑏=1.042, the thickness of piezoelectric layer is 0.002, the thickness of each layer is assumed to be equivalent. The reference values of temperature, Young’s modulus, and thermal expansion coefficient are 𝑇₀=200 K, 𝐸₀=300 GPa, and 𝛼₀=4.9×10⁻⁵ K⁻¹, respectively.

In this paper, the functionally graded finite cylinder made of mullite and molybdenum is analyzed. The thermoelastic properties of mullite and molybdenum are given as follows [11, 19]: 𝐸_𝑚=330GPa,𝜇_𝑚=0.3;𝜆_𝑚𝛼=138W/mK;_𝑚=4.9×10⁻⁶𝐸/K_𝑐=330GPa,𝜇_𝑐=0.3;𝜆_𝑐𝛼=138W/mK;_𝑐=4.9×10⁻⁶/K. (26)

The material properties of piezoelectric layer (cadmium selenite) are summarized as 𝐶₁₁=83.6GPa,𝐶₂₂=𝐶₃₃𝐶=74.1GPa₂₃=45.2GPa,𝐶₁₂=𝐶₁₃𝐶=39.3GPa,₆₆=13.17GPa,𝑒₁=0.347C/m²,𝑒₂=𝑒₃=−0.16C/m²,𝑒₅=𝑒₆=−0.138C/m²,𝜂₁=9.03×10⁻¹¹C²/Nm²,𝜂₂=𝜂₃=8.25×10⁻¹¹C²/Nm²,𝑃₁=−2.94×10⁻⁶C/m²𝑑K,₁=−3.92×10⁻¹²C/N,𝜆=8.6W/mK. (27)

And it is assumed that the cylindrical shell is subjected to the following nonumiform thermal and mechanical loadings on its outer surfaces: 𝑇_𝑎(𝜃,𝑧)=0at𝑅_𝑎𝑇=0.95_𝑏(𝜃,𝑧)=200×sin[3𝜃]×sin𝜋𝑧6at𝑅_𝑏𝑞=1.04_𝑎(𝜃,𝑧)=0at𝑅_𝑎𝑞=0.95_𝑏(𝜃,𝑧)=−10⁸×sin[3𝜃]×sin𝜋𝑧6at𝑅_𝑏=1.042. (28)

Figures 2 to 8 illustrate the numerical results of dimensionless temperature, displacement, and stresses of functionally graded cylindrical shell with piezoelectric layer.

(a)

(b)

Figure 2 

Dimensionless temperature distribution: (a) at section 𝜃=𝜋/6, (b) 𝑅=1.042.

Figure 2 shows the numerical results of dimensionless temperature distributions. Due to the graded variation of the material properties along the radial direction, the temperature reduced in a nonlinear form through thickness direction. For the thermal load is a function of sin, the maximum temperature in each layer is located along the axial 𝑧=3.

Figure 3 shows the axial displacement; the magnitude of axial displacement becomes larger near the outer surface. Figures 4 to 12 show the numerical results of dimensionless radial, hoop, axial, and shear stresses, respectively. The radial stress increases from the inner surface to outer surface nonlinearly; the radial stress is compressive for the inner heating and pressure loads.

(a)

(b)

Figure 3 

Dimensionless axial displacement distribution: (a) at section 𝜃=𝜋/6, (b) 𝑅=1.042.

(a)

(b)

Figure 4 

Distribution of dimensionless radial stress: (a) 𝑍=3.0, 𝜃=𝜋/6, (b) 𝜃=𝜋/6.

Figure 5(a) shows the variation of circumferential stress on the thickness direction at point 𝑍=3.0, 𝜃=𝜋/6. Due to the assumed thermomechanical loadings, the circumferential stress is compressive stress at the inner surface and is tensile stress at the outer surface. Because of the mismatch material properties, the circumferential stress is not continuous across the interfaces. Due to the assumed thermomechanical loading, the distribution of circumferential stress is nonuniform. It can be seen that maximum values of the circumferential stress occur at the same surface where the temperature loadings are applied. Similar distributions of circumferential stress can be obtained at other sections in the thickness direction. For the sake of brevity, the distributions in other thickness section are omitted here. Figure 6(a) shows the distributions of axial stress on the variation of thickness direction at 𝑍=3 and 𝜃=𝜋/6. in Figure 6(b), due to the assumed boundary condition and thermomechanical loading, the axial stress is tensile stress at the inner surface and varies from compressive to tensile stress, which is similar to circumferential stress.

(a)

(b)

Figure 5 

Distribution of dimensionless circumferential stress: (a) 𝑍=3.0, 𝜃=𝜋/6, (b) 𝜃=𝜋/6.

(a)

(b)

Figure 6 

Distribution of dimensionless axial stress: (a) at point 𝑍=3.0, 𝜃=𝜋/6. (b) 𝜃=𝜋/6.

Figures 7 to 9 show the distributions of three shear stresses at different sections. It is seen that the maximum values of all three shear stresses are smaller than those of normal stresses. Due to the assumed thermo-mechanical loadings, distributions of the three shear stresses show different distribution. According to the boundary condition and material properties, the maximum shear stresses 𝜎_𝑅𝜃 and 𝜎_𝑅𝑍 appear at the middle radius, while the shear stress 𝜎_𝜃𝑍 varies from compressive stress to tensile stress along its length. The maximum shear stress 𝜎_𝑅𝑍 appears at the middle along the length and the middle thickness. Figures 10, 11, and Figure 12 show the variation of electric potential and displacement along the thickness; the electric potential distribution is similar to the temperature distribution. The electric displacement varied nonlinearly.

Figure 7 

Distribution of dimensionless shear stress 𝜎𝜃𝑍 (𝑅=1.04).

(a)

(b)

Figure 8 

Distribution of dimensionless shear stress 𝜎𝑅𝑍: (a) 𝑍=3.0, (b) at point 𝑍=3.0, 𝜃=𝜋/6.

Figure 9 

Distribution of dimensionless shear stress (𝑍=3).

(a)

(b)

Figure 10 

Distribution of dimensionless electric potential: (a) at section 𝑍=3, (b) 𝑅=1.04.

(a)

(b)

Figure 11 

Distribution of dimensionless electric displacement 𝐷𝑅: (a) at section 𝑅=1.04, (b) 𝑍=3.

(a)

(b)

Figure 12 

Distribution of dimensionless electric displacement 𝐷𝜃: (a) at section 𝑅=1.04, (b) 𝑍=3.

5. Conclusions

The piezothermoelastic analysis for a functionally graded cylindrical shell with piezoelectric layer is presented in this paper. The cylindrical shell is simply supported at four end edges and subjected to thermomechanical loadings on its outer surfaces. The piezoelectric layers are polarized along radial direction as a sensor. An exact 3D analytical solution is derived by using multilayered approximate approach and series solving method. Referred to the presented solution, other temperature boundary conditions can be solved by suitable forms of series solutions to satisfy both temperature and simply supported boundary conditions. It is also worth noting that the trigonometric series, which are used to separate variables, are only suitable for simply supported at four end edges. For other supported boundary conditions, another suitable series form must be considered to satisfy the related boundary conditions. For different functionally graded material, this analytical method can also be used.

Acknowledgment

This project is supported by the National Natural Science Foundation of China (Grant nos. 51375396 and 11202164).