Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction

doi:10.1007/s10483-022-2853-9

Abstract

Abstract: The dielectric elastomer (DE) is an important intelligent soft material widely used in soft actuators, and the dynamic response of the DE is highly nonlinear due to the material properties. In the DE, electrostriction denotes the deformation-dependent permittivity. In the present study, we formulate the nonlinear dynamic governing equations of the DE membrane considering the electrostriction effect. The free vibration and parametric excitation of the DE membrane with different geometric sizes are calculated. The free vibration bifurcations induced by the initial location and the voltage are both discussed according to an energy-based approach. The amplitude-frequency characteristics and bifurcation diagrams of parametric excitation are also given. The results show that electrostriction decreases the free vibration amplitude and increases the frequency, but it has less influence on the parametric excitation oscillation frequency and decreases the parametric excitation amplitude except when the membrane resonates. The initial location and the applied voltage can induce the snap-through instability of the free vibration. A large geometric size will lead to a much lower resonance frequency. The resonance amplitudes increase while the resonance frequencies decrease with the increase in the applied voltage. The critical voltage of snap-through instability for the parametric excitation is larger than that for the free vibration one.

Key words: dielectric elastomer (DE), nonlinear dynamics, electrostriction, bifurcation, electroelasticity

2010 MSC Number:

74H45
65P40

Yaode YIN, Demin ZHAO, Jianlin LIU, Zengyao XU. Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 793-812.

TrendMD

References

[1] PELRINE, R., KORNBLUH, R., PEI, Q., and JOSEPH, J. High-speed electrically actuated elastomers with strain greater than 100%. Science, 287(5454), 836-839(2000)
[2] AN, L., WANG, F., CHENG, S., LU, T., and WANG, T. J. Experimental investigation of the electromechanical phase transition in a dielectric elastomer tube. Smart Materials and Structures, 24(3), 035006(2015)
[3] GUO, Y., LIU, L., LIU, Y., and LENG, J. Review of dielectric elastomer actuators and their applications in soft robots. Advanced Intelligent Systems, 3(10), 2000282(2021)
[4] LAI, Z., WANG, S., ZHU, L., ZHANG, G., WANG, J., YANG, K., and YURCHENKO, D. A hybrid piezo-dielectric wind energy harvester for high-performance vortex-induced vibration energy harvesting. Mechanical Systems and Signal Processing, 150, 107212(2021)
[5] SUO, Z. Theory of dielectric elastomers. Acta Mechanica Solida Sinica, 23(6), 549-578(2010)
[6] FOX, J. W. and GOULBOURNE, N. C. On the dynamic electromechanical loading of dielectric elastomer membranes. Journal of the Mechanics and Physics of Solids, 56(8), 2669-2686(2008)
[7] GU, G. Y., GUPTA, U., ZHU, J., ZHU, L. M., and ZHU, X. Modeling of viscoelastic electromechanical behavior in a soft dielectric elastomer actuator. IEEE Transactions on Robotics, 33(5), 1263-1271(2017)
[8] LI, G., CHEN, X., ZHOU, F., LIANG, Y., XIAO, Y., CAO, X., ZHANG, Z., ZHANG, M., WU, B., YIN, S., XU, Y., FAN, H., CHEN, Z., SONG, W., YANG, W., PAN, B., HOU, J., ZOU, W., HE, S., YANG, X., MAO, G., JIA, Z., ZHOU, H., LI, T., QU, S., XU, Z., HUANG, Z., LUO, Y.,XIE, T., GU, J., ZHU, S., and YANG, W. Self-powered soft robot in the Mariana Trench. nature, 591(7848), 66-71(2021)
[9] GARNELL, E., ROUBY, C., and DOAR, O. Dynamics and sound radiation of a dielectric elastomer membrane. Journal of Sound and Vibration, 459, 114836(2019)
[10] LINNEBACH, P., RIZZELLO, G., and SEELECKE, S. Design and validation of a dielectric elastomer membrane actuator driven pneumatic pump. Smart Materials and Structures, 29(7), 075021(2020)
[11] DAI, H. and WANG, L. Nonlinear oscillations of a dielectric elastomer membrane subjected to in-plane stretching. Nonlinear Dynamics, 82(4), 1709-1719(2015)
[12] GOULBOURNE, N., MOCKENSTURM, E., and FRECKER, M. A nonlinear model for dielectric elastomer membranes. Journal of Applied Mechanics, 72(6), 899-906(2005)
[13] SUO, Z., ZHAO, X., and GREENE, W. H. A nonlinear field theory of deformable dielectrics. Journal of the Mechanics and Physics of Solids, 56(2), 467-486(2008)
[14] DORFMANN, A. and OGDEN, R. W. Nonlinear electroelasticity. Acta Mechanica, 174(3), 167-183(2005)
[15] ZHAO, X. and SUO, Z. Electrostriction in elastic dielectrics undergoing large deformation. Journal of Applied Physics, 104(12), 123530(2008)
[16] ASK, A., MENZEL, A., and RISTINMAA, M. Electrostriction in electro-viscoelastic polymers. Mechanics of Materials, 50, 9-21(2012)
[17] ASK, A., MENZEL, A., and RISTINMAA, M. Modelling of viscoelastic dielectric elastomers with deformation dependent electric properties. Procedia IUTAM, 12, 134-144(2015)
[18] GEI, M., COLONNELLI, S., and SPRINGHETTI, R. The role of electrostriction on the stability of dielectric elastomer actuators. International Journal of Solids and Structures, 51(3), 848-860(2014)
[19] LI, B., CHEN, H., and ZHOU, J. Electromechanical stability of dielectric elastomer composites with enhanced permittivity. Composites Part A:Applied Science and Manufacturing, 52, 55-61(2013)
[20] KUMAR, A. and PATRA, K. Proposal of a generic constitutive model for deformation-dependent dielectric constant of dielectric elastomers. Engineering Science and Technology, an International Journal, 24(6), 1347-1360(2021)
[21] FOO, C. C., CAI, S. Q., KOH, S. J. A., BAUER, S., and SUO, Z. G. Model of dissipative dielectric elastomers. Journal of Applied Physics, 111(3), 034102(2012)
[22] LU, T., MA, C., and WANG, T. Mechanics of dielectric elastomer structures:a review. Extreme Mechanics Letters, 38, 100752(2020)
[23] WANG, F., LU, T., and WANG, T. J. Nonlinear vibration of dielectric elastomer incorporating strain stiffening. International Journal of Solids and Structures, 87, 70-80(2016)
[24] ZHAO, X., KOH, S. J. A., and SUO, Z. Nonequilibrium thermodynamics of dielectric elastomers. International Journal of Applied Mechanics, 3(2), 203-217(2011)
[25] WANG, Z. and HE, T. Electro-viscoelastic behaviors of circular dielectric elastomer membrane actuator containing concentric rigid inclusion. Applied Mathematics and Mechanics (English Edition), 39(4), 547-560(2018) https://doi.org/10.1007/s10483-018-2318-8
[26] ZHANG, J., CHEN, H., and LI, D. Nonlinear dynamical model of a soft viscoelastic dielectric elastomer. Physical Review Applied, 8(6), 064016(2017)
[27] ZHAO, D., YIN, Y., and LIU, J. A fractional finite strain viscoelastic model of dielectric elastomer. Applied Mathematical Modelling, 100, 564-579(2021)
[28] ZHU, J., CAI, S., and SUO, Z. Resonant behavior of a membrane of a dielectric elastomer. International Journal of Solids and Structures, 47(24), 3254-3262(2010)
[29] ZHANG, J. and CHEN, H. Voltage-induced beating vibration of a dielectric elastomer membrane. Nonlinear Dynamics, 100(3), 2225-2239(2020)
[30] SHARMA, A. K., ARORA, N., and JOGLEKAR, M. M. DC dynamic pull-in instability of a dielectric elastomer balloon:an energy-based approach. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences, 474(2211), 20170900(2018)
[31] HEIDARI, H., ALIBAKHSHI, A., and AZARBONI, H. R. Chaotic motion of a parametrically excited dielectric elastomer. International Journal of Applied Mechanics, 12(3), 2050033(2020)
[32] CHEN, F., ZHU, J., and WANG, M. Y. Dynamic electromechanical instability of a dielectric elastomer balloon. Europhysics Letters, 112(4), 47003(2015)
[33] LV, X., LIU, L., LIU, Y., and LENG, J. Dynamic performance of dielectric elastomer balloon incorporating stiffening and damping effect. Smart Materials and Structures, 27(10), 105036(2018)
[34] KHURANA, A., KUMAR, A., SHARMA, A. K., and JOGLEKAR, M. M. Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators. Nonlinear Dynamics, 104(2), 1227-1251(2021)
[35] JOGLEKAR, M. M. An energy-based approach to extract the dynamic instability parameters of dielectric elastomer actuators. Journal of Applied Mechanics, 81(9), 091010(2014)
[36] ZHU, J., CAI, S., and SUO, Z. Nonlinear oscillation of a dielectric elastomer balloon. Polymer International, 59(3), 378-383(2010)
[37] ALIBAKHSHI, A. and HEIDARI, H. Analytical approximation solutions of a dielectric elastomer balloon using the multiple scales method. European Journal of Mechanics-A/Solids, 74, 485-496(2019)
[38] WANG, Y., ZHANG, L., and ZHOU, J. Incremental harmonic balance method for periodic forced oscillation of a dielectric elastomer balloon. Applied Mathematics and Mechanics (English Edition), 41(3), 459-470(2020) https://doi.org/10.1007/s10483-020-2590-7
[39] TANG, D., LIM, C. W., HONG, L., JIANG, J., and LAI, S. K. Analytical asymptotic approximations for large amplitude nonlinear free vibration of a dielectric elastomer balloon. Nonlinear Dynamics, 88(3), 2255-2264(2017)
[40] LIU, F. and ZHOU, J. Shooting and arc-length continuation method for periodic solution and bifurcation of nonlinear oscillation of viscoelastic dielectric elastomers. Journal of Applied Mechanics, 85(1), 011005(2017)
[41] FRIED, E. An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity. Journal of the Mechanics and Physics of Solids, 50(3), 571-582(2002)
[42] HOSSAIN, M., VU, D. K., and STEINMANN, P. Experimental study and numerical modelling of VHB 4910 polymer. Computational Materials Science, 59, 65-74(2012)
[43] NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, John Wiley & Sons, New York (2008)
[44] KEPLINGER, C., LI, T., BAUMGARTNER, R., SUO, Z., and BAUER, S. Harnessing snapthrough instability in soft dielectrics to achieve giant voltage-triggered deformation. Soft Matter, 8(2), 285-288(2012)
[45] LI, T., KEPLINGER, C., BAUMGARTNER, R., BAUER, S., YANG, W., and SUO, Z. Giant voltage-induced deformation in dielectric elastomers near the verge of snap-through instability. Journal of the Mechanics and Physics of Solids, 61(2), 611-628(2013)
[46] LI, Z., WANG, Y., FOO, C. C., GODABA, H., ZHU, J., and YAP, C. H. The mechanism for large-volume fluid pumping via reversible snap-through of dielectric elastomer. Journal of Applied Physics, 122(8), 084503(2017)