Waves and vibrations in a finitely deformed electroelastic circular cylindrical tube

Abstract

<jats:p>In two recent papers, conditions for which axisymmetric incremental bifurcation could arise for a circular cylindrical tube subject to axial extension and radial inflation in the presence of an axial load, internal pressure and a radial electric field were examined, the latter being effected by a potential difference between compliant electrodes on the inner and outer radial surfaces of the tube. The present paper takes this work further by considering the incremental deformations to be time-dependent. In particular, both the axisymmetric vibration of a tube of finite length with appropriate end conditions and the propagation of axisymmetric waves in a tube are investigated. General equations and boundary conditions governing the axisymmetric incremental motions are obtained and then, for purposes of numerical evaluation, specialized for a Gent electroelastic model. The resulting system of equations is solved numerically and the results highlight the dependence of the frequency of vibration and wave speed on the tube geometry, applied deformation and electrostatic potential. In particular, the bifurcation results obtained previously are recovered as a special case when the frequency vanishes. Specification of an incremental potential difference in the present work ensures that there is no incremental electric field exterior to the tube. Results are also illustrated for a neo-Hookean electroelastic model and compared with those previously obtained for the case in which no incremental potential difference (or charge) is specified and an external field is required.</jats:p>

Bifurcation analysis of residually stressed neo-Hookean and Ogden electroelastic tubes

“…We follow the steps in [11,14,16] and arrange the equations in the form y = My, (5.23) with the prime indicating differentiation with respect to r, and M the 6 × 6 matrix…”

“…The electroelastic moduli tensors are listed in index form and then specialized to both neo-Hookean and Mooney-Rivlin electroelastic materials. Dorfmann & Ogden [16] use time-dependent increments in the deformation and in the electric displacement to investigate the axisymmetric vibrations and the propagation of axisymmetric waves in a thick-walled tube. Shmuel & deBotton [17] investigate the propagation of waves in a finitely deformed tube subject to a radial electric field for different tube geometries for fixed axial stretch and zero axial load.…”

The effect of an exterior electric field on the instability of dielectric plates

“…Advanced theories to describe the main electro-mechanical phenomena observed in DEs have been developed over the years, including nonlinear deformation [14,15], vibration [16][17][18], wrinkling instability [19][20][21] and wave propagation [22][23][24]. Most of these studies focused on electro-elastic plates actuated by flexible electrodes.…”

Nonlinear response and axisymmetric wave propagation in functionally graded soft electro-active tubes

“…Since Pelrine et al (1998) proposed an SEA tube actuator, there has been an increasing interest in studying the nonlinear static and dynamic responses of this particular structural configuration subject to biasing fields (Dorfmann and Ogden, 2006;Zhu et al, 2010b;Son and Goulbourne, 2010;Melnikov and Ogden, 2016;Bortot, 2018). Based on the theory of nonlinear electro-elasticity and its linearized incremental theory developed by Ogden (2006, 2010a), considerable efforts have been devoted to elucidating the effects of biasing fields on the structural stability of SEA tubes (Zhu et al, 2010b;An et al, 2015;Su et al, 2018Su et al, , 2020, elastic wave propagation (Chen and Dai, 2012;Shmuel and deBotton, 2013;Shmuel, 2015;Su et al, 2016;Wu et al, 2017), and tunable vibrations and oscillations (Son and Goulbourne, 2010;Sarban et al, 2011;Bortot, 2018;Dorfmann and Ogden, 2020;Zhu et al, 2020). Note that recent advances in the study of instabilities, tunable phononic crystals, and acoustic/vibration control based on SEA materials have been reviewed in the papers by Zhao et al (2016), Dorfmann and Ogden (2019) and Wang et al (2020).…”