The low degree of freedom model

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Literature Review

Although the first Dielectric Elastomer Actuator (DEA) was demonstrated by Roentgen in 1876, they have only existed in their modern form for approximately 12 years. During that time there has been substantial worldwide effort focused on understanding the basic principles, developing new materials, exploring possible configurations, and applications. DEA are ideal for the creation of bio-inspired networks coordinated using distributed sensing and control. It was identified in the previous chapter that the creation of these networks is being held back by a lack of modelling tools for device design, proof that DEA self-sensing strategies can underpin control of arrays, and sensor and driver circuitry that does not limit applications through bulk or rigidity. Thus this chapter explores these limitations, presents a critical review of the field of DEA as applied to networks with distributed sensing and control, and sets the stage for the rest of the thesis.

Brief history of DEA

Dielectric Elastomer Actuators (DEA) were introduced in their modern form by researchers at SRI international in 1998 [20]. The principle of electrostatic stresses giving rise to deformation was already well established at this point and had been investigated by Roentgen over 100 years previously in 1886 [51]. Electrostatic actuators also had a long pedigree and enjoyed widespread use, especially for MEMS applications. However the real breakthrough of the SRI team was the idea that the active stress of electrostatic actuators could be improved 10 to 100 fold by coupling compliant electrodes to a flexible dielectric [52]. This was a huge improvement which finally made electrostatic actuators, in the form of DEA, viable as a true artificial muscle technology.
There was initially some discussion as to the true origin of the actuation stress. Electrostriction due to non-isotropic dielectric was proposed but it soon became apparent that its contribution was small and the actuation was largely electrostatic [53, 54]. A compressive mechanical „Maxwell pressure‟ (equation (2-1)) became the standard interpretation of electromechanical coupling for DEA and was validated by a number of researchers, notably Pelrine [20], Kofod [27], and Wissler [41]. In the classic equation below εo and εr are the absolute and relative dielectric permtivities respectively and E is the electric field strength.
2 EP roMaxwell   (2-1)
There has been some concern from Suo et al. as well as McMeeking and Landis that this interpretation of Maxwell pressure as a mechanical body force can be misleading [55, 56]. The distinction between electrical and mechanical forces in dielectrics is somewhat arbitrary, and it pays to remember that actuation stress originates from minimization of the free electrical energy in the system. This is an important point, and to help illustrate it we can reproduce Kofod‟s derivation of the Maxwell pressure [27] with a slight variation. This derivation clearly shows that whilst the body force interpretation of Maxwell pressure is useful, the actuation stress does in fact arise directly from energy considerations. Equation (2-2) gives an expression for the electrostatic energy density (Uel) in an ideal parallel plate capacitor with flexible electrodes; t is the thickness, V is the applied voltage, and ε is the permittivity. Equation (2-3) gives the total energy (W) in the capacitor found by multiplying the electrostatic energy density with the volume; A is the area and ψ is the volume (At) which remains constant as the dielectric is assumed to be volumetrically incompressible.
As the basic mechanisms underlying DEA operation became apparent, research turned towards failure modes. Plante and Dubowsky presented a comprehensive overview of the failure characteristics of DEA and showed that DEA fail in three main ways: mechanical failure, dielectric breakdown, and electromechanical instability (which they called pull-in) [57]. They applied their analysis to design bistable actuators that were protected viscoelastically from failure by high strain rates [58]. Lochmatter showed that actuators can be made stable against electromechanical instability by using constant charge control instead of constant voltage [59]. With a constant charge the system always has one unique stable point, with constant voltage the system can become unstable.
In parallel with these theoretical developments, a significant worldwide effort to develop better dielectric materials has emerged. Sommer-Larsen and Larsen [60], and Kofod et al. [61] showed that the ideal DEA membrane should be soft to minimise the amount of work done to strain the actuator, have a high dielectric constant to increase sensitivity, and have a high breakdown strength to produce a large maximum Maxwell pressure. Many materials were developed and are still in development to meet these needs [60-74]. However, despite these advances no material has yet surpassed the overall performance benchmark set by one of the first materials discovered: VHB 4910/4905 (VHB).
VHB is a highly viscous hyper-elastic acrylic elastomer commercially available from 3M that was shown by the SRI team to be capable of passive area strains of 2500% and active strains of 380% [66]. VHB has the largest energy density for a DEA membrane reported to date, is convenient to use, and is highly adhesive (VHB stands for Very High Bonding). VHB does have significant limitations including a proprietary formulation and fixed thicknesses, a tendency to creep to failure under long term stress, high viscous losses during operation, and high rates of „infant mortality‟ due to inclusions or defects in the material [75]. However, whilst there are many new materials in development, these materials as yet have not beaten VHB across the board in terms of strain, stress, ease of use, and price point.
Electrode development was equally important, even though the first DEA didn‟t require electrodes; Roentgen demonstrated that rubber would expand in plane when electrons were sprayed onto its surface [51]. Keplinger et al. recently applied the same principal to create transparent VHB actuators that exhibit massive actuations without risk of electro-mechanical instability [76]. However, despite large potential for applications such as lenses or valves, electron spraying is slow, inefficient, and requires electron emitters to function. The limitations of electron spraying really highlight the advantages of compliant electrodes.
Carpi et al. showed that the effects of different compliant electrodes on actuator performance are significant, both mechanically and electrically [77]. The electrode must be as soft as possible, maintain conductivity over large strains, be thin, and adhere well to the membrane. Just like for dielectrics, there are many electrode materials available [74, 78-85]. Different electrodes technologies all have different advantages and disadvantages. For example; carbon grease is easy to use but messy [78], ion implantation is good for micro-actuators but doesn‟t scale up well [80-83], and carbon nano-tube electrodes can be made transparent but due to manufacturing limitations are expensive (although dropping in price) [79].
Many different configurations of DEA have been developed over the last 12 years [42, 44, 57, 58, 66, 86-107], but there have been very few attempts to develop DEA-based biomimetic distributed actuator arrays. Menon, Carpi, and De Rossi took the first steps with their overview of natural distributed actuator systems and identification of DEA based peristaltic conveyors and pumps as being ideal devices for space applications [28, 29]. They conducted preliminary studies showing that placing DEA in a vacuum had limited effect on actuator performance, modelled buckling actuators suitable for peristaltic conveyance, and modelled and built a single tubular actuator unit suitable for a peristaltic pump. However, to the best of the author‟s knowledge there has been no practical demonstration of a biomimetic DEA pump or conveyor, or any ideas on how such a system might be controlled.
Perhaps the most successful example of DEA being used in a bio-inspired distributed actuator array is the artificial annelid robot developed by Jung et al. [30]. The robot is inspired by annelid worms that are able to move across varied and challenging terrain using lengthwise waves of contraction to move the robot forward (vermiculation). The South Korean team made their robot out of 8 circular segments of PCB each with 12 buckling mode DEA spaced in pairs around the disk (6 each side). The buckling DEA of one disk pressed against the DEA of the next thus the robot could contract or bend depending on the driving signals applied. Their robot was 45 mm long and 20mm in diameter, able to move forward at 0.056 body lengths per second. However, DEA on the robot were collectively controlled in front and rear groups, unlike their annelid inspirations there was no distribution of control and sensing throughout their device, and the robot was still substantially rigid in the radial direction due to the 8 PCB disks. Nevertheless, the annelid robot is the largest (n = 96) fabricated biomimetic network of DEA reported to date.
Kofod et. al. invented dielectric elastomer minimum energy structure (DEMES) bending actuators [36, 108, 109]. DEMES are formed by adhering a pre-stretched DEA to a thin plastic frame which, after release, curls up as the strain energy in the membrane equalises with the bending energy in the frame (see Figure 2-1). When Maxwell pressure is applied, the energy equilibrium shifts causing the structure to move. DEMES provide a method for creating bending actuators from otherwise planar DEA, minimise the need for bulky prestretch supports in the device, and have a natural voltage limit where they become flat. They are ideally suited to arrays of artificial cilia, a capability that will be explored in this thesis.

 Modelling of DEA

Dielectric elastomer actuators exhibit time varying hyper-visco-elastic incompressible material behaviours, coupled nonlinear electro-mechanics, high aspect ratios, and large displacements. DEA based on VHB are especially known for these behaviours and it follows that developing VHB based DEA devices for biomimetic networks is non-intuitive and challenging.
Whilst there is no substitute for rapid prototyping and experimentation, this approach can be time consuming and frustrating. A second limitation of the experimental approach is that some device failure modes are very difficult to measure and draw insight from. For example if a device failed during operation it is difficult to see if this was caused by poor manufacture or design choices, and if the failure mode was membrane tear, electrical breakdown, or electromechanical instability. In addition it can be difficult to see where the failure occurred, especially in the event of fire or crumpling. Thus there is a strong need for simulation tools to aid in the design of DEA devices.
There has been extensive research worldwide to create simulation tools that describe material behaviors, electromechanical coupling, and failure modes [27, 33, 41-43, 45, 47, 110, 111]. Some groups have applied these fundamental techniques to device design, for example Kofod et. al. presented a simple analytical model of DEMES based on system level energy balancehe geometry consisted of a rectangular frame bent into a semicircle by the tension in rectangular piece of VHB attached end to end. The model equated a Gaussian statistical model expression for the free energy of the elastic membrane to an expression of the bending energy in the frame and showed how the equilibrium position shifted with an applied voltage. Moscardo et al. presented an analytical model of DEA spring roll actuators formulated as an energy balance between the spring, membrane, and electrostatics [37]. To describe electromechanical coupling they applied a Gaussian statistical model based on the theoretical description of membrane free energy developed by Suo, Zhao, and Greene [40]. Their spring roll model was applied to analyse failure modes and optimize the actuator. He, Zhao, and Suo applied the same membrane model to analytically describe the performance and failure characteristics of out-of-plane “universal muscle actuator” DEA [34].
These analytical approaches provide understanding of the fundamental behaviour of specific devices, but they have several flaws in common when applied to real world design. Firstly, real device geometries are typically complex and possessed of heterogeneous mechanical and electrical fields, thus analytical models cannot easily or automatically be used to describe their behaviour. Secondly, the strain energy functions chosen to model membranes cannot describe the complete design space occupied by VHB due to limited maximum stretches, and a lack of viscoelasticity. To address the first concern, some groups have turned to finite element modelling so that complex geometries can be described. This has been done in two main ways:
The first method is to take the energy balancing described by the analytical approaches and generalise it to the underlying media. Partial derivatives of a local free energy function can then be taken to find mechanical and electrical fields. This approach was taken by Zhao and Suo where they implemented their nonlinear theory of deformable dielectrics [40] as a user defined material in the commercial FEA package ABAQUS [49]. They demonstrated the power of the approach by applying their model to the simulation of electromechanical instability in diaphragm actuators, wrinkling in expanding planar actuators, and the simulation of DEMES. However, they chose arbitrary material properties and did not experimentally validate their results.
The second method is based around implementing electromechanical coupling by applying the Maxwell pressure as a mechanical stress either against or within the material. The most sophisticated example of this approach was taken by Son and Goulborne [112]. They developed the constitutive equations for electromechanical coupling based around the simplifying assumptions of a membrane, i.e. plane stress, zero shear, through thickness homogeneity, and high aspect ratio. They then simulated a simple planar actuation mechanism inspired by bat wings and experimentally validated their results with some success. However, both the Son and Goulborne ABAQUS implementation, as well as the Zhao and Suo implementation had several limitations in common:
1. Both implementations were based around strain energy functions unable to describe either the visco-elastic or large strain response of VHB adequately. 2. Both used solid brick continuum elements and neither took advantage of the ability of ABAQUS to describe DEA using membrane elements, although Son and Goulborne did use membrane assumptions to simplify their constitutive laws. Membrane elements have fewer degrees of freedom than brick elements and an order of magnitude less membrane elements are required than brick to describe a given high aspect ratio device.
There is therefore an outstanding need for a rapid and robust simulation method capable of describing the behaviour of VHB based DEA devices. Such a method would build on previous work in the field and use a strain energy function that can describe VHB adequately (see Wissler‟s Arruda-Boyce / Prony model discussed below); make full use of a membrane element formulation to be fast and robust; be experimentally validated; and sufficiently accurate on real, complex, and arbitrary devices.
Wissler provided a good constitutive law for VHB 4910/4905 in his PhD thesis [41]. In his work he fitted a range of strain energy functions to biaxial and uniaxial test data for VHB 4910/4905 at different rates and for different relaxation curves to give models with as large validity as possible. He identified the Arruda-Boyce model (equation (2-7)) augmented with the Prony series (equation (2-8)) as giving the best visco-hyper-elastic description of VHB.


The Arruda-Boyce model relates the strain energy in the material to the first stretch invariant1 and is micromechanical, i.e. based on statistical mechanics, rather than a black box approach. The advantages of micromechanical models can be seen in that the Arruda-Boyce model has only two parameters, μ and λm, and that it provides a good prediction of material behaviour outside of the range of training data. Physically μ can be thought of as the initial modulus of an elastomer, and λm as the stretch at which polymer chains in the material lock [113]. The Prony series, equation (2-8), consists of summed exponential decay curves each with amplitudes (gk) and time constants (tk). Despite its success, Wissler‟s model requires a mechanical pressure to implement actuation stress (i.e. actuation stress is not a fundamental property of the constitutive law) and may be inaccurate for a given application as it has been optimised for general use

Table of Contents
1.1 Background and motivation
1.2 Research objectives
1.3 Thesis outline and contributions
1.4 Chapter summary
Literature Review
2.1 Brief history of DEA
2.2 Modelling of DEA
2.3 Self-sensing and driver circuits
2.4 Chapter summary
LDOF Modelling of DEA
3.1 Introduction
3.2 The low degree of freedom model
3.3 Analytical verification
3.4 Experimental validation
3.5 Case study
3.6 Chapter summary
FEA for Design of Self-Sensing DEA
4.1 Introduction
4.2 Materials and methods
4.3 Results
4.4 Discussion
4.5 Chapter summary
Travelling Waves for DEA
5.1 Introduction
5.2 Materials and methods
5.3 Results
5.4 Discussion
5.5 Applications
5.6 Chapter summary
Dielectric Elastomer Switches
6.1 Introduction
6.2 The DES concept
6.3 Development of a switching material
6.4 Experimental demonstrations
6.5 Speculative applications
6.6 Discussion
6.7 Chapter summary
7.1 Summary
7.2 Impact and contributions
7.3 Outlook and future work
7.4 Chapter summary
7.5 Publications
8.1 Mathematical derivation of the SEF
8.2 UHYPER/USDFLD implementation
8.3 DEMES development by virtual experimentation
8.4 Description of supplementary movies

Simulation, Fabrication, and Control of Biomimetic Actuator Arrays

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