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On the Influence of the Coupled Invariant in Thermo-Electro-Elasticity

  • Markus Mehnert
  • Tiphaine Mathieu-Pennober
  • Paul Steinmann
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

Electro-active polymers (EAPs) are a comparatively new class of smart materials that can change their properties and undergo large deformations as a result of an external electric excitation. These characteristics make them promising candidates in a wide range of applications, for example in sensor and actuator technology. As the experimental testing is both expensive and time consuming, simulation methods are developed in order to predict the material behavior. These simulations are based on well established energy formulation that are amended by additional coupling terms, often times in form of an invariant description. While the form of the purely mechanical and purely electric invariant quantities does not vary among the contributions of the electro-mechanical community, two different formulations for the coupling invariant can be found. In this contribution we demonstrate the influence of the selected coupling invariant on the material response. Therefore a thermo-electromechanically coupled constitutive model is derived based on the frequently used total energy approach.We devise the relevant constitutive equations starting from the basic laws of thermodynamics. Two distinctively different non-homogeneous boundary value problems are solved analytically in order to demonstrate the influence of the selected coupling invariant.

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Notes

Acknowledgements

The authors acknowledge the funding within the DFG project No. STE 544/52-1 and by the ERC advanced grant MOCOPOLY.

References

  1. Ask A, Menzel A, Ristinmaa M (2012) Electrostriction in electro-viscoelastic polymers. Mechanics of Materials 50:9–21Google Scholar
  2. Bar-Cohen Y (2002) Electroactive polymers: current capabilities and challenges. In: SPIE’s 9th Annual International Symposium on Smart Structures and Materials, International Society for Optics and Photonics, pp 1–7Google Scholar
  3. Bar-Cohen Y (2004) Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges, vol 136. SPIE pressGoogle Scholar
  4. Bland D (1956) Elastoplastic thick-walled tubes of work-hardening material subject to internal and external pressures and to temperature gradients. Journal of the Mechanics and Physics of Solids 4(4):209–229Google Scholar
  5. Bodelot L, Pössinger T, Danas K, Triantafyllidis N, Bolzmacher C (2016) Magnetorheological elastomers: Experimental and modeling aspects. In: Mechanics of Composite and Multi-functional Materials, Volume 7, Springer, pp 251–256Google Scholar
  6. Böse H, Fuß E (2014) Novel dielectric elastomer sensors for compression load detection. In: SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring, International Society for Optics and Photonics, pp 905,614–905,614Google Scholar
  7. Brochu P, Pei Q (2010) Advances in dielectric elastomers for actuators and artificial muscles. Macromolecular Rapid Communications 31(1):10–36Google Scholar
  8. Bustamante R (2009a) Transversely isotropic non-linear electro-active elastomers. Acta Mechanica 206(3):237–259Google Scholar
  9. Bustamante R (2009b) A variational formulation for a boundary value problem considering an electro-sensitive elastomer interacting with two bodies. Mechanics Research Communications 36(7):791–795Google Scholar
  10. Bustamante R (2010) Transversely isotropic nonlinear magneto-active elastomers. Acta Mechanica 210(3-4):183–214Google Scholar
  11. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis 13(1):167–178Google Scholar
  12. Danas K, Triantafyllidis N (2014) Instability of a magnetoelastic layer resting on a non-magnetic substrate. Journal of the Mechanics and Physics of Solids 69:67–83Google Scholar
  13. Danas K, Kankanala SV, Triantafyllidis N (2012) Experiments and modeling of iron-particle-filled magnetorheological elastomers. Journal of the Mechanics and Physics of Solids 60(1):120–138Google Scholar
  14. Dorfmann A, Ogden R (2003) Magnetoelastic modelling of elastomers. European Journal of Mechanics-A/Solids 22(4):497–507Google Scholar
  15. Dorfmann A, Ogden R (2004) Nonlinear magnetoelastic deformations of elastomers. Acta Mechanica 167(1):13–28Google Scholar
  16. Dorfmann A, Ogden R (2005a) Nonlinear electroelasticity. Acta Mechanica 174(3-4):167–183Google Scholar
  17. Dorfmann A, Ogden RW (2005b) Some problems in nonlinear magnetoelasticity. Zeitschrift für angewandte Mathematik und Physik (ZAMP) 56(4):718–745Google Scholar
  18. Dorfmann L, Ogden RW (2014) Nonlinear mechanics of soft fibrous materials. SpringerGoogle Scholar
  19. Elahinia M, Vertechy R, Berselli G, Parenti Castelli V, Bergamasco M (2013) Continuum thermoelectro-mechanical model for electrostrictive elastomers. Journal of Intelligent Material Systems and Structures 24(6):761–778Google Scholar
  20. Epstein M, Maugin GA (1990a) Energy-momentum tensor and J-integral in electrodeformable bodies. International Journal of Applied Electromagnetics in Materials 2(2):141–145Google Scholar
  21. Epstein M, Maugin GA (1990b) Inhomogeneities, Eshelby’s tensor and J-integral in electroelasticity. In: Hsieh RKT (ed) Mechanical Modelling of New Electromagnetic Materials, pp 253–258Google Scholar
  22. Erbts P, Hartmann S, Düster A (2015) A partitioned solution approach for electro-thermo-mechanical problems. Archive of Applied Mechanics 85(8):1075–1101Google Scholar
  23. Eringen AC (1963) On the foundations of electroelastostatics. International Journal of Engineering Science 1(1):127–153Google Scholar
  24. Eringen AC (1980) Mechanics of Continua. Robert E. Krieger Publishing Co., Huntington, NYGoogle Scholar
  25. Eringen AC, Maugin GA (2012) Electrodynamics of Continua I: Foundations and Solid Media. Springer Science & Business MediaGoogle Scholar
  26. Griffiths DJ (1989) Introduction to Electrodynamics, 2nd edn. Prentice Hall, Englewood CliffsGoogle Scholar
  27. Holzapfel GA (2000) Nonlinear Solid Mechanics, vol 24. Wiley, ChichesterGoogle Scholar
  28. Holzapfel GA, Ogden RW (2010) Constitutive modelling of arteries. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 466(2118):1551–1597Google Scholar
  29. Holzapfel GA, Simo JC (1996) Entropy elasticity of isotropic rubber-like solids at finite strains. Computer Methods in Applied Mechanics and Engineering 132(1-2):17–44Google Scholar
  30. Hossain M, Steinmann P (2013) More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study. Journal of the Mechanical Behavior of Materials 22(1-2):27–50Google Scholar
  31. Kankanala SV, Triantafyllidis N (2004) On finitely strained magnetorheological elastomers. Journal of the Mechanics and Physics of Solids 52(12):2869–2908Google Scholar
  32. Koh SJA, Keplinger C, Li T, Bauer S, Suo Z (2011) Dielectric elastomer generators: How much energy can be converted? IEEE/ASME Transactions on mechatronics 16(1):33–41Google Scholar
  33. Kost A (1994) Numerische Methoden in der Berechnung elektromagnetischer Felder. SpringerGoogle Scholar
  34. Kovetz A (2000) Electromagnetic Theory. Oxford University Press, OxfordGoogle Scholar
  35. Lu S, Pister K (1975) Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. International Journal of Solids and Structures 11(7-8):927–934Google Scholar
  36. Maugin GA (1976) A continuum theory of deformable ferrimagnetic bodies. I. Field equations. Journal of Mathematical Physics 17(9):1727–1738Google Scholar
  37. Maugin GA (1993) Material Inhomogeneities in Elasticity, vol 3. CRC PressGoogle Scholar
  38. Maugin GA (2013) Continuum Mechanics of Electromagnetic Solids, vol 33. ElsevierGoogle Scholar
  39. Maugin GA, Eringen AC (1972) Deformable magnetically saturated media. I. Field equations. Journal of Mathematical Physics 13(2):143–155Google Scholar
  40. Maugin GA, Eringen AC (1977) On the equations of the electrodynamics of deformable bodies of finite extent. Journal de Mécanique 16:101–147Google Scholar
  41. Mehnert M, Hossain M, Steinmann P (2016) On nonlinear thermo-electro-elasticity. Proc R Soc A 472(2190)Google Scholar
  42. Merodio J, Ogden RW (2002) Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Archives of Mechanics 54(5-6):525–552Google Scholar
  43. Miehe C (1995) Entropic thermoelasticity at finite strains. aspects of the formulation and numerical implementation. Computer Methods in Applied Mechanics and Engineering 120(3-4):243–269Google Scholar
  44. Ogden R (1972) Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 326(1567):565–584Google Scholar
  45. O’Halloran A, O’Malley F, McHugh P (2008) A review on dielectric elastomer actuators, technology, applications, and challenges. Journal of Applied Physics 104(7):9Google Scholar
  46. Pelrine R, Kornbluh R, Kofod G (2000a) High-strain actuator materials based on dielectric elastomers. Advanced Materials 12(16):1223–1225Google Scholar
  47. Pelrine R, Kornbluh R, Pei Q, Joseph J (2000b) High-speed electrically actuated elastomers with strain greater than 100%. Science 287(5454):836–839Google Scholar
  48. Pössinger T, Bolzmacher C, Bodelot L, Triantafyllidis N (2014) Influence of interfacial adhesion on the mechanical response of magneto-rheological elastomers at high strain. Microsystem Technologies 20(4-5):803–814Google Scholar
  49. Rajagopal KR, Huang YN (1994) Finite circumferential shearing of nonlinear solids in the context of thennoelasticity. IMA Journal of Applied Mathematics 53(2):111–125Google Scholar
  50. Rivlin RS (1948) Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 241(835):379–397Google Scholar
  51. Rivlin RS (1949a) Large elastic deformations of isotropic materials. V. The problem of flexure. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 195(1043):463–473Google Scholar
  52. Rivlin RS (1949b) Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 242(845):173–195Google Scholar
  53. Santapuri S (2012) Unified continuum modeling of fully coupled thermo-electro-magnetomechanical behavior, with applications to multifunctional materials and structures. PhD thesis, The Ohio State UniversityGoogle Scholar
  54. Santapuri S, Lowe RL, Bechtel SE, Dapino MJ (2013) Thermodynamic modeling of fully coupled finite-deformation thermo-electro-magneto-mechanical behavior for multifunctional applications. International Journal of Engineering Science 72:117–139Google Scholar
  55. Spencer AJM (2013) Part III. Theory of invariants. In: Eringen AC (ed) Continuum Physics, vol 1, pp 239–353Google Scholar
  56. Steinmann P (2011) Computational nonlinear electro-elasticity—getting started—. In: Mechanics and Electrodynamics of Magneto-and Electro-elastic Materials, Springer, pp 181–230Google Scholar
  57. Steinmann P, Hossain M, Possart G (2012) Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Archive of Applied Mechanics 82(9):1183–1217Google Scholar
  58. Trimarco C, Maugin GA (2001) Material mechanics of electromagnetic solids. In: Configurational Mechanics of Materials, Springer, pp 129–171Google Scholar
  59. Vertechy R, Berselli G, Parenti Castelli V, Vassura G (2010) Optimal design of lozenge-shaped dielectric elastomer linear actuators: mathematical procedure and experimental validation. Journal of Intelligent Material Systems and Structures 21(5):503–515Google Scholar
  60. Vertechy R, Fontana M, Papini GPR, Forehand D (2014) In-tank tests of a dielectric elastomer generator for wave energy harvesting. In: SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring, International Society for Optics and Photonics, pp 90,561G–90,561GGoogle Scholar
  61. Vogel F (2015) On the Modeling and Computation of Electro- and Magneto-active Polymers. Lehrstuhl für Technische Mechanik, Universität Erlangen-NürnbergGoogle Scholar
  62. Vogel F, Göktepe S, Steinmann P, Kuhl E (2014) Modeling and simulation of viscous electro-active polymers. European Journal of Mechanics-A/Solids 48:112–128Google Scholar
  63. Voltairas P, Fotiadis D, Massalas C (2003) A theoretical study of the hyperelasticity of electrogels. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 459(2037):2121–2130Google Scholar
  64. Vu D, Steinmann P, Possart G (2007) Numerical modelling of non-linear electroelasticity. International Journal for Numerical Methods in Engineering 70(6):685–704Google Scholar
  65. Vu DK (2014) A study on nonlinear electro-elastostatics: Theory and numerical simulation. Habilitation, Friedrich-Alexander University of Erlangen-Nürnberg: Erlangen, Bayern, GermanyGoogle Scholar
  66. Wang CC (1970) A new representation theorem for isotropic functions: An answer to Professor G.F. Smith’s criticism of my papers on representations for isotropic functions. Archive for Rational Mechanics and Analysis 36(3):166–197Google Scholar
  67. Wriggers P (2008) Nonlinear Finite Element Methods. Springer Science & Business MediaGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Markus Mehnert
    • 1
  • Tiphaine Mathieu-Pennober
    • 2
  • Paul Steinmann
    • 1
  1. 1.Chair of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Ecole PolytechniqueParisFrance

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