Anomalous bulging behaviors of a dielectric elastomer balloon under internal pressure and electric actuation

Accepted Manuscript

Anomalous bulging behaviors of a dielectric elastomer balloon under internal pressure and electric actuation Fangfang Wang , Chao Yuan , Tongqing Lu , T.J. Wang PII: DOI: Reference:

S0022-5096(17)30031-5 10.1016/j.jmps.2017.01.021 MPS 3060

To appear in:

Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

8 January 2017 31 January 2017 31 January 2017

Please cite this article as: Fangfang Wang , Chao Yuan , Tongqing Lu , T.J. Wang , Anomalous bulging behaviors of a dielectric elastomer balloon under internal pressure and electric actuation, Journal of the Mechanics and Physics of Solids (2017), doi: 10.1016/j.jmps.2017.01.021

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ACCEPTED MANUSCRIPT

Anomalous bulging behaviors of a dielectric elastomer balloon under internal pressure and electric actuation Fangfang Wang, Chao Yuan, Tongqing Lu, T. J. Wang State Key Laboratory for Strength and Vibration of Mechanical Structures, Department

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Xi’an 710049, China

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of Engineering Mechanics, School of Aerospace Engineering, Xi’an Jiaotong University,

Keywords: Dielectric elastomer, Anomalous bulging instability, Irregular bulging shape, Electromechanical coupling



Corresponding authors: [email protected], [email protected]

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Abstract When a clamped membrane of elastomer is subject to a lateral pressure, it bulges into a hemispherical balloon. However, for a clamped membrane of dielectric elastomer (DE) under a lateral pressure as well as a voltage through the thickness, it may bulge into a regular hemispherical balloon or an irregular shape. This work focuses on the anomalous bulging behaviors (i.e. the irregular bulging shape) of a DE balloon under

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electromechanical coupling loading. The full set of the equilibrium configurations of the DE balloon is theoretically derived within the framework of thermodynamics, based on which we find that with the increase of the applied voltage, the pressure-volume relationship changes from the single-N shape for the case of purely mechanical loading

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to a double-N shape, where five or more equilibrium configurations exist including both regular and irregular bulging shapes. Through stability analysis we find the anomalous bulging is a common behavior for the DE balloon under electromechanical coupling loading and all types of irregular bulging shapes can be achieved by following carefully

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designed loading paths. Besides, the irregular bulging region usually has the largest local strain which may initiate the failure of DE devices. Guided by the theoretical analysis, we

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conducted experiments on a DE balloon under the internal pressure and electrical actuation. Typical irregular shapes were successfully observed and the entire evolution

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of the shape changing agrees very well with theoretical predictions. These findings enrich understandings of highly nonlinear behaviors for soft materials under

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electromechanical coupling loading.

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1. Introduction Rubber-like materials are easy to undergo nonlinear elastic large deformation subject to comparatively small external mechanical loadings. From the microscopic view, rubber-like materials consist of networks of randomly-oriented, long chain molecules with sparse crosslinks (Boyce and Arruda, 2000). After being stretched, the randomly orientated molecule network becomes preferentially oriented. This underlying structure

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enables them to exhibit high degree of deformability. Upon further stretching, the deformability of the material will be increasingly restricted by the extension limit since the finite contour length of the molecular chains is reached. This phenomenon can be represented by the load-displacement curve with a tangent of gradual decrease followed

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by an obvious increase.

Pressurized rubber balloons are a large family of common structures in practical applications, such as vehicle tires. Due to the high nonlinearity of rubber, when we inflate a rubber balloon, less strength will be cost to blow it larger after the balloon is

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inflated to a certain size. When the stretch of the rubber balloon approaches the extension limit, the force needed to blow it further increases sharply. These phenomena

and the stretch.

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can be clearly reflected from the N-shaped relationship between the internal pressure

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This non-monotonic load-displacement relation is closely related to a number of shape bifurcations (or structural instabilities) observed in inflated rubber balloons with

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various initial configurations including circular membrane (Rivlin and Saunders, 1951; Treloar, 1944), spherical balloon (Alexander, 1971; Haughton and Ogden, 1978;

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Needleman, 1977), and tubular balloon (Haughton and Ogden, 1979; Kyriakides and Chang, 1990, 1991). Treloar (Treloar, 1944) conducted bulge tests to measure the principal strains of the inflated circular membrane subject to an internal pressure. On blowing the air, the initially flat circular membrane was gradually inflated into a hemispherical balloon and no shape bifurcation was observed during the inflating process. The bulge test has now become a popular way (Joye et al., 1972; Schmidt and Carley, 1975) to determine the material characteristics of a rubber membrane. On inflating a tubular balloon, the localized aneurysm bulging is usually observed at a

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critical pressure in a symmetric manner (Kyriakides and Chang, 1990, 1991; Pamplona et al., 2006). Further inflation may result in the axial enlargement of the bulged sections at the expense of unbulged sections. On inflating an initial spherical rubber balloon, experiments (Alexander, 1971) show that the spherical balloon may bifurcate into a pear shape through localized thinning near one of the poles . Stability analysis was conducted to verify that the pear-shaped configuration was stable under mass control (Fu and Xie,

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2014). However, up to now, no secondary bulging instabilities (a localized bulge on a global bulging shape) have been observed for a rubber balloon under pure internal pressure loading.

Rubble-like materials are also widely used as insulators. Rubber as a capacitor can

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sustain charges subject to an applied voltage. In Röntgen's early experiment (Röntgen, 1880), he sprayed electric charges on a pre-stretched natural rubber stripe and observed a few percent of length change induced by the Columbic force compressing the membrane along the thickness direction. When the electrical field on the rubber

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membrane is high and the elastic modulus is low, the electrically induced deformation can be very large. It was reported that greater than 100% strain was electrically actuated

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in this kind of rubber called dielectric elastomer (DE) (Pelrine et al., 2000). Since then, various applications including soft robots (Koh et al., 2016; Lu et al., 2016; Shian et al.,

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2015), adaptive optics (Carpi et al., 2011; Shian et al., 2013; Wei et al., 2014), sensors (Lee et al., 2016; Zhang et al., 2016), bio-engineering devices (Akbari and Shea, 2012),

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generators (Bortot et al., 2015; McKay et al., 2015; Moretti et al., 2015; Tutcuoglu and Majidi, 2014) have been demonstrated, owing to the excellent attributes of large

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deformation, fast response and high energy density of DEs. The new record for the huge voltage-induced deformation has reached 2200% strain in area (An et al., 2015). When a DE membrane is subject to a voltage, the Columbic force of attraction thins down the membrane, and the decreased thickness amplifies the electric field. This positive feedback can induce the electromechanical instability (Stark and Garton, 1955; Zhao and Suo, 2007). Considering the strain stiffening effect at large deformation due to the extension limit of polymer chains, the voltage-deformation relation ultimately exhibits a N-shaped curve (Suo, 2010; Wang et al., 2016; Zhao et al., 2007).

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Analogous to rubber balloons, the N-shaped feature for DE leads to various kinds of bifurcation behaviors for DE balloons subject to a pressure and a voltage (An et al., 2015; Liang and Cai, 2015; Lu et al., 2015; Xie et al., 2016). A spherical DE balloon under the internal pressure and voltage was theoretically predicted to bifurcate into a nonspherical pear-shaped configuration (Liang and Cai, 2015; Xie et al., 2016). An et al (An et al., 2015; Lu et al., 2015) designed an experiment to demonstrate the electrically induced

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bifurcation in a DE tubular balloon. Similar to the purely mechanical case, a localized bulging section in coexistence with the unbulged section was observed at the transition voltage followed by a steady propagation of the bulged section. In Li's experiment (Li et al., 2013) where a circular elastomer membrane mounted on an air chamber was subject

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to internal pressure and voltage, a bifurcation configuration of local bulging was observed on the top region of the inflated DE membrane. Moreover, wrinkles along the longitudinal direction were observed at the side edge of the bulged region. The observed shape bifurcation in the electrical bulge test is novel compared to the purely mechanical

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bugle test, where there is no bifurcation at all during the entire inflating process (Joye et al., 1972; Rivlin and Saunders, 1951; Schmidt and Carley, 1975; Treloar, 1944). Indeed,

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the bulging behaviors of DE balloons under electromechanical coupling loading are fruitful, to the best of our knowledge, the previous literatures were far from enough to

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give a clear understanding.

This work aims to understand the mechanisms for the anomalous bulging behaviors

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induced by electromechanical coupled loading. We find that these irregular bulging shapes are essentially states of equilibrium which are unstable under voltage and

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pressure control condition but become stable when the electromechanical coupling loading paths are specially designed. We theoretically predict that by tuning the loading path, various irregular bulging shapes can be achieved. Our experimental results verified our predictions remarkably well. The paper is organized as follows. Section 2 derives the governing equations of an initially flat DE membrane mounted on a chamber undergoing inhomogeneous inflating deformation from one side when subject to an internal pressure and a voltage across the thickness. In Section 3 the ideal dielectric elastomer material constitutive model is

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incorporated into the governing equations. Section 4 gives some notes for the numerical calculation. Section 5 numerically solves the governing equations to obtain the equilibrium states. Section 6 analyzes the stability of the coexistent configurations by comparing the potential energy under two types of electromechanical loading methods. Section 7 carries out a series of experiments to observe three typical configurations as

theoretical predictions. Section 8 gives the conclusion. 2. Governing equations

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well as a coexistence situation, and compares the observed phenomena with the

Consider a DE balloon subject to an internal pressure and a voltage across the

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thickness. The cross-sections of a DE membrane sandwiched between two compliant electrodes in several states are shown in Fig. 1. In the reference state, the flat circular DE membrane of thickness H and radius A is subject to no force or voltage. Each material particle in the membrane is labeled by its distance R from the center O (Fig. 1a). In the pressurized state, the membrane is pre-stretched in-plane to a fixed boundary

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with the radius a and subject to a differential pressure p from one side of the membrane (Fig. 1b). In the actuated state, the membrane further deforms under a

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voltage  through the thickness as well as the pressure p . The inflated DE balloon under electromechanical coupling loading may finally stabilize at a regular shape (Fig.1c)

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or an irregular shape (Fig.1d). The balloon is assumed to be thin-walled and the fields

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along the thickness of the membrane are assumed to be homogeneous. The volume enclosed by the balloon is denoted by v . The charges accumulated on the two surfaces

 r, z 

at the deformed state is established with the origin

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are Q . A coordinate system

O being fixed at the center of the original plane. The deformation fields are specified by

the radius r  R  , the height z  R  , and the thickness h  R  . The slope of the balloon at material particle R is denoted by   R  (Fig.1c and 1d). The deformation profile of the balloon along the r-direction is assumed to be axisymmetric, but no prerequisites are made for the profile along the  -direction.

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Fig. 1 The cross-sections of a DE membrane sandwiched between two compliant electrodes

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with negligible stiffness in several states. (a) The reference state, the circular membrane is flat without applying force or voltage. (b) The pressurized state, the membrane is mounted on a chamber with a circular opening, and subject to an internal pressure. In the actuated

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state, the membrane is subject to a pressure as well as a voltage through the thickness, and

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can deform into (c) a regular shape or (d) an irregular shape.

Consider an annular material element between two circles of material particles with

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radius R and R  dR in the reference state. In the actuated state, the circle of material particles of radius R becomes a circle of radius r  R  with a height of z  R  , and the other circle of material particles, radius R  dR , becomes a circle of radius

r  R  dR  with a height of z  R  dR  . The flat annular material element of width dR in the reference state deforms to a tridimensional annulus of width ds 

 dr 

2

  dz  . 2

The longitudinal stretch, the latitudinal stretch, and the thickness stretch are respectively calculated by

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1 

ds r h , 2  , 3  . dR R H

(1)

Based on geometrical relationship, we have (2)

dr  1 cos  . dR

(3)

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dz  1 sin  , dR

The elastomer is taken to be incompressible i.e. 123  1 . The true electric field along the thickness of the balloon is

   1  R  2  R  . h  R H

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E  R 

(4)

The true electric displacement is D  R  , and the amount of charge on either electrode is the integral of the electric displacement over the surface area of the balloon, A

Q   212 DRdR .

(5)

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0

The volume enclosed by the balloon can be calculated as 0

A dz dR   122 sin  R 2dR . 0 dR

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v    r2 A

(6)

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The DE balloon, along with the mechanisms applying the voltage and the internal pressure, constitutes a composite thermodynamic system. We suppose the composite

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thermodynamic system exchanges energy with the rest of the world by heat, but is maintained at a constant temperature. The thermodynamic state of the composite can be

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represented by three independent variables 1  R  , 2  R  and D  R  . The viscosity of the material is neglected. At the equilibrium state, the variation of free energy of the DE balloon  F equals

to the sum of the mechanical work p v done by the internal pressure and the electrical work  Q done by the power source:

 F  p v   Q . The free energy of the DE balloon is

(7)

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(8)

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where W  1 , 2 , D  is the Helmholtz free energy density of the balloon defined by the total free energy of an element in the deformed state divided by the volume of the element.

governing equations (Lu et al., 2013):

p22 R , s1  2 H sin  p2 R H sin 

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s1 

W  1 , 2 , D  2 D  1 H

2 R d    1  , 2sin  dR  

W  1 , 2 , D  , D and

s2 

W  1 , 2 , D  1D  2 H

(10)

(11)

denote

the

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where

(9)

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s2 

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Considering independent variations on 1 , 2 and  D , we obtain the

nominal stresses. Eqs. (9) and (10) indicate the force balance along longitudinal and

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latitudinal directions and Eq. (11) indicates the electric balance. The true stresses along the longitudinal and latitudinal directions can be calculated by  1  1s1 and  2  2 s2 .

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The above governing equations are valid for an arbitrary material model specified by the

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free energy function W  1 , 2 , D  .

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3. Material model

To calculate the electromechanical coupling deformation of the DE balloon, the

constitutive model of ideal dielectric elastomer is adopted (Zhao et al., 2007). The model assumes that the dielectric behavior of an elastomer is liquid-like and unaffected by deformation. To be specific, the true electric displacement D is linear with respect to the true electric field E ,

D  E ,

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where  is the permittivity of the elastomer and taken to be a constant independent of

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electric displacement and mechanical deformation. Substituting Eq. (12) into Eq. (11) and integrating with respect to D , we obtain

W  1 , 2 , D   Wstretch  1 , 2  

D2 , 2

(13)

where the integration Wstretch  1 , 2  represents the free energy associated with the

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stretching of the elastomer. Thus for the ideal dielectric elastomers, the free energy of the elastomer W  1 , 2 , D  is a combination of the elastic energy and the electrostatic energy. Inserting Eq. (4) into (12), the true electric displacement D can be expressed as

D  R   1  R  2  R 

 . H

(14)

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To describe the free energy associated with the mechanical stretching, the Gent model (Gent, 1996, 2005) is employed, in which the effect of strain stiffening of the elastomer is taken into consideration. The strain stiffening characteristic can be explained from the microscopic picture: The elastomer consists of polymer chains, each

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of which has a finite contour length. When the elastomer is subject to loads, the polymer chains are elongated. With the increase of the loads, the end-to-end distance of each

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polymer chain may approach the finite contour length, which corresponds to the macroscopic picture that the elastomer approaches a limiting stretch and behaves as the

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strain-stiffening effect. Based on the Gent model, the elastic energy density is calculated by

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Wstretch  1 , 2  = 

 J lim

  2  22  1222  3  ln 1  1 , 2 J lim  

(15)

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where  is the shear modulus and J lim is a material constant related to the limiting stretch.

In the current work the DE material used in the experiment is VHB4910 (3M

company) and the comparisons in Section 7 are qualitative. Therefore, knowing the specific values of the dimensional quantities, such as the shear modulus

 and the

permittivity of the elastomer  are not necessary. For reference, the shear modulus of VHB4910 can be selected within the range   30kPa ~ 60kPa due to viscosity and the

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dielectric constant  is usually taken as 4 1011 F / m approximately (Huang et al., 2012; Kofod et al., 2003; Lu et al., 2015; Lu et al., 2012; Qiang et al., 2012; Zhu et al., 2012). The material constant J lim is taken as J lim  270 in this paper (Li et al., 2013).

4. Notes for numerical calculation

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The inhomogeneous deformation of the DE balloon can be calculated by using the governing equations in Section 2 and the material model in Section 3. Combining Eqs. (1), (3), (9) and (10), we obtain

d s2 sin  12 p ,   dR s1R s1H

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  s  d 1  s    s1  s2 cos   1  1 cos   2    R 1  . 2 dR    1 

(16)

(17)

At the apex of the balloon, R  0 , symmetry requires that

r  0  0 ,   0   .

(18)

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At the edge of the balloon, R  A , the fixed boundary dictates that

z  A  0 .

(19)

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r  A  a ,

The deformation of the inflated DE balloon is governed by four differential

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equations Eqs. (2), (3), (16), and (17), along with the boundary conditions Eqs. (18) and (19).

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The shooting method (Ma and Wang, 2003) is used to solve the two-point boundary-value problem as an initial-value problem. Eq. (18) gives two known initial

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values ( r  0  and   0  ), and we need to guess two unknown initial values z  0  and

1  0  to satisfy two target conditions Eq. (19). Since the right side of Eq. (2) does not

explicitly contain any functions of z , we introduce a new parameter z  z  0   z and Eq. (2) is rewritten as

dz  1 sin  , dR and the second equation of Eq. (19) becomes

(20)

ACCEPTED MANUSCRIPT z  0   0 and z  A  z  0  .

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By using the new parameter, the number of the known initial values increases to three ( r  0  ,   0  and z  0  ), leaving us only one unknown initial value 1  0  to guess. The guessed initial value 10 is selected from 1 to 12 with the minimum interval of 0.00025 to insure accuracy. Once the two external loading parameters p and 

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are prescribed, the four ordinary differential equations can be numerically integrated with the three known initial conditions and the guessed value 10 to obtain r  R  ,

  R  , 1  R  , and z  R  . The integration procedure is iterated until the solved r  A

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satisfies the boundary condition r  A  a given by Eq. (19). z  R  can be recalculated by z  R   z  A  z  R  . In our integration process, the singularity at R  0 is avoided by selecting the initial values at R0  0.0001A instead of R0  0 .

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5. Equilibrium configurations

The equilibrium configurations of the DE balloon under electromechanical coupling

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loads are numerically calculated in this section. It is desirable to use global variables to describe state, such as volume enclosed by the balloon v that is work-conjugated with

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pressure p , and total charges on the balloon Q that is work-conjugated with voltage

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 . We can plot the equilibrium state in the p  v curve under a fixed  , or   Q curve under a fixed p . The pressure, voltage, volume, and charge are normalized by







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pA   H  ,  H   , v  A3  , and Q  A2 



in the theoretical part.

For certain configurations, e.g. equal biaxial loading condition, when the DE

membrane is pre-stretched, the electromechanical instability is often suppressed (Huang et al., 2012). In this work, we focus on the irregular configuration associated with electromechanical instability. Therefore, we consider a representative case where the initial in-plane pre-stretch is 1, namely a  A , and the pre-stretch of the balloon is only induced by the initial internal pressure p . With this small initial pre-stretch, the

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electromechanical instability is easier to trigger.

Fig. 2 The pressure vs volume relation curves at several voltages. The points marked by A

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and B indicate the bifurcation pressures under the voltage 

H

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   0.12 , and the

points marked by C, D, E, and F indicate the bifurcation pressures under the voltage





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 H    0.15 .

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Fig. 2 shows the p  v curves at several voltage levels. We find that at different voltage levels, the number of equilibrium states will change. Under a low voltage, e.g.





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 H    0.12 , the pressure-volume relation follows a N-shaped curve with two

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bifurcation points, a peak point A and a valley point B . Under this condition, at a low pressure level there is only one equilibrium state, at an intermediate pressure level there are three equilibrium states, and at a high pressure level one equilibrium state again. Under a high voltage, e.g. 

H



   0.15 , a localized N-shaped curve appears in

the falling segment of the global N-shaped pressure-volume curve. The four bifurcation points are marked as C , D , E , and F . Under this condition, the number of equilibrium states changes by 1, 3, 5, 3, 1 as the pressure level increases. Under a higher

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voltage, e.g. 

H



   0.18 , the bifurcation points are too close to be identified in

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the figure.

Fig. 3 (a) The pressure vs volume curve under the voltage 

H



   0.18 .

The

corresponding profiles of equilibrium configurations under different pressures: (b)

pA   H  =0.2 , (c) pA   H  =0.6 , (d) pA   H  =1.6 , (e) pA   H  =1.68 and (f) pA   H  =1.8 . In the regions marked by red line, the calculated stresses are negative.

Fig. 3 shows the calculated configuration profiles under the voltage of

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



 H    0.18 and several representative pressures. Under a small pressure, e.g.

pA   H  =0.2 , there is only one intersection point in Fig. 3a and the DE balloon has only one equilibrium state: an inflated hemispherical regular shape with small deformation (Fig. 3b). When the pressure becomes larger, e.g. pA   H  =0.6 , there are three intersection points in Fig. 3a and three equilibrium states exist: a regular shape

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with small deformation, a regular shape with large deformation and a mushroom-cloud like irregular shape (Fig. 3c). When the pressure increases to pA   H  =1.6 , the balloon has five possible equilibrium states: three regular bulging shapes (left figure in Fig. 3d), and two irregular shapes with localized bugles in the center (right figure in Fig.

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3d). Under the pressure of pA   H  =1.68 , the aforementioned two irregular shapes disappear while the three regular bulging shapes remain (Fig. 3e). When the pressure reaches pA   H  =1.8 , the balloon has only one equilibrium state again: an inflated

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hemispherical regular shape with large deformation (Fig. 3f). Parts of the profiles are marked in red color, indicating the computed stresses along the latitudinal direction in

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this region are negative, which may induce wrinkles along the longitudinal direction. One of the predicted irregular configurations in Fig. 3d is consistent with the irregular

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shape observed in previous experiment (Li et al., 2013). Comparing with configuration profiles under other voltage levels we can find that when the p  v curves in Fig. 2

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exhibit a double-N shape, the equilibrium configurations in the descending segment of the global N-shaped curve correspond to the irregular shapes with localized bulge and

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the equilibrium configurations in the two ascending segments correspond to the regular shapes of small deformation or large deformation. The unusual equilibrium curves under specific electromechanical coupling loading conditions are the key to form the irregular shapes. In particular, under pure mechanical loading without voltage, only regular shapes can be observed, consistent with a commonly practiced inflation experiment.

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Fig. 4 The stretch and stress distributions for three representative configurations: the small bulge (a-e), the large bulge (f-j), and the regular shape (k-o). The points marked by B and C

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denote the bulgy particles and the concave particles, respectively. The regions marked by the

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red line denote the areas with negative stress.

We take three representative shapes (Fig. 3d irregular shape, Fig. 3c irregular shape,

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Fig. 3c regular shape) as examples to calculate the inhomogeneous deformation/stress fields. Fig. 4 shows the stretch and stress distributions along the longitudinal and latitudinal direction. The turning points along the profiles are of interest and are marked as point B (Bulge) and point C (Concave). For the case of regular shape (Fig. 4k-4o), the stretches and stresses smoothly change as the normalized radius varies from 0 to 1. For the cases of irregular shape with a small bulge (Fig. 4a-4e) or the large bulge (Fig. 4f-4j), the stretches and stresses undergo a sharp change from point B to point C. The parts with a negative stress  2 are marked in red.

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Inspecting the strain and stress distributions, we can see there are no intrinsic differences between the small bulge profile and the large bulge profile. The irregular shapes can be regarded as the combination of a segment with large deformation in the central region and a segment with small deformation in the surrounding region. When the proportion of the segment with small deformation is large, it forms the irregular shape with a small bulge; when the proportion of the segment with large deformation is

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large, it forms the irregular shape with a large bulge. Combinations of the large deformation region and the small deformation region in different proportions yield different irregular shapes. Based on this analogy, the regular shape of small deformation can be considered as a special case without central large deformation region, while the

surrounding small deformation region. 6. Stability analysis

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regular shape of large deformation can be considered as a special case without the

Multiple equilibrium configurations including regular and irregular shapes may

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exist under the same electro-mechanical coupling loads. However, only the most stable configuration can be observed in a real experiment. According to thermodynamics, the

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most stable configuration minimizes the Helmholtz free energy of the system at a constant temperature. The calculation of the Helmholtz free energy as well as the

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determination of stability depends on the loading paths, including voltage control,

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charge control, pressure control, volume control, and some mixed control (Liang and Cai, 2015). In this work, we focus on two kinds of loading paths, which are relatively easy to

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realize in an experimental setup. One is voltage and pressure control mode: control the voltage across the thickness of the DE balloon and control the internal pressure inside the balloon. The other is voltage and mass control mode: control the voltage and control the number of gas molecules inside the balloon.

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Fig. 5 (a) The pressure vs volume relation curve under the voltage  The corresponding Helmholtz free energy vs the pressure.

H



   0.18 . (b)

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For the voltage and pressure control mode, in which the applied voltage and pressure are independent of the deformation, the Helmholtz free energy  of the composite thermodynamic system is the sum of the free energy of the DE balloon F and the potential energy of the mechanisms that apply voltage and the differential pressure,

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  F  Q  pv .



(22)



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The Helmholtz free energy  is normalized by   A2 H . As a representative,





we focus on one loading path in which the voltage is fixed at  H    0.18 and

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the pressure is ramped up. The equilibrium curve in Fig. 5a is divided into three regions:

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from the starting point to the peak (black), from the valley to the end (blue), and the region in between (red). Under different pressure levels, there are possibly multiple

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equilibrium solutions, each of which corresponds to a value of Helmholtz free energy. The relationship between the Helmholtz free energy and the pressure is plotted in Fig. 5b. We find that the Helmholtz free energy of the equilibrium configurations in the falling segment of the global N-shaped pressure-volume curve in Fig. 5a (red region) is always higher than those of the two rising segments (black and blue region). As analyzed in Section 5, the irregular shapes can only exist in the falling segment. Therefore, we can predict that for the voltage and pressure control mode the equilibrium configurations of regular shape are more stable than the equilibrium configurations of irregular shape. If

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we control to ramp up the pressure or the voltage gradually, the DE balloon will first stabilize with regular shape of small deformation until snap-through instability occurs, then stabilize with regular shape of large deformation. This predicted result can hardly be observed in real experiments since the snap-through instability dramatically decreases the thickness of the membrane and increases the electrical field, which is

equilibrium states with irregular shape cannot be observed.

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usually followed by the failure of electrical breakdown. Thus under this loading path, the

The other electro-mechanical loading path is the voltage and mass control mode where the applied voltage and the number of gas molecules inside the DE balloon are independent of the deformation. This mode can be achieved by mounting the DE

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membrane on a pressure chamber of volume vc . A number of gas molecules are pumped into the space and the DE membrane is inflated into a balloon of volume v0 with an initial differential pressure p0 . Then the DE balloon and the chamber are sealed to form

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a closed system, and the number of the gas molecules remains unchanged in the subsequent loading procedure. When the voltage through the thickness is gradually

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applied, the DE balloon further deforms to a volume of v and the differential pressure becomes p . Assuming the gas molecules enclosed by the chamber and the balloon obey

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the ideal gas law, we have

NkT   p  patm  v  vc    p0  patm  v0  vc  ,

(23)

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where N is the number of the gas molecules, k is the Boltzmann constant, T is the

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temperature of the gas, patm is the atmospheric pressure. According to Eq. (23), the number of the gas molecules N is determined by the initial differential pressure p0 , the atmospheric pressure patm and the volume of chamber vc . And for the given number of gas molecules N under isothermal conditions, the pressure p  patm shows a negative correlation with the volume v  vc . As the applied voltage increases, the volume of the balloon v increases and the pressure p decreases. Therefore, a special loading path depends on the dimensionless initial inputs: the initial pressure

ACCEPTED MANUSCRIPT p0 A   H  , the atmospheric pressure patm A   H  , and the volume of chamber

vc  A3  . The Helmholtz free energy  for such a composite thermodynamic system consists of the free energy of the DE balloon F , the potential energy of the mechanisms that apply voltage, the atmospheric pressure patm , and the enclosed gas, v

  F  Q   pdv .

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(24)

v0

In calculating the integral in Eq. (24), we use the condition of ideal gas law to replace

constant terms.

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v v  vc NkT and thus obtain   pdv  patm v  NkT ln by dropping v 0 v0  vc v  vc

p  patm with

We design three dimensionless loading paths (dashed lines in Fig. 6a) to obtain the three representative equilibrium configurations shown in Fig. 6c, 6e and 6g. The dimensionless

atmospheric

pressure

 A  3

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dimensionless chamber volumes vc

is

taken

as

patm A   H   80.414 . The

for the three cases are respectively chosen

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as 6.52, 142.60 and 1018.60. The dimensionless initial pressures and the corresponding volumes of the balloon

 p A   H  , v  A  0

3

0

for the three loading paths are set as

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(1.860, 0.7041), (1.913, 0.9375), and (1.840, 0.6701). Along the three loading paths, the voltage is gradually increases while the internal pressure changes following the

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constraint Eq. (23).

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Fig. 6 The DE balloon is actuated under the voltage and mass control mode. (a) The loading

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paths denoted by the three dash lines are named as Case 1, Case 2 and Case 3. Along these three paths, the gas enclosed by the balloon obeys ideal gas law. (b, c) The pressurized state without voltage and the actuated state under a certain number of enclosed ideal gas for Case 1. (d, e) The pressurized state without voltage and the actuated state for Case 2. (f, g) The pressurized state without voltage and the actuated state for Case 3.

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As the voltage increases, the DE balloon undergoes a series of deformations following the most stable configuration. For Case 1 with a small volume chamber, upon

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increasing the voltage, the DE balloon may transform from a regular shape (Fig. 6b) into a cup cover-like irregular shape with a small embossment in the center of the bottom

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part with small deformation (Fig. 6c). For Case 2 with an intermediate volume chamber, the DE balloon may transform from a regular shape (Fig. 6d) into an irregular

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mushroom cloud shape (Fig. 6e). For Case 3 with a large volume chamber, no irregular

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shape appears upon increasing the voltage but undergoes snap-through instability; the DE balloon snaps through from a regular shape of small volume (Fig. 6f) to a regular shape of large volume (Fig. 6g). The origins of deformation for the three cases are analyzed as follows.

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Fig. 7 Following the loading path of Case 1, (a) the applied voltage vs volume of the DE balloon, (b) the Helmholtz free energy vs the voltage, (c) the configurations under four representative voltages: 0, a little higher than

1 , a little lower than  3 , and  m .

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Following the loading path of Case 1, the voltage induced volume change is plotted in Fig. 7a and the Helmholtz free energy of the system at different voltage levels is

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plotted in Fig. 7b. At certain voltage levels, there are multiple equilibrium states, corresponding to multiple values of free energy. The curve in Fig. 7a consists of three

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segments: an ascending segment LA from   0 to the maximum applied voltage

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 m  0.195 , a descending segment LB from  3 to 1 , and finally an ascending segment LC . At the voltage level  3 , the curve bifurcates into two branches (black

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solid and red dotted). When the voltage is in the range of  ~  0, 1  , the DE balloon remains the regular shape with small deformation (Fig. 7c-1). The unique equilibrium state is the most stable state. When the voltage is in the range of  ~  1 , 3  , there are three equilibrium states. By comparing the free energy of the three states in Fig 7b, we find that there is a critical voltage level  2 , below which the free energy of the equilibrium on LA segment is the lowest and above which the free energy of the

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equilibrium on LC segment is the lowest. The profiles of the three equilibrium states are plotted in Fig. 7c-2 (voltage level a little higher than 1 ) and Fig. 7c-3 (voltage level a little lower than  3 ), where the solid lines represent the most stable configurations. The free energy of the equilibrium state on LB is higher than those equilibrium states

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on LA and LC except in a region near  3 , as shown in the amplified inset of Fig. 7b. Fig. 7c-4 shows the profile under the voltage level  m , where the irregular shape is more stable.

However, in a real loading path where the voltage is gradually increased, the

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deformation may not always follow the most stable configuration due to energy barrier. For example, when monotonically loaded to a voltage at  ~   2 , 3  , although the state on LC is the most stable, the membrane may still stabilize at the configuration of

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LA since the state on LA has a localized minimum in free energy. When the voltage level approaches  3 , the free energy of the state on LB becomes smaller than the

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value of that on LA . In this case, as the state on LA is the most unstable, the profile

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LA cannot be maintained and will transform to LB , in the center of which an exceedingly small area sticks out (Fig. 7c-3), and subsequently snap-through to the state due to the flaw sensitivity. Combining the stability analysis and the

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on LC

consideration of energy barrier, when subject to a monotonically increasing voltage, one

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of the possible deformation history may follow the sequence that the shape of the DE balloon barely changes (Fig. 7c-2 LA ) until an exceedingly small region in the center of the balloon bulges out (Fig. 7c-3 LB ), followed by a snap-through growth of localized bulging at the same voltage (Fig. 7c-3 LC ).

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Fig. 8 Following the loading path of Case 2, (a) the applied voltage vs volume of the DE balloon, (b) the Helmholtz free energy vs the voltage, (c) the configurations under four

1 , a little lower than  2 , and  3 .

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representative voltages: 0,

For the loading path of Case 2, the voltage-volume curve is N-shaped, with an

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ascending segment LA , a descending segment LB and a second ascending segment

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LC . There is a critical voltage level 1 between the peak point and the valley point of the N-shaped curve (Fig. 8a). For a voltage lower than 1 , the free energy of the

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equilibrium state on LA is the lowest, while for a voltage greater than 1 , the free

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energy of the equilibrium state on LC is the lowest (Fig. 8b). When the applied voltage is gradually increased, the DE balloon will snap-through near the peak point (voltage level  2 ) due to the energy barrier. This kind of snap-through instability has been extensively discussed in literature (Lu et al., 2012). The profiles with solid lines in Fig. 8c show the most stable states under different voltage levels. The deformation history may follow the sequence (Fig. 8c-2 LA ), (Fig. 8c-3 LC ), (Fig. 8c-4 LC ). For the loading path of Case 3, the descending segment and the second ascending

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segment is broken. Similar to Case 2, the critical voltage to differentiate the magnitude of free energy is marked as 1 and the voltage level of the peak point is  2 . We find that all the profiles for the equilibrium states on three segments are of regular shape (Fig. 9c). When the applied voltage is gradually increased, snap-through instability occurs near the peak point, following the sequence (Fig. 9c-1 LA ), (Fig. 9c-2 LA ), (Fig. 9c-3

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LC ), (Fig. 9c-4 LC ).

Fig. 9 Following the loading path of Case 3, (a) the applied voltage vs volume of the DE

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balloon, (b) the Helmholtz free energy vs the voltage, (c) the configurations under four

1 , a little lower than  2 , and  3 .

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representative voltages: 0,

7. Experiment and discussion

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In order to verify the theoretical predictions, we conducted experiments for a dielectric elastomer balloon under the electro-mechanical coupling loads. The experimental setup of a DE balloon subject to an internal pressure and a voltage through the thickness is drawn in Fig. 10. A transparent DE membrane (3M VHB 4910) with the original thickness of H  1mm was mounted on an air chamber of volume vc with a circular opening of radius A  25mm . The exposed top and bottom surfaces of the DE membrane were uniformly coated with conductive carbon grease (MG Chemical

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846-80G) as electrodes. The experiment was carried out at room temperature and under

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the atmospheric pressure.

Fig. 10 (a) Schematic of the experimental setup. The DE balloon subject to an internal pressure and a voltage across the thickness. (b) Picture of the components of the

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experimental setup: ①-The DE balloon, ②-The pressure chamber, ③-The high voltage source, ④-The pressure gauge.

We used the voltage and mass control mode. Air was pumped into the space sealed by the DE membrane and the chamber through a regulating valve. The regulating valve

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was then closed to fix the number of gas molecules enclosed by the chamber and the

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inflated balloon. The internal differential pressure in the chamber was measured by a pressure gauge. Subsequently, the voltage through the balloon thickness was applied by a high voltage power source (Trek Model 610E) connected to the electrodes. The

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electrically actuated deformation process of the DE balloon was recorded by a video

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camera. To verify the theoretical predictions under the three loading paths in Section 6,

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three chamber volumes 0.32L, 1.32L and 22L were selected to carry out the experiments.

Fig. 11 Electrically-actuated deformation process for the case with a small size of chamber. (a) Theoretical predictions, (b) experimental pictures.

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Fig. 11 shows the deformation process for the small chamber with volume 0.32L. The shape of the DE balloon barely changed under the voltage ranging from 0kV to 7.02kV (Fig. 11b-2). When the voltage was increased to 7.06kV, the apex of the balloon became sharp, and an exceedingly small region in the center of the balloon bulged out (Fig. 11b-3). While maintaining this voltage, the localized bulge grew larger immediately

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accompanied by the appearance of wrinkles along the longitudinal direction at the side edge of the bulged region. Our theory predicts that the DE balloon finally stabilizes at a largely bulged profile (Fig. 11a-4). In the experiment, however, with the growth of the bulged region, the apex region became thinner and thinner and the electric field became

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higher and higher. As a result, we observed the failure by electrical breakdown during the growth of the bulge. Fig. 11b-4 shows a snapshot with a clear burning spark

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associated with the electrical breakdown.

Fig. 12 Electrically-actuated deformation process for the case with an intermediate size of

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chamber. (a) Theoretical predictions, (b) experimental pictures.

Fig. 12 shows the deformation process for the chamber with volume 1.32L. As the

applied voltage increased, the DE balloon firstly elongated along the height direction and the upper portion of the balloon gradually became thinner and then bulged out (Fig. 12b-3). Upon further increasing the voltage, the bulged region expanded larger while the height of the balloon decreased a little (Fig. 12b-4). During the bulging process, wrinkles along the longitudinal direction were observed at the side edge of the bulged region

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before final electric breakdown (Fig. 12b-4). The experimental phenomena shown in Fig.

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12b are in good agreement with the theoretical predictions shown in Fig. 12a.

Fig.13 Electrically-actuated deformation process for the case with a large size of chamber. (a)

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Theoretical predictions, (b) experimental pictures.

Fig. 13 shows the deformation process for the chamber with volume 22L. As the theory predicts, no irregular shapes were observed during the whole actuation process. Before the applied voltage reached 3.3kV, the DE balloon showed a regular shape of

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small deformation. Upon increasing the voltage to 3.3kV and maintaining for a while, the DE balloon suddenly snapped from the regular shape of small deformation (Fig.

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13b-2) to the regular shape of large deformation (Fig. 13b-3). The balloon became even larger with further increasing the applied voltage to 5.05kV (Fig. 13b-4). Wrinkles along

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the longitudinal direction were observed at the side edge of the large balloon before the final electric breakdown. Again, the experimental phenomena shown in Fig. 13b are in

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good agreement with the theoretical predictions shown in Fig. 13a. Although the qualitative comparisons between the theory and the experiment are

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remarkably good, the quantitative comparisons are rather difficult due to the following reasons: first, when the dielectric elastomer is mechanically stretched or electrically actuated to an extremely large stretch, the two-parameter elastic Gent model is far less accurate to capture the deformation history (An et al., 2015; Li et al., 2013). Second, it is reported that the dielectric constant possibly depends on the temperature, the pre-stretch, and the deformation state in a complex manner (Jean-Mistral et al., 2010; Qiang et al., 2012). Third, the dielectric material VHB used in the experiment exhibits

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significant viscous effects, which influences the accuracy of the recorded deformation

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curves (Keplinger et al., 2008; Kollosche et al., 2015).

Fig. 14 (a) Two DE balloons subject to the same electro-mechanical loadings can respectively deform through two paths marked by two dashed lines in the pressure-volume plane. (b) The initial almost identical configurations of the DE balloon without applied voltage. (c-e) With the increasing applied voltage, the left balloon deforms along Path 1 to smaller and smaller

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regular shapes, while the right balloon deforms along Path 2 to bigger and bigger irregular

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shapes.

As analyzed in Section 5, there may exist multiple equilibrium states including

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regular and irregular shapes at certain pressure and voltage levels. Therefore, we designed an experiment to observe the coexisting equilibrium configurations

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simultaneously. Two identical membranes were mounted on a chamber with two circular openings of the same radius A  25mm . They were connected with the common

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chamber and electrically connected in parallel by wires, ensuring the internal pressure and voltage of the two balloons to be always equal (Fig. 14). Before applying voltage we applied the internal pressure to the level approaching the peak point (Fig. 14a). Under this condition, two DE balloons exhibited the similar shapes (Fig. 14b). Upon applying an increasing voltage, the DE balloon on the right became larger and transformed into an irregular shape while the balloon on the left became smaller and maintained the regular shape. During the whole actuation process, the two equilibrium configurations were coexistent under the same electro-mechanical coupling loading (Fig. 14b-e). Taking

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the two balloons as a whole, we can regard the different deformation profiles of the two balloons as a bifurcation qualitatively indicated by the arrows in Fig. 14a. The right arrow indicated the balloon on the right deformed along Path 2 where equilibrium configurations of irregular shapes existed. The balloon on the left deformed along Path 1 where the equilibrium configurations were of regular shapes with small deformation.

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8. Conclusion The bulging instability of a DE balloon subject to an internal pressure and a voltage across the thickness is studied. We establish the equilibrium equations for the DE balloon under electromechanical loading. We find that with the increase of the applied

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voltage, the pressure-volume relationship varies from the single-N shape to double-N shape. The irregular shapes can only be found under a high applied voltage where the pressure-volume curve becomes double N-shaped. We calculate the stretch and stress distributions for the irregular shapes and find that all the irregular shapes share the common characteristic that the local bulging region undergoes large deformation while

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the rest region remains small deformation. Some regions suffering the negative stress along the latitudinal direction exhibit wrinkles along the longitudinal direction. We

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conduct stability analysis by comparing the free energy of all the coexisting equilibrium configurations for two types of loading paths. It is predicted that for the voltage and

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pressure control mode, irregular configurations are always less stable than the regular

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ones and therefore cannot be observed; for the voltage and mass control mode, irregular bulging shapes can be triggered under specifically designed loading paths. In our

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experiment, we observed three typical irregular configurations by varying the chamber volume as well as the initial pressure to specify loading paths, consistent with the theoretical predictions. Finally, two coexistent equilibrium configurations under the same voltage and pressure were observed in the experiment, further confirming that the configuration of irregular shape was one single equilibrium state analogous to the configuration of regular shape.

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Acknowledgment This work was supported by the NSFC (No. 11402185). References

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