Acta Mechanica Solida Sinica

, Volume 25, Issue 5, pp 530–541 | Cite as

Waves in Pre-Stretched Incompressible Soft Electro Active Cylinders: Exact Solution

  • Weiqiu Chen
  • Huihui Dai


This paper studies wave propagation in a soft electroactive cylinder with an underlying finite deformation in the presence of an electric biasing field. Based on a recently proposed nonlinear framework for electroelasticity and the associated linear incremental theory, the basic equations governing the axisymmetric wave motion in the cylinder, which is subjected to homogeneous pre-stretches and pre-existing axial electric displacement, are presented when the electroactive material is isotropic and incompressible. Exact wave solution is then derived in terms of (modified) Bessel functions. For a prototype model of nonlinear electroactive material, illustrative numerical results are given. It is shown that the effect of pre-stretch and electric biasing field could be significant on the wave propagation characteristics.

Key words

electroelastic waves finite pre-stretch soft electroactive cylinder dispersion relation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cauchy, A.L., Exercices de Mathématique, vol. 2. Paris: Bure Freres, 1827.Google Scholar
  2. [2]
    Biot, M.A., The influence of initial stress on elastic waves. Journal of Applied Physics, 1940, 11(8): 522–530.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Hayes, M. and Rivlin, R.S., Propagation of a plane wave in an isotropic elastic material subjected to pure homogeneous deformation. Archive for Rational Mechanics and Analysis, 1961, 8(1): 15–22.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Chadwick, P. and Jarvis, D.A., Surface waves in a pre-stressed elastic body. Proceedings of the Royal Society of London (A), 1979, 366: 517–536.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Ogden, R.W. and Sotiropoulos, D.A., The effect of pre-stress on the propagation and reflection of plane waves in incompressible elastic solids. IMA Journal of Applied Mathematics, 1997, 59: 95–121.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Guz, A.N., Elastic waves in bodies with initial (residual) stresses. International Applied Mechanics, 2002, 38(1): 23–59.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Akbarov, S.D., Recent investigations on dynamic problems for an elastic body with initial (residual) stresses (review). International Applied Mechanics, 2007, 43(12): 3–27.CrossRefGoogle Scholar
  8. [8]
    Kushnir, V.P., Longitudinal waves in a continuous transversally isotropic cylinder with initial stresses. International Applied Mechanics, 1974, 10(7): 775–778.Google Scholar
  9. [9]
    Haughton, D.M., Wave speeds in rotating elastic cylinders at finite deformation. Quarterly Journal of Mechanics and Applied Mathematics, 1982, 35(1): 125–139.CrossRefGoogle Scholar
  10. [10]
    Belward, J.A. and Wright, S.J., Small-amplitude waves with complex wave numbers in a prestressed cylinder of Mooney material. Quarterly Journal of Mechanics and Applied Mathematics, 1987, 40(3): 383–399.CrossRefGoogle Scholar
  11. [11]
    Akbarov, S.D., Kepceler, T. and Egilmez, M.M., Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder. Journal of Sound and Vibration, 2011, 330: 4519–4537.CrossRefGoogle Scholar
  12. [12]
    Dai, H.H. and Peng, X.C., Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves. Wave Motion, 2011, 48(8): 761–772.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Pelrine, R., Kornbluh, R., Pei, Q.B. and Joseph, J., High-speed electrically actuated elastomers with strain greater than 100%. Science, 2000, 287: 836–839.CrossRefGoogle Scholar
  14. [14]
    Suo, Z., Theory of dielectric elastomers. Acta Mechanica Solida Sinica, 2010, 23(6): 549–578.CrossRefGoogle Scholar
  15. [15]
    Li, M., Xiao, J., Wu, J., Kim, R.H., Kang, Z., Huang, Y. and Rogers, A., Mechanics analysis of two-dimensionally prestrained elastomeric thin film for stretchable electronics. Acta Mechanica Solida Sinica, 2010, 23(6): 592–599.CrossRefGoogle Scholar
  16. [16]
    Fukada, E., Piezoelectricity in polymers and biological materials. Ultrasonics, 1968, 6: 229–234.CrossRefGoogle Scholar
  17. [17]
    Liu, Y.M., Zhang, Y.H., Chow, M.J., Chen, Q.N. and Li, J.Y., Biological ferroelectricity uncovered in aortic walls by piezoresponse force microscopy. Physical Review Letters, 2012, 108: 078103.CrossRefGoogle Scholar
  18. [18]
    Yang, J.S. and Hu, Y.T., Mechanics of electroelastic bodies under biasing fields. Applied Mechanics Reviews, 2004, 57(3): 173–189.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Chai, J.F. and Wu, T.T., Propagation of surface waves in a prestressed piezoelectric materials. Journal of the Acoustical Society of America, 1996, 100: 2112–2122.CrossRefGoogle Scholar
  20. [20]
    Liu, H., Kuang, Z.B. and Cai, Z.M., Propagation of Bleustein-Gulyaev waves in a prestressed layered piezoelectric structure. Ultrasonics, 2003, 41: 397–405.CrossRefGoogle Scholar
  21. [21]
    Hu, Y.T., Yang, J.S. and Jiang, Q., Surface waves in electrostrictive materials under biasing fields. Zeitschrift für angewandte Mathematik und Physik, 2004, 55: 678–700.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Qian, Z., Jin, F., Kishimoto, K. and Wang, Z., Effect of initial stress on the propagation behavior of SH-waves in multilayered piezoelectric composite structures. Sensors and Actuators A: Physical, 2004, 112: 368–375.CrossRefGoogle Scholar
  23. [23]
    Singh, B., Wave propagation in a prestressed piezoelectric half-space. Acta Mechanica, 2010, 211(3–4): 337–344.CrossRefGoogle Scholar
  24. [24]
    Dorfmann, A. and Ogden, R.W., Electroelastic waves in a finitely deformed electroactive material. IMA Journal of Applied Mathematics, 2010, 75: 603–636.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Du, J., Jin, X., Wang, J. and Zhou, Y., SH wave propagation in a cylindrically layered piezoelectric structure with initial stress. Acta Mechanica, 2007, 191(1–2): 59–74.CrossRefGoogle Scholar
  26. [26]
    Abd-alla, A.N., Al-sheikh, F. and Al-Hossain, A.Y., Effect of initial stresses on dispersion relation of transverse waves in a piezoelectric layered cylinder. Material Science and Engineering B, 2009, 162: 147–154.CrossRefGoogle Scholar
  27. [27]
    Dorfmann, A. and Ogden, R.W., Nonlinear electroelasticity. Acta Mechanica, 2005, 174: 167–183.CrossRefGoogle Scholar
  28. [28]
    Dorfmann, A. and Ogden, R.W., Nonlinear electroelastic deformations. Journal of Elasticity, 2006, 82: 99–127.MathSciNetCrossRefGoogle Scholar
  29. [29]
    Dorfmann, A. and Ogden, R.W., Nonlinear electroelastostatics: Incremental equations and stability. International Journal of Engineering Science, 2010, 48(1): 1–14.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Ding, H.J., Chen, W.Q. and Zhang, L.C., Elasticity of Transversely Isotropic Materials. Dordrecht: Springer, 2006.zbMATHGoogle Scholar
  31. [31]
    Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Body. Moscow: Mir Publishers, 1981.zbMATHGoogle Scholar
  32. [32]
    Ding, H.J., Chen, W.Q., Guo, Y.M. and Yang, Q.D., Free vibrations of piezoelectric cylindrical shells filled with compressible fluid. International Journal of Solids and Structures, 1997, 34(16): 2025–2034.CrossRefGoogle Scholar
  33. [33]
    Mindlin, R.D. and McNiven, H.D., Axially symmetric waves in elastic rods. Journal of Applied Mechanics, 1960, 27: 145–151.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Dai, H.H. and Huo, Y., Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods. Acta Mechanica, 2002, 157: 97–112.CrossRefGoogle Scholar
  35. [35]
    Folkow, P.D. and Mauritsson, K., Dynamic higher-order equations for finite rods. Quarterly Journal of Mechanics and Applied Mathematics, 2010, 63(1): 1–21.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2012

Authors and Affiliations

  1. 1.Soft Matter Research Center (SMRC)Zhejiang UniversityHangzhouChina
  2. 2.Department of Engineering MechanicsZhejiang UniversityHangzhouChina
  3. 3.Department of MathematicsCity University of Hong KongKowloon, Hong KongChina

Personalised recommendations