Stacked Dielectric Elastomer

Thin membrane structures can assume complex shapesunder the influence of boundary geometry. A simple exam-ple is a soap film supported within a non-planar boundarythat assumes a shape minimizing the film’s stored elasticstrain and gravitational energy. Kofod and co-workers53,54

have taken this a step further, by coupling a stretched DEmembrane to a flexible plastic frame. The resulting self-configuring electro-active structure can be used for smallmachines. Examples include a mini-gripper54 that openswhen the voltage is on (Fig. 10), or an array of touch-sensitive actuators for a conveyor, as developed by O’Brienet al.55 The latter device is described further in Sec. III.

In all of the examples cited above, workers have usedthe in-plane electro-active extension or expansion strain of

DE for actuation. To achieve muscle-like contraction, wecan also take advantage of DE thickness reduction. Thispresents some interesting engineering challenges associatedwith the stacking of many thin layers of muscle. Severalmethods for manufacturing such actuators have been pro-posed. Carpi et al.56 produced a multi-level contractile actua-tor consisting of an elastomer helix electroded on top andbottom and sealed with a thin layer of silicone (Fig. 11) thatwas capable of contractile strains of 5% for an electric fieldof 14 MV/m. The group has also produced an easier to fabri-cate, silicone sealed, stack actuator built up from a foldedsingle layer of an electroded silicone dielectric57 (Fig. 11).Folded actuators 85 mm high and 25 mm wide were pro-duced from folded layers that were 0.5-0.8 mm thick.

FIG. 12. (a) Schematic of the stack actuator design by Kovacs et al.58 (b) Stack actuators in action. Reprinted with permission from Ref. 58.

FIG. 13. (Top) Schematic of three-stageautomated process for producing multi-layerpatterned devices. (Bottom) The vibrotactiledisplay. Demonstrator with highlighted actu-ator elements and contact areas. Reprintedwith permission from Lotz et al.62

041101-8 Anderson et al. J. Appl. Phys. 112, 041101 (2012)

Downloaded 23 Aug 2013 to 128.237.174.168. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

Thin membrane structures can assume complex shapesunder the influence of boundary geometry. A simple exam-ple is a soap film supported within a non-planar boundarythat assumes a shape minimizing the film’s stored elasticstrain and gravitational energy. Kofod and co-workers53,54

have taken this a step further, by coupling a stretched DEmembrane to a flexible plastic frame. The resulting self-configuring electro-active structure can be used for smallmachines. Examples include a mini-gripper54 that openswhen the voltage is on (Fig. 10), or an array of touch-sensitive actuators for a conveyor, as developed by O’Brienet al.55 The latter device is described further in Sec. III.

In all of the examples cited above, workers have usedthe in-plane electro-active extension or expansion strain of

DE for actuation. To achieve muscle-like contraction, wecan also take advantage of DE thickness reduction. Thispresents some interesting engineering challenges associatedwith the stacking of many thin layers of muscle. Severalmethods for manufacturing such actuators have been pro-posed. Carpi et al.56 produced a multi-level contractile actua-tor consisting of an elastomer helix electroded on top andbottom and sealed with a thin layer of silicone (Fig. 11) thatwas capable of contractile strains of 5% for an electric fieldof 14 MV/m. The group has also produced an easier to fabri-cate, silicone sealed, stack actuator built up from a foldedsingle layer of an electroded silicone dielectric57 (Fig. 11).Folded actuators 85 mm high and 25 mm wide were pro-duced from folded layers that were 0.5-0.8 mm thick.

FIG. 12. (a) Schematic of the stack actuator design by Kovacs et al.58 (b) Stack actuators in action. Reprinted with permission from Ref. 58.

FIG. 13. (Top) Schematic of three-stageautomated process for producing multi-layerpatterned devices. (Bottom) The vibrotactiledisplay. Demonstrator with highlighted actu-ator elements and contact areas. Reprintedwith permission from Lotz et al.62

041101-8 Anderson et al. J. Appl. Phys. 112, 041101 (2012)

Downloaded 23 Aug 2013 to 128.237.174.168. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

Sensors and Actuators A 155 (2009) 299–307

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical

journa l homepage: www.e lsev ier .com/ locate /sna

Stacked dielectric elastomer actuator for tensile force transmission

G. Kovacs ∗, L. Düring, S. Michel, G. TerrasiEmpa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Mechanical System Engineering, Uberlandstrasse 129, 8600 Dubendorf, Switzerland

a r t i c l e i n f o

Article history:Received 26 February 2009Received in revised form 25 May 2009Accepted 30 August 2009Available online 8 September 2009

Keywords:Active structuresElectro-active polymers (EAPs)Soft dielectric EAPsContractive tension force actuator

a b s t r a c t

This paper presents a novel approach for active structures driven by soft dielectric electro-active polymers(EAPs), which can perform contractive displacements at external tensile load. The active structure iscomposed of an array of equal segments, where the dielectric films are arranged in a pile-up configuration.The proposed active structure has the capability of exhibiting uniaxial contractive deformations, whilebeing exposed to external tensile forces. The serial arrangement of active segments has one contractingdegree of freedom in the thickness direction of the dielectric EAP film layers.

Due to the envisaged tension force transmission capability, special attention is paid to the electrodedesign which is of paramount importance with regard to functionality of the actuator. A compliant elec-trode system with anisotropic deformation properties is presented based on nano scale carbon powder.In experiments, the free deformation as well as the contractive motion under external tensile loading ofseveral actuator configurations with different setups is characterized. These involve the study of varioussizes and numbers of stacked film layers as well as different electrode designs.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In the last decade, the interest in “smart materials”, whichrespond to external stimuli by changing their shape or size, hasessentially increased. In particular soft dielectric EAPs as muscle-like actuators, a subgroup of the electro-active polymers (EAPs),have attracted much interest in recent years due to their outstand-ing active deformation potential [1–4]. Soft dielectric EAPs consistof a thin elastomer film, which is coated on both sides with compli-ant electrodes (Fig. 1, left). When applying a DC high voltage U (inthe range of several kV) to this compliant capacitor, the electrodessqueeze the elastomeric dielectric in the thickness direction (elec-trode pressure, Pequivalent [5]), and thus the nearby incompressiblefilm expands in the planar direction (Fig. 1, right).

Pequivalent = ε0 · εr ·(

Ud

)2(1)

Thereby ε0 is the free-space dielectric permittivity(ε0 = 8.85 × 10−12 F/m), εr is the relative permittivity of thedielectric material and d represents the thickness of the dielectricfilm. As soon as the voltage is switched off and the electrodes areshort-circuited the film deforms back to its initial state.

So far, a variety of different types of dielectric elastomer (DE)actuators have demonstrated the versatile capabilities of this actu-ator technology (e.g. [6,7]). Due to their intrinsic compliance and

∗ Corresponding author. Tel.: +41 0 44 823 4063; fax: +41 0 44 823 4011.E-mail address: gabor.kovacs@empa.ch (G. Kovacs).

unique deformation potential, soft dielectric EAPs are promisingfor compliant, lightweight structures in the macro-scale, which canperform continuous displacements. The actuation of the dielectricelastomer films can be used in two different ways. Based on theprinciple of operation, two directions are possible to perform workagainst external loads (Fig. 2):

• Work in the planar directions (expanding actuator): Under electri-cal activation of a DE basic unit the film expands in the x, y plane(Fig. 1) and can thus work against external pressure loads in bothplanar directions x and y.

• Work in the thickness direction (contractile actuator): Under elec-trical activation the electrodes squeeze the DE film in thethickness direction (z) (Fig. 1). Thus, the actuator can work againstexternal tensile loads acting in the direction of the electric fieldlines of the compliant capacitor.

With regard to the elastomers used as the dielectric in DE actu-ators, the best overall performances have been shown by mainlysilicone and acrylic films, such as commercially available VHB 4910from 3 M [8,9]. The dielectric film, made from acrylic VHB 4910especially for DE actuators, is strongly pre-strained in planar direc-tions in order to reduce the thickness and therefore the requiredactivation voltage level. To maintain the DE film in this biaxiallypre-strained state a support structure is needed. To enable therequired displacements this support structure must offer the cor-responding mechanical degrees of freedom (DOF). Obviously, inpre-strained DE actuators the design of the support structure isone of the key issues and causes many design constraints.

0924-4247/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.sna.2009.08.027

Sensors and Actuators A 155 (2009) 299–307

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical

journa l homepage: www.e lsev ier .com/ locate /sna

Stacked dielectric elastomer actuator for tensile force transmission

G. Kovacs ∗, L. Düring, S. Michel, G. TerrasiEmpa, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Mechanical System Engineering, Uberlandstrasse 129, 8600 Dubendorf, Switzerland

a r t i c l e i n f o

Article history:Received 26 February 2009Received in revised form 25 May 2009Accepted 30 August 2009Available online 8 September 2009

Keywords:Active structuresElectro-active polymers (EAPs)Soft dielectric EAPsContractive tension force actuator

a b s t r a c t

This paper presents a novel approach for active structures driven by soft dielectric electro-active polymers(EAPs), which can perform contractive displacements at external tensile load. The active structure iscomposed of an array of equal segments, where the dielectric films are arranged in a pile-up configuration.The proposed active structure has the capability of exhibiting uniaxial contractive deformations, whilebeing exposed to external tensile forces. The serial arrangement of active segments has one contractingdegree of freedom in the thickness direction of the dielectric EAP film layers.

Due to the envisaged tension force transmission capability, special attention is paid to the electrodedesign which is of paramount importance with regard to functionality of the actuator. A compliant elec-trode system with anisotropic deformation properties is presented based on nano scale carbon powder.In experiments, the free deformation as well as the contractive motion under external tensile loading ofseveral actuator configurations with different setups is characterized. These involve the study of varioussizes and numbers of stacked film layers as well as different electrode designs.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In the last decade, the interest in “smart materials”, whichrespond to external stimuli by changing their shape or size, hasessentially increased. In particular soft dielectric EAPs as muscle-like actuators, a subgroup of the electro-active polymers (EAPs),have attracted much interest in recent years due to their outstand-ing active deformation potential [1–4]. Soft dielectric EAPs consistof a thin elastomer film, which is coated on both sides with compli-ant electrodes (Fig. 1, left). When applying a DC high voltage U (inthe range of several kV) to this compliant capacitor, the electrodessqueeze the elastomeric dielectric in the thickness direction (elec-trode pressure, Pequivalent [5]), and thus the nearby incompressiblefilm expands in the planar direction (Fig. 1, right).

Pequivalent = ε0 · εr ·(

Ud

)2(1)

Thereby ε0 is the free-space dielectric permittivity(ε0 = 8.85 × 10−12 F/m), εr is the relative permittivity of thedielectric material and d represents the thickness of the dielectricfilm. As soon as the voltage is switched off and the electrodes areshort-circuited the film deforms back to its initial state.

So far, a variety of different types of dielectric elastomer (DE)actuators have demonstrated the versatile capabilities of this actu-ator technology (e.g. [6,7]). Due to their intrinsic compliance and

∗ Corresponding author. Tel.: +41 0 44 823 4063; fax: +41 0 44 823 4011.E-mail address: gabor.kovacs@empa.ch (G. Kovacs).

unique deformation potential, soft dielectric EAPs are promisingfor compliant, lightweight structures in the macro-scale, which canperform continuous displacements. The actuation of the dielectricelastomer films can be used in two different ways. Based on theprinciple of operation, two directions are possible to perform workagainst external loads (Fig. 2):

• Work in the planar directions (expanding actuator): Under electri-cal activation of a DE basic unit the film expands in the x, y plane(Fig. 1) and can thus work against external pressure loads in bothplanar directions x and y.

• Work in the thickness direction (contractile actuator): Under elec-trical activation the electrodes squeeze the DE film in thethickness direction (z) (Fig. 1). Thus, the actuator can work againstexternal tensile loads acting in the direction of the electric fieldlines of the compliant capacitor.

With regard to the elastomers used as the dielectric in DE actu-ators, the best overall performances have been shown by mainlysilicone and acrylic films, such as commercially available VHB 4910from 3 M [8,9]. The dielectric film, made from acrylic VHB 4910especially for DE actuators, is strongly pre-strained in planar direc-tions in order to reduce the thickness and therefore the requiredactivation voltage level. To maintain the DE film in this biaxiallypre-strained state a support structure is needed. To enable therequired displacements this support structure must offer the cor-responding mechanical degrees of freedom (DOF). Obviously, inpre-strained DE actuators the design of the support structure isone of the key issues and causes many design constraints.

0924-4247/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.sna.2009.08.027

Folded dielectric elastomer actuators

(a) (b)

Figure 12. Axial contraction of a prototype folded actuator withrectangular cross section: (a) rest state; (b) activation.

0 1 2 3 4 5 6 7 8 9 10 11 12-16-15-14-13-12-11-10

-9-8-7-6-5-4-3-2-1

Axi

al s

trai

n [%

]

Electric field [V/µm]

0 1 2 3 4 5 6 7 8 9 10 11 12

-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1

Rectangular cross-sectionNo pre-compression

Figure 13. Electric field dependence of the axial strain of a prototypefolded actuator with rectangular cross section and nopre-compression (a fitting curve is inserted as a guide for the eye).

(not lower than 0.5 mm). Thinner elastomers may behavedifferently, even in consideration of the stress experienced bythe material around the folds.

As another point, a sort of ‘optimization’ of the ratiobetween the electroded area of the cross section with respectto its total extension is considered of fundamental importance.For a definite geometry of the cross section, this implies areduction of the inactive area to the limit electrically permitted(in order to avoid lateral electric discharges). However,if the geometry of the cross section is not constrained bythe application, the use of a rectangular shape seems to bepreferable. In fact, this permits us to limit the inactive portionsto a couple of lateral bands (increasing the above-mentionedarea ratio). Therefore, for the same value of the active area, arectangular cross section seems to be better than a circular one.As an example, figure 12 presents a contraction of a prototypesample of such an actuator. The efficacy of this elementaryconfiguration is shown by the preliminary data reported infigure 13. They show that strains greater than those of a circularactuator can be achieved.

These preliminary data encourage further investigationsaimed at identifying optimized dimensional parameters, so as

to obtain the most from this simple and promising new type ofcontractile actuator.

6. Conclusions

The concept of a new dielectric elastomer actuator withcontractile linear properties has been described. The proposedconfiguration permits us to obtain a monolithic compactstructure, functionally equivalent to a multi-layer stack butwith continuous electrodes. Silicone-made prototypes havebeen demonstrated. The simplicity of fabrication of this newtype of actuator may facilitate the short-term diffusion ofcontractile devices, to be readily employed for a broad rangeof potential applications.

References

[1] Bar-Cohen Y (ed) 2004 Electroactive Polymer (EAP) Actuatorsas Artificial Muscles. Reality, Potential, and Challenges2nd edn (Bellingham, WA: SPIE Press)

[2] Madden J D W et al 2004 Artificial muscle technology:physical principles and naval prospects IEEE J. Ocean. Eng.29 706–28

[3] Ashley S 2003 Artificial muscles Sci. Am. (October) 52–9[4] Pelrine R E, Kornbluh R D and Joseph J P 1998 Electrostriction

of polymer dielectrics with compliant electrodes as a meansof actuation Sensors Actuators A 64 77–85

[5] Pelrine R, Kornbluh R, Pei Q and Joseph J 2000 High-speedelectrically actuated elastomers with strain greater than100% Science 287 836–9

[6] Pelrine R, Kornbluh R and Kofod G 2000 High-strain actuatormaterials based on dielectric elastomers Adv. Mater.12 1223–5

[7] Pelrine R, Kornbluh R, Joseph J, Heydt R, Pei Q andChiba S 2000 High-field deformation of elastomericdielectrics for actuators Mater. Sci. Eng. C 11 89–100

[8] Pei Q, Pelrine R, Stanford S, Kornbluh R andRosenthal M 2003 Electroelastomer rolls and theirapplication for biomimetic walking robots Synth. Met.135/136 129–31

[9] Pei Q, Rosenthal M, Stanford S, Prahlad H and Pelrine R 2004Multiple-degrees-of-freedom electroelastomer roll actuatorsSmart Mater. Struct. 13 N86–92

[10] Kofod G, Sommer-Larsen P, Kornbluh R and Pelrine R 2003Actuation response of polyacrylate dielectric elastomersJ. Intell. Mater. Syst. Struct. 14 787–93

[11] Schlaak H, Jungmann M, Matysek M and Lotz P 2005 Novelmultilayer electrostatic solid-state actuators with elasticdielectric Smart Structures and Materials 2005:Electroactive Polymer Actuators and Devices, Proc. SPIE5759 121–33

[12] Carpi F and De Rossi D 2004 Dielectric elastomer cylindricalactuators: electromechanical modelling and experimentalevaluation Mater. Sci. Eng. C 24 555–62

[13] Carpi F, Migliore A, Serra G and De Rossi D 2005 Helicaldielectric elastomer actuators Smart Mater. Struct.14 1210–6

[14] Carpi F and De Rossi D 2005 Attuatore, sensore e generatore apolimeri elettroattivi in configurazione ripiegata ItalianPatent Application No PI/2005/A/000095

[15] Carpi F and De Rossi D 2005 Improvement ofelectromechanical actuating performances of a siliconedielectric elastomer by dispersion of titanium dioxidepowder IEEE Trans. Dielectr. Electr. Insul. 12 835–43

S305

Folded dielectric elastomer actuators

(a) (b)

Figure 12. Axial contraction of a prototype folded actuator withrectangular cross section: (a) rest state; (b) activation.

0 1 2 3 4 5 6 7 8 9 10 11 12-16-15-14-13-12-11-10

-9-8-7-6-5-4-3-2-1

Axi

al s

trai

n [%

]

Electric field [V/µm]

0 1 2 3 4 5 6 7 8 9 10 11 12

-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1

Rectangular cross-sectionNo pre-compression

Figure 13. Electric field dependence of the axial strain of a prototypefolded actuator with rectangular cross section and nopre-compression (a fitting curve is inserted as a guide for the eye).

(not lower than 0.5 mm). Thinner elastomers may behavedifferently, even in consideration of the stress experienced bythe material around the folds.

As another point, a sort of ‘optimization’ of the ratiobetween the electroded area of the cross section with respectto its total extension is considered of fundamental importance.For a definite geometry of the cross section, this implies areduction of the inactive area to the limit electrically permitted(in order to avoid lateral electric discharges). However,if the geometry of the cross section is not constrained bythe application, the use of a rectangular shape seems to bepreferable. In fact, this permits us to limit the inactive portionsto a couple of lateral bands (increasing the above-mentionedarea ratio). Therefore, for the same value of the active area, arectangular cross section seems to be better than a circular one.As an example, figure 12 presents a contraction of a prototypesample of such an actuator. The efficacy of this elementaryconfiguration is shown by the preliminary data reported infigure 13. They show that strains greater than those of a circularactuator can be achieved.

These preliminary data encourage further investigationsaimed at identifying optimized dimensional parameters, so as

to obtain the most from this simple and promising new type ofcontractile actuator.

6. Conclusions

The concept of a new dielectric elastomer actuator withcontractile linear properties has been described. The proposedconfiguration permits us to obtain a monolithic compactstructure, functionally equivalent to a multi-layer stack butwith continuous electrodes. Silicone-made prototypes havebeen demonstrated. The simplicity of fabrication of this newtype of actuator may facilitate the short-term diffusion ofcontractile devices, to be readily employed for a broad rangeof potential applications.

References

[1] Bar-Cohen Y (ed) 2004 Electroactive Polymer (EAP) Actuatorsas Artificial Muscles. Reality, Potential, and Challenges2nd edn (Bellingham, WA: SPIE Press)

[2] Madden J D W et al 2004 Artificial muscle technology:physical principles and naval prospects IEEE J. Ocean. Eng.29 706–28

[3] Ashley S 2003 Artificial muscles Sci. Am. (October) 52–9[4] Pelrine R E, Kornbluh R D and Joseph J P 1998 Electrostriction

of polymer dielectrics with compliant electrodes as a meansof actuation Sensors Actuators A 64 77–85

[5] Pelrine R, Kornbluh R, Pei Q and Joseph J 2000 High-speedelectrically actuated elastomers with strain greater than100% Science 287 836–9

[6] Pelrine R, Kornbluh R and Kofod G 2000 High-strain actuatormaterials based on dielectric elastomers Adv. Mater.12 1223–5

[7] Pelrine R, Kornbluh R, Joseph J, Heydt R, Pei Q andChiba S 2000 High-field deformation of elastomericdielectrics for actuators Mater. Sci. Eng. C 11 89–100

[8] Pei Q, Pelrine R, Stanford S, Kornbluh R andRosenthal M 2003 Electroelastomer rolls and theirapplication for biomimetic walking robots Synth. Met.135/136 129–31

[9] Pei Q, Rosenthal M, Stanford S, Prahlad H and Pelrine R 2004Multiple-degrees-of-freedom electroelastomer roll actuatorsSmart Mater. Struct. 13 N86–92

[10] Kofod G, Sommer-Larsen P, Kornbluh R and Pelrine R 2003Actuation response of polyacrylate dielectric elastomersJ. Intell. Mater. Syst. Struct. 14 787–93

[11] Schlaak H, Jungmann M, Matysek M and Lotz P 2005 Novelmultilayer electrostatic solid-state actuators with elasticdielectric Smart Structures and Materials 2005:Electroactive Polymer Actuators and Devices, Proc. SPIE5759 121–33

[12] Carpi F and De Rossi D 2004 Dielectric elastomer cylindricalactuators: electromechanical modelling and experimentalevaluation Mater. Sci. Eng. C 24 555–62

[13] Carpi F, Migliore A, Serra G and De Rossi D 2005 Helicaldielectric elastomer actuators Smart Mater. Struct.14 1210–6

[14] Carpi F and De Rossi D 2005 Attuatore, sensore e generatore apolimeri elettroattivi in configurazione ripiegata ItalianPatent Application No PI/2005/A/000095

[15] Carpi F and De Rossi D 2005 Improvement ofelectromechanical actuating performances of a siliconedielectric elastomer by dispersion of titanium dioxidepowder IEEE Trans. Dielectr. Electr. Insul. 12 835–43

S305

IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) S300–S305 doi:10.1088/0964-1726/16/2/S15

Folded dielectric elastomer actuatorsFederico Carpi, Claudio Salaris and Danilo De Rossi

Interdepartmental Research Centre ‘E Piaggio’, School of Engineering, University of Pisa,via Diotisalvi, 2, I-56100 Pisa, Italy

E-mail: f.carpi@ing.unipi.it

Received 20 June 2006, in final form 18 July 2006Published 9 March 2007Online at stacks.iop.org/SMS/16/S300

AbstractPolymer-based linear actuators with contractile ability are currentlydemanded for several types of applications. Within the class of dielectricelastomer actuators, two basic configurations are available today for such apurpose: the multi-layer stack and the helical structure. The first consists ofseveral layers of elementary planar actuators stacked in series mechanicallyand parallel electrically. The second configuration relies on a couple ofhelical compliant electrodes alternated with a couple of helical dielectrics.The fabrication of both these configurations presents some specificdrawbacks today, arising from the peculiarity of each structure. Accordingly,the availability of simpler solutions may boost the short-term use ofcontractile actuators in practical applications. For this purpose, a newconfiguration is here described. It consists of a monolithic structure made ofan electroded sheet, which is folded up and compacted. The resulting deviceis functionally equivalent to a multi-layer stack with interdigitated electrodes.However, with respect to a stack the new configuration is advantageously notdiscontinuous and can be manufactured in one single phase, avoidinglayer-by-layer multi-step procedures. The development and preliminarytesting of prototype samples of this new actuator made of a siliconeelastomer are presented here.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Several types of polymers present intrinsic high compliance,lightness, high processability and the usual low cost.These properties, in addition to different types of activefunctionalities they can show, are frequently regarded asadvantageous for the use of such materials for specificactuation tasks. In particular, actuators based on electroactivepolymers (EAP) offer attractive performances in these andother respects [1, 2]. They are being studied for a very broadspectrum of potential fields of application, such as robotics,biomedical engineering, automotive, aerospace, etc. However,at present a large number of such areas of application wouldbenefit from new reliable actuators having linear contractileproperties. For instance, they may find challenging uses as‘artificial muscles’ [1, 2]. Prosthetics, orthotics, robotics andbio-robotics are just a few examples of the general areas ofpotential application for such devices.

Within the broad EAP family, actuators based on dielectricelastomers represent one of the most performing technologies.

They consist of insulating soft elastomers subjected to highelectric fields, applied via compliant electrodes [3–5]. Withrespect to different EAP actuators, they can exhibit very largedeformations at medium–high stresses. The electromechanicalresponse is mainly due to an electrostatic compression, whicharises from interactions among the electrode free charges(Maxwell stress effect). In particular, when a thin film ofa dielectric elastomer is sandwiched between two compliantelectrodes charged by a high voltage difference, a squeezing ofthe material at constant volume is achieved [3–5].

Several research efforts are today being focused ondielectric elastomer actuation. Accordingly, a quick wideningof the affordable range of applications is expected in thenext few years. A large number of different types ofdielectric elastomer actuators have been demonstrated so far.Most notable examples include planar devices, rolls, tubes,stacks, diaphragms, extenders, bimorph and unimorph benders,etc [3–12]. Nevertheless, despite such a variety of devices,very few solutions are currently available to obtain linearcontractile actuators. A first type of approach adopts auxiliary

0964-1726/07/020300+06$30.00 © 2007 IOP Publishing Ltd Printed in the UK S300

IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) S300–S305 doi:10.1088/0964-1726/16/2/S15

Folded dielectric elastomer actuatorsFederico Carpi, Claudio Salaris and Danilo De Rossi

Interdepartmental Research Centre ‘E Piaggio’, School of Engineering, University of Pisa,via Diotisalvi, 2, I-56100 Pisa, Italy

E-mail: f.carpi@ing.unipi.it

Received 20 June 2006, in final form 18 July 2006Published 9 March 2007Online at stacks.iop.org/SMS/16/S300

AbstractPolymer-based linear actuators with contractile ability are currentlydemanded for several types of applications. Within the class of dielectricelastomer actuators, two basic configurations are available today for such apurpose: the multi-layer stack and the helical structure. The first consists ofseveral layers of elementary planar actuators stacked in series mechanicallyand parallel electrically. The second configuration relies on a couple ofhelical compliant electrodes alternated with a couple of helical dielectrics.The fabrication of both these configurations presents some specificdrawbacks today, arising from the peculiarity of each structure. Accordingly,the availability of simpler solutions may boost the short-term use ofcontractile actuators in practical applications. For this purpose, a newconfiguration is here described. It consists of a monolithic structure made ofan electroded sheet, which is folded up and compacted. The resulting deviceis functionally equivalent to a multi-layer stack with interdigitated electrodes.However, with respect to a stack the new configuration is advantageously notdiscontinuous and can be manufactured in one single phase, avoidinglayer-by-layer multi-step procedures. The development and preliminarytesting of prototype samples of this new actuator made of a siliconeelastomer are presented here.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Several types of polymers present intrinsic high compliance,lightness, high processability and the usual low cost.These properties, in addition to different types of activefunctionalities they can show, are frequently regarded asadvantageous for the use of such materials for specificactuation tasks. In particular, actuators based on electroactivepolymers (EAP) offer attractive performances in these andother respects [1, 2]. They are being studied for a very broadspectrum of potential fields of application, such as robotics,biomedical engineering, automotive, aerospace, etc. However,at present a large number of such areas of application wouldbenefit from new reliable actuators having linear contractileproperties. For instance, they may find challenging uses as‘artificial muscles’ [1, 2]. Prosthetics, orthotics, robotics andbio-robotics are just a few examples of the general areas ofpotential application for such devices.

Within the broad EAP family, actuators based on dielectricelastomers represent one of the most performing technologies.

They consist of insulating soft elastomers subjected to highelectric fields, applied via compliant electrodes [3–5]. Withrespect to different EAP actuators, they can exhibit very largedeformations at medium–high stresses. The electromechanicalresponse is mainly due to an electrostatic compression, whicharises from interactions among the electrode free charges(Maxwell stress effect). In particular, when a thin film ofa dielectric elastomer is sandwiched between two compliantelectrodes charged by a high voltage difference, a squeezing ofthe material at constant volume is achieved [3–5].

Several research efforts are today being focused ondielectric elastomer actuation. Accordingly, a quick wideningof the affordable range of applications is expected in thenext few years. A large number of different types ofdielectric elastomer actuators have been demonstrated so far.Most notable examples include planar devices, rolls, tubes,stacks, diaphragms, extenders, bimorph and unimorph benders,etc [3–12]. Nevertheless, despite such a variety of devices,very few solutions are currently available to obtain linearcontractile actuators. A first type of approach adopts auxiliary

0964-1726/07/020300+06$30.00 © 2007 IOP Publishing Ltd Printed in the UK S300

Dielectric: BJB TC-5005 Silicone/PDMS Conductor: CAF 4 Rhodorsil Silicone + Vulcan XC R72 Carbon Powder

Dielectric: VHB 4910 Acrylic Elastomer infused w/ a penetrating network of 1,6-hexanediol diacrylate for enhanced dielectric breakdown strength Conductor: Ketjenblack 600 (Akzo Nobel) Carbon Powder

u0 = Λsin(ω0t) u0 ε

time, t

Energy Harvesting (Damping)

ΔC

Actuation(Amplification)

V

“Stacked” Soft-Matter Capacitor

•  Electromechanical coupling: C/C0 ~ λ2

•  Harmonic excitation at base •  Stready state ΔC and λ controlled

by Λ, |ωn ω0|, rubber viscosity η, and electrostatic damping from Maxwell Stress

•  Phase of electrostatic “loading” matters o w/ contraction ⇒amplification o w/ stretch ⇒ damping

A Supply Electrostatic Charge

B Stretch to Length L

L0

A0

z ∈ [0, L0]

C Remove Charge

F F

L

D Remove Load

L

F̂ F̂

Energy Harvesting

36

B

L

F̂

D

F

L0Length

Force

F0A

C

C

Q0

Φ0

A

Voltage

Charge B

Φ

L0

F0 F0

Viscoelasticity

λ

1

λ* = λ :σ =C1 λ2 −1λ

$

%&

'

()

*+,

-./

λ

t

σ

t

“Creep”

When loaded with a dead weight P, most elastomers stretch over time:

L0λ(t) L0

σ =λPA0

Viscoelasticity

Maxwell Model: dλdt

=1Edσdt+ση

λ

t

σ

t

λ

1

dλdt

≈1Edσdt

⇒σ = E λ −1( )

dλdt

≈ση⇒ λ = λ* +

σηt − t0( )

λ =σηtf − t0( )

Viscoelasticity

Kelvin-Voigt Model: σ = E λ −1( )+ηdλdt

σ

t

λλ

1

t

λ =1+ σ0E1− e−E t−t0( )/η{ }

λ =1+ λ f −1( )e−E t−tf( )/η

Viscoelasticity

For a Neo-Hookean Solid subject to unixaxial loading and Maxwell Stress,

σ3 −σM = C1 λ2 −1λ

$

%&

'

()

Including the viscoelasticity term, this becomes

σ =C1 λ2 −1λ

$

%&

'

()+

εΦ2

λ2h02 +η

dλdt

0 5 10 15 20 25 301

1.005

1.01

1.015

1.02

1.025

1.03

1.035

time (s)

λ

Φ = 0 Φ = 5 kV Φ = 0

σM = εE2 =

εΦ2

λ2t02

HW 3

λ

Stacked Capacitor

m

L0 + x(t) L0

Kovacs et al. (EMPA Switzerland)

24-673 HW 3: Soft Conductors and Dielectrics Due: 3/2/15

4 Stacked DEA. As in lecture, consider a stacked DEA with an initial radius R0 = 1cm length L0 = 10 cm, and dielectric gap h0 = 0.1 mm, The DEA contains N = L0/h0 = 1000 capacitive elements and the thickness and rigidity of the capacitive electrodes will be ignored.

Let the dielectric be treated as a Neo-Hookean solid with an added Maxwell stress and viscosity term

σ = 2C1 λ2 −1λ

$

%&

'

()+ε

Φλh0

$

%&

'

()

2

+ηdλdt

with modulus E = 1 MPa, dielectric constant εr = 2, and viscosity η = 1 MPa-s. [4 points]

a) Suppose that at time t = 0s, the DEA is loaded with a tensile deadweight P = 10 N. Plot λ vs. t over the domain 0s ≤ t ≤ 10s .

b) Now suppose that starting at time t = 10s, a voltage Φ = 5 kV is applied for 10s. Reconstruct the plot below for λ vs. time t over the domain 0s ≤ t ≤ 30s .

05

1015

20

01

23

450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Initial Pressure (kPa)Voltage (kV)

d2Π

/dλ2

(J)

0 5 10 15 20 25 300.98

0.99

1

1.01

1.02

1.03

1.04

time (s)

λ

HW 3

24-673 HW 3: Soft Conductors and Dielectrics Due: 3/2/15

4 Stacked DEA. As in lecture, consider a stacked DEA with an initial radius R0 = 1cm length L0 = 10 cm, and dielectric gap h0 = 0.1 mm, The DEA contains N = L0/h0 = 1000 capacitive elements and the thickness and rigidity of the capacitive electrodes will be ignored.

Let the dielectric be treated as a Neo-Hookean solid with an added Maxwell stress and viscosity term

σ = 2C1 λ2 −1λ

$

%&

'

()+ε

Φλh0

$

%&

'

()

2

+ηdλdt

with modulus E = 1 MPa, dielectric constant εr = 2, and viscosity η = 1 MPa-s. [4 points]

a) Suppose that at time t = 0s, the DEA is loaded with a tensile deadweight P = 10 N. Plot λ vs. t over the domain 0s ≤ t ≤ 10s .

b) Now suppose that starting at time t = 10s, a voltage Φ = 5 kV is applied for 10s. Reconstruct the plot below for λ vs. time t over the domain 0s ≤ t ≤ 30s .

05

1015

20

01

23

450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Initial Pressure (kPa)Voltage (kV)

d2Π

/dλ2

(J)

0 5 10 15 20 25 300.98

0.99

1

1.01

1.02

1.03

1.04

time (s)

λ

24-673 HW 3: Soft Conductors and Dielectrics Due: 3/2/15

4 Stacked DEA. As in lecture, consider a stacked DEA with an initial radius R0 = 1cm length L0 = 10 cm, and dielectric gap h0 = 0.1 mm, The DEA contains N = L0/h0 = 1000 capacitive elements and the thickness and rigidity of the capacitive electrodes will be ignored.

Let the dielectric be treated as a Neo-Hookean solid with an added Maxwell stress and viscosity term

σ = 2C1 λ2 −1λ

$

%&

'

()+ε

Φλh0

$

%&

'

()

2

+ηdλdt

with modulus E = 1 MPa, dielectric constant εr = 2, and viscosity η = 1 MPa-s. [4 points]

a) Suppose that at time t = 0s, the DEA is loaded with a tensile deadweight P = 10 N. Plot λ vs. t over the domain 0s ≤ t ≤ 10s .

b) Now suppose that starting at time t = 10s, a voltage Φ = 5 kV is applied for 10s. Reconstruct the plot below for λ vs. time t over the domain 0s ≤ t ≤ 30s .

05

1015

20

01

23

450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Initial Pressure (kPa)Voltage (kV)

d2Π

/dλ2

(J)

0 5 10 15 20 25 300.98

0.99

1

1.01

1.02

1.03

1.04

time (s)

λ

24-673 HW #3 Solution 2/22/16

function xdot = DEA(t,x) global L0 R0 h0 eta P C1 A = pi*R0^2/x; xdot = (P/A - 2*C1*(x^2 - 1/x))/eta;

b) function prob2b_alt global L0 R0 h0 eta P C1 eps V L0 = 0.1; R0 = 0.01; h0 = 0.1e-3; eta = 1e6; P = 10; E = 1e6; C1 = E/6; eps = 2*8.85e-12; V = 5e3; t0 = 0; t1 = 10; t2 = 20; tf = 30; [T1,lambda1] = ode45(@DEA1,[t0 t1],1); N1 = length(lambda1); [T2,lambda2] = ode45(@DEA2,[t1 t2],lambda1(N1)); N2 = length(lambda2); [T3,lambda3] = ode45(@DEA1,[t2 tf],lambda2(N2)); plot(T1,lambda1,'k-',T2,lambda2,'k-',T3,lambda3,'k-') xlabel('time (s)') ylabel('\lambda') %%%%%%%%%%%

0 2 4 6 8 101

1.005

1.01

1.015

1.02

1.025

1.03

1.035

time (s)

λ

m Mass

Vibrating Structure Elastomer (E, h, A, L)

m

u(t)

L(t)

y(t)

m y

F −F ≡my

u = u0 sin ωt( )y = u+L

L = L0 + x = λL0y 0( ) = L0y 0( ) = 0

⇒ y = L0 + u + x

σ = Eε+ηεKelvin-Voigt Solid: ε =xL0

ε =xL0

k c

F ≈ σA0 =EA0L0

x+ ηA0L0x

−F ≡my⇒ x+ Fm= −u

y 0( ) = L0 ⇒ x 0( ) = 0y 0( ) = 0⇒ x 0( ) = −u0ω

x+ cmx+ kmx = u0ω

2 sin ωt( )

Linearized Model

ωn =km

It is convenient to define the following:

x+ 2ζωn x+ωn2x = u0ω

2 sin ωt( )

Damping Ratio:

Natural Frequency:

ζ =c

2 mk

Solution: x = xc + xp

“complementary solution”

“particular solution”

x 0( ) = 0x 0( ) = −u0ω

Natural Frequency

Recall that for a spring mass system, the natural frequency is

ωn =km

For a Hookean solid, k = EA0/L0

L0 = 10 cm R0 = 1 cm h0 = 0.1 mm u0 = 1 mm

E = 1 MPa η = 100 Pa-s m = 10 mg εr = 2

ωn = 560.5 Hz

Solution Summary

X = u0ω2

ωn2 −ω2( )

2+ 2ζωnω( )

2

x =u0ω

2 sin ωt + φ( )

ωn2 −ω2( )

2+ 2ζωnω( )

2

X1 =u0ω

2 ωn2 −ω2( )

ωn2 −ω2( )

2+ 2ζωnω( )

2X2 =

−u0ω2 2ζωnω( )

ωn2 −ω2( )

2+ 2ζωnω( )

2

x =A1eωn −ζ+iωd( )t +A2e

ωn −ζ−iωd( )t +X1sin ωt( )+X2 cos ωt( )

At steady state, this converges to

where

is the steady state amplitude

ωd = 1− ζ2

A2 =λ1X2 − u0ω−ωX1

λ2 − λ1A1 =

λ2X2 − u0ω−ωX1λ1 − λ2

λ1,2 =ωn −ζ± ζ2 −1( )

and

φ = nπ+ tan−1 X2X1

$

%&

'

()

Complementary Solution

xc + 2ζωn xc +ωn2xc = 0

xc = Aieλit∑

⇒ Aieλit λ i

2 + 2ζωnλ i +ωn2{ }∑ = 0

For this to be satisfied for arbitrary Ai and t,

λ i2 + 2ζωnλ i +ωn

2 = 0

This only has two linearily independent solutions. Therefore in general,

xc =A1eλ1t +A2e

λ2t where λ1,2 =ωn −ζ± ζ2 −1( )

Particular Solution

We postulate that xp has the form

xp = X1sin ωt( )+X2 cos ωt( )Substituting xc + xp into the governing ODE and noting

xc + 2ζωn xc +ωn2xc = 0,

it follows that xp + 2ζωn xp +ωn2xp = u0ω

2 sin ωt( )Substituting the expression for xp into the above ODE,

−X1ω2 sin ωt( )−X2ω2 cos ωt( ){ }

+2ζωn X1ωcos ωt( )−X2ωsin ωt( ){ }+ωn

2 X1 sin ωt( )+X2 cos ωt( ){ }= u0ω2 sin ωt( )

Rearranging terms,

−X1ω2 − 2ζωnωX2 +ωn

2X1{ }sin ωt( )+ −X2ω

2 + 2ζωnωX1 +ωn2X2{ }cos ωt( )

= u0ω2 sin ωt( )

⇒ωn2 −ω2( )X1 − 2ζωnωX2 = u0ω2

2ζωnωX1 + ωn2 −ω2( )X2 = 0

%

&'

('

X1 =u0ω

2 ωn2 −ω2( )

ωn2 −ω2( )

2+ 2ζωnω( )

2X2 =

−u0ω2 2ζωnω( )

ωn2 −ω2( )

2+ 2ζωnω( )

2

Noting that sin(A+B) = sin(A)cos(B) + cos(A)sin(B), it follows that xp can also be expressed as

xp = Xsin ωt +φ( ),

where X = X12 +X2

2 =u0ω

2

ωn2 −ω2( )

2+ 2ζωnω( )

2

is the amplitude, and

φ = nπ+ tan−1 X2X1

$

%&

'

() is the phase shift.

Boundary Conditions

Lastly, A1 and A2 are determined by applying the boundary conditions x 0( ) = 0 x 0( ) = −u0ω

x =A1eλ1t +A2e

λ2t +X1sin ωt( )+X2 cos ωt( )x = λ1A1e

λ1t +λ2A2eλ2t +ωX1 cos ωt( )−ωX2 sin ωt( )

x 0( ) = 0⇒ A1 +A2 +X2 = 0⇒ A1 = − A2 +X2( )

x 0( ) = −u0ω⇒−λ1 A2 +X2( )+λ2A2 +ωX1 = −u0ω

⇒ A2 =λ1X2 − u0ω−ωX1

λ2 −λ1

Solution

The coefficients, A1, A2, X1, and X2 are real. However, for an underdamped system (ζ < 1) λ1 and λ2 will be imaginary.

x =A1eωn −ζ+iωd( )t +A2e

ωn −ζ−iωd( )t +X1sin ωt( )+X2 cos ωt( )

ωd = 1−ζ2 .

= e−ζωnt A1eiωnωdt +A2e

−iωnωdt{ }+X1sin ωt( )+X2 cos ωt( )where

The first term vanishes as t approaches ∞, and so it is known as the “transient solution.” The solution converges to the last two terms (particular solution), which is known as the “steady state solution.”

The smaller the damping, the longer it takes to reach steady state and the larger the steady state amplitude X.

Once the steady state solution is reached, the length of the viscoelastic solid will change with amplitude

X = u0ω2

ωn2 −ω2( )

2+ 2ζωnω( )

2

Clearly, this is maximized when the excitation frequency ω of the external (base) vibration matches the natural frequency ωn of the system: Xmax = u0/2ζ. Interestingly, X approaches zero when ω is small. This means that the elastomer doesn’t deform and the mass vibrates with the base. When ω gets large, X approaches u0. This is not intuitive and should be examined in more detail.

0 0.5 1 1.5 2 2.5 30

1

2

3

4X/u 0

ω/ωn

Xmax/u0 = 1/2ζ

ω = ωn

ω = 0.1ωn

x ≈ 0 ∀t

y ≈ L0 + u0 sin ωt( )

0 10 20 30 40−2

0

2

4

6

8

10

12

t (milliseconds)m

illim

eter

s

xuy

0 5 10 15 20 25 30 35 40−5

0

5

10

15

t (milliseconds)

milli

met

ers

xuy

ω = ωn

x→ u02ζsin ωnt −

π2

&

'(

)

*+ y→ L0 + u0 sin ωnt( )+

u02ζsin ωnt −

π2

&

'(

)

*+

∴y→ L0 +Ysin ωnt −φ( )Y = u0 1+ 2ζ( )

−2

φ = tan−1 1 2ζ( )

= L0 + u0 sin ωnt( )−u02ζcos ωnt( )

0 0.5 1 1.5 2 2.5 3 3.5 4−5

0

5

10

15

t (milliseconds)

milli

met

ers

xuy

ω = 5ωn

x ≈ −u0 sin ωnt( )

y ≈ L0

X1 ≈ −u0 X2 ≈ 0 ⇒

Displacements are equal-and-opposite à mass remains stationary

σ = 2C1 λ2 −1λ

$

%&

'

()+η

dλdt+ε

Φλh0

$

%&

'

()

2

Kelvin-Voigt Solid:

λ =LL0

dλdt

=1L0dLdt

y 0( ) = L0 ⇒λ 0( ) =1

y 0( ) = 0⇒ λ 0( ) =1− u0ωL0

F = σA0λ

= 2C1A0 λ −1λ2

$

%&

'

()+

ηA0λdλdt+εA0λ3h0

2 Φ2

−F ≡m!!y =m ddt

λL0 + u{ }

⇒ mL0!!λ −mu0ω2 sin ωt( )+ 2C1A0 λ −

1λ2

⎛

⎝⎜

⎞

⎠⎟+

ηA0λ!λ+

εA0h02λ3

Φ2 = 0

λ = u0ω2

L0sin ωt( )− εA0

mL0h02λ3

Φ2 −2C1A0mL0

λ −1λ2

&

'(

)

*+−

ηA0mL0λ

λODE:

BCs:

Nonlinear Model

Code

Comparison: No Voltage, ω = ωn

0 0.05 0.1 0.15 0.2 0.25−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t (sec)

Dis

plac

emen

t (m

)

(Solid) Nonlinear (Dashed) Linear

Φ =12Φ0 1+ sin ωt −ψ( ){ }

Assume a sinusoidal applied voltage with amplitude Φ0 , frequency ω, and phase ψ:

Power in: Pin = nΦq =Lh0

"

#$

%

&'Φ

ddt

εAhΦ

"

#$

%

&'

=Lh0

!

"#

$

%&Φ

ddt

εA0λ2h0

Φ*+,

-./

∴Pin =εA0L0h02λ2

Φ12ωcos ωt −ψ( )− 2

λΦλ

()*

+,-

Pavg =εA0L0h02Tλ2

Φ Φ− 2λΦλ

%&'

()*t0

t0+T

∫ dtAverage Power:

Period of Oscillation: T = 2π ωn

=εA0L0h02λ2

Φ Φ− 2λΦλ

%&'

()*

Φ0 = 1 kV, ω = ωn

0 0.02 0.04 0.06 0.08 0.1 0.12−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t (sec)

Dis

plac

emen

t (m

)

(black) ψ = 0 (blue) ψ = π

Maxwell Stress has modest influence on deformation.

0 0.02 0.04 0.06 0.08 0.1 0.12−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

t (sec)

Dis

plac

emen

t (m

)Φ0 = 5 kV, ω = ωn

(black) ψ = 0 Pavg = 35 W *out of phase w/ excitation (blue) ψ = π Pavg = 56 W **in phase w/ excitation(dashed) Φ0 = 0

Φ0 = 1 kV, ψ = 0, ω = ωn

0 0.02 0.04 0.06 0.08 0.1 0.12−15

−10

−5

0

5

10

15

t (sec)

P in

W

0 0.02 0.04 0.06 0.08 0.1 0.12−15

−10

−5

0

5

10

15

t (sec)

P in

W

Harvested

Supplied

Pavg = –1 W

ψ = π

Pavg = 1 W

~10 µW/mm3

Energy Harvested

Energy Consumed