A Semi-Analytical Finite Layer Method for Analyzing the 3D Coupled Electro-Mechanical Behavior of Exponentially Graded Piezoelectric Circular Hollow Microscale Cylinders

Abstract

Within the framework of consistent couple-stress theory (CCST), we develop a semi-analytical finite layer method (FLM) to investigate the three-dimensional (3D) coupled electro-mechanical behavior of an exponentially graded (EG) piezoelectric circular hollow microscale cylinder under simply supported boundary conditions. The microscale cylinder is subjected to mechanical loads and electric voltages and is placed under closed-circuit surface conditions on its outer and inner surfaces. Using the principle of stationary potential energy, we first derive a 3D Galerkin weak formulation for this study. We divide the microscale cylinder into n_l layers and select each layer’s elastic displacements and electric potential as the primary variables. We then incorporate a layer-wise generalized displacement model into the weak formulation to develop the semi-analytical FLM. The novelty of our method lies in its ability to accurately determine the electric and elastic field variables induced in the microscale cylinder. This feature has not been explored in previous research. We rigorously validate our method’s accuracy by comparing its solutions for EG piezoelectric circular hollow macroscale cylinders with the corresponding 3D solutions reported in the literature, with the material length-scale parameter set to zero. We also examine the impact of several key factors on the coupled electro-mechanical behavior of the microscale cylinder, including the radius-to-thickness ratio, inhomogeneity index, piezoelectricity, and material length-scale parameter, which appear to be significant.

1. Introduction

Because of their direct and reverse piezoelectric effects, piezoelectric materials can convert mechanical and electrical energy into each other to achieve the functions of measuring a structure’s deformations after applying mechanical loads and suppressing a structure’s deformations after applying electric voltages. As a result, piezoelectric materials have been employed as components in sensors, controllers, actuators, resonators, and micro-electro-mechanical systems [1,2,3], which are widely used in cutting-edge industries, including aerospace, automotive, biotechnology, high-rise buildings, and submarines [4,5,6].

Functionally graded (FG) structures typically consist of two-phase materials, blended according to the volume fractions of the constituents that vary in the FG structure’s thickness direction to achieve desired structural properties, including high strength and stiffness, outstanding thermal resistance, and excellent corrosion resistance [7,8]. Accordingly, in response to the popularity of FG and exponentially graded (EG) piezoelectric structures, considerable attention has been devoted within the classical continuum mechanics (CCM) framework to developing novel computational methods for coupled electromechanical analyses of these structures [9].

The FG/EG piezoelectric structures mentioned above are also becoming increasingly miniaturized due to rapid advances in materials and manufacturing technology. As we know, the coupled electro-mechanical behavior of microscale structures differs from that of macroscale structures due to microstructure-dependent effects. Traditional CCM-based computational methods for analyzing the coupled electro-mechanical behavior of macroscale structures are not applicable to microscale structures in practice. Subsequently, several higher-order non-CCM-based theories were developed, including the strain gradient theory [10,11], the couple stress theory (CST) [12,13,14], Eringen’s nonlocal elasticity theory [15,16], doublet mechanics [17], the micropolar elasticity theory [18], and the consistent/modified couple stress theory (CCST/MCST) [19,20,21]. Because our formulation is based on the CCST, we focus the following literature review on the development of the CCST/MCST and its application to the examination of the coupled electro-mechanical behavior of FG/EG piezoelectric microscale plates and cylinders, particularly the stress and deformation behaviors.

Between 1962 and 1964, Toupin [22], Mindlin and Tiersten [23], and Koiter [24] successively developed several CSTs for an elastic microscale body. Because these CSTs are similar, they are collectively referred to as the original CST. To account for rotational kinematics, the original CST requires two material length-scale parameters, in addition to two elastic coefficients, to analyze an elastic isotropic body. Unfortunately, Eringen [25] criticized the original CST model as indeterminate, meaning the number of primary variables in the original CST exceeds the number of momentum and force equilibrium equations. This disadvantage handicapped the original CST’s development and application for a long time. To overcome this shortcoming, Hadjesfandiari and Dargush [26,27] and Yang et al. [28] developed the CCST and the MCST, respectively, by imposing skew-symmetry and symmetry on the couple-stress tensor. As a result, only one material length-scale parameter, combined with two elastic coefficients, is required in the CCST and the MCST.

Within the CCST/MCST framework, several studies have examined the mechanical behavior of FG/EG microscale plates and cylinders. Babadi and Beni [29] used Love’s classical shell theory, accounting for flexoelectricity, to investigate the coupled magneto-electro-mechanical behavior of an FG cylindrical nanocylinder, and to determine and discuss the flexoelectric effect on its free vibration. Based on the three-dimensional (3D) MCST, Wei and Qing [30] investigated various mechanical behaviors of an axisymmetric bi-directional FG circular and annular microplate, including static buckling, free vibration, stress, and deformation. Finally, Wu and Lin [31] and Wu and Hu [32] incorporated a unified displacement model into the CCST to develop a unified theory for studying the mechanical behavior of FG/EG piezoelectric/elastic microplates. Beni et al. [33] and Zeighampour and Shojaeian [34] incorporated Donnell’s thin-shell displacement model into the MCST to examine the free vibration characteristics of an FG circular hollow microcylinder and the static buckling behavior of an FG circular hollow sandwich microcylinder, respectively. Razavi et al. [35] employed the CCST to investigate the free vibration characteristics of an FG piezoelectric circular hollow nanoscale cylinder. Based on the MCST, Beni et al. [36] developed a first-order shear deformation theory (FSDT) to analyze the free vibration characteristics of an FG circular hollow microcylinder and presented Navier’s solutions for the hollow microcylinder’s lowest natural frequency. The influence of the microstructure-dependent effect on these natural frequency solutions was examined. Arefi [37] developed a microstructure-dependent refined shear deformation theory (RSDT) to examine the stress and deformation behaviors of a doubly curved piezoelectric sandwich microshell. Wang et al. [38] developed a unified higher-order shear deformation theory (HSDT) to analyze the free vibration of an FG circular hollow microcylinder. In Wang et al.’s formulation, a unified kinematic model was used to represent the distributions of transverse shear deformations along the hollow microcylinder’s thickness direction. As a result, various MCST-based shear deformation theories can be recovered by assigning specific functions in their formulation. Within a new couple-stress theory framework, Zhang et al. [39] developed an 18-degrees-of-freedom triangular quasi-conforming element to investigate the microstructure-dependent effect on the mechanical behavior of microscale structures.

After a rigorous review of the published literature, we find that most studies on the mechanical behavior of FG/EG piezoelectric circular hollow microcylinders adopt an approximate two-dimensional (2D) shear deformation theory based on the MCST, whereas fewer use the 3D CCST/MCST. As a result, 3D effects on the coupled electromechanical behavior of these microcylinders are often ignored in the existing literature, including 3D piezoelectric and couple-stress effects, as well as zig-zag deformation and thickness-stretching effects. To address these shortcomings, Wu and Lu [40] and Wu and Hsu [41] presented Hermitian C¹ and Lagrangian C⁰ semi-analytical finite layer methods (FLMs), respectively, within the 3D CCST framework to investigate the mechanical behavior of an FG/EG elastic/piezoelectric microplate. In this article, we aim to accelerate the convergence of previously published semi-analytical FLMs and extend their practical application from FG/EG microplates to FG/EG circular hollow microcylinders. First, within the 3D CCST framework, we derive a Galerkin weak formulation for the current semi-analytical FLM using the principle of stationary potential energy. We divide the microcylinder into n_l layers, represent each layer’s primary variable in the in-surface domain using a double Fourier series, and interpolate it in the thickness direction using Hermitian C² polynomial functions. Then, we employ the current semi-analytical FLM to analyze the 3D stress and deformation behavior of an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions subjected to electromechanical loads. For comparison, we modify our semi-analytical FLM to analyze an EG piezoelectric circular hollow macroscale cylinder with the material length-scale parameter set to zero, since no relevant 3D solutions for an EG piezoelectric circular hollow microcylinder have been documented in the literature. Finally, we use the current semi-analytical FLM to conduct a parametric analysis, in which the impact of several key factors on the coupled electro-mechanical behavior of an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions is examined. The factors considered particularly important are the radius-to-thickness ratio, the inhomogeneity index, piezoelectricity, and the material length-scale parameter.

2. The Basic Equations of the 3D CCST

Hadjesfandiari [27] developed the 3D CCST to analyze a dielectric microbody’s coupled electro-mechanical behavior, which accounts for 3D piezoelectric and couple-stress tensor effects, in addition to the 3D force-stress tensor effect. In the 3D CCST, the force-stress tensor (${\sigma }_{ij}$), the skew-symmetric couple-stress tensor (${\mu }_{ij}$), and the electric displacement tensor (D_k) are induced at a material point when the dielectric microscale body deforms. The force-stress tensor induced in a loaded microscale body is generally asymmetric due to the couple-stress tensor effect. Hadjesfandiari thus decomposed the force-stress tensor into a skew-symmetric part (${\sigma }_{\left[ij\right]}$) and a symmetric part (${\sigma }_{\left(ij\right)}$), such that ${\sigma }_{ij}={\sigma }_{\left(ij\right)}+{\sigma }_{\left[ij\right]}$.

In the 3D CCST formulation, Hadjesfandiari deduced the force-stress tensor’s skew-symmetric part in terms of the couple-stress tensor as follows [27]:

$${\sigma }_{[ji]}=-\left(1/2\right)\left({\mu }_i{,}_j-{\mu }_j{,}_i\right),$$

(1)

where ${\mu }_k={\mu }_{ji}=-{\mu }_{ij}$, $i\ne j\ne k$, and the subscripts i, j, and k permute in a natural order.

Hadjesfandiari also expressed the strain energy density functional of a dielectric microscale body as a function of the skew-symmetric part of the curvature tensor (${\kappa }_k$), the strain tensor (${\epsilon }_{ij}$), and the electric field tensor (E_k), for which ${\kappa }_k={\kappa }_{ji}=-{\kappa }_{ij}$. The skew-symmetric part of the curvature tensor is conjugated with the skew-symmetric couple-stress tensor (${\mu }_{ij}$); the strain tensor is symmetric and conjugated with the symmetric part of the force-stress tensor (${\sigma }_{\left(ij\right)}$); and the electric field tensor is conjugated with the electric displacement tensor (D_k). Thus, the strain energy stored in a dielectric microscale body occupying a volume Ω can be expressed as follows [27]:

$$U_s={\displaystyle {\int }_{\mathrm{\Omega }}\left[\left(1/2\right){\sigma }_{\left(ij\right)} {\epsilon }_{ij}-{\mu }_k{\kappa }_k-\left(1/2\right)D_k{E}_k\right]} d\mathrm{\Omega },$$

(2)

where ${\epsilon }_{ij}=\left(u_i{,}_j+u_j{,}_i\right)/2,$ and $u_i$ is the displacement tensor; ${\kappa }_k=\left(1/2\right) \mathrm{curl} {\theta }_k$, where ${\theta }_k$ denotes the rotation tensor, and ${\theta }_k={\theta }_{ji}=\left(u_j{,}_i-u_i{,}_j\right)/2$; and $E_k=-\Phi {,}_k$, where $\Phi$ is the electric potential.

3. The Semi-Analytical FLM

In this work, we aim to develop a semi-analytical FLM to analyze the 3D stress and deformation behavior of an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions. The microcylinder of interest is subjected to electromechanical loads and is placed under closed-circuit surface conditions on its outer and inner surfaces. A schematic diagram of the microcylinder under a sinusoidally distributed mechanical load is shown in Figure 1. In our formulation, the microcylinder is divided into n_l layers, each with equal thickness unless otherwise specified. A cylindrical coordinate system (i.e., x, $\theta$, and r coordinates) is placed at the center of the cross-section at the left-hand side of the microcylinder. A global thickness coordinate, $\zeta$, and a set of local thickness coordinates, $z_m (m=1, 2, \dots , n_l)$, are placed on the microcylinder’s and each layer’s mid-surfaces, respectively. The thickness of the microcylinder is defined as h, and the thickness of its mth layer is defined as h_m, such that ${\displaystyle \sum _{m=1}^{n_l}{h}_m= }h.$ The variables L and R represent the microcylinder’s length and mid-surface radius, respectively. The relationship between the global thickness coordinate and the radial coordinate is $r=R+\zeta$. In addition, the relationship between the local thickness coordinate and the global coordinate of the mth layer can be expressed as $\zeta ={\overline{\zeta }}_m+z_m$, where ${\overline{\zeta }}_m=\left({\zeta }_m+{\zeta }_{m-1}\right)/2$. The variables ${\zeta }_{m-1} \mathrm{and} {\zeta }_m$ are defined as the global thickness coordinates, measured from the microcylinder’s mid-surface to the bottom and top surfaces of the mth layer, respectively.

3.1. Generalized Displacement Model

Within the current semi-analytical FLM framework, the generalized displacement model for each layer is expressed as follows:

$$u_{ x}^{(m)}\left(x, \theta , z_m\right)={\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)}} d_{ ui}^{(m)},$$

(3)

$$u_{ \theta }^{(m)}\left(x, \theta , z_m\right)={\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)}} d_{ vi}^{(m)},$$

(4)

$$u_{ r}^{(m)}\left(x, \theta , z_m\right)={\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)}} d_{ wi}^{(m)},$$

(5)

$${\Phi }^{(m)}\left(x, \theta , z_m\right)={\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)}} d_{ \varphi i}^{(m)},$$

(6)

where ${\psi }_i^{(m)}=\left\{\begin{array}{ccc}{\psi }_{3i-2}^{(m)}& {\psi }_{ 3i-1}^{(m)}& {\psi }_{ 3i}^{(m)}\end{array}\right\}$, and the variables ${\psi }_{ i}^{(m)}$ ($i=1, 2, \dots , 3n_d$) denote the interpolation functions, which are provided in Appendix A; $d_{ ui}^{(m)}={\left\{\begin{array}{ccc}{u}_{ i}^{(m)}& {\theta }_{ ui}^{(m)}& {\kappa }_{ ui}^{(m)}\end{array}\right\}}^T$, $d_{ vi}^{(m)}={\left\{\begin{array}{ccc}{v}_{ i}^{(m)}& {\theta }_{ vi}^{(m)}& {\kappa }_{ vi}^{(m)}\end{array}\right\}}^T$, $d_{ wi}^{(m)}={\left\{\begin{array}{ccc}{w}_{ i}^{(m)}& {\theta }_{ wi}^{(m)}& {\kappa }_{ wi}^{(m)}\end{array}\right\}}^T,$ and $d_{ \varphi i}^{(m)}={\left\{\begin{array}{ccc}{\varphi }_{ i}^{(m)}& {\theta }_{ \varphi i}^{(m)}& {\kappa }_{ \varphi i}^{(m)}\end{array}\right\}}^T,$ in which the variables ${\theta }_{ fi}^{(m)}$ and ${\kappa }_{ fi}^{(m)}$ are defined as ${\theta }_{ fi}^{(m)}=d{f}_{ i }^{(m)}/d{z}_m$ and ${\kappa }_{ fi}^{(m)}=d^2{f}_{ i }^{(m)}/d{z}_m^2$, and $f_{ i }^{(m)}=u_{ i}^{(m)}, v_{ i}^{(m)}, w_{ i}^{(m)}, \mathrm{and} {\varphi }_{ i}^{(m)}$, which represent the first- and second-order derivatives of elastic displacements and the electric potential; the symbol n_d represents the total number of a typical layer element’s nodal surfaces, and the value of n_d is set at two in this article.

The linear constitutive equations for orthotropic piezoelectric materials are expressed as follows [27]:

The relationship between the generalized force stresses and strains is [27]

$$\left\{\begin{array}{c}{\sigma }_{ \left(xx\right)}^{(m)}\\ {\sigma }_{ \left(\theta \theta \right)}^{(m)}\\ {\sigma }_{\left( rr\right)}^{(m)}\\ {\sigma }_{\left(\theta r\right)}^{(m)}\\ {\sigma }_{\left(xr\right)}^{(m)}\\ {\sigma }_{\left(x\theta \right)}^{(m)}\end{array}\right\}=\left[\begin{array}{cccccc}{c}_{11}^{(m)}& c_{12}^{(m)}& c_{13}^{(m)}& 0& 0& 0\\ c_{12}^{(m)}& c_{22}^{(m)}& c_{23}^{(m)}& 0& 0& 0\\ c_{13}^{(m)}& c_{23}^{(m)}& c_{33}^{(m)}& 0& 0& 0\\ 0& 0& 0& c_{44}^{(m)}& 0& 0\\ 0& 0& 0& 0& c_{55}^{(m)}& 0\\ 0& 0& 0& 0& 0& c_{66}^{(m)}\end{array}\right] \left\{\begin{array}{c}{\epsilon }_{ xx}^{(m)}\\ {\epsilon }_{ \theta \theta }^{(m)}\\ {\epsilon }_{ rr}^{(m)}\\ {\gamma }_{\theta r}^{(m)}\\ {\gamma }_{xr}^{(m)}\\ {\gamma }_{x\theta }^{(m)}\end{array}\right\}-\left[\begin{array}{ccc}0& 0& e_{31}^{(m)}\\ 0& 0& e_{32}^{(m)}\\ 0& 0& e_{33}^{(m)}\\ 0& e_{24}^{(m)}& 0\\ e_{15}^{(m)}& 0& 0\\ 0& 0& 0\end{array}\right] \left\{\begin{array}{c}{E}_{ x}^{(m)}\\ E_{ \theta }^{(m)}\\ E_{ r}^{(m)}\end{array}\right\},$$

(7)

where $c_{ij}^{(m)} (i, j=1–6)$ represent the elastic coefficients; and $e_{ jk}^{(m)} (j=1–3, k=1–6)$ represent the piezoelectric coefficients.

The relationship between the electric displacements and generalized strains is [27]

$$\left\{\begin{array}{c}{D}_{ x}^{(m)}\\ D_{ \theta }^{(m)}\\ D_{ r}^{(m)}\end{array}\right\}=\left[\begin{array}{cccccc}0& 0& 0& 0& e_{15}^{(m)}& 0\\ 0& 0& 0& e_{24}^{(m)}& 0& 0\\ e_{31}^{(m)}& e_{32}^{(m)}& e_{33}^{(m)}& 0& 0& 0\end{array}\right] \left\{\begin{array}{c}{\epsilon }_{ xx}^{(m)}\\ {\epsilon }_{ \theta \theta }^{(m)}\\ {\epsilon }_{ rr}^{(m)}\\ {\gamma }_{\theta r}^{(m)}\\ {\gamma }_{xr}^{(m)}\\ {\gamma }_{x\theta }^{(m)}\end{array}\right\}+\left[\begin{array}{ccc}{\eta }_{11}^{(m)}& 0& 0\\ 0& {\eta }_{22}^{(m)}& 0\\ 0& 0& {\eta }_{33}^{(m)}\end{array}\right] \left\{\begin{array}{c}{E}_{ x}^{(m)}\\ E_{ \theta }^{(m)}\\ E_{ r}^{(m)}\end{array}\right\},$$

(8)

where ${\eta }_{kk}^{(m)} (k=1–3)$ represents the dielectric permeability coefficients.

The relationship between the couple stresses and anti-symmetric parts of the curvatures is [27]

$$\left\{\begin{array}{c}{\mu }_{ x}^{(m)}\\ {\mu }_{ \theta }^{(m)}\\ {\mu }_{ r}^{(m)}\end{array}\right\}=-\left({\displaystyle \frac{1}{2}}\right)\left[\begin{array}{ccc}{b}_{11}^{(m)}& 0& 0\\ 0& b_{22}^{(m)}& 0\\ 0& 0& b_{33}^{(m)}\end{array}\right] \left\{\begin{array}{c}{\kappa }_{ x}^{(m)}\\ {\kappa }_{ \theta }^{(m)}\\ {\kappa }_r^{(m)}\end{array}\right\},$$

(9)

where the symbols $b_{kk}^{(m)} (k=1–3)$ denote the material length-scale coefficients given in Equation (9). They are defined as $b_{ 11}^{(m)}=16 G_{32}^{(m)} l_1^2$, $b_{ 22}^{(m)}=16 G_{13}^{(m)} l_2^2,$ and $b_{ 33}^{(m)}=16 G_{ 21}^{(m)} l_3^2,$ where the symbol $l_{ i}^{(m)}$ denotes the material length-scale parameter associated with the kj-surface of the mth layer and the symbol $G_{ ij}^{(m)}$ represents the shear modulus related to the ij-surface of the mth layer. When an isotropic material is considered, the coefficients $b_{ kk}^{(m)} (k=1–3)$ reduce to $b_{ 11}^{(m)}=b_{ 22}^{(m)}=b_{ 33}^{(m)}=16 G{l}^2$.

The strain–displacement relationships for the mth layer are given by [42]

$${\epsilon }_{ xx}^{(m)}={\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)}{d}_{ ui}^{(m)}{,}_x},$$

(10)

$${\epsilon }_{ \theta \theta }^{(m)}={\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(1/r\right){\psi }_{ i}^{(m)}} d_{ vi }^{(m)}{,}_{\theta }+{\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(1/r\right)} {\psi }_{ i}^{(m)} d_{ wi }^{(m)},$$

(11)

$${\epsilon }_{ rr}^{(m)}={\displaystyle \sum _{i=1}^{n_d} \left(D{\psi }_{ i}^{(m)}\right)} d_{ wi}^{(m)},$$

(12)

$${\gamma }_{ xr}^{(m)}={\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D{\psi }_{ i}^{(m)}\right)} d_{ ui}^{(m)}+ {\displaystyle {\displaystyle \sum _{i=1}^{n_d}} {\psi }_{ i}^{(m)}} d_{ wi}^{(m)}{,}_x,$$

(13)

$${\gamma }_{ \theta r}^{(m)}={\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D{\psi }_{ i}^{(m)}\right)} d_{ vi}^{(m)}-\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} {\psi }_{ i}^{(m)}} d_{ vi}^{(m)}+\left(1/r\right) {\displaystyle {\displaystyle \sum _{i=1}^{n_d}} {\psi }_{ i}^{(m)}} d_{ wi}^{(m)}{,}_{\theta },$$

(14)

$${\gamma }_{ x\theta }^{(m)}=\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} {\psi }_{ i}^{(m)}} d_{ ui}^{(m)}{,}_{\theta }+ {\displaystyle {\displaystyle \sum _{i=1}^{n_d}} {\psi }_{ i}^{(m)}} d_{ vi}^{(m)}{,}_x,$$

(15)

where m = 1, 2, …, n_l; the commas denote partial differentiation with respect to the suffix variables, and $D {\psi }_{ i}^{(m)}=d \left({\psi }_{ i}^{(m)}\right)/dr=d \left({\psi }_{ i}^{(m)}\right)/d\zeta =d \left({\psi }_{ i}^{(m)}\right)/d{z}_m.$

The relationships between the electric field components and the electric potential in the mth layer are expressed as follows [27]:

$$E_{ x}^{(m)}=-{\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)} d_{ \varphi i}^{(m)}{,}_x},$$

(16)

$$E_{ \theta }^{(m)}=-\left(1/r\right){\displaystyle \sum _{i=1}^{n_d} {\psi }_{ i}^{(m)} d_{ \varphi i}^{(m)}{,}_{\theta }},$$

(17)

$$E_{ r}^{(m)}=-{\displaystyle \sum _{i=1}^{n_d} \left(D {\psi }_{ i}^{(m)}\right) d_{ \varphi i}^{(m)}}.$$

(18)

The relationships between the skew-symmetric part of the curvature and the elastic displacements for the mth layer are expressed as follows [27]:

$$\begin{array}{c}{\kappa }_{ x}^{(m)}=\left(1/4\right)\left[-\left(1/r^2\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }{\psi }_{ i}^{(m)} d_{ ui}^{(m)}{,}_{\theta \theta }-\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }\left(D{\psi }_{ i}^{(m)}\right) d_{ ui}^{(m)}-{\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }\left(D^2{\psi }_{ i}^{(m)}\right) d_{ ui}^{(m)}\right.\\ \left. +\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }{\psi }_{ i}^{(m)} d_{ vi}^{(m)}{,}_{x\theta }+\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }{\psi }_{ i}^{(m)} d_{ wi}^{(m)}{,}_x+ {\displaystyle {\displaystyle \sum _{i=1}^{n_d}} }\left(D{\psi }_{ i}^{(m)}\right) d_{ wi}^{(m)}{,}_x\right],\end{array}$$

(19)

$$\begin{array}{c}{\kappa }_{ \theta }^{(m)}=\left(1/4\right)\left[\left(1/r\right){\displaystyle \sum _{i=1}^{n_d}{\psi }_{ i}^{(m)} d_{ ui}^{(m)}{,}_{x\theta }}-\right.{\displaystyle {\displaystyle \sum _{i=1}^{n_d}}{\psi }_{ i}^{(m)}} d_{ vi}^{(m)}{,}_{xx}-\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D{\psi }_{ i}^{(m)}\right) }{d}_{ vi}^{(m)}-{\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D^2{\psi }_{ i}^{(m)}\right)d_{ vi}^{(m)} }\\ \left.+\left(1/r^2\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}{\psi }_{ i}^{(m)} }{d}_{ vi}^{(m)}-\left(1/r^2\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}{\psi }_{ i}^{(m)} }{d}_{ wi}^{(m)}{,}_{\theta }+\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D{\psi }_{ i}^{(m)}\right) }{d}_{ wi}^{(m)}{,}_{\theta }\right],\end{array}$$

(20)

$$\begin{array}{c} {\kappa }_{ r}^{(m)}=\left(1/4\right)\left[{\displaystyle \sum _{i=1}^{n_d}\left(D{\psi }_{ i}^{(m)}\right)d_{ui}^{(m)}{,}_x+ }\right.\left(1/r^2\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}{\psi }_i^{(m)} d_{vi}^{(m)}{,}_{\theta }+}\left(1/r\right){\displaystyle {\displaystyle \sum _{i=1}^{n_d}}\left(D{\psi }_i^{(m)}\right)d_{vi}^{(m)}{,}_{\theta }}\\ \left.-{\displaystyle \sum _{i=1}^{n_d}{\psi }_{ i}^{(m)} d_{wi}^{(m)}{,}_{xx}-\left(1/r^2\right){\displaystyle \sum _{i=1}^{n_d}{\psi }_{ i}^{(m)} d_{wi}^{(m)}{,}_{\theta \theta }}}\right],\end{array}$$

(21)

where $D^2{\psi }_{ i}^{(m)}=d^2{\psi }_{ i}^{(m)}/d{r}^2=d^2{\psi }_{ i}^{(m)}/d{\zeta }^2=d^2{\psi }_{ i}^{(m)}/d{z}_m^2$.

3.2. The Principle of Stationary Potential Energy

The Euler–Lagrange equations for the current semi-analytical FLM can be derived from the principle of stationary potential energy [43]. The potential energy functional for the microcylinder is given by

$${\mathrm{\Pi }}_p={\displaystyle \sum _{m=1}^{n_l} }{U}_s^{(m)}-W,$$

(22)

where the symbols $U_{ s}^{(m)}$ and W, respectively, represent the strain energy functional of the mth layer of the microcylinder and the work done by the electromechanical loads. They are given as follows:

$$\begin{array}{c}{U}_{ s}^{(m)}=\left(1/2\right){\displaystyle {\int }_{{\zeta }_{m-1}}^{{\zeta }_m} {\displaystyle {\iint }_{\mathrm{\Omega }} \left[{\sigma }_{\left(xx\right)}^{(m)} {\epsilon }_{ xx}^{(m)}+{\sigma }_{\left(\theta \theta \right)}^{(m)} {\epsilon }_{ \theta \theta }^{(m)}+{\sigma }_{\left(rr\right)}^{(m)} {\epsilon }_{rr}^{(m)}+{\sigma }_{\left(xr\right)}^{(m)} {\gamma }_{xr}^{(m)}+{\sigma }_{\left(\theta r\right)}^{(m)} {\gamma }_{\theta r}^{(m)}+{\sigma }_{\left(x\theta \right)}^{(m)} {\gamma }_{ x\theta }^{(m)}\right.}}\\ \left.-D_{ x}^{(m)}{E}_{ x}^{(m)}-D_{ \theta }^{(m)}{E}_{ \theta }^{(m)}-D_{ r}^{(m)}{E}_{ r}^{(m)}-2 {\mu }_{ x}^{(m)} {\kappa }_{ x}^{(m)}-2 {\mu }_{ \theta }^{(m)} {\kappa }_{ \theta }^{(m)}-2 {\mu }_{ r}^{(m)} {\kappa }_{ r}^{(m)}\right] r dx d\theta d\zeta ,\end{array}$$

(23)

$$W={\displaystyle {\iint }_{{\mathrm{\Omega }}^{+}}{\overline{q}}^{+}}{u}_{\zeta }^{+} \left(R+h/2\right) dx d\theta +{\displaystyle {\iint }_{{\mathrm{\Omega }}^{-}}{\overline{q}}^{-}}{u}_{\zeta }^{-} \left(R-h/2\right) dx d\theta$$

(24)

where the symbol $\mathrm{\Omega }$ denotes the $x\theta -$ surface domain of the microcylinder. The symbols ${\mathrm{\Omega }}^{-}$ and ${\mathrm{\Omega }}^{+}$ represent the microcylinder’s inner and outer surface domains, respectively. The symbols ${\overline{q}}^{-} \mathrm{and} {\overline{q}}^{+}$ are the external loads applied to the microcylinder’s inner and outer surfaces. The symbols $u_{ \zeta }^{-} \mathrm{and} u_{ \zeta }^{+}$ represent the elastic displacements of the microcylinder’s inner and outer surfaces, respectively.

By applying the principle of stationary potential energy, we obtain the following equations:

$$\begin{array}{c} \delta {\mathrm{\Pi }}_p={\displaystyle \sum _{m=1}^{n_l} }\left(\delta U_s^{(m)}\right)-\delta W\\ =0,\end{array}$$

(25)

where

$$\delta U_{ s}^{(m)}={\displaystyle {\iint }_{\mathrm{\Omega }}{\displaystyle {\int }_{{\zeta }_{m-1}}^{{\zeta }_m} \left\{{\left(\delta {\mathbf{\epsilon }}_n^{(m)}\right)}^T{\mathbf{\sigma }}_{ n}^{(m)}+{\left(\delta {\mathbf{\epsilon }}_{ s}^{(m)}\right)}^T{\mathbf{\sigma }}_{ s}^{(m)}-{\left(\delta E^{(m)}\right)}^T{D}^{(m)}-2{\left(\delta {\mathbf{\kappa }}^{(m)}\right)}^T{\mathbf{\mu }}^{(m)}\right\}} } r dx d\theta d\zeta ,$$

(26)

$$\delta W={\displaystyle {\iint }_{{\mathrm{\Omega }}^{+}}{\overline{q}}^{+}\delta }{u}_{\zeta }^{+} \left(R+h/2\right) dx d\theta +{\displaystyle {\iint }_{{\mathrm{\Omega }}^{-}}{\overline{q}}^{-}}\delta u_{\zeta }^{-} \left(R-h/2\right) dx d\theta$$

(27)

$${\mathbf{\epsilon }}_{ n}^{(m)}={\left[\begin{array}{ccc}{\epsilon }_{ xx}^{(m)}& {\epsilon }_{ \theta \theta }^{(m)}& {\epsilon }_{ rr}^{(m)}\end{array}\right]}^T=B_{ 1}^{(m)} u^{(m)}+B_{ 2}^{(m)} w^{(m)},$$

(28)

$${\mathbf{\epsilon }}_{ s}^{(m)}={\left[\begin{array}{ccc}{\gamma }_{\theta r}^{(m)}& {\gamma }_{xr}^{(m)}& {\gamma }_{x\theta }^{(m)}\end{array}\right]}^T=B_3^{(m)} u^{(m)}+B_4^{(m)} w^{(m)}$$

(29)

$${{\rm E}}^{(m)}={\left[\begin{array}{ccc}{E}_{ x}^{(m)}& E_{ \theta }^{(m)}& E_r^{(m)}\end{array}\right]}^T=-B_{ 5}^{(m)} {\mathbf{\Phi }}^{(m)},$$

(30)

$${\mathbf{\kappa }}^{(m)}={\left[\begin{array}{ccc}{\kappa }_{ x}^{(m)}& {\kappa }_{ \theta }^{(m)}& {\kappa }_r^{(m)}\end{array}\right]}^T=B_{ 6}^{(m)} u^{(m)}+B_{ 7}^{(m)} w^{(m)},$$

(31)

$$u^{(m)}=\left[\begin{array}{c}{d}_{ ui }^{(m)}\\ d_{ vi}^{(m)}\end{array}\right],$$

(32)

$$w^{(m)}=\left[d_{ wi }^{(m)}\right],$$

(33)

$${\mathbf{\Phi }}^{(m)}=\left[d_{ \varphi i}^{(m)}\right],$$

(34)

$$\begin{array}{ll}{\mathbf{\sigma }}_{ n}^{(m)}& ={\left[\begin{array}{ccc}{\mathrm{\sigma }}_{\left(x x\right)}^{(m)}& {\mathrm{\sigma }}_{\left(\theta \theta \right)}^{(m)}& {\mathrm{\sigma }}_{\left(rr\right)}^{(m)}\end{array}\right]}^T\\ & =Q_{ cn}^{(m)} {\mathbf{\epsilon }}_{ n}^{(m)}-Q_{ en}^{(m)} E^{(m)}\\ & =Q_{ cn}^{(m)}{B}_1^{(m)} u^{(m)}+Q_{ cn}^{(m)}{B}_{ 2}^{(m)} w^{(m)}+Q_{ en}^{(m)} B_{ 5}^{(m)} {\mathbf{\Phi }}^{(m)},\end{array}$$

(35)

$$\begin{array}{ll}{\mathbf{\sigma }}_{ s}^{(m)}& ={\left[\begin{array}{ccc}{\mathrm{\sigma }}_{ \left(\theta r\right)}^{(m)}& {\mathrm{\sigma }}_{ \left(xr\right)}^{(m)}& {\mathrm{\sigma }}_{ \left(x\theta \right)}^{(m)}\end{array}\right]}^T\\ & =Q_{ cs}^{(m)} {\mathbf{\epsilon }}_{ s}^{(m)}-Q_{ es}^{(m)} E^{(m)}\\ & =Q_{ cs}^{(m)} B_{ 3}^{(m)} u^{(m)}+Q_{ cs}^{(m)} B_{ 4}^{(m)} w^{(m)}+Q_{ es}^{(m)} B_{ 5}^{(m)} {\mathbf{\Phi }}^{(m)},\end{array}$$

(36)

$$\begin{array}{ll}{D}^{(m)}& ={\left[\begin{array}{ccc}{D}_{ x}^{(m)}& D_{ \theta }^{(m)}& D_{ r}^{(m)}\end{array}\right]}^T\\ & ={\left(Q_{ en}^{(m)}\right)}^T{\mathbf{\epsilon }}_{ n}^{(m)}+{\left(Q_{ es}^{(m)}\right)}^T{\mathbf{\epsilon }}_{ s}^{(m)}+Q_{ \eta }^{(m)} E^{(m)}\\ & =\left[{\left(Q_{ en}^{(m)}\right)}^T{B}_{ 1}^{(m)}+{\left(Q_{ es}^{(m)}\right)}^T{B}_{ 3}^{(m)}\right] u^{(m)}+\left[{\left(Q_{ en}^{(m)}\right)}^T{B}_{ 2}^{(m)}+{\left(Q_{ es}^{(m)}\right)}^T{B}_{ 4}^{(m)}\right] w^{(m)}-Q_{ \eta }^{(m)} B_{ 5}^{(m)} {\mathbf{\Phi }}^{(m)},\end{array}$$

(37)

$$\begin{array}{ll}{\mathbf{\mu }}^{(m)}& ={\left[\begin{array}{ccc}{\mu }_{ x}^{(m)}& {\mu }_{ \theta }^{(m)}& {\mu }_{ r}^{(m)}\end{array}\right]}^T\\ & =-\left(1/2\right)Q_{ b}^{(m)} {\mathbf{\kappa }}^{(m)}\\ & =-\left(1/2\right) \left(Q_{ b}^{(m)}{B}_{ 6}^{(m)} u^{(m)}+Q_{ b}^{(m)}{B}_{ 7}^{(m)} w^{(m)}\right),\end{array}$$

(38)

$$Q_{ cn}^{(m)}=\left[\begin{array}{ccc}{c}_{ 11}^{(m)}& c_{ 12}^{(m)}& c_{ 13}^{(m)}\\ c_{ 12}^{(m)}& c_{ 22}^{(m)}& c_{ 23}^{(m)}\\ c_{ 13}^{(m)}& c_{ 23}^{(m)}& c_{ 33}^{(m)}\end{array}\right],$$

(39)

$$Q_{ cs}^{(m)}=\left[\begin{array}{ccc}{c}_{ 44}^{(m)}& 0& 0\\ 0& c_{ 55}^{(m)}& 0\\ 0& 0& c_{ 66}^{(m)}\end{array}\right],$$

(40)

$$Q_{ en}^{(m)}=\left[\begin{array}{ccc}0& 0& e_{ 31}^{(m)}\\ 0& 0& e_{ 32}^{(m)}\\ 0& 0& e_{ 33}^{(m)}\end{array}\right],$$

(41)

$$Q_{ es}^{(m)}=\left[\begin{array}{ccc}0& e_{ 24}^{(m)}& 0\\ e_{ 15}^{(m)}& 0& 0\\ 0& 0& 0\end{array}\right],$$

(42)

$$Q_{ \eta }^{(m)}=\left[\begin{array}{ccc}{\eta }_{ 11}^{(m)}& 0& 0\\ 0& {\eta }_{ 22}^{(m)}& 0\\ 0& 0& {\eta }_{ 33}^{(m)}\end{array}\right],$$

(43)

$$Q_{ b}^{(m)}=\left[\begin{array}{ccc}{b}_{ 11}^{(m)}& 0& 0\\ 0& b_{ 22}^{(m)}& 0\\ 0& 0& b_{ 33}^{(m)}\end{array}\right],$$

(44)

$$B_1^{(m)}=\left[\begin{array}{cc}{\psi }_{ i}^{(m)}{\partial }_x& 0\\ 0& \left({\psi }_{ i}^{(m)}/r\right){\partial }_{\theta }\\ 0& 0\end{array}\right],$$

(45)

$$B_{ 2}^{(m)}=\left[\begin{array}{c}0\\ {\psi }_{ i}^{(m)}/r\\ D{\psi }_{ i}^{(m)}\end{array}\right],$$

(46)

$$B_{ 3}^{(m)}=\left[\begin{array}{cc}0& D{\psi }_{ i}^{(m)}-\left({\psi }_{ i}^{(m)}/r\right)\\ D{\psi }_{ i}^{(m)}& 0\\ \left({\psi }_{ i}^{(m)}/r\right){\partial }_{\theta }& {\psi }_{ i}^{(m)}{\partial }_x\end{array}\right],$$

(47)

$$B_{ 4}^{(m)}=\left[\begin{array}{c}\left({\psi }_{ i}^{(m)}/r\right) {\partial }_{\theta }\\ {\psi }_{ i}^{(m)} {\partial }_x\\ 0\end{array}\right],$$

(48)

$$B_{ 5}^{(m)}=\left[\begin{array}{c}{\psi }_{ i}^{(m)} {\partial }_x\\ \left({\psi }_{ i}^{(m)}/r\right) {\partial }_{\theta }\\ D{\psi }_{ i}^{(m)}\end{array}\right],$$

(49)

$$B_{ 6}^{(m)}=\left(1/4\right)\left[\begin{array}{cc}-\left[\left({\psi }_{ i}^{(m)}/r^2\right){\partial }_{\theta \theta }+\left(1/r\right)D{\psi }_{ i}^{(m)}+D^2{\psi }_{ i}^{(m)}\right]& \left({\psi }_{ i}^{(m)}/r\right){\partial }_{x\theta }\\ \left({\psi }_{ i}^{(m)}/r\right){\partial }_{x\theta }& -\left[{\psi }_{ i}^{(m)}{\partial }_{xx}+\left(1/r\right)D{\psi }_{ i}^{(m)}+D^2{\psi }_{ i}^{(m)}-\left({\psi }_{ i}^{(m)}/r^2\right)\right]\\ \left(D{\psi }_{ i}^{(m)}\right){\partial }_x& \left({\psi }_{ i}^{(m)}/r^2\right){\partial }_{\theta }+\left(D{\psi }_{ i}^{(m)}/r\right){\partial }_{\theta }\end{array}\right],$$

(50)

$$B_{ 7}^{(m)}=\left(1/4\right)\left[\begin{array}{c}\left({\psi }_{ i}^{(m)}/r\right) {\partial }_x+\left(D{\psi }_{ i}^{(m)}\right) {\partial }_x\\ -\left({\psi }_{ i}^{(m)}/r^2\right) {\partial }_{\theta }+\left(D{\psi }_{ i}^{(m)}/r\right) {\partial }_{\theta }\\ -\left[{\psi }_{ i}^{(m)}{\partial }_{xx}+\left({\psi }_{ i}^{(m)}/r^2\right){\partial }_{\theta \theta }\right]\end{array}\right],$$

(51)

where $i=1, 2,\cdots , n_d$, and $m=1, 2,\cdots , n_l$.

3.3. Surface and Edge Boundary Conditions and Continuity Conditions

Following Hadjesfandiari [27], the surface conditions of the microcylinder are specified as follows:

Case 1 (Sensors):

$$\begin{array}{l}\left\{{\sigma }_{ rx}^{(n_l)}\left(x, \theta , h/2\right) {\sigma }_{ r\theta }^{(n_l)}\left(x, \theta , h/2\right) {\sigma }_{rr}^{(n_l)}\left(x, \theta , h/2\right) {\Phi }^{(n_l)}\left(x, \theta , h/2\right) {\mu }_{ x}^{\left(n_l\right)}\left(x, \theta , h/2\right) {\mu }_{ \theta }^{\left(n_l\right)}\left(x, \theta , h/2\right)\right\}\\ =\left\{0 0 {\overline{q}}_{ r}^{+} 0 0 0\right\} \mathrm{on} \mathrm{the} \mathrm{outer} \mathrm{surface}, \end{array}$$

(52)

$$\begin{array}{l}\left\{{\sigma }_{ rx}^{(1)}\left(x, \theta , -h/2\right) {\sigma }_{ r\theta }^{(1)}\left(x, \theta , -h/2\right) {\sigma }_{rr}^{(1)}\left(x, \theta , -h/2\right) {\Phi }^{(1)}\left(x, \theta , -h/2\right) {\mu }_{ x}^{\left(1\right)}\left(x, \theta , -h/2\right) {\mu }_{\theta }^{\left(1\right)}\left(x, \theta , -h/2\right)\right\}\\ =\left\{0 0 {\overline{q}}_{ r}^{-} 0 0 0\right\} \mathrm{on} \mathrm{the} \mathrm{inner} \mathrm{surface}.\end{array}$$

(53)

Case 2 (Actuators):

$$\begin{array}{l}\left\{{\sigma }_{ rx}^{(n_l)}\left(x, \theta , h/2\right) {\sigma }_{ r\theta }^{(n_l)}\left(x, \theta , h/2\right) {\sigma }_{rr}^{(n_l)}\left(x, \theta , h/2\right) {\Phi }^{(n_l)}\left(x, \theta , h/2\right) {\mu }_{ x}^{(n_l)}\left(x, \theta , h/2\right) {\mu }_{ \theta }^{(n_l)}\left(x, \theta , h/2\right)\right\}\\ =\left\{0 0 0 {\overline{\Phi }}^{+} 0 0\right\} \mathrm{on} \mathrm{the} \mathrm{outer} \mathrm{surface},\end{array}$$

(54)

$$\begin{array}{l}\left\{{\sigma }_{ rx}^{(1)}\left(x, \theta , -h/2\right) {\mathrm{\sigma }}_{ r\theta }^{(1)}\left(x, \theta , -h/2\right) {\sigma }_{rr}^{(1)}\left(x, \theta , -h/2\right) {\Phi }^{(1)}\left(x, \theta , -h/2\right) {\mu }_{ x}^{(1)}\left(x, \theta , -h/2\right) {\mu }_{ \theta }^{(1)}\left(x, \theta , -h/2\right)\right\}\\ =\left\{0 0 0 {\overline{\Phi }}^{-} 0 0\right\} \mathrm{on} \mathrm{the} \mathrm{inner} \mathrm{surface}.\end{array}$$

(55)

For the simply supported boundary conditions, the elastic displacements ($u_{ \theta }^{(m)} \mathrm{and} u_{ r}^{(m)}$), traction stress (${\sigma }_{ xx}^{(m)}$), electric potential (${\Phi }^{(m)}$), and couple stresses ${\mu }_{ \theta }^{(m)} \mathrm{and} {\mu }_{ r}^{(m)}$ are set to zero, and are expressed in the following form [30]:

$$u_{ \theta }^{(m)}=u_{ r}^{(m)}={\sigma }_{ xx}^{(m)}={\Phi }^{(m)}={\mu }_{ \theta }^{(m)}={\mu }_{ r}^{(m)}=0 \mathrm{at} x=0, x=L \mathrm{for} m=1, 2,\dots , n_l,$$

(56)

As mentioned above, this work artificially divides the microscale cylinder into n_l layers; the continuity conditions at the interfaces between adjacent layers are expressed as follows:

$$F^{(m+1)}\left(\zeta =-h_{m+1}/2\right)=F^{(m)}\left(\zeta =h_m/2\right),$$

(57)

where F represents the field variables, including $u_x, u_{\theta }, u_r, {\sigma }_{rx}, {\sigma }_{r\theta }, {\sigma }_{rr}, {\mu }_x,$$\mathrm{and} {\mu }_{\theta },$ and $m=1, 2,\dots , \left(n_l-1\right)$.

3.4. The Finite Layer Element Equations

As mentioned above, the primary variables of each layer are expanded as a double Fourier series in the in-surface domain, and the simply supported boundary conditions provided in Equation (30) are satisfied exactly. These variables are therefore expressed in the following form:

$$\left[u_{ i}^{(m)} {\theta }_{ ui}^{(m)} {\kappa }_{ ui}^{(m)}\right]={\displaystyle \sum _{\widehat{m}=1}^{\infty }{\displaystyle \sum _{\widehat{n}=0}^{\infty } \left[{\left(u_{ \widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\theta }_{ u\widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\kappa }_{ u\widehat{m}\widehat{n}}^{(m)}\right)}_i\right]}} \cos \tilde{m} x \cos \tilde{n} \theta ,$$

(58)

$$\left[v_{ i}^{(m)} {\theta }_{ vi}^{(m)} {\kappa }_{ vi}^{(m)}\right]={\displaystyle \sum _{\widehat{m}=1}^{\infty }{\displaystyle \sum _{\widehat{n}=0}^{\infty } \left[{\left(v_{ \widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\theta }_{ v\widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\kappa }_{ v\widehat{m}\widehat{n}}^{(m)}\right)}_i\right]}} \sin \tilde{m} x \sin \tilde{n} \theta ,$$

(59)

$$\left[w_{ i}^{(m)} {\theta }_{ wi}^{(m)} {\kappa }_{ wi}^{(m)}\right]={\displaystyle \sum _{\widehat{m}=1}^{\infty }{\displaystyle \sum _{\widehat{n}=0}^{\infty } \left[{\left(w_{ \widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\theta }_{ w\widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\kappa }_{ w\widehat{m}\widehat{n}}^{(m)}\right)}_i\right]}} \sin \tilde{m} x \cos \tilde{n} \theta ,$$

(60)

$$\left[{\varphi }_{ i}^{(m)} {\theta }_{ \varphi i}^{(m)} {\kappa }_{ \varphi i}^{(m)}\right]={\displaystyle \sum _{\widehat{m}=1}^{\infty }{\displaystyle \sum _{\widehat{n}=0}^{\infty } \left[{\left({\varphi }_{ \widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\theta }_{ \varphi \widehat{m}\widehat{n}}^{(m)}\right)}_i {\left({\kappa }_{ \varphi \widehat{m}\widehat{n}}^{(m)}\right)}_i\right]}} \sin \tilde{m} x \cos \tilde{n} \theta ,$$

(61)

where $\tilde{m}=\widehat{m} \pi /L$ and $\tilde{n}=\widehat{n}$. The symbols $\widehat{m}$ and $\widehat{n}$ represent the half- and full-wave numbers in the x and $\theta$ directions, respectively, and are either zero or a positive integer.

Substituting Equations (58)–(61) into Equation (25) and employing the principle of stationary potential energy, we can obtain the element equations for a typical mth layer in the following form:

$$\left[\begin{array}{ccc}{K}_{\mathrm{I} \mathrm{I}}^{(m)}& K_{\mathrm{I} \mathrm{II}}^{(m)}& K_{\mathrm{I} \mathrm{III}}^{(m)}\\ K_{\mathrm{II} \mathrm{I}}^{(m)}& K_{\mathrm{II} \mathrm{II}}^{(m)}& K_{\mathrm{II} \mathrm{III}}^{(m)}\\ K_{\mathrm{III} \mathrm{I}}^{(m)}& K_{\mathrm{III} \mathrm{II}}^{(m)}& K_{\mathrm{III} \mathrm{III}}^{(m)}\end{array}\right] \left[\begin{array}{c}{\tilde{u}}^{(m)}\\ {\tilde{w}}^{(m)}\\ {\tilde{\mathbf{\Phi }}}^{(m)}\end{array}\right]=\left[\begin{array}{c}0\\ q^{\left(m\right)}\\ 0\end{array}\right],$$

(62)

where m = 1, 2, …, n_l. $q^{(m)}={\delta }_{m n_l}{\delta }_{i n_d} {\overline{q}}_{\widehat{m}\widehat{n}}^{+}+{\delta }_{m1}{\delta }_{i 1} {\overline{q}}_{\widehat{m}\widehat{n}}^{-},$ and the detailed expression of $K_{k l}^{(m)}$ (k, l = I, II, and III) is provided in Appendix B, for which the applied mechanical loads and electric voltages are also expressed as ${\overline{q}}_{ r}^{\pm }={\displaystyle \sum _{\widehat{m}=1}^{\infty } {\displaystyle \sum _{\widehat{n}=0}^{\infty } }{\overline{q}}_{ \widehat{m}\widehat{n}}^{\pm }} \sin \tilde{m}x \cos \tilde{n}\theta ,$ and ${\overline{\Phi }}^{\pm }={\displaystyle \sum _{\widehat{m}=1}^{\infty } {\displaystyle \sum _{\widehat{n}=0}^{\infty } }{\overline{\varphi }}_{ \widehat{m}\widehat{n}}^{\pm }} \sin \tilde{m}x \cos \tilde{n}\theta$, and ${\delta }_{kl}=\left\{\begin{array}{l}1 \mathrm{when} k=l\\ 0 \mathrm{when} k\ne l\end{array}\right.$.

By assembling each layer’s element stiffness matrix, we can obtain the microcylinder’s structural equations, expressed as follows:

$$\left[\begin{array}{ccc}{K}_{11}& K_{12}& K_{13}\\ K_{21}& K_{22}& K_{23}\\ K_{31}& K_{32}& K_{33}\end{array}\right] \left[\begin{array}{c}\tilde{u}\\ \tilde{w}\\ \tilde{\mathbf{\Phi }}\end{array}\right]=\left[\begin{array}{c}0\\ q\\ 0\end{array}\right].$$

(63)

By incorporating the surface conditions (i.e., Equations (52)–(55)) into Equation (63), we can determine all nodal surface values of elastic displacements and electric potentials. As a result, the secondary variables at each nodal surface can be obtained using the relevant equations. The through-thickness distributions of each primary and secondary variable for specific sensor and actuator cases will be presented in the following section. The effects of key factors on the hollow microcylinder’s stress and deformation behaviors will be examined, including the thickness-to-radius ratio, inhomogeneity index, piezoelectricity, and the material length-scale parameter.

4. Illustrative Examples

Because no 3D solutions for the stress and deformation behavior of a simply supported EG piezoelectric circular hollow microcylinder are documented in the literature, we modify the current semi-analytical FLM to analyze EG piezoelectric circular hollow macroscale cylinders for comparison and validation, setting the material length-scale parameter to zero. Additionally, we use the current semi-analytical FLM to conduct a parametric analysis, during which several key factors affecting the deformations, force and couple stresses, electric displacements, and electric potential of the loaded microcylinder will be examined. The relevant material properties of the microcylinder considered here are listed in

Table 1. Elastic, piezoelectric, and dielectric coefficients of piezoelectric materials.

Moduli	PVDF (Kapuria et al. [44])	PZT-4 (Zhong and Shang [45])
c11 (GPa)	3.0	139
c22 (GPa)	3.0	139
c33 (GPa)	3.0	115
c12 (GPa)	1.5	77.8
c13 (GPa)	1.5	74.3
c23 (GPa)	1.5	74.3
c44 (Gpa)	0.75	25.6
c55 (Gpa)	0.75	25.6
c66 (Gpa)	0.75	30.6
e24 (C/m2)	0.0	12.7
e15 (C/m2)	0.0	12.7
e31 (C/m2)	−0.15 × 10−2	−5.2
e32 (C/m2)	0.285 × 10−1	−5.2
e33 (C/m2)	−0.51 × 10−1	15.1
η11 (C2/Nm2)	0.1062 × 10−9	6.46 × 10−9
η22 (C2/Nm2)	0.1062 × 10−9	6.46 × 10−9
η33 (C2/Nm2)	0.1062 × 10−9	5.62 × 10−9
l (m) a	8.8 × 10−6	8.8 × 10−6
$\widehat{l}$ (m) a	17.6 × 10−6	17.6 × 10−6

^a The symbols l and $\widehat{l}$ denote the material length-scale parameters for the CCST and MCST, respectively. PVDF: Polyvinylidene Difluoride; PZT: Lead Zirconium Titanate.

.

4.1. Validation and Comparison Studies

This section considers the coupled electro-mechanical problems of a single-layer, axisymmetric, homogeneous, piezoelectric circular hollow macroscale cylinder (

Table 2. The elastic and electric field variables at crucial positions of single-layer axisymmetric piezoelectric circular hollow cylinders under lateral loads.

R/h	Theories	$-{\overline{\mathit{u}}}_{\mathit{x}}\left(0, \frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{u}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{D}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{D}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{x}\mathit{x}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{x}\mathit{x}}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{\theta }\mathit{\theta }}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{\theta }\mathit{\theta }}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{r}\mathit{x}}\left(0, 0\right)$	$-{\overline{\mathit{\sigma }}}_{\mathit{r}\mathit{r}}\left(\frac{\mathit{L}}{2}, 0\right)$	$-\overline{\mathit{\varphi }} \left(\frac{\mathit{L}}{2}, 0\right)$
4	Hermitian C2 FLM (nl = 1)	0.43729	1.04655	0.63721	0.81996	0.34774	−0.38417	1.04155	1.23225	0.07125	0.58587	1.58097
	Hermitian C2 FLM (nl = 2)	0.43729	1.04655	0.63721	0.81996	0.34778	−0.38421	1.04156	1.23224	0.07125	0.58587	1.58100
	Asymptotic solutions (Wu et al. [46])	0.4373	1.0466	0.6372	0.8200	0.3478	−0.3842	1.042	1.232	0.0712	0.5859	1.581
	3D solutions (Kapuria et al. [47])	0.4373	1.0466	0.6372	0.8200	0.3478	−0.3842	1.042	1.232	0.0712	0.5859	1.581
10	Hermitian C2 FLM (nl = 1)	0.42819	1.01726	0.71591	0.79131	0.34732	−0.36234	1.01327	1.09055	0.06957	0.53463	1.58695
	Hermitian C2 FLM (nl = 2)	0.42819	1.01726	0.71591	0.79131	0.34732	−0.36234	1.01327	1.09054	0.06957	0.53463	1.58695
	Asymptotic solutions (Wu et al. [46])	0.4282	1.0173	0.7159	0.7913	0.3474	−0.3623	1.013	1.091	0.0696	0.5346	1.587
	3D solutions (Kapuria et al. [47])	0.4282	1.0173	0.7159	0.7913	0.3473	−0.3623	1.013	1.091	0.0696	0.5346	1.587
100	Hermitian C2 FLM (nl = 1)	0.42474	1.00167	0.76166	0.76931	0.34483	−0.34634	1.00112	1.00892	0.06786	0.50347	1.58366
	Hermitian C2 FLM (nl = 2)	0.42474	1.00167	0.76166	0.76931	0.34483	−0.34634	1.00112	1.00892	0.06786	0.50347	1.58366
	Asymptotic solutions (Wu et al. [46])	0.4247	1.0017	0.7617	0.7693	0.3448	−0.3463	1.001	1.009	0.0679	0.5035	1.584
	3D solutions (Kapuria et al. [47])	0.4247	1.002	0.7617	0.7693	0.3448	−0.3463	1.001	1.009	0.0679	0.5035	1.584

and

Table 3. The elastic and electric field variables at crucial positions of single-layer axisymmetric piezoelectric circular hollow cylinders under lateral potential.

R/h	Theories	${\overline{\mathit{u}}}_{\mathit{x}}\left(0, \frac{\mathit{h}}{2}\right)$	${\overline{\mathit{u}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	${\overline{\mathit{D}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	${\overline{\mathit{D}}}_{\mathit{r}}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	${\overline{\mathit{\sigma }}}_{\mathit{x}\mathit{x}}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	${\overline{\mathit{\sigma }}}_{\mathit{x}\mathit{x}}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	${\overline{\mathit{\sigma }}}_{\mathit{\theta }\mathit{\theta }}\left(\frac{\mathit{L}}{2}, \frac{\mathit{h}}{2}\right)$	${\overline{\mathit{\sigma }}}_{\mathit{\theta }\mathit{\theta }}\left(\frac{\mathit{L}}{2}, -\frac{\mathit{h}}{2}\right)$	${\overline{\mathit{\sigma }}}_{\mathit{r}\mathit{x}}\left(0, 0\right)$	${\overline{\mathit{\sigma }}}_{\mathit{r}\mathit{r}}\left(\frac{\mathit{L}}{2}, 0\right)$	$\overline{\mathit{\varphi }} \left(\frac{\mathit{L}}{2}, 0\right)$
4	Hermitian C2 FLM (nl = 1)	0.19377	−0.63721	−53.9941	−68.0172	−0.09738	0.12349	0.45624	−0.54903	−0.21576	−0.12375	0.52845
	Hermitian C2 FLM (nl = 2)	0.19377	−0.63721	−53.9939	−68.0175	−0.09742	0.12354	0.45620	−0.54897	−0.21577	−0.12376	0.52845
	Asymptotic solutions (Wu et al. [46])	0.1938	−0.6372	−53.99	−68.02	−0.0974	0.1235	0.4562	−0.5490	−0.2158	−0.1238	0.528
	3D solutions (Kapuria et al. [47])	0.1938	−0.6372	−53.49	−68.02	−0.0974	0.1235	0.4562	−0.5490	−0.2158	−0.1238	0.528
10	Hermitian C2 FLM (nl = 1)	0.15614	−0.71591	−57.4170	−63.2631	−0.11982	0.13076	0.45702	−0.49342	−0.24578	−0.11850	0.51197
	Hermitian C2 FLM (nl = 2)	0.15614	−0.71591	−57.4170	−63.2631	−0.11982	0.13076	0.45702	−0.49342	−0.24578	−0.11850	0.51197
	Asymptotic solutions (Wu et al. [46])	0.1561	−0.7159	−57.42	−63.26	−0.1198	0.1308	0.4570	−0.4934	−0.2458	−0.1185	0.512
	3D solutions (Kapuria et al. [47])	0.1561	−0.7159	−57.42	−63.28	−0.1198	0.1308	0.4570	−0.4934	−0.2458	−0.1185	0.512
100	Hermitian C2 FLM (nl = 1)	0.13032	−0.76166	−59.9128	−60.5131	−0.13133	0.13244	0.45973	−0.46335	−0.25896	−0.11538	0.50123
	Hermitian C2 FLM (nl = 2)	0.13032	−0.76166	−59.9128	−60.5131	−0.13133	0.13244	0.45973	−0.46335	−0.25896	−0.11538	0.50123
	Asymptotic solutions (Wu et al. [46])	0.1303	−0.7617	−59.91	−60.51	−0.1313	0.1324	0.4597	−0.4634	−0.2590	−0.1154	0.501
	3D solutions (Kapuria et al. [47])	0.1303	−0.7617	−59.91	−60.51	−0.1313	0.1324	0.4597	−0.4634	−0.2590	−0.1154	0.501

). The macroscale cylinder in Table 2 and Table 3 is composed of polyvinylidene difluoride (PVDF), which is polarized along the radial direction. Following Kapuria et al. [44], Table 1 gives PVDF’s piezoelectric, elastic, and dielectric properties. The formulation for analyzing an axisymmetric, circular hollow macroscale cylinder can be obtained by setting the circumferential wavenumber ($\widehat{n}$) to zero. Mechanical loads (${\overline{q}}_{ r}^{+} =q_0 \sin \pi x/L$ and ${\overline{q}}_{ r}^{-}=0 {\mathrm{N}/\mathrm{m}}^2$) or electric voltages (${\overline{\Phi }}^{+}={\varphi }_0 \sin \pi x/L$, ${\overline{\Phi }}^{-}=0 \mathrm{V}$) which are applied to the macroscale cylinder’s outer and inner surfaces are considered. The dimensionless variables are defined in the same form as in Kapuria et al. [44] and are given as follows.

In the cases of applied mechanical loads,

$$\left({\overline{u}}_x, {\overline{u}}_r\right)=\left(u_x/h, u_r/h\right)/\left( \left|q_0\right| S^2/E_T\right),$$

(64)

$$\left({\overline{D}}_x, {\overline{D}}_r\right)=\left[10 D_x S/\left(\left|q_0\right| \left|d_T\right|\right), D_r/\left(\left|q_0\right| S \left|d_T\right|\right)\right],$$

(65)

$$\left({\overline{\sigma }}_{xx}, {\overline{\sigma }}_{\theta \theta }, {\overline{\sigma }}_{rr}, {\overline{\sigma }}_{rx}\right)=\left({\sigma }_{xx}, {\sigma }_{\theta \theta }/S, {\sigma }_{rr}, {\sigma }_{rx}S\right)/\left|q_0\right|,$$

(66)

$$\overline{\varphi }=1000 \Phi E_T \left|d_T\right|/\left(h \left| q_0\right|\right).$$

(67)

In the cases of applied voltages,

$$\left({\overline{u}}_x, {\overline{u}}_r\right)=\left[u_x/\left({\varphi }_0 S \left|d_T\right|\right), u_r/\left({\varphi }_0 S \left|d_T\right|\right)\right],$$

(68)

$$\left({\overline{D}}_x, {\overline{D}}_r\right)=\left[h D_x S/\left(100 {\varphi }_0 E_T {\left|d_T\right|}^2\right), h D_r/\left({\varphi }_0 E_T {\left|d_T\right|}^2\right)\right],$$

(69)

$$\left({\overline{\sigma }}_{xx}, {\overline{\sigma }}_{\theta \theta }\right)=\left[h S {\sigma }_{xx}/\left({\varphi }_0 E_T \left|d_T\right|\right), h S {\sigma }_{\theta \theta }/\left({\varphi }_0 E_T \left|d_T\right|\right)\right]$$

(70)

$$\left({\overline{\sigma }}_{rr}, {\overline{\sigma }}_{rx}\right)=\left[h S^2{\sigma }_{rr}/\left({\varphi }_0 E_T \left|d_T\right|\right), 10 h S^2{\sigma }_{rx}/\left({\varphi }_0 E_T \left|d_T\right|\right)\right]$$

(71)

$$\overline{\varphi }=\Phi /{\varphi }_0 ,$$

(72)

and $S=R/h$, $E_T=2.0 \mathrm{GPa}$, and $\left|d_T\right|=30 \times {10}^{-12} {\mathrm{CN}}^{-1}$.

Table 2 and Table 3 present the elastic and electric field variables induced at specific positions of a single-layered, axisymmetric, homogeneous piezoelectric circular hollow macroscale cylinder. They show that the current semi-analytical FLM solutions converge rapidly, and that convergence is achieved when n_l = 2. The convergent solutions closely agree with those of Wu et al. [46] using the perturbation method and with those of Kapuria et al. [47] using the state-space method. Because the semi-analytical FLM performs well, we use it in the following parametric analysis.

4.2. Parametric Analysis

In this section, we conduct a parametric analysis of the coupled electro-mechanical behavior of an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions. The microcylinder is subjected to mechanical loads (${\overline{q}}_{ r}^{+} =q_0 \sin \left(\pi x/L\right) \cos \left(2\theta \right),$ and ${\overline{q}}_{ r}^{-}=0 {\mathrm{N}/\mathrm{m}}^2$) and electric voltages (${\overline{\Phi }}^{+} ={\varphi }_0 \sin \left(\pi x/L\right) \cos \left(2\theta \right)$, and ${\overline{\Phi }}^{-}=0 \mathrm{V}$). The material properties of the microcylinder of interest are assumed to follow an exponential law that varies in the thickness direction and are expressed in the following form:

$$m_{ij}\left(\zeta \right) =m_{ij}^b e^{{\kappa }_e\left[\left(1/2\right)+\left(\zeta /h\right)\right]},$$

(73)

where $m_{ ij}^b$ denotes the hollow microcylinder’s material properties, defined as those of PZT-4 [45] in this work and listed in Table 1.

The dimensionless variables are defined as follows:

In the cases of applied mechanical loads,

$$\left({\overline{u}}_x, {\overline{u}}_{\theta }, {\overline{u}}_r\right)=\left(u_x, u_{\theta }, u_r\right) \left[c_{ 11}^{ b}/\left(q_0 S^2 h\right)\right],$$

(74)

$$\left({\overline{\sigma }}_{xx}, {\overline{\sigma }}_{\theta \theta }, {\overline{\sigma }}_{rr}, {\overline{\sigma }}_{rx}, {\overline{\sigma }}_{r\theta }, {\overline{\sigma }}_{x\theta }\right)=\left({\sigma }_{xx}, {\sigma }_{\theta \theta }, {\sigma }_{rr}, {\sigma }_{rx}, {\sigma }_{r\theta }, {\sigma }_{x\theta }\right) / q_0,$$

(75)

$$\left({\overline{D}}_x, {\overline{D}}_{\theta }, {\overline{D}}_r\right)=\left(D_x, D_{\theta }, D_r\right)\left[ c_{ 11}^{ b}/ \left(q_0 S e_{ 33}^{ b}\right)\right]$$

(76)

$$\overline{\varphi }=\left(\Phi e_{ 33}^{ b}\right)/ \left(q_0 h \right),$$

(77)

$$\left({\overline{\mu }}_x, {\overline{\mu }}_{\theta }, {\overline{\mu }}_r\right)=\left({\mu }_x, {\mu }_{\theta }, {\mu }_r\right)/ \left(q_0 h\right),$$

(78)

In the cases of applied voltages,

$$\left({\overline{u}}_x, {\overline{u}}_{\theta }, {\overline{u}}_r\right)=\left(u_x, u_{\theta }, u_r\right)\left[ c_{ 11}^{ b}/\left({\varphi }_0 S e_{ 33}^{ b}\right)\right],$$

(79)

$$\left({\overline{\sigma }}_{xx}, {\overline{\sigma }}_{\theta \theta }, {\overline{\sigma }}_{rr}, {\overline{\sigma }}_{rx}, {\overline{\sigma }}_{r\theta }, {\overline{\sigma }}_{x\theta }\right)=\left({\sigma }_{xx}, {\sigma }_{\theta \theta }, {\sigma }_{rr}, {\sigma }_{rx}, {\sigma }_{r\theta }, {\sigma }_{x\theta }\right) \left[ h/\left({\varphi }_0 e_{ 33}^{ b}\right)\right],$$

(80)

$$\left({\overline{D}}_x, {\overline{D}}_{\theta }, {\overline{D}}_r\right)=\left( D_x, D_{\theta }, D_r \right)\left[\left(c_{ 11}^{ b} h\right)/\left({\varphi }_0 e_{ 33}^{ b} e_{ 33}^{ b}\right)\right],$$

(81)

$$\overline{\phi }=\Phi /{\phi }_0,$$

(82)

$$\left({\overline{\mu }}_x, {\overline{\mu }}_{\theta }, {\overline{\mu }}_r\right)=\left({\mu }_x, {\mu }_{\theta }, {\mu }_r\right)/ \left({\varphi }_0 e_{ 33}^{ b}\right),$$

(83)

where $\left(u_x, {\sigma }_{rx}, D_x, {\mu }_x\right)$ are at the positions of $\left(x=0, \theta =0\right)$, $\left(u_{\theta }, {\sigma }_{r\theta }, D_{\theta }, {\mu }_{\theta }\right)$ at those of $\left(x=L/2, \widehat{n}\theta =\pi /2\right)$, $\left(u_r, {\sigma }_{xx}, {\sigma }_{\theta \theta }, {\sigma }_{rr}, \Phi , D_r, {\mu }_r\right)$ at those of $\left(x=L/2, \theta =0\right)$, and ${\sigma }_{x\theta }$ at $\left(x=0, \widehat{n}\theta =\pi /2\right)$.

Figure 2 and Figure 3 show the distributions of elastic displacements (i.e., ${\overline{u}}_x$ and ${\overline{u}}_r$), transverse shear and normal stresses (i.e., ${\overline{\sigma }}_{rx} \mathrm{and} {\overline{\sigma }}_{rr}$), in-plane normal and shear stresses (i.e., ${\overline{\sigma }}_{xx} \mathrm{and} {\overline{\sigma }}_{x\theta }$), electric potential ($\overline{\Phi }$), electric displacement (${\overline{D}}_r$), and couple stresses (i.e., ${\overline{\mu }}_x \mathrm{and} {\overline{\mu }}_r$) along the thickness direction induced in the microcylinder when it is subjected to mechanical loads and electric voltages, respectively. The l/h ratio is set to 0.25, 0.5, and 0.75 for the couple stresses and to 0, 0.25, and 0.5 for the other variables. Following Lam et al. [48], the material length-scale parameter is provided as $l=8.8\times {10}^{-6} \mathrm{m}.$ Other parameters are ${\kappa }_e=2,$ R/h = 5, and L/R = 5.

Figure 2a–f show that the elastic displacements and force-stress components decrease as the l/h ratio increases. This is because the material length-scale parameter stiffens the microcylinder, resulting in reduced deformations, in-surface stresses, and transverse stresses as the l/h ratio increases. Additionally, the results in Figure 2a–j show that, under applied mechanical loads (sensor cases), the material length-scale parameter significantly affects the through-thickness distributions of elastic and electrical variables. However, the results in Figure 3a–j show that, under applied voltages (actuator cases), the material length-scale parameter significantly affects the through-thickness distributions of elastic variables more than those of electric variables. It is noted that the results in Figure 2e–g,i, as well as Figure 3e–g,i, show that the surface conditions of the hollow microcylinder given in Equations (28) and (29) are precisely satisfied.

Figure 4a–j show variations in the through-thickness distributions of elastic and electric variables induced in the microcylinder subjected to mechanical loads, with the inhomogeneity index ${\kappa }_e$ being 0, 1.5, and 3. The material and geometric parameters are given as $l=8.8\times {10}^{-6} \mathrm{m},$ $l/h=0.2,$ R/h = 5, and L/R = 5. It can be seen in Figure 4a–f that the distributions of the elastic displacements (i.e., ${\overline{\mu }}_x \mathrm{and} {\overline{\mu }}_r$), in-plane normal and shear stresses (i.e., ${\overline{\sigma }}_{xx} \mathrm{and} {\overline{\sigma }}_{x\theta }$), transverse shear stresses (${\overline{\sigma }}_{rx}$), and transverse normal stresses (${\overline{\sigma }}_{rr}$) in the thickness direction appear to be linear, linear, parabolic, and higher-order polynomial functions, respectively, for a homogeneous microcylinder (${\kappa }_e=0$). The distributions of the electric variables ($\overline{\Phi }$ and ${\overline{D}}_r$) and couple stresses (${\overline{\mu }}_x \mathrm{and} {\overline{\mu }}_r$) in the thickness direction appear to be higher-order polynomial functions. However, the distributions of in-surface and transverse stresses, electric potentials, electric displacements, and couple stresses in the thickness direction appear to be higher-order polynomial functions for a nonhomogeneous microcylinder (${\kappa }_e\ne 0$). The variations in the induced elastic and electric variables along the thickness direction for a nonhomogeneous microcylinder are more pronounced than those for a homogeneous microcylinder. The higher the inhomogeneity index, the more pronounced the variations in the induced elastic and electric variables will be along the thickness direction.

Figure 5a–j show variations in the distributions of elastic and electric variables through the thickness of the microcylinder under mechanical loads, with R/h ratios of 4, 10, and 20. The material and geometric parameters are given as $l=8.8\times {10}^{-6} \mathrm{m},$ l/h = 0.2, ${\kappa }_e=2,$ and L/R = 5. It can be seen in Figure 5a–f that the elastic deformations, in-surface stresses, and transverse stresses induced in the thick microcylinder (R/h = 4) are smaller than those induced in the thin microcylinder (R/h = 20). This is because a decrease in the R/h ratio increases the overall stiffness of the hollow microcylinder, which reduces the induced elastic deformations, in-plane stresses, and transverse stresses.

Figure 6a–f show variations in the distributions of elastic and electric variables along the thickness direction induced in the circular hollow microcylinder under mechanical loads for three sets of different values of the piezoelectric coefficients and the l/h ratio, which are $e_{ij}=0 \mathrm{and} l/h=0;$ $e_{ij}\ne 0 \mathrm{and} l/h=0;$ and $e_{ij}\ne 0 \mathrm{and} l/h=0.5$. The geometric and material parameters are given as $R/h=10,$ $L/R=10,$ and ${\kappa }_e=2$. It can be seen in Figure 6a,b that the elastic displacements induced in the microcylinders when the piezoelectricity effect and the material length-scale parameter effect are considered are smaller than the elastic displacements induced in the microcylinder without considering these effects. This is because the piezoelectric and material length-scale parameters stiffen the microcylinder, resulting in smaller elastic displacements. Figure 6a–f show that the piezoelectric effect affects displacements more significantly than the in-surface and transverse stresses. In the case of l/h = 0.5 and ${\kappa }_e=2,$ the material length-scale parameter effect on the displacements and stresses induced in the microcylinder is more significant than the piezoelectric effect.

5. Conclusions

This work developed a semi-analytical Hermitian C² FLM within the 3D CCST framework, which we used to examine the coupled electro-mechanical behavior of an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions. In the current semi-analytical FLM formulation, we divided the microcylinder into n_l layers and selected the elastic displacements and electric potential as the primary variables. We applied the principle of stationary potential energy to derive the element equations for each layer and assembled them into the microcylinder’s structural equations. Implementation of the current semi-analytical FLM showed that its solutions converged rapidly compared with the relevant solutions of the 3D piezoelectricity theory for EG piezoelectric circular hollow macroscale cylinders reported in the literature. In addition, we conducted a parametric analysis to assess the impact of key factors on elastic deformations, force and couple stresses, electric displacements, and the electric potential induced in the microcylinder. The critical factors were the radius-to-thickness ratio, the inhomogeneity index, the piezoelectricity, and the material length-scale parameter. Some concluding remarks drawn from this parametric analysis are provided as follows:

• The current semi-analytical FLM for analyzing EG piezoelectric circular hollow microcylinders could be reduced to one for analyzing EG piezoelectric circular hollow macroscale cylinders by setting the material length-scale parameter to zero. Our semi-analytical FLM could also be reduced to the form used to analyze EG elastic circular hollow microcylinders by ignoring the piezoelectric effect. Furthermore, the current semi-analytical FLM was validated for accuracy and convergence by comparing its solutions with relevant 3D solutions reported in the literature.
• An increase in the material length-scale parameter stiffened the microcylinder, decreasing its elastic deformation, transverse stresses, and in-surface stresses.
• The results for the applied-load (sensor) cases showed that the material length-scale parameter significantly affected the distributions of elastic and electric variables along the thickness direction. However, the results for the applied-voltage (actuator) cases showed that the material length-scale parameter significantly affected the distributions of elastic variables along the thickness direction, as compared with those of electric variables.
• The variations in the elastic and electric variables along the thickness direction for a nonhomogeneous microcylinder were more pronounced than those for a homogeneous microcylinder. The higher the inhomogeneity index, the more pronounced the variations in the elastic and electric variables along the thickness direction.
• An increase in the thickness of the microcylinder stiffened it, decreasing its elastic deformation, transverse stresses, and in-surface stresses.
• The piezoelectric effect stiffened the microcylinder, resulting in decreased displacements and in-surface and transverse stresses within the microcylinder.
• In the case of l/h = 0.5 and ${\kappa }_e=2,$ the material length-scale parameter’s effect on the displacements and stresses induced in the microcylinder was more significant than the piezoelectric effect.

To the best of our knowledge, no 3D solutions have been reported for the elastic deformations, transverse and in-surface stresses, electric displacements, electric potential, and couple stresses induced in an EG piezoelectric circular hollow microcylinder under simply supported boundary conditions. Therefore, our semi-analytical FLM solutions presented in this article provide a reference for assessing approximate 2D solutions obtained using advanced, refined microstructure-dependent shear deformation shell theories. The through-thickness variations in the elastic and electric fields also provide a reference for the kinematic and kinetic assumptions when developing a new microstructure-dependent, advanced, and refined shear deformation theory.

It is noted that the material properties used in this work are suitable for the macroscale structure and should change as the structure shrinks to the microscale due to microstructural effects. Although these material properties are not well documented in the literature, this effect must be accounted for in future studies to ensure satisfactory results. The CCST/MCST is suitable for microscale structural analysis. When the structure shrinks below the microscale, the effects of dilatational and deviatoric strain gradients become significant, and the consistent/modified strain-gradient theory would be a better alternative. Additionally, as mentioned before, the current semi-analytical FLM uses a double Fourier series to simulate variations in each field variable in the in-surface domain and a Lagrange polynomial to interpolate its variation in the thickness direction. Therefore, it is suitable only for analyzing FG piezoelectric hollow microcylinders under simply supported boundary conditions. The computational cost of the current FLM is much lower than that of some 2D and 3D numerical methods because it involves numerical calculations only in the thickness domain.