Voltage-controlled instability transitions and competitions in a finitely deformed dielectric elastomer tube

Abstract

This work compares the transition and competitive mechanism between three types of instabilities of an incompressible dielectric tube: wrinkling, pull-in instability and electric breakdown. We also see how to select one type of instability mode on demand. First, we investigate the finite response and the wrinkling of a tube subject to a combination of applied radial voltage, torsion and axial force (or stretch). We use the surface impedance matrix method to determine the wrinkling threshold, and obtain the corresponding two-dimensional pattern shape of wrinkled surface. Second, we look at illustrative numerical calculations for ideal Mooney-Rivlin dielectrics and study the effects of actuation methods, electric voltage, torsion and geometrical parameters on the three types of instabilities. Results show that the deformation of the solid will influence the true electric field in the solid, and induce competitive effects between the applied voltage and the mechanical loading. We find that in addition to the expected contractile buckling, buckling may also occur in extension in an electrically actuated dielectric tube, a departure from the purely elastic wrinkling. Moreover, the electro-elastic behavior of the DE elastomer can be enhanced by introducing torsion. We also find that large stable actuation can be achieved and that the wrinkling pattern can be selected on demand in the tube by finely tuning the actuation, voltage, torsion and geometry, without encountering material failure.

Introduction

Dielectric elastomers (DEs) have received extensive attention due to their promising applications in biomedical engineering, soft robots, adaptive optics, high-performance actuators and sensors, etc. (Brochu, Pei, 2012, Duduta, Hajiesmaili, Zhao, Wood, Clarke, 2019, O’Halloran, O’malley, McHugh, 2008, Rasmussen, 2012). Pelrine, Kornbluh, and Joseph (1998) first proposed the so-called tubular DE actuator consisting of a thin-walled cylindrical elastomeric tube sandwiched between two compliant electrodes on its inner and outer faces. With the application of a voltage through the electrodes, the interposed tube wall is squeezed, causing a radial expansion and an axial elongation (Cameron, Szabo, Johnstone, Massey, Leidner, 2008, Carpi, De Rossi, 2004). By cyclically activating and de-activating the DE tube, the inner volume of the tube can be changed repeatedly to control the inlet and outlet of gases or liquids. One potential use of such mechanism is for manufacturing large-volume pumps (Brown, Lai, 2009, Wang, Li, Qin, Caddy, Yap, Zhu, 2018). Tubular DE actuators can be manufactured through well-established industrial processes, such as extrusion and flexible dip-coating techniques. (Arora, Ghosh, Muth, 2007, Stoyanov, Kofod, Gerhard, 2008). Compared with planar DE actuators, tubular DE actuators are less bulky and more versatile for applications (Cameron, Szabo, Johnstone, Massey, Leidner, 2008, Stoyanov, Kofod, Gerhard, 2008, Zhu, Stoyanov, Kofod, Suo, 2010).

Instability analysis for DE devices is of considerable theoretical and industrial significance. However, the coupled partial differential equations governing the wrinkling behavior of DEs are difficult to solve, even numerically, due to geometric/physical non-linearities and multi-physics coupling. Robust numerical strategies should be developed for solving the resulting dispersion equations.

From a practical viewpoint, DEs may suffer from more failure modes than purely elastic elastomers, such as pull-in (snap-through) instability (Pelrine, Kornbluh, Pei, Joseph, 2000, Zhao, Suo, 2007, Zurlo, Destrade, DeTommasi, Puglisi, 2017), electric breakdown (Dissado, Fothergill, 1992, Stark, Garton, 1955, Zhang, Wang, Godaba, Khoo, Zhang, Zhu, 2017), buckling instability (Bertoldi, Gei, 2011, Gei, Colonnelli, Springhetti, 2014, Goshkoderia, Rudykh, 2017), localized necking (Fu, Dorfmann, Xie, 2018a, Fu, Xie, Dorfmann, 2018b) and bulging (Che, Lu, Wang, 2017, Lu, An, Li, Yuan, Wang, 2015, Wang, Yuan, Lu, Wang, 2017), which pose clear limitations on developing DE devices. Moreover, there are complicated transitions and competitions between these failure modes, and thus a comprehensive comparison is required for reliability analysis.

On the other hand, these instability modes can be seen as beneficial: hence, the electromechanical coupling behavior of DEs can generate complex 3D patterns, useful to control surface shapes (Pang, Cheng, Zhao, Guo, Ji, Li, Liang, Xu, Song, Zhang, Xu, Sang, Huang, Li, Zhang, 2020, Wang, Gossweiler, Craig, Zhao, 2014); similarly, the large actuation induced by the pull-in instability is eagerly pursued in DE actuators to produce giant changes in surface area (Huang et al., 2012).

Compared with planar DE devices, instability analysis in tubular devices is complicated by geometrical complexity and finite deformation inhomogeneity. Singh (1966) first studied the static response of a DE tube under radial electric field. Dorfmann, Ogden, 2005, Dorfmann, Ogden, 2006 investigated the azimuthal shear, extension and inflation responses of a DE tube subject to a radial electric field. Later, they specialized the boundary value problem to the Gent dielectric model in a review article (Dorfmann & Ogden, 2017) to illustrate numerically the influence of the applied electric field on the deformation of the tube. Zhu et al. (2010) studied the finite deformation of a pre-stressed DE tube and analyzed the snap-through instability using the so-called Hessian approach. They indicated that it can be enhanced by applying pre-stress along the length. Note that the Hessian approach cannot predict inhomogeneous wrinkling-type instabilities, which requires incremental analysis. Shmuel and deBotton (2013) and Wu, Su, Chen, and Zhang (2017) studied axisymmetric and circumferential waves in a finitely deformed DE tube subject to a radial voltage, respectively. Note that the vanishing of the wave speed corresponds to the threshold of wrinkling instability. Su, Zhou, Chen, and Lü (2016a); Su, Wang, Zhang, and Chen (2016b) examined non-axisymmetric waves and the wrinkling instability of an incompressible DE tube. They obtained analytical dispersion equations in terms of Bessel functions by assuming a homogeneous finite pre-deformation. Bortot and Shmuel (2018) and Melnikov and Ogden (2018) studied the prismatic and axisymmetric 2D wrinkling of a DE tube subject to a radial voltage and an axial pre-stretch. Lu et al. (2015) investigated electromechanical bulging instability in a finitely deformed DE tube. They obtained the equilibrium and stability governing equations from the first and second variations of the free energy of the thermodynamic system. Their tube is considered thin and the mechanical and electric fields are assumed to be homogeneous to simplify the analysis.

With this paper, we propose a theoretical analysis of finite deformation and the associated 3D wrinkling-type instability (or buckling) of an incompressible DE tube subject to the combined action of a radial electrical voltage and mechanical loads. We consider not only the wrinkling instability, but also the instabilities due to pull-in (snap-through) and to electric breakdown, and examine the transition and competition between these failures. Additionally, we study the effect of actuation methods on the nonlinear response and the stability of the tube.

The originality and innovative aspects of the current paper include

• Presenting a 3D wrinkling analysis for a DE tube subject to a radially inhomogeneous electric field, an axial force (or stretch), and twisting moments applied at the top and bottom faces. Here we note that most of the existing work so far is concerned with 2D wrinkling instability of DE tubes, while a 3D analysis reveals more information about the various modes actually selected in the instability process, especially for a tube with finite length.

• Investigating the extensional wrinkling mode in a stretched DE tube, which does not occur in purely elastic solids with Mooney-Rivlin or neoHookean strain energy. So far only contractile wrinkling has been considered for DE tubes and not the extensional wrinkling mode, which may occur in stretched DE films, see Fig. 1 and Su, Broderick, Chen, and Destrade (2018).

• Revealing the competitive effects between voltage and mechanical loadings, and investigating the influence of actuation method on the finite deformation and instabilities of the tube.

• Investigating the transition and competitive mechanism between failure modes in a DE tube. Instabilities in DE elastomers have been well investigated. Here we find scenarios where one instability triggers another, and propose a rational approach for designing stable structural patterns on demand and obtaining stable large actuation deformations of DE tubes.

The paper is structured as follows. In Section 2, we derive the governing equations for a finitely deformed DE tube subject to a radial voltage, internal pressure, an axial pre-stretch and a torsion, based on the nonlinear theory of electro-elasticity developed by Dorfmann, Ogden, 2005, Dorfmann, Ogden, 2016. Note that there are some other electroelasitc models proposed, for example, by Yang and Hu (2004), Romeo (2011), Poya et al. (2018), etc. We then formulate the linearized incremental equations of motion in Section 3. To solve the incremental boundary value problem, we use the surface impedance matrix method, a robust numerical procedure for deriving the threshold for the onset of the instability.

In Section 4, we illustrate the actuation methods of the tube and specialize the formulas obtained in Sections 2 and 3 to ideal Mooney-Rivlin dielectric models. Here two problems corresponding to two different loading paths are considered. In the first problem, the tube is subject to a fixed radial voltage and a torsion where the top and bottom ends of the tube move under the action of an axial force F: the force-tuning problem. In the second problem, the tube is subject to a fixed axial pre-stretch, a torsion created by moments applied at the top and bottom faces and a variable radial voltage: the voltage-tuning problem.

The numerical analysis for each problem is detailed in Sections 5 and 6, respectively, where the influences of actuation methods, geometrical parameters of the tube, electric field and mechanical loading on the transition and competition of instabilities are investigated. Moreover, we find the condition for the occurrence of extensional buckling in the solid. Finally in Section 7, we draw some conclusions.