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Nonlinear response of an electroelastic spherical shell. (English) Zbl 1423.74305
Summary: In this paper the theory of nonlinear electroelasticity is used to examine radial deformations of a thick-walled spherical shell of soft dielectric material with compliant electrodes on its inner and outer surfaces combined with an internal pressure. A general expression for the dependence of the pressure on the deformation and a potential difference between the electrodes, or uniform surface charge distributions on the electrodes, is obtained in respect of a general incompressible isotropic electroelastic energy function. To illustrate the behavior of the shell specific forms of energy functions are used for numerical purposes, for either fixed potential difference or fixed charge distribution. Explicit general results are given for the limiting case of a thin-walled (or membrane) shell. Depending on the elastic part of the energy function (as distinct from that part which involves the electric field) different behaviors are obtained, including non-monotonicity of the pressure-radius relationship reminiscent of that known in the purely elastic situation, but moderated by the presence of an electric field.

MSC:
74F15 Electromagnetic effects in solid mechanics
74B20 Nonlinear elasticity
74K25 Shells
Software:
Mathematica
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[1] Akkas, N., On the dynamic snap-out instability of inflated non-linear spherical membranes, International Journal of Non-Linear Mechanics, 13, 177-183, (1978) · Zbl 0384.73048
[2] Alexander, H., Tensile instability of initially spherical balloons, International Journal of Engineering Science, 9, 151-162, (1971)
[3] Beatty, M. F., Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues - with examples, Applied Mechanics Reviews, 40, 1699-1734, (1987)
[4] Bustamante, R.; Dorfmann, A.; Ogden, R. W., On electric body forces and Maxwell stresses in nonlinearly electroelastic solids, International Journal of Engineering Science, 47, 1131-1141, (2009) · Zbl 1213.74123
[5] deBotton, G.; Bustamante, R.; Dorfmann, A., Axisymmetric bifurcations of thick spherical shells under inflation and compression, International Journal of Solids and Structures, 50, 403-413, (2013)
[6] Dorfmann, A.; Ogden, R. W., Nonlinear electroelasticity, Acta Mechanica, 174, 167-183, (2005) · Zbl 1066.74024
[7] Dorfmann, A.; Ogden, R. W., Nonlinear electroelastic deformations, Journal of Elasticity, 82, 99-127, (2006) · Zbl 1091.74014
[8] Dorfmann, A.; Ogden, R. W., Nonlinear theory of electroelastic and magnetoelastic interactions, (2014), Springer New York · Zbl 1291.78002
[9] Edmiston, J.; Steigmann, D. J., Analysis of nonlinear electroelastic membranes, (Ogden, R. W.; Steigmann, D. J., Mechanics and electrodynamics of magneto- and electro-elastic materials, CISM Courses and Lectures, Vol. 527, (2011), Springer Wien-New York), 153-180
[10] Eringen, A. C.; Maugin, G. A., Electrodynamics of continua I: foundations and solid media, (1990), Springer New York
[11] Fu, Y. B.; Xie, Y. X., Stability of pear-shaped configurations bifurcated from a pressurized spherical balloon, Journal of the Mechanics and Physics of Solids, 68, 33-44, (2014) · Zbl 1328.74044
[12] Gent, A. N., A new constitutive relation for rubber, Rubber Chemistry and Technology, 69, 59-61, (1996)
[13] Green, A. E.; Adkins, J. E., Large elastic deformations and non-linear continuum mechanics, (1960), Oxford University Press · Zbl 0090.17501
[14] Haughton, D. M.; Ogden, R. W., On the incremental equations in non-linear elasticity II. bifurcation of pressurized spherical shells, Journal of the Mechanics and Physics of Solids, 26, 111-138, (1978) · Zbl 0401.73077
[15] Keplinger, C.; Li, T.; Baumgartner, R.; Suo, Z.; Bauer, S., Harnessing snap-through instability in soft dielectrics to achieve giant voltage-triggered deformation, Soft Matter, 8, 285-288, (2012)
[16] Li, T.; Keplinger, C.; Baumgartner, R.; Bauer, S.; Yang, W.; Suo, Z., Giant voltage-induced deformation in dielectric elastomers near the verge of snap-through instability, Journal of the Mechanics and Physics of Solids, 61, 611-628, (2013)
[17] Mockensturm, E. M.; Goulbourne, N., Dynamic response of dielectric elastomers, International Journal of Non-Linear Mechanics, 41, 388-395, (2006)
[18] Needleman, A., Inflation of spherical rubber balloons, International Journal of Solids and Structures, 13, 409-421, (1977)
[19] Ogden, R. W., Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids, Proceedings of the Royal Society of London A, 326, 565-584, (1972) · Zbl 0257.73034
[20] Ogden, R. W.; Steigmann, D. J., Mechanics and electrodynamics of magneto- and electro-elastic materials, CISM Courses and Lectures, Vol. 527, (2011), Springer Wien-New York · Zbl 1209.74005
[21] Rudykh, S.; Bhattacharya, K.; deBotton, G., Snap-through actuation of thick-wall electroactive balloons, International Journal of Non-Linear Mechanics, 47, 206-209, (2012)
[22] Treloar, L. R.G., Stress-strain data for vulcanized rubber under various types of deformation, Transactions of the Faraday Society, 40, 59-70, (1944)
[23] Wolfram Research, Inc. (2013). Mathematica, Version 9.0, Champaign, Illinois.
[24] Zhao, X.; Wang, Q., Harnessing large deformation and instabilities of soft dielectrics: theory, experiment, and application, Applied Physics Reviews, 1, 021304, (2014)
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