The surface effect on axisymmetric wave propagation in piezoelectric cylindrical shells

Abstract

Based on the surface piezoelectricity theory and first-order shear deformation theory, the surface effect on the axisymmetric wave propagating in piezoelectric cylindrical shells is analyzed. The Gurtin–Murdoch theory is utilized to get the nontraditional boundary conditions and constitutive equations of the surface, in company with classical governing equations of the bulk, from which the basic formulations are obtained. Numerical results show that the surface layer has a profound effect on wave characteristics in nanostructure at a higher mode.

Introduction

With the flourishing development and growing needs of nanoscience and nanotechnology in the 21 century, the surface effect on the mechanics of nanomaterials has received an ever-increasing interest.^1–5 Methods for simulating size-dependent phenomenon at nano-scales include the continuum model, experimental method, and molecular method. But among the various research tools, the continuum surface model is a good alternative to better understand the surface effect on mechanical behavior of the nano structures. As everyone knows, the effect of the surface stress can be neglected in the classical theory. However, the elastic behavior of nano structures whose characteristic length is in nanometer range has been beyond capture since the continuum elasticity contains no intrinsic length scale.

The generalized continuum model was first presented by Gurtin and Murdoch ⁶ (GM theory). In this theory, the equilibrium and constitutive equations of the bulk material are the same as those in classical elasticity, but with nontraditional boundary conditions to ensure the dynamic balance of the surface. Thus, within GM theory, the boundary conditions and constitutive relations of the surface, along with classical governing equations of the bulk, constitute the basic formulations of any nano-sized body with surface effect. The GM theory has been widely used to study the mechanical responses of nano structures. For instance, Liu and Rajapakse ⁷ discussed the static and dynamic responses in nano beams, Assadi et al. ⁸ analyzed the size-dependent dynamic response in nano plates. Detailed introduction concerning the continuum model with surface effect can refer to reviews.^9,10

To study the surface effect on piezoelectric structures, Huang and Yu ¹¹ added the electric field variables into the surface energy density expression, creatively extended the GM theory to the piezoelectric bodies, and established the vital theoretical basis for studying the mechanical behavior of piezoelectric nano structures. Based on the model proposed by Huang and Yu, Pan et al. ¹² completed the phenomenological continuum theory of surface piezoelectricity to consider the linear superficial interplay between electricity and elasticity; Yan and Jiang^13,14 performed the static bending, vibration, and buckling analysis in piezoelectric nano plates with surface effect.

The surface effect on the propagation of elastic waves was first analyzed by Gurtin and Murdoch.^15–17 These studies show that the wave propagation behavior is extremely sensitive to the surface material properties: Rayleigh wave becomes dispersive due to the surface stress; however, in the classical theory, the wave velocity is independent on frequency (i.e. nondispersion). Recently, many researchers investigated the wave motion in nano structures. Chakraborty ¹⁸ studied Lamb wave propagating in micro-nano elastic structures with flat surface/interface, and the special negative dispersion phenomenon was observed. Li and Lee ¹⁹ discussed Love wave propagating in a nano-coating. Liu et al. ²⁰ performed the SH wave problem in nano films, and a special symmetric mode was found due to the surface effect. Chen ^21,22 treated the effect of surface layer on BG wave, and Love and Rayleigh–Lamb wave propagating in piezoelectric structures. Zhang et al.^23,24 explored the surface effect on SH wave propagating in piezoelectric nano plates.

Research on surface effect in piezoelectric nano structures mostly focused on plate structures. However, piezoelectric nano-tubes or nano-shells also play an important role in engineering science. To the author’s best knowledge, the relevant research on wave propagation in piezoelectric nano-shells with surface effect is very limited, except the work of Zhang et al., ²⁵ where only the free vibration problem was considered, and some typos can be found. On the basis of existing literature data, we carried out studies in an effort to axisymmetric wave propagation in the piezoelectric cylindrical shells. Numerical examples will show the effect of surface parameters on the dispersion curves, phase velocity curves, and frequency variations.

Basic equations

In the classical piezoelectricity theory, the constitutive equations, the strain–displacement, and electric potential relations, the equations of motion and conservation of charge are, respectively

$$T_{\mathrm{ij}}=c_{\mathrm{ijkl}}{S}_{\mathrm{kl}}-e_{\mathrm{kij}}{E}_k,D_i=e_{\mathrm{ikl}}{S}_{\mathrm{kl}}+{\epsilon }_{\mathrm{ik}}{E}_k$$

(1)

$$S_{\mathrm{ij}}=\frac{1}{2}({u_i|}_j+{u_j|}_i),E_i=-{\varphi |}_i$$

(2)

$${T_{\mathrm{ij}}|}_j=\rho {\stackrel{··}{u}}_i,{D_i|}_i=0$$

(3)

where $T_{\mathrm{ij}}$ , $S_{\mathrm{ij}}$ , $u_i$ , $D_i$ , $E_i$ , and $\varphi$ are stresses, strains, displacements, electric displacements, electric field, and electric potential, respectively, where material parameters $c_{\mathrm{ijkl}}$ , $e_{\mathrm{kij}}$ , ${\epsilon }_{\mathrm{ij}}$ , and $\rho$ are elastic, piezoelectric, dielectric constants, and density of the bulk, respectively. The tensor symbol $|{}_i$ denotes the covariant derivative in this article. We adopt the subscripts $i,j,k\text{and}l$ range over 1, 2, and 3, and $a,b,c,\text{and}d$ range over 1 and 2 in the entire article.

Unlike the classical piezoelectricity, the constitutive relations ¹¹ and the equations of motion ⁶ for the surface layer are written as

$$T_{\mathrm{ab}}^s=T_{\mathrm{ab}}^0+c_{\mathrm{abcd}}^s{S}_{\mathrm{cd}}-e_{\mathrm{kab}}^s{E}_k,D_a^s=D_a^0+e_{\mathrm{acd}}^s{S}_{\mathrm{cd}}+{\epsilon }_{\mathrm{ak}}^s{E}_k$$

(4)

$${T_{\mathrm{ia}}^s|}_a+T_{i3}={\rho }^s{\stackrel{··}{u}}_i,D_i^s|{}_i=0$$

(5)

where $T_{\mathrm{ab}}^s$ and $D_a^s$ are the surface stresses and electric displacements, and $T_{\mathrm{ab}}^0$ and $D_a^0$ are the surface residual stresses and electric displacements, $c_{\mathrm{abcd}}^s$ , $e_{\mathrm{kab}}^s$ , ${\epsilon }_{\mathrm{ak}}^s$ , and ${\rho }^s$ are surface elastic, surface piezoelectric, surface dielectric constants, and surface mass density, respectively. The geometric equations and electric potential relations (equations (2)) are also valid for surface layer.

According to Huang and Yu, ¹¹ the residual part is constant in general whose space derivative is 0. Hence, the residual stresses and electrical displacements of the surface layer will be omitted in the following for simplicity. So our main concern is how the surface piezoelectricity affects the dynamic behavior of the piezoelectric cylindrical nano-shells.

Formulation of two-dimensional theory

Figure 1 shows the geometry and coordinate system of an infinite-long thin shell with thickness $2h$ and outer radius $R$ . The orthogonal coordinate system ( $x_1,x_2,x_3$ ) is fixed at the middle surface of the cylindrical shell. $x_1$ is the axial direction, $x_2$ the circumferential direction, and $x_3$ the radial direction.

Figure 1.

A cylindrical piezoelectric shell.

Based on the first-order theory, the mechanical displacements are

$$u_a=u_a^{(0)}+x_3{u}_a^{(1)},u_3=u_3^{(0)}+x_3{u}_3^{(1)}+x_3^2{u}_3^{(2)}$$

(6)

where $u_a^{(0)}$ , $u_3^{(0)}$ , and $u_a^{(1)}$ are the extensional, flexural, and thickness-shear components, respectively, and $u_i^{(p)}=u_i^{(p)}(x_1,x_2,t)$ , ( $p=0,1,2$ ). Though the thickness stretch or contraction components $u_3^{(1)}$ and $u_3^{(2)}$ are included in equation (6), they can be eliminated through stress relaxation process. The strains and electrical potential are in the form of

$$S_{\mathrm{ij}}=S_{\mathrm{ij}}^{(0)}+x_3{S}_{\mathrm{ij}}^{(1)},\varphi ={\varphi }^{(0)}+x_3{\varphi }^{(1)}$$

(7)

With the basic field quantities written as equations (6) and (7), the governing equations of the shell can be obtained by integrating over the thickness direction, as follows

$$\begin{array}{c}\multicolumn{1}{c}{T_{\mathrm{ab}}^{(0)}{|{}_a+[T_{3b}]}_{-h}^h={\rho }^{(0)}{\stackrel{··}{u}}_b^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_b^{(1)}}\\ \multicolumn{1}{c}{T_{a3}^{(0)}|{}_a+{[T_{33}]}_{-h}^h={\rho }^{(0)}{\stackrel{··}{u}}_3^{(0)}}\end{array}$$

(8)

$$\begin{array}{cc}\multicolumn{1}{c}{}& T_{\mathrm{ab}}^{(1)}|{}_a-T_{3b}^{(0)}+{[x_3{T}_{3b}]}_{-h}^h={\rho }^{(1)}{\stackrel{··}{u}}_b^{(0)}+{\rho }^{(2)}{\stackrel{··}{u}}_b^{(1)}\hfill \\ \multicolumn{1}{c}{}& \{T_{\mathrm{ab}}^{(0)},T_{a3}^{(0)},T_{33}^{(0)}\}=\underset{-h}{\overset{h}{\int }}\{T_{\mathrm{ab}},T_{a3},T_{33}\}d{x}_3,\hfill \\ \multicolumn{1}{c}{T_{\mathrm{ab}}^{(1)}=\underset{-h}{\overset{h}{\int }}{T}_{\mathrm{ab}}{x}_{3}d{x}_3,{\rho }^{(n)}=\underset{-h}{\overset{h}{\int }}\rho x_3^{n}d{x}_3}\end{array}$$

(9)

where (n = 0, 1, 2), ${[·]}_{-h}^h$ represents the equivalent force term applied by the surface layer at the bulk boundaries $x_3=\pm h$ .

We utilized the effective boundary conditions to incorporate the surface effect. The effective boundary conditions are, in fact, taking the effect of the surface layer as external forces exerting on the bulk. At the surfaces $x_3=\pm h$ , the equilibrium equations are ²⁵

$$\begin{array}{c}\multicolumn{1}{c}{T_{\mathrm{ia}}^s|{}_a-T_{3i}={\rho }^s{\stackrel{··}{u}}_i^{(0)},x_3=h}\\ \multicolumn{1}{c}{T_{\mathrm{ia}}^s|{}_a+T_{3i}={\rho }^s{\stackrel{··}{u}}_i^{(0)},x_3=-h}\end{array}$$

(10)

The substitution of equation (10) into equation (8) gets

$$\begin{array}{cc}\multicolumn{1}{c}{}& T_{\mathrm{ab}}^{(0)}|{}_a+[T_{\mathrm{ab}}^s|{}_a(h)+T_{\mathrm{ab}}^s|{}_a(-h)]\hfill \\ \multicolumn{1}{c}{={\rho }^{(0)}{\stackrel{··}{u}}_b^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_b^{(1)}+{\rho }^s[{\stackrel{··}{u}}_b(h)+{\stackrel{··}{u}}_b(-h)]}\\ \multicolumn{1}{c}{}& T_{a3}^{(0)}|{}_a+[T_{a3}^s|{}_a(h)+T_{a3}^s|{}_a(-h)]\hfill \\ \multicolumn{1}{c}{={\rho }^{(0)}{\stackrel{··}{u}}_3^{(0)}+{\rho }^s[{\stackrel{··}{u}}_3(h)+{\stackrel{··}{u}}_3(-h)]}\\ \multicolumn{1}{c}{}& T_{\mathrm{ab}}^{(1)}|{}_a+h[T_{\mathrm{ab}}^s|{}_a(h)-T_{\mathrm{ab}}^s|{}_a(-h)]-T_{3b}^{(0)}\hfill \\ \multicolumn{1}{c}{={\rho }^{(1)}{\stackrel{··}{u}}_b^{(0)}+{\rho }^{(2)}{\stackrel{··}{u}}_b^{(1)}-{\rho }^{s}h[{\stackrel{··}{u}}_b(-h)-{\stackrel{··}{u}}_b(h)]}\end{array}$$

(11)

The effective resultant forces are described as

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{T}}_{\mathrm{ai}}^{(0)}=T_{\mathrm{ai}}^{(0)}+T_{\mathrm{ai}}^s(h)+T_{\mathrm{ai}}^s(-h)}\\ \multicolumn{1}{c}{{\hat{T}}_{\mathrm{ab}}^{(1)}=T_{\mathrm{ab}}^{(1)}+h[T_{\mathrm{ab}}^s(h)-T_{\mathrm{ab}}^s(-h)]}\end{array}$$

(12)

Using the continuity conditions of the displacements at $x_3=\pm h$ , equation (11) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{T}}_{\mathrm{ab}}^{(0)}|{}_a=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_b^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_b^{(1)}}\\ \multicolumn{1}{c}{{\hat{T}}_{a3}^{(0)}|{}_a=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_3^{(0)}}\end{array}$$

(13)

$${\hat{T}}_{\mathrm{ab}}^{(1)}|{}_a-T_{3b}^{(0)}={\rho }^{(1)}{\stackrel{··}{u}}_b^{(0)}+({\rho }^{(2)}+2{\rho }^s{h}^2){\stackrel{··}{u}}_b^{(1)}$$

Similarly, the charge equations are derived

$$\begin{array}{cc}\multicolumn{1}{c}{}& D_a^{(0)}|{}_a+D_3^s(h)-D_3^s(-h)=0\hfill \\ \multicolumn{1}{c}{}& D_a^{(1)}|{}_a-D_3^{(0)}+h[D_3^s(h)+D_3^s(-h)]=0\hfill \end{array}$$

(14)

Now the geometric and electric potential equations (7), the equations of motion (13), and the charge equations (14) are obtained, meanwhile revisions on the constitutive relations are also needed. Since the object investigated is an ultra-thin shell, the stress relaxation approximation of vanishing normal stress $T_{33}\approx 0$ is assumed. Then the reduced constitutive relations yield

$$T_{\mathrm{ij}}={\overline{c}}_{\mathrm{ijkl}}{S}_{\mathrm{kl}}-{\overline{e}}_{\mathrm{kij}}{E}_k,D_i={\overline{e}}_{\mathrm{ikl}}{S}_{\mathrm{kl}}+{\overline{\epsilon }}_{\mathrm{ik}}{E}_k$$

(15)

where ${\overline{c}}_{\mathrm{ijkl}}=c_{\mathrm{ijkl}}-c_{\mathrm{ij}33}{c}_{33\mathrm{kl}}/c_{3333}$ , ${\overline{e}}_{\mathrm{ikl}}=e_{\mathrm{ikl}}-e_{i33}{c}_{33\mathrm{kl}}/c_{3333}$ , and ${\overline{\epsilon }}_{\mathrm{ik}}={\epsilon }_{\mathrm{ik}}+e_{i33}{e}_{k33}/c_{3333}$ .

Now the general two-dimensional basic equations are all presented in tensor form; the definite equations will predict the only solution with given boundary conditions. In the subsequent sections, all field quantities undeclared have to be treated as the same physical quantities.

Basic equations of cylindrical shells with surface effect

For axisymmetric wave propagating in piezoelectric cylindrical shells, one considers $u_2=0$ and ${()}_{,2}=0$ in the general equations of section “Formulation of two-dimensional theory,” yielding the geometric and electric potential relations ²⁵

$$S_{11}^{(0)}=u_{1,1}^{(0)},S_{22}^{(0)}=\frac{u_3^{(0)}}{R}$$

$$2S_{13}^{(0)}=u_1^{(1)}+u_{3,1}^{(0)},S_{11}^{(1)}=u_{1,1}^{(1)}$$

(16)

$$E_1=-{\varphi }_{,1}^{(0)}-x_3{\varphi }_{,1}^{(1)},E_3=-{\varphi }^{(1)}$$

(17)

The governing equations of the cylindrical piezoelectric shell are presented as

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{T}}_{11,1}^{(0)}+{\hat{T}}_{12,2}^{(0)}/R=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_1^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_1^{(1)}}\\ \multicolumn{1}{c}{{\hat{T}}_{12,1}^{(0)}+{\hat{T}}_{22,2}^{(0)}/R+{\hat{T}}_{23}^{(0)}/R=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_2^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_2^{(1)}}\\ \multicolumn{1}{c}{{\hat{T}}_{13,1}^{(0)}+{\hat{T}}_{23,2}^{(0)}/R-{\hat{T}}_{22}^{(0)}/R=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_3^{(0)}}\\ \multicolumn{1}{c}{{\hat{T}}_{11,1}^{(1)}+{\hat{T}}_{12,2}^{(1)}/R-T_{13}^{(0)}={\rho }^{(1)}{\stackrel{··}{u}}_1^{(0)}+({\rho }^{(2)}+2{\rho }^s{h}^2){\stackrel{··}{u}}_1^{(1)}}\\ \multicolumn{1}{c}{{\hat{T}}_{12,1}^{(1)}+{\hat{T}}_{22,2}^{(1)}/R-T_{23}^{(0)}={\rho }^{(1)}{\stackrel{··}{u}}_2^{(0)}+({\rho }^{(2)}+2{\rho }^s{h}^2){\stackrel{··}{u}}_2^{(1)}}\end{array}$$

(18)

where ${\hat{T}}_{12}^{(0)}={\hat{T}}_{12}^{(1)}={\hat{T}}_{23}^{(0)}=0$ , and hence, the second and fifth equations are abandoned due to the axisymmetric assumption.

The charge equations are written as

$$\begin{array}{cc}\multicolumn{1}{c}{}& D_{1,1}^{(0)}+D_1^{(0)}/R+D_{2,2}^{(0)}/R+D_3^s(h)-D_3^s(-h)=0\hfill \\ \multicolumn{1}{c}{}& D_{1,1}^{(1)}+D_1^{(1)}/R+D_{2,2}^{(1)}/R-D_3^{(0)}+h[D_3^s(h)-D_3^s(-h)]=0\hfill \end{array}$$

(19)

where $D_2^{(0)}=D_2^{(1)}=0$ .

The effective resultant forces can be determined by combining equations (9), (12), and (15), and the substitution of the resultant forces into equation (18) yields

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\overline{c}}_{11}^{(0)}+2c_{11}^s)u_{1,11}^{(0)}+{\overline{c}}_{11}^{(1)}{u}_{1,11}^{(1)}+({\overline{c}}_{12}^{(0)}+2c_{12}^s)u_{3,1}^{(0)}/R\hfill \\ \multicolumn{1}{c}{+({\overline{e}}_{31}^{(0)}+2e_{31}^s){\varphi }_{,1}^{(1)}=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_1^{(0)}+{\rho }^{(1)}{\stackrel{··}{u}}_1^{(1)}}\\ \multicolumn{1}{c}{}& -({\overline{c}}_{12}^{(0)}+2c_{12}^s)u_{1,1}^{(0)}/R+({\overline{c}}_{55}^{(0)}+2c_{55}^s-{\overline{c}}_{12}^{(1)}/R)u_{1,1}^{(1)}\hfill \\ \multicolumn{1}{c}{+({\overline{c}}_{55}^{(0)}+2c_{55}^s)u_{3,11}^{(0)}-({\overline{c}}_{11}^{(0)}+2c_{11}^s)u_3^{(0)}/R^2}\\ \multicolumn{1}{c}{}& +({\overline{e}}_{15}^{(0)}+2e_{15}^s){\varphi }_{,11}^{(0)}+{\overline{e}}_{15}^{(1)}{\varphi }_{,11}^{(1)}-({\overline{e}}_{31}^{(0)}+2e_{31}^s){\varphi }^{(1)}/R\hfill \\ \multicolumn{1}{c}{=({\rho }^{(0)}+2{\rho }^s){\stackrel{··}{u}}_3^{(0)}}\end{array}$$

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\overline{c}}_{11}^{(1)}{u}_{1,11}^{(0)}+({\overline{c}}_{11}^{(2)}+2c_{11}^s{h}^2)u_{1,11}^{(1)}-{\overline{c}}_{55}^{(0)}{u}_1^{(1)}\hfill \\ \multicolumn{1}{c}{+({\overline{c}}_{12}^{(1)}/R-{\overline{c}}_{55}^{(0)})u_{3,1}^{(0)}}\\ \multicolumn{1}{c}{}& -{\overline{e}}_{15}^{(0)}{\varphi }_{,1}^{(0)}+({\overline{e}}_{31}^{(1)}-{\overline{e}}_{15}^{(1)}){\varphi }_{,1}^{(1)}={\rho }^{(1)}{\stackrel{··}{u}}_1^{(0)}\hfill \\ \multicolumn{1}{c}{+({\rho }^{(2)}+2{\rho }^s{h}^2){\stackrel{··}{u}}_1^{(1)}}\end{array}$$

(20)

Similarly, the governing equations of the electric field can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\overline{e}}_{15}^{(0)}{u}_{1,1}^{(1)}+{\overline{e}}_{15}^{(0)}{u}_1^{(1)}/R+{\overline{e}}_{15}^{(0)}{u}_{3,11}^{(0)}\hfill \\ \multicolumn{1}{c}{+{\overline{e}}_{15}^{(0)}{u}_{3,1}^{(0)}/R-{\overline{\epsilon }}_{11}^{(0)}{\varphi }_{,11}^{(0)}-{\overline{\epsilon }}_{11}^{(0)}{\varphi }_{,1}^{(0)}/R}\\ \multicolumn{1}{c}{}& -{\overline{\epsilon }}_{11}^{(1)}{\varphi }_{,11}^{(1)}-{\overline{\epsilon }}_{11}^{(1)}{\varphi }_{,1}^{(1)}/R+D_3^s(h)-D_3^s(-h)=0\hfill \end{array}$$

$$\begin{array}{cc}\multicolumn{1}{c}{}& -{\overline{e}}_{31}^{(0)}{u}_{1,1}^{(0)}+({\overline{e}}_{15}^{(1)}-{\overline{e}}_{31}^{(1)})u_{1,1}^{(1)}+{\overline{e}}_{15}^{(1)}{u}_1^{(1)}/R\hfill \\ \multicolumn{1}{c}{-{\overline{e}}_{31}^{(0)}{u}_3^{(0)}/R+{\overline{e}}_{15}^{(1)}{u}_{3,11}^{(0)}+{\overline{e}}_{15}^{(1)}{u}_{3,1}^{(0)}/R}\\ \multicolumn{1}{c}{}& -{\overline{\epsilon }}_{11}^{(1)}{\varphi }_{,11}^{(0)}-{\overline{\epsilon }}_{11}^{(1)}{\varphi }_{,1}^{(0)}/R+{\overline{\epsilon }}_{33}^{(0)}{\varphi }^{(1)}-{\overline{\epsilon }}_{11}^{(2)}{\varphi }_{,11}^{(1)}\hfill \\ \multicolumn{1}{c}{-{\overline{\epsilon }}_{11}^{(2)}{\varphi }_{,1}^{(1)}/R+h[D_3^s(h)+D_3^s(-h)]=0}\end{array}$$

(21)

For open circuit, the electric displacements are equal to 0 at the surfaces $x_3=\pm h$ , that is, $D_3^s(\pm h)=0$ . For short circuit, $\varphi (\pm h)=0$ and $\varphi ={\varphi }^{(0)}+x_3{\varphi }^{(1)}$ , thus one can deduce ${\varphi }^{(0)}={\varphi }^{(1)}=0$ .

For wave propagating in axial direction, one assumes

$${\left\{\begin{array}{ccccc}{u}_1^{(0)}\hfill & u_1^{(1)}\hfill & u_3^{(0)}\hfill & {\varphi }^{(0)}\hfill & {\varphi }^{(1)}\hfill \end{array}\right\}}^T={\left\{\begin{array}{ccccc}{\overline{u}}_1^{(0)}\hfill & {\overline{u}}_1^{(1)}\hfill & {\overline{u}}_3^{(0)}\hfill & {\overline{\varphi }}^{(0)}\hfill & {\overline{\varphi }}^{(1)}\hfill \end{array}\right\}}^T{e}^{i(k{x}_1-\omega t)}$$

(22)

where $k$ is the wave number in $x_1$ direction, and $\omega$ is the angular frequency.

Substituting equation (22) into equations (20) and (21), one obtains the frequency equations for open circuit condition

$$\begin{array}{cc}\multicolumn{1}{c}{}& [({\rho }^{(0)}+2{\rho }^s){\omega }^2-({\overline{c}}_{11}^{(0)}+2c_{11}^s)k^2]{\overline{u}}_1^{(0)}\hfill \\ \multicolumn{1}{c}{+({\rho }^{(1)}{\omega }^2-{\overline{c}}_{11}^{(1)}{k}^2){\overline{u}}_1^{(1)}+({\overline{c}}_{12}^{(0)}+2c_{12}^s)\mathrm{ik}{\overline{u}}_3^{(0)}/R}\\ \multicolumn{1}{c}{}& +({\overline{e}}_{31}^{(0)}+2e_{31}^s)\mathrm{ik}{\overline{\varphi }}^{(1)}=0\hfill \\ \multicolumn{1}{c}{}& -({\overline{c}}_{12}^{(0)}+2c_{12}^s)\mathrm{ik}{\overline{u}}_1^{(0)}/R+({\overline{c}}_{55}^{(0)}+2c_{55}^s-{\overline{c}}_{12}^{(1)}/R)\mathrm{ik}{\overline{u}}_1^{(1)}\hfill \\ \multicolumn{1}{c}{+[({\rho }^{(0)}+2{\rho }^s){\omega }^2-({\overline{c}}_{55}^{(0)}+2c_{55}^s)k^2}\\ \multicolumn{1}{c}{}& -({\overline{c}}_{11}^{(0)}+2c_{11}^s)/R^2]{\overline{u}}_3^{(0)}-({\overline{e}}_{15}^{(0)}+2e_{15}^s)k^2{\overline{\varphi }}^{(0)}\hfill \\ \multicolumn{1}{c}{+[-{\overline{e}}_{15}^{(1)}{k}^2-({\overline{e}}_{31}^{(0)}+2e_{31}^s)/R]{\overline{\varphi }}^{(1)}=0}\\ \multicolumn{1}{c}{({\rho }^{(1)}{\omega }^2-{\overline{c}}_{11}^{(1)}{k}^2){\overline{u}}_1^{(0)}}\\ \multicolumn{1}{c}{+[({\rho }^{(2)}+2{\rho }^s{h}^2){\omega }^2-({\overline{c}}_{11}^{(2)}+2c_{11}^s{h}^2)k^2-{\overline{c}}_{55}^{(0)}]{\overline{u}}_1^{(1)}}\\ \multicolumn{1}{c}{}& +({\overline{c}}_{12}^{(1)}/R-{\overline{c}}_{55}^{(0)})\mathrm{ik}{\overline{u}}_3^{(0)}-{\overline{e}}_{15}^{(0)}\mathrm{ik}{\overline{\varphi }}^{(0)}+({\overline{e}}_{31}^{(1)}-{\overline{e}}_{15}^{(1)})\mathrm{ik}{\overline{\varphi }}^{(1)}=0\hfill \\ \multicolumn{1}{c}{}& {\overline{e}}_{15}^{(0)}(\mathrm{ik}+1/R){\overline{u}}_1^{(1)}+{\overline{e}}_{15}^{(0)}(-k^2+\mathrm{ik}/R){\overline{u}}_3^{(0)}\hfill \\ \multicolumn{1}{c}{+{\epsilon }_{11}^{(0)}(k^2-\mathrm{ik}/R){\overline{\varphi }}^{(0)}+{\epsilon }_{11}^{(1)}(k^2-\mathrm{ik}/R){\overline{\varphi }}^{(1)}=0}\\ \multicolumn{1}{c}{}& -{\overline{e}}_{31}^{(0)}\mathrm{ik}{\overline{u}}_1^{(0)}+[({\overline{e}}_{15}^{(1)}-{\overline{e}}_{31}^{(1)})\mathrm{ik}+{\overline{e}}_{15}^{(1)}/R]{\overline{u}}_1^{(1)}\hfill \\ \multicolumn{1}{c}{+(-{\overline{e}}_{31}^{(0)}/R-{\overline{e}}_{15}^{(1)}{k}^2+{\overline{e}}_{15}^{(1)}\mathrm{ik}/R){\overline{u}}_3^{(0)}}\\ \multicolumn{1}{c}{}& +({\overline{\epsilon }}_{11}^{(1)}{k}^2-{\overline{\epsilon }}_{11}^{(1)}\mathrm{ik}/R){\overline{\varphi }}^{(0)}+({\overline{\epsilon }}_{33}^{(0)}+{\overline{\epsilon }}_{11}^{(2)}{k}^2-{\overline{\epsilon }}_{11}^{(2)}\mathrm{ik}/R){\overline{\varphi }}^{(1)}=0\hfill \end{array}$$

The frequency equations for short circuit condition are omitted here for the sake of simplicity.

Results and discussion

In order to validate the present analysis for cylindrical shells, the comparisons with the free vibration problem presented by Lam and Qian ²⁶ and Liu et al. ²⁷ are shown in Table 1. Unlike the wave propagation problem, the cylindrical shells are laminated with limited length L. The stacking sequence of the laminated shell is [90/0/90], and the non-dimensional frequency parameter is ${\Omega }^2=\rho {\omega }^2{L}^4/E_2{h}^2$ . It can be seen that the present values are consistent with those of other references even for a short and thick cylindrical shell, which verifies the validity of the present analysis. It is worth mentioning that the first-order shear deformation theory is also utilized in Lam and Qian ²⁶ and Liu et al. ²⁷

Table 1.

Comparison of the first natural frequencies of the lami- nated composite cylindrical shell with simply supported boundary conditions (m = 1, h/R = 0.2, $E_1=40E_2,G_{12}=G_{13}=$ $0.6E_2$ , $G_{23}=0.5E_2$ ).

Theory	L/R = 1, n = 2	L/R = 2, n = 2
Present	10.045	17.787
Lam and Qian 26	10.0446	17.7861
Liu et al. 27	10.1141	17.3073

On the basis of previous formulation, we studied the axisymmetric wave propagating in piezoelectric shells in the presence of surface effect. In the following examples, the piezoelectric cylindrical shell composed of $\text{BaTi}{\text{O}}_3$ is considered. The material properties are given as follows ²⁸ : c ₁₁ = 166 GPa, c ₁₂ = 77 GPa, c ₁₃ = 78 GPa, c ₃₃ = 162 GPa, c ₄₄ = 43 GPa, $e_{15}=11.6\text{C}/{\text{m}}^2$ , $e_{31}=-4.4\text{C}/{\text{m}}^2$ , $e_{33}=18.6\text{C}/{\text{m}}^2$ , ${\epsilon }_{11}=11.2\text{C}/(\text{Vm})$ , ${\epsilon }_{33}=12.6\text{C}/(\text{Vm})$ , and the mass density $\rho =5800\text{kg}/{\text{m}}^3$ . ²⁹ For the surface material properties, we use a simple but reasonable approach proposed by Gurtin and Murdoch ⁶ and Zhang et al. ²³ The surface material constants are equal to the corresponding bulk material constants multiplied by the material intrinsic length $h_0$ .

In Figures 2–6, the dimensionless frequency, wave number, and phase velocity are adopted: ${\Omega }^2=\rho {\omega }^2{h}^2/{\overline{c}}_{11}$ , $K=\mathrm{kh}$ , $C=\Omega /K$ , and geometry parameter is $R=0.2\text{m}$ . Figure 2 shows the phase velocity curves for electric short-circuited conditions, with fixed $h=50\text{nm}$ . In Figure 2, the effect of surface stress on the phase velocity is investigated, and the first two spectra are displayed for h ₀ = 0, 10, and 20 nm. It can be seen that the phase velocity increases/decreases monotonically with the frequency. The first branches for three different cases almost overlap together, while the higher branches decrease drastically with increasing $\Omega$ at lower frequencies. When $\Omega$ is larger than 1.5, both two branches approach to constant values. Hence, one can conclude that the branches are dispersive at lower frequencies and become nondispersive at higher frequencies. And distinguishingly, the surface layer affects predominantly the higher mode in low-frequency range: the surface effect tends to decrease the frequency.

Figure 2.

Phase velocity curves for short circuit ( $h=50\text{nm}$ ).

Figure 3.

Dispersion curves for short circuit ( $h=50\text{nm}$ ).

Figure 4.

$\Omega$ versus $\log (h/h_0)$ for open circuit ( $h=50\text{nm}$ , $K=2$ ).

Figure 5.

$\Omega$ versus $\log (h/h_0)$ for short circuit ( $h=50\text{nm}$ , $K=2$ ).

Figure 6.

$\Delta =(\Omega -{\Omega }_0)/{\Omega }_0$ versus $\log (h/h_0)$ for open circuit ( $h=50\text{nm}$ , $K=2$ ).

Figure 3 presents the dispersion curves for short circuit with $h=50\text{nm}$ , in which blue, red, and black dotted curves represent h ₀ = 0, 10, and 20 nm, respectively. The branches for real wave number correspond to the propagating modes, while those for imaginary wave number correspond to the evanescent waves or nonpropagating modes, that is, a disturbance that will spatially decrease in amplitude (see $e^{i(k{x}_1-\omega t)}$ in equation (22)). The branches for purely imaginary wave number change from a near semicircle to a flat semiellipse due to the surface effect. For each $h_0$ , two branches are exhibited in the computation interval for real wave number. Similar to Figure 2, the higher curves for different $h_0$ are diverse from each other, while the lowest branches are indistinguishable. One can deduce that the strengthening of surface effect (increasing $h_0$ ) makes two branches approach closer, and the mode transformation (an intersection zone) ³⁰ is more likely to happen, which may be useful in nano sensors. So the surface effect must be taken into consideration when nano-sized structures are analyzed.

The frequencies for K = 0 are known as cutoff frequencies in the elastodynamics, at which some of the evanescent waves (purely imaginary branches) start propagating (purely real branches) and all the points of the waveguide vibrate in phase. It is seen that the first branch of propagating wave in Figure 3 has a zero and a nonzero cutoff frequency, and a backward wave occurs in the first branch. ³⁰ For a given mode, the cutoff frequencies are decreased with the increase of $h_0$ (see inset of Figure 3).

Figures 4 and 5 present the curves of frequency $\Omega$ versus $\log (h/h_0)$ for electric open and short circuits, respectively. The blue solid line and red dash line represent the case with or without surface effect, respectively. It is clearly displayed that the dimensionless frequency at the nano-scale is size dependent: when $h/h_0$ is larger than 100, the surface effect is weak and can be neglected, but when $h/h_0$ decreases, the surface effect affects the frequency drastically and cannot be ignored anymore.

The curves of relative difference $\Delta =(\Omega -{\Omega }_0)/{\Omega }_0$ versus $\log (h/h_0)$ are plotted in Figure 6 for open circuit, where $\Omega$ is the frequency considering surface effect, and ${\Omega }_0$ is the frequency without surface effect. Similar to Figures 4 and 5, when $h/h_0$ is small, such as 10, the relative difference is up to 12%, so the surface effect plays a vital role in nano-scale shells.

Conclusion

We carried out a research of surface effect on axisymmetric wave propagating in piezoelectric nano-shells. The results for axisymmetric wave propagation show that the surface effect has a considerable effect on wave dispersion spectra at the nano-scale, making the first two branches approach closer. Further investigation has revealed that when the thickness of the shell is comparable to that of the surface layer, the surface effect would play a significant role in dynamic behavior of the structure. These findings will provide a solid theoretical basis for optimal design of piezoelectric nano structures.