voltage-induced wrinkling in a constrained annular...

Kai LiDepartment of Civil Engineering,

Anhui Jianzhu University,

Hefei 230601, Anhui, China;

Department of Mechanical and

Aerospace Engineering,

University of California, San Diego,

La Jolla, CA 92093

Wanfang WuDepartment of Mechanical and



La Jolla, CA 92093

Ziyang JiangDepartment of Mechanical and



La Jolla, CA 92093

Shengqiang Cai1Department of Mechanical and



La Jolla, CA 92093

e-mail: [email protected]

Voltage-Induced Wrinklingin a Constrained AnnularDielectric Elastomer FilmWrinkles can be often observed in dielectric elastomer (DE) films when they are sub-jected to electrical voltage and mechanical forces. In the applications of DEs, wrinkleformation is often regarded as an indication of system failure. However, in some scenar-ios, wrinkling in DE does not necessarily result in material failure and can be even con-trollable. Although tremendous efforts have been made to analyze and calculate a varietyof deformation modes in DE structures and devices, a model which is capable of analyz-ing wrinkling phenomena including the critical electromechanical conditions for theonset of wrinkles and wrinkle morphology in DE structures is currently unavailable. Inthis paper, we experimentally demonstrate controllable wrinkling in annular DE filmswith the central part being mechanically constrained. By changing the ratio between theinner radius and outer radius of the annular films, wrinkles with different wavelength canbe induced in the films when externally applied voltage exceeds a critical value. To ana-lyze wrinkling phenomena in DE films, we formulate a linear plate theory of DE filmssubjected to electromechanical loadings. Using the model, we successfully predict thewavelength of the voltage-induced wrinkles in annular DE films. The model developed inthis paper can be used to design voltage-induced wrinkling in DE structures for differentengineering applications. [DOI: 10.1115/1.4038427]

1 Introduction

A dielectric elastomer (DE) can deform when it is under theaction of electrical field or mechanical forces. Because of theadvantages of easy fabrication, low cost, excellent deformability,and electromechanical robustness, DEs have been recently inten-sively explored and developed to a variety of structures withdiverse functions [1,2]. For instance, DE films with differentshapes and sizes have been fabricated to harvest energy from dif-ferent sources such as ocean wave [3], motions of human-beings,wind, and even combustion [4]. DEs have also been used as artifi-cial muscles in designing walking robots [5], programmable grip-pers [6], camouflage devices [7], and antifouling systems [8]. Inaddition to those, transparent DE loudspeakers [9], planar DErotary motors [10], and nonlinear DE strain gauges [11] have alsobeen successfully made recently.

In most of the applications, large deformation in DEs can beubiquitously observed. General three-dimensional (3D) modelsfor the finite deformation of DEs under the actions of an arbitraryfield of electrical potential and forces have been formulated bydifferent researchers [12–18]. Numerous phenomena associatedwith the electromechanical coupling in DEs have been success-fully analyzed using these developed models, such as the pull-ininstability in a DE membrane sandwiched by two compliantelectrodes [19–21], voltage-induced creasing and cratering insta-bilities in a constrained DE layer [22], giant deformation andshape bifurcation in DE balloons [23,24], and instabilities in lay-ered soft dielectrics [25,26].

Because large deformation in a DE requires relatively largeelectric field, thin DE films are frequently adopted in most

applications. As a consequence, multiple wrinkles can be oftenobserved in the experiments in DE structures when they are sub-jected to electromechanical loading. Wrinkle formation has beenoften regarded as one of the failure mechanisms in DE devices[27–32]. In the last few years, it has been shown that in certainconditions, wrinkle formation in DE structures can be reversibleand leads to no damage of the material [27,33–38]. Moreover, thewrinkles in DE structures may provide additional functions whichcannot be easily obtained otherwise.

To predict the critical conditions of wrinkling in a DE filmunder different electromechanical loading, the film is usuallyassumed to be under plane stress condition. Because DE films arethin and can bear little compression, the external electromechani-cal loading which leads to the loss of tension at any point of theDE film is commonly adopted to represent the critical conditionfor the wrinkle formation [39–44] or failure of the structure. Rea-sonable agreements between the predictions and experimentalmeasurements of the conditions for the onset of wrinkles in DEmembranes have been obtained in several different studies[33,41,45,46].

Nevertheless, though the loss of tension can be used to estimatethe critical conditions for the wrinkle formation in a DE film,analyses based on plane stress assumption cannot provide addi-tional information about wrinkle morphology such as wavelengthof wrinkles. Considering the recent interests in harnessing insta-bilities of soft active structures to achieve novel functions, a theo-retical method which can accurately predict the critical conditionsof wrinkling and the morphology of wrinkles in a DE film sub-jected to electromechanical loading is highly desired.

Most current electromechanical constitutive models of DE filmsare based on membrane assumption. Specifically, stress in a DEfilm is assumed to be uniform along its thickness direction, and itsbending stiffness is completely ignored. Such assumptions greatlysimplify the way of computing stress/stretch field and electricalfield in DE structures under different loading conditions. The

1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in the

JOURNAL OF APPLIED MECHANICS. Manuscript received October 17, 2017; finalmanuscript received November 4, 2017; published online November 22, 2017.Editor: Yonggang Huang.

Journal of Applied Mechanics JANUARY 2018, Vol. 85 / 011007-1Copyright VC 2018 by ASME

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computational results based on the membrane assumption oftenagree well with experiments. However, the onset of wrinkles in afilm is a result of competition between its bending energy andstretching energy. Therefore, taking account of the bendingenergy of a DE film becomes critical in analyzing the wrinkle for-mation and its morphology.

A general formulation of plate theory for DE films subjected toelectromechanical loading is extremely challenging, which mainlystems from large deformation of the material, coupling betweenmechanics and electrical field, and complex geometries. Todevelop a model for predicting the critical condition of wrinklinginstability, instead of establishing a general plate theory for DEthin films, we analyze the problem with a small three-dimensionaldisplacement field associated with bending deformation super-posed onto a finite two-dimensional deformation state of a DEfilm. Because the additional three-dimensional displacement fieldis small and we only focus on the bending of the DE film, we pre-scribe the form of additional displacement field by followingKirchhoff assumptions of linear plate theory.

Wrinkling in an annular membrane, caused by surface tension[47], inhomogeneous growth [48,49], and mechanical force[50,51], has been intensively studied in the past. As a demonstra-tion of the application of our model, in this paper, we analyze thewrinkle formation in an annular DE film with internal constraintas shown in Fig. 1.

2 Experiment

In the experiment, we use laser cutter to cut a circular DE film(VHB 4905 purchased from 3M company (Maplewood, MN)) and

paint carbon grease on both surfaces of the DE film as compliantelectrode. To avoid electrical arcing across free edge of the film,we intentionally leave an annular gap near the edge of the filmunpainted as shown in Fig. 1. We next glue a circular acrylateplate with different diameters on the center of the DE film to con-strain its deformation. The adhesion between VHB film and theacrylate plate is strong enough, and no debonding and slidinghave been observed in our experiments. Finally, we apply an elec-trical voltage across the thickness of the annular DE film withgradually increasing the magnitude. When the voltage is highenough, wrinkles can be clearly observed in the film. Dependingon the size of the central rigid plate, wrinkles with different wave-length may appear as shown in Fig. 1. If the voltage in the experi-ments is not too high (e.g., below 10 kV), the formation andannihilation of wrinkles can repeat many times with the voltagebeing turned on and off.

3 Model and Formulation

3.1 A Constrained Annular DE Film Subjected to anElectrical Voltage. Figure 2 sketches an annular DE film withclamped inner boundary. In the undeformed state, the thickness ofthe film is denoted by H. The inner radius and outer radius of theannular plate are denoted by A and B, respectively. In the actuatedstate, a voltage U is applied between the two surfaces of the film.When the voltage is small, the DE film deforms axisymmetricallyand maintains flat configuration (Fig. 2(b)). The hoop stress in theDE film is compressive, while the radial stress is always tensile.With increasing the voltage, the compressive hoop stressincreases, and finally, wrinkles form in the DE film as shown in

Fig. 1 Experimental photos of voltage-induced wrinkles in a constrained annular DE filmwith different ratios between the inner radius A and outer radius B: (a) A/B 5 0.6, (b) A/B 5 0.7,and (c) A/B 5 0.8. The wavenumber of wrinkling mode increases with increasing radius ratio.

Fig. 2 Schematics of an annular DE film constrained by an inner circular rigid plate. The innerradius and outer radius of the annular DE film without deformation are denoted by A and B,respectively. In the experiment, an electrical voltage U is applied across the thickness of theDE film. Three different states of the annular DE film are sketched: (a) undeformed state, (b)deformed state without wrinkles, and (c) wrinkled state.

011007-2 / Vol. 85, JANUARY 2018 Transactions of the ASME


Figs. 1 and 2(c). In the following, we first analyze the deformationof DE film without wrinkles.

In the reference state, we label each material particle by itsradial coordinate R in the interval ðA;BÞ. In a deformed state, thematerial particle R takes the position of coordinate rðRÞ. The func-tion rðRÞ describes the deformed state of the DE film. The radialstretch and the hoop stretch can be calculated as

kr ¼dr

dR(1)

kh ¼r

R(2)

The DE is assumed to be incompressible, so that the stretch inthickness direction is kz ¼ 1=krkh.

The electrical field E in the DE film is along the normal direc-tion of the film and relates to the voltage U as

E ¼ Uh

(3)

where h is the thickness of the film in the deformed state, soh ¼ Hkz.

The equation of force balance in the annular DE film is givenby

d

dR

rrrkr

� �� 1

R

rhhkh� rrr

kr

� �¼ 0 (4)

In this paper, we adopt ideal dielectric elastomer model, whichassumes electrical permittivity e is a constant and unaffected bythe deformation and electrical field [16]. Meanwhile, neo-Hookean model is adopted here to describe the hyperelasticity ofthe material. As a consequence, we have the following constitu-tive equation:

rrr ¼ lðk2r � k�2r k�2h Þ � eE2 (5)

rhh ¼ lðk2h � k�2r k

�2h Þ � eE2 (6)

where the first term in Eqs. (5) and (6) originates from the elastic-ity of the elastomer, and the second term is known as Maxwellstress.

From the equilibrium equation (4), we can obtain the first deriv-ative of krðRÞ

dkrdR¼

rhhkh� rrr

krþ kh � krð Þ

@

@kh

rrrkr

� �

R@

@kr

rrrkr

� � (7)

First derivative of khðRÞ in Eq. (2) gives that

dkhdR¼ kr � kh

R(8)

The ordinary differential equations (ODEs) of Eqs. (7) and (8)can be solved numerically with the following two boundaryconditions:

rðAÞ ¼ A (9)

rrrðBÞ ¼ 0 (10)

3.2 Linear Stability Analysis of Voltage-Induced Wrinklesin a DE Film. As discussed earlier, voltage-induced compressivehoop stress in the annular DE film may result in wrinkling. To

investigate the formation and morphology of wrinkles, we firstformulate a linear plate theory for a DE film under electromechan-ical loading.

To derive the governing equations for the deflection of a DEfilm, we first need to obtain the relationship between the stressand deflection of the film subjected to electromechanicalloadings. For the purpose of clarity and simplicity, we firstderive the stress–deflection relationship of a DE film in a Carte-sian coordinate, and then, we transfer the results to a polarcoordinate.

We consider an element of a DE film subjected to a homogene-ous electrical field along the thickness direction. The deformationgradient of the film from initial undeformed state denoted by Bi toa flat state denoted by B0 is given by

F0 ¼kx 0 0

0 ky 0

0 0 kz

2664

3775 (11)

where kx, ky, and kz are the principle stretches in three orthogonaldirections x, y, and z, where x and y are two perpendicular direc-tions in the plane of the film, and z is the direction perpendicularto the film.

Next, we assume the displacement field associated with wrin-kling deformation can be represented by u, v, and w in the threeorthogonal directions x, y, and z. The deformation gradient fromthe predeformed state B0 to wrinkled state B1 can be given by

F1 ¼

1þ @u@x

@u

@y

@u

@z

@v

@x1þ @v

@y

@v

@z

@w

@x

@w

@y1þ @w

@z

266666664

377777775

(12)

Therefore, the deformation gradient from initial state Bi towrinkled state B1 can be calculated as

F ¼ F1 � F0 ¼

kx 1þ@u

@x

� �ky@u

@ykz@u

@z

kx@v

@xky 1þ

@v

@y

� �kz@v

@z

kx@w

@xky@w

@ykz 1þ

@w

@z

� �

2666666664

3777777775(13)

The corresponding left Cauchy-Green strain tensor is

B ¼ F � FT

¼

k2x 1þ 2@u

@x

� �k2x@v

@xþ k2y

@u

@yk2x@w

@xþ k2z

@u

@z

k2x@v

@xþ k2y

@u

@yk2y 1þ 2

@v

@y

� �k2y@w

@yþ k2z

@v

@z

k2x@w

@xþ k2z

@u

@zk2y@w

@yþ k2z

@v

@zk2z 1þ 2

@w

@z

� �

26666666664

37777777775

(14)

Based on ideal dielectric elastomer model, Cauchy stress in aDE film can be decomposed into elastic part and Maxwell stress,namely,

r ¼ rela þ rm (15)

Journal of Applied Mechanics JANUARY 2018, Vol. 85 / 011007-3


where Maxwell stress is a tensor and can be represented by

rm ¼

� 12eE2 0 0

0 � 12eE2 0

0 01

2eE2

2666664

3777775 (16)

and elastic stress rela is given by

rela ¼ F@Wstretch@F

� pI (17)

where Wstretch is the strain energy density of the elastomer, and p is the hydrostatic pressure for the incompressibility of the elastomer.We adopt neo-Hookean model in the current paper, so Eq. (17) can be written explicitly as

rela ¼ lB� pI (18)

Substituting Eqs. (16) and (18) into Eq. (15) leads to

r ¼

�p� eE2

2þ lk2x 1þ 2

@u

@x

� �l k2x

@v

@xþ k2y

@u

@y

� �l k2x

@w

@xþ k2z

@u

@z

� �

l k2x@v

@xþ k2y

@u

@y

� ��p� eE

2

2þ lk2y 1þ 2

@v

@y

� �l k2y

@w

@yþ k2z

@v

@z

� �

l k2x@w

@xþ k2z

@u

@z

� �l k2y

@w

@yþ k2z

@v

@z

� ��pþ eE

2

2þ lk2z 1þ 2

@w

@z

� �

2666666664

3777777775

(19)

We can further rewrite Eq. (19) as

@u

@x¼ 2rxx þ 2pþ eE

2

4lk2x� 1

2(20a)

@v

@y¼ 2ryy þ 2pþ eE

2

4lk2y� 1

2(20b)

@w

@z¼ 2rzz þ 2p� eE

2

4lk2z� 1

2(20c)

k2x@v

@xþ k2y

@u

@y¼ rxy

l(20d)

k2x@w

@xþ k2z

@u

@z¼ rxz

l(20e)

k2y@w

@yþ k2z

@v

@z¼ ryz

l(20f )

The incompressibility condition of the DE film requires that

@u

@xþ @v@yþ @w@z¼ 0 (21)

A combination of Eqs. (20a)–(20c) and Eq. (21) gives that

p ¼3lk2xk

2y � k2yrxx � k2xryy � k4xk4yrzz � k4xk4yeE2

k2x þ k2y þ k4xk4y� eE

2

2(22)

Substituting Eq. (22) into Eqs. (20a)–(20f), we can furtherobtain

@u

@x¼

3lk2xk2y � k

2yrxx � k

2xryy � k

4xk

4yrzz � k

4xk

4yeE

2

2lk2x k2x þ k2y þ k4xk4y

� � � 12

(23a)

@v

@y¼

3lk2xk2y � k2xryy � k2yrxx � k4xk4yrzz � k4xk4yeE2

2lk2y k2x þ k2y þ k4xk4y

� � � 12

(23b)

@w

@z¼

3lk4xk4y � k2xk4yrxx � k4xk2yryy � k2xk2y k2x þ k2y

� �rzz � k6xk6yeE2

2l k2x þ k2y þ k

4xk

4y

� �

�k2xk

2yeE

2

2l� 1

2(23c)

k2x@v

@xþ k2y

@u

@y¼ rxy

l(23d)

k2x@w

@xþ k2z

@u

@z¼ rxz

l(23e)

k2y@w

@yþ k2z

@v

@z¼ ryz

l(23f )

A combination of Eqs. (23a)–(23f) and force balance equationsgives a complete set of governing equations for both additionaldisplacement field and stress field in a DE. However, without fur-ther simplifications, these equations are very difficult to solve. As



discussed previously, in this paper, we only focus on additionalbending deformation in a DE film, so we will follow Kirchhoff’shypotheses [52] to further simplify the problem. Kirchhoff’shypotheses developed for linear plate theory state that the straightlines, initially normal to the middle plane of a plate before bend-ing, remain straight and normal to the middle surface during bend-ing, and the length of such plate elements does not change.Mathematically, we have the following equations:

@w

@z¼ 0 (24a)

@w

@xþ @u@z¼ 0 (24b)

@w

@yþ @v@z¼ 0 (24c)

The three new equations ((24a)–(24c)) originating from Kirchh-off’s hypotheses make the problem overdetermined. Therefore, itis necessary to drop three equations. Following conventional lin-ear plate theory [52], Eqs. (23c), (23e), and (23f) are discarded.Moreover, the stress normal to the middle plane of the plate isassumed to be negligible compared with other stress components,i.e., rzz ¼ 0. Finally, the nonzero stress components in the DEfilm can be expressed as

rxxryyrxy

8><>:

9>=>; ¼

l k2x � k�2x k�2y� �

� eE2

l k2y � k�2x k

�2y

� �� eE2

0

8>>><>>>:

9>>>=>>>;

þ

2l k�2x k�2y þ k2x

� � @u@xþ 2lk�2x k�2y

@v

@y

2l k�2x k�2y þ k

2y

� � @v@yþ 2lk�2x k

�2y

@u

@x

l k2x@v

@xþ k2y

@u

@y

� �

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(25)

Integrating Eqs. (24a)–(24c), we can obtain the displacement ofa material point in the DE film with distance z from the middleplane

uzxuzy

� �¼

�z @w@x

�z @w@y

8>>><>>>:

9>>>=>>>;

(26)

where w ¼ wðx; yÞ is independent on z and denotes the deflectionof the DE film.

Substituting Eq. (26) into Eq. (25), we can further obtain thestress components of a material point in the DE film with distancez from the middle plane

rzxxrzyyrzxy

8>><>>:

9>>=>>; ¼

l k2x � k�2x k�2y� �

� eE2

l k2y � k�2x k

�2y

� �� eE2

0

8>>><>>>:

9>>>=>>>;

� zC11 C12 0

C21 C22 0

0 0 2C66

264

375

@2w

@x2

@2w

@y2

@2w

@x@y

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(27)

where C11 ¼ 2lðk�2x k�2y þ k2xÞ, C22 ¼ 2lðk�2x k�2y þ k2yÞ, C12 ¼C21 ¼ 2lk�2x k

�2y , and 2C66 ¼ lðk

2x þ k

2yÞ.

In our model, we assume that the thickness of the DE film issmall and the electrical field does not vary along the thicknessdirection. Then, the bending moments Mxx, Myy, and twistingmoment Mxy can be integrated as

Mxx

Myy

Mxy

8><>:

9>=>; ¼

ðh=2�h=2

rzxxrzyyrzxy

8>><>>:

9>>=>>;zdz

¼ � H3

12k3xk3y

C11 C12 0

C21 C22 0

0 0 2C66

264

375

@2w

@x2

@2w

@y2

@2w

@x@y

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

(28)

In a polar coordinate, the bending moments and twisting momentcan be similarly written as

Mrr

Mhh

Mrh

8><>:

9>=>; ¼ �

H3

12k3r k3h

Crr Crh 0

Chr Chh 0

0 0 2Css

264

375

@2w

@r2

1

r

@w

@rþ 1

r2@2w

@h2

1

r

@2w

@r@h� 1

r2@w

@h

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;(29)

where Crr ¼ 2lðk�2r k�2h þ k2r Þ, Chh ¼ 2lðk�2r k�2h þ k2hÞ, Crh ¼Chr ¼ 2lk�2r k

�2h , and 2Css ¼ lðk

2r þ k

2hÞ. Therefore, in a polar

coordinate, we have

Mrr ¼ � Drr@2w

@r2þ Drh

1

r

@w

@rþ 1

r2@2w

@h2

� �� (30a)

Mhh ¼ � Dhh1

r

@w

@rþ 1

r2@2w

@h2

� �þ Dhr

@2w

@r2

� (30b)

Mrh ¼ Mhr ¼ �2Dss1

r

@2w

@r@h� 1

r2@w

@h

� �(30c)

where Drr ¼ lH3ðk�5r k�5h þ k

�1r k

�3h Þ=6, Dhh ¼ lH3ðk

�5r k

�5h

þk�3r k�1h Þ=6, Drh ¼ Dhr ¼ lH3k�5r k�5h =6, and Dss ¼ lH3ðk�1r k�3hþk�3r k�1h Þ=24.

In the normal direction, force balance condition requires that

@2Mrr@r2

þ 2r

@Mrr@rþ 2

r

@2Mrh@r@h

þ 2r2@Mrh@h� 1

r

@Mhh@rþ 1

r2@2Mhh

@h2

þNrr@2w

@r2þ 2Nrh

1

r

@2w

@r@h� 1

r2@w

@h

� �þNhh

1

r

@w

@rþ 1

r2@2w

@h2

� �¼ 0

(31)

where Nrr ¼ rrrh, Nhh ¼ rhhh, and Nrh ¼ rrhh are the membraneforces.

On the inner circular boundary, the DE film is clamped,namely,

w ¼ 0 (32)

@w

@r¼ 0 (33)



On the outer circular boundary, the bending moment is zeroand vertical shear force is zero, namely,

Mrr ¼ 0 (34)

@Mrr@rþ 2

r

@Mrh@hþMrr �Mhh

r¼ 0 (35)

A combination of Eqs. (30a)–(30c) and (31) with boundaryconditions (Eqs. (32)–(35)) sets an eigenvalue problem. Weassume the deflection of the DE film follows:

w ¼ f ð�rÞcos ðkhÞ (36)

where �r ¼ r=B, f ð�rÞ is a single variable function, and k is wave-number of wrinkle.

Inserting Eq. (36) into Eqs. (30) and (31), we get the followinghomogeneous ODE:

d4f

d�r4þ 2

�r

d3f

d�r3

� ��Drrþ 2

d3f

d�r3þ 1

�r

d2f

d�r2

� �d �Drrd�rþ d

2f

d�r2d2 �Drrd�r2

þ 1�r2

1

�r

df

d�r� d

2f

d�r2þ k

2� 2�r2

k2f

� ��Dhh�

1

�r2df

d�r� k

2f

�r

� �d �Dhh

d�r

þ 2k2

�r2� f

�r2þ 1

�r

df

d�r� d

2f

d�r2

� ��Drhþ

2k2

�r2f

�r� df

d�r

� �d �Drhd�r

þ 1�r�k

2f

�rþ df

d�r

� �d2 �Drhd�r2

þ 4k2

�r2� f

�r2þ 1

�r

df

d�r� d

2f

d�r2

� ��Dss

þ 4k2

�r2f

�r� df

d�r

� �d �Dssd�r� 12

�H2krkh

�rrrd2f

d�r2þ �rhh

�r

df

d�r� k

2f

�r

� �� ¼ 0

(37)

The corresponding homogeneous boundary conditions are

f j�r¼AB ¼ 0 (38a)

Fig. 3 Distributions of radial stretch and hoop stretch in a constrained annular DE film forseveral different voltages and ratios between its inner radius A and outer radius B: (a) and (b)A/B 5 0.1, (c) and (d) A/B 5 0.6, and (e) and (f) A/B 5 0.9. The highest voltages in the figurescorrespond to the critical voltage of inducing pull-in instability in the DE film.



df

d�r

�r¼AB¼ 0 (38b)

�Drrd2f

d�r2þ �Drh

1

�r

df

d�r� k

2

�r2f

� ��

�r¼bB ¼ 0 (38c)

1

�r

d2f

d�r2þ d

3f

d�r3

� ��Drr þ

1

�r2f

�r� df

d�r

� ��Dhh þ

k2

�r2f

�r� df

d�r

� ��Drh

"

þ4k2

�r2f

�r� df

d�r

� ��Dss þ �

k2f

�r2þ f

�r

df

d�r

� �d �Drhd�rþ d

2f

d�r2d �Drrd�r

�r¼ bB ¼ 0(38d)

In Eqs. (37) and (38), we define the following dimensionlessquantities: dimensionless thickness: �H ¼ H=B; dimensionless mem-brane stresses: �rrr ¼ rrr=l and �rhh ¼ rhh=l; dimensionless bend-ing/twisting stiffness: �Drr ¼ 2ðk�5r k

�5h þ k

�1r k

�3h Þ; �Dhh ¼ 2ðk

�5r k

�5h

þk�3r k�1h Þ; �Drh ¼ 2k�5r k�5h , and �Dss ¼ ðk�1r k�3h þ k�3r k�1h Þ=2: Inthe equations above, dimensionless thickness �H and the ratiobetween inner radius and outer radius of the annular DE film A=Bare the only two system parameters which can be varied in experi-ments. The membrane stresses in the radial and hoop directions: �rrrand �rhh, and the bending/twisting stiffness both depend on the elec-trical field E or voltage U, which can be regarded as loading parame-ters for the system. For a given set of parameters: �H and A=B, wecan solve the eigenvalue problem in Eq. (37) with boundary condi-tions (38) numerically by using the function bvp4c in MATLAB. Thecharacteristic equation determines the critical condition, namely,voltage, for the onset of wrinkles, and the associated eigenvectorsgive the mode of wrinkling.

Fig. 5 Distributions of radial stress and hoop stress in the DE film without wrinkles, for sev-eral voltages and radius ratios: (a) and (b) A/B 5 0.1, (c) and (d) A/B 5 0.6, and (e) and (f)A/B 5 0.9. The radial stress in the film is tensile, while the hoop stress is compressive. Thecompressive hoop stress may wrinkle the DE film.

Fig. 4 Dependence of the critical voltage of inducing pull-ininstability of the annular DE film on the ratio between its innerradius and outer radius. When the ratio between inner radiusand outer radius approaches zero, the annular film becomes afree-standing film; when the ratio approaches one, the deforma-tion state of the film is pure-shear.



4 Results and Discussion

Using the shooting method, we solve the ODEs of Eqs. (7) and(8) associated with the boundary conditions (9) and (10) for a con-strained annular DE film without wrinkle formation. Figure 3plots the distributions of radial stretch and hoop stretch in theannular DE film, for several different voltages and ratios betweeninner and outer radius of the film. With increasing the voltage,both radial stretch and hoop stretch increase.

As shown in Fig. 3, when the voltage is larger than a criticalvalue, no equilibrium solution can be found, which corresponds topull-in instability of the DE film. Pull-in instability is one of themost important electromechanical failure modes in DE structures[19–21]. The critical voltage for the pull-in instability increaseswith the increase in the ratio A/B as plotted in Fig. 4. When theratio A/B approaches zero, the annular DE film behaves like afree-standing film. It has been shown in previous studies that thecritical voltage for the pull-in instability of a free-standing neo-

Hookean DE film is Upull�in=ðHffiffiffiffiffiffiffil=e

pÞ ¼ 0:69 [19]. When the

ratio A/B approaches one, the deformation state in the annular DEfilm is close to pure-shear. It can also be easily shown that for aDE film with voltage-induced pure-shear deformation, the normal-

ized voltage for pull-in instability is Upull-in=ðHffiffiffiffiffiffiffil=e

pÞ ¼ 1. Both

two limiting scenarios discussed earlier have been plotted inFig. 4.

Figure 5 plots the distribution of radial stress and hoop stress inthe annular DE film without wrinkle formation for several differ-ent radius ratios and applied voltages. The results show that theradial stress in the DE film is tensile, while the hoop stress in thefilm is compressive. For a given radius ratio, both radial stress andhoop stress in the DE film increase with increasing the voltage. Inaddition, the radial stress in the DE film monotonically decreasesfrom the inner boundary to the outer boundary as shown inFigs. 5(a), 5(c), and 5(e). The distribution of the hoop stress is lit-tle bit more intricate. For small radius ratio, the hoop stress in theDE film monotonically increases from the inner boundary to outer

boundary. However, for big radius ratio, the hoop stress in the DEfilm may monotonically decrease from the inner boundary to theouter boundary when the voltage is large as shown in Fig. 5(f).

When the voltage is high enough, compressive hoop stress caninduce wrinkles in the annular DE film as shown in our experi-ments (Fig. 1). By solving the eigenvalue problem formulated inSec. 3, we can calculate the critical voltage for the onset of wrin-kles with different wavelengths in the DE film for different valuesof A/B. Our calculation results are plotted in Fig. 6. For a givenvalue of A/B and wavenumber k of wrinkles, the voltage for theonset of wrinkles decreases with decreasing the thickness ofthe film, namely, H/B. The results suggest that lower voltage isneeded to trigger wrinkle formation in thinner DE films. Fordifferent values of A/B, the wrinkling mode which needs the low-est critical voltage is also different. For instance, in Fig. 6(a), forA/B¼ 0.1, the wrinkling mode with k¼ 2 needs the lowestvoltage; in Fig. 6(b), for A/B¼ 0.3, the critical mode is k¼ 3.Figures 6(c) and 6(d) show that the critical modes are k¼ 4 andk¼ 6 or 7 for A/B¼ 0.5 and 0.7, respectively. In Figs. 6(a)–6(d),we also plot the voltage for inducing pull-in instability in the DEfilm. For a thick DE film, the critical voltage for wrinkling may beeven larger than the critical voltage for pull-in instability. There-fore, for thick DE films, the wrinkles form after pull-in instability,which can be consequently regarded as an indication of materialfailure.

In Fig. 7, we plot the dependence of critical wrinkling mode onthe ratio between the inner radius and outer radius of the DE filmfor two different film thicknesses: H/B¼ 0.02 and 0.005, respec-tively. The result shows that the wavenumber of the critical wrin-kling mode increases with increasing the value of A/B. The trendagrees well with the wrinkling instability observed in a con-strained annular film induced by differential swelling or plasticdeformation [48,50]. The results also show that for a given radiusratio, the DE film with different thickness may also have differentcritical wrinkling modes. For the ratio: A/B¼ 0.75, the criticalwrinkling mode of the DE film with thickness H/B¼ 0.02 is k¼ 7,

Fig. 6 Dependence of critical voltage for inducing wrinkling instability on the thickness ofthe DE film, for several radius ratios and wrinkling modes: (a) A/B 5 0.1, (b) A/B 5 0.3, (c)A/B 5 0.5, and (d) A/B 5 0.7. For larger film thickness, the voltage for inducing pull-in instabil-ity is smaller than the voltage for inducing wrinkling in the film.



while k¼ 8 for the film with H/B¼ 0.005. However, it is notedthat when A/B is close to one, thin plate theory adopted in this arti-cle will not be valid anymore.

In the experiment, it is usually challenging to precisely measurethe critical voltage for the onset of wrinkles in a DE film. How-ever, the wavenumber of wrinkles in the DE film can be easilymeasured as shown in Fig. 1. The experimental results of thewavenumber of wrinkles in annular DE films for three differentvalues of A/B are plotted together with our predictions as shownin Fig. 7. The wavenumber of wrinkles in the experiments is con-sistently larger than the predictions. This difference can be under-stood as follows: as described in Sec. 2 and also shown in Fig. 1,to avoid electrical arcing, a gap near the free edge of the annularDE film is intentionally not coated with electrode. Due to theadditional constraint from the gap which is not considered inour model, stretching free energy penalty (compared to bendingenergy penalty) in the film is actually larger than the prediction.Therefore, in the experiments, wavenumber of wrinkles in thefilm is larger than predictions to reduce stretching energy penalty.

Based on our calculation, we construct a phase diagram for aconstrained DE annular film with the thickness H/B¼ 0.02 asshown in Fig. 8. Depending on the radius ratio and magnitude ofvoltage, the DE film may stay in one of the three phases: flat,wrinkling, and pull-in instability. It can be seen from Fig. 8 thatthe critical voltage for triggering wrinkling in the DE film firstdecreases and then increases with increasing the ratio betweeninner radius and outer radius. When A/B¼ 0.4, the required volt-age to induce wrinkles in the film is the lowest.

5 Conclusions

In this paper, we formulate a linear plate theory for a DE filmunder electromechanical loading with small three-dimensionaldeformation field superposed onto a finite two-dimensionaldeformation. Based on the theory, we investigate the formationand morphology of wrinkles in an annular DE film subjected toa voltage and clamped on its inner radius. The theoretical pre-dictions of wrinkling in the DE film agree well with our experi-mental observations. Furthermore, we show that for certainranges of ratio between inner radius and outer radius of the DEfilm, the critical voltage of inducing pull-in instability of thesystem can be lower than the critical voltage of inducing wrin-kles in the film. As a result, we construct a phase diagram withthree different regions depending on the magnitude of voltageand the ratio between inner radius and outer radius of an annularDE film: (1) the region corresponding to stable and flat configu-ration of the film; (2) wrinkling region; and (3) the region forpull-in instability. The results obtained in this paper and themethodology developed here will be useful for the future designof DE structures.

Funding Data

� Anhui Provincial Universities Natural Science Research Pro-ject (Grant Nos. KJ2015A046 and KJ2016A147).

� National Natural Science Foundation of China (Grant Nos.11402001, 11472005, and 51408005).

� National Science Foundation (Grant No. CMMI-1538137).

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Fig. 8 A phase diagram of a constrained annular DE film withthickness H/B 5 0.02. Depending on the ratio between the innerradius and outer radius of the film and the magnitude of appliedvoltage, the film may stay in a flat and stable phase, wrinklingphase, or pull-in instability phase. The boundary between theflat phase and wrinkling phase of the DE film is given by thedependence of critical voltage for wrinkling on its radius ratioA/B. For the annular DE film with the radius ratio: A/B 5 0.4, therequired voltage to induce wrinkles in the film is the lowest.

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voltage-induced wrinkling in a constrained annular...

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