Abstract We report on the frequency dependent behavior ofdielectric elastomer actuators (DEA). The introduced smartmaterial actuators consist of 3M™’s elastomer VHB™4905(9469) and a compliant, sputtered copper electrode on eachside. The presented experiments on these compounds con-tain the active tuning of their resonance frequency and theirapplication as acoustic actuators. We are able to decrease themembranes’ eigenfrequency by 30% with an electrical off-set potential. Alternatively, if an alternating signal is applied,sound pressure levels up to 130 dB in an enclosed volume of28 ccm are achieved. In order to verify the results, a numer-ical simulation is introduced incorporating the two physicalfields involved: electrical and mechanical.
1 Introduction
Smart materials have been known for decades. A plethoraof high technology mechatronic devices are based on piezo-electric [1] or similar smart materials [2]. The advantagesof materials exhibiting piezoelectricity are large stressesand a sudden step response. On the other hand, they im-ply some disadvantages since they are rigid, exhibit onlysmall strains, and often show a strongly non-linear even hys-teretic behavior [3]. To overcome the limitations of these
K. Hochradel (�) · S.J. Rupitsch · A. Sutor · R. LerchChair of Sensor Technology, Friedrich-Alexander-University ofErlangen-Nuremberg, Paul-Gordan Str. 3-5, 91052 Erlangen,Germanye-mail: [email protected]
D.K. Vu · P. SteinmannChair of Applied Mechanics, Friedrich-Alexander-University ofErlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen,Germany
rigid materials, many researchers focus on novel soft andbendable alternatives such as EMFi (Electro-Mechanical-Film) materials [4–6]. EMFi materials offer good perfor-mance in sensing, but exhibit only poor behavior for actu-ation and an insufficient temperature stability [7]. Dielectricelastomers [8] extend the variety of flexible smart materi-als that show entirely different mechanical properties sincethey feature large strains [9] for actuation and high temper-ature stability [10]. Roentgen discovered in 1880 the stretchof a low modulus polymer with electrodes due to an ap-plied electrical potential [11]. It was not until 1984 that re-searchers tried to utilize this effect to create a novel classof smart materials. The first experiments revealed static de-formations up to 3% [12]. In 2000, the group of Kornbluhobtained area expansions of 380% [9] which evoked a re-search boom for dielectric elastomer actuators (DEA). Lat-est research results demonstrate area strains of 1692% [13].These actuators consist of a thin elastomer film between twocompliant electrodes. They are simple, cheap, lightweight,and feature high efficiency [14, 15] as well as a high dielec-tric strength.
The majority of researchers focuses on the quasistatic de-formation of DEAs with respect to applications such as ar-tificial muscles [14] or tactile displays [16]. However, thedynamic performance [17] of dielectric elastomer actuatorsis crucial for sound radiating applications [18, 19] or tun-able resonators [20]. This contribution extends the researchin this field with its main issue of the frequency dependentperformance of circular dielectric elastomer membranes.Therefore, we carry on Dubois’ idea [20] of a decreasingresonance frequency by means of an applied electrical po-tential. Yet, the elastomer VHB™4905 (9469) is utilized in-stead of electroactive polymers. In addition to the voltagecontrol of the actuator’s resonance frequency, its applica-bility as an acoustic actuator [18, 19] is evaluated. First of
all, we briefly introduced the developed fabrication of circu-lar dielectric elastomer membranes coated with copper elec-trodes. Using these DEA membranes, we were able to de-termine several material parameters of VHB™4905 (9469)such as the relative permittivity since the values publishedby various researchers [21–23] show a discrepancy from 3.0up to 6.0.
In Sect. 2, we initially introduce the fabrication and theproperties of the dielectric elastomer actuator. Experimentson them are presented in Sect. 3, starting with the experi-mental set-up, and ending with the results for the differentinvestigations. In order to confirm the experimental results,we perform in Sect. 4 a numerical computation based on thefinite element approach. Finally, we draw a conclusion re-garding the results of this research.
2 Fabrication and properties of the elastomer actuator
The elastomers VHB™4905 and 9469 by 3M™ exhibit a lowYoung’s modulus (approximately 1.8 MPa) and are, there-fore, difficult to handle. This property as well as their ad-hesive characteristics make reproducible experimental re-sults hardly achievable. In order to ensure constant qual-ity DEA samples, several manufacturing steps and a repro-ducible process to coat the elastomer with conductive elec-trodes are necessary.
At the beginning, the elastomer is attached to an alu-minum frame and stretched (see Fig. 1) since its initialthickness of 500 µm obstructs the effects of the electro-mechanical coupling. After stretching the elastomer to athickness smaller than 100 µm, the soft and adhesive elas-tomer is fixed between two circular acrylic glass framesto improve handling. These process steps are crucial andhave to be done with meticulous precision and in a dust-freeworkspace to receive constant quality membranes with a re-producible thickness. A compliant electrode on each side ofthe clamped membrane completes the dielectric elastomeractuator. For the compliant electrodes, most researchers usegraphite or silver greases which are brushed onto the elas-tomer. This technique is not applicable for this study sincethese electrodes are not reproducible, no defined layer thick-ness is achievable, and the mechanical properties of greasedominate due to its weight. In order to receive lightweightand continuous quality electrodes, copper is deposited onthe elastomer with a radio frequency magnetron sputter pro-cess [24]. The sputtering has to be done with minimumpower to reduce the thermal load, since the elastomer with-stands high temperatures only for short terms. Additionally,multiple pauses are necessary to coat elastomer membranesthinner than 20 µm with copper. The developed sputter pro-cess consists of 5×30 s sputtering at 50 W with a 30 s pause
Fig. 1 Prestress of the initial elastomer to thin down the material; fix-ation between two acrylic glass frames; bottom: two DEA membraneswith different electrodes (left: copper; right: graphite grease)
in between. Combining all process steps, we are able to fab-ricate DEA membranes with different diameters and a min-imum thickness of 8 µm. The sputtered electrodes are keptas thin as possible (<70 nm), to minimize the electrodes’influence on the mechanical behavior of the DEA.
A comparison and verification of the DEA’s qualityand reproducibility is realized by determining the thick-ness of the elastomer layer with nanofocus’ confocal micro-scope µsurf explorer and by measuring the DEA’s electricalimpedance with HP’s impedance analyzer 4194A. Combin-ing both measurements, we are able to determine the relativepermittivity of the elastomer.
The dielectric constant εr yields 3.1 which correspondswith several publications [9, 10] as well as the newly re-leased manufacturer description. Furthermore, an impor-tant quantity for the following experiments is the dielectricstrength of the elastomer. To prevent the measurement de-vices from damage, we must avoid the occurrences of unex-pected breakdowns. The experimentally determined break-down field strength amounts to approximately 100 MV/m(dielectric strength of air is approximately 1 MV/m).
3 Dynamic experiments on the dielectric elastomeractuator
3.1 Measurement set-up
The DEA membrane is clamped between two acrylic glassframes (Fig. 1) and mounted onto a cylindrical chamber with
Dynamic performance of dielectric elastomers utilized as acoustic actuators 533
Fig. 2 Mounting of the dielectric elastomer membrane onto a cylin-dric volume; mechanical excitation of DEA with a pistonphone; mea-surement of DEA’s displacement with optical displacement sensor;possibility to determine the pressure in the closed chamber
a volume of 28 ccm (see Fig. 2). A pistonphone generatesair pressure to mechanically excite the actuator membrane.The piston is mounted on top of TIRA’s vibration systemS50018, which allows us to generate static as well as alter-nating pressures up to 20 kHz. This excitation pressure canbe measured with the integrated pressure sensor. The micro-phone (Sennheiser KE-4-211-2) is also used to determinethe sound pressure radiated by the elastomer actuator. Thedeflection of the DEA is measured with Micro-Epsilon’s op-tical displacement sensor optoNCDT 2220-20LL which isbased on the triangulation principle. The displacement sen-sor limits the measurements to frequencies below 2000 Hzand to a resolution of 0.3 µm. The set-up allows investiga-tion for different excitations of the DEA membrane as it of-fers electrical as well as mechanical excitations both, staticand alternating.
3.2 Mechanical properties of the DEA membrane
The majority of researchers focuses on the quasistatic defor-mation of elastomers regarding applications such as artifi-cial muscles [14]. Therefore, most experiments aim on thestrongly nonlinear behavior of the viscoelastic elastomers.Yet, the main research issue of this contribution is the acqui-sition of the DEA’s dynamic characteristics. For high fre-quencies, respectively small displacements, we can assumea linear relation between the excitation and the membrane’sdisplacement. That is because the maximum deflection of acircular clamped membrane can be approximated in the lin-
Fig. 3 Deflection of a circular clamped membrane; linear relation be-tween excitation pressure and displacement
Fig. 4 Mechanical behavior of a DEA with a radius of 15 mm; si-nusoidal excitation frequency f = 1 Hz; displacement of membrane’scenter; linear relation between pressure and caused displacement
ear case by [25]
x = pr4
64D(1)
with
D = Yz3
12(1 − ν2). (2)
Accordingly, the displacement x depends on the pressure p,the radius of the membrane r , the thickness z, Young’s mod-ulus Y and the Poisson ratio ν (see Fig. 3). Hence, we obtaina linear relation of displacement and applied pressure
p ∼ x. (3)
With a view to proving the linear mechanical properties,we excite the membrane with an alternating pressure (p̂ =250 Pa, f = 1 Hz) and record the DEA’s displacement. Thelow frequency of 1 Hz prevents any appearance of time de-lays since we are far off the resonance frequency. The exper-imental result is illustrated in Fig. 4. The alternating pressureof 250 Pa causes a maximum displacement of 300 µm. Whenplotted in one graph, they yield an almost linear relation. Theslight hysteresis will be neglected.
534 K. Hochradel et al.
3.3 Voltage control of the resonance frequency
Numerous mechatronic devices demand for a constant orcontrollable resonance frequency (e.g., micromirros [26]).The eigenfrequency of plates or membranes depends basi-cally on the geometry, the material and, if present, on theprestress. In order to modify the resonance frequency, differ-ent techniques are common (e.g., tuning the spring constantby an electrostatic force), which share one disadvantage: thedevice to control the natural frequency is larger than the ac-tual resonator. Therefore, Dubois [20] investigated the volt-age control of the resonance frequency on micromechani-cal membranes with polymers as dielectric material betweentwo compliant electrodes. His membranes had diameters of2–4 mm at a thickness of 35 µm. He achieved a resonancedrop up to 40% of membranes made of PDMS Sylgard184 [20]. Similar investigations for elastomers, especiallyVHB™4905, are missing.
This decrease of the resonance frequency is caused by thecoupling of the electric field and the mechanical deforma-tion of the elastomer. An applied electrical potential inducesthe Maxwell stress σM
σM = −1
2ε0εr
V 2e
z2, (4)
σM ∼ E2 (5)
which depends on the applied voltage Ve and the thick-ness z, alternatively on the electrical field strength E. TheMaxwell stress acts on both electrodes and causes them tomove closer. Thus, the elastomer between the electrodes isthinned down. Since the elastomer is incompressible, theDEA enlarges in radial direction which causes a decreaseof the fabrication prestress (Fig. 1), and hence a decrease ofthe membranes resonance frequency.
Experimentally, we determine the resonance by a fre-quency sweep of the excitation pressure while its ampli-tude is kept constant. The caused displacement of the mem-brane’s center is displayed in Fig. 5. The resonance isreached at 370 Hz for a 30 mm diameter membrane witha thickness of 35 µm. After applying a constant electricalpotential of 3 kV, the eigenfrequency drops about 22% to290 Hz. The phase shift between excitation pressure and dis-placement confirms the decrease since the phase amounts to−90◦ at the resonance frequency.
According to (4), the resonance decrease only dependson the thickness of the elastomer layer if the voltage is keptconstant. Figure 6 shows the resonance shift of two DEAmembranes which features a different thickness while thesame constant electrical potential of 2 kV is applied. The168 µm thick membrane reaches its resonance at 230 Hz andthe electrical potential causes a drop of 4.3% to 220 Hz. Incontrast to the thick membrane, the resonance of the 26 µm
Fig. 5 Decrease of the resonance frequency due to an electrical poten-tial of 3 kV (sample: ∅ 30 mm, thickness 35 µm)
DEA drops 28%. This can be ascribed to the larger electricalfield intensity for the thin membrane.
3.4 Dielectric elastomer as acoustic actuator
The second issue in this contribution is the behavior of thedielectric elastomer membrane while excited with an alter-nating, but unipolar, electrical potential. The characteristicsof the membrane’s oscillation is crucial for various appli-cations such as acoustic actuators [18]. The idea is to cre-ate a constant pressure inside the chamber to inflate the ini-tial membrane ① in order to reach a working point ② (seeFig. 7a). An applied sinusoidal electrical potential initiatesan oscillation of the DEA ③ by means of the varying pre-stress. The oscillation of the DEA membrane is shown inFig. 7b. This graphic illustrates the compliance of the oscil-lation frequency and the frequency of unipolar, electrical ex-citation. Yet, experiments showed that the oscillation shapeof the membrane is almost independent of the static pres-sure. Even without any static pressure, the DEA oscillatesbecause the actuator is not entirely symmetric. An increas-ing static pressure (working point) just causes a larger oscil-lation amplitude while frequency and phase are steady.
Thus, the frequency response of the mechanical displace-ment, as displayed in Fig. 8, was measured without staticpressure inside the chamber ( �→ working point is ①). Nev-ertheless, the center’s displacement reaches a maximum of220 µm at 700 Hz. For various DEA membranes, the ampli-tudes reach several hundred micrometer, even for frequen-
Dynamic performance of dielectric elastomers utilized as acoustic actuators 535
Fig. 6 Eigenfrequency decrease of two different DEA membranes(∅ 30 mm) by means of a constant electrical potential of 2 kV; bothmeasurements with identical boundary conditions
cies above 1000 Hz large amplitudes are possible. These re-sults strengthen the idea to apply the device as an acousticactuator.
With a view to investigating a possible application in thefield of acoustics, we measure the sound pressure inside theenclosed chamber (volume is 28 ccm; Fig. 2). The resultsof both, a displacement and a sound pressure measurement,are shown in Fig. 9. The recorded sound pressure level andthe displacement present a nearly identical behavior. Themaximum sound pressure level (SPL) in resonance is ap-proximately 121 dB (32 Pa). Yet, we are able to maximizethe sound pressure to 70 Pa (128 dB) by fabricating thinnerDEA membranes. In addition to the high peak pressure, weachieve an almost constant sound pressure level of 94 dBover a frequency band of 300 Hz. However, these resultswere obtained without any optimizations regarding acous-tic applications. Furthermore, the results can be optimizedby structuring the copper electrodes with photo lithographyand etching technology as well as by changing the mem-brane’s geometry or fabricating an array [18] of actuator
Fig. 7 Alternating displacement of the DEA membrane at the workingpoint by means of a sinusoidal, unipolar electrical excitation
Fig. 8 Repeated measurements of the alternating displacement dueto sinusoidal electrical excitation of 2.8 kV (pp); no constant pres-sure inside the chamber �→ working point is ① (sample: VHB™4905,∅ 30 mm, thickness z = 35 µm)
membranes. Hence, dielectric elastomer actuators show highpotential regarding their application as acoustic actuators.
These optimizations are not limited to the frequency re-sponse; we are also able to vary the sound pressure levelinside the chamber. The quadratic relation between the driv-
536 K. Hochradel et al.
Fig. 9 Pressure and displacement caused by an alternating high volt-age of 2 kV (pp); no constant pressure inside the chamber �→ workingpoint is ① (sample: VHB™9469, ∅ 12 mm, thickness z = 27 µm)
Fig. 10 Pressure of an electrically excited sample between 400 V and800 V; no constant pressure inside the chamber �→ working point is ①(sample: VHB™4905, ∅ 12 mm, thickness z = 31 µm)
ing coupling quantity, the Maxwell stress, and the appliedvoltage (4)
σM ∼ V 2e (6)
allows us to influence the sound pressure radiated by themembrane. Figure 10 shows the acoustic pressure amplitudefor three measurements using different voltages on the samesample. Again, two resonances appear: one at 450 Hz, a sec-ond at 880 Hz. All three measurements show an identicalfrequency response and just vary in amplitude. In order toproof the quadratic relation, we determine the differences inamplitude of each measurement and compare them to thetheoretical value. According to (6), the results of these mea-
Table 1 Multipliers (7) between four different measurements; experi-ments prove the theoretical quadratic relation between Maxwell stressand voltage
400 V–600 V 600 V–800 V 800 V–1000 V
kemc theoretical 2.25 1.78 1.56
kemc measured 2.10 1.79 1.66
surements should differ by a factor of
kemc = V 2e1
V 2e2
. (7)
Table 1 shows the comparison of the theoretical quadratic re-lation between excitation voltage and caused Maxwell stressas well as the measured relation between excitation voltageand radiated sound pressure. The measured results match thepredicted quadratic relation.
3.5 Tuning of the acoustic actuator
The previous sections discussed the decreasing resonanceof DEA membranes as well as their application as acousticactuators. Furthermore, we briefly introduced several pos-sibilities to optimize the acoustic actuator regarding its fre-quency response and its radiated sound pressure level. Here,we show how the decreasing resonance frequency can be de-ployed to actively tune the frequency response of the acous-tic actuator. As in the previous section, we excite the actua-tor membrane with an alternating, but unipolar, electric po-tential of 2 kV (pp). Yet this time, we add an offset to varythe resonance frequency. The results of this combination aredisplayed in Fig. 11. The initial frequency response of thesound pressure level radiated by the DEA is decreased byan amount of 50 Hz. Thus, we are able to change the reso-nance frequency of the acoustic actuator just by varying theexcitation offset.
4 Numerical verification
Computational modeling is an important step toward the de-sign and fabrication of novel actuators. Therefore, we intro-duce a numerical verification of the experiments discussedin Sect. 3.3. As the elastomer features strongly viscoelas-tic and, in particular, hyperelastic characteristics, the mod-eling requires a mechanically non-linear ansatz. Numerousresearch groups work on novel modeling techniques [27–31] or use existing models (e.g., the Ogden model [32])to describe the nonlinearities of viscoelastic materials. Yet,Sect. 3.2 showed that the displacement caused by the al-ternating excitation pressure is linear due to the small de-flections. The linear relation is crucial since the deployed
Dynamic performance of dielectric elastomers utilized as acoustic actuators 537
Fig. 11 Combination of the decreasing eigenfrequency and the ap-plication as acoustic actuator; �→ acoustic actuator with tunable reso-nances (sample: ∅ 12 mm, thickness z = 31 µm)
numerical computation does not consider the nonlinearitiesof hyperelastic materials. Thus, we are able to simulate thedecreasing resonance of Sect. 3.3 by utilizing an appro-priate numerical computation based on the finite elementmethod [33]. The performed numerical simulation solves thecoupled problem between the electrical and the mechanicalfield. Prior to this, the stretching of the fabrication (Fig. 1)has to be solved. Thus, the simulation is divided into two se-quences, the first one (i) a static, mechanical field problemcomputing the mechanical prestress of the manufacturing,the second one (ii) solving the transient, coupled field prob-lem. Figure 12 illustrates the simulation sequence.
The numerical simulation features an inaccuracy: the ge-ometry is not updated from sequence (i) to (ii). Accordingly,the thinning of the membrane due to the fabrication pre-stretch (Fig. 1) is disregarded and the geometry is modeledfor both sequences with the final metrics of the DEA mem-brane. Hence, the applied prestress of the first simulationsequence is arbitrary to match the unloaded, simulated withthe measured resonance. Accordingly, we had to modify theexcitation pressure of (ii) in order to match the simulatedand the measured displacement for an electrical excitationof 0 V. That is why, the results displayed in Fig. 13 showa perfect match of the resonances and displacements for avoltage of 0 V. Figure 13 shows the results of a measure-ment and a simulation (same thickness of the membranes).The simulation confirms the experimental results of the de-creasing eigenfrequency.
5 Conclusion
This research investigated the dynamic performance of di-electric elastomer actuators. The elastomers used for the in-
Fig. 12 Sequence of the numerical simulation
Fig. 13 Comparison of measurement and a corresponding simula-tion (sample: VHB™4905, ∅ 30 mm, thickness z = 42 µm); prestressadapted to match simulation and measurement for 0 V
538 K. Hochradel et al.
troduced experiments were 3M™’s VHB™4905 and 9469.We reported on the fabrication of the developed DEA mem-branes with a diameter of 30 mm (12 mm), a minimumthickness of 8 µm and sputtered copper electrodes on eachside. The experiments on these DEA membranes containedthe tuning of their resonance frequency, their applicationas acoustic actuators and the combination of both. The de-crease of the resonance frequency by 30% due to an elec-trical offset potential of this actuator membrane has beendemonstrated. A numerical computation solving the coupledfield problem verified these results. Furthermore, we demon-strated the applicability of DEA membranes as acoustic ac-tuators. A sound pressure level of 130 dB in a 28 ccm vol-ume has been achieved as the actuator membrane was ex-cited with an alternating, but unipolar, electrical potentialof 2 kV. The radiated sound pressure level as well as theresonances were modified by increasing the applied electricpotential, respectively, by adding an offset to the alternatingelectrical potential.
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