Electrostatically tunable axisymmetric vibrations of soft electro-active tubes

Due to their unique electromechanical coupling properties, soft electro-active (SEA) resonators are actively tunable, extremely suitable, and practically important for designing the next-generation acoustic and vibration treatment devices. In this paper, we investigate the electrostatically tunable axisymmetric vibrations of SEA tubes with different geometric sizes. We consider both axisymmetric torsional and longitudinal vibrations for an incompressible SEA cylindrical tube under inhomogeneous biasing fields induced by radial electric voltage and axial pre-stretch. We then employ the state-space method, which combines the state-space formalism in cylindrical coordinates with the approximate laminate technique, to derive the frequency equations for two separate classes of axisymmetric vibration of the tube subjected to appropriate boundary conditions. We perform numerical calculations to validate the convergence and accuracy of the state-space method and to illuminate that the axisymmetric vibration characteristics of SEA tubes may be tuned significantly by adjusting the electromechanical biasing fields as well as altering the tube geometry. The reported results provide a solid guidance for the proper design of tunable resonant devices composed of SEA tubes


Introduction
Compared with traditional piezoelectric materials, soft electroactive (SEA) materials, besides exhibiting the exotic capability of high-speed electrical actuation with strains greater than 100% [1], also possess many other excellent electromechanical properties such as low actuation voltage, high fracture toughness and high energy density [2,3]. These characteristics therefore have received considerable academic and industrial interest, and found widespread applications ranging from actuators, sensors and energy harvesters to biomedical and flexible electronic devices [4][5][6][7][8]. It is generally accepted that electric stimuli can affect the electromechanical characteristics of SEA materials in a rapid and reversible way, which in turn provides an effective approach to tune the vibration and wave characteristics of SEA structures and devices. Consequently, SEA materials can be ideally applied to the manufacturing of high-performance vibration and wave devices such as tunable resonators and acoustic/elastic waveguides [9][10][11][12][13].
Strong nonlinearity and electromechanical coupling of SEA materials are two important aspects in developing a general continuum mechanics framework. Early development of the nonlinear theory of electroelasticity can be tracked back to the seminal works of Toupin [14,15] for static and dynamic analyses of finitely deformed elastic dielectrics. Later on, Tiersten [16] extended Toupin's formulations to further incorporate thermal effects and developed a thermo-electro-elastic coupled theory by applying the laws of continuum physics to a well-defined macroscopic model. Due to the development of various smart materials and structures as well as their extensive application prospects, a general nonlinear continuum theory for electro-magneto-mechanical couplings has been regularly reformulated since 1980s [17,18]. During this period, particular efforts have been made on finite element simulations of multifield coupling problems and micromechanics analysis of smart composites. In recent decades, the appearance of SEA materials capable of large deformations on the market [19][20][21][22] has again promoted a re-interpretation, improvement and applications of nonlinear electroelasticity [23][24][25][26][27]. It is noted that a nonlinear continuum framework accounting for the nonlinear interaction between mechanical and electromagnetic fields, which is well documented in the monograph by Dorfmann and Ogden [27], is 3 appropriate for the analysis of electroactive and magnetoactive materials undergoing large deformations [28][29][30][31].
In many practical applications, the performance of intelligent systems composed of SEA materials usually depends on the biasing fields induced by, for instance, pre-stretch, internal pressure and electric stimuli. On the one hand, the biasing fields may result in instability and even failure of the systems [30,[32][33][34]. On the other hand, they can be exploited to actively control waves and vibrations in SEA devices. For example, experiment on the lightweight push-pull acoustic transducer consisting of dielectric elastomer (DE) films for sound generation in advanced audio systems [35] showed that the push-pull driving can suppress harmonic distortion. Hosoya et al. [36] fabricated and investigated a hemispherical breathing mode loudspeaker using a DE actuator, while Lu et al. [37] experimentally demonstrated an electrostatically tunable duct silencer using external control signals. Earlier, Dubois [38] used an electric biasing field to tune the resonance frequency of dielectric electroactive polymer (DEAP) membranes which requires no external actuators or variable elements, and observed a 77% resonant frequency reduction from the initial value. Moreover, Zhang et al. [39] put forward a vibration damper and achieved vibration attenuation by applying alternating oppositely phased voltages to a DE actuator. Consequently, tunable SEA resonators are extremely suitable for the next-generation acoustic treatment devices.
To investigate how the biasing fields influence the small-amplitude dynamic characteristics of SEA structures, different versions of linearized incremental theories [17,18,[40][41][42][43] based on the nonlinear electroelasticity theory have been established in the literature by adopting either the Lagrangian description or the updated Lagrangian description as well as in terms of different energy density functions. By introducing three configurations to describe the general motion of an electroelastic body, Wu et al. [44] compared in detail different versions of nonlinear electroelasticity theory and associated linearized incremental theory, identified the similarities and differences between them, and concluded that these seemingly various theories are in principle equivalent without any essential difference.
Following the theory of nonlinear electroelasticity and its associated linearized incremental theory developed by Dorfmann and Ogden [40], much effort has been devoted in 4 recent years to investigating the effects of biasing fields on small-amplitude wave propagation characteristics in SEA materials, such as bulk waves and different types of guided waves [12,13,28,[45][46][47][48]. More recently, the state-space method (SSM) was employed by Mao et al. [10] and Wang et al. [49] to explore the electrostatically tunable free vibration behaviors of SEA balloons and of multilayered electroactive plates, respectively.
Numerical results in both papers proved that the SSM is a highly effective method for the analysis of SEA structures with inhomogeneous biasing field or multilayered configuration.
The purpose of the present study is to shed light on the effects of inhomogeneous biasing field and tube geometrical size on axisymmetric free vibrations of incompressible SEA cylindrical tubes. Both axisymmetric torsional and longitudinal vibrations (hereafter abbreviated as T vibrations and L vibrations) are considered. The biasing field is generated by applying an electric voltage difference between the two electrodes on inner and outer tube surfaces respectively, in addition to the pre-stretch in the axial direction (see Figure 1). The SSM proposed by Wu et al. [46] for the analysis of circumferential guided waves in SEA tubes is utilized here to tackle the problem of inhomogeneous biasing field. Deformed configuration after activation generated by a combined action of radial electric voltage and axial pre-stretch z  and related geometrical sizes.
This paper is organized as follows. Using nonlinear electroelasticity theory [23,27], 5 Section 2 briefly reviews the basic formulations governing the nonlinear axisymmetric deformation and inhomogeneous biasing fields in SEA tubes characterized by a neo-Hookean ideal dielectric model. Based on the linearized incremental theory [40], Section 3 provides the governing equations and the state-space formalism in cylindrical coordinates for the incremental fields. For the generalized rigidly supported conditions, Section 4 derives the frequency equations for the two types of axisymmetric vibrations of SEA tubes with the help of the approximate laminate technique. We conduct numerical calculations in Section 5 to first, validate the convergence and accuracy of the proposed SSM for axisymmetric vibrations, and then elucidate the effects of electromechanical biasing field and tube geometry on the axisymmetric vibration characteristics. A conclusive summary is provided in Section 6 and some related mathematical expressions or derivations are presented in Appendices A-D.

Nonlinear axisymmetric deformation of an SEA tube
For better understanding the derivations of governing equations for the nonlinear axisymmetric deformation and the superimposed small-amplitude vibration in an SEA tube, the general nonlinear electroelasticity theory and its associated linearized incremental theory are briefly reviewed in Appendix A. The detailed formulations can be found in the work by Dorfmann and Ogden [23,27,40].
The nonlinear axisymmetric deformation of an SEA tube subjected to radial electric field, internal/external pressures and axial pre-stretch has already been provided elsewhere [45-46, 48, 50-51]. In this section, we just briefly review the basic equations and expressions when the SEA tube coated with electrodes on both the inner and outer surfaces is subjected to a radial voltage as well as an axial pre-stretch.
As displayed in Figure 1, the inner and outer radii as well as the length of the tube are specified as A , B and L , respectively, in the undeformed configuration with the thickness where / mm I     , with  being the total energy density function.
Since the deformation is axisymmetric and also invariant along the axis, all the physical quantities are independent of the coordinates  and z . As a result, Faraday's law (A.1) 3 is satisfied automatically, and Gauss's law (A.1) 2 where () Qa and () Qb are the total free surface charges on the inner and outer surfaces of the deformed SEA tube and satisfy ( ) is the electric potential difference between the inner and outer surfaces. Moreover, by inserting Eq. (9) 1 into Eq. (6) 2 , conducting the integration from a to b, and assuming that both the inner and outer surfaces are traction-free, we find that In a similar way, the radial normal stress can be found as * d ( ) .
After rr  is obtained analytically or numerically from Eq. (12) for a specific energy density function, the circumferential (   ) and axial ( zz  ) normal stresses can be derived by 8 Eq. 3 / 2 / 2 , where  denotes the shear modulus of the SEA material in the absence of biasing fields and  is the dielectric constant of the ideal dielectric material, independent of the deformation.
For the neo-Hookean ideal dielectric model, the explicit expressions of the physical variables related to the nonlinear axisymmetric deformation have been provided by Zhu et al.

Incremental equations and state-space formalism
To describe the time-dependent incremental motion accompanied by an incremental electric field in the finitely deformed SEA tube, the incremental governing equations given in Appendix A.2 are written in the cylindrical coordinates   ,, rz  in this section. Then we reproduce the state-space formalism for the incremental fields which has been presented in Wu et al. [46].
It can be seen from Eq. (A.6) 2 that the incremental electric field 0 is curl-free and thus an incremental electric potential  can be introduced such that 0 grad  . Its components in the cylindrical coordinates are Accordingly, the incremental Gauss's law (A.6) 3 and the incremental equations of motion (A.6) 1 can be written, respectively, as In addition, the incremental displacement gradient tensor H can be written as The incremental incompressibility condition (A.10) in the cylindrical coordinates thus can be expressed as (20) According to Eqs. (16) and (19) 331 12 33   03333  0331 13  44  02323   47  02332  55  01313  0133 35  58  01331  0133 35   66  01212  0122 26  77  03232  88  03131  0133 35   69  01221  0122 26  99  02121  0122 26 , , , in which the non-zero components of the instantaneous electroelastic moduli tensors 0 , 11 0  and 0 for the axisymmetric deformation of SEA tubes subjected to a radial electric displacement field have been derived by Wu et al. [46]. Their explicit expressions can be found in Appendix B of Ref. [46]. Note that adjusting the electromechanical biasing fields may alter the effective material properties of SEA tubes, which will generate large effects on the superimposed dynamic behavior.
It is obvious that the biasing fields are radially inhomogeneous when subjected to a radial electric voltage, which makes the effective material parameters depend on the radial coordinate r . Consequently, the resulting incremental governing equations are a system of coupled partial differential equations with variable coefficients, which are difficult to solve analytically or even numerically via the conventional displacement-based method. Therefore, the state-space method (SSM) [46,52,53] combining the state-space formalism with the approximate laminate technique is adopted in this paper to derive the frequency equations of the axisymmetric vibrations of SEA tubes.
The basic incremental governing equations (17)- (18) and (20) (25) and M is an 88  system matrix, with its four 44  sub-matrices presented in Appendix B.

Axisymmetric vibrations of an
It is apparent from equations (26) and (27) Figure 6). Note that the cylindrically breathing mode characterized by the sole radial displacement r u is a special mode of the L vibrations (see Figure 5(a)), which needs to be dealt with separately.
Assume the deformed SEA tube (see Fig. 1(c)) is subject to the generalized rigidly supported (GRS) conditions [53] at the two ends. Moreover, we suppose that the electric inductions in the surrounding vacuum near the tube ends are negligible so that the zero incremental electric displacement condition applies at the tube ends. Thus, the incremental mechanical and electric boundary conditions are For the harmonic axisymmetric free vibrations of the SEA tube, we assume that  (29) where i1  is the imaginary unit,  is the circular frequency of vibration, are the dimensionless radial and axial coordinates in the deformed configuration,  and  are material constants described in Section 2, and n is the axial mode number. Note that the circumferential mode number is equal to zero for the axisymmetric vibrations. According to Eqs. (21) 5,7 -(22) 3 and (29) where   and is the dimensionless circular frequency. It is evident from Eqs. (30) and (31) where   are the global transfer matrices of sixth-order ( 1 k  ) and second-order ( 2 k  ), through which the state variables at the inner and outer surfaces are connected.

Frequency equations
Assuming that the inner and outer surfaces of the SEA tube are traction-free and that the where / rr DD   and / pp   are defined in Section 2.

Numerical results and discussions
Our goal is to study the axisymmetric vibration characteristics of SEA tubes, and in particular investigate how its resonant frequencies are affected by the electromechanical biasing fields (i.e., the axial pre-stretch z  and the dimensionless radial electric voltage V ) and by the tube geometry (i.e., the inner-to-outer radius ratio  and the length-to-thickness ratio / LH ).  z = 0.75 Figure 2: The circumferential stretch a  at the outer surface versus the electric voltage V for SEA tubes with different axial pre-stretches z  and outer-to-inner radius ratios 1   .
Based on the nonlinear governing equation (14) for the neo-Hookean ideal dielectric model, we obtain the axisymmetric response curves in Figure 2, which displays the variations of the circumferential stretch a  at the outer surface with the dimensionless voltage V for different axial pre-stretches z  and inner-to-outer radius ratios  . Note that the inverse of  (i.e., the outer-to-inner radius ratio) is chosen to be 1 1.1    , 2 and 5 , corresponding to thin-, medium-and thick-walled tubes, respectively. The curves of a  versus V for different combinations of z  and 1   reveal a monotonically increasing variation trend, which means physically that the tube expands in the radial direction. We find that no axisymmetric solution exists and that the SEA tube collapses when the voltage exceeds a critical value, which we call the electromechanical instability voltage EMI V . There, the balance between the compressive force caused by the radial electric voltage and the mechanical resistance force cannot be maintained [45,46]

Validation of the state-space method
As stated in Sections 3 and 4, the state-space method (SSM) combining the state-space formalism with the approximate laminate technique is an analytical but approximate method.
It is necessary to validate its convergence and accuracy for the axisymmetric vibrations of SEA tubes.
For the convergence analysis, Tables 2-5 Tables 2 and 4, while Tables 3 and 5 correspond to the results for the thin and slender tubes. Obviously, the results based on the SSM show an excellent convergence rate with increasing layer number, and thus we are satisfied that we can obtain accurate resonant frequencies with an arbitrary precision via the present SSM.     When there is no electric voltage applied to the tube, the deformation of the pre-stretched SEA tube is homogeneous, i.e., the biasing fields become homogeneous without the application of voltage. Therefore, the exact resonant frequencies for the superimposed pre-stretch has no effect on the resonant frequencies of the T vibration for the neo-Hookean hyperelastic model; they are only determined by the outer-to-inner radius ratio and the length-to-thickness ratio. In summary, the superior convergence rate and the excellent agreement with the exact solutions demonstrate that the obtained numerical results based on the SSM are highly accurate.

Effect of the electromechanical biasing fields
In this subsection, we focus on how the electromechanical biasing fields influence the resonant frequencies of the two classes of axisymmetric vibration of SEA tubes. Without loss of generality, the tube geometry is taken as for an arbitrary energy function, which is independent of the applied voltage as verified in Appendix D. Furthermore, the first-order vibration frequency grows monotonously and linearly with increasing axial mode number. However, the second-order vibration frequency declines with the applied voltage and grows nonlinearly with the increasing axial mode number, as shown in Figure 4(b).
The first-order L vibration modes corresponding to different axial mode numbers 0 n  , 1, 2 and 6 are presented in Figure 5. The breathing mode in Figure 5(a) has the sole radial displacement component r u that is independent of z and  . According to Eqs. (C.17) and (C. 19), the radial displacement for the breathing mode is an inversely proportional function 22 of the radius. Therefore, the displacement at the inner surface is larger than that at the outer surface and the circumferential gridlines become sparse from the inner surface to the outer surface. Figures 5(b-d) show the L vibration modes for non-zero axial mode numbers ( 0 n  ) with both the radial and axial displacement components r u and z u . The SEA tube for these modes vibrates in the form of trigonometric function in the axial direction, which conforms well with the formal solutions (29) 1 . Obviously, the axial mode number 0 n  is equal to the integer multiple of the half-wave number.  and (D.9)), and the vibration is a rotation of each cross-section as a whole about its center, which is analogous to the torsional waves in an isotropic elastic cylinder [54]. Thus, the gridlines still distribute uniformly in the cross-section. But the torsional displacement varies according to the trigonometric function in the axial direction (see the right view of Figure   6(a)). For the second-order vibration mode in Figure 6  increasing voltage [10]. In particular, when the voltage approaches the critical value cr V , the global stiffness reduces rapidly so that barrelling instabilities [52,55] occur in the SEA tube. This is why the frequencies of modes 1 n  and 2 n  in Figures 7(b-c) decrease gently at first and then dramatically. Moreover, we find that the larger axial pre-stretch the tube is subjected to, the lower critical voltage the tube may withstand.
For the breathing mode 0 n  in Figure 7 Table 1 for the axisymmetric deformation.  Figure 7(d), we see that the critical voltage decreases to a minimum at 1 n  and then increases monotonically when the applied axial pre-stretch is over approximately 1.6. Therefore, for a higher axial pre-stretch, the SEA tube undergoes the barrelling instability first at 1 n  . If the axial pre-stretch is less than 1.6, then the SEA tube will have the barrelling instability at a higher axial mode number. 25 For the T vibration with 1 n  , Figure 8 displays the variations of the first two resonant frequencies with the applied voltage for different axial pre-stretches. Compared with the curves of L vibrations in Figure 7, the frequency variation trend for the T vibration in Figure   8 is quite unique. Specifically, the first-order resonant frequency (the lower curves shown in In short, the dependence of the resonant frequency of the axisymmetric vibrations on the electromechanical biasing fields provides a possibility to tune the small-amplitude free vibrations of SEA tubes.

Effect of the tube geometry
Now we turn to explore how the tube geometry including the length-to-thickness ratio / LH and the outer-to-inner radius ratio 1   influences the axisymmetric vibrations of the 26 SEA tube. In this subsection, the dimensionless electric voltage is always taken as 0.2 V  and the tube is divided into 120 thin sublayers to guarantee convergence and accuracy. Apparently, the length-to-thickness ratio makes no difference to the vibration frequency; this is because / LH disappears from the frequency equation for the breathing mode 0 n  according to Eq. (32) 1 . Nonetheless, the outer-to-inner radius ratio 1   and the axial pre-stretch z  still have an influence on the vibration frequency for a non-zero voltage. Specifically, for a fixed axial pre-stretch, the thicker the SEA tube is, the larger resonant frequency we obtain. Additionally, increasing the axial pre-stretch lowers the resonant 27 frequency. However, the variation gap between the axial pre-stretch and the vibration frequency depends on 1   . The vibration frequency is barely affected by the axial pre-stretch for a thin tube ( 1 1.1    ), but it is remarkably decreased by the axial pre-stretch for a thick tube ( 1 5    ).
For the L vibration mode 1 n  , Figures 9(b)  . Generally, the vibration frequency gradually decreases with increasing / LH and the vibration frequency of the thick tube is higher than that of the thin tube. In addition, we see from reduces to zero when / LH approaches a critical value 2.0391, which corresponds to the barrelling instability with the axial mode number 1 n  . Thus, the axisymmetric instability occurs more easily for a slender tube subjected to the axial compression. When it comes to the T vibration with 1 n  , Figure 10  inversely proportional function of the length-to-thickness ratio / LH. Therefore, the frequency is independent of the axial pre-stretch and the outer-to-inner radius ratio, as shown in Figure 10(a). According to Appendix D, the torsional displacement is distributed linearly along the radial direction. In addition, the resonant frequency tends to zero for an infinite SEA tube, which means physically that the longer tube achieves torsional instability more easily. For the second-order frequency depicted in Figure 10(b), a thicker SEA tube ( 1 5    ) results in a higher resonant frequency, especially when /1 LH  . Besides, the frequency is hardly affected by the axial pre-stretch for a low voltage ( 0.2 V  ), but for a higher voltage, increasing the axial pre-stretch will lead to a lower resonant frequency, which is analogous to the phenomena observed in Figure 8.

Conclusions
In this work, we conducted an analytical study of the small-amplitude axisymmetric 1) The proposed SSM is a highly accurate and efficient method for studying the axisymmetric vibrations of SEA tubes under inhomogeneous biasing fields. 29 2) The manipulation of axisymmetric vibration behaviors of the neo-Hookean SEA tubes is feasible by tuning the electromechanical biasing fields except for the lowest torsional mode with linear displacement distribution along the radial direction.
3) By varying the tube geometrical size, the resonant frequencies of different modes for the neo-Hookean SEA tubes could be readily adjusted, except for the breathing mode and the lowest torsional mode that are independent of the length-to-thickness ratio and the outer-to-inner radius ratio, respectively.
This work provides not only a novel method (SSM) to derive the frequency equations of three-dimensional free vibrations of SEA cylindrical structures, but also demonstrates the electrostatic tunability of resonant frequency of SEA tubes with various geometrical sizes.
The present investigation clearly indicates that it is feasible to use biasing fields to tune the  where grad  Hu is the incremental displacement gradient tensor, p is the incremental Lagrange multiplier, and 0 , 0  and 0 are, respectively, the fourth-, third-and second-order tensors, which are referred to as instantaneous electroelastic moduli tensors. In 32 component form, 0 , 0  and 0 are given by Additionally, the incremental incompressibility condition can be written as div tr 0.