## Abstract

Wave propagation in hollow dielectric elastomer cylinders is studied. The quasi-static deformation of the tube owing to a combination of radial electric field and mechanical loading is determined first. Two combinations are accounted for, one at which the tube is free to expand in the axial direction, and another at which the tube is axially pre-stretched and restricted from elongating. Subsequently, longitudinal axisymmetric incremental motions are superposed on the underlying state. The governing equations in the tube and in the surrounding space are formulated and a numerical procedure is used in order to solve the resulting set of equations. The fundamental mode in the frequency spectrum is determined for thin, intermediate and thick wall tubes. The influences of the tube geometry, the mechanical pre-stretch and particularly the electric bias field are examined. An important observation is the ability to manipulate the propagation of the waves by adjusting the electromechanical bias field. This infers the use of dielectric elastomers in tubular configurations as active waveguides or isolators by a proper tuning of the electrostatic stimuli.

## 1. Introduction

The class of smart materials entitled *dielectric elastomers* (DEs) draws growing attention in virtue of their low cost, light weight, ability to sustain large strains and change their electromechanical properties in response to electric stimuli [1–4]. DEs are essentially highly deformable capacitors consisting of a soft elastomer material coated with compliant electrodes. Applying an electric field to the electrodes generates attractive forces between them, thus squeezing the elastomer along the direction between the electrodes and elongating it in the transverse ones. Mini-robots, biomedical equipment and fine industrial devices are among the various mechanisms based on this idea ([5], and reference therein).

Only recently applications of DEs as waveguides and isolators have been addressed. Dorfmann & Ogden [6] showed how the velocity of surface waves is rendered when applying a bias electric field to an electroelastic half-space. To the best of the authors knowledge, Gei *et al*. [7] were the first to demonstrate how wave propagation can be manipulated when changing the geometry of a DE plate by applying an electric field in a periodic manner. Shmuel *et al*. [8] showed the strong influence of the electrostatic bias fields on the propagation of generalized Lamb waves in DE layers. The ability to electrostatically control the attenuation of waves using DE constituents in periodic structures was demonstrated for laminates by Shmuel & deBotton [9], and for fibre composites by Shmuel [10]. A tubular configuration was first considered by Pelrine *et al*. [11], when a cylindrical DE tube was subjected to an electric potential drop between two compliant electrodes coated on the internal and external lateral surfaces. The application of electrostatic field resulted in squeezing of the tube in the radial direction and elongating it in the axial direction. Motivated by these results, this work investigates electroelastic longitudinal waves in actuated and pre-stretched DE hollow cylinders. Special attention is given to the influence of the bias fields on the way these waves propagate. We note that in the purely elastic case longitudinal motions in elastic tubes were mathematically treated by Ghosh [12]. Mirsky & Herrmann [13] derived the exact frequency equation and evaluated the dispersion curve for the fundamental longitudinal mode.

The paper is constructed as follows. Relying on the theoretical framework provided in Baesu *et al*. [14], McMeeking & Landis [15], Dorfmann & Ogden [16], deBotton *et al*. [17] and Dorfmann & Ogden [6], §2 revisits the equations required to describe the static, dynamic and incremental behaviours of DEs. Using this framework, §3 analyses the static response of a hollow DE cylinder to an electrostatic excitation. Two loading configurations are analysed: an expansion-free loading path and a pre-stretched tube. The mathematical formulation for superposed longitudinal axisymmetric motions on the actuated tube is provided in §4. The resultant system of coupled differential equations cannot be solved analytically. Accordingly, extending a method used for bifurcation problems [18–20], a numerical procedure for tackling this problem is introduced to investigate the relation among the frequencies, velocities and the lengths of the waves. Section 5 is dedicated to a numerical study of the dependence of the fundamental mode on the geometry of the tube, the boundary conditions and pre-stretch, and most importantly on the bias electrostatic field. To this end, results are determined for representative values of wall thicknesses, electric bias fields and mechanical pre-stretches. Section 6 summarizes the highlights and main results of the work.

## 2. Dynamics and incremental equations of elastic dielectrics

Consider an electroelastic solid body occupying a region with boundary ∂*Ω*_{0} in its stress-free reference configuration. The surrounding vacuum occupies the region . Let *Ω* denote the deformed configuration of the body, with ∂*Ω* being its corresponding boundary. The mapping of its material points from their referential position **X**∈*Ω*_{0} to their current position **x**∈*Ω* is given by a continuous and twice differential vector field ** χ**, such that

**x**=

**(**

*χ***X**,

*t*). The velocity and acceleration of the material points are

**v**=

*χ*_{,t}and

**a**=

*χ*_{,tt}, respectively. The spatial derivative of

**with respect to**

*χ***X**defines the deformation gradient

**F**=

**∂**

**/**

*χ***∂**

**X**=∇

_{X}

**. Its determinant is a positive quantity owing to material impenetrability. The infinitesimal material line d**

*χ***X**, area

**N**d

*A*and volume d

*V*elements in the vicinity of

**X**are mapped to the current configuration via d

**x**=

**F**d

**X**,

**n**d

*a*=

*J*

**F**

^{−T}

**N**d

*A*and d

*v*=

*J*d

*V*, respectively, where

**N**and

**n**are unit vectors normal to d

*A*and d

*a*, respectively. As measures of the deformation the right and left Cauchy–Green strain tensors

**C**=

**F**

^{T}

**F**and

**b**=

**FF**

^{T}are used.

In the absence of mechanical body forces, the equations of motion read
2.1where ** σ** is the ’total’ stress tensor [16], incorporating both mechanical and electrical contributions. The balance of angular momentum implies that the total stress tensor is symmetric.

Let **e** and **d** denote the electric field and electric displacement field, respectively. These fields are related in free space via the vacuum permittivity *ϵ*_{0} such that **d**^{⋆}=*ϵ*_{0}**e**^{⋆}, where herein and henceforth a star superscript denotes quantities outside the body. The relation between **d** and **e** within the dielectric solid is given in terms of a constitutive law.

The governing equations of electrostatics are
2.2where we assume the absence of a free body charge in the ideal dielectric and consider a quasi-electrostatic approximation. The second of equation (2.2) motivates the derivation of **e** as the gradient of an electrostatic potential.

To complete the description of the electroelastic system, the following jump conditions across the boundary ∂*Ω* are recalled
2.3where *w*_{e} is the surface charge density, and
2.4is the Maxwell stress outside the body, **I** is the identity tensor, and **t**_{m} is a prescribed mechanical traction. The total traction on a deformed area element is the a sum of **t**_{m} and the electrical traction **t**_{e}=*σ*^{⋆}**n** induced by Maxwell stress.

A Lagrangian formulation of the problem that is derived by appropriate *pull-back* operations is often found to be advantageous. To this end, define the ’total’ first Piola–Kirchhoff stress, Lagrangian electric displacement and electric fields via
2.5respectively. These fields are to satisfy the Lagrangian forms of the governing equations (2.1) and (2.2), namely
2.6Consider an arbitrary **F**^{⋆} in the surrounding space satisfying **F**^{⋆}=**F**(**I**+**M**⊗**N**) at ∂*Ω*_{0}, where **M** is a tangent vector to the boundary [21]. With this definition **P**^{⋆},**D**^{⋆} and **E**^{⋆}, the Lagrangian counterparts of *σ*^{⋆},**d**^{⋆} and **e**^{⋆}, respectively, are derived by using equation (2.5) with **F**^{⋆}. These fields are used to formulate the referential equivalents to the jump conditions in equation (4.22), namely,
2.7where **t**_{M}d*A*=**t**_{m}d*a* and *w*_{E}d*A*=*w*_{e}d*a*.

Following Dorfmann & Ogden [16], the usage of an *augmented energy-density functionΨ*, with the independent variables **F** and **D**, is adopted to derive the total first Piola–Kirchhoff stress and the Lagrangian electric field via
2.8The first of equation (2.8) is rendered when considering incompressible solids to
2.9where *p*_{0} is a Lagrange multiplier induced by the kinematic constraint. The latter can be calculated only from the boundary conditions together with the equations of motion.

Following the formulation of Dorfmann & Ogden [6], we consider incremental time-dependent elastic and electric displacements and which are superimposed on the static deformed configuration *Ω*(** χ**). The superposed dot notation is used to denote incremental quantities. The

*push-forward*of the increments in the first Piola–Kirchhoff stress, the Lagrangian electric displacement and the electric fields are denoted, respectively,

**, and . These are defined via the transformations 2.10These fields are to satisfy the incremental governing equations 2.11The incremental kinematic constraint for incompressible solids reads 2.12where is the displacement gradient. Linearization of the incompressible material constitutive equations in the increments yields 2.13and 2.14where . The components of the electromechanical moduli and are 2.15where 2.16We refer to , and as the referential reciprocal dielectric tensor, electroelastic coupling tensor and elasticity tensor, respectively. , and are their Eulerian counterparts.**

*Σ*## 3. Dielectric elastomer tube subjected to radial electric loading

Different formulations for the static response of an elastic dielectric tube to an electromechanical loading has been addressed recently by Carpi & De Rossi [22] and Mockensturm & Goulbourne [23]. Herein, we make use of the solution of Singh & Pipkin [24] to account for different boundary conditions and specialize it to a specific constitutive behaviour.

Consider an incompressible infinitely long tube with initial inner and outer radii *R*_{A} and *R*_{B}, respectively, such that at the reference its thickness is *H*=*R*_{B}−*R*_{A}, as illustrated in figure 1. The tube is subjected to a radial electric displacement field by placing electrodes at its inner and outer circumferences which are being free of mechanical traction. Maintaining its tubular shape and satisfying incompressibility, the deformation is given by the mapping
3.1where (*R*,*Θ*,*Z*) and (*r*,*θ*,*z*) are the referential and current cylindrical coordinate systems, respectively. Here, the constant *A* is the reciprocal of the axial stretch, and the constant *B* is related to the radial stretch. Mapping (3.1) lends itself to the following representation of the deformation gradient in cylindrical coordinates
3.2where (•)′≡(•)_{,R} and . For convenience, we set the basis unit vectors to be , and adopt the notation
3.3

The electric displacement field in the current configuration is
3.4where *q*_{A} is the charge per unit length on the interior electrode at *r*(*R*_{A})≡*r*_{A}.

We assume that the constitutive behaviour of the tube is characterized by the *incompressible dielectric neo-Hookean* (DH) model
3.5where *I*_{1}=tr(**C**), *I*_{5e}=**D**⋅**C****D**, and *μ* and *ϵ*=*ϵ*_{0}*ϵ*_{r} are the shear modulus and the dielectric constant, respectively, with *ϵ*_{r} being the relative dielectric constant. Consequently, the total stress is
3.6and the current electric displacement field and the electric field are related via the isotropic linear relation
3.7

Equilibrium equations along *θ* and *z* reveal that *p*_{0} is a function of *r* alone. The equilibrium equation in the *r*-direction
3.8yields
3.9where *P* is an integration constant, and the normalized quantity is introduced. Boundary conditions are required for determining the constants *A*,*B* and *P*. The tube is free of mechanical traction on its inner and outer circumferences, and hence
3.10where *r*_{B}≡*r*(*R*_{B}). Henceforth, we distinguish between two loading paths.

*A traction-free tube:* the top and bottom bases of the tube are free to move in response to the applied electric excitation such that the resultant axial force vanishes. Thus, the integration of the axial component of the stress over the base of the hollow cylinder is
3.11Equation (3.11) together with equation (3.10) constitute a set of three equations for the unknowns *A*,*B* and *P*. With this expression we implicitly neglect the so-called fringing effect of the electric field outside the tube.

*A pre-stretched tube:* the tube is mechanically pre-stretched along the *z*-axis and subsequently the circumferential electrodes are charged. Along this loading path *A* is prescribed by the displacement boundary condition, and equation (3.10) is sufficient for determining *B* and *P*.

From a practical viewpoint the electric potential between the two electrodes is an important controllable parameter. Hence, a connection of the above formulation with this quantity is established via the relation 3.12that yields 3.13We also define the normalized quantity 3.14which, upon usage of equation (3.13), becomes 3.15

Note that beyond a critical value of , there is no solution to the static problem. This value depends on the thickness of the tube wall and the loading path. From a mathematical viewpoint, when exceeding a critical the resultant value of *σ*_{rr} cannot vanish at *r*_{A} and *r*_{B}, and thus equation (3.10) fails to hold. The physical interpretation is as follows. The radial stress *σ*_{rr} is composed of a compressive stress induced by the electric field and a stress resulting from the mechanical response of the solid. Their sum is required to satisfy equilibrium and pertinent traction-free conditions across the circumferential boundaries. When a critical magnitude of is applied, the compressive stress induced by the electric field is too large for the mechanical part to balance and the tube collapses.

A preliminary step before addressing the wave-propagation problem is to determine the linearized constitutive tensors. For the DH model, equation (2.15) leads to
3.16where *δ*_{ij} are the components of the Kronecker delta. With respect to the principal axes of the deformation described in equation (3.1), the non-zero components according to equation (3.16) are
3.17
3.18
3.19
3.20
and
3.21and we recall that *d*_{r},*λ*_{r}, and *λ*_{θ} are functions of *r*.

## 4. Longitudinal axisymmetric motions superimposed on a finitely deformed dielectric elastomer tube

Consider a harmonic excitation superimposed on the underlying configuration of the tube detailed in the previous section. Restricting attention to axisymmetric motions, the electromechanical fields considered are independent of the coordinate *θ*. Moreover, focusing on longitudinal waves, the incremental displacements have only two components, namely . Accordingly, the displacement gradient specializes to
4.1with the incompressibility constraint
4.2Derivation of from an incremental electric potential *φ*(*r*,*z*,*t*) such that , guarantees that the incremental Faraday’s law is *a priori* satisfied. The connection reveals that as anticipated. Consequently, the incremental Gauss equation reduces to
4.3

The associated non-zero incremental stress components are
4.4
4.5
4.6
4.7
and
4.8In cylindrical coordinates, the corresponding incremental equations of motion along *r* and *z*, respectively, are
4.9and
4.10where the equation along *θ* is identically satisfied.

The excitation of the tube induces electric fields outside the solid which must be accounted for too. These fields are to satisfy Laplace equation for the incremental electric potential, together with an appropriate decaying condition outside the tube at , and a finite value at *r*=0 at the centre of the internal hollow section of the tube. Accordingly, we have that
4.11where
4.12which yields
4.13and
4.14where *I*_{0} and *K*_{0} are the zero order modified Bessel functions of the first and second kind, respectively, and *n*_{1} and *n*_{2} are constants.

Inside the tube a solution to the governing equations in the form
4.15
4.16
4.17
and
4.18is sought. Substitution of equations (4.15) and (4.16) into equation (4.2) yields
4.19Equation (4.19) is used to eliminate *s* from the remaining equations. Equation (4.10) is used to express *q* in terms of *g*, *f* and their derivatives. Differentiation of the resulting expression for *q* with respect to *r* enables to eliminate *q*′ from equation (4.9). Equation (4.3) together with the revised form of equation (4.9) constitute two coupled ordinary linear differential equations of order 2 in *f* and order 4 in *g*.

The vector
4.20is used to formulate an equivalent set of six coupled linear differential equations of order one in the form
4.21with *A*_{ij}(*r*) being the coefficient of *y*_{j} in the *i*-th equation. Equations (4.21) are subjected to appropriate jump conditions at the internal and the external circumferences. In total, there are eight jump conditions as follows:
4.22

Equations (4.21) cannot be solved analytically, not even in the limit of mechanical bifurcation which corresponds to waves with vanishing velocity. To tackle this problem, we use the following numerical procedure. First, for any pair (*ω*,*k*), equations (4.21) are solved numerically for the initial-value problems
4.23for the six cases *m*=1,…,6. The pertinent solutions are denoted **y**^{(m)} and form a basis for a solution to equations (4.21) such that
4.24where *η*_{m} are six constants. The numerical solutions were obtained via the default solver of the commercial code Wolfram Mathematica 8. Solutions are pairs of (*ω*,*k*) for which the appropriate jump conditions at the boundaries of the tubes are satisfied. In terms of **y**, *φ*^{⋆}_{i} and *φ*^{⋆}_{o} given in equations (4.13), (4.14) and (4.24), this means determining a non-trivial set of eight constants *η*_{1},…,*η*_{6}, *n*_{1} and *n*_{2} that satisfy the jump conditions in equation (4.22). The corresponding mathematical requirement is a vanishing 8×8 determinant of the coefficients of the constants *η*_{1},…,*η*_{6}, *n*_{1} and *n*_{2}. The proposed numerical procedure is essentially an extension of a method used in associated bifurcation problems [18–20,25].

## 5. Numerical investigation of the fundamental mode

Our goal is to characterize the dispersion relation, and particularly to explore how it is influenced by the thickness of the tube wall, the pre-stretch and the bias electric field. To this end, we consider three DE tubes with different representative thicknesses whose electromechanical properties are
5.1The initial inner radii of the three tubes are *R*_{A}=1 (mm), with referential thicknesses *H*=0.1, 1 and 4 (mm). These values correspond to referential ratio of thickness to mean tube radius of , and , respectively. Henceforth, these will be referred to as the thin, the intermediate and the thick wall tubes, respectively.

The solution detailed in the previous section yields infinite number of curves in the frequency–wavenumber plane. Of special interest is the first fundamental mode which is most important from a practical viewpoint. Accordingly, we determine the associated curve under different electromechanical loading modes. For conciseness, we present the following results in terms of the non-dimensional quantities and where is the bulk shear wave velocity in the unstretched isotropic elastic dielectric.

Figure 2*a*, *b* and *c* displays the normalized velocities as functions of the normalized wavenumber for the thin, intermediate and thick wall tubes, respectively. Herein and henceforth the solid curves, the dashed curves and the dashed curves with diamond, triangle and square marks correspond to the bias normalized electric potentials and 1, respectively. Bearing in mind that not in all cases the tube can withstand all of these electrostatic bias fields, certain figures lack the curves associated with the higher values of .

The case corresponds to the purely mechanical problem. The pertinent curves agree with the solution of Mirsky & Herrmann [13]. In particular, in the limit of long waves as the velocity is the same as for a solid rod, whereas in the limit of short waves, the velocity attains Rayleigh surface wave velocity .

The influence of the bias field is evident: as increases a monotonic rise of the velocity is exhibited in both limits of short and long waves. Analogous observation was reported by Shmuel *et al*. [8] in the context of wave propagation in actuated DE layers. The behaviours of the curves in-between the two limits depend mainly on the current thickness of the tube wall *h*=*r*_{B}−*r*_{A}. If the thickness is small enough, there is a segment at which the wavelengths are small compared with the mean radius of the tube , but still large in comparison with *h*. Freely speaking, the waves are too small to be affected by the curvature of the tube, yet large enough to be affected by its boundaries. Thus, within this segment the behaviour is expected to be reminiscent of the one observed in connection with a flat DE plate. The corresponding trend is a rapid decrease of the velocity from the limit to the fundamental antisymmetric curve of a plate near the origin at which the velocity is low [8]. Subsequently, the velocity increases towards the modified surface velocity of an actuated DE half-space. Recalling that actuation of the tube results in enlargement of and thinning of *h*, this trend becomes more evident with the enhancement of . For instance, the ratio of a thin wall tube changes from 0.095 at the reference at , to 0.035 at . Accordingly, the velocity drop is at approximately in this actuated case, versus in the purely mechanical case. At the opposite end, consider the referential configuration of the thick wall tube with , implying that the thickness is larger than the mean radius. When actuated at , the ratio changes to . Because *h* is not small in comparison with , there is no range in which is small compared with , but yet large in comparison with *h*. Thus, in this case, we observe a monotonous decrease of the velocity from the long waves to the short waves limit.

Figure 2*d*, *e* and *f* displays the normalized frequencies as functions of the normalized wavenumber for the thin, intermediate and thick wall tubes, respectively. The curves admit monotonous behaviours as functions of , even along the moderate slope in the interval in figure 2*d*. This implies that higher frequencies give rise to shorter waves of the fundamental mode. This observation can be summarized as follows. Each frequency excites waves with a single wavelength propagating in the fundamental mode, and enhancement of the excitation frequency results in shorter waves. This trend is independent of the actuation voltage. Nonetheless, we note that the magnitude of the fundamental frequencies do depend on the applied electric field.

Figures 3, 4 and 5 correspond to the cases of axially constraint tubes with pre-stretches and 2, respectively. As before, panels (*a*), (*b*) and (*c*) display the normalized velocities as functions of the normalized wavenumber for the thin, intermediate, and thick wall tubes, respectively, and panels (*d*)–(*f*) the corresponding normalized frequencies. Before we proceed, we note that for the thick wall tube pre-stretched at *λ*_{z}=2, the numerical procedure followed failed to converge to a physically sound solution. This is probably due to the difficulty associated with the numerical search for a vanishing determinant (e.g. [26,27]). Specifically, the velocity determined in the short wavelength limit was incorrect at . Instead, curves for are presented in figure 5.

In a manner reminiscent of the one observed for the traction-free tube, in the limit of long waves the velocity exhibits a monotonous rise as a function of the electric potential. However, a reversed trend is observed in the limit of short waves, as the velocity decays monotonically with . Moreover, there is a threshold value at which the velocity in the limit of surface waves vanishes, such that
5.2This situation is recognized as a loss of stability in the limiting case of a dielectric half-space that is associated with compressive loadings. Thus, the fixed boundaries in the axial direction prevent the electrostatic forces from elongating the tube, and hence result in compressive axial stresses along the tube. Accordingly, the stabilizing effect of the pre-stretch is evident as higher values of are required for a stability loss at *λ*_{z}=2, whereas lower values are sufficient at . Moreover, the pre-compressed thin wall tube is already in an unstable state even without electrostatic actuation (e.g. the solid curve in figure 3*a*).

Beyond the velocity vanishes at finite wavenumbers. We adopt the notation for the cut-off wavenumber at which the velocity vanishes at a certain . The corresponding cut-off wavelength is the minimal wavelength at which waves can propagate within the actuated tube. The value of decreases monotonically with higher values of . This observation hints at a possible use of the electrostatic actuation as a mechanism to annihilate certain wavelengths. The thickness of the wall has a stabilizing effect, and the thicker the wall the higher the electric potential required to attain a vanishing wave velocity.

Addressed next are the curves describing the normalized frequencies as functions of the normalized wavenumber. Contrary to the monotonicity of the curves observed for the traction-free tube, here the slope of certain curves turns negative at some wavelength. At the branches for which there exist (e.g. the curves in figure 3*d*), the slope remains negative until the curve attains a zero ordinate at the cut-off wavenumber. At the branches with no prohibited range of wave propagation, the slope changes its trend and turns positive once again, e.g. the curves associated with and in figure 3*e*. This hints at the possibility that several waves, that correspond to the fundamental branch, with different lengths and velocities may propagate simultaneously in response to a single excitation frequency. Taking the frequency as the independent variable, shorter and slower waves emerge when the excitation frequency is increased.

Analysis of the magnitude of the dynamic displacements in the radial direction is carried out next. During the above study it was observed how the electric potential and the geometry have influence the realization of the displacements. The thin wall tube exhibits a richer behaviour in virtue of the existence of an intermediate range of wavelengths at which , as discussed earlier. Accordingly, a representative case of a traction-free thin wall tube subjected to is examined in details. Figure 6 displays the corresponding radial displacements as functions of the radial coordinate normalized by the value of the displacement at *r*_{B}. Specifically, figure 6*a*,*b* and *c* corresponds to wavenumbers and 8, respectively. In the case, which corresponds to the limit of long waves, the displacements shown in figure 6*a* demonstrate a linear dependency on the radial coordinate. This is expected on grounds of the similarity of the behaviour in this limit with that of a rod. The second wavenumber is in the interval of lengths that tends to follow the antisymmetric plate mode, i.e. the same magnitude and direction of the displacement with respect to the mid-plane of the plate. Accordingly, the displacements in figure 6*b* exhibit an antisymmetric-like distribution with respect to the analogue to the mid-plane, that is, the cylindrical surface with a mean radius . The wavenumber corresponds to the limit of short waves, hence a realization of two distinct regions of waves with larger amplitudes near the wall boundaries as shown in figure 6*f*. This is reminiscent of the propagation of Rayleigh surface waves along the inner and outer circumferences.

For completeness, we recall that during the numerical evaluations higher modes were found too, but their investigation is beyond the scope of this work. We mention that under the condition of some of these modes are recognized as symmetric plate modes, as inferred by [28] in the study of the purely elastic case.

## 6. Concluding remarks

The propagation of waves in finitely deformed DE tubes is addressed in response to the growing interest and potential functionality of these smart materials. First, the static deformation of a DE tube whose electromechanical behaviour is characterized by the DH model is determined when subjected to a radial electric field. Two different loading configurations are accounted for: in the first, the tube is free to expand in the axial direction, and in the second, the tube is axially held at a pre-stretched state.

Subsequently, longitudinal axisymmetric incremental motions are superposed on the underlying configuration. The corresponding system of coupled differential equations that governs the motion of the axisymmetric waves within the tube is introduced, together with the appropriate relations for the electrostatic waves outside the solid, and the corresponding jump conditions at the inner and outer circumferences. The resulting set of equations cannot be solved analytically. To treat this mathematical complexity an extension of a numerical procedure commonly used in bifurcation problems is introduced and used.

The dispersion relations characterizing the first fundamental mode in the frequency spectrum were examined for typical electromechanical properties of DEs. To explore the influence of the geometry of the hollow cylinder on the dispersion curves, three different ratios of thickness to mean radius were examined corresponding to thin, intermediate and thick wall tubes. The dependency of the curves on the bias electric field and mechanical pre-stretch was investigated by considering various intensities of the applied electric potential and different axial stretches.

It was demonstrated how in the limit of long waves the velocity increases monotonically with the applied electric field in both types of the loading states. At the short waves limit enhancement of the electric potential led to a higher velocity in the free tube, whereas in the axially constraint ones a reversed trend was observed. The behaviour of the curves in-between the two limits is dominated by the relative thickness of the tube wall. A certain threshold value of electric bias field at which the velocity vanishes in the surface waves limit was detected. This situation is associated with loss of stability of a dielectric half-space. Higher magnitudes of electric field were required to attain the onset of instability when the tube was pre-stretched. Changes in the thickness of the tube wall resulted in a similar effect on the required electric excitation. When the magnitude of the electric potential exceeded the threshold, the velocity vanished at finite wavelengths, suggesting that waves with a shorter length cannot propagate in the actuated tube.

We conclude by emphasizing that the ability to manipulate wave propagation in DE tubes by adjusting the electromechanical bias fields was demonstrated. This result, together with earlier works by Gei *et al*. [7], Shmuel *et al*. [8], Shmuel & deBotton [9] and Shmuel [10] for different DE configurations, infer the use of DEs as active waveguides and isolators by means of a proper electrostatic actuation.

## Acknowledgements

This work was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (grant 1246/11).

- Received February 3, 2013.
- Accepted March 28, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.