Abstract
This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney–Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Pelrine, R., Sommer-Larsen, P., Kornbluh, R., Heydt, R., Kofod, G., Pei, Q., Gravesen, P.: Applications of dielectric elastomer actuators. In: Bar-Cohen, Y. (Ed) Smart Structures and Materials 2001: Electroactive Polymer Actuators and Devices, Proceedings of SPIE, Vol. 4329, pp. 335–349 (2001)
Zhu, J., Stoyanov, H., Kofod, G., Suo, Z.: Large deformation and electromechanical instability of a dielectric elastomer tube actuator. J. Appl. Phys. 108, 074113 (2010)
Melnikov, A., Ogden, R.W.: Finite deformations of an electroelastic circular cylindrical tube. Z. Angew. Math. Phys. 67, 140 (2016)
Skala, D.P.: Modified equations of rubber elasticity applied to the inflation mechanics of a thick-walled rubber cylinder. Rubber Chem. Technol. 43, 745–757 (1970)
Charrier, J.M., Li, Y.K.: Large elastic deformation theories and unstable deformations of rubber structures: thick tubes. Trans. Soc. Rheol. 21, 301–325 (1977)
An, L., Wang, F., Cheng, S., Lu, T., Wang, T.J.: Experimental investigation of the electromechanical phase transition in a dielectric elastomer tube. Smart Mater. Struct. 24, 035006 (2015)
Haughton, D.M., Ogden, R.W.: Bifurcation of inflated circular cylinders of elastic material under axial loading–I. Membrane theory for thin-walled tubes. J. Mech. Phys. Solids 27, 179–212 (1979)
Haughton, D.M., Ogden, R.W.: Bifurcation of inflated circular cylinders of elastic material under axial loading–II. Exact theory for thick-walled tubes. J. Mech. Phys. Solids 27, 489–512 (1979)
Su, Y., Zhou, W., Chen, W., Lu, C.: On buckling of a soft incompressible electroactive hollow cylinder. Int. J. Solids Struct. 97–98, 400–416 (2016)
Bortot, E., Shmuel, G.: Prismatic bifurcations of soft dielectric tubes. Int. J. Eng. Sci. 124, 104–114 (2018)
Shmuel, G., deBotton, G.: Axisymmetric wave propagation in finitely deformed dielectric elastomer tubes. Proc. R. Soc. Lond. A 469, 20130071 (2013)
Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity. Acta Mech. 174, 167–183 (2005)
Melnikov, A.: Bifurcation of thick-walled electroelastic cylindrical and spherical shells at finite deformation. Ph.D. Thesis, University of Glasgow (2016)
Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity: incremental equations and stability. Int. J. Eng. Sci. 48, 1–14 (2010)
Dorfmann, L., Ogden, R.W.: Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer, New York (2014)
Ogden, R.W.: Non-linear Elastic Deformations. Dover Publications, New York (1997)
MATLAB version 9.3.0.713579 (R2017b). The MathWorks Inc., Natick, Massachusetts (2017)
Ogden, R.W.: Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A 326, 565–584 (1972)
Zhu, Y., Luo, X.Y., Ogden, R.W.: Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure. Int. J. Solids Struct. 45, 3410–3429 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Melnikov, A., Ogden, R.W. Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field. Z. Angew. Math. Phys. 69, 60 (2018). https://doi.org/10.1007/s00033-018-0954-5
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-018-0954-5