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

Authors: Yong-gang Kang; Zhong-qi Wang; Gongnan Xie
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 ofN 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 lengthl, internal radius ra, external radius rb, 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 rc, and the radius of the interface between the ith and (i+1)th layer is ri. 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 Ta(θ,z) and Tb(θ,z) on the inner and outer surface. The temperatures at end edges keep zero. Nonuniform pressure loadings qa(θ,z) and  qb(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 ofN 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
(1)E(r)=VmEm+(1-Vm)EC,μ(r)=Vmμm+(1-Vm)μC,λ(r)=Vmλm+(1-Vm)λC,α(r)=Vmαm+(1-Vm)αC,
where the subscripts m and c represent the metal and ceramic, respectively. Vm is the volume fraction in the functionally graded material:
(2)Vm(r)=(r-rarb-ra)p.
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 theith layer can be expressed as follows.


### 2.1. Conduction Equations and Equilibrium Equations

The heat conduction equation can be expressed as(3)(∂2∂r2+1r∂∂r+1r2∂2∂θ2+∂2∂z2)Ti=0,
where T¯i is the temperature in the ith layer. While the equilibrium equation of the piezoelectric functionally graded cylindrical shell is
(4)∂σrri∂r+1r∂σrθi∂θ+∂σrzi∂z+σrri+σθθir=0,∂σrθi∂r+1r∂σθθi∂θ+∂σθzi∂z+2σrθir=0,∂σrzi∂r+1r∂σθzi∂θ+∂σzzi∂z+σrzir=0,the electrostatics equation can be written as(5)∂Dr∂r+Drr+1r∂Dθ∂θ+∂Dz∂z=0.The Constitutive equation can be(6){σrriσθθiσzziσθziσrziσrθi}=[C11iC12iC13i000C21iC22iC23i000C13iC23iC33i000000C44i000000C55i000000C66i]{εrriεθθiεzziεθziεrziεrθi}-{βriTiβθiTiβziTi000}-(e100e200e30000000e50e60){ErEθEz},
where
(7)βri=(C11i+C12i+C13i)αiβθi=(C12i+C22i+C23i)αiβzi=(C13i+C23i+C33i)αi2i=1,2,3,…,N+1,In the case of the stress-strain relations of functionally graded cylindrical shell (i=1 to N), the terms ek and Ek on the right side in (5) vanished. And Ckli, β are elastic constants and thermoelastic coefficients.The constitutive equations for the electric field are(8)Dr=e1εrr+e2εθθ+e3εzz+η1Er+P1T1,Dθ=e6εrθ+η2Eθ,Dz=e5εrz+η3Ez,(9)εrri=∂ui∂r;εθθi=1r(∂vi∂θ+ui);εzzi=∂wi∂z;εrθi=1r∂ui∂θ+∂vi∂r-∂νir;εθzi=∂vi∂z+1r∂wi∂θ;εzri=∂ui∂z+∂wi∂r;Er=-∂ϕ∂r;Eθ=-1r∂ϕ∂θ;i=1,2,3,…,N+1.


### 2.2. Boundary and Continuity Conditions

Consider the following:(10a)θ=0,θ0:Ti=0(10b)z¯=0,lr0:Ti=0(10c)r¯=r1r0:T1=Ta(θ,z¯)(10d)r¯=rN+2r0:TN+1=Tb(θ,z¯)(10e)r¯=ri+1r0:Ti=Ti+1;λi∂Ti∂r=λi+1∂Ti+1∂r,i=1,2,3,…,N,where λi and λi+1 are the thermal dimensionless conductivity parameter in ith and i+1th layer, respectively. Ta(θ,z) and Tb(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 as(11a)θ=0,θ0:ui=wi=0;σθθi=σrθi=σθzi=0;ϕ=0,i=1,2,3,…,N+1,(11b)z=0,lr0:ui=vi=0;σzzi=σrzi=σθzi=0;ϕ=0,i=1,2,3,…,N+1,(11c)r=r1r0:σr,1=qa(θ,z);σrz,1=σrθ,1=0(11d)r=rN+1r0:ϕ=0,(11e)r=rN+2r0:Dr=0;σrθ,N=σrz,N=0;σrr,N+1=qb(θ,z)r=rir0:σrri=σrr,i+1;σrθi=σrθ,i+1;σrzi=σrz,i+1;(11f)ui=ui+1;vi=vi+1;wi=wi+1,2222222i=1,2,3,…,N+1+1.To simplify the solving process of the 3D piezothermomechanical problem, the following dimensionless variables are introduced:(12)r¯=rr0,z¯=zr0,L¯=lr0,αk¯=αkα0,λk¯=λkλ0,kk¯=kkk0,Tk¯=TT0,Ta¯=TaT0,Tb¯=TbT0,q¯a=qaα0Y0T0,q¯b=qbα0Y0T0,σ¯kli=σkliα0Y0T0,ε¯kli=εkliα0T0,{u¯i,v¯i,w¯i}={ui,vi,wi}α0r0T0,α¯ki=αkiα0,C¯kli=CkliY0,E¯k=Ek|d1|α0T0;d¯k=Dkα0T0Y0|d1|,ϕ¯k=ϕ|d1|α0T0r0,e¯k=ekY0|d1|,η¯k=ηkY0|d1|2,P¯1=P1α0Y0|d1|,
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 σkl,i are the stress components, εkl,i are the strain components, (ui, vi, wi) are the displacement components, Ckl,i are the elastic stiffness constants, Dk are the electric displacement components, ek are the piezoelectric coefficients, ηk are the dielectric constants, P1 is the pyroelectric constant, and d1 is the piezoelectric modulus.The dimensionless displacement-expressed equilibrium equations can be expressed as(13a)[C¯11i(∂2∂r¯2+1r¯∂∂r¯)-C¯66i∂2r¯2∂θ+C¯55i∂2∂z¯2-C¯22ir¯2]ui+1r¯(C¯12i+C¯66i)∂2v¯i∂r¯∂θ-1r¯2(C¯22i+C¯66i)∂v¯i∂θ+(C¯13i+C¯55i)∂2w¯i∂r¯∂z¯+(C¯13i-C¯23i)1r¯∂w¯i∂z¯-β¯ri∂T¯i∂r¯-(β¯ri-β¯θi)T¯ir¯+e¯1∂2ϕ¯∂r¯2+e¯6∂2ϕ¯r¯2∂θ2+e¯5∂2ϕ¯∂z¯2+(e¯1-e¯2)1r¯∂ϕ¯∂r¯=0,(13b)(C¯12i+C¯66i)1r¯∂2u¯i∂r¯∂θ+(C¯22i+C¯66i)1r¯2∂u¯i∂θ+C¯66i(∂2v¯i∂r¯2+1r¯∂v¯i∂r¯-v¯ir¯2)+C¯22i1r¯2∂2v¯i∂θ2+C¯44i∂2v¯i∂z¯2+(C¯23i+C¯44i)1r¯∂2w¯i∂θ∂z¯-β¯θi1r¯∂T¯i∂θ+(e¯2+e¯6)1r¯∂2ϕ¯∂r¯∂θ=0,(13c)(C¯13i+C¯55i)∂2u¯i∂r¯∂z¯+(C¯23i+C¯55i)1r¯∂u¯i∂z¯+(C¯23i+C¯44i)1r¯∂2v¯i∂θ∂z¯+C¯55i∂2w¯i∂r¯2+C¯55i1r¯∂w¯i∂r¯+C¯44i∂2w¯ir¯2∂θ2+C¯33i∂2w¯i∂z¯2-β¯z¯i∂T¯i∂z¯+(e¯3+e¯6)∂2ϕ¯∂r¯∂z¯+e¯51r¯∂ϕ¯∂z¯=0.The dimensionless displacement-expressed electrostatic equation can be expressed as(14)e¯1∂2u¯i∂r¯2+(e¯1+e¯2)1r¯∂u¯i∂r¯+e6¯1r¯2∂2u¯i∂θ2+e5¯∂2u¯i∂z¯2+(e¯2+e¯6)1r¯∂2v¯i∂r∂θ-e¯61r¯2∂v¯i∂θ+(e¯3+e¯5)∂2w¯i∂r¯∂z¯+e¯31r¯∂w¯i∂z¯-η1¯(∂2ϕ¯∂r¯2+1r¯∂ϕ¯∂r¯)-η¯21r¯2∂2ϕ¯∂θ¯2-η¯3∂2ϕ¯∂z¯2+p¯1(∂T¯i∂r¯+1r¯T¯i)=0.
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 as(15a)Ti¯(r¯,θ·z¯)=∑n=1∞∑m=1∞T¯nmi(r¯)sin(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15b)ui¯(r¯,θ·z¯)=∑n=1∞∑m=1∞u¯nmi(r¯)sin(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15c)vi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞v¯nmi(r¯)cos(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15d)wi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞w¯nmi(r¯)sin(aθ)cos(bz¯)222222222i=1,2,3,…,N+1,(15e)ϕi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞ϕ¯nmi(r¯)sin(aθ)sin(bz¯)22222222222222222i=N+1,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
(16)d2T¯nmidr¯2+1r¯dT¯nmidr¯-a2r¯2T¯nmi-b2T¯nmi=0,22222222222222222i=1,2,3,…,N+1,(17a)[C¯11i(d2dr¯2+1r¯ddr¯)-a2C¯66i1r¯2-b2C¯55i-C¯22i1r¯2]u¯nmi-a(C¯12i+C¯66i)1r¯dv¯nmidr¯+a(C¯22i+C¯66i)1r¯2v¯nmi-b[(C¯13i+C¯55i)ddr¯+(C¯13i-C¯23i)1r¯]w¯nmi-β¯ridT¯nmidr¯-(β¯ri-β¯θi)T¯nmir¯+e¯1d2ϕ¯nmidr¯2-a2e¯61r¯2ϕ¯nmi-b2e¯5ϕ¯nmi+(e¯1-e¯2)1r¯dϕ¯nmidr¯=0,(17b)a[(C¯12i+C¯66i)1r¯ddr¯+(C¯22i+C¯66i)1r¯2]u¯nmi+C¯66i(d2dr¯2+1r¯ddr¯-1r¯2)v¯nmi-a2C¯22i1r¯2v¯nmi-b2C¯44iv¯nmi-ab(C¯23i+C¯44i)1r¯w¯nmi-aβ¯θi1r¯T¯nmi+a(e¯2+e¯6)1r¯ϕ¯nmi=0,(17c)b[(C¯13i+C¯55i)ddr¯+(C¯23i+C¯55i)1r¯]u¯nmi-ab(C¯23i+C¯44i)1r¯v¯nmi+C¯55id2w¯nmidr¯2+(C¯55i1r¯ddr¯-α2C¯44i1r¯2-b2C¯33i)w¯nmi-bβ¯z¯iT¯nmi+b[(e¯3+e¯6)ddr¯+e¯51r¯]ϕ¯nmi=0,(18)[e¯1d2dr¯2+(e¯1+e¯2)1r¯ddr¯-a2e6¯1r¯2-b2e5¯]u¯nmi-a[(e¯2+e¯6)1r¯ddr¯-e¯61r¯2]v¯nmi-be3¯1r¯w¯nmi-b(e¯3+e¯5)dw¯nmidr¯-[η¯1(d2dr¯2+1r¯ddr¯)-a2η¯21r¯2-b2η¯3]ϕ¯nmi+p1¯(ddr¯+1r¯)T¯nmi=0.Furthermore, substituting the trigonometric series (15a) to (15e) into boundary and continuous conditions (10c) to (10e) and (11c) to (11f), we can obtain(19a)r¯=r1r0:T¯nm1=T¯nma,(19b)r¯=rN+2r0:T¯nm,N+1=T¯nmb,(19c)r¯=ri+1r0:T¯nmi=T¯nm,i+1;λ¯idT¯nmidr¯=λ¯i+1∂T¯nm,i+1∂r¯(20a)r¯=r1r0:C¯111du¯nm1dr¯+C¯121u¯nm1r¯-aC¯121v¯nm1r¯-bC¯131w¯nm1-β¯r1T¯nm1=q¯nmabu¯nm1+dw¯nm1dr¯=0,au¯nm1r¯+dv¯nm1dr¯-v¯nm1r¯=0(20b)r¯=rN+2r0:ϕ¯nm,N+1=0,e¯1du¯nmidr¯+e¯2r¯(u¯nmi-av¯nmi)-be¯3w¯nmi-η¯1dϕ¯nmidr¯+p¯1T¯nmi=0C¯11idu¯nmidr¯+C¯12i(-ar¯v¯nmi+u¯nmi)-bC¯13iw¯nmi-β¯riT¯nmi+e¯1dϕ¯nmidr¯=q¯nmbC¯66i(ar¯u¯nmi+dv¯nmidr¯-v¯nmir¯)+ae¯6r¯ϕ¯nmi=0,C¯55i(dw¯nmidr¯+bu¯nmi)+be¯5ϕ¯nmi=0(20c)r¯=ri+1r0:u¯nmi=u¯nm,i+1,v¯nmi=v¯nm,i+1,w¯nmi=w¯nm,i+1C¯11idu¯nmidr¯+C¯12i(-ar¯v¯nmi+u¯nmi)-bC¯13iw¯nmi-β¯riT¯nmi=C¯11,i+1du¯nm,i+1dr¯+C¯12,i+1(-ar¯v¯nm,i+1+u¯nm,i+1)-bC¯13,i+1w¯nm,i+1-β¯r,i+1T¯nm,i+1C¯66i(ar¯u¯nmi+dv¯nmidr¯-v¯nmir¯)=C¯66,i+1(ar¯u¯nm,i+1+dv¯nm,i+1dr¯-v¯nm,i+1r¯)C¯55i(dw¯nmidr¯+bu¯nmi)=C¯55,i+1(dw¯nm,i+1dr¯+bu¯nm,i+1).Using the orthogonality of trigonometric functions, we can obtain(21){T¯nmaT¯nmbq¯nmaq¯nmb}=4r0θ0l∫0θ0∫0l/r0{T¯a(θ,z¯)T¯b(θ,z¯)q¯a(θ,z¯)q¯b(θ,z¯)}sin(aθ)sin(bz¯)dθdz¯.


## 2.1. Conduction Equations and Equilibrium Equations

The heat conduction equation can be expressed as(3)(∂2∂r2+1r∂∂r+1r2∂2∂θ2+∂2∂z2)Ti=0,
where T¯i is the temperature in the ith layer. While the equilibrium equation of the piezoelectric functionally graded cylindrical shell is
(4)∂σrri∂r+1r∂σrθi∂θ+∂σrzi∂z+σrri+σθθir=0,∂σrθi∂r+1r∂σθθi∂θ+∂σθzi∂z+2σrθir=0,∂σrzi∂r+1r∂σθzi∂θ+∂σzzi∂z+σrzir=0,the electrostatics equation can be written as(5)∂Dr∂r+Drr+1r∂Dθ∂θ+∂Dz∂z=0.The Constitutive equation can be(6){σrriσθθiσzziσθziσrziσrθi}=[C11iC12iC13i000C21iC22iC23i000C13iC23iC33i000000C44i000000C55i000000C66i]{εrriεθθiεzziεθziεrziεrθi}-{βriTiβθiTiβziTi000}-(e100e200e30000000e50e60){ErEθEz},
where
(7)βri=(C11i+C12i+C13i)αiβθi=(C12i+C22i+C23i)αiβzi=(C13i+C23i+C33i)αi2i=1,2,3,…,N+1,In the case of the stress-strain relations of functionally graded cylindrical shell (i=1 to N), the terms ek and Ek on the right side in (5) vanished. And Ckli, β are elastic constants and thermoelastic coefficients.The constitutive equations for the electric field are(8)Dr=e1εrr+e2εθθ+e3εzz+η1Er+P1T1,Dθ=e6εrθ+η2Eθ,Dz=e5εrz+η3Ez,(9)εrri=∂ui∂r;εθθi=1r(∂vi∂θ+ui);εzzi=∂wi∂z;εrθi=1r∂ui∂θ+∂vi∂r-∂νir;εθzi=∂vi∂z+1r∂wi∂θ;εzri=∂ui∂z+∂wi∂r;Er=-∂ϕ∂r;Eθ=-1r∂ϕ∂θ;i=1,2,3,…,N+1.


## 2.2. Boundary and Continuity Conditions

Consider the following:(10a)θ=0,θ0:Ti=0(10b)z¯=0,lr0:Ti=0(10c)r¯=r1r0:T1=Ta(θ,z¯)(10d)r¯=rN+2r0:TN+1=Tb(θ,z¯)(10e)r¯=ri+1r0:Ti=Ti+1;λi∂Ti∂r=λi+1∂Ti+1∂r,i=1,2,3,…,N,where λi and λi+1 are the thermal dimensionless conductivity parameter in ith and i+1th layer, respectively. Ta(θ,z) and Tb(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 as(11a)θ=0,θ0:ui=wi=0;σθθi=σrθi=σθzi=0;ϕ=0,i=1,2,3,…,N+1,(11b)z=0,lr0:ui=vi=0;σzzi=σrzi=σθzi=0;ϕ=0,i=1,2,3,…,N+1,(11c)r=r1r0:σr,1=qa(θ,z);σrz,1=σrθ,1=0(11d)r=rN+1r0:ϕ=0,(11e)r=rN+2r0:Dr=0;σrθ,N=σrz,N=0;σrr,N+1=qb(θ,z)r=rir0:σrri=σrr,i+1;σrθi=σrθ,i+1;σrzi=σrz,i+1;(11f)ui=ui+1;vi=vi+1;wi=wi+1,2222222i=1,2,3,…,N+1+1.To simplify the solving process of the 3D piezothermomechanical problem, the following dimensionless variables are introduced:(12)r¯=rr0,z¯=zr0,L¯=lr0,αk¯=αkα0,λk¯=λkλ0,kk¯=kkk0,Tk¯=TT0,Ta¯=TaT0,Tb¯=TbT0,q¯a=qaα0Y0T0,q¯b=qbα0Y0T0,σ¯kli=σkliα0Y0T0,ε¯kli=εkliα0T0,{u¯i,v¯i,w¯i}={ui,vi,wi}α0r0T0,α¯ki=αkiα0,C¯kli=CkliY0,E¯k=Ek|d1|α0T0;d¯k=Dkα0T0Y0|d1|,ϕ¯k=ϕ|d1|α0T0r0,e¯k=ekY0|d1|,η¯k=ηkY0|d1|2,P¯1=P1α0Y0|d1|,
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 σkl,i are the stress components, εkl,i are the strain components, (ui, vi, wi) are the displacement components, Ckl,i are the elastic stiffness constants, Dk are the electric displacement components, ek are the piezoelectric coefficients, ηk are the dielectric constants, P1 is the pyroelectric constant, and d1 is the piezoelectric modulus.The dimensionless displacement-expressed equilibrium equations can be expressed as(13a)[C¯11i(∂2∂r¯2+1r¯∂∂r¯)-C¯66i∂2r¯2∂θ+C¯55i∂2∂z¯2-C¯22ir¯2]ui+1r¯(C¯12i+C¯66i)∂2v¯i∂r¯∂θ-1r¯2(C¯22i+C¯66i)∂v¯i∂θ+(C¯13i+C¯55i)∂2w¯i∂r¯∂z¯+(C¯13i-C¯23i)1r¯∂w¯i∂z¯-β¯ri∂T¯i∂r¯-(β¯ri-β¯θi)T¯ir¯+e¯1∂2ϕ¯∂r¯2+e¯6∂2ϕ¯r¯2∂θ2+e¯5∂2ϕ¯∂z¯2+(e¯1-e¯2)1r¯∂ϕ¯∂r¯=0,(13b)(C¯12i+C¯66i)1r¯∂2u¯i∂r¯∂θ+(C¯22i+C¯66i)1r¯2∂u¯i∂θ+C¯66i(∂2v¯i∂r¯2+1r¯∂v¯i∂r¯-v¯ir¯2)+C¯22i1r¯2∂2v¯i∂θ2+C¯44i∂2v¯i∂z¯2+(C¯23i+C¯44i)1r¯∂2w¯i∂θ∂z¯-β¯θi1r¯∂T¯i∂θ+(e¯2+e¯6)1r¯∂2ϕ¯∂r¯∂θ=0,(13c)(C¯13i+C¯55i)∂2u¯i∂r¯∂z¯+(C¯23i+C¯55i)1r¯∂u¯i∂z¯+(C¯23i+C¯44i)1r¯∂2v¯i∂θ∂z¯+C¯55i∂2w¯i∂r¯2+C¯55i1r¯∂w¯i∂r¯+C¯44i∂2w¯ir¯2∂θ2+C¯33i∂2w¯i∂z¯2-β¯z¯i∂T¯i∂z¯+(e¯3+e¯6)∂2ϕ¯∂r¯∂z¯+e¯51r¯∂ϕ¯∂z¯=0.The dimensionless displacement-expressed electrostatic equation can be expressed as(14)e¯1∂2u¯i∂r¯2+(e¯1+e¯2)1r¯∂u¯i∂r¯+e6¯1r¯2∂2u¯i∂θ2+e5¯∂2u¯i∂z¯2+(e¯2+e¯6)1r¯∂2v¯i∂r∂θ-e¯61r¯2∂v¯i∂θ+(e¯3+e¯5)∂2w¯i∂r¯∂z¯+e¯31r¯∂w¯i∂z¯-η1¯(∂2ϕ¯∂r¯2+1r¯∂ϕ¯∂r¯)-η¯21r¯2∂2ϕ¯∂θ¯2-η¯3∂2ϕ¯∂z¯2+p¯1(∂T¯i∂r¯+1r¯T¯i)=0.
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 as(15a)Ti¯(r¯,θ·z¯)=∑n=1∞∑m=1∞T¯nmi(r¯)sin(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15b)ui¯(r¯,θ·z¯)=∑n=1∞∑m=1∞u¯nmi(r¯)sin(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15c)vi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞v¯nmi(r¯)cos(aθ)sin(bz¯)222222222i=1,2,3,…,N+1,(15d)wi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞w¯nmi(r¯)sin(aθ)cos(bz¯)222222222i=1,2,3,…,N+1,(15e)ϕi¯(r¯,θ·z¯)=∑n=1∞∑m=1∞ϕ¯nmi(r¯)sin(aθ)sin(bz¯)22222222222222222i=N+1,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
(16)d2T¯nmidr¯2+1r¯dT¯nmidr¯-a2r¯2T¯nmi-b2T¯nmi=0,22222222222222222i=1,2,3,…,N+1,(17a)[C¯11i(d2dr¯2+1r¯ddr¯)-a2C¯66i1r¯2-b2C¯55i-C¯22i1r¯2]u¯nmi-a(C¯12i+C¯66i)1r¯dv¯nmidr¯+a(C¯22i+C¯66i)1r¯2v¯nmi-b[(C¯13i+C¯55i)ddr¯+(C¯13i-C¯23i)1r¯]w¯nmi-β¯ridT¯nmidr¯-(β¯ri-β¯θi)T¯nmir¯+e¯1d2ϕ¯nmidr¯2-a2e¯61r¯2ϕ¯nmi-b2e¯5ϕ¯nmi+(e¯1-e¯2)1r¯dϕ¯nmidr¯=0,(17b)a[(C¯12i+C¯66i)1r¯ddr¯+(C¯22i+C¯66i)1r¯2]u¯nmi+C¯66i(d2dr¯2+1r¯ddr¯-1r¯2)v¯nmi-a2C¯22i1r¯2v¯nmi-b2C¯44iv¯nmi-ab(C¯23i+C¯44i)1r¯w¯nmi-aβ¯θi1r¯T¯nmi+a(e¯2+e¯6)1r¯ϕ¯nmi=0,(17c)b[(C¯13i+C¯55i)ddr¯+(C¯23i+C¯55i)1r¯]u¯nmi-ab(C¯23i+C¯44i)1r¯v¯nmi+C¯55id2w¯nmidr¯2+(C¯55i1r¯ddr¯-α2C¯44i1r¯2-b2C¯33i)w¯nmi-bβ¯z¯iT¯nmi+b[(e¯3+e¯6)ddr¯+e¯51r¯]ϕ¯nmi=0,(18)[e¯1d2dr¯2+(e¯1+e¯2)1r¯ddr¯-a2e6¯1r¯2-b2e5¯]u¯nmi-a[(e¯2+e¯6)1r¯ddr¯-e¯61r¯2]v¯nmi-be3¯1r¯w¯nmi-b(e¯3+e¯5)dw¯nmidr¯-[η¯1(d2dr¯2+1r¯ddr¯)-a2η¯21r¯2-b2η¯3]ϕ¯nmi+p1¯(ddr¯+1r¯)T¯nmi=0.Furthermore, substituting the trigonometric series (15a) to (15e) into boundary and continuous conditions (10c) to (10e) and (11c) to (11f), we can obtain(19a)r¯=r1r0:T¯nm1=T¯nma,(19b)r¯=rN+2r0:T¯nm,N+1=T¯nmb,(19c)r¯=ri+1r0:T¯nmi=T¯nm,i+1;λ¯idT¯nmidr¯=λ¯i+1∂T¯nm,i+1∂r¯(20a)r¯=r1r0:C¯111du¯nm1dr¯+C¯121u¯nm1r¯-aC¯121v¯nm1r¯-bC¯131w¯nm1-β¯r1T¯nm1=q¯nmabu¯nm1+dw¯nm1dr¯=0,au¯nm1r¯+dv¯nm1dr¯-v¯nm1r¯=0(20b)r¯=rN+2r0:ϕ¯nm,N+1=0,e¯1du¯nmidr¯+e¯2r¯(u¯nmi-av¯nmi)-be¯3w¯nmi-η¯1dϕ¯nmidr¯+p¯1T¯nmi=0C¯11idu¯nmidr¯+C¯12i(-ar¯v¯nmi+u¯nmi)-bC¯13iw¯nmi-β¯riT¯nmi+e¯1dϕ¯nmidr¯=q¯nmbC¯66i(ar¯u¯nmi+dv¯nmidr¯-v¯nmir¯)+ae¯6r¯ϕ¯nmi=0,C¯55i(dw¯nmidr¯+bu¯nmi)+be¯5ϕ¯nmi=0(20c)r¯=ri+1r0:u¯nmi=u¯nm,i+1,v¯nmi=v¯nm,i+1,w¯nmi=w¯nm,i+1C¯11idu¯nmidr¯+C¯12i(-ar¯v¯nmi+u¯nmi)-bC¯13iw¯nmi-β¯riT¯nmi=C¯11,i+1du¯nm,i+1dr¯+C¯12,i+1(-ar¯v¯nm,i+1+u¯nm,i+1)-bC¯13,i+1w¯nm,i+1-β¯r,i+1T¯nm,i+1C¯66i(ar¯u¯nmi+dv¯nmidr¯-v¯nmir¯)=C¯66,i+1(ar¯u¯nm,i+1+dv¯nm,i+1dr¯-v¯nm,i+1r¯)C¯55i(dw¯nmidr¯+bu¯nmi)=C¯55,i+1(dw¯nm,i+1dr¯+bu¯nm,i+1).Using the orthogonality of trigonometric functions, we can obtain(21){T¯nmaT¯nmbq¯nmaq¯nmb}=4r0θ0l∫0θ0∫0l/r0{T¯a(θ,z¯)T¯b(θ,z¯)q¯a(θ,z¯)q¯b(θ,z¯)}sin(aθ)sin(bz¯)dθdz¯.


## 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]:
(22)T¯nmi(r¯)=∑k=1∞Aki(s)(r¯-1)ki=1,N+1u¯nmi(r¯)=∑k=1∞Bki(s)(r¯-1)ki=1,N+1v¯nmi(r¯)=∑k=1∞Cki(s)(r¯-1)ki=1,N+1w¯nmi(r¯)=∑k=1∞Dki(s)(r¯-1)ki=1,N+1ϕ¯nmi(r¯)=∑k=1∞Fki(s)(r¯-1)ki=N+1.Substituting series (22) into (16) to (18), comparing the coefficient of (r¯-1)k, we can obtain the following recurrence equation:
(23)-(k+2)(k+1)Ak+2,i=(2k+1)(k+1)Ak+1,i+k2Aki-(a2+b2)Aki-2b2k¯iλ¯iAk-1,i-b2k¯iλ¯iAk-2,i(24a)-(k+2)(k+1)Bk+2,i=(2k+1)(k+1)Bk+1,i+(k2-C¯22i+b2C¯55i+a2C¯66iC¯11i)Bki-2b2C¯55iC¯11iBk-1,i-b2C¯55iC¯11iBk-2,i-a(C¯12i+C¯66i)C¯11i(k+1)Ck+1,i+a(C¯22i+C¯66i)C¯11iCki-ak(C¯12i+C¯66i)C¯11iCki-b(C¯13i+C¯55i)C¯11i(k+1)Dk+1,i-2bk(C¯13i+C¯55i)C¯11iDk,i-b(C¯13i-C¯23i)C¯11iDk,i-β¯riC¯11i(k+1)Ak+1,i-b[(k-1)(C¯13i+C¯55i)C¯11i+(C¯13i-C¯23i)C¯11i]Dk-1,i-(2k+1)β¯ri-β¯θiC¯11iAk,i-kβ¯ri-β¯θiC¯11iAk-1,i+(k+2)(k+1)e¯1C¯11iFk+2,i+(k+1)[(2k+1)e¯1-e¯2]C¯11iFk+1,i+1C¯11i(k2e¯1-ke¯2-a2e¯6)Fk,i-b2e¯5C¯11iFk,i-b2e¯5C¯11i(2Fk-1,i+Fk-2,i)(24b)-(k+2)(k+1)Ck+2,i=(2k+1)(k+1)Ck+1,i+(k2-a2C¯22i+b2C¯44iC¯66i-1)Ck,i-2b2C¯44iC¯66iCk-1,i-b2C¯44iC¯66iCk-2,i+aC¯12i+C¯66iC¯66i(k+1)Bk+1,i+akC¯12i+C¯66iC¯66iBk,i+aC¯22i+C¯66iC¯66iBk,i-abC¯23i+C¯44iC¯66i(Dki+Dk-1,i)-aβ¯θiC¯66i(Aki+Ak-1,i)-ae¯2+e¯6C¯66i[(k+1)Fk+1,i+kFki](24c)-(k+2)(k+1)Dk+2,i=(2k+1)(k+1)Dk+1,i+(k2-a2C¯44i+b2C¯33iC¯55i)Dki-2b2C¯33iC¯55iDk-1,i-b2C¯33iC¯55iDk-2,i+b(C¯13i+C¯55i)C¯55i(k+1)Bk+1,i+bC¯55i[2k(C¯13i+C¯55i)+(C¯23i+C¯55i)]Bk,i-ab(C¯23i+C¯44i)C¯55i(Ck,i+Ck-1,i)-bβ¯ziC¯55i(Ak,i+2Ak-1,i+Ak-2,i)+(2k+1)be¯5C¯55iFk,i+2bke¯3C¯55iFk,i+bC¯55i[(k-1)e¯3+ke¯5]Fk-1,i(24d)(k+2)(k+1)Fk+2,i=-(2k+1)(k+1)Fk+1,i-(k2+a2η¯2η¯1+b2η¯3η¯1)Fki+b2η¯3η¯1(2Fk-1,i+Fk-2,i)+p¯1η¯1[(k+1)Ak+1,i+(2k+1)Ak,i+kAk-1,i]+e¯1η¯1(k+2)(k+1)Bk+2,i+e¯1η¯1(2k+1)(k+1)Bk+1,i+e¯2η¯1(k+1)Bk+1,i+1η¯1(e¯1k2+e¯2k-a2e¯6-b2e¯5)Bk,i-2b2e¯5η¯1Bk-1,i-a(e¯2+e¯6)η¯1(k+1)Ck+1,i-b2e¯5η¯1Bk-2,i-ae¯2k+ae¯6(k-1)η¯1Ck,i-b(e¯3+e¯5)η¯1(k+1)Dk+1,i-1η¯1be¯3(2k+1)Dk,i-2be¯5kη¯1Dk,i-bke¯3η¯1Dk-1,i-b(k-1)e¯5η¯1Dk-1,i,where A-1,i=A-2,i=0. From the recurrence equations (23) and (24a) to (24d), we can see that all coefficients Aki, Bki, Cki, Dki, and Fki in Taylor’s series can be briefly expressed by Aoi, A1i, Boi, B1i; Coi, C1i; Doi, D1i; Foi, and F1i: (25a)Aki=ξ1ikA0i+ξ2ikA1i;(25b)Bki=ζ1ikB0i+ζ2ikB1i+ζ3ikC0i+ζ4ikC1i+ζ5ikD0i+ζ6ikD1i+ζ7ikF0i+ζ8ikF1i+ζ9ikA0i+ζ10ikA1i,(25c)Cki=L1ikB0i+L2ikB1i+L3ikC0i+L4ikC1i+L5ikD0i+L6ikD1i+L7ikF0i+L8ikF1i+L9ikA0i+L10ikA1i,(25d)Dki=ρ1ikB0i+ρ2ikB1i+ρ3ikC0i+ρ4ikC1i+ρ5ikD0i+ρ6ikD1i+ρ7ikF0i+ρ8ikF1i+ρ9ikA0i+ρ10ikA1i,(25e)Fki=γ1ikB0i+γ2ikB1i+γ3ikC0i+γ4ikC1i+γ5ikD0i+γ6ikD1i+γ7ikF0i+γ8ikF1i+γ9ikA0i+γ10ikA1i,where the coefficients ξ1ik, ξ2ik; ζ1ik~ζ10ik; L1ik~L10ik; ρ1ik~ρ10ik and γ1ik~γ10ik 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¯).WhereA0i, A1i; B0i, B1i; C0i, C1i; D0i, D1i; F0i and F1i 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 solutionsT¯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 dimensionsL=6, θ0=π/3, Ra=0.95, and Rb=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]:
(26)Em=330GPa,μm=0.3;λm=138W/mK;αm=4.9×10-6/KEc=330GPa,μc=0.3;λc=138W/mK;αc=4.9×10-6/K.The material properties of piezoelectric layer (cadmium selenite) are summarized as(27)C11=83.6GPa,C22=C33=74.1GPaC23=45.2GPa,C12=C13=39.3GPa,C66=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/m2K,d1=-3.92×10-12C/N,λ=8.6W/mK.And it is assumed that the cylindrical shell is subjected to the following nonumiform thermal and mechanical loadings on its outer surfaces:(28)Ta(θ,z)=0atRa=0.95Tb(θ,z)=200×sin[3θ]×sin[πz6]atRb=1.04qa(θ,z)=0atRa=0.95qb(θ,z)=-108×sin[3θ]×sin[πz6]atRb=1.042.Figures2 to 8 illustrate the numerical results of dimensionless temperature, displacement, and stresses of functionally graded cylindrical shell with piezoelectric layer.Dimensionless temperature distribution: (a) at sectionθ=π/6, (b) R=1.042.
(a)
(b)Figure2 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.Figure3 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.Dimensionless axial displacement distribution: (a) at sectionθ=π/6, (b) R=1.042.
(a)
(b)Distribution of dimensionless radial stress: (a)Z=3.0, θ=π/6, (b) θ=π/6.
(a)
(b)Figure5(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.Distribution of dimensionless circumferential stress: (a)Z=3.0, θ=π/6, (b) θ=π/6.
(a)
(b)Distribution of dimensionless axial stress: (a) at pointZ=3.0, θ=π/6. (b) θ=π/6.
(a)
(b)Figures7 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 σRθ and σRZ appear at the middle radius, while the shear stress σθZ varies from compressive stress to tensile stress along its length. The maximum shear stress σRZ 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 stressσθZ (R=1.04).Distribution of dimensionless shear stressσRZ: (a) Z=3.0, (b) at point Z=3.0, θ=π/6.
(a)
(b)Figure 9
Distribution of dimensionless shear stress (Z=3).Distribution of dimensionless electric potential: (a) at sectionZ=3, (b) R=1.04.
(a)
(b)Distribution of dimensionless electric displacementDR: (a) at section R=1.04, (b) Z=3.
(a)
(b)Distribution of dimensionless electric displacementDθ: (a) at section R=1.04, (b) Z=3.
(a)
(b)


## 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.



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*Source: 947693-2013-12-14.xml*