Exact solutions for the electromechanical responses of a dielectric nano-ring

Abstract

Based on an extended linear theory for dielectrics, this work presents exact solutions for the electromechanical responses of a dielectric nano-ring subjected to mechanical and electrical loads. By incorporating terms involving the strain gradient and the electric field gradient into the electric Gibbs free energy, both the direct and converse flexoelectric effects can be captured. The general solution to the differential governing equation is a linear combination of modified Bessel functions and the electromechanical fields are obtained by solving boundary value problems. The influences of the material surfaces and the electric circuit conditions on the electromechanical coupling behavior of the dielectric nano-ring have also been considered. It is found that the flexoelectricity and the surface effect on the electromechanical fields are substantial, and they are affected by the size of the ring structure. Moreover, the flexoelectric effect is sensitive to the material length scales introduced by the new theory and the electric circuit conditions. From simulation results, it is also suggested that nano-scaled electromechanical coupling devices could be built based on dielectric materials through flexoelectricity, which thereby opens up new perspectives for nano-technology.

Keywords

Direct and converse flexoelectric effects dielectric nano-ring electromechanical coupling size-dependent behaviors surface effect

Introduction

The development of micro- and nano-technologies enables the fabrication of complex miniaturized systems, that is, micro- and nano-electromechanical systems (MEMS and NEMS). One of the fundamental mechanisms that makes the systems functional is the electromechanical coupling effect. Due to the inherent electromechanical coupling or piezoelectricity, nano-structured piezoelectric materials are the most promising material candidates for developing transduction devices in MEMS and NEMS (Eom and Trolier-McKinstry, 2012; Wang, 2007). However, piezoelectricity is restricted only to certain materials with non-centrosymmetric crystal structures and is severely compromised at high temperatures (Gubinyi et al., 2008). In addition, electric fatigue may occur in piezoelectric devices associated with repeated electrical cycling, leading to the degradation of ferroelectric properties such as a significant decrease in remnant and saturated polarizations (Jiang et al., 1994). Piezoelectric ceramics based on lead zirconate and lead titanate also have environmental concerns over their lead component (Cross, 2004). Consequently, it is desirable to have more material choices that can be applied to MEMS and NEMS under extreme and special needs conditions. Fortunately, the recent surge of scientific interest in a somewhat understudied electromechanical coupling, flexoelectricity, may help such a choice of functional materials and micro- and nano-scale elements.

Flexoelectricity, referring to a linear dielectric polarization in response to non-uniform strains or strain gradients (the direct flexoelectric effect; Fu et al., 2006). Materials exhibiting the direct flexoelectric effect also show the converse flexoelectric effect, that is, the coupling between the induced mechanical stress or strain and the polarization gradient (Kogan, 1964). However, it has been very difficult to measure the flexoelectric effects since they are rather weak at the macro-scale. In the 2000s, inspired by the prediction that the flexoelectric coefficient of a material is proportional to its dielectric constant (Tagantsev, 1986), Ma and Cross (2001a,b, 2005, 2006) conducted a series of experiments to measure the flexoelectric coefficients of high-permittivity materials such as relaxor and ferroelectric ceramics. They confirmed the theoretical prediction of large flexoelectric coefficients and thus large flexoelectric effects in such materials. As a result, there has been an increasing interest in the study of flexoelectricity. The strain gradient induced polarization in single crystals of paraelectric SrTiO₃ has later been measured by Zubko et al. (2007). Li et al. (2013) measured the flexoelectric coefficient in BZN/Ag composite, and very recently, Lu et al. (2016) measured the flexoelectric coefficient of piezoelectric polymers, that is, polyvinylidene fluoride. Maranganti and Sharma (2009) reported estimates for the flexoelectric constants for certain representative materials from both density-functional theory and empirical shell models. Hong et al. (2010) attempted a first-principles calculation of flexoelectric constants of BaTiO₃. The presence of flexoelectricity has also been found in studying the polarization hysteresis curves (Lee et al., 2011), rotation of polarization (Catalan et al., 2011) in thin films and the critical phase transition temperature of nano-wires and thin pills (Eliseev et al., 2009). Additionally, Majdoub et al. (2008) and Huang et al. (2011) investigated the scaling effect of flexoelectricity and it was found that the effective piezoelectric coefficient and electrical energy density are significantly enhanced due to flexoelectricity, suggesting that nano-scaled dielectric materials hold a promise of potential applications in MEMS and NEMS. Since flexoelectricity occurs in all 32 crystallographic point groups, unlike piezoelectricity which exists only in 20 non-centrosymmetric point groups, it can be exploited to design pseudo-piezoelectric materials that can be applied to MEMS and NEMS. For example, non-piezoelectric barium strontium titanate-based composites have been shown to yield effective piezoelectric coefficients comparable to those of commercial piezoelectric ceramics due to the local strain gradients and flexoelectricity (Chu et al., 2009). Recently, Bhaskar et al. (2016) fabricated a silicon-compatible thin-film cantilever nano-actuator with a single flexoelectrically active layer of strontium titanate; its actuation is comparable to that of state-of-the-art piezoelectric bimorph nano-cantilevers. Flexoelectricity may also offer unprecedented possibilities for energy harvesting at the micro- and nano-scale. As an example, Deng et al. (2014) theoretically demonstrated a flexoelectric energy harvester based on a simple symmetric thin beam under base excitation, which is not possible in piezoelectric energy harvesting.

Since miniaturized beams, plates, rings and cylinders made of electromechanical materials can serve as essential building blocks for MEMS and NEMS, it is of great importance to investigate the flexoelectric responses of these basic structures. Pioneered by the physical and mathematical formulations for elastic dielectrics with flexoelectricity (Hu and Shen, 2009; Maranganti et al., 2006; Hu and Shen, 2010), a volume of meaningful theoretical work has been conducted in this area. For instance, based on continuum mechanics models, Yan and Jiang (2013a,b) discussed the influence of flexoelectricity on the size-dependent electroelastic responses of piezoelectric nano-beams with different boundary conditions and the static and dynamic responses of a simply supported piezoelectric nano-beam. Liang et al. (2014) solved the static bending problem of a cantilever beam to illustrate the effects of surface and flexoelectricity. Li et al. (2014) solved the static bending and free vibration problems of a three-layer micro-beam including a flexoelectric dielectric layer based on a piezoelectric couple stress theory developed by Hadjesfandiari (2013). The first-order strain gradient effects in micro-piezoelectric-bimorph power harvesters have been examined by including the first-order gradient terms in the energy density function (Hu et al., 2011; Wang et al., 2012). Yang et al. (2015) explored the influence of flexoelectricity on the electromechanical coupling behavior of a simply supported piezoelectric nano-plate by using the Kirchhoff plate theory. Hu et al. (2013) carried out a study on the spatially distributed flexoelectric signals on circular rings. Mao and Purohit (2014) investigated the electromechanical fields of a beam and a circular cylinder with a central hole under external loadings by combining a theory of strain gradient elasticity and classical electrostatics. The effect of flexoelectricity on the electroelastic fields of a hollow piezoelectric nano-cylinder was also analyzed (Yan and Jiang, 2015). These investigations contribute to the understanding of flexoelectricity and its role in the electromechanical coupling of the nano-structured dielectrics. However, these studies focused on the direct flexoelectric effect only by ignoring the terms involving polarization or electric field gradients. Considering both the direct and converse flexoelectric effects, the exact solutions for the displacement and the electric potential fields in a dielectric nano-beam and an elastic beam integrated with a flexoelectric nano-actuator layer were obtained by Ray (2014, 2016). The electroelastic responses, vibration and buckling behaviors of a piezoelectric nanoplate/nanofilm have also been studied with the two electromechanical coupling effects (Liang et al., 2016; Zhang et al., 2014). Recently, Yan (2016) investigated the size-dependent bending and vibration behaviors of a clamped piezoelectric circular nano-plate with the consideration of direct and converse flexoelectric effects as well as the surface effect. It should be mentioned that to make the problem mathematically tractable, these recent studies still omitted some coupling terms that were stated in the theoretical frameworks of Maranganti et al. (2006) and Hu and Shen (2009, 2010), for example, the strain gradient and strain gradient coupling term in (Liang et al., 2016; Ray, 2014, 2016; Zhang et al., 2014) and some third- and fifth-order coupling terms in (Liang et al., 2016; Yan, 2016; Zhang et al., 2014). Therefore, an exact and rigorous analysis of the flexoelectric effect on the nano-structured dielectrics is still essential despite plenty of research that has been performed in the field. It should be mentioned that the size-dependent behaviors of a dielectric cantilever subjected to a force at the free end and a voltage across the thickness were captured based on a reformulated flexoelectric theory by splitting the strain gradient tensor into mutually independent parts (Li et al., 2015).

In the current work, based on the extended linear theory for elastic dielectrics with flexoelectricity, exact solutions of a dielectric nano-ring under mechanical and electrical loads are presented without any assumptions and simplifications. Since it has been widely recognized that surface effect also plays a vital role in the mechanical and physical behaviors of elastic and piezoelectric nano-materials (Huang and Yu, 2006; Miller and Shenoy, 2000), its influence on the electromechanical behaviors of the dielectric nano-ring has also been examined. Simulation results will be conducted to show the mechanical and electrical fields of the dielectric nano-ring with varying sizes, material properties and loading conditions.

Governing equations and the general solutions

Based on an extended linear theory for dielectrics, the general expression for the bulk electric Gibbs free energy density function $U_{b}$ depends on the conventional strain and electric field and their gradients. For isotropic dielectrics, $U_{b}$ can be written as (Hu and Shen, 2009)

\begin{array}{l} U_{b} = - \frac{1}{2} a_{k l} E_{k} E_{l} - \frac{1}{2} b_{i j k l} V_{i j} V_{k l} + \frac{1}{2} c_{i j k l} ε_{i j} ε_{k l} \\ - e_{i j k l} ε_{i j} V_{k l} - μ_{i j k l} η_{i j k} E_{l} + \frac{1}{2} g_{i j k l m n} η_{i j k} η_{l m n} \end{array}

(1)

in which $a_{kl} = a δ_{kl}$ and $c_{ijkl} = c_{12} δ_{ij} δ_{kl} + c_{44} (δ_{ik} δ_{jl} + δ_{il} δ_{jk})$ are the dielectric and elastic tensors, respectively. $b_{ijkl}$ , $e_{ijkl}$ , $μ_{ijkl}$ and $g_{ijklmn}$ are the new material constants due to the introduction of the strain gradient and the electric field gradient into the electric Gibbs free energy. Particularly, $μ_{ijkl}$ and $e_{ijkl}$ are the direct and converse flexoelectric tensors, respectively, and $μ_{ijkl} = - e_{ijkl} = f_{1} δ_{ij} δ_{kl} + f_{2} (δ_{ik} δ_{jl} + δ_{il} δ_{jk})$ according to the work by Le Quang and He (2011) and Ray (2014), with $f_{1}$ and $f_{2}$ being two flexoelectric constants. $b_{ijkl} = b_{1} δ_{ij} δ_{kl} + b_{2} δ_{ik} δ_{jl} + b_{3} δ_{il} δ_{jk}$ represents the higher-order coupling between the electric field gradient and the electric field gradient with $b_{1} - b_{3}$ being material constants (Maranganti et al., 2006). $g_{ijklmn}$ represents the purely non-local elastic effect and relates to the strain gradient elasticity theory, which could be expressed as $g_{ijklmn} = l_{1}^{2} [c_{12} δ_{ij} δ_{lm} + c_{44} (δ_{il} δ_{jm} + δ_{im} δ_{jl})]$ with $l_{1}$ being a newly introduced material length scale with the dimension of length (Mao and Purohit, 2014). In addition, $ε_{ij}$ , $η_{ijk}$ , $E_{i}$ and $V_{ij}$ are the strain, strain gradient, electric field and electric field gradient tensors, respectively, which are defined as

ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i}); η_{i j k} = u_{i, j k}; E_{i} = - ϕ_{, i}; V_{i j} = E_{i, j}

(2)

with $u_{i}$ and $ϕ$ denoting the displacement vector and electric potential, respectively. The coma notation is adopted in this work to represent the derivatives. The constitutive relations can be expressed in terms of $U_{b}$ as (Hu and Shen, 2009)

σ_{ij} = \frac{\partial U_{b}}{\partial ε_{ij}}; τ_{ijm} = \frac{\partial U_{b}}{\partial η_{ijm}}; D_{i} = - \frac{\partial U_{b}}{\partial E_{i}}; Q_{ij} = - \frac{\partial U_{b}}{\partial V_{ij}}

(3)

in which $σ_{ij}$ and $D_{i}$ are the conventional stress and electric displacement field tensors, and $τ_{ijm}$ and $Q_{ij}$ are the higher-order stress and the electric quadrupole, respectively. According to Hu and Shen (2009), the governing equations can be written as

{(σ_{ij} - τ_{ijm, m})}_{, j} = 0; {(D_{i} - Q_{ij, j})}_{, i} = 0

(4)

The problem considered is a dielectric circular nano-ring with inner and outer radii being $R_{i}$ and $R_{o}$ . Under closed circuit condition, the ring is subject to an electric potential V, as seen from Figure 1. Additionally, internal and external pressures $p_{i}$ and $p_{o}$ are applied to the ring. This is a very classic problem, just like calculating the stress in an elastic ring in linear elasticity, and calculating the capacitance and the electrical fields in a circular cylindrical shell in electrostatics or piezoelectricity. The solution of this simple geometry in this work offers not only a direct comparison to that from classical elasticity, strain gradient elasticity and theory with the direct flexoelectric effect only, but also emphasizes the roles of converse flexoelectricity and surface effect in the electromechanical behavior of nano-scaled dielectrics. Polar coordinates $(r, θ)$ are adopted to describe the problem for convenience. Due to axial symmetry, the only relevant displacement and electric field components are $u_{r} = u_{r} (r)$ and $E_{r} = E_{r} (r)$ , respectively. By using the relation $ε_{rr} = u_{r, r}$ , $ε_{θ θ} = u_{r} / r$ , $η_{rrr} = u_{r, rr}$ , $η_{r θ θ} = η_{θ θ r} = u_{r, r} / r - u_{r} / r^{2}$ , $V_{rr} = E_{r, r}$ and $V_{θ θ} = E_{r} / r$ , the bulk constitutive relations given by equation (3) can be reduced to

\begin{matrix} σ_{rr} = c_{11} u_{r, r} + c_{12} \frac{u_{r}}{r} + f' E_{r, r} + f_{1} \frac{E_{r}}{r} \\ σ_{θ θ} = c_{12} u_{r, r} + c_{11} \frac{u_{r}}{r} + f_{1} E_{r, r} + f' \frac{E_{r}}{r} \\ τ_{rrr} = - f' E_{r} + [c_{11} u_{r, rr} + c_{12} (\frac{u_{r, r}}{r} - \frac{u_{r}}{r^{2}})] l_{1}^{2} \\ τ_{θ θ r} = - f_{1} E_{r} + [c_{12} u_{r, rr} + c_{11} (\frac{u_{r, r}}{r} - \frac{u_{r}}{r^{2}})] l_{1}^{2} \\ τ_{θ r θ} = - f_{2} E_{r} + 2 c_{44} (\frac{u_{r, r}}{r} - \frac{u_{r}}{r^{2}}) l_{1}^{2} \\ D_{r} = a E_{r} + f' (u_{r, rr} + \frac{u_{r, r}}{r} - \frac{u_{r}}{r^{2}}) \\ Q_{rr} = b' E_{r, r} + b_{1} \frac{E_{r}}{r} - f' u_{r, r} - f_{1} \frac{u_{r}}{r} \\ Q_{θ θ} = b_{1} E_{r, r} + b' \frac{E_{r}}{r} - f_{1} u_{r, r} - f' \frac{u_{r}}{r} \end{matrix}

(5)

with $c_{11} = c_{12} + 2 c_{44}$ , $f' = f_{1} + 2 f_{2}$ and $b' = b_{1} + b_{2} + b_{3}$ . Substituting equations (5) into equation (4), the governing equations for the dielectric ring can be written as

\begin{matrix} c_{11} (u_{r, rr} + \frac{u_{r, r}}{r} - \frac{u_{r}}{r^{2}}) + 2 f' (E_{r, rr} + \frac{E_{r, r}}{r} - \frac{E_{r}}{r^{2}}) \\ - c_{11} l_{1}^{2} (u_{r, rrrr} + 2 \frac{u_{r, rrr}}{r} - 3 \frac{u_{r, rr}}{r^{2}} + 3 \frac{u_{r, r}}{r^{3}} - 3 \frac{u_{r}}{r^{4}}) = 0 \end{matrix}

(6)

and

\begin{matrix} a (E_{r, r} + \frac{E_{r}}{r}) - b' (E_{r, rrr} + 2 \frac{E_{r, rr}}{r} - \frac{E_{r, r}}{r^{2}} + \frac{E_{r}}{r^{3}}) \\ + 2 f' (u_{r, rrr} + 2 \frac{u_{r, rr}}{r} - \frac{u_{r, r}}{r^{2}} + \frac{u_{r}}{r^{3}}) = 0 \end{matrix}

(7)

Let $p (r) = E_{r, r} + E_{r} / r$ and $q (r) = u_{r, rr} + u_{r, r} / r - u_{r} / r^{2}$ . The above equations can be rewritten as

c_{11} q + 2 f' p_{, r} - c_{11} l_{1}^{2} (q_{, rr} + \frac{q_{, r}}{r} - \frac{q}{r^{2}}) = 0

(8)

ap - b' (p_{, rr} + \frac{p_{, r}}{r}) + 2 f' (q_{, r} + \frac{q}{r}) = 0

(9)

From equation (8), the derivative of p with respect to r (i.e. $p_{, r}$ ) can be expressed in terms of q and its derivatives (i.e. $q_{, r}$ and $q_{, rr}$ ). Then if we differentiate $p_{, r}$ with respect to r, we can get the expressions of $p_{, rr}$ and further $p_{, rrr}$ . Next, substitute $p_{, r}, p_{, rr}$ and $p_{, rrr}$ into the equation resulting from taking the derivative of equation (9), and the following decoupled equation can be finally obtained as

(\frac{d^{2}}{d r^{2}} + \frac{d}{r d r} - \frac{1}{r^{2}} - α) (\frac{d^{2}}{d r^{2}} + \frac{d}{r d r} - \frac{1}{r^{2}} - β) q = 0

(10)

If we take $f' = - l_{f} \sqrt{a c_{11}}$ and $b' = {al}_{2}^{2}$ with $l_{f}$ and $l_{2}$ being length scale parameters associated with the flexoelectricity and the electric field gradient, $α$ and $β$ in the above equation can be expressed by $α = (k + \sqrt{k^{2} - 4 a^{2} l_{1}^{2} l_{2}^{2}}) / 2 {al}_{1}^{2} l_{2}^{2}$ and $β = (k - \sqrt{k^{2} - 4 a^{2} l_{1}^{2} l_{2}^{2}}) / 2 {al}_{1}^{2} l_{2}^{2}$ with $k = a (l_{1}^{2} + l_{2}^{2} + 4 l_{f}^{2})$ . The solution of equation (10) can be expressed as

\begin{matrix} q = c_{1} I_{1} (\sqrt{α} r) + c_{2} K_{1} (\sqrt{α} r) + c_{3} I_{1} (\sqrt{β} r) \\ + c_{4} K_{1} (\sqrt{β} r) \end{matrix}

(11)

where $c_{1} - c_{4}$ are unknown constants, and $I_{1}$ and $K_{1}$ are the first-order modified Bessel functions of the first- and second-kind, respectively. Accordingly, the displacement field $u_{r}$ , the electric field $E_{r}$ and the electric potential $ϕ$ can be expressed as

\begin{matrix} u_{r} = c_{1} \frac{I_{1} (\sqrt{α} r)}{α} + c_{2} \frac{K_{1} (\sqrt{α} r)}{α} \\ + c_{3} \frac{I_{1} (\sqrt{β} r)}{β} + c_{4} \frac{K_{1} (\sqrt{β} r)}{β} + c_{5} r + \frac{c_{6}}{r} \end{matrix}

(12)

\begin{matrix} E_{r} = - \frac{c_{11}}{2 f'} [c_{1} \frac{(1 - l_{1}^{2} α) I_{1} (\sqrt{α} r)}{α} \\ + c_{2} \frac{(1 - l_{1}^{2} α) K_{1} (\sqrt{α} r)}{α} + c_{3} \frac{(1 - l_{1}^{2} β) I_{1} (\sqrt{β} r)}{β} \\ + c_{4} \frac{(1 - l_{1}^{2} β) K_{1} (\sqrt{β} r)}{β}] + c_{7} r + \frac{c_{8}}{r} \end{matrix}

(13)

and

\begin{matrix} ϕ = \frac{c_{11}}{2 f'} [c_{1} \frac{(1 - l_{1}^{2} α) I_{0} (\sqrt{α} r)}{α \sqrt{α}} - c_{2} \frac{(1 - l_{1}^{2} α) K_{0} (\sqrt{α} r)}{α \sqrt{α}} \\ + c_{3} \frac{(1 - l_{1}^{2} β) I_{0} (\sqrt{β} r)}{β \sqrt{β}} - c_{4} \frac{(1 - l_{1}^{2} β) K_{0} (\sqrt{β} r)}{β \sqrt{β}}] \\ - c_{7} \frac{r^{2}}{2} - c_{8} \ln (r) + c_{9} \end{matrix}

(14)

where $c_{5} - c_{9}$ are constants to be determined. It should be mentioned that the constants $c_{1} - c_{9}$ are not linearly independent and $c_{7}$ can be written in terms of $c_{1} - c_{4}$ if considering equation (9). The remaining constants will be determined from the boundary conditions.

Boundary conditions with the consideration of surface effect

Due to the inherent large surface-area-to-volume ratio exhibited by typical nano-structures, surface effects are believed to play a significant role in the electromechanical coupling properties of nano-scaled dielectrics. As stated in the linear surface elasticity model developed by Gurtin and Murdoch (1975), the surface is modeled as a thin layer with negligible thickness adhering to the bulk and the equilibrium of surface is governed by the generalized Young–Laplace equations. In addition, in the presence of the flexoelectric effect, the mechanical and electrical boundary conditions were not coincident with the conventional ones. According to Hu and Shen (2009), the traction and higher-order traction boundary conditions can be expressed as

\begin{matrix} T_{rr} (R_{i}) = - p_{i} + \frac{σ_{θ θ}^{s} (R_{i})}{R_{i}} \\ T_{rr} (R_{o}) = - p_{o} - \frac{σ_{θ θ}^{s} (R_{o})}{R_{o}} \\ τ_{rrr} = 0 at (r = R_{i}, R_{o}) \end{matrix}

(15)

in which $T_{rr} = σ_{rr} - σ_{rrr, r} + 2 σ_{θ r θ} / r - σ_{rrr} / r + σ_{θ θ r} / r$ (Yan and Jiang, 2015). It is noted that the same expression is obtained for an elastic material based on the strain gradient elasticity theory (Sadeghi et al., 2012). In addition, $σ_{θ θ}^{s}$ is the surface circumferential stress, which can be expressed as

σ_{θ θ}^{s} = σ_{0} + c_{12}^{s} ε_{rr} + c_{11}^{s} ε_{θ θ}

(16)

with $σ_{0}$ being the residual surface stress, and $c_{11}^{s}$ and $c_{12}^{s}$ being the surface elastic constants. The electrical boundary conditions depend on the electric circuit conditions, which are written as

\begin{matrix} \begin{matrix} ϕ = 0 at (r = R_{i}) \\ ϕ = V at (r = R_{o}) \end{matrix}; closed circuit condition \\ {\tilde{D}}_{r} = 0 at (r = R_{i}, R_{o}); open circuit condition \end{matrix}

(17)

with ${\tilde{D}}_{r} = D_{r} - Q_{rr, r} + Q_{θ θ} / r - Q_{rr} / r$ . For both electric circuit conditions, the electric quadrupole on the boundaries are

Q_{rr} = 0 at (r = R_{i}, R_{o})

(18)

After applying these boundary conditions, all the previous unknown constants can be determined. Thus, the mechanical and electrical fields of the dielectric nano-ring with both direct and converse flexoelectricity as well as the surface effect are solved.

It should be mentioned that if considering the direct flexoelectricity only, the terms associated with the electric field gradient and electric field gradient coupling and the strain and electric field gradient coupling in equation (1) are omitted, or the coefficients $b_{ijkl}$ and $e_{ijkl}$ can be regarded as zeros. Then the governing equation (10) can be reduced to

q^{df} - l_{0}^{2} (\frac{d^{2}}{d r^{2}} + \frac{d}{dr} - \frac{1}{r^{2}}) q^{df} = 0

(19)

with $l_{0} = \sqrt{l_{1}^{2} + l_{f}^{2}}$ , and the solution of the above equation is

q^{df} = c_{1}^{df} I_{1} (r / l_{0}) + c_{2}^{df} K_{1} (r / l_{0})

(20)

with $c_{1}^{df}$ and $c_{2}^{df}$ being the unknown constants. Correspondingly, the mechanical and electrical fields of the dielectric nano-ring with direct flexoelectricity can be determined. If further neglecting the strain gradient and the electric field coupling term in equation (1), the theory for dielectrics based on strain gradient elasticity is obtained. In the following section, comparisons among the results from these different theories will be given.

Results and discussion

In this section, simulation results are presented to demonstrate the electromechanical behavior of the dielectric nano-ring. Under the closed circuit condition, the electric potential, the electric field and the polarization along the radial direction of the dielectric ring from the current theory with flexoelectricity and surface effect are plotted in Figures 2(a)–(c), respectively. In these figures, $l_{1} = 0.2 R_{i}, l_{f} = 0.5 l_{1}, v = 0.3, χ = 1$ , $R_{o} = 2 R_{i}$ and $p_{o} = 2 p_{i}$ are chosen following Mao and Purohit (2014), in which v is Poisson’s ratio and $χ$ is related to a by $a = ε_{0} (χ + 1)$ with $ε_{0} = 8.85 \times 10^{- 12} {Fm}^{- 1}$ being the vacuum permittivity. In addition, $c_{11}^{s} = 10 N m^{- 1}$ , $σ_{0} = 1 N m^{- 1}$ and $l_{2} = l_{1} = l$ are taken when the surface effect and the flexoelectricity are considered. To show the results quantitatively, the Young’s modulus of the bulk material is taken as $E = 50 GPa$ , besides, $V = 0.1 V$ , $R_{i} = 20 nm$ , $p_{i} = 30 MPa$ , $b_{1} = c_{12} b' / c_{11}$ and $c_{12}^{s} = c_{12} c_{11}^{s} / c_{11}$ are assumed. The results calculated based on the pure elasticity, the strain gradient elasticity and the theory with direct flexoelectricity for a dielectric material are also provided for comparison. From these figures, it is seen that strain gradient elasticity provided the same solutions as those from the pure elasticity, indicating the non-local elastic effect has no influence on the electrical fields of the dielectric structure. It is also seen that the results from the theory with direct flexoelectricity deviate from the classical solution; however, the deviations are much smaller than those from the theories with flexoelectricity and with both flexoelectricity and surface effect. The results suggest that, in addition to the strain gradient induced direct flexoelectricity, the electric field gradient induced converse flexoelectricity and the surface effect may play a significant role in the electromechanical behavior of dielectric materials. As seen from the insets that show the corresponding curves from pure elasticity/strain gradient elasticity and the theory with direct flexoelectricity, the trends of the electric potentials and the electric fields along the radial direction of the ring are consistent well with those in Mao and Purohit (2014), which gave the normalized results. However, a concave instead of a convex curve as illustrated in Mao and Purohit (2014) for polarization with direct flexoelectricity is observed. The discrepancy may due to the different definitions of $l_{f}$ in the calculation since the coupling between the electric field and the strain gradient is considered for direct flexoelectricity in the current work while the coupling between the polarization field and the strain gradient was considered in the reference work. Additionally, Figures 3(a)–(c) plot the mechanical fields along the radial direction of the dielectric ring. Similar to those presented in Mao and Purohit (2014), it is observed that the magnitude of the displacement and the hoop strain are reduced due to the non-local elastic effect and the direct flexoelectric effect, and the variation of the radial strain becomes smooth due to the flexoelectric effect. However, it is found from these figures that the surface effect is more distinct and dominates the mechanical fields of the dielectric nano-ring within the selected values of the surface elasticity and the residual surface stress.

To see how the flexoelectricity and surface effect vary with the size of the investigated structure, Figures 4(a) and (b) plot the normalized polarization and radial strain at $r = 1.5 R_{i}$ versus the inner radius $R_{i}$ , in which $P_{r}^{0}$ and $ε_{rr}^{0}$ are obtained by using pure elasticity theory. The dependence of the surface effect on the residual surface stress $σ_{0}$ is also examined by assuming various values of $σ_{0}$ in the calculation. The other parameters for simulation are the same as those provided in Figure 2. As indicated by the solid lines, the flexoelectric effect on the polarization is appreciable while the influence of flexoelectricity on the radial strain is much smaller and can only be observed when $R_{i}$ is relatively small (e.g. $R_{i} = 20 nm$ ). In addition, it is seen that the normalized polarization is approaching to a constant as $R_{i}$ increases, and such a constant is not necessary to be 1 since the influence of flexoelectricity here is equivalent to an extra electrical load source through sophisticated electromechanical coupling (Yan and Jiang, 2015). It is also found that the surface effect is important in determining the electromechanical fields of the dielectric nano-ring, particularly as the value of $σ_{0}$ is relatively large ( $σ_{0} = 1.0 N / m$ ) and $R_{i}$ becomes relatively small ( $R_{i} = 20 nm$ ). With the increase of $R_{i}$ , surface effect reduces significantly with normalized polarization with flexoelectricity and surface effect approaching to that with flexoelectricity only. However, the radial strain is still less than $80 %$ of the classical result when $R_{i} = 200 nm$ and $σ_{0} = 1.0 N / m$ , indicating the significant influence of surface effect on the mechanical field of the dielectric nano-ring. From these two figures, it can be concluded that it is necessary to consider both flexoelectricity and surface effect in investigating the electromechanical coupling behavior of the dielectric nanostructures.

It is also important to investigate the dependence of the electromechanical fields on the material length scales. As examples, the distributions of polarization and radial strain with the consideration of flexoelectricity when $l / R_{i} = 0.2, 0.05$ and 0.02 and $l_{f} / l = 0.5$ are plotted in Figures 5(a) and (b), respectively. The solutions from pure elasticity are also given for comparison, and the material, geometrical parameters and the loading conditions are the same as those provided in Figure 2. It is shown clearly from these figures that the polarization and radial strain from the current theory highly depend on the length to inner radius ratios $l / R_{i}$ , that is, the larger the ratio, the more significant the flexoelectric effect. It is also worth special attention that there exist polarization and strain jumps between the current solutions and the classical ones at and near the edges of the structure even at very small $l / R_{i}$ . This phenomenon can be interpreted as a type of boundary behavior that is typical for a gradient theory (Yan and Jiang, 2015; Yang et al., 2004; Zhang et al., 2014). Next, the polarization and radial strain distributions for various $l_{f} / l$ are plotted when the ratio $l / R_{i}$ is fixed to be 0.2, as exhibited in Figures 6(a) and (b). It is observed that the flexoelectricity on the polarization and the radial strain diminishes with the decrease of $l_{f} / l$ as expected. When $l_{f} / l = 0.01$ , the polarization is almost the same as that predicted by pure elasticity/with strain gradient elasticity while the strain predicted by strain gradient elasticity is completely recovered from the current theory with the flexoelectric effect.

The influence of the electric circuit condition on the electromechanical responses of the dielectric nano-ring is presented in Figure 7. Under the open circuit condition, mechanical loads $p_{i} = 10 Mpa$ and $p_{o} = 2 p_{i}$ are applied while $V = 0.1 V$ is prescribed when the structure is connected to a source of voltage. Under the closed circuit condition (or electrical loading), electrical fields (e.g. polarization) will be induced in a dielectric structure whereas nonzero mechanical fields (e.g. radial strain) exist under open circuit condition (or mechanical loading) from the insight of pure elasticity. Evidently, conventional dielectric materials possess no electromechanical coupling or piezoelectricity. However, both electrical and mechanical fields having the same magnitude as that of the conventional fields are produced under either electric circuit condition due to the flexoelectricity, as seen from the current figure. It is therefore proved that dielectric materials can produce electric polarizations despite being non-piezoelectric at the nano-scale and flexoelectricity provides a viable route for developing actuation and sensing building blocks in NEMS/MEMS based on dielectric nanostructures.

Conclusions

In this paper, the electromechanical responses of a dielectric nano-ring have been investigated based on the extended linear theory for dielectrics considering both the direct and converse flexoelectric effects. Exact solutions of the mechanical and electrical fields are obtained, which are in the form of a linear combination of modified Bessel functions and functions of r. Applying boundary conditions that take into account the surface effect and the electric circuit conditions, the electromechanical fields of the dielectric nano-ring are solved. It should be mentioned that there are very few, if any, known analytical solutions of the boundary value problems in the theory of flexoelectricity, in which both the direct and converse flexoelectricity, the surface effect and the non-local elastic effect are included. Simulation results from the current theory, theories of pure elasticity, strain gradient elasticity and the one with direct flexoelectricity only are compared and it is found that the converse flexoelectric effect and surface effect, which are often neglected in the study of flexoelectricity, play a significant role in the size-dependent electromechanical behavior of dielectric materials and thus should be considered. In addition, the length scales of the materials and the electric circuit conditions have significant influence on the electromechanical fields. The solutions from this study could also provide some insights into the electroelastic fields near point defects in nano-scaled flexoelectric materials and the results could be helpful for understanding the electromechanical coupling of flexoelectric structures at the nano-scale, the knowledge of which is still limited in comparison to that of the conventional piezoelectric ones.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant number 11502084) and the Fundamental Research Funds for the Central Universities, HUST (Grant number 2015QN139).

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