Extreme Mechanics Letters 13 (2017) 116–125

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Extreme Mechanics Letters

journal homepage: www.elsevier.com/locate/eml

Role of nonlinear elasticity in mechanical impedance tuning ofannular dielectric elastomer membranesA. Cugnoa,b, S. Palumboa, L. Deseri a,d,e,f,g, M. Fraldi b,c, C. Majidi f,h,⇤aCivil, Environmental & Mechanical Engineering, University of Trento, 38123 Trento, Italy

bStructures for Engineering and Architecture, University of Napoli Federico II, 80125 Napoli, Italy

cInterdisciplinary Research Center for Biomaterials, University of Napoli Federico II, 80125 Napoli, Italy

dMechanical & Materials Science, University of Pittsburgh, Pittsburgh, PA 15261, USA

eMechanical, Aerospace & Civil Engineering, Brunel University London, Uxbridge, UB8 3PH, UK

fMechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

gNanomedicine, The Methodist Hospital Research Institute, MS B-490, Houston, TX 77030, USA

hCivil & Environmental Engineering, Carnegie Mellon University, Pittsburgh, 15213 PA, USA

a r t i c l e i n f o

Article history:

Received 18 November 2016Available online 6 March 2017

Keywords:

Dielectric elastomerMembrane theoryNonlinear elasticityAerospace structures

a b s t r a c t

We use finite elasticity to examine the behavior of a lightweight mechanism for rapid, reversible, andlow-power control of mechanical impedance. The device is composed of a central shaft suspended byan annular membrane of prestretched dielectric elastomer (DE), which is coated on both sides with aconductive film. Applying an electrical field across the thickness of the membrane, attractive Coulombicforces (so-called ‘‘Maxwell stresses’’) are induced that (i) squeeze the annulus, (ii) relieve the membranestress, and (iii) reduce the mechanical resistance of the elastomer to out-of-plane deflection. Thisvariable stiffness architecture was previously proposed by researchers who performed an experimentalimplementation and demonstrated a 10⇥ change in stiffness. In this manuscript, we generalize thisapproach to applications in aerospace and robotics by presenting a complete theoretical analysis thatestablishes a relationship between mechanical impedance, applied electrical field, device geometry, andthe constitutive properties of the dielectric elastomer. In particular, we find that the stiffness reductionunder applied voltage is non-linear. Such decay ismost significantwhen theMaxwell stress is comparableto the membrane prestress. For this reason, both the prestretch level and the hyperelastic properties ofthe DE membrane have a critical influence on the impedance response.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

In current aerospace applications, activemechanical impedancecontrol typically requires clutches, brakes, transmissions, andother hardware that depend on motors and hydraulics. Whileadequate for large conventional systems, these variable stiffnessmechanisms may be challenging to implement in smaller, col-lapsible, or structurally reconfigurable systems that require con-tinuous stiffness change or complex triggering. For these emerg-ing applications, rigidity-tuning hardware should be replacedwiththin, lightweight, elastic materials and composites that are ca-pable of modulating their stiffness through direct electrical op-eration. Potential approaches include electrostatic activation of

⇤ Corresponding author.E-mail address: cmajidi@andrew.cmu.edu (C. Majidi).

electro-active polymers (EAPs) and electrical (Joule) heating ofthermally-responsive materials. The latter has been demonstratedwith phase-change of low-melting point metal alloys [1,2], glasstransition of shape memory polymer [1,3,4], and softening of con-ductive thermoplastic elastomers [5].While promising for applica-tions that involve only intermittent operation, thermally-activatedrigidity tuning is typically too slow (⇠0.01–1 Hz) and energeticallycostly (⇠0.1–10 W) for dynamical systems that require high cy-cling rates. Instead, recent efforts in aerospace and space researchhave focused on EAPs that couple shape and stiffness with rapidand low-power electrostatic activation [6–8].

Here, we use finite elasticity to examine a particularlypromising class of EAP architectures that can be used for robustmechanical impedance control. Referring to Fig. 1, the structureis composed of a central shaft (or ‘‘shuttle’’) that is suspended byan annular membrane of prestretched dielectric elastomer (DE).

http://dx.doi.org/10.1016/j.eml.2017.03.0012352-4316/© 2017 Elsevier Ltd. All rights reserved.

A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125 117

Both sides of the DE film are coated with a conductive fluid(e.g. liquid metal alloy), grease (silicone oil mixed with carbonpowder), or elastomer (silicone rubbermixedwith carbon powder)that remains conductive as the DE film is stretched. Connectingthe film to a power supply and applying a voltage drop � acrossthe thickness induces a Coulombic attraction, sometimes referredto as ‘‘Maxwell stress’’, that squeezes the membrane and relievesits prestretch. This stress reduction results in the lowering of themechanical resistance of the shuttle to deformations in the out-of-plane direction.

The idea of a variable-stiffness device with EAPs was firstintroduced by Pelrine, Kornbluh, et al. [9,10]. They performedexperimental tests to show the effect of the voltage on thestiffness of a framed prestretched planar acrylic film, obtaininga reduction of the stiffness up to 10⇥ under an applied voltagedrop of 6 kV. Variable stiffness with DE membranes has alsobeen proposed by Carpi et al. [11], who explore applications inhand rehabilitation. The design examined in this study (Fig. 1)was first proposed for voltage-controlled stiffness tuning byDastoor & Cutkosky [12]. Their implementation showed a 7–10⇥reduction in stiffness and exhibited electromechanical couplingthat was consistent with theoretical predictions based on alinearized model. While inadequate for modeling large strains orelectromechanical instabilities [13], the linear theory enabled theauthors to identify the important role of elastomer stiffness andprestretch in the coupling between applied voltage andmechanicalimpedance. Building on this work, Orita and Cutkosky [14] testeda multi-layer diaphragm device and performed a nonlinear FEAstudywith the aim of identifying sources of failure and approachesfor failure mitigation. The design in Fig. 1 has also been employedfor replicating human pulse signal by means of a model-basedrobust control [15]. For applications in acoustics, Lu et al. [16] usevoltage-controlled stiffness tuning to alter the resonance peaks ofa membrane-based silencer. Applying different external voltagesenabled a maximum resonance frequency shift of 59.5 Hz andallowed the silencer to adjust its absorption of target noisewithoutany addition of mechanical part. Related analysis by [17] showsthat the sound transmission band-gap of the membrane filter canbe tuned by adjusting the voltage applied to the membrane.

In addition to the analyses in [12] and [14], the mechanics ofthe annular DE membrane have been examined by He et al. [18].They treated the membrane as a Neo-Hookean solid and foundnumerical solutions to the governing balance equations, whichincluded the Maxwell stress associated with � . More recently,Melnikov and Ogden [19] performed a theoretically analysis ona related system composed of a tubular DE shell subject toa combination of (radial) electric field, internal air pressure,and axial mechanical loading. They used a nonlinear theory ofelectroelasticity that allowed them to predict a loss of tension

associated with variations in electrical field.In order to understand the underlying mechanics of the system

originally presented by Dastoor [12] and draw practical insightsfor future aerospace applications, we must perform a morecomplete electromechanical analysis. As described in the followingsection (Section 2), the theory incorporates the kinematics offinite deformation, the nonlinear constitutive properties of ahyperelastic solid, and the electrical enthalpy induced by voltage-controlled electrostatic field. Analysis is performed by treatingthe DE film as a hyperelastic membrane and using variationaltechniques to determine the electro-elastostatic configuration thatminimizes the potential energy. This approach is adapted fromthe methods presented in [20] for examining the axisymmetricdeformation of an annular membrane [21] and builds on the fieldequations for DE actuators and transducers previously presentedin [22–27]. Here, the dielectric is treated as either a Neo-Hookean or (two-parameter) Ogden solid, although the theory

can be generalized to any constitutive law for an incompressiblehyperelastic material.

In order to obtain an algebraic relationship between the stiff-ness and the applied voltage,weperforma linear incremental anal-ysis with respect to (w.r.t.) the prestretched configuration. Boththe numerical results of the nonlinearmodel and the linearized ap-proximation suggest an electromechanical coupling that is consis-tent with the previous experimental observations [12]. We reviewthese in Section 3 and discuss in Section 4 how the theory leadsto new insights about the influence of the prestretch and appliedvoltage on mechanical response.

2. The electro-mechanical model

As shown in Fig. 1, the device is composed of an outer rigidframe with inner circular opening of radius Re that is attachedto a cylindrical bar of radius Ri through a prestretched annularDE membrane. Prior to stretching, the membrane has an innerradius Ri, outer radius Ro = Re/�p and thickness H . The membranestress is controlled by applying a drop voltage� across compliantelectrodes coated to the surfaces of the dielectric film. Suchelectromechanical coupling arises from the Maxwell stress tensorthat is generated by the internal electrostatic field. As for anyelastic film in tension, this change in membrane stress will leadto a change in the mechanical resistance to out-of-plane (z-axis)deflection.

2.1. Kinematics

Referring to Fig. 1, the current (deformed) configuration⌦ canbe obtained from a deformation mapping � = �0 � �p that iscomposed as follows: (i) a uniform biaxial (pre)stretch xp = �p(X),which maps the body from the reference configuration ⌦0 to theintermediate one ⌦p, and (ii) a vertical deflection x = �0(xp)induced by the prescribed shaft displacement u, which maps thebody to ⌦ . For convenience, three different coordinate systems(COOS) and orthonormal bases are used to represent points in thebody:

1. Cylindrical COOS in ⌦0, which describes material points inthe reference stress-free configuration, spanned by the triad{Es, E✓ , E3},

2. Cylindrical COOS in⌦p with bases {es, e✓ , e3},3. Curvilinear COOS in⌦ with covariant bases {el, et , en} that are

tangent to the coordinate lines.

The deformation gradient of themapping� can bemultiplicativelydecomposed as F = F0Fp. Here, Fp = diag{�p, �p, ��2

p} is the

deformation gradient due to the prestretchwhile F0 corresponds tothe out-of-plane deflection. The transverse stretch ��2

pis obtained

by the incompressibility constraint det Fp = 1 and gives theintermediate membrane thickness hp = H/�2

p.

The deformation mapping x = �0(xp), adapted from themembrane theory previously presented in [20], has the followingform:

�0(xp) = xp � x3e3 + u0 + (x3 + q) en, (1)

where u0 = u0(s, ✓) is the displacement of a point withcoordinates (s, ✓) on the midplane (for which x3 = 0), en is theunit vector normal to the deformed surface, and q(s, ✓ , x3) is thenormal component of the displacement of points away from themidplane related to the deformed configuration. By definition, thefunction q (and its partial derivatives w.r.t. s and ✓ ) must vanish onthe midplane:

q(x3 = 0) = @q

@s

���x3=0

= 1s

@q

@✓

���x3=0

= 0. (2)

118 A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125

Fig. 1. Sketch of the diaphragmdevice inwhich an outer rigid framewith inner circular opening of radius Re is attached to a cylindrical bar of radius Ri through a prestretched(�p) annular DE membrane with initial thickness H and prestretched thickness hp = H/�2

p. In the membrane it is possible to induce a drop voltage � by means of two

compliant electrodes. (a) Reference configuration, top view above and lateral view in the bottom; (b) Intermediate configuration, top view above and lateral view in thebottom; (c) Lateral view of the current configuration characterized by a prescribed pulling out displacement u; (d) Sketch of the device in the current configuration.

The covariant bases at a point on a surface parallel to the midplanein the current configuration may be expressed as

e0s= es + @

@su0, e0

✓ = e✓ + 1s

@

@✓u0

and en = e0s⇥ e0

✓

ke0s⇥ e0

✓k.

(3)

Noting that, for the membrane in the prestretched configuration,hp becomes small w.r.t. the annular width, the deformationgradient F0 can be assumed homogeneous along the thickness andthus approximated as follows [20]:

F0 ' (F0)x3=0 = I � e3 ⌦ e3 + ru0 + �n3en ⌦ e3, (4)

where �n3 = (1 + @q/@x3) is hence independent from x3.Because we are interested in the response of the device to anormal displacement imposed on the internal frame (see Fig. 1),we consider the deformation as axisymmetric. The resultingdisplacement u0 is then a function of the coordinate s and nodisplacement occurs in the ✓ direction:

u0 = us(s)es + u3(s)e3 and q = q(s). (5)

Substituting the expressions for Fp and F0 into F, it is possible towrite the right Cauchy–Green deformation tensor [28] as follows:

C = FTF = diag

(

�2p

⇥(u0

s+ 1)2 + (u0

3)2⇤ ,

�2p(us + s)2

s2,�2n3

�4p

)

, (6)

where the principal stretches are simply the square roots ofthe diagonal elements. Applying the incompressibility constraintyields the following expression for �n3:

�n3 = s

(us + s)

q�u0s+ 1

�2 +�u

03�2 . (7)

2.2. Constitutive relations

To capture the nonlinear mechanics of the membrane, we treatit as an incompressible Ogden solid with two elastic coefficients.The constitutive relationship between the Cauchy stress and theprinciple stretches can be obtained from the strain energy densityfunction . For a selected pair of exponents, has the form [29]:

= 12µ1��2s+ �2✓ + �23 � 3

�+ 1

4µ2��4s+ �4✓ + �43 � 3

�, (8)

where µ1 and µ2 represent coefficients of elasticity. In order toconverge to theHooke’s law at small strains, the elastic coefficientsshould satisfy the identity µ = µ1 + 2µ2, where µ is theelastic shear modulus measured for infinitesimal deformations.The principal components of the Cauchy stress tensor are

�i = �i

✓@

@�i

◆+ p = µ1�

2i+ µ2�

4i+ p, (9)

where i 2 {s, ✓ , 3} correspond to the principal directions and p isthe Lagrangianmultiplier that represents the hydrostatic pressure.The unknown p is determined by balancing �3 with the Maxwellstress:

�M = �" �2

(�3H)2, (10)

where " is the electrical permittivity, H is the natural thickness,and �3 is the transverse stretch [25]. Alternatively, the influenceof applied voltage on mechanical deformation can be modeled byadding electrical enthalpy Ue to the potential energy functional.For a membrane with uniform electric field through its thickness,Ue can be computed by the following area integral in the current(deformed) placement:

Ue =Z

⌦

� d⌦. (11)

Here � = �"�2/2h2 is the electrical enthalpy density and can beadded to to obtain a Lagrangian density for the potential energyfunctional⇧ .

A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125 119

2.3. Variational analysis

The total potential energy written w.r.t. the intermediate(prestretched) configuration⌦p is expressed as:

⇧ =Z

Re

Ri

L ds, (12)

where the Lagrangian density L = 2⇡shp( + � ) corresponds tothe potential energy per unit width of each concentric ring formingthe annulus. The explicit expression for L is obtained from Eqs.(6), (7), (8) and (11), and is reported in detail in Appendix A.1.At static equilibrium, ⇧ must be minimized w.r.t. the functions{us, u

0s, u0

3}, which implies the following stationary conditions(i.e. Euler–Lagrange equations):

@L

@us

� @

@s

@L

@u0s

= 0 and@

@s

@L

@u03

= 0 (13)

(see Appendix A.1 for the complete expressions). The solutions tothe governing equations in (13) must satisfy the following set ofboundary conditions:

us(s)|s=Ri= 0 us(s)|s=Re

= 0 u3(s)|s=Ri= u

u3(s)|s=Re= 0, (14)

where u is the prescribed displacement of the inner shaft.Once the solution to (13) with boundary conditions (14) is ob-

tained, the out-of-plane stiffness K3 of the device can be evaluatedfor different values of the applied voltage. This is done by applyingCastigliano’s theorem (or the Crotti’s theorem in the generalized con-text of nonlinear elasticity) to establish a relationship between theshaft displacement u and the corresponding reaction force F . Theforce F is obtained from the following derivative of the total poten-tial energy evaluated for the extremized potential⇧⇤ = ⇧(u⇤

0) atstatic equilibrium:

F = @⇧⇤

@ u. (15)

Finally, the effective spring stiffness K3 corresponds to the slope ofthe force–displacement curve at u = 0:

K3 = @F

@ u

����u=0

. (16)

It is worth noting that K3 depends on the following set ofparameters: {�p,�, µ1, µ2, Ri, Re} so, at least in principle, itis possible to tune the stiffness of the device not only byvarying prestretch, geometry and constitutive characteristics ofthematerial, but also by exploiting the electromechanical coupling,i.e. by changing the drop voltage between the two sides of themembrane.

2.4. Approximate solution

In order to obtain a closed-form approximation that relates theout-of-plane stiffness K3 and the voltage� , we use a small-on-large

strategy. This is applicable by assuming that the displacementsassociated with the mapping �0 : ⌦p ! ⌦ are relativelysmall, thus employing a first order incremental approach. Thekinematics introduced in the previous section is such that afurther deformation is superimposed on the highly prestretched(intermediate) configuration ⌦p through the prescription of thedisplacement u. Therefore, under the hypothesis that u is relativelysmall, and, consequently, the current configuration is not far fromthe intermediate configuration, a linear approximation of thekinematics can be used to predict incremental variations of thesystem response.

For sake of clarity, a displacement u0 can be defined by scalingu0 in (5) by a quantity ⌘ ⌧ 1, namely

u0 = ⌘u0 = ⌘ (uses + u3e3) , (17)

and used in place of u0 in the sequel. By performing a Taylorexpansion of the Euler–Lagrange equations in (13) and keepingonly first order terms w.r.t. ⌘, one obtains a linearizationfor deformations from the intermediate configuration. Next, bysubstituting relation (17) in (A.1), Eq. (13) implies the following‘‘scaled’’ set of Euler–Lagrangian equations:

u00s+ 1

su

0s� 1

s2us = 0 and u

003 + 1

su

03 = 0. (18)

Note that, as expected in axisymmetric problems encountered inlinear elasticity, the first order approximation of the problem doesnot depend on the constitutive behavior of the material whenthe boundary conditions are completely prescribed in terms ofdisplacement.

As shown in (18) the system takes the form of a set of Eu-ler–Cauchy differential equations that, with reference to boundaryconditions in (14), leads to the following analytic solutions:

us(s) = 0 and u3(s) = log(s) � log(Re)

log(Ri) � log(Re)u. (19)

The force in (15) can be thus expressed, after some algebraic ma-nipulations, as follows:

Fl = 2⇡"�2 �2p

H log⇣

Ri

Re

⌘ u + gF (20)

where gF = gF (�p, u, µ1, µ2, Ri, Re,H, ) (see details in Ap-pendix A.2) is a suitable function introduced – for the sake of sim-plicity – to highlight the 2nd-order dependence of Fl on the dropvoltage� . Furthermore, an analytic scaled relationship among theout-of-plane stiffness at u = 0, the drop voltage, the prestretchand the geometrical and constitutive parameters is obtained:

Kl = 2⇡"�2 �2p

H log⇣

Ri

Re

⌘ + gK , (21)

where the function gK = @gF/@ u|u=0 = gK (�p,µ1, µ2, Ri, Re,H).

3. Results

The boundary value problem described by (13) and (14) issolved numerically using the bvp4c finite difference solver inMATLABTM 2015b (The Mathworks, Inc.). The numerical plots ofthe full-nonlinear version and the analytic linearized-scaled versionin (19) are obtained using MathematicaTM 10 (Wolfram Research,Inc.). Results are obtained for values of parameters presented inTable 1. These are based on values previously reported for the softpolyacrylate [30] (VHB 4910; 3M) used as the dielectric membraneand the device dimensions reported in [12]. It should be notedthat there is significant variation in the stiffness of polyacrylateelastomers, which can have an elastic modulus ranging from⇠0.1 to 1 MPa [1,31,32]. Polydimethylsiloxane (PDMS) is anotherpopular dielectric elastomer that exhibits a similar range ofstiffness. This includes commercially available silicones likeSylgard 184 (Dow-Corning), Smooth-Sil 950 (Smooth-On), andEcoFlex 00-30 (Smooth-On), which were recently characterizedfor applications in soft-matter engineering by Case, White, &Kramer [33].

As with polyacrylate, silicone elastomers can be engineeredto exhibit a variety of nonlinear stress–strain responses. This

120 A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125

Fig. 2. Displacements u3 and us , normalized w.r.t. the width of the annulus w, plotted versus the radial abscissa s normalized w.r.t. the external radius Re , for values ofthe prescribed displacement u equal to (a) 1 mm, (b) 5 mm, (c) 30 mm. The results are obtained for values of the parameters reported in Table 1, prestretch �p = 6 andapplied drop voltage� = 1 kV. For the constitutive model, the following three pairs of elastic coefficients were selected: (NH) µ1 = µ and µ2 = 0, which corresponds to aNeo-Hookean solid; (OG1) µ1 = 0 and µ2 = µ/2; (OG2) µ1 = µ/2 and µ2 = µ/4.

Table 1Parameters adopted in the simulations.

Physical parameter Symbol Value Unit

Radius of shuttle Ri 2.75 mmRadius of circular frame Re 12.5 mmNatural membrane thickness H 1 mmElastic shear modulus [30] µ 73 kPaRelative permittivity "r 3.21 –

ranges from the strain softening behavior consistent with a Neo-Hookeanmodel to the stiffening or sequential softening–stiffeningresponses that can be captured by an Ogden model. To accountfor these different constitutive properties, we considered thefollowing three pairs of elastic coefficients: (NH) µ1 = µ andµ2 = 0, which corresponds to a Neo-Hookean solid; (OG1)µ1 = 0and µ2 = µ/2; (OG2) µ1 = µ/2 and µ2 = µ/4.

In Fig. 2 the algebraic approximation (solid line) obtained fromEq. (19) is compared with the results of the numerical analysis(markers). The displacements us = us(s) and u3 = u3(s) arenormalizedw.r.t. the annularwidthw = Re�Ri andplotted againstthe normalized coordinate s/Re for a prestretch �p = 6. The resultsof the nonlinear analyses are plotted for all the three constitutivemodels: NH (circle); OG1 (triangle); OG2 (rhombus). The plots havebeen obtained considering three different values of the externalimposed displacement u = 1, 5, 30 mm for sub-figures (a), (b),and (c), respectively, for a fixed drop voltage of � = 1 kV. Itis worth highlighting that the amplitude of the displacement us

is always more than two orders of magnitude lower than thedisplacement u3, as expected from the analytic solution in (19). Asshown in Fig. A.9 in the Appendix A.3, in the case of a Neo-Hookeansolid, this difference is even greater. Furthermore, the numericalsolution for u3 is close to the analytic solution if the amplitudeof the imposed displacement is relatively low. Fig. 3 shows thenormalized response force F/Fr versus displacement u/w for theanalytic approximation (20) and numerical analysis performed onthe three pairs of Ogden parameters. The force Fr is computed asthemaximum force predicted by the analytic model at u/w = 0.5:(NH; Fig. 3(a)) Fr = 1.36 N; (OG1; Fig. 3(b)) Fr = 33.54 N, (OG2;Fig. 3(c)) Fr = 17.45 N. Results are plotted for four different valuesof voltage (� = 0, 2, 4 and 6 kV). As shown in Fig. 3 the analyticapproximation from Eq. (20) is in very good agreement with thenumerical solutions when u is relatively low. As u increases, somedeviation is observed at higher displacements and voltages.

Next, Fig. 4 shows how the analytic solutions for the out-of-plane stiffness scale with voltage. Results are shown for the threeconstitutive models NH (Fig. 4(a)), OG1 (Fig. 4(b)), OG2 (Fig. 4(c))in the limit as u goes to zero. The stiffness values are normalizedto its K0 evaluated when � = 0 for each of the three selectedprestretches: (NH) K0 = (302.93 N/m)(1 � ��6

p); (OG1) K0 =

(151.46 N/m)(�2p� ��10

p); (OG2) K0 = (151.46 N/m)(1 � ��6

p) +

(75.73 N/m)(�2p

� ��10p

). Consistently with the results in Fig. 3,

significant voltage-induced softening response is only expected forthe Neo-Hookean solid. For the other constitutive models, greatervoltage or prestretch is required to observe a comparable stiffnesschange. This influence may be better examined by comparingthe internal stress in the intermediate configuration in the radialdirection.

Fig. 5 presents a combined plot of the radial stress normalizedw.r.t. the membrane stress �0 at � = 0 versus the appliedvoltage for the three constitutive models. In all cases, a prestretchof �p = 6 is selected and the correspondingmembrane stresses are�0 = 2.63, 47.30, and 24.97 MPa for the NH, OG1, and OG2 solids,respectively. Although the prestretch induces different levels ofresidual stress, all the curves exhibit similar trends and profiles,i.e. �s / ��2. The key difference among the three cases is foundas the voltage tends to a critical value (i.e.�⇤) atwhich the stiffnessapproaches zero, that is: �⇤ ' 8, 35, and 26 kV for the NH, OG1,and OG2 solids, respectively. It is important to note that, as shownin Fig. 5, not all the plotted values are physically realizable. Thisis due to dielectric breakdown, which occurs when the appliedvoltage exceeds the ability of the dielectric to hold charge and it isno longer an insulator. Fig. 6 contains an estimate of the voltages atwhich breakdownwill occur for VHB 4910 (3M), which is a popularpolyacrylate used in DE actuators and transducers [34] used in [12]to perform the experiment.

4. Discussion

As shown in Fig. 4, it is apparent that themechanical impedanceof the variable stiffness device is strongly affected by appliedvoltage, prestretch, and nonlinear constitutive properties of thedielectric. Even for materials exhibiting the same shear modulusat infinitesimal strain, high order strain softening or stiffening canlead to dramatic differences in the electromechanical response.This sensitivity arises from the fact that stiffness tuning is onlypronounced when the electrostatic Maxwell stress is comparableto the residual membrane stress of the prestretched film. This isclear in Fig. 5, which shows that reduction is most pronouncedwhen the value of the drop voltage induces a Maxwell stress witha magnitude compensating the amount of prestress (for largervoltages, the predicted stiffness is negative and likely correspondsto an electromechanical instability [13]).

In general, it is observed that for a Neo-Hookean solid, theanalytic and numerical solutions are in very good agreement. Inparticular, the numerical solution for us is extremely small (exceptfor u = 30 mm) and there is virtually no discrepancy in theprediction for u3. Even for the OG1 and OG2 solids, results fromthe analytic solution are found to be adequate to approximatethe nonlinear profile of the membrane so long as the verticaldisplacement is moderate (u 5 mm).

In Fig. 3, it is evident that the out-of-plane response of thedevice to a pulling out displacement is strongly dependent on the

A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125 121

Fig. 3. Analytic (solid) and numeric (dotted) response force F normalized w.r.t. its maximum value versus the prescribed displacement u normalized w.r.t. the annularwidth w, for different values of voltage (� = 0, 2, 4, 6 kV) and for the three constitutive models: (a) NH (µ1 = µ and µ2 = 0); (b) OG1 (µ1 = 0 and µ2 = µ/2); (c) OG2(µ1 = µ/2 and µ2 = µ/4). The results are obtained for values of the parameters reported in Table 1 and prestretch �p = 6.

Fig. 4. Analytic prediction of the out-of-plane stiffness of the device Kl normalized w.r.t. its value K0 evaluated for � = 0 versus the applied voltage � for different valuesof prestretch (�p = 4, 6, 8), for different values of voltage (� = 0, 2, 4, 6 kV) and for the three constitutive models: (a) NH (µ1 = µ and µ2 = 0) which correspondsto K0 = (302.93 N/m)(1 � ��6

p); (b) OG1 (µ1 = 0 and µ2 = µ/2) which corresponds to K0 = (151.46 N/m)(�2

p� ��10

p); (c) OG2 (µ1 = µ/2 and µ2 = µ/4) which

corresponds to K0 = (151.46 N/m)(1 � ��6p

) + (75.73 N/m)(�2p� ��10

p). The results are obtained for values of the parameters reported in Table 1.

Fig. 5. Radial stress �s normalized w.r.t. the membrane stress �0 at � = 0versus the applied voltage for the three constitutive models: (a) NH (µ1 = µand µ2 = 0) which corresponds to �0 = 2.63 MPa; (b) OG1 (µ1 = 0 andµ2 = µ/2) which corresponds to �0 = 47.30 MPa; (c) OG2 (µ1 = µ/2 andµ2 = µ/4) which corresponds to 24.97 MPa, highlighting the region in whichthe dielectric breakdown occurs (gray region). The critical voltage�⇤ , at which thestress vanishes, is' 8, 35, 26 kV for the NH, OG1, and OG2model, respectively. Theresults are obtained for values of the parameters reported in Table 1 and prestretch�p = 6.

constitutive behavior chosen to model the membrane. In the caseof the Neo-Hookean solid, there is a pronounced relaxing effect,with a behavior comparable with the experiments by [12], thatcorresponds to a lower slope of the force–displacement curve withincreasing voltage. It is important to note that this effect is notsignificant in the other two constitutive models. Such a differenceillustrates the importance of performing an accurate experimentalcharacterization of the polymer’s constitutive properties. For thereader’s convenience, force in Eq. (20) versus the normalizeddisplacement of the device for the three constitutive models ispresented in Fig. 7. This is displayed in comparison with the strainresponse in terms of first Piola–Kirchhoff stress normalized w.r.t.the tangent shear modulus versus stretch, in the case of uni-axialstress state. The presence of the parameter µ2, as well known,induces a stiffening effect with an increase of the deformation thatapparently interferes with the tunable stiffness coupling.

Fig. 6. Estimate of the breakdown voltage for the polyacrylate VHB 4910 (3M),for increasing value of prestretch �p , experimentally obtained by [34] when theelastomer sheet is bi-axially (pre)stretched.

To examine how it might be possible to maximize the tuningeffect, it is helpful to express Eq. (21) for the special case of a Neo-Hookean solid with u = 0, namely:

KNH = 2⇡H↵

"

µ

1 � 1�6p

!

� "�2 �2p

H2

#

, (22)

where↵ = log(Re/Ri). This shows in a simpleway that the stiffnessof the device corresponds to the difference in contributions frommechanical prestretch and electrical enthalpy. As shown in Fig. 8,prestretch can have an important role in the stiffness change.In fact, although there is a saturation effect in the first termwith the brackets, a quadratic dependence in the second termwith an increase of �p can be identified. A less pronounced(but non-negligible) effect can be related to a reduction of theinitial thickness H and the reduction of the shear modulus of themembrane. As with �p, such parameters can be adjusted in orderto influence electromechanical response.

122 A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125

Fig. 7. Comparison between (a) the strain response of the device in terms of tensile force F versus the prescribed displacement u normalized w.r.t. the annular width wand (b) the strain response in terms of First Piola–Kirchhoff stress P normalized w.r.t. the tangent shear modulus (µ) versus stretch (�) in a case of uni-axial stress state. Theresults are obtained for values of the parameters reported in Table 1, drop voltage� = 3 kV and prestretch �p = 6. For the constitutive model, the following three pairs ofelastic coefficients were selected: (NH) µ1 = µ and µ2 = 0, which corresponds to a Neo-Hookean solid; (OG1) µ1 = 0 and µ2 = µ/2; (OG2) µ1 = µ/2 and µ2 = µ/4.

Fig. 8. Analytic stiffness response K specialized for the NHmodel, given by Eq. (22),normalized w.r.t. its value Kmax ' 303 N/m versus the prestretch �p , for differentvalues of the applied voltage (� = 0, 4, 6 kV). The results are obtained for valuesof the parameters reported in Table 1.

5. Conclusions

The variation of the out-of-plane stiffness of an annular DEmembrane in response to an applied voltage has been analyticallymodeled. The theory takes into account the constitutive nonlin-earity associatedwith large prestretch and electromechanical cou-pling from Maxwell stress. We show that the increase of voltageinduces a nonlinear softening effect (/ ��2) that is significantlymore pronounced when the Maxwell stress is comparable to theamount of prestress. The analytic approximation of these valuescritically depends on the constitutive response of the membrane.This is due to the important role of the residual membrane stresswhen high prestretch �p is applied. Among the three models cho-sen to perform the simulation, for the cases of OG1 and OG2 a sig-nificant reduction of the stiffness occurs for voltages exceeding theexperimentally estimated breakdown threshold. On the contrary,this is not the case for the NH solid and, thus, it is possible – atleast in principle – to have a drastic reduction of the out-of-planestiffness if the value of the imposed voltage is sufficiently close to acritical value�⇤. This suggests that a membrane with constitutiveproperties similar to that of a Neo-Hookean solid behavior mightallow for the largest changes in effective stiffness.

Acknowledgment

C.M. acknowledges support from the NASA Early Career FacultyAward (NNX14AO49G).

Appendix

A.1. Lagrangian

The energy density per unit length expressed in (12), consider-ing the Eqs. (8) and (11) takes the form:

L(us, u0s, u0

3, s)

= 12⇡Hs

8><

>:�

2�2"�4p(us + s)2

h�u

0s+ 1

�2 +�u

03�2i

H2s2

+ µ2

8><

>:s4

�8p(us + s)4

h�u0s+ 1

�2 +�u

03�2i2

+�4p(us + s)4

s4+ �4

p

h�u

0s+ 1

�2 +�u

03�2i2 � 3

9>=

>;

+ 2µ1

8<

:2�2p+ s

2

�4p(us + s)2

h�u0s+ 1

�2 +�u

03�2i

+�2pu2s

s2+ �2

p

�u

0s

�2 + �2pu

0s+ 2

2�2pus

s+ �2

p

�u

03�2 � 3

9=

;

9>=

>;(A.1)

To minimize L, since it is function of (us, u0s, u3), the following

set of Euler–Lagrangian equations have to be imposed equal tozero:@L

@us

� @

@s

@L

@u0s

=2⇡"�2 (s + us)

ns

h��u

03�2 +

�u

0s

�2 + u0s+ su

00s

i� us

�u

0s� su

00s+ 1

�o�2p

Hs2

+ H⇡

2�2p

8>>>><

>>>>:

�4µ1

�u

0s+ 1

�(

�6p� s

2

(s+us)2h(u0

3)2+(u0

s+1)2i2

)

�4p

+ sµ2

8><

>:

4 (s + us)3 �4

p

s4� 4s4

(s + us)5 �8

p

h�u

03�2 +

�u0s+ 1

�2i2

9>=

>;

A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125 123

+4sµ1

((s+us)

4�6ps2

� s2(u0

3)2

h(u0

3)2+(u0

s+1)2i2 � (u0

ss+s)2h(u0

3)2+(u0

s+1)2i2

)

(s + us)3 �4

p

�4µ2

�u

0s+ 1

� ⇢�12p

h�u

03�2 +

�u

0s+ 1

�2i4 � s4

(s+us)4

�

�8p

h�u

03�2 +

�u0s+ 1

�2i3

� 4sµ1

�4p

8><

>:u

00s�6p� 4s2u0

3�u

0s+ 1

�u

003

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

� 2s2�u

03�2

u00s

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3 � 2�u

0ss + s

�2u

00s

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

+ s

�u

0s+ su

00s+ 1

�

(s + us)2h�

u03�2 +

�u0s+ 1

�2i2 � 4s�u

0s+ 1

�2 �u

0s+ su

00s+ 1

�

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

� 4s2�u

0s+ 1

� ⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

+ 12s2�u

0s+ 1

�3 ⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i4

+ 12s2�u

03�2 �

u0s+ 1

� ⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i4

� 2s2�u

0s+ 1

�2

(s + us)3h�

u03�2 +

�u0s+ 1

�2i2 + s

�u

0s+ 1

�

(s + us)2h�

u03�2 +

�u0s+ 1

�2i2

+ 4s2�u

0s+ 1

�4

(s + us)3h�

u03�2 +

�u0s+ 1

�2i3 + 4s2�u

03�2 �

u0s+ 1

�2

(s + us)3h�

u03�2 +

�u0s+ 1

�2i3

� 4s�u

03�2 �

u0s+ 1

�

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

9>=

>;� 4sµ2

�8p

8><

>:

h�u

03�2 +

�u

0s+ 1

�2iu

00s�12p

+�u

0s+ 1

� ⇥2u0

3u003 + 2

�u

0s+ 1

�u

00s

⇤�12p

� s4u

00s

(s + us)4h�

u03�2 +

�u0s+ 1

�2i3 + 6s4�u

0s+ 1

� ⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)4h�

u03�2 +

�u0s+ 1

�2i4

+ 4s4�u

0s+ 1

�2

(s + us)5h�

u03�2 +

�u0s+ 1

�2i3 � 4s3�u

0s+ 1

�

(s + us)4h�

u03�2 +

�u0s+ 1

�2i3

9>=

>;

9>>>=

>>>;

(A.2)

and

@

@s

@L

@u03

= �2⇡✏�2 (s + us) �

2p

⇥u

03�2u0

ss + s � us

�+ s (s + us) u

003⇤

Hs2

+ 2H⇡�10p

8><

>:µ1u

03

8><

>:�6p� s

2

(s + us)2h�

u03�2 +

�u0s+ 1

�2i2

9>=

>;�4p

+ sµ1�4p

8><

>:u

003�

6p+ s

2u

003

(s + us)2h�

u03�2 +

�u0s+ 1

�2i2

� 6s2�u

03�2

u003

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3 � 2�u

0ss + s

�2u

003

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

� 4su03�u

0s+ 1

� �u

0s+ su

00s+ 1

�

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3 � 4s2u03⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

+ 12s2�u

03�3 ⇥

u03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i4

+ 6u03�u

0ss + s

�2 ⇥2u03u

003 + 2

�u

0s+ 1

�u

00s

⇤

(s + us)2h�

u03�2 +

�u0s+ 1

�2i4

+ 2su03

(s + us)2h�

u03�2 +

�u0s+ 1

�2i2 � 2s2u03�u

0s+ 1

�

(s + us)3h�

u03�2 +

�u0s+ 1

�2i2

� 4s�u

03�3

(s + us)2h�

u03�2 +

�u0s+ 1

�2i3

+ 4s2u03�u

0s+ 1

�3

(s + us)3h�

u03�2 +

�u0s+ 1

�2i3

+ 4s2�u

03�3 �

u0s+ 1

�

(s + us)3h�

u03�2 +

�u0s+ 1

�2i3

9>=

>;

+µ2u

03

⇢�12p

h�u

03�2 +

�u

0s+ 1

�2i4 � s4

(s+us)4

�

h�u

03�2 +

�u0s+ 1

�2i3

+ sµ2

8><

>:

h�u

03�2 +

�u

0s+ 1

�2iu

003�

12p

+ u03⇥2u0

3u003 + 2

�u

0s+ 1

�u

00s

⇤�12p

� s4u

003

(s + us)4h�

u03�2 +

�u0s+ 1

�2i3 + 6s4u03⇥u

03u

003 +

�u

0s+ 1

�u

00s

⇤

(s + us)4h�

u03�2 +

�u0s+ 1

�2i4

� 4s3u03

(s + us)4h�

u03�2 +

�u0s+ 1

�2i3

+ 4s4u03�u

0s+ 1

�

(s + us)5h�

u03�2 +

�u0s+ 1

�2i3

9>=

>;

9>=

>;.

(A.3)

A.2. gF

The function gF (�p, u, µ1, µ2, R1, R2,H, ) introduced in Eq. (20)for sake of clarity in order to isolate the dependence of the force onthe drop voltage takes the form:

gF = H⇡ u

2↵4�u2 + ↵2R2

e

�2 �u2 + ↵2R2

i

�2�10p

⇥⇢2⇢

� log�u2 + ↵2

R2e

� hu4R4e+ u

2 �u2 + 2↵2

R2e

�R2iR2e

+�u2 + ↵2

R2e

�2R4i

iµ2↵

4 + log�u2 + ↵2

R2i

�

⇥hu4R4e+ u

2 �u2 + 2↵2

R2e

�R2iR2e

+�u2 + ↵2

R2e

�2R4i

iµ2↵

4

� log✓u2 + ↵2

R2e

u2 + ↵2R2i

◆ �u2 + ↵2

R2e

�2 �u2 + ↵2

R2i

�2�4pµ1

+ log✓u2 + ↵2

R2i

u2 + ↵2R2e

◆u4 ⇥3R2

eR2i↵4

124 A. Cugno et al. / Extreme Mechanics Letters 13 (2017) 116–125

Fig. A.9. Displacements us , normalized w.r.t. the width of the annulus w, plotted versus the radial abscissa s normalized w.r.t. the external radius Re , for values of theprescribed displacement u equal to (a) 1 mm, (b) 5 mm, (c) 30 mm. The results are obtained for values of the parameters reported in Table 1, prestretch �p = 6 and applieddrop voltage � = 1 kV. For the constitutive model, the following three pairs of elastic coefficients were selected: (NH) µ1 = µ and µ2 = 0, which corresponds to aNeo-Hookean solid; (OG1) µ1 = 0 and µ2 = µ/2; (OG2) µ1 = µ/2 and µ2 = µ/4.

+ 2u2 �R2e+ R

2i

�↵2 + u

4⇤µ2

�↵2

+ 1R2eR2i

⇢2u6

R2e

�u2 + ↵2

R2e

�2µ2�

12p

+↵4R6i

⇢2↵2

R2e

�u2 + ↵2

R2e

�

⇥⇥2↵�u2 + ↵2

R2e

��6p� u

2⇤µ1�4p

+h�2

�u2 + ↵2

R2e

�2 �u2 � 2↵3

R2e

��12p

� 4↵4u2R4e� 3↵2

u4R2e

iµ2

�

+ u4R2i

n2↵3

R2e

�u2 + ↵2

R2e

�

⇥⇥2u2�6

p+ ↵R2

e

�2↵�6

p+ 1

�⇤µ1�

4p

+n

� 2�u2 + ↵2

R2e

�2 ⇥u2 � 2↵2(↵ + 1)R2

e

⇤

⇥ �12p

+ 3↵6R6e+ 2↵4

u2R4e

oµ2

o

+ 2↵2u2R4i

⇢↵2

R2e

h4↵�u2 + ↵2

R2e

�2�6p� u

4 + ↵4R4e

i

⇥ µ1�4p+⇢ �

u2 + ↵2

R2e

�2 ⇥↵2(4↵ + 1)R2

e� 2u2⇤

⇥ �12p

+ 2↵6R6e� ↵2

u4R2e

�µ2

���. (A.4)

A.3. us

The component of the displacement in the s-direction is hereshown in Fig. A.9.

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