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Universal relations for nonlinear electroelastic solids. (English) Zbl 1111.74014
Summary: Electro-sensitive elastomers are materials that can support large elastic deformations under the influence of an electric field. There has been growing interest recently in their applications as so-called “smart materials”. This paper is devoted to the derivation of universal relations in the context of nonlinear theory of electroelasticity that underpins such applications. Universal relations are equations relating the components of stress, the electric variables and the deformation that are independent of the constitutive law for a family of materials. For the general constitutive equations of an isotropic electroelastic material derived from a free energy function and for some special cases of these equations, we obtain universal relations, the word “universal” being relative to the considered class or subclass of constitutive laws. These universal relations are then applied to some controllable states (homogeneous and non-homogeneous) in order to highlight some examples that may be useful from the point of view of experimental characterization of the material properties. Additionally, we examine the (non-controllable) problem of helical shear of a circular cylindrical tube in the presence of a radial electric field, and we find that a nonlinear universal relation that has been obtained previously [R. W. Ogden et al., Q. J. Mech. Appl. Math. 26, 23–41 (1973)] for an elastic material also holds when the electric field is applied.

##### MSC:
 74F15 Electromagnetic effects in solid mechanics 74A20 Theory of constitutive functions in solid mechanics 74B20 Nonlinear elasticity
##### Keywords:
free energy function; helical shear; radial electric field
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##### References:
 [2] Costen, R. C., Su, J., Harrison, J. S.: Model for bending actuators that use electrostrictive graft elastomers. In: Proc. SPIE Vol. 4329, Smart Structures and Materials 2001: Electroactive Polymer Actuators and Devices (Bar-Cohen, J., ed.), pp. 436–444. SPIE Press 2001. [10] Rivlin, R. S.: Some applications of elasticity theory to rubber engineering. In: Proc. Rubber Technology Conf., London (Dawson, T. R., ed.), pp. 1–8. Cambridge: Heffer 1948 (appeared also in: Collected papers of Ronald S. Rivlin, vol. 1 (Barenblatt, G. I., Joseph, D. D., eds.), pp. 9–16. Berlin: Springer 1996) · Zbl 0032.08904
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