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**Chapter 2 Electromechanical Behavior of Dielectric Elastomers**

This chapter explores the electromechanical actuation and sensing abilities of dielectric elastomers. It begins by discussing the basic driving mechanisms behind the DE‟s unique capabilities. It then presents the constitutive relations that describe its basic behavior and discusses the different components of a DE. In order to better understand the electromechanical behavior of DEs, uniaxial actuation tests are conducted on a planar sample with constant loads applied. The experiment is compared to theoretical results. Finally, an out-of-plane sensing model is developed which equates the peak stretch in a clamped DE membrane to its capacitance.

**Principle of Operation**

In the previous chapter, the concept of dielectric elastomers and the relation to other smart materials with similar characteristics was introduced. This section takes this a step further to explain how DEs operate. Although these particular smart materials are primarily known for their actuation, they can also be used as sensors. In both cases, they are comprised of the same basic components: an elastomeric dielectric material coated with compliant electrodes on the major surfaces. As actuators, Coulombic forces caused by the application of an electric field create an attraction between the electrodes. This attraction generates thickness compressive stresses and planar tensile stresses in the elastomer. Due to the material‟s high compliance and incompressibility, these stresses are enough to cause appreciable constriction in the thickness direction, which then causes expansion in the in-plane direction. This behavior is demonstrated in Figure 3 (left figure). As sensors, the capacitance of the specimen can be used to measure strain. When a load is applied, the thickness of the DE decreases and the surface area increases, causing an increase in capacitance. By measuring this change in capacitance, the strain can be calculated. A schematic of a DE sensor is shown in Figure 3 (right figure). Capacitive sensing, while very applicable for DEs, is not a new sensing concept. Dargahi et al. design a piezoelectric-capacitive sensor to provide tactile feedback in minimally invasive surgery for loads between 0 to 1 Newton [32]. In work conducted by Young, a capacitance pressure sensor is designed for high-temperature industrial, automotive, and aerospace applications [33]. Another capacitive pressure sensor is designed by Park and Gianchandani which consists of a skirt extending outward from the roof of a sealed cavity. Most capacitive sensors designed with a sealed cavity have difficulty transferring the electrical leads outside the cavity, however the skirt allows the pick-off capacitance to be located outside the cavity, hence eliminating this problem [34]. Both of the sensors developed in [33] and [34] measure pressures in the kPa range. Finally, Toth and Goldenberg use capacitance readings to infer the strain (up to 60%) of a DE planar membrane caused by the application of a voltage [35]. In the current work, the DE sensor is in a membrane configuration and is not actuated, but used solely as a sensor. In other words, the capacitance change is used to measure strains caused by applied (mechanical) surface tractions and not by an applied voltage.

**Constitutive Model**

The previous section described the mechanism which causes the large strains that dielectric elastomers are known for, as well as the method by which these devices can be used as sensors. This was however only a physical explanation and does not present a relationship between voltage and or mechanically induced stress and the strain experienced by the elastomer. In order to obtain these relationships, the constitutive relations for dielectric elastomer actuators and sensors must be established.

Constitutive relations are essential in modeling any system‟s response. For structural analysis, they equate the stresses in a medium to its strains and take into consideration the fact that various materials may behave differently even though they share the same mass and geometry. Constitutive relations can take on many forms, depending on the type of material being modeled. The type of material used in the current work is hyperelastic, incompressible, and homogeneous. Over the years, many strain-energy functions have been developed to predict the constitutive response for these types of materials. Most widely used are the Mooney-Rivlin model and the Ogden model. For the particular type of material used in this research, the Mooney-Rivlin model captures the stress-strain response up to a strain of approximately 300%, while the Ogden model typically captures the stress-strain response up to 700% [36]. Because of the Ogden model‟s superior fit to the stress-strain data, it was used in this work.

The Ogden strain-energy function differs from the Mooney-Rivlin and other strain-energy functions in that it is a linear combination of strain invariants. It is given as [37]

where *m** _{r}* and

*a*

*are material parameters,*

_{r}*l*

*(i=1,2,3) are the principle stretches, and summation over*

_{i}*r*is implied. The number of terms in the summation is typically dependent on the material and loading conditions. For simple uniaxial extension, a two term model is sufficient. The stretch is defined as final length divided by initial length. From this strain-energy function, the principle stresses become

*t*

_{3}

*=*

*l*

_{3}(

*m*

_{1}

*l*

^{a}_{3}

^{1}

^{–}

^{1}

*+*

*m*

_{2}

*l*

^{a}_{3}

^{2}

^{–}

^{1}) –

*p*

where

*p*is the unknown hydrostatic pressure. These make up the constitutive relations for a hyperelastic, incompressible material.

While these constitutive relations correctly relate stress and strain for a hyperelastic incompressible material, for DEs they must be augmented to account for the stress incurred from the application of an electric field. To this end, Goulbourne postulated that the Cauchy stress in the membrane in the presence of the electric field is equal to the sum of the mechanical stress and the electrical stress [36]: applied voltage,

*l*

_{3}is the stretch in the thickness direction, and

*h*is the original membrane thickness. Note here that the

_{o}*l*

_{3}

*h*term in the denominator is equivalent to the instantaneous material thickness. For a sufficiently thin membrane, dividing the voltage by this gives the instantaneous electric field. The Maxwell stress is proportional to the electric field squared. The ramifications of this nonlinear relationship between stress and applied voltage will be apparent throughout the results presented in this thesis.

Substituting Equations 2 and 4 into Equation 3, the principle stresses become which make up the constitutive relations for a dielectric elastomer actuator. The polarity of the Maxwell stress is such that it is positive only in the direction of the electric field. For DE configurations considered in this work, the applied field is in the thickness direction. Thus, for this simple applied electric field, the only positive stress contribution is in the

*t*

_{3}direction, and the Maxwell stress contributions take the form as shown in Equation 5. These equations will be used to model the behavior studied in Section 2.5. To obtain the material constants, the Ogden strain-energy function must be fitted to experimental data. This is done in the next section which discusses the types of materials used for DEs as well as the particular type used for the work done in this thesis.

**Dielectric Materials**

To restate what was stated in the introduction, dielectric elastomers are made of an elastomeric dielectric material coated on either side by compliant electrodes. This section will discuss the defining characteristics of the dielectric material and the materials that are currently in use.

There are a limited number of materials that can be used for DEs. The material must be elastomeric, soft and flexible enough to experience appreciable strains when actuated, but at the same time stiff enough to handle the loads applied to it. This being the case, there is no “ideal” material for DEs, as certain applications may call for more force and less strain or vice versa, which would change the characteristics of the desired DE. Currently, there are two main elastomers that have been employed for DEs: silicone and polyacrylate. Some examples of silicones used are NuSil CF19-2186 [4] and Wacker Elastosil RT 625[38]. The polyacrylate used is VHB and is a double sided tape made by 3M®. It has the benefits of being relatively inexpensive, coming ready to use, and having a large electrostatic effect. It does however have its drawbacks. For example, it must be prestretched before use, and is viscoelastic in nature. Silicone on the other hand is a very elastic material which makes its response time much faster than VHB, but requires a multistep fabrication process and does not have as large an electrostatic effect. For the work presented in this paper, VHB 4905 and VHB 4910 were used as the dielectric material, the difference between the two being that VHB 4905 is half a millimeter thick while VHB 4910 is a millimeter thick. Both come in a large roll, 8” wide and 36 yards long.

In order to use the constitutive relations derived in Section 2.2, the material constants for the polyacrylic must be obtained. This is done by conducting a uniaxial extension test on a sample of the material and fitting the hyperelastic Ogden model to the experimental data. A paper published by Ogden in 2004 is a helpful guide in the fitting process [39]. The uniaxial tests for the polyacrylic were done on an Instron machine, with samples ¼” wide and 1” long. Eight samples were tested and the average of all tests was computed. Figure 4 shows the results from these tests. The Mathematica function “NonlinearRegress” was used to fit the Ogden model to the experiment (see Appendix A). As can be seen, the Ogden model provides an excellent fit for the entire stress-strain curve. The material constants taken from Mathematica are shown in Table 1.

Chapter 1 Introduction

1.1 Motivation

1.2 Dielectric Elastomers

1.3 DE Membranes

1.4 Background

1.5 Scope of Thesis

Chapter 2 Electromechanical Behavior of Dielectric Elastomers

2.1 Principle of Operation

2.2 Constitutive Model

2.3 Dielectric Materials

2.4 Electrode Materials

2.5 Constant Load Experiment

2.6 Capacitance-Pole stretch model

Chapter 3 Static Experiments

3.1 Methods

3.2 Actuation Experiments

3.3 Sensing Experiments

Chapter 4 Dynamic Experiments

4.1 Methods

4.2 Dynamic Mechanical Loading Experiments

4.3 Dynamic Electrical Loading Experiments

Chapter 5 Summary and Conclusions

5.1 Quasi-Static Experiments

5.2 Dynamic Experiments

References

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Electromechanical Characterization of the Static and Dynamic Response of Dielectric Elastomer Membranes