Analytical models for circular and spherical dielectric elastomers

Zhang Hui Dai Min Zhang Zhisheng Xia Zhijie

(School of Mechanical Engineering, Southeast University, Nanjing 211189, China)

Abstract：In order to imitate skin characteristics, a dielectric elastomer (DE) membrane coated with flexible electrodes is applied with high voltage, which can lead to wrinkles and other phenomena. To develop soft-actuated air vehicles and other equipment, lightweight gas is pumped into a DE spherical shell to generate controllable flight movements. According to experimental phenomena and data, the calculation models of phase transitions on circular DE films are built. Meanwhile, the deformation characteristics of different DE (acrylic polymer and rubber) spherical actuators combined with helium are compared. The peak pressure inside a rubber balloon is greater than that of a VHB (acrylic polymer) balloon shell, but the limit stretch of rubber is much smaller. By taking advantages of this phenomenon, large deformations of a VHB spherical shell can be realized at an actuated state. Moreover, multi-layer spherical DE shells can achieve larger voltage-induced volume change than monolayer ones. The research indicates that pre-stretching is one of the key factors to induce phase transitions between flat, wrinkled and bulging regions on circular DE films, and the internal pressure determines the electromechanical performance of balloon actuators.

Key words：dielectric elastomer; circular frame; balloon actuator; large deformation; phase transition

Soft active materials are utilized to manufacture soft machines, and they emerge as an exciting engineering field. Dielectric elastomer actuators (DEAs) are a kind of soft active materials which can deform in response to voltage[1-2], and they are used in a variety of innovative applications, such as medical devices[3], and unmanned flight systems[4], etc. Suitably designed DEAs usually cause giant voltage-induced deformation and beauty phase transitions prior to electric breakdown[5]. Yu et al.[6]studied the propagation of wrinkles in dielectric elastomers after observing the skin and feet of animals. Huang and Suo[7]theoretically analyzed the transition in DE membranes under uniaxial loads. Lu and Suo[8]studied the electromechanical phase transition between bulged and unbulged states in a dielectric tube. Recently, Godaba et al.[9]observed wrinkle patterns in DE membranes, and the flat and wrinkled regions coexisted.

Recently, DEAs coupled with electrodes and lighter gas have been developed as soft pumps and robots. Zhu et al.[10]studied a thin-walled DE balloon and analyzed its nonlinear oscillation. Chen et al.[11]analyzed the dynamic performance of DE balloon actuators driven by a pressure and a voltage. Zhang et al.[12]presented balloon actuators based on DEs, and the initial volume was important for large deformation.

This paper investigates analytical models for circular DE membranes, and studies dynamic patterns on their surfaces. The performance of DE balloon actuators coupled with helium gas is also proposed, and the mass change can be controlled for application in soft-actuated air vehicles. The purpose of this research is to optimize the parameters and theoretical models of DEs, and to give some guidelines for intelligent soft robots and devices.

1 Equations for Deformation

Fig.1 illustrates the experimental setup for observing the phase transitions on a circular DE film. Voltage is programmed with the software of Labview and amplified by a high voltage amplifier (10/40 A). Cameras are employed to capture the side view of DE actuators. A circular membrane is axisymmetric in its configuration, as shown in Fig.1, consisting of an active region with electrodes and a passive part. The voltage is then applied to the active region which expands into a new state of equilibrium. In the case of the pre-stretchλpre=1, the volt-age will induce compressive stresses in the passive region, and the thin membrane can experience out-of-plane bulging due to the loss of tension (LT).

1— Circular DE actuator; 2— Camera to take pictures and videos of the side view; 3—Computer; 4—BNC 2120 junction box; 5—High voltage amplifier

In the actuated state, the circular active part undergoes homogeneous, equal-biaxial deformation, which is different from the passive region.λrandλθare the stretches in the radial and hoop directions, respectively. Using geometry, ∂λθ/∂R=λr/R-λθ/R.Ris the reference radius of an arbitrary point in the active region, andris the deformed radius of the same particle during the actuated state.

The stretch ratio of the radius isλi, namelyλi=ri+1/ri. The function related toλiandλpreis expressed by

(1)

(2)

Fig.2Schematic of a DE spherical shell. (a) Original state; (b) Pre-stretched state; (c) Actuation state; (d) Deformed state of the element in a DE film subjected to tri-axial stresses

(3)

The fixed amount of gas is enclosed. Combining the ideal gas law, the internal pressure of a DE balloon actuator is

(4)

Then, the pressure decreases with the increase in the volume caused by the voltage, and it is described as

(5)

The condition of dielectric breakdown is expressed[13]asEDB=30.6λ1.13×106. The voltage corresponding toEDBisΦDB=EDBHλ-2.

Fig.3 shows that DE balloon actuators are characterized by the desired nonlinear pressure-prestretch relationshipλ1=r1/R. The curves follow the trends predicted by analytical models. The original radiusRis 3.50, 7.25 and 12.75 cm, respectively. In Fig.3(a), the stretch limitJlim=175 and shear modulusμ=35 kPa are used. Electric breakdown always occurs at about Ф = 6 kV. From Fig.3(b), the parametersμ=350 kPa, andJlim=35. Electric breakdown commonly occurs at about Ф=3 kV. The original thicknesses of the rubbers are chosen to be 0.25, 0.4 and 0.47 mm, respectively. It is found that the peak pressure in a rubber balloon is greater than that of a VHB spherical shell. The limit stretch of natural rubber is much smaller than that of the VHB material.

(a)

(b)

Fig.3Pressurep1in gas pumping process as a function of pre-stretchλ1. (a) Pressure change in VHB 4910 balloon shells; (b) Pressure change in natural rubber balloons

Fig.4 plots the gas law for numbers of molecules within a rubber balloon and a VHB balloon, respectively. The rhombus curve represents the number of gas moles in a rubber balloon, and the dotted line shows that in a VHB 4910 spherical shell. With the same original diameter (7 cm) and pre-stretch, the initial thickness of VHB membrane is 1 mm, for rubber it is 0.25 mm. As a result, the gas amount in the rubber balloon is more than that in the VHB balloon, because VHB is a softer material.

Fig.4 Relationship between the number of gas moles and pre-stretch

2 Discussion

Phase transitions are observed on circular DE membranes (the original thicknessH=1 mm) with different pre-stretchλpre. In the pre-stressed state, thin membranes are flat, as shown in Fig.5(a) and Fig.6(a). When the voltage is relatively small, the membrane keeps flat and elongates around the frame. As the voltage further increases, the tensile stress in the passive region decreases gradually. At a critical voltage, bulging (see Fig.5(c),λpre=1) or wrinkles (see Fig.6(c),λpre=4) appears. Wrinkled regions can spread over the entire membrane within a short time.

(a) (b) (c)

Fig.5The phase transition from the flat state to bulging state under different voltages. (a) 0 kV; (b) 7 kV; (c) 9.5 kV

(a) (b) (c)

Fig.6The phase transition from the flat state to wrinkle state under different voltages. (a) 0 kV; (b) 5.5 kV; (c) 6.6 kV

For DE balloons, when the voltage is applied to double-layer spherical membranes, the volume change is larger than that of monolayer DE balloons. A new method is also used to obtain a large deformation of a DE balloon. After dielectric breakdown of the inner rubber, we continue to apply the voltage to the double-layer spherical VHB membranes and the balloon further expands its volume without gas leakage. This process can increase the mass change and it is controlled in the flight robot system[14].

3 Conclusions

1) This paper investigates the calculation models for the analysis of circular and spherical DE membranes. Phase transitions among flat, bulging and wrinkles can be theoretically predicted and experimentally observed in a narrow voltage range prior to electrical failure.

2) The analytic models of soft balloon actuators combining free energy and the ideal gas law can provide guidance for experiments. Larger deformations of multi-layer DE balloons are also explored. The spherical DE membranes can be utilized in a giant deformation to produce the controlling force, and they have potential application for soft flight robots.

3) It is hoped that the proposed theory and results will help the design of DEAs and maximize their actuation performance.