Advertisement

Nonlinear Constitutive Modeling of Electroelastic Solids

  • Luis DorfmannEmail author
  • Raymond Ogden
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 262)

Abstract

In this chapter, the equations governing the mechanical behavior of electroelastic solids capable of finite deformations are summarized with particular reference to the development of constitutive equations describing the electromechanical interactions in soft dielectric materials. Following a brief summary of some background from continuum mechanics and nonlinear elasticity theory, the equations of electrostatics are given in Eulerian form and then re-cast in Lagrangian form. The electroelastic constitutive equations are based on the existence of a so-called total energy function, which may be regarded as a function of the Lagrangian form of either the electric field or electric displacement as the independent electric variable, together with the deformation gradient. For each form of the total energy function, corresponding expressions for the (total) nominal and Cauchy stress tensors are provided, both in full generality and for their isotropic specializations for unconstrained and incompressible materials. The general formulas are then applied to the basic problem of homogeneous deformation of a rectangular plate, and illustrated by the choice of a simple specific example of constitutive equation. As a further illustration, the theory is applied to the analysis of a nonhomogeneous deformation of a circular cylindrical tube subject to an axial force, a torsional moment, and a radial electric field generated by a potential difference between flexible electrodes covering its inner and outer curved surfaces.

References

  1. 1.
    Bar-Cohen Y (2002) Electro-active polymers: current capabilities and challenges. In: Bar-Cohen Y (ed) Proceedings of the 4th electroactive polymer actuators and devices (EAPAD) conference, 9th smart structures and materials symposium. San Diego. SPIE Publishers, Bellingham, WA, pp 1–7Google Scholar
  2. 2.
    Brochu P, Pei Q (2010) Advances in dielectric elastomers for actuators and artificial muscles. Macromol Rapid Commun 31:10–36CrossRefGoogle Scholar
  3. 3.
    Bustamante R, Dorfmann A, Ogden RW (2009) Nonlinear electroelastostatics: a variational framework. Z Angew Math Phys 60:154–177MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bustamante R, Dorfmann A, Ogden RW (2009) On electric body forces and Maxwell stresses in nonlinearly electroelastic solids. Int J Eng Sci 47:1131–1141MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mech 174:167–183CrossRefGoogle Scholar
  6. 6.
    Dorfmann A, Ogden RW (2006) Nonlinear electroelastic deformations. J Elast 82:99–127MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dorfmann A, Ogden RW (2010) Nonlinear electroelasticity: incremental equations and stability. Int J Eng Sci 48:1–14CrossRefGoogle Scholar
  8. 8.
    Dorfmann A, Ogden RW (2010) Electroelastic waves in a finitely deformed electroactive material. IMA J Appl Math 75:603–636MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dorfmann L, Ogden RW (2014) Instabilities of an electroelastic plate. Int J Eng Sci 77:79–101MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dorfmann L, Ogden RW (2014) Nonlinear theory of electroelastic and magnetoelastic interactions. Springer, New YorkCrossRefGoogle Scholar
  11. 11.
    Dorfmann L, Ogden RW (2018) The effect of deformation dependent permittivity on the elastic response of a finitely deformed dielectric tube. Mech Res Commun 93:47–57CrossRefGoogle Scholar
  12. 12.
    Dorfmann L, Ogden RW (2017) Nonlinear electroelasticity: materials, continuum theory and applications. Proc R Soc A 473:20170311Google Scholar
  13. 13.
    Dorfmann L, Ogden RW (2019) Instabilities of soft dielectrics. Philso Trans R Soc A 377:20180077MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goulbourne NC (2009) A mathematical model for cylindrical, fiber reinforced electro-pneumatic actuators. Int J Solids Struct 46:1043–1052CrossRefGoogle Scholar
  15. 15.
    Melnikov A, Ogden RW (2016) Finite deformations of an electroelastic circular cylindrical tube. Z Angew Math Phys 67:140Google Scholar
  16. 16.
    Melnikov A, Ogden RW (2018) Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field. Z Angew Math Phys 69:60Google Scholar
  17. 17.
    Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). SpringerGoogle Scholar
  18. 18.
    Merodio J, Ogden RW (2019) Finite deformation elasticity theory. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). SpringerGoogle Scholar
  19. 19.
    Pelrine R, Kornbluh R, Pei QB, Joseph J (2000) High-speed electrically actuated elastomers with strain greater than 100%. Science 287:836–839CrossRefGoogle Scholar
  20. 20.
    Rivlin RS (1949) Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Philso Trans R Soc A 242:173–195MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rogers JA (2013) A clear advance in soft actuators. Science 341:1243314CrossRefGoogle Scholar
  22. 22.
    Spencer AJM (1971) Theory of Invariants. In: Eringen AC (ed) Continuum physics, vol 1. Academic, New York, pp 239–353CrossRefGoogle Scholar
  23. 23.
    Su YP, Broderick HC, Chen WQ, Destrade M (2018) Wrinkles in soft dielectric plates. J Mech Phys Solids 119:298–318MathSciNetCrossRefGoogle Scholar
  24. 24.
    Suo Z (2010) Theory of dielectric elastomers. Acta Mech Solida Sin 23:549–578CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of EngineeringTufts UniversityMedfordUSA
  2. 2.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

Personalised recommendations