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ZJUS-A
Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering)  2016, Vol. 17 Issue (1): 1-21    DOI: 10.1631/jzus.A1500125
    
Mechanics of dielectric elastomers: materials, structures, and devices*
Feng-bo Zhu1,2,Chun-li Zhang1,2,3,Jin Qian1,2,3,†(),Wei-qiu Chen1,2,3
1 Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
2 Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China
3 Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou 310027, China
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Abstract  

Dielectric elastomers (DEs) respond to applied electric voltage with a surprisingly large deformation, showing a promising capability to generate actuation in mimicking natural muscles. A theoretical foundation of the mechanics of DEs is of crucial importance in designing DE-based structures and devices. In this review, we survey some recent theoretical and numerical efforts in exploring several aspects of electroactive materials, with emphases on the governing equations of electromechanical coupling, constitutive laws, viscoelastic behaviors, electromechanical instability as well as actuation applications. An overview of analytical models is provided based on the representative approach of non-equilibrium thermodynamics, with computational analyses being required in more generalized situations such as irregular shape, complex configuration, and time-dependent deformation. Theoretical efforts have been devoted to enhancing the working limits of DE actuators by avoiding electromechanical instability as well as electric breakdown, and pre-strains are shown to effectively avoid the two failure modes. These studies lay a solid foundation to facilitate the use of DE materials, structures, and devices in a wide range of applications such as biomedical devices, adaptive systems, robotics, energy harvesting, etc.

Key words： Artificial muscle      Smart material      Dielectric elastomer (DE)      Electromechanical coupling      Constitutive law      Viscoelasticity      Electromechanical instability      Actuation     
Received: 15 April 2015      Published: 06 January 2016
Fund:  the National Natural Science Foundation of China(No. 11321202);the Zhejiang Provincial Natural Science Foundation of China(No. LR16A020001)
Corresponding Authors: Jin Qian     E-mail: jqian@zju.edu.cn
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Wei-qiu Chen
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Feng-bo Zhu,Chun-li Zhang,Jin Qian,Wei-qiu Chen. Mechanics of dielectric elastomers: materials, structures, and devices*. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 2016, 17(1): 1-21.

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http://www.zjujournals.com/xueshu/zjus-a/10.1631/jzus.A1500125     OR     http://www.zjujournals.com/xueshu/zjus-a/Y2016/V17/I1/1

Fig. 1 Working principle of DE actuators (a) Schematics of a membrane of DE with coated electrodes on both sides; (b) By applying an electric voltage, the electrostatic force compresses the DE membrane in the thickness direction and expands it in the membrane plane. The DE membrane recovers its original configuration when the voltage is removed; (c) An elastomer film at its reference state, with randomly distributed polymer dipoles; (d) Polarization of an ideal DE, in which the polymer dipoles are polarized freely; (e) A non-ideal DE under large deformation. The stress field is perpendicular to the direction of electric field, which impedes the polarization of dipoles (Anderson et al., 2012; Foo et al., 2012a)
Energy function Function form Strain Reference
neo-Hookean model ${W_s} = \frac{G}{2}(\lambda _1^2 + \lambda _2^2 + \lambda _1^{ - 2}\lambda _2^{ - 2} - 3)$ 30%–40% (Dollhofer et al., 2004; Zhao and Suo, 2007; Wang H. et al., 2013)
Gent model ${W_s} = - \frac{{G{J_{\lim }}}}{2}\log \left( {1 - \frac{{\lambda _1^2 + \lambda _2^2 + \lambda _1^{ - 2}\lambda _2^{ - 2} - 3}}{{{J_{\lim }}}}} \right)$ 30%–155% (Huang and Suo, 2012; Lu et al., 2012; Qu et al., 2012; Zhou et al., 2013; 2014a)
Mooney-Rivlin model $\begin{gathered} {W_s} = {C_1}({I_1} - 3) + {C_2}({I_2} - 3), \hfill \\ {I_1} = \lambda _1^2 + \lambda _2^2 + \lambda _3^2,\;{I_2} = \lambda _1^2\lambda _2^2 + \lambda _2^2\lambda _3^2 + \lambda _3^2\lambda _1^2 \hfill \\ \end{gathered} $ 
200% (Liu et al., 2010)
Yeoh model $\begin{gathered} {W_s} = \frac{{{C_1}}}{2}({I_1} - 3) + \frac{{{C_2}}}{2}{({I_1} - 3)^2} + \frac{{{C_3}}}{2}{({I_1} - 3)^3}, \hfill \\ {I_1} = \lambda _1^2 + \lambda _2^2 + \lambda _3^2 \hfill \\ \end{gathered} $ >200% (Wissler and Mazza, 2005)
Ogden model ${W_s} = \sum\limits_{p = 1}^N {\frac{{{\mu _p}}}{{{\alpha _p}}}} (\lambda _1^{{\alpha _p}} + \lambda _2^{{\alpha _p}} + \lambda _1^{ - {\alpha _p}}\lambda _2^{ - {\alpha _p}} - 3)$ 100%–400% (Goulbourne, 2011; Proulx et al., 2011; Qu and Suo, 2012)
Table 1 Strain energy density functions adopted in DE modeling
Fig. 2 A rheological model of DEs The viscoelasticity model consisting of two components in parallel: one being a spring α, and the other including another spring β and a dashpot η. An arbitrary deformation can be decomposed into two parts by postulating an intermediate state of deformation, which is achieved by elastic relaxation of spring β from the current state (Hong, 2011; Zhao et al., 2011; Wang Y.Q. et al., 2013)
Fig. 3 Three types of DE transducers and the influence of pre-stretch on electromechanical instability (a) and (b) Three types of dielectrics, depending on where the two curves ϕ(λ) and ϕB (λ) intersect; (c) Electromechanical response of a DE membrane with or without pre-stretch
Property Natural muscles DEs
Energy density 150 J/kg (peak to 300 J/kg) (Boy et al., 2013) 3400 J/kg (Brochu and Pei, 2010)
Frequencies and response time 10 Hz, 10 ms (Meijer et al., 2001) 10 kHz, 0.1 ms (Sheng et al., 2013)
Maximum strain 100% linear (Brochu and Pei, 2010) 1692% areal (Li T.F. et al., 2013)
Efficiency – 60%–90% (Brochu and Pei, 2010)
Stroke 20%–40% linear (peak to 100%) 10%–100% linear (peak to 300%) (O’Halloran et al., 2008)
Actuation mechanism Molecular motors Maxwell force
Power source ATP Electricvoltage (1–10 kV) (Shankar et al., 2007)
Environment Wet and nutrition Insulation
Structure Bundled muscle cells, self-assembly Sandwiched membrane, grouped tensegrity and electrodes arrays (Bauer et al., 2014)
Size 10 μm–1 m (Magid and Law, 1985) 100 μm–0.1 m (Brochu and Pei, 2010)
Fabrication Tissue engineering Lithography, machining
Typical material Mammal muscle cells, molecules connected by extracellular matrices 3M-VHB, PDMS, Silicon (Li T.F. et al., 2012a; Wang J. et al., 2014)
Modulus 20–2000 kPa (Morrow et al., 2010) 20–2000 kPa (Sheng et al., 2012)
Viscoelasticity 0.1–100 s (van Loocke et al., 2008; 2009) 0.1–100 s (Zhang et al., 2014a)
Fatigues and life time 107 cycles (van Loocke et al., 2009) 107 cycles (Rosset et al., 2009)
Density 1.06 kg/L (Kornbluh et al., 2002) 0.94 kg/L (Shankar et al., 2007)
Table 2 Actuation performance: natural muscles vs. DEs
Fig. 4 Various DE actuators and their applications (a) An eye-camera system made of DE; (b) A bulging DE actuator; (c) A microfluidic valve that is dielectrically actuated; (d) A unimorph DE actuator; (e) A multi-finger DE gripper; (f) A stacking DE actuator; (g) A tubular DE actuator; (h) A swelling DE balloon
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