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An electro-viscoelastic micromechanical model with non-constant relaxation time

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Abstract

An electro-viscoelastic constitutive model based on non-constant relaxation time is proposed in this paper. We start by deriving the viscoelastic model from the well-known generalized Maxwell model (GMM) provided with stress–stretch-dependent relaxation time. Here, we consider relaxation time no longer as a constant but a variable that depends on dissipation or viscous process in polymer chains due to electroelastic stress–stretch. Comparison with the electro-viscoelastic experiments shows that our proposed model can capture well dielectric elastomer viscoelastic behaviors under time- and rate-dependent electric loadings for different pre-stretch values.

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References

  1. Iannarelli, A., Ghaffarian Niasar, M., Ross, R.: Frequency-independent breakdown strength of dielectric elastomers under ac stress. Appl. Phys. Lett. 115(9), 092904 (2019)

    Article  Google Scholar 

  2. Zhou, J., Jiang, L., Khayat, R.E.: Investigation on the performance of a viscoelastic dielectric elastomer membrane generator. Soft Matter 11(15), 2983–2992 (2015)

    Article  Google Scholar 

  3. Srivastava, A.K., Basu, S.: Modelling the performance of devices based on thin dielectric elastomer membranes. Mech. Mater. 137, 103136 (2019)

    Article  Google Scholar 

  4. Plante, J., Dubowsky, S.: On the properties of dielectric elastomer actuators and their design implications. Smart Mater. Struct. 16(2), 227 (2007)

    Article  Google Scholar 

  5. Giousouf, M., Kovacs, G.: Dielectric elastomer actuators used for pneumatic valve technology. Smart Mater. Struct. 22(10), 104010 (2013)

    Article  Google Scholar 

  6. Bergström, J.S., Boyce, M.C.: Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids 46(5), 931–954 (1998)

    Article  MATH  Google Scholar 

  7. Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35(26–27), 3455–3482 (1998)

    Article  MATH  Google Scholar 

  8. Huber, N., Tsakmakis, C.: Finite deformation viscoelasticity laws. Mech. Mater. 32(1), 1–18 (2000)

    Article  MATH  Google Scholar 

  9. Behera, S.K., Kumar, D., Sarangi, S.: Modeling of electro-viscoelastic dielectric elastomer: a continuum mechanics approach. Eur. J. Mech. A Solids 90, 104369 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mehnert, M., Hossain, M., Steinmann, P.: Experimental and numerical investigations of the electro-viscoelastic behavior of vhb 4905tm. Eur. J. Mech. A Solids 77, 103797 (2019)

    Article  Google Scholar 

  11. Eder-Goy, D., Zhao, Y., Xu, B.-X.: Dynamic pull-in instability of a prestretched viscous dielectric elastomer under electric loading. Acta Mech. 228(12), 4293–4307 (2017)

    Article  MathSciNet  Google Scholar 

  12. Nedjar, B.: A finite strain modeling for electro-viscoelastic materials. Int. J. Solids Struct. 97, 312–321 (2016)

    Article  Google Scholar 

  13. Bortot, E., Denzer, R., Menzel, A., Gei, M.: Analysis of viscoelastic soft dielectric elastomer generators operating in an electrical circuit. Int. J. Solids Struct. 78, 205–215 (2016)

    Article  Google Scholar 

  14. Vogel, F., Göktepe, S., Steinmann, P., Kuhl, E.: Modeling and simulation of viscous electro-active polymers. Eur. J. Mech. A Solids 48, 112–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hong, W.: Modeling viscoelastic dielectrics. J. Mech. Phys. Solids 59(3), 637–650 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wissler, M., Mazza, E.: Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators. Sens. Actuators A 134(2), 494–504 (2007)

    Article  Google Scholar 

  17. Chen, Y., Kang, G., Yuan, J., Hu, Y., Li, T., Qu, S.: An electro-mechanically coupled visco-hyperelastic-plastic constitutive model for cyclic deformation of dielectric elastomers. Mech. Mater. 150, 103575 (2020)

    Article  Google Scholar 

  18. Hossain, M., Vu, D.K., Steinmann, P.: Experimental study and numerical modelling of vhb 4910 polymer. Comput. Mater. Sci. 59, 65–74 (2012)

    Article  Google Scholar 

  19. de Gennes, P.-G.: Reptation of a polymer chain in the presence of fixed obstacles. J. Chem. Phys. 55(2), 572–579 (1971)

    Article  Google Scholar 

  20. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)

    Google Scholar 

  21. Simo, J.C.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60(2), 153–173 (1987)

    Article  MATH  Google Scholar 

  22. Holzapfel, G.A., Simo, J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int. J. Solids Struct. 33(20–22), 3019–3034 (1996)

    Article  MATH  Google Scholar 

  23. Holzapfel, G.A.: On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures. Int. J. Numer. Methods Eng. 39(22), 3903–3926 (1996)

    Article  MATH  Google Scholar 

  24. Waluyo, S.: Stress dependent relaxation time in large deformation. Struct. Eng. Mech. 61(3), 317–323 (2017)

    Article  Google Scholar 

  25. Haupt, P., Lion, A.: On finite linear viscoelasticity of incompressible isotropic materials. Acta Mech. 159(1), 87–124 (2002)

    Article  MATH  Google Scholar 

  26. Itskov, M., Khiêm, V.N., Waluyo, S.: Electroelasticity of dielectric elastomers based on molecular chain statistics. Math. Mech. Solids 1081286518755846 (2018)

  27. Cohen, N., Menzel, A., et al.: Towards a physics-based multiscale modelling of the electro-mechanical coupling in electro-active polymers. Proc. R. Soc. A 472, 20150462 (2016)

    Article  Google Scholar 

  28. Cohen, N.: A generalized electro-elastic theory of polymer networks. J. Mech. Phys. Solids 110, 173–191 (2018)

    Article  MathSciNet  Google Scholar 

  29. Waluyo, S.: An electroelastic constitutive model for dielectric elastomers based on the Langevin statistic and its instability characteristics. In: Mechanics of Advanced Materials and Structures, pp. 1–24 (2021)

  30. Plagge, J., Ricker, A., Kröger, N., Wriggers, P., Klüppel, M.: Efficient modeling of filled rubber assuming stress-induced microscopic restructurization. Int. J. Eng. Sci. 151, 103291 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Coleman, B.D., Gurtin, M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47(2), 597–613 (1967)

    Article  Google Scholar 

  32. Valanis, K.C.: Irreversible Thermodynamics of Continuous Media: Internal Variable Theory, vol. 77. Springer, Vienna (1972)

    MATH  Google Scholar 

  33. Bach, A., Rasmussen, H.K., Hassager, O.: Extensional viscosity for polymer melts measured in the filament stretching rheometer. J. Rheol. 47(2), 429–441 (2003)

    Article  Google Scholar 

  34. Malkin, A.Y., Arinstein, A., Kulichikhin, V.: Polymer extension flows and instabilities. Prog. Polym. Sci. 39(5), 959–978 (2014)

    Article  Google Scholar 

  35. Bergström, J., Boyce, M.: Constitutive modeling of the time-dependent and cyclic loading of elastomers and application to soft biological tissues. Mech. Mater. 33(9), 523–530 (2001)

    Article  Google Scholar 

  36. Linder, C., Tkachuk, M., Miehe, C.: A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity. J. Mech. Phys. Solids 59(10), 2134–2156 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Davidson, J.D., Goulbourne, N.C.: Microscopic mechanisms of the shape memory effect in crosslinked polymers. Smart Mater. Struct. 24(5), 055014 (2015)

    Article  Google Scholar 

  38. Herrmann, L.: A numerical procedure for viscoelastic stress analysis. In: Seventh Meeting of ICRPG Mechanical Behavior Working Group, Orlando, FL (1968)

  39. Taylor, R.L., Pister, K.S., Goudreau, G.L.: Thermomechanical analysis of viscoelastic solids. Int. J. Numer. Methods Eng. 2(1), 45–59 (1970)

    Article  MATH  Google Scholar 

  40. Dorfmann, A., Ogden, R.: Nonlinear electroelasticity. Acta Mech. 174(3–4), 167–183 (2005)

    Article  MATH  Google Scholar 

  41. Blythe, A.R., Bloor, D.: Electrical Properties of Polymers. Cambridge University Press, New York (2005)

    Google Scholar 

  42. Pelrine, R.E., Kornbluh, R.D., Joseph, J.P.: Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation. Sens. Actuators A 64(1), 77–85 (1998)

    Article  Google Scholar 

  43. Suo, Z., Zhao, X., Greene, W.H.: A nonlinear field theory of deformable dielectrics. J. Mech. Phys. Solids 56(2), 467–486 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. McMeeking, R.M., Landis, C.M.: Electrostatic forces and stored energy for deformable dielectric materials. J. Appl. Mech. 72(4), 581–590 (2005)

    Article  MATH  Google Scholar 

  45. Wissler, M., Mazza, E.: Electromechanical coupling in dielectric elastomer actuators. Sens. Actuators A 138(2), 384–393 (2007)

    Article  Google Scholar 

  46. Kofod, G.: The static actuation of dielectric elastomer actuators: how does pre-stretch improve actuation? J. Phys. D Appl. Phys. 41(21), 215405 (2008)

    Article  Google Scholar 

  47. Zhou, J., Jiang, L., Khayat, R.E.: A micro-macro constitutive model for finite-deformation viscoelasticity of elastomers with nonlinear viscosity. J. Mech. Phys. Solids 110, 137–154 (2018)

    Article  MathSciNet  Google Scholar 

  48. Yarali, E., Baniasadi, M., Bodaghi, M., Baghani, M.: 3d constitutive modeling of electro-magneto-visco-hyperelastic elastomers: a semi-analytical solution for cylinders under large torsion-extension deformation. Smart Mater. Struct. 29(8), 085031 (2020)

    Article  Google Scholar 

  49. Kollosche, M., Kofod, G., Suo, Z., Zhu, J.: Temporal evolution and instability in a viscoelastic dielectric elastomer. J. Mech. Phys. Solids 76, 47–64 (2015)

    Article  Google Scholar 

  50. Hossain, M., Vu, D.K., Steinmann, P.: A comprehensive characterization of the electro-mechanically coupled properties of vhb 4910 polymer. Arch. Appl. Mech. 85(4), 523–537 (2015)

    Article  Google Scholar 

  51. Zurlo, G., Destrade, M., DeTommasi, D., Puglisi, G.: Catastrophic thinning of dielectric elastomers. Phys. Rev. Lett. 118(7), 078001 (2017)

    Article  Google Scholar 

  52. Kofod, G., Sommer-Larsen, P., Kornbluh, R., Pelrine, R.: Actuation response of polyacrylate dielectric elastomers. J. Intell. Mater. Syst. Struct. 14(12), 787–793 (2003)

    Article  Google Scholar 

  53. Ghosh, K., Lopez-Pamies, O.: On the two-potential constitutive modeling of dielectric elastomers. Meccanica 56, 1505–1521 (2021)

    Article  MathSciNet  Google Scholar 

  54. Gent, A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69(1), 59–61 (1996)

    Article  Google Scholar 

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Acknowledgements

We would like to thank the financial support from the Universitas Jenderal Soedirman (UNSOED) Indonesia, for the period of 2020–2021.

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  1. Department of Industrial Engineering, Universitas Jenderal Soedirman, Jl. Mayjen Sungkono Km.5, Purbalingga, Jawa Tengah, 53373, Indonesia

    Sugeng Waluyo

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Waluyo, S. An electro-viscoelastic micromechanical model with non-constant relaxation time. Acta Mech 233, 4505–4522 (2022). https://doi.org/10.1007/s00707-022-03344-x

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