Elasticity solutions of functionally graded piezoelectric plates under electric fields in cylindrical bending

Abstract

This paper presents elasticity solutions for functionally graded piezoelectric plates under electric fields in cylindrical bending. Based on the generalized Mian and Spencer plate theory, the assumption of the material parameters which can vary along the thickness direction of the plate in an arbitrary fashion is kept; however, the materials are extended from elastic materials to piezoelectric materials. The electric potential function is constructed following the forms of the displacement functions in Mian and Spencer plate theory. The essential idea of Mian and Spencer plate theory (J Mech Phys Solids 46:2283–2295, 1998) is that the three-dimensional elasticity equations for inhomogeneous materials can be obtained by two-dimensional solution for homogeneous materials by straightforward substitutions. Through rigorous derivation, the corresponding elasticity solutions of cylindrical bending of functionally graded piezoelectric plates under electric fields are obtained. In the numerical examples, the accuracy of the present solutions is verified and the responses of plates subjected to electrical potential difference and electrical displacement are investigated, respectively.

1 Introduction

Piezoelectric materials have been considered as promising candidates for actuators and sensors applied in micro-electro-mechanical and smart systems due to their intrinsic coupling of the electric field and the electric field [1]. Traditional piezoelectric devices are designed as laminated forms of piezoelectric materials, which easily cause interfacial debonding and high-stress concentrations under mechanical or electric loadings. Inspired by the concept of functionally graded materials, functionally graded piezoelectric materials (FGPMs) have been developed, whose composition and properties can be designed smoothly and continuously in the preferred orientation [1, 2].

Currently, numerical, semi-analytic and analytic methods have been proposed to investigate the mechanical/piezoelectric responses of FGPM plates [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Semi-analytical methods, combined analytical methods with numerical methods, have been developed for the static, free vibration and buckling analyses of FGPM plates with various boundary conditions [4,5,6]. For example, semi-analytical solutions were elaborated based on the extended Stroh-like formalism for 3D static problems of electro-elastic mono- and multilayered plates, where mechanical and electrical forces were applied on the top and/or bottom layer and may take arbitrary forms [5]; Wu and Ding [6] employed a unified formulation of the finite layer method, based on the Reissner mixed variational theorem, for the coupled electro-elastic analysis of the simply-supported FGPM plates under electro-mechanical loads. As for numerical methods, such as the differential quadrature (DQ), finite element (FE) and meshless methods, which have been adopted for more complex problems of FGPM plates [8,9,10]. For instants, geometrically nonlinear static response of functionally graded piezoelectric plate under mechanical and electrical loads was studied by using Mesh-less method [9]. Radial point interpolation method (RPIM) was used to create the shape function and approximate the field variables. The first-order shear deformation plate theory (FSDT) and Von Karman strains were used to model the nonlinear behavior of plate; Kumar and Harsha [10] executed a modal analysis of functionally graded piezoelectric material rectangular plate to find the natural frequency with different mode shapes by using the FE Technique where the piezoelectric material was considered to be graded along the thickness according to the simplified power-law.

Although the analytical methods are limited to the relatively simple problems, the availability of analytical solutions with the highest accuracy is inherently of much importance to serve as a benchmark for numerical and semi-analytical modeling [11]. Based on the three-dimensional (3D) piezoelectricity theory and the state space approach, Zhong and Shang [12] obtained an exact solution of a simply-supported FGPM rectangular plate with exponent-law distribution under the sinusoidal mechanical and electric loadings. Li et al. [13, 14] and Zhao et al. [15] adopted the direct displacement method to perform a 3D analysis of the transversely isotropic FGPM circular plates subjected to a uniform mechanical loading, electric potential difference and electric displacement on the upper and/or lower surfaces, respectively. Li and Pan [16] investigated the static bending problems of a simply supported FGPM microplate with considering the size effect, and the material properties were varied through the thickness according to a power law. Liu and Wang [17] obtained 3D analytical solutions for the structural instability of a simply supported orthotropic piezoelectric rectangular microplate by means of a linear perturbation analysis. An analytical solution was presented by Ghafarollahi and Shodja [18] for the scattering of transverse surface waves by a homogeneous piezoelectric fiber contained in a functionally graded piezoelectric half-space with exponentially varying electromechanical properties using a multipole expansion method. Zenkour and Hafed [19] presented the piezoelectric effect on the bending of simply supported FGPM plate by using a simple quasi-3D sinusoidal shear deformation theory, in which material properties were assumed to vary exponentially in the thickness direction. Besides, Bachir et al. [20, 21] developed a refined shear deformation theory with four unknown functions to obtain the elastic solutions of the thermo-mechanical bending response for the functionally graded plate. As the reviewed papers mentioned above, it is worth noting that due to the mathematical difficulties, not much work has been done to obtain elasticity solutions of FGPM plates based on piezoelectricity theory.

Mian and Spencer [22] developed an ingenious theory for deriving 3D analytical solutions for isotropic FGM plates with tractions-free surfaces. In this theory, the material properties are assumed to vary arbitrarily in the thickness direction of plates. The origins of this method can be traced back to the classical solutions by Michell [23] for plane stress of moderately thick elastic plates. Yang et al. [24, 25] extended above Mian and Spencer plate theory to study transversely isotropic FGM plates subjected to uniform loads applied on the top and bottom surfaces. In this paper, we further extend the generalized Mian and Spencer plate theory for deriving elasticity solutions of the equations of linear piezoelasticity for FG piezoelectric materials, which are no longer the elastic materials and reveal the innovation of the present work. This is not a trivial extension because the elastic deformation and electric field are coupled for piezoelectric materials. To illustrate the procedure of this extension, functionally graded piezoelectric plates under electric fields in cylindrical bending are investigated and the corresponding elasticity solutions are presented. The proposed theory will be helpful for the further investigation on related boundary value problems of FGPM plates.

2 Basic equations

In a rectangular Cartesian coordinate (x, y, z), consider an orthotropic and FGPM plate of uniform thickness h, shown in Fig. 1a, with the xy plane at mid-plane ($z=0)$ and z-axis perpendicular to the mid-plane. It is noteworthy that the material parameters are varied arbitrarily along the z-axis. As shown in Figs. 1b, c, two typical external electric loadings are applied, respectively, that is, a uniform electric potential difference ($\overline{\Phi }_{2}$ at $z=h / 2$ and $\overline{\Phi }_{1}$ at $z={-h} / 2)$ and electric displacement $D_{\textrm{0}}$ are uniformly distributed on the top and/or bottom surfaces the of plate. When the plate is in a state of cylindrical bending, the geometrical dimension of the plate along the y-axis is assumed to be far longer than the other two directions such that $0\le x\le l \quad -\infty<y<\infty$ and $-h / 2\le z\le h / 2$ and then, the three-dimensional (3D) plate problem can be simplified to a two-dimensional (2D) plane problem where all.

Fig. 1

a 3D FGPM plate and coordinate system; 2D FGPM plates subjected to an (b) electric potential difference and c electric displacement

In the absence of body forces, the basic equations for piezoelasticity are given in the following [24]:

$$\begin{aligned}{} & {} \sigma _{x,x} +\tau _{xz,z} =0, \quad \tau _{xz,x} +\sigma _{z,z} =0. \end{aligned}$$

(1)

$$\begin{aligned}{} & {} D_{x,x} +D_{z,z} =0, \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \sigma _{x} =c_{11} u_{,x} +c_{13} w_{,z} +e_{31} \phi _{,z}, \quad \sigma _{z} =c_{13} u_{,x} +c_{33} w_{,z} +e_{33} \phi _{,z}, \quad \tau _{zx} =c_{55} \left( {u_{,z} +w_{,x} } \right) +e_{15} \phi _{,x}, \end{aligned}$$

(3)

$$\begin{aligned}{} & {} D_{x} =e_{15} \left( {u_{,z} +w_{,x} } \right) -\lambda _{11} \phi _{,x}, \quad D_{z} =e_{31} u_{,x} +e_{33} w_{,z} -\lambda _{33} \phi _{,z} \end{aligned}$$

(4)

where the comma denotes differentiation with respect to the indicated variable; $\sigma _{x}$, $\sigma _{z}$ and $\tau _{xz}$ are the stress components; u and w are the displacement components; $D_{x}$ and $D_{z}$ are the electric displacements; $\phi$ is the electric potential; $c_{ij}$, $e_{ij}$ and $\lambda _{ij}$ are the elastic, piezo-electric and dielectric coefficients, respectively. It is noteworthy that these elastic coefficients are functions of z for FGPMs, and particularly become constant coefficients for homogeneous materials.

Based on the generalized Mian and Spencer plate theory [22], we seek solutions of the displacements as follows:

$$\begin{aligned}{} & {} u\left( {x,z} \right) =\bar{{u}}\left( x \right) +F\left( z \right) \bar{{u}}_{,xx} +A\left( z \right) \bar{{w}}_{,x} +B\left( z \right) \bar{{w}}_{,xxx}, \nonumber \\{} & {} w\left( {x,z} \right) =\bar{{w}}\left( x \right) +G\left( z \right) \bar{{u}}_{,x} +C\left( z \right) \bar{{w}}_{,xx} +D\left( z \right) , \end{aligned}$$

(5)

where $A\left( z \right) ,\;B\left( z \right) ,\;C\left( z \right) ,\;D\left( z \right) ,\;F\left( z \right) ,\;G\left( z \right)$ are the unknown functions to be determined Assume functions $\bar{{u}}=\bar{{u}}\left( x \right)$ and $\bar{{w}}=\bar{{w}}\left( x \right)$ are the mid-plane displacements of the plate and hence, we have $A\left( 0 \right) =0$, $B\left( 0 \right) =0$, $C\left( 0 \right) =0$, $D\left( 0 \right) =0$, $F\left( 0 \right) =0$,$G\left( 0 \right) =0$ from Eq. (5).

Following the above expression forms of the displacements, it is assumed that the electric potential function has the following form:

$$\begin{aligned} \phi \left( {x,z} \right) =\phi _{1} \left( z \right) \bar{{u}}_{,x} +\phi _{2} \left( z \right) \bar{{w}}_{,xx} +\phi _{0} \left( z \right) , \end{aligned}$$

(6)

where $\phi _{0} \left( z \right) , \phi _{1} \left( z \right)$, $\phi _{2} \left( z \right)$ are the unknown functions to be determined

Substituting Eqs. (5)–(6) into Eqs. (3)–(4), the expressions of the stress components and electric displacement components are rewritten as:

$$\begin{aligned} \sigma _{x}= & {} \bar{{u}}_{,x} \left( {c_{11} +c_{13} {G}'\text{+e }_{\text{31 }} {\phi }'_{\text{1 }} } \right) +\bar{{w}}_{,xx} \left( {c_{11} A+c_{13} {C}'+e_{31} {\phi }'_{2} } \right) +Fc_{11} \bar{{u}}_{,xxx} +Bc_{11} \bar{{w}}_{,xxxx} \nonumber \\{} & {} +{D}'c_{13} \text{+e }_{\text{31 }} {\phi }'_{\text{0 }}, \nonumber \\ \sigma _{z}= & {} \bar{{u}}_{,x} \left( {c_{13} +c_{33} {G}'\text{+ }e_{33} {\phi }'_{1} } \right) +\bar{{w}}_{,xx} \left( {c_{13} A+c_{33} {C}'+e_{33} {\phi }'_{2} } \right) +Fc_{13} \bar{{u}}_{,xxx} +Bc_{13} \bar{{w}}_{,xxxx} \nonumber \\{} & {} +{D}'c_{33} +e_{33} {\phi }'_{0}, \nonumber \\ \tau _{zx}= & {} \bar{{u}}_{,xx} \left[ {c_{55} \left( {{F}'+G} \right) +e_{15} \phi _{1} } \right] +\bar{{w}}_{,xxx} \left[ {c_{55} \left( {{B}'+C} \right) +e_{15} \phi _{2} } \right] +\bar{{w}}_{,x} \left[ {c_{55} \left( {{A}'+1} \right) } \right] , \end{aligned}$$

(7)

$$\begin{aligned} D_{x}= & {} \bar{{u}}_{,xx} \left[ {e_{15} \left( {{F}'+G} \right) -\lambda _{11} \phi _{1} } \right] +\bar{{w}}_{,xxx} \left[ {e_{15} \left( {{B}'+C} \right) -\lambda _{11} \phi _{2} } \right] +\bar{{w}}_{,x} \left[ {e_{15} \left( {{A}'+1} \right) } \right] ,\nonumber \\ D_{z}= & {} \bar{{u}}_{,x} \left( {e_{31} +e_{33} {G}'-\lambda _{33} {\phi }'_{1} } \right) +\bar{{w}}_{,xx} \left( {e_{31} A+e_{33} {C}'-\lambda _{33} {\phi }'_{2} } \right) +e_{31} F\bar{{u}}_{,xxx} +e_{31} B\bar{{w}}_{,xxxx} \nonumber \\{} & {} +e_{33} {D}'-\lambda _{33} {\phi }'_{0}. \end{aligned}$$

(8)

Substituting Eqs. (7)–(8) into Eqs. (1)–(2) and provided that:

$$\begin{aligned}{} & {} \left[ {c_{55} \left( {{A}'+1} \right) } \right] ^{\prime }=0, \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \left( {c_{13} +c_{33} {G}'\text{+ }e_{33} {\phi }'_{1} } \right) ^{\prime }=0, \end{aligned}$$

(10)

$$\begin{aligned}{} & {} \left( {e_{31} +e_{33} {G}'-\varepsilon _{33} {\phi }'_{1} } \right) ^{\prime }\text{= }0, \end{aligned}$$

(11)

$$\begin{aligned}{} & {} c_{55} \left( {{A}'+1} \right) +\left( {c_{13} A+c_{33} {C}'+e_{33} {\phi }'_{2} } \right) ^{\prime }=0, \end{aligned}$$

(12)

$$\begin{aligned}{} & {} e_{15} \left( {{A}'+1} \right) +\left( {e_{31} A+e_{33} {C}'-\varepsilon _{33} {\phi }'_{2} } \right) ^{\prime }\text{= }0, \end{aligned}$$

(13)

$$\begin{aligned}{} & {} c_{11} +c_{13} {G}'\text{+ }e_{31} {\phi }'_{1} +\left( {c_{55} \left( {{F}'+G} \right) +e_{15} \phi _{1} } \right) ^{\prime }=c_{55} \kappa _{1}, \end{aligned}$$

(14)

$$\begin{aligned}{} & {} c_{11} A+c_{13} {C}'+e_{31} {\phi }'_{2} +\left( {c_{55} \left( {{B}'+C} \right) +e_{15} \phi _{2} } \right) ^{\prime }=c_{55} \kappa _{2}, \end{aligned}$$

(15)

$$\begin{aligned}{} & {} \bar{{u}}_{,xxx} =\kappa _{3}, \end{aligned}$$

(16)

$$\begin{aligned}{} & {} \bar{{w}}_{,xxxx} =\kappa _{4}. \end{aligned}$$

(17)

where “$^{\prime }$” denotes differentiation with respect to z, and $\kappa _{i} ( i \quad =$ 1,2,3,4) are arbitrary constant to be determined.

Accordingly, Eqs. (1)–(2) are rewritten as:

$$\begin{aligned}{} & {} \kappa _{1} \kappa _{3} +\kappa _{2} \kappa _{4} =0. \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \left[ {c_{55} \left( {{F}'+G} \right) +\left( {Fc_{13} } \right) ^{\prime }\text{+ }e_{15} \phi _{1} } \right] \kappa _{3} +\left[ {c_{55} \left( {{B}'+C} \right) +\left( {c_{13} B} \right) ^{\prime }\text{+ }e_{15} \phi _{2} } \right] \kappa _{4} \text{+ }\left( {c_{33} {D}'\text{+ }e_{33} {\phi }'_{0} } \right) ^{\prime }=0, \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \left[ {e_{15} \left( {{F}'+G} \right) +\left( {Fe_{31} } \right) ^{\prime }-\varepsilon _{11} \phi _{1} } \right] \kappa _{3} +\left[ {e_{15} \left( {{B}'+C} \right) +\left( {e_{31} B} \right) ^{\prime }-\varepsilon _{11} \phi _{2} } \right] \kappa _{4} \text{+ }\left( {e_{33} {D}'-\varepsilon _{33} {\phi }'_{0} } \right) ^{\prime }=0.\nonumber \\ \end{aligned}$$

(20)

The mid-plane displacement components can be given by integrating Eqs. (16)–(17), respectively:

$$\begin{aligned} \bar{{w}}\left( x \right)= & {} \frac{1}{24}\kappa _{4} x^{4}+\frac{1}{6}C_{1} x^{3}+\frac{1}{2}C_{2} x^{2}+C_{3} x+C_{4}, \end{aligned}$$

(21)

$$\begin{aligned} \bar{{u}}\left( x \right)= & {} \frac{1}{6}\kappa _{3} x^{3}-\frac{1}{2}\frac{\kappa _{2} }{\kappa _{1} }C_{1} x^{2}+C_{5} x+C_{6}. \end{aligned}$$

(22)

where $C_{\textrm{i}}(i=1, 2, \ldots , 6)$ are integral constants to be determined by the edge boundary conditions at $x=0,l$

3 Under electric potential case

3.1 Determination of the displacement functions and electric potential functions

When only electric potential is applied, the boundary conditions on the top and bottom surfaces of the plate can be expressed as:

$$\begin{aligned} z= & {} -h / 2: \quad \sigma _{z} =0, \quad \tau _{zx} =0, \quad \phi \text{= }\overline{\Phi }_{1} \end{aligned}$$

(23)

$$\begin{aligned} z= & {} h / 2: \quad \sigma _{z} =0, \quad \tau _{zx} =0, \quad \phi \text{= }\overline{\Phi }_{2} \end{aligned}$$

(24)

The expression of the electric potential function in Eq. (6) is updated by introducing Eqs. (21)–(22):

$$\begin{aligned} \phi \left( {x,z} \right) =\frac{\kappa _{3} \phi _{1} \left( z \right) +\kappa _{4} \phi _{2} \left( z \right) }{2}x^{2}-\left[ {\frac{\kappa _{2} }{\kappa _{1} }\phi _{1} \left( z \right) -\phi _{2} \left( z \right) } \right] C_{1} x+\phi _{1} \left( z \right) C_{5} +\phi _{2} \left( z \right) C_{2} +\phi _{0} \left( z \right) . \end{aligned}$$

(25)

Substituting Eq. (7) and Eq. (25) into Eqs. (23)–(24) leads to the following differential equations:

$$\begin{aligned}{} & {} {A}'\left( {\pm h / 2} \right) +1=0, \end{aligned}$$

(26)

$$\begin{aligned}{} & {} c_{13} \left( {\pm h / 2} \right) +c_{33} \left( {\pm h / 2} \right) {G}'\left( {\pm h / 2} \right) \text{+ }e_{33} \left( {\pm h / 2} \right) {\phi }'_{1} \left( {\pm h / 2} \right) =0, \end{aligned}$$

(27)

$$\begin{aligned}{} & {} c_{13} \left( {\pm h / 2} \right) A\left( {\pm h / 2} \right) +c_{33} \left( {\pm h / 2} \right) {C}'\left( {\pm h / 2} \right) \text{+ }e_{33} \left( {\pm h / 2} \right) {\phi }'_{2} \left( {\pm h / 2} \right) =0, \end{aligned}$$

(28)

$$\begin{aligned}{} & {} c_{13} \left( {\pm h / 2} \right) \left[ {\kappa _{3} F\left( {\pm h / 2} \right) +\kappa _{4} B\left( {\pm h / 2} \right) } \right] +c_{33} \left( {\pm h / 2} \right) {D}'\left( {\pm h / 2} \right) \text{+ }e_{33} \left( {\pm h / 2} \right) {\phi }'_{0} \left( {\pm h / 2} \right) =0 \end{aligned}$$

(29)

$$\begin{aligned}{} & {} c_{55} \left( {\pm h / 2} \right) \left[ {{F}'\left( {\pm h / 2} \right) +G\left( {\pm h / 2} \right) } \right] +e_{15} \left( {\pm h / 2} \right) \phi _{1} \left( {\pm h / 2} \right) \text{= }0 \end{aligned}$$

(30)

$$\begin{aligned}{} & {} c_{55} \left( {\pm h / 2} \right) \left[ {{B}'\left( {\pm h / 2} \right) +C\left( {\pm h / 2} \right) } \right] +e_{15} \left( {\pm h / 2} \right) \phi _{2} \left( {\pm h / 2} \right) \text{= }0 \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \kappa _{3} \phi _{1} \left( {\pm h / 2} \right) +\kappa _{4} \phi _{2} \left( {\pm h / 2} \right) \text{= }0 \end{aligned}$$

(32)

$$\begin{aligned}{} & {} \phi _{1} \left( {-h / 2} \right) C_{5} +\phi _{2} \left( {-h / 2} \right) C_{2} +\phi _{0} \left( {-h / 2} \right) =\overline{\Phi }_{1} \end{aligned}$$

(33)

$$\begin{aligned}{} & {} \phi _{1} \left( {h / 2} \right) C_{5} +\phi _{2} \left( {h / 2} \right) C_{2} +\phi _{0} \left( {h / 2} \right) \text{= }\overline{\Phi }_{2} \end{aligned}$$

(34)

Integrating Eqs. (9)–(15) and (19)–(20) by virtue of Eqs. (26)–(34), the expressions of functions of variable z in Eqs. (5) and (6) can be determined:

$$\begin{aligned}{} & {} A\left( z \right) =-z \end{aligned}$$

(35)

$$\begin{aligned}{} & {} B\left( z \right) =k_{2} f_{5} \left( z \right) -\Phi _{2} f_{6} \left( z \right) +H_{4} f_{7} \left( z \right) +B_{1} \left( z \right) +B_{0}, \end{aligned}$$

(36)

$$\begin{aligned}{} & {} C\left( z \right) =C_{1} \left( z \right) +H_{4} a_{0}^{0} \left( z \right) +C_{0}, \end{aligned}$$

(37)

$$\begin{aligned}{} & {} D\left( z \right) =-\kappa _{3} \left[ {D_{00} \left( z \right) H_{2} +D_{01} \left( z \right) } \right] -\kappa _{4} \left[ {D_{00} \left( z \right) H_{4} +D_{02} \left( z \right) } \right] +a_{0}^{0} \left( z \right) H_{7} +D_{0} \end{aligned}$$

(38)

$$\begin{aligned}{} & {} F\left( z \right) =k_{1} f_{5} \left( z \right) -\Phi _{1} f_{6} \left( z \right) +H_{2} f_{7} \left( z \right) +F_{1} \left( z \right) +F_{0}, \end{aligned}$$

(39)

$$\begin{aligned}{} & {} G\left( z \right) =G_{1} \left( z \right) +H_{2} a_{0}^{0} \left( z \right) +G_{0}, \end{aligned}$$

(40)

$$\begin{aligned}{} & {} \phi _{1} \left( z \right) =\phi _{10} \left( z \right) -H_{2} b_{0}^{0} \left( z \right) +\Phi _{1} \end{aligned}$$

(41)

$$\begin{aligned}{} & {} \phi _{2} \left( z \right) =\phi _{20} \left( z \right) -H_{4} b_{0}^{0} \left( z \right) +\Phi _{2}, \end{aligned}$$

(42)

$$\begin{aligned}{} & {} \phi _{0} \left( z \right) =-\kappa _{3} \left[ {\phi _{00} \left( z \right) H_{2} +\phi _{01} \left( z \right) } \right] -\kappa _{4} \left[ {\phi _{00} \left( z \right) H_{4} +\phi _{02} \left( z \right) } \right] -b_{0}^{0} \left( z \right) H_{7} +\Phi _{0}, \end{aligned}$$

(43)

where the expressions of functions including $a_{0}^{0} \left( z \right)$, $b_{0}^{0} \left( z \right)$, $F_{1} \left( z \right)$, $B_{1} \left( z \right)$, $f_{i} \left( z \right) \left( {i=5,6,7} \right)$, $D_{0i} \left( z \right) \left( {i=0,1,2} \right)$, $\phi _{0i} \left( z \right) \left( {i=0,1,2} \right)$ are listed in “Appendix A”; $B_{0}$, $C_{0}$, $D_{0}$, $F_{0}$, $G_{0}$, $\Phi _{1}$, $\Phi _{2}$, $\Phi _{0}$, $H_{2}$, $H_{4}$ and $H_{7}$ are integration constants, in which $H_{2}$, $H_{4}$ and $H_{7}$ can be expressed according to Eqs. (8), (10) and (11):

$$\begin{aligned}{} & {} H_{2} =e_{31} +e_{33} {G}'\left( z \right) -\lambda _{33} {\phi }'_{1} \left( z \right) , \quad H_{4} =e_{31} A+e_{33} {C}'\left( z \right) -\lambda _{33} {\phi }'_{2} \left( z \right) ,\\{} & {} H_{7} \text{= }\kappa _{3} e_{31} \left( {{-h} / 2} \right) F\left( {{-h} / 2} \right) +\kappa _{4} e_{31} \left( {-h / 2} \right) B\left( {-h / 2} \right) +e_{33} \left( {-h / 2} \right) {D}'\left( {-h / 2} \right) -\lambda _{33} \left( {-h / 2} \right) {\phi }'_{0} \left( {-h / 2} \right) \end{aligned}$$

3.2 Determination of the remaining integral constants

Integrating Eqs. (14)–(15) gives:

$$\begin{aligned} \kappa _{1}= & {} \frac{\int \limits _{-h/2}^{h/2} {c_{11} +c_{13} {G}'\text{+ }e_{31} {\phi }'_{1} d\xi } }{\int \limits _{-h/2}^{h/2} {c_{55} d\xi } }\text{= }\frac{h_{0}^{0} \left( {h / 2} \right) -H_{2} f_{0}^{0} \left( {h / 2} \right) }{g_{5} \left( {h / 2} \right) }, \nonumber \\ \kappa _{2}= & {} \frac{\int \limits _{-h/2}^{h/2} {c_{11} A+c_{13} {C}'+e_{31} {\phi }'_{2} d\xi } }{\int \limits _{-h/2}^{h/2} {c_{55} d\xi } }\text{= }\frac{-h_{1}^{0} \left( {h / 2} \right) -H_{4} f_{0}^{0} \left( {h / 2} \right) }{g_{5} \left( {h / 2} \right) } \end{aligned}$$

(44)

where $g_{5} \left( z \right) \text{= }\int \limits _{-h/2}^z {c_{55} d\xi }$, and $h_{i}^{j} \left( z \right) \left( {i,j=0,1} \right)$ is given in “Appendix A”.

Substituting Eqs. (41)–(42) into Eq. (32) leads to:

$$\begin{aligned}{} & {} \kappa _{3} \Phi _{1} +\kappa _{4} \Phi _{2} \text{= }0, \end{aligned}$$

(45)

$$\begin{aligned}{} & {} \kappa _{3} \left[ {\phi _{10} \left( {h / 2} \right) -H_{2} b_{0}^{0} \left( {h / 2} \right) } \right] +\kappa _{4} \left[ {\phi _{20} \left( {h / 2} \right) -H_{4} b_{0}^{0} \left( {h / 2} \right) } \right] \text{= }0. \end{aligned}$$

(46)

Integrating Eq. (19) and using Eq. (29) with ${z=h} / 2$, we can obtain the following expression:

$$\begin{aligned} \kappa _{3} \int \limits _{-h / 2}^{h / 2} {\left[ {c_{55} \left( {{F}'+G} \right) \text{+ }e_{15} \phi _{1} } \right] } d\xi +\kappa _{4} \int \limits _{-h / 2}^{h / 2} {\left[ {c_{55} \left( {{B}'+C} \right) \text{+ }e_{15} \phi _{2} } \right] } d\xi =0. \end{aligned}$$

(47)

Substituting Eqs. (36)–(43) into above Eq. (47) leads to

$$\begin{aligned} \kappa _{3} \left[ {-h_{0}^{1} \left( {h /2} \right) +H_{2} f_{0}^{1} \left( {h / 2} \right) } \right] +\kappa _{4} \left[ {h_{1}^{1} \left( {h / 2} \right) +H_{4} f_{0}^{1} \left( {h / 2} \right) } \right] \text{= }0. \end{aligned}$$

(48)

Substituting Eq. (44) into (18), Eqs. (46)–(48) become:

$$\begin{aligned}{} & {} \left[ {f_{1}^{0} \left( {h / 2} \right) f_{0}^{0} \left( {h / 2} \right) -b_{0}^{0} \left( {h / 2} \right) h_{1}^{0} \left( {h / 2} \right) } \right] H_{2} +\left[ {f_{0}^{0} \left( {h / 2} \right) f_{0}^{0} \left( {h / 2} \right) -b_{0}^{0} \left( {h / 2} \right) h_{0}^{0} \left( {h / 2} \right) } \right] H_{4} \nonumber \\{} & {} \quad =f_{1}^{0} \left( {h / 2} \right) h_{0}^{0} \left( {h / 2} \right) -f_{0}^{0} \left( {h / 2} \right) h_{1}^{0} \left( {h / 2} \right) , \end{aligned}$$

(49)

$$\begin{aligned}{} & {} \left[ {h_{1}^{1} \left( {h / 2} \right) f_{0}^{0} \left( {h / 2} \right) -f_{0}^{1} \left( {h / 2} \right) h_{1}^{0} \left( {h / 2} \right) } \right] H_{2} +\left[ {h_{0}^{1} \left( {h / 2} \right) f_{0}^{0} \left( {h / 2} \right) -f_{0}^{1} \left( {h / 2} \right) h_{0}^{0} \left( {h / 2} \right) } \right] H_{4} \nonumber \\{} & {} \quad =h_{1}^{1} \left( {h / 2} \right) h_{0}^{0} \left( {h / 2} \right) -h_{0}^{1} \left( {h / 2} \right) . \end{aligned}$$

(50)

Thus, constants $H_{2}$ and $H_{4}$ can be determined by solving these two simultaneous equations.

Furthermore, we define the mean values of electric displacement along x-axis at $x=0,l$ as [13,14,15]

$$\begin{aligned} D_{x}^{*} \text{= }\frac{1}{h}\int \limits _{-h/2}^{h/2} {D_{x} } dz=0. \end{aligned}$$

(51)

We can arrive at the following expressions from Eq. (51):

$$\begin{aligned} \Phi _{1}= & {} \frac{1}{\int \limits _{-h / 2}^{h / 2} {\lambda _{11} } d\xi }\int \limits _{-h / 2}^{h / 2} {\left\{ {e_{15} \left[ {{F}'\left( z \right) +G\left( z \right) } \right] -\lambda _{11} \left[ {\phi _{10} \left( z \right) -H_{2} b_{0}^{0} \left( z \right) } \right] } \right\} } d\xi , \nonumber \\ \Phi _{2}= & {} \frac{1}{\int \limits _{-h / 2}^{h / 2} {\lambda _{11} } d\xi }\int \limits _{-h / 2}^{h / 2} {\left\{ {e_{15} \left[ {{B}'\left( z \right) +C\left( z \right) } \right] -\lambda _{11} \left[ {\phi _{20} \left( z \right) -H_{4} b_{0}^{0} \left( z \right) } \right] } \right\} } d\xi \end{aligned}$$

(52)

By considering the fact that $\phi \left( {0,0} \right) =\phi \left( {l,0} \right)$ for the FGPM plate in this case, we obtain the following equation:

$$\begin{aligned} \frac{1}{2}\left[ {\kappa _{3} \phi _{1} \left( 0 \right) +\kappa _{4} \phi _{2} \left( 0 \right) } \right] l^{2}-C_{1} \left[ {\frac{\kappa _{2} }{\kappa _{1} }\phi _{1} \left( 0 \right) -\phi _{2} \left( 0 \right) } \right] l\text{= }0. \end{aligned}$$

(53)

Accordingly, we can obtain the expressions of $\kappa _{3}$ and $\kappa _{4}$ by using Eqs. (18) and (53):

$$\begin{aligned} \kappa _{3} \text{= }\frac{\text{2 }}{l}\frac{\kappa _{2} }{\kappa _{1} }C_{1}, \quad \kappa _{4} \text{= }-\frac{\text{2 }}{l}C_{1}. \end{aligned}$$

(54)

Introducing Eqs. (41)–(43) into Eqs. (33)–(34), we have the following equation:

$$\begin{aligned}{} & {} \frac{\text{2 }}{l}\left\{ {\frac{\kappa _{2} }{\kappa _{1} }\left[ {\phi _{01} \left( {h / 2} \right) } \right] -\phi _{02} \left( {h / 2} \right) } \right\} C_{1} -\left[ {f_{1}^{0} \left( {h / 2} \right) +H_{4} b_{0}^{0} \left( {h / 2} \right) } \right] C_{2} \nonumber \\{} & {} \quad +\left[ {f_{0}^{0} \left( {h / 2} \right) -H_{2} b_{0}^{0} \left( {h / 2} \right) } \right] C_{5} -b_{00} \left( {h / 2} \right) H_{7} =\bar{{\Phi }}_{2} -\bar{{\Phi }}_{1}. \end{aligned}$$

(55)

The six integral constants $C_{\textrm{i}}(i=1, 2, \ldots , 6)$ arising from Eq. (21)–(22) can be determined by the edge boundary conditions at $x =$ 0,l, which there are simply supported (S), clamped (C) and free (F) conditions:

$$\begin{aligned}{} & {} {\textbf {S}}:\qquad \bar{{u}}=0, \quad \bar{{w}}=0, \quad M_{x} =0, \end{aligned}$$

(56)

$$\begin{aligned}{} & {} {\textbf {C}}:\qquad \bar{{u}}=0, \quad \bar{{w}}=0, \quad \bar{{w}}_{,x} =0, \end{aligned}$$

(57)

$$\begin{aligned}{} & {} {\textbf {F}}:\qquad N_{x} =0, \quad M_{x} =0, \quad Q_{x} =0, \end{aligned}$$

(58)

where the expressions of the resultant forces and moments are given in “Appendix B”. There are totally 4 different kinds of combination of Eqs. (56)–(58), namely SS, CC, SC, CF, in which the first letter denotes the conditions at $x =$ 0    and the second signifies those at $x = l$. For any kind of the above-mentioned boundary conditions, we have six equations from boundary conditions and a supplementary equation from Eq. (55) which are just enough to determine constants: $C_{\textrm{i}}(i=1, 2, \ldots , 6)$ and $H_{\textrm{7}}$ completely.

Therefore, the elasticity solutions of FGPM plates under the external electric potentials can be obtained from Eqs. (3)–(5).

4 Under electric displacement case

4.1 Determination of the displacement and electric potential functions

When an electric displacement is applied on the top surface of the plate, the boundary conditions on the top and bottom surfaces are proposed as

$$\begin{aligned}{} & {} z=-h / 2:\qquad \qquad \sigma _{z} =0, \quad \tau _{zx} =0, \quad D_{z} =0 \end{aligned}$$

(59)

$$\begin{aligned}{} & {} z=h / 2:\qquad \qquad \sigma _{z} =0, \quad \tau _{zx} =0, \quad D_{z} =-D_{0} \end{aligned}$$

(60)

Similarly, substituting the related stress and electric displacement components in Eqs. (7)–(8) into Eqs. (59)–(60), and in view of Eqs. (26)–(31), the following equations can be gotten:

$$\begin{aligned}{} & {} e_{31} \left( {\pm h / 2} \right) +e_{33} \left( {\pm h / 2} \right) {G}'\left( {\pm h / 2} \right) -\lambda _{33} \left( {\pm h / 2} \right) {\phi }'_{1} \left( {\pm h / 2} \right) \text{= }0, \end{aligned}$$

(61)

$$\begin{aligned}{} & {} e_{31} \left( {\pm h / 2} \right) A\left( {\pm h / 2} \right) +e_{33} \left( {\pm h / 2} \right) {C}'\left( {\pm h / 2} \right) -\lambda _{33} \left( {\pm h / 2} \right) {\phi }'_{2} \left( {\pm h / 2} \right) \text{= }0, \end{aligned}$$

(62)

$$\begin{aligned}{} & {} e_{31} \left( {-h / 2} \right) F\left( {-h / 2} \right) \kappa _{3} +e_{31} \left( {-h / 2} \right) B\left( {{-h} / 2} \right) \kappa _{4} +e_{33} \left( {-h / 2} \right) {D}'\left( {-h / 2} \right) \nonumber \\{} & {} \quad -\lambda _{33} \left( {-h / 2} \right) {\phi }'_{0} \left( {-h / 2} \right) \text{= }0, \end{aligned}$$

(63)

$$\begin{aligned}{} & {} e_{31} \left( {+h / 2} \right) F\left( {+h / 2} \right) \kappa _{3} +e_{31} \left( {+h / 2} \right) B\left( {{+h} / 2} \right) \kappa _{4} +e_{33} \left( {+h / 2} \right) {D}'\left( {+h / 2} \right) \nonumber \\{} & {} \quad -\lambda _{33} \left( {+h / 2} \right) {\phi }'_{0} \left( {+h / 2} \right) =-D_{0}. \end{aligned}$$

(64)

Based on some of Eqs. (9)–(15), and Eqs. (19)–(20), (26)–(31) and (61)–(64), the displacement and electric potential functions can be easily deduced:

$$\begin{aligned}{} & {} A\left( z \right) =-z \end{aligned}$$

(65)

$$\begin{aligned}{} & {} B\left( z \right) =\int \limits _{-h/2}^z {\frac{1}{c_{55} }\left[ {B_{10} \left( \xi \right) -e_{15} \phi _{2} \left( \xi \right) } \right] -C\left( \xi \right) } d\xi +B_{0}, \end{aligned}$$

(66)

$$\begin{aligned}{} & {} C\left( z \right) =-\int \limits _{-h/2}^z {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( \xi \right) } d\xi +C_{0}, \end{aligned}$$

(67)

$$\begin{aligned}{} & {} D\left( z \right) =-\int \limits _{-h/2}^z {\frac{\lambda _{33} \left[ {D_{10} \left( \xi \right) +D_{00} \left( \xi \right) } \right] +e_{33} \left[ {D_{20} \left( \xi \right) +D_{01} \left( \xi \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} d\xi +D_{0}, \end{aligned}$$

(68)

$$\begin{aligned}{} & {} F\left( z \right) =\int \limits _{-h/2}^z {\frac{1}{c_{55} }\left[ {F_{10} \left( \xi \right) -e_{15} \phi _{1} \left( \xi \right) } \right] -G\left( \xi \right) } d\xi +F_{0}, \end{aligned}$$

(69)

$$\begin{aligned}{} & {} G\left( z \right) =-\int \limits _{-h/2}^z {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }} d\xi +G_{0}, \end{aligned}$$

(70)

$$\begin{aligned}{} & {} \phi _{1} \left( z \right) =\int \limits _{-h/2}^z {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi } +\Phi _{1}, \end{aligned}$$

(71)

$$\begin{aligned}{} & {} \phi _{2} \left( z \right) =\int \limits _{-h/2}^z {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( \xi \right) d\xi } +\Phi _{2}, \end{aligned}$$

(72)

$$\begin{aligned}{} & {} \phi _{0} \left( z \right) =\int \limits _{-h/2}^z {\frac{c_{33} \left[ {D_{20} \left( \xi \right) +D_{01} \left( \xi \right) } \right] -e_{33} \left[ {D_{10} \left( \xi \right) +D_{00} \left( \xi \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} d\xi +\Phi _{0}, \end{aligned}$$

(73)

where the expressions of functions $F_{10} \left( z \right)$, $B_{10} \left( z \right)$, $D_{10} \left( z \right)$, $D_{00} \left( z \right)$, $D_{20} \left( z \right)$ and $D_{01} \left( z \right)$ are listed in “Appendix C”; $B_{0}$, $C_{0}$, $D_{0}$, $F_{0}$, $G_{0}$, $\Phi _{1}$, $\Phi _{2}$ and $\Phi _{0}$ are integration constants which revealed in “Appendix C”.

4.2 Determination of the remaining integral constants

The determination of $\kappa _{1}$ and $\kappa _{2}$ is similar to the electric potential case with integrating Eqs. (10)–(11), while $\kappa _{3}$ and $\kappa _{4}$ can be obtained via the following equations which can be obtained by integrating Eqs. (19)–(20) with the aid of Eqs. (63)–(64):

$$\begin{aligned}{} & {} \kappa _{3} \int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{F}'+G} \right) +e_{15} \phi _{1} } \right] } d\xi +\kappa _{4} \int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{B}'+C} \right) +e_{15} \phi _{2} } \right] } d\xi =0 \end{aligned}$$

(74)

$$\begin{aligned}{} & {} \kappa _{3} \int \limits _{-h/2}^{h / 2} {\left[ {e_{15} \left( {{F}'+G} \right) -\lambda _{11} \phi _{1} } \right] } d\xi +\kappa _{4} \int \limits _{-h/2}^{h / 2} {\left[ {e_{15} \left( {{B}'+C} \right) -\lambda _{11} \phi _{2} } \right] } d\xi \text{= }D_{0} \end{aligned}$$

(75)

Considering the boundary conditions at $x =$ 0,l (as shown in Eqs. (56)–(58)), six integral constants, $C_{\textrm{i}}(i=1, 2, \ldots , 6)$ can be determined by six equations provided by any two kind of boundary conditions. Accordingly, the elasticity solutions of FGPM plate under the external electric displacement can be obtained completely from Eqs. (3)–(5).

5 Numerical results and discussion

To illustrate the present elasticity solutions, numerical examples are given in this section to reveal the electrical responses of FGPM plates subjected to a uniform electric potential difference and electric displacement, respectively. Unless stated otherwise, the thickness of FGPM plate $h=$0.15 m, the width $l \quad =$1 m, and $x=l / 2$ in this work. Note that the material parameters can be varied arbitrarily along the thickness direction in this paper

5.1 Verification

The verification of the proposed analytical solutions is performed via a homogeneous piezoelectric plate subjected to a uniform electric potential difference. We compare the present results with a numerical model which are established by ABAQUS software via finite element method (FEM). The 8-node reduced integration element (C3D8R) is adopted to mesh the plate, and the material property of piezoelectric material PZT-4 is listed in Table 1. An incremental electric potential difference (1V) is applied between the top and bottom surfaces of the plate to obtain the piezoelectric responses.

Table 1 Constants of some typical piezoelectric materials

Taking CC and SS boundary conditions as typical examples to prove the accuracy of the present analytical solutions for brevity. Figure 2 shows the distributions of the dimensionless deflection $\bar{{W}}(\bar{{W}}=W / h)$ and normal stress component $\bar{{\sigma }}_{x}$ ($\bar{{\sigma }}_{x} ={\sigma _{x} } / {c_{11}^{0} })$ along the thickness direction of the plate. It is obviously that the present solutions are in excellent agreement with FEM results. As shown in Table 2, the absolute errors of the dimensionless deflection $\bar{{W}}$ and normal stress $\bar{{\sigma }}_{x}$ are less than 3.42% and 2.60%, respectively. These results indicate that the present analytical solution is valid and accuracy in predicting the piezoelectric responses of plates.

Fig. 2

Comparison of dimensionless a deflection $\overline{W}$and b stress component $\bar{{\sigma }}_{x}$ between the present and FEM results

Table 2 Comparison of $\overline{W}$and $\bar{{\sigma }}_{x}$ at top and bottom surfaces

5.2 Under electric potential case

We now consider a FGPM plate subjected to a uniform electric potential difference with $\overline{\Phi }_{1} =0$ and $\overline{\Phi }_{1} =1$V and with material properties varied exponentially along the z-direction [18]:

$$\begin{aligned} c_{ij} \left( z \right) =c_{ij}^{0} e^{\left[ {{\left( {z+h / 2} \right) } / h} \right] p}, \quad e_{ij} \left( z \right) =e_{ij}^{0} e^{\left[ {{\left( {z+h / 2} \right) } / h} \right] p}, \quad \lambda _{ij} \left( z \right) =\lambda _{ij}^{0} e^{\left[ {{\left( {z+h / 2} \right) } / h} \right] p}, \quad i,j=1,2,...,6, \end{aligned}$$

(76)

where p is the gradient index to characterize the degree of material inhomogeneity. In Particular, the FGPM will degrade into the homogenous piezoelectric material if $p=0$. $c_{ij}^{0}$, $e_{ij}^{0}$ and $\lambda _{ij}^{0}$ are the elastic, piezoelectric and dielectric properties of PZT-4 [14] (given in Table 1) at the bottom surface ($z={-h} / 2)$ of plate, respectively. Without loss of generality, CS boundary condition is considered in this case. Moreover, the following dimensionless quantities are introduced:

$$\begin{aligned} \bar{{W}}=\frac{W}{h}, \bar{{\sigma }}_{x} =\frac{\sigma _{x} }{c_{11}^{0} }, \quad \bar{{\Phi }}^{*}=\frac{\phi }{\left( {\overline{\Phi }_{2} -\overline{\Phi }_{1} } \right) }, \quad \bar{{D}}_{z} =\frac{D_{z} }{\sqrt{c_{11}^{0} \varepsilon _{33}^{0} } }. \end{aligned}$$

(77)

Figure 3a plots the variations of the dimensionless electric potential $\overline{\Phi }^{*}$ with the p changed from -10 to 10 along the thickness direction of the plate. It is found that $\overline{\Phi }^{*}$ changes in a linear way when $p=0$. As for the FGPMs, $\overline{\Phi }^{*}$ exhibits an obvious nonlinear behavior. Affected by the distribution characteristics of exponential function in Eq. (76), $\overline{\Phi }^{*}$ firstly increases sharply and then, shows a slowly increase when $p>0$, while $\overline{\Phi }^{*}$ firstly show a relatively gentle rising and then, increases significantly when $p<0$. Moreover, these trends are becoming apparent with the increased absolute value of p. The dimensionless electric displacement component $\bar{{D}}_{z}$ for various values of p along the thickness direction of plate is shown in Fig. 3b. It is observed that the distributions of $\bar{{D}}_{z}$ basically keep constant along the thickness direction of the plate, and the absolute value is increasing with p changed from -10 to 10.

Fig. 3

Distribution of dimensionless a electric potential $\overline{\Phi }^{*}$ and b electric displacement component $\overline{D}_{z}$ along the thickness direction of plate

When it comes to the elastic field, the through-the-thickness distributions of the dimensionless deflection $\overline{W}$ are presented in Fig. 4a. It is easy to observe that $\overline{W}$ tends to zero at the mid-plane of plate regardless of p. $\overline{W}$ shows a linear trend for homogeneous materials and has an evident nonlinear change for FGPMs. When $p>0$, the maximum deflection occurs on the bottom surface of the FGPM plate; when $p<0$, the maximum value of the absolute deflection achieved on the top surface. It is also obvious that $\overline{W}$ is central symmetric for the same absolute value of p. Figure 4b displays the variation of dimensionless normal stress $\bar{{\sigma }}_{x}$ along the thickness direction of plate. The distribution pattern of $\bar{{\sigma }}_{x}$ is the same as that of $\bar{{D}}_{z}$, which means that $\bar{{\sigma }}_{x}$ keeps constant along the thickness direction of plate and the absolute value increases with p.

Fig. 4

Dimensionless a deflection $\overline{W}$ and b stress component $\bar{{\sigma }}_{x}$ along the thickness direction of plate

5.3 Under electric displacement case

Next, we consider a FGPM plate subjected to an external electric displacement $D_{0} \text{= }0.01{\text{ C }} / {m^{2}}$ uniformly distributed on the top surface of plate. The material properties vary in a power law manner [15]:

$$\begin{aligned} c_{ij} \left( z \right)= & {} c_{ij}^{B} \left( {1 / 2-z / h} \right) ^{p}+c_{ij}^{C} \left[ {1-\left( {1 / 2-z / h} \right) ^{p}} \right] ,\nonumber \\ e_{ij} \left( z \right)= & {} e_{ij}^{B} \left( {1 / 2-z / h} \right) ^{p}+e_{ij}^{C} \left[ {1-\left( {1 / 2-z / h} \right) ^{p}} \right] ,\nonumber \\ \lambda _{ij} \left( z \right)= & {} \lambda _{ij}^{B} \left( {1 / 2-z / h} \right) ^{p}+\lambda _{ij}^{C} \left[ {1-\left( {1 / 2-z / h} \right) ^{p}} \right] , \quad i,j=1,2,...,6, \end{aligned}$$

(78)

where $c_{ij}^{B}$, $e_{ij}^{B}$, $\lambda _{ij}^{B}$, $c_{ij}^{C}$, $e_{ij}^{C}$ and $\lambda _{ij}^{C}$ are material coefficients of BaTiO$_{\textrm{3}}$ and CoFe$_{\textrm{2}}$O$_{\textrm{4}}$(listed in Table 1) [27]. The gradient index $p=$0 corresponds to BaTiO$_{\textrm{3}}$ material. We take a FGPM plate with CF boundary condition as a typical example in this case, and a set of dimensionless variables are introduced for the sake of display:

$$\begin{aligned} \bar{{U}}=\frac{u}{h}, \quad \bar{{W}}=\frac{W}{h}, \quad \bar{{\sigma }}_{x} =\frac{\sigma _{x} }{c_{11}^{C} }, \quad \bar{{\sigma }}_{z} =\frac{\sigma _{z} }{c_{11}^{C} }, \quad \bar{{\Phi }}^{*}=\frac{\phi }{h}\sqrt{\frac{\lambda _{33}^{C} }{c_{33}^{C} }}, \quad \bar{{D}}_{z} =\frac{D_{z} }{D_{0} }. \end{aligned}$$

(79)

Figure 5 shows the variations of the dimensionless displacements $\overline{W}$and $\overline{U}$ along the thickness direction of the plate. The material gradient index p is taken for 0, 5 and 10. In contrast to the case of the homogeneous piezoelectric material ($p=$0) where $\overline{W}$ appears linear distribution along thickness direction in Fig. 5a, it becomes increasingly nonlinear distribution with increased p. Besides, it can be found that the value of $\overline{W}$ increases withp. The dimensionless in-plane displacement $\overline{U}$ shows a linear variation across the thickness direction in Fig. 5b. The maximal absolute value of $\overline{U}$ occurs at the top surface of the plate, which also increases with p.

Fig. 5

Dimensionless displacements a $\overline{W}$ and b $\overline{U}$ along the thickness direction of plate

Distributions of the dimensionless stresses $\bar{{\sigma }}_{x}$ and $\bar{{\sigma }}_{z}$ along the thickness direction of the plate are illustrated in Fig. 6. It can be observed that $\bar{{\sigma }}_{x}$ basically presents a linear distribution through the thickness direction, while $\bar{{\sigma }}_{z}$ exhibit obvious nonlinear change characteristics. The maximal absolute values of $\bar{{\sigma }}_{x}$ and $\bar{{\sigma }}_{z}$ are gained on the top surface and in the upper half of the plates, respectively. We can see that $\bar{{\sigma }}_{z}$ on the top and bottom surfaces of plate exactly meet the boundary conditions in Eqs. (59)–(60). Moreover, compared with $\bar{{\sigma }}_{x}$, $\bar{{\sigma }}_{z}$ is a high-order small quantity which is consistent with that under mechanical loading.

Fig. 6

Dimensionless stress component a $\bar{{\sigma }}_{x}$ and b $\bar{{\sigma }}_{z}$ along the thickness direction of plate

As illustrated in Fig. 7a, the dimensionless electric displacement $\overline{D}_{z}$ is nonlinear along the thickness direction for FGPMs as compared with a linear distribution for the homogeneous piezoelectric material ($p=0)$. Moreover, the curve of the dimensionless electric potential $\overline{\Phi }^{*}$ is plotted in Fig. 7b which shows a nonlinear change along the thickness direction of the plate.

Fig. 7

Dimensionless a electric displacement component $\overline{D}_{z}$ and b electric potential $\overline{\Phi }^{*}$ along the thickness direction of plate

Furthermore, the effects of the boundary conditions, including SS, CC, CS and CF, on the thickness distributions of $\overline{W}$, $\bar{{\sigma }}_{x}$ and $\overline{\Phi }^{*}$ are revealed in Figs. 8a–c with $p=10$. As shown in Fig. 8a, $\overline{W}$ has negative values for the SS, CC and CS cases and obtains positive value for the CF case. Besides, $\overline{W}$ gains the maximal absolute value for the SS boundary condition. As for $\bar{{\sigma }}_{x}$ in Fig. 8b, the maximal absolute values appear on the top surface of the FGPM plates in which the CF plate has the maximum value. As shown in Fig. 8c, it is obvious that the distributions of $\overline{\Phi }^{*}$ exhibit mainly no difference for all boundary conditions within the upper part of the FGPM plates, while there is a significant difference in the lower part of the plates.

Fig. 8

Dimensionless a displacement $\overline{W}$, b stress component $\bar{{\sigma }}_{x}$ and c electric potential $\overline{\Phi }^{*}$ along the thickness direction of plate with $p=10$

6 Conclusions

Elasticity solutions of FGPM plates under electric fields in cylindrical bending are presented in this paper. The main conclusions are as follows:

• 1. The generalized Mian and Spencer plate theory is extended by constructing the electric potential functions referring to the forms of the displacement components, such that the boundary conditions on the top and bottom surfaces of plate are exactly satisfied;

• 2. The present analytical solutions are verified through comparison with the FEM results with the aid of ABAQUS software, which shows an excellent agreement;

• 3. In the numerical examples, a uniformly distributed electrical potential difference and electrical displacement are applied on the top and bottom surfaces of plate, respectively. It is concluded that the gradient index of material p, the edge boundary conditions and the distribution model of material properties have a remarkable influence on the elastic and electric fields in the FGPM plates;

• 4. Since the present theory can deal with FGPM plates with arbitrary distribution model of material properties and arbitrary edge boundary conditions, it indicates that the electro-mechanical responses of FGPM plates under electric fields can be actively controlled and optimized by regulating above-mentioned influencing factors.

It should also be emphasized that the present theory is within the framework of piezoelectricity. The proposed piezoelectricity solutions can serve as good benchmarks for the study of problem in the paper based on various simplified plate theories or numerical methods.

Appendices

Appendix A: Expressions of the introduced functions related to variable z and integral constants in electric potential case

$$\begin{aligned} g_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }} d\xi ,\qquad \left( {i,n=0,1} \right) \nonumber \\ h_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\left( {c_{11} -c_{13} \frac{c_{13} \lambda _{33} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }-e_{31} \frac{c_{13} e_{33} -c_{33} e_{31} }{e_{33}^{2} +c_{33} \lambda _{33} }} \right) } d\xi ,\quad \left( {i,n=0,1} \right) ,\nonumber \\ f_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ a_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ b_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ f_{5} \left( z \right)= & {} \int \limits _{-h / 2}^z {\frac{g_{5} \left( \xi \right) }{c_{55} }} d\xi , \quad f_{6} \left( z \right) =\int \limits _{-h / 2}^z {\frac{e_{15} }{c_{55} }} d\xi ,\nonumber \\ f_{7} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{f_{0}^{0} \left( \xi \right) +e_{15} b_{0}^{0} \left( \xi \right) }{c_{55} }-\left[ {a_{0}^{0} \left( \xi \right) -a_{0}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ f_{16} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {f_{0}^{0} \left( \xi \right) +e_{15} b_{0}^{0} \left( \xi \right) } \right] +b_{0}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ f_{17} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {h_{0}^{0} \left( \xi \right) +e_{15} f_{0}^{0} \left( \xi \right) } \right] +f_{0}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ f_{18} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {h_{1}^{0} \left( \xi \right) +e_{15} f_{1}^{0} \left( \xi \right) } \right] +f_{1}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ k_{31}^{*} \left( z \right)= & {} c_{13} \left[ {F_{1} \left( z \right) +F_{0} } \right] -h_{0}^{1} \left( z \right) , \quad k_{32}^{*} \left( z \right) =e_{31} \left[ {F_{1} \left( z \right) +F_{0} } \right] -f_{17} \left( z \right) ,\nonumber \\ k_{33}^{*} \left( z \right)= & {} c_{13} f_{7} \left( z \right) +f_{0}^{1} \left( z \right) , \quad k_{41}^{*} \left( z \right) =c_{13} \left[ {B_{1} \left( z \right) +B_{0} } \right] +h_{1}^{1} \left( z \right) ,\nonumber \\ k_{42}^{*} \left( z \right)= & {} e_{31} \left[ {B_{1} \left( z \right) +B_{0} } \right] +f_{18} \left( z \right) , \quad k_{44}^{*} \left( z \right) =e_{31} f_{7} \left( z \right) +f_{16} \left( z \right) ,\nonumber \\ B_{1} \left( z \right)= & {} -\int \limits _{-h / 2}^z {\left\{ {\frac{1}{c_{55} }\left[ {h_{1}^{0} \left( \xi \right) +e_{15} f_{1}^{0} \left( \xi \right) } \right] -\left[ {g_{1}^{0} \left( z \right) -g_{1}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ C_{1} \left( z \right)= & {} g_{1}^{0} \left( z \right) \quad D_{00} \left( z \right) =\int \limits _{-h / 2}^z {\left[ {\frac{k_{33}^{*} \left( \xi \right) \lambda _{33} +k_{44}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ D_{01} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{31}^{*} \left( \xi \right) \lambda _{33} +k_{32}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi , \quad D_{02} \left( z \right) =\int \limits _{-h / 2}^z {\left[ {\frac{k_{41}^{*} \left( \xi \right) \lambda _{33} +k_{42}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ F_{1} \left( z \right)= & {} -\int \limits _{-h / 2}^z {\left\{ {\frac{1}{c_{55} }\left[ {h_{0}^{0} \left( \xi \right) +e_{15} f_{0}^{0} \left( \xi \right) } \right] -\left[ {g_{0}^{0} \left( \xi \right) -g_{0}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ G_{1} \left( z \right)= & {} -g_{0}^{0} \left( z \right) , \quad \phi _{10} \left( z \right) =f_{0}^{0} \left( z \right) , \quad \phi _{20} \left( z \right) =-f_{1}^{0} \left( z \right) , \nonumber \\ \phi _{00} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{33}^{*} \left( \xi \right) e_{33} -k_{44}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \phi _{01} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{31}^{*} \left( \xi \right) e_{33} -k_{32}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \phi _{02} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{41}^{*} \left( \xi \right) e_{33} -k_{42}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \end{aligned}$$

(A-1)

$$\begin{aligned} B_{0}= & {} -k_{2} f_{5} \left( 0 \right) +\Phi _{2} f_{6} \left( 0 \right) -H_{4} f_{7} \left( 0 \right) -B_{1} \left( 0 \right) ,\nonumber \\ C_{0}= & {} -g_{1}^{0} \left( 0 \right) -H_{4} a_{0}^{0} \left( 0 \right) ,\nonumber \\ D_{0}= & {} \kappa _{3} \left[ {D_{00} \left( z \right) H_{2} +D_{01} \left( z \right) } \right] +\kappa _{4} \left[ {D_{00} \left( z \right) H_{4} +D_{02} \left( z \right) } \right] -a_{0}^{0} \left( z \right) H_{7},\nonumber \\ F_{0}= & {} -k_{1} f_{5} \left( 0 \right) +\Phi _{1} f_{6} \left( 0 \right) -H_{2} f_{7} \left( 0 \right) -F_{1} \left( 0 \right) ,\nonumber \\ G_{0}= & {} g_{0}^{0} \left( 0 \right) -H_{2} a_{0}^{0} \left( 0 \right) , \quad \Phi _{0} \text{= }\bar{{\Phi }}_{1} -\left( {\Phi _{2} C_{2} +\Phi _{1} C_{5} } \right) . \end{aligned}$$

(A-2)

Appendix B: Expressions of stress resultants and moments

$$\begin{aligned} N_{x}= & {} \int \limits _{-h/2}^{h/2} {\sigma _{x} dz=} N_{1} \bar{{u}}_{,x} +N_{3} \bar{{w}}_{,xx} +N_{5} \bar{{u}}_{,xxx} +N_{7} \bar{{w}}_{,xxxx} +N_{0} \nonumber \\ M_{x}= & {} \int \limits _{-h/2}^{h/2} {\sigma _{x} zdz} =M_{1} \bar{{u}}_{,x} +M_{3} \bar{{w}}_{,xx} +M_{5} \bar{{u}}_{,xxx} +M_{7} \bar{{w}}_{,xxxx} +M_{0} \nonumber \\ Q_{x}= & {} \int \limits _{-h/2}^{h/2} {\tau _{zx} dz=Q_{1} \bar{{u}}_{,xx} +Q_{2} \bar{{w}}_{,xxx} } \end{aligned}$$

(B-1)

where

$$\begin{aligned} N_{1}= & {} \int \limits _{-h/2}^{h/2} {\left( {c_{11} +c_{13} {G}'+e_{31} \phi _{1}^{\prime }} \right) } dz,N_{3} =\int \limits _{-h/2}^{h/2} {\left( {c_{11} A+c_{13} {C}'+e_{31} \phi _{2}^{\prime }} \right) } dz,\nonumber \\ N_{5}= & {} \int \limits _{-h/2}^{h/2} {c_{11} F} dz,N_{7} =\int \limits _{-h/2}^{h/2} {c_{11} B} dz,N_{0} =\int \limits _{-h/2}^{h/2} {c_{13} {D}'+e_{31} {\phi }'_{0} } dz,\nonumber \\ M_{1}= & {} \int \limits _{-h/2}^{h/2} {z\left( {c_{11} +c_{13} {G}'+e_{31} \phi _{1}^{\prime }} \right) } dz,M_{3} =\int \limits _{-h/2}^{h/2} {z\left( {c_{11} A+c_{13} {C}'+e_{31} \phi _{2}^{\prime }} \right) } dz,\nonumber \\ M_{5}= & {} \int \limits _{-h/2}^{h/2} {c_{11} zF} dz,M_{7} =\int \limits _{-h/2}^{h/2} {c_{11} zB} dz,M_{0} =\int \limits _{-h/2}^{h/2} {\left( {c_{13} {D}'+e_{31} {\phi }'} \right) z} dz,\nonumber \\ Q_{1}= & {} \int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{F}'+G} \right) +e_{15} \phi _{1} } \right] } dz,Q_{2} =\int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{B}'+C} \right) +e_{15} \phi _{2} } \right] } dz, \end{aligned}$$

(B-2)

Appendix C: Expressions of the introduced functions related to variable z and integral constants in electric displacement case

$$\begin{aligned} F_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left[ {c_{55} \kappa _{1} -c_{11} -c_{13} {G}'\left( \xi \right) -e_{31} {\phi }'_{1} \left( \xi \right) } \right] } d\xi , \nonumber \\ B_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left[ {c_{55} \kappa _{2} -c_{11} A\left( \xi \right) -c_{13} {C}'\left( \xi \right) -e_{31} {\phi }'_{2} \left( \xi \right) } \right] } d\xi ,\nonumber \\ D_{00}= & {} c_{13} \left( {-h / 2} \right) \left[ {\kappa _{3} F\left( {-h / 2} \right) +\kappa _{4} B\left( {-h / 2} \right) } \right] , \nonumber \\ D_{01}= & {} e_{31} \left( {-h / 2} \right) \left[ {\kappa _{3} F\left( {-h / 2} \right) +\kappa _{4} B\left( {-h / 2} \right) } \right] ,\nonumber \\ D_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left\{ {\kappa _{3} \left[ {c_{55} \left( {{F}'+G} \right) +\left( {Fc_{13} } \right) ^{\prime }+e_{15} \phi _{1} } \right] +\kappa _{4} \left[ {c_{55} \left( {{B}'+C} \right) +\left( {c_{13} B} \right) ^{\prime }+e_{15} \phi _{2} } \right] } \right\} } d\xi ,\nonumber \\ D_{20} \left( z \right)= & {} \int \limits _{-h/2}^z {\left\{ {\kappa _{3} \left[ {e_{15} \left( {{F}'+G} \right) +\left( {Fe_{31} } \right) ^{\prime }-\lambda _{11} \phi _{1} } \right] +\kappa _{4} \left[ {e_{15} \left( {{B}'+C} \right) +\left( {e_{31} B} \right) ^{\prime }-\lambda _{11} \phi _{2} } \right] } \right\} } d\xi . \end{aligned}$$

(C-1)

$$\begin{aligned} B_{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{1}{c_{55} }\left[ {B_{10} \left( z \right) -e_{15} \phi _{2} \left( z \right) } \right] -C\left( z \right) } \right\} } dz,\nonumber \\ C_{0}= & {} \int \limits _{-h/2}^0 {\left[ {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( z \right) } \right] } dz,\nonumber \\ D_{0}= & {} \int \limits _{-h/2}^0 {\left\{ {\frac{\lambda _{33} \left[ {D_{10} \left( z \right) +D_{00} \left( z \right) } \right] +e_{33} \left[ {D_{20} \left( z \right) +D_{01} \left( z \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} \right\} } dz,\nonumber \\ F_{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{1}{c_{55} }\left[ {F_{10} \left( z \right) -e_{15} \phi _{1} } \right] -G\left( z \right) } \right\} } dz, \quad G_{0} =\int \limits _{-h/2}^0 {\left( {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right) } dz,\nonumber \\ \Phi _{1}= & {} -\int \limits _{-h/2}^0 {\left( {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right) dz}, \quad \Phi _{2} \text{= }-\int \limits _{-h/2}^0 {\left[ {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( z \right) } \right] dz},\nonumber \\ \Phi _{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{c_{33} \left[ {D_{20} \left( z \right) +D_{01} \left( z \right) } \right] -e_{33} \left[ {D_{10} \left( z \right) +D_{00} \left( z \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} \right\} } dz. \end{aligned}$$

(C-2)