1 Introduction

Functionally graded materials (FGMs) are composed of two various materials,for which thecontinuous transition form one to the other is in a specific gradient. Continuous variations ofthe material parameters are usually assumed to be in power law or exponential functions alongthe radial direction. FGMs are so beneficial in high-technology industries such as aircraft andaerospace because of possessing properties of both their constitutive materials and possibilityto define the final desired specifications. The combination of the FGMs and the piezoelectricmaterials produces functionally graded piezoelectric materials (FGPMs) with more sophisti-cated applications. Piezoelectric materials due to their direct and inverse effects have differentapplications as sensors and actuators. The primary goal in designing such materials is to takeadvantage of the desirable properties^{[1, 2, 3]}. Although a large number of studies have been de-veloped in the case of symmetric behavior^{[4, 5]} of FGMs,there are not enough researches innon-axisymmetric fields. Displacement and stress distributions under different loads are typicalresults presented in these research works.

Electro-magneto-elastic behaviors of functionally graded piezoelectric solid cylinder andsphere were studied by Dai et al.^[4] using the infinitesimal theory of electro-magneto-elasticity.Alibeigloo^[1],Alibeigloo and Chen^[5],and Alibeigloo and Nouri^[6] used the state space methodto obtain thermo-elastic behavior of functionally graded cylindrical shell bonded to thin piezo-electric layers. Tutuncu and Temel^[7] analyzed some FGM structures like cylinders,disks,andspheres using the plane elasticity theory and the complementary functions method. Their novelapproach reduces the boundary value problem to an initial-value problem which can be solvedaccurately by one of the many efficient methods such as the Runge-Kutta approach. Arefi andRahimi^[8] also developed a three-dimensional multi-field formulation of a functionally gradedpiezoelectric thick shell of revolution by the tensor analysis. Chen and Shi^[9] applied the theoryof elasticity using the Airy stress function to obtain the exact solution of cylinder subjectedto thermal and electric loadings. For comparison purposes,numerical results were also carriedout. With this regard,Akbarzadeh et al.^[10] used the hybrid Fourier-Laplace transform methodto evaluate dynamic response of a simply supported functionally graded rectangular plate sub-jected to a lateral thermo-mechanical loading. Using the theory of electro-thermo-elasticity,Daiet al.^[11] obtained the analytical solution of FGPM hollow cylinder rotating about its axis ata constant angular velocity. Peng and Li^[12] also presented the distribution of thermal stressesand radial displacement numerically by converting the resulting boundary value problem to aFredholm integral equation. Dumir et al.^[13] considered separation of variables to obtain anexact piezoelastic solution for an infinitely long,simply-supported,orthotropic,piezoelectric,radially polarized,and circular cylindrical shell panel in a cylindrical bending under pressureand electrostatic excitation. They applied the trigonometric Fourier series for displacementsand electric potential to satisfy the boundary conditions. The electro-thermo-mechanical so-lution of a radially polarized cylindrical shell was also presented by Dube et al.^[14] using theseparation of variables technique.

In the case of non-axisymmetric loadings,some studies can be named. Tokovyy and Ma^[15]carried out a solution based upon the direct integration method for a radially inhomogeneoushollow cylinder under non-axisymmetric conditions. Poultangari et al.^[16] investigated ana-lytically functionally graded hollow spheres under non-axisymmetric thermo-mechanical loads.They used the Legendre polynomials and the system of Euler differential equations to solvethe Navier equations. Shao et al.^[17] obtained analytical solutions of time-dependent temper-ature and thermo-mechanical stresses through the functionally graded hollow cylinder undernon-axisymmetric mechanical and transient thermal loads by the complex Fourier series andLaplace transform techniques. Jabbari et al.^[18] also presented the general solution using theseparation of variables and complex Fourier series for functionally graded hollow cylinders undernon-axisymmetric and steady-state thermo-mechanical loads. For functionally graded eccentricand non-axisymmetrically loaded circular cylinders,Batra and Nie^[19] investigated plane straindeformations analytically. They used Frobenius series to solve the resulting fourth-order ordi-nary differential equations.

Yang and Gao^[20] developed two-dimensional analysis of thermal stresses in a functionallygraded plate with a circular hole based on the complex variable method. Jafari-Fesharaki etal.^{[21, 22]} also applied the separation of variables and complex Fourier series to get stress andstrain fields through an FGPM cylinder under non-axisymmetric electro-mechanical loads.

In this research work,two-dimensional (r,θ) distributions of displacement and stress in athick walled FGPM cylinder placed in a uniform electric field and under non-axisymmetricthermo-mechanical loads are evaluated analytically for different boundary conditions. Eventu-ally,a number of examples are considered to check the solution. 2 Statement of problem

Consider a thick walled functionally graded piezoelectric cylinder with a and b as the innerand outer radii,respectively. All the material properties,except Poisson’s ratio (ν) which isassumed to be constant,are approximated to obey the same power law through the thickness.These parameters are Young’s modulus and the thermal expansion,dielectric,pyroelectric,and piezoelectric coefficients,respectively,

As mentioned before,to check the solution procedure,some examples will be carried out. The material chosen for this purpose is PZT-4,and its constants are^[23] 3 Derivation of basic formulations

To govern the basic equations,in the absence of body forces,consider the equilibrium andMaxwell equations in the cylindrical coordinates (r,θ),respectively,as^[5]

Here,σ_ij are the stress tensor components,and D_i are the electrical displacement components.The strain-displacement and constitutive equations are expressed as^[24] where U and V are the displacement components in the radial and circumferential directions,respectively. φ is the electric potential,and T is the thermal distribution in the thickness of cylinder. λ and μ are Lame’ coefficients as follows: Moreover,the electrical displacements D_r and D_θ are represented as^[5] Equation (3) after replacement from Eqs. (4)−(7) and (1) becomes as below. All of the constants in these three partial differential equations are explained in Appendix A. 4 Solution procedure

To solve Eq. (8),U,V ,and φ are considered to be in the complex Fourier series form as

where U_n(r) and V_n(r) are the coefficients of the complex Fourier series of U(r,θ) and V (r,θ), respectively,i.e., Due to the considered symmetric condition,the electric field is independent of the circumfer-ential direction. Therefore,Eq. (9c) is valid only for n = 0,and Eq. (8c) will be excluded from the system of Eq. (8) for n ≠ 0.

The steady state heat conduction problem has already been solved by Jabbari et al.^[18] and is in the form of the complex Fourier series as follows:

After substitution of Eqs. (9a),(9b),and (10) into Eqs. (8a) and (8b) and omitting the term exp (inθ),Eq. (8) can be written as The above two equations yield a system of ordinary differential equations having two general and particular solutions which may be written in the following forms: A general solution is conducted with the assumptions of The replacement of Eq. (14) into Eq. (12) expresses In order to find A and B and to use them for calculating nontrivial solutions of U_n^g(r) and V_n^g(r),the determinant of coefficients in the system of Eq. (15) must be set to be zero. It is dueto dependency of these two equations which must express eventually two related distributions for U and V ,i.e., The concluded fourth-order equation has four roots of ζ_n1 to ζ_n4 which give the general solution as follows: where N_nj is the relation between the constants A_nj and B_nj and is obtained from Eq. (15) as The particular solutions of Eq. (12) are assumed in the forms of Substituting Eq. (19) and the second equation of Eq. (11) into Eq. (12) yields where the constants h₁,h₂,· · · ,h₁₂ are presented in Appendix A. Equating the coefficients of the identical powers gives two systems of algebraic equations expressing H_n1 to H_n4. Rewriting Eq. (13),the complete solutions for U_n(r) and V_n(r) then become For n = 0,the relations between U − V and φ − V are lost,and the system of Eq. (8) becomes just an uncoupled equation in terms of V and two other coupled equations as follows. Therefore,it is necessary to obtain the separate solution for n = 0. Inserting n = 0 into Eqs. (9) and (11) and substituting them in Eq. (8),one obtains It is clear that the previous process should be repeated in order to reach the solutions for n = 0 as The uncoupled equation,the second equation of Eq. (22),is an Euler equation,which can be solved by the assumption of V₀ = r^ζ0 , For the general solution of remained equations of (22),the same procedure as done for obtaining Eq. (17) must be performed. By assumption of U₀^g = A₀r^ζ0 and φ₀^g = C₀r^ζ0 ,the general solution for n = 0 would be where Similarly,the particular solutions of the first and third equations of Eq. (22) are supposed in the forms of Substituting the above equations in two pointed equations of Eq. (22) leads to The used constants of h_0i are also presented in Appendix A. H₀₁,H₀₂,H₀₅,and H₀₆ are obtained by equating the coefficients of the identical powers. Now,the solutions for U(r,θ),V (r,θ),and φ₀(r) are completed and can be written as Substituting these relations into Eqs. (4) and (5) gives the strain and stress distributions in the functionally graded piezoelectric hollow cylinder. The coefficients K_i are also stated in Ap-pendix A. 5 Results and discussion

The analytical solution can be checked for some examples. The first example is adoptedto compare the results of the proposed solution with the data reported by Jafari-Fesharaki etal.^[21]. A hollow cylinder with the inner and outer radii equal to 0.5 m and 2 m,respectively,and made of PZT-4 is considered for this example. The assumed boundary conditions are alsolisted in Table 1. In this order,the temperature for current analysis at inner and outer radiimust be equated to zero. Figure 1 depicts the identical variation of the electric potential forvarious values of material inhomogeneity (β) with the graph presented in Fig. 2 of Ref. ^[21].

For the second and third examples,the previously mentioned material parameters of PZT-4 are taken into account. Furthermore,the internal and external radii of functionally gradedpiezoelectric hollow cylinder are also assumed to be 0.5 m and 0.8 m,respectively,and the power-law index of material is chosen as β = 1. At the second example,the effects of two-dimensional electro-mechanical fields are shown,while for the third one,the thermo-electromechanical be-havior of the cylinder is discussed. The considered boundary conditions for these examples arealso expressed in Table 1. These conditions are selected in a manner to monitor the effects oftemperature better. It is also examined that after 35 numbers of terms the series expansion isconverged.

Figures 2 and 3 show the distributions of the radial component of electric displacement forboth examples,and it is clear that temperature can decrease its amplitude. While for thecircumferential components,temperature does not have such an effect,as shown in Figs. 4 and5. Due to the assumed boundary conditions,radial stresses have a harmonic pattern at theinner surface of cylinder and continue to decrease along the radial direction. These variationscan be witnessed in both Figs. 6 and 7 ,and clearly no significant discrepancies exist whentemperature enhances. Figures 8 and 9 also represent the shear stress components through thecylinder. The behavior of this stress component has turned from the inner surface,which isconstant because of considered boundary conditions,to harmonic pattern at outer layers.

As previously mentioned about the temperature effects on the radial electric displacement,this parameter has the same effect on the circumferential stress and decreases the harmonicmanner of this stress (see Figs. 10 and 11). Figures 12 and 13 show a harmonic pattern forthe radial displacements in Examples 2 and 3,respectively. Although the cylinder has radialand circumferential stresses at its outer surface,there are not any displacement in that sectionwhich is in accord with the boundary conditions. Moreover,the same variations for displacement components in the circumferential direction can be observed (see Figs. 14 and 15). Knowing theFGPM cylinder behavior and its responses against different field exposures makes one capableof applying these structures more beneficial in different industries as sensors and actuators.

6 Conclusions

The exact two-dimensional solution of a functionally graded piezoelectric hollow cylinderunder non-axisymmetric loads,based on the direct method using the complex Fourier series,is conducted in current research work. Generality of this method and its possibility to exertdifferent electric,thermal,and mechanical fields without any mathematical limitation are ofgreat importance. Obtaining the displacement and stress distributions through the cylindermakes it possible to control and optimize applications of such parts better in different industrialcases. Appendix A