Snap-through actuation of thick-wall electroactive balloons

Solution to the problem of a spherical balloon made out of an electroactive polymer which is subjected to coupled mechanical and electrical excitations is determined. It is found that for certain material behaviors instabilities that correspond to abrupt changes in the balloon size can be triggered. This can be exploited to electrically control different actuation cycles as well as to use the balloon as a micropump. & 2011 Published by Elsevier Ltd. Electroactive polymers (EAP) are materials that change their size and shape in response to electrostatic excitation. Roughly speaking, the electrostatically induced Maxwell stress results in the deformation of the material. Various EAP based actuators have been considered in the past [1–7]. A major limitation of these materials originates in the need for relatively large electric fields [8]. However, the usage of instability phenomena [9–13] may reduce the intensity of the required electric field. Accordingly, we analyze the response of EAP balloons and demonstrate that a relatively small electrostatic field can be used as a trigger for large deformations. We follow the theory of nonlinear electroelasticity [14,15], where the total stress is s ij 1⁄4 s ðcÞ ij þs ðmÞ ij : ð1Þ Here, s ij is the Cauchy mechanical stress, s ðmÞ ij 1⁄4 ee0EiEj ðe0=2Þ EnEndij is the Maxwell stress induced by the electric field E, e is a dielectric modulus, and e0 is the vacuum permittivity. We assume that e1⁄4 6, which is typical for common polymers, is constant [16]. We adopt a quite general constitutive law for incompressible isotropic materials, in which the principal stresses are s k 1⁄4 XN p 1⁄4 1 mpl ap k q, ð2Þ where lk are the principal stretches, q is an arbitrary hydrostatic pressure, mp are shear moduli, and ap are material constants. In the case N1⁄41 with a1 1⁄4 2, the model (2), which is commonly denoted as Ogden model, reduces to the neo-Hookean one. With N1⁄43 an excellent correlation with experimental data for Elsevier Ltd. ical Engineering, Ben-Gurion eBotton). elastomers is revealed [17]. Accordingly, we assume the following typical values for the elastic constants of soft polymers [17] m1 1⁄4 6:3 10 5 Pa, m2 1⁄4 1:2 10 3 Pa, m3 1⁄4 1 10 4 Pa, a1 1⁄4 1:3, a2 1⁄4 5, and a3 1⁄4 2. In the vicinity of the reference configuration this polymer behaves like a neo-Hookean material with shear modulus m1⁄4 2 P3 p 1⁄4 1 mpap 1⁄4 4:225 10 5 Pa. Consider a spherical balloon made out of a dielectric elastomer with inner Ri and outer Ro radii. Here and thereafter, the notations ð Þi and ð Þo are used to specify quantities at the inner and outer radii, respectively. The thickness of the balloon wall is H1⁄4 Ro Ri. The inner and outer surfaces of the balloon wall are covered with thin electrodes with negligible elastic modulus [1]. With these electrodes electric field is induced across the wall. The balloon can be inflated with inner pressure Pi, and electrically exited with electric potential jo between the two electrodes. The associated boundary conditions are sðtÞ rr ðriÞ 1⁄4 Pi, s ðtÞ rr ðroÞ 1⁄4 0, jðriÞ 1⁄4 0, jðroÞ 1⁄4jo, ð3Þ where r is the radius in the deformed configuration. In spherical coordinate system the principal stretch ratios are lr 1⁄4 dr dR 1⁄4 l , ly 1⁄4 lf 1⁄4 r R 1⁄4 l, ð4Þ where lðRÞ 1⁄4 ð1þðRo=RÞðl3o 1ÞÞ . Maxwell equations reduce to Laplace equation for the electrical potential j, which is solved in the deformed configuration. The components of the electric field E rj satisfying the electrostatic boundary conditions in (3) are Er 1⁄4 jorori ri ro 1 r2 , Ey 1⁄4 Ef 1⁄4 0: ð5Þ 0.0 0.3 0.6 0.9 1.2 1.5 o 0.4 0.2 0.1 0.05 E 0 = 0.0