Abstract
In this paper we present a first account towards the modeling of electro-elastic plates as electro-elastic material surfaces. A complete direct approach is developed without the need to involve the three-dimensional formulation. In particular, we consider the case of electrostrictive polymer plates, in which ponderomotive forces as well as constitutive coupling by means of electrostriction are accounted for. We propose a rational formulation for the augmented free energy of electro-elastic material surfaces incorporating electrostriction by a multiplicative decomposition of the surface stretch tensor and an additive decomposition of the surface curvature tensor into elastic and electrical parts. Moreover, we show that concepts such as the total stress, the electrostatic stress and the mechanical stress exist also for electro-elastic material surfaces. Results computed within the framework of this complete direct approach are compared to results based on a numerical integration of the plane stress three-dimensional augmented free energy. A very good agreement is found.
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References
Toupin, R.A.: The elastic dielectric. J. Ration. Mech. Anal. 5(6), 849–915 (1956)
Pao, Y.H.: Electromagnetic forces in deformable continua. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 4, pp. 209–306. Pergamon Press, Oxford (1978)
Prechtl, A.: Eine Kontinuumstheorie elastischer Dielektrika. Teil 1: Grundgleichungen und allgemeine Materialbeziehungen (in German). Archiv für Elektrotechnik 65(3), 167–177 (1982)
Prechtl, A.: Eine Kontinuumstheorie elastischer Dielektrika. Teil 2: Elektroelastische und elastooptische Erscheinungen (in German). Archiv für Elektrotechnik 65(4), 185–194 (1982)
Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988)
Gao, Z., Tuncer, A., Cuitiño, A.: Modeling and simulation of the coupled mechanical-electrical response of soft solids. Int. J. Plast. 27(10), 1459–1470 (2011)
Pelrine, R.E., Kornbluh, R.D., Joseph, J.P.: Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation. Sens. Actuators A Phys. 64, 77–85 (1998)
Bar-Cohen, Y.: Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges. SPIE, Bellingham, WA (2004)
Zäh, D., Miehe, C.: Multiplicative electro-elasticity of electroactive polymers accounting for micromechanically-based network models. Comput. Methods Appl. Mech. Eng. 286, 394–421 (2015)
Dorfmann, A., Ogden, R.W.: Nonlinear electroelasticity. Acta Mech. 174, 167–183 (2005)
Vu, D.K., Steinmann, P., Possart, G.: Numerical modelling of non-linear electroelasticity. Int. J. Numer. Methods Eng. 70, 685–704 (2007)
Skatulla, S., Sansour, C., Arockiarajan, A.: A multiplicative approach for nonlinear electro-elasticity. Comput. Methods Appl. Mech. Eng. 245–246, 243–255 (2012)
Staudigl, E., Krommer, M., Humer, A.: Modeling of dielectric elastomers accounting for electrostriction by means of a multiplicative decomposition of the deformation gradient tensor. In: Altenbach, H., Carrera, E., Kulikov, G. (eds.) Analysis and Modelling of Advanced Structures and Smart Systems. Springer, Vienna (2018)
Humer, A., Krommer, M.: Modeling of piezoelectric materials by means of a multiplicative decomposition of the deformation gradient. Mech. Adv. Mater. Struct. 22, 125–135 (2015)
Altenbach, H., Zhilin, P.A.: A general theory of elastic simple shells (in Russian). Uspekhi Mekhaniki 11, 107–148 (1998)
Opoka, S., Pietraszkiewicz, W.: On modified displacement version of the non-linear theory of thin shells. Int. J. Solids Struct. 46(17), 3103–3110 (2009)
Eliseev, V.V., Vetyukov, Y.: Finite deformation of thin shells in the context of analytical mechanics of material surfaces. Acta Mechanica 209(1–2), 43–57 (2010)
Vetyukov, Y.: Finite element modeling of Kirchhoff-Love shells as smooth material surfaces. ZAMM 94, 150–163 (2014a)
Altenbach, H., Eremeyev, V.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 12(49), 1294–1301 (2011)
Vetyukov, Y., Staudigl, E., Krommer, M.: Hybrid asymptotic-direct approach to finite deformations of electromechanically coupled piezoelectric shells. Acta Mech. 229(2), 953–974 (2018)
Vetyukov, Y.: Nonlinear Mechanics of Thin-Walled Structures: Asymptotics. Direct Approach and Numerical Analysis. Springer, Vienna, NewYork (2014)
Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells. Cambridge University Press (2005)
Klinkel, S., Zwecker, S., Mueller, R.: A solid shell finite element formulation for dielectric elastomers. J. Appl. Mech. 80, 021026-1–021026-11 (2013)
Bishara, D., Jabareen, M.: An optimal solid-shell finite element for modeling dielectric elastomers. In: Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry, 6 pp. (2017)
Staudigl, E., Krommer, M., Vetyukov, Y.: Finite deformations of thin plates made of dielectric elastomers: modeling, numerics and stability. J. Intell. Mater. Syst. Struct. 19 (2017) (published online October 2017)
Krommer, M., Staudigl-Hansy, E.: A complete direct approach to modeling of dielectric elastomer plates as material surfaces. In Metveenko, V., Krommer, M., Belyaev, A.K., Irschik, H. (eds.) Dynamics and Control of Advanced Structures and Machines: Contributions from the 3rd International Workshop, Perm, Russia. Springer Nature (2019)
Naghdi, P.: The theory of shells and plates. In: Flügge, S., Truesdell, C. (eds.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Berlin (1972)
Altenbach, H., Eremeyev, V.A.: Cosserat-type shells. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 541. Springer, Vienna (2013)
Diaconu, I., Dorohoi, D.O.: Properties of polyurethane thin films. J. Optoelectron. Adv. Mater. 7(2), 921–924 (2005)
Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)
Breslavsky, I.D., Amabili, M., Legrand, M.: Nonlinear vibrations of thin hyperelastic plates. J. Sound Vib. 333(19), 4668–4681 (2014)
Koiter, W.T.: On the nonlinear theory of thin elastic shells. Proc. Koninklijke Ned. Akad. van Wet. B 69, 1–54 (1966)
Duong, T.X., Roohbakhshan, F., Sauer, R.A.: A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Comput. Methods Appl. Mech. Eng. 316, 43–83 (2017)
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Krommer, M., Hansy-Staudigl, E. (2019). A Complete Direct Approach to Modeling of Electrostrictive Polymer Plates as Electro-elastic Material Surfaces. In: Altenbach, H., Irschik, H., Matveenko, V. (eds) Contributions to Advanced Dynamics and Continuum Mechanics. Advanced Structured Materials, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-030-21251-3_9
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