Soft dielectric elastomer tubes in an electric field

Abstract

Soft dielectric elastomers with high relative permittivity, very low modulus and high electric breakdown strength have emerged as promising materials for various applications as sensors, actuators and in energy harvesting and soft robotics. We study the intricate deformation behaviour of a soft dielectric elastomer tube of finite length and closed ends, that carries a dead load, is internally pressurised by an injected fluid and has a high electric potential applied across its walls. As for soft tubes in the absence of a potential, this electro-hyperelastic problem involving very large deformations, exhibits a multitude of possibilities, including homogeneous deformation, inhomogeneous bifurcation, snap-through instabilities and post bifurcation behaviour in the form of propagation of axisymmetric bulges. We develop a coupled Finite Element procedure for the situation where the domains of the mechanical and electrostatic problems coincide. The procedure can handle volume flow rate controlled electro-elastic problems. We use it to study the many aspects of the deformation behaviour and limitations placed by competing failure mechanisms on practical utilisation of the large areal strains and axial actuations that can be produced. If electrical breakdown can be avoided by ingenious design of loading sequences, the large volume increase in the tube due to the triggering of instabilities can be harnessed and has far-reaching technological implications.

Introduction

When a thin-walled hyperelastic tube is internally pressurised with an imposed axial stretch, homogeneous axisymmetric solutions where the cylinder inflates as a cylinder (which we will call ‘trivial solutions’ from now on), exhibit a non-monotonic variation of the internal pressure with radial stretch for many strain energy functions. The internal pressure exhibits a maximum and a minimum (except for strain energy functions like Mooney Rivlin; see, Corneliussin and Shield, 1961), when plotted against the monotonically increasing radius or, equivalently, the volume of pressurising fluid inside the tube (Shield, 1971, Haughton and Ogden, 1979). However, in reality, due to imperfections in the geometry of the tube, close to the pressure maximum, the tube undergoes a bifurcation to an inhomogeneous ‘bulged’ state (Haughton and Ogden, 1979, Kyriakides and Chang, 1990, Kyriakides and Chang, 1991). The point at which the bulge appears depends on the length of the tube and the extent of axial stretch. In tubes that are closed at the ends, the bifurcation occurs between the pressure maximum and minimum (Haughton and Ogden, 1979). The appearance of the bulge is accompanied by a large drop in pressure when the volume flow rate of the pressurised fluid is controlled. Following this, the bulge first grows in radius up to a point and subsequently extends in the axial direction at an almost constant propagation pressure, which can be predicted using the pressure–volume plots from the trivial solution by an elegant and simple procedure called ‘Maxwell construction’ (Chater and Hutchinson, 1984). The physics of the phenomenon closely parallels that of necking in solids (Tvergaard and Needleman, 1980). Again, for tubes of finite length, experiments by Kyriakides and Chang (1991) show that once the bulge reaches the ends, the pressure rises again and the tube inflates in a homogeneous manner.

Complete understanding of the homogeneous to inhomogeneous axisymmetric bifurcation and post-bifurcation deformation behaviour of internally pressurised and axially stretched soft tubes have been obtained more than two decades ago. In recent times, the idea of applying a high electric potential difference across the thickness of a cylindrical tube (in addition to axially stretching and internally pressurising it) made up of a soft material with high dielectric permittivity and electric breakdown strength, has led to interesting technological possibilities.

In some sense, a potential difference applied across the thickness modulates the pressure–volume response of the cylindrical tube. Irrespective of the strain energy density function of the material, the pressure required to quasi-statically increase the volume of a cylindrical tube is decreased by the application of a potential difference across the thickness (Melnikov and Ogden, 2016). However, the qualitative nature of the pressure–volume response remains almost unchanged when a potential difference is applied across the wall thickness. Depending on the strain energy density function of the material, pressure can be a non-monotonic or monotonic function of volume. Further, depending on the potential difference and axial stretch applied, the internally pressurised tube can undergo bifurcation to an inhomogeneous axisymmetric mode when the radius reaches a critical value (Dorfmann and Ogden, 2019). Apart from the homogeneous to inhomogeneous axisymmetric bifurcation, nonlinear analysis of unpressurised tubes subject to only potential (Zhu et al., 2010), underlines the possibility of electromechanical instability at rather low actuation strains, especially when the tube is thin and unstretched.

In fact, experiments on bulge propagation in tubes of dielectric elastomers show (An et al., 2015, Lu et al., 2015) that bulges can be made to propagate along the length in at least two ways. Fluid can be pushed into the tube till a bulge initiates and then, the bulge can be made to propagate only by applying a potential difference across the thickness. A significant part of the propagation occurs at a constant value of the potential difference (An et al., 2015). The bulge can also be made to propagate at a constant pressure and potential difference if the flow of fluid is continued (Lu et al., 2015). By applying potential across the thickness, it is possible to create situations where bulged and unbulged sections of the tube coexist, the entire length of the tube is bulged or the tube deforms in an homogeneous manner.

The idea of using a cylindrical tube made up of a dielectric elastomer (DE) with conducting electrodes coated on the inner and outer surfaces as an actuator was first proposed by Pelrine et al. (1998). Prototypes of such actuators with (eg. Arora et al., 2007) and without (eg. Sarban et al., 2011) embedded fibres have been demonstrated. Among many possible applications of cylindrical dielectric elastomer tubes, a promising one, that was recognised early in its development, is as artificial muscles. Nonlinear elastic analysis of internally pressurised dielectric elastomer tubes reinforced by helically wound inextensible fibres (Goulbourne, 2009) show that – quite like the more popular pneumatic muscles (the so called McKibben actuator) – they too are capable of contracting axially and applying significant axial contractile forces. Dielectric elastomer tubes are also used in harvesting energy (Chiba et al., 2008), say, from waves in the ocean or flowing water. Large axial displacements are desirable when the tube is used as an actuator. On the other hand, large areal strains generated by extensive bulging may be useful when utilising the tubular DE to harvest energy.

In this paper, we study the onset of bifurcation and post bifurcation deformation of an internally pressurised, axially stretched DE tube of finite length that is subjected simultaneously to an electric potential across its thickness and a rate of fluid flow into it. Supplementing earlier experimental and theoretical work on this problem, we use nonlinear coupled Finite Element (FE) analyses conducted within the framework of finite deformation electro-hyperelasticity. The purpose of the work is to extend the theoretical analyses on the onset of axisymmetric bifurcation presented in Melnikov and Ogden (2016) and Dorfmann and Ogden (2019) with a view to understand the roles of material behaviour, multiple failure modes and loading sequences on the post bifurcation behaviour. We start with a tube of finite length with an imperfection at the mid section to seed the bifurcation. We avoid situations leading to electromechanical instabilities and allow only axisymmetric bulging and bulge propagation. Though we consider the generic problem of a soft cylindrical tube under electro-mechanical loading, the numerical framework presented herein should aid in designing practical devices aimed at actuation or energy harvesting.

Coupled Finite Element analyses of soft solids, especially in situations where the material is nearly incompressible and extremely large deformations are expected, requires a number of careful considerations. The theoretical framework and FE formulation are detailed in Sections 2 Mechanics of soft dielectric elastomers in an electric field, 3 Finite Element formulation. Numerical aspects specific to the problem being solved are given in Section 4. Results are discussed in Section 5 and conclusions in 6.

The present work builds upon and draws from, the theoretical framework laid down by Melnikov and Ogden (2016) and Dorfmann and Ogden (2019) as well as the experimental results of An et al. (2015) and Lu et al. (2015). The salient contributions of this work are the following:

• Based on an existing theoretical framework for finite deformation electro-hyperelasticity, we have developed a fully coupled Finite Element scheme that handles near incompressible situations (see, Section 3.2). Moreover, the simulations on soft cylindrical tubes have been conducted by controlling volume flow rate of the pressurising fluid. This allows us to extend the simulations further than achieved before (Sections 2 Mechanics of soft dielectric elastomers in an electric field, 3 Finite Element formulation).

• We probe the post-bifurcation behaviour of the soft tubes under given volume flow rates and applied electric fields. Previous theoretical studies have been concerned mainly with conditions governing the onset of bifurcation (Sections 5.2–5.4).

• The significant effect of finite extensibility of the material of the tube on its post-bifurcation behaviour is elucidated (Section 5.2).

• The ways in which competing failure mechanisms can disrupt the large post-bifurcation deformations is demonstrated (Section 5.3).