Deformation and Stability of Dielectric Elastomer Films

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DEFORMATION AND STABILITY OF DIELECTRIC ELASTOMER

FILMS USED IN ACTUATORS

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Chuan Zeng

August, 2020 DEFORMATION AND STABILITY OF DIELECTRIC ELASTOMER

FILMS USED IN ACTUATORS

Chuan Zeng

Dissertation

Approved: Accepted:

______Advisor Department Chair Dr. Xiaosheng Gao Dr. Sergio Felicelli

______Committee Member Interim Dean of College Dr. Gregory Morscher Dr. Craig Menzemer

______Committee Member Dean of the Graduate School Dr. Kwek-Tze Tan Dr. Marnie Saunders

______Committee Member Date Dr. Ernian Pan

______Committee Member Dr. Lingxing Yao

ABSTRACT

Dielectric elastomers belong to the group of electroactive polymers which are considered as “smart” soft materials. Dielectric elastomers easily produce actuated strain greater than 100% when responding to electric stimuli. Dielectric elastomers have broad applications in many areas. However, there remain some issues in the application of dielectric elastomers. Modeling and numerical simulations are very good techniques to help the design and improvement of the performance of dielectric elastomer transducers.

This dissertation is concerned with modeling and numerical simulations of the electromechanical coupling behavior of the dielectric elastomer as actuators. The numerical models of the actuator configurations studied in this work come from important prototypes in experiments. The effect of the material properties and the configuration parameters on the performance of the actuators were analyzed.

First, the deformation dependent behavior of the permittivity of dielectric elastomers was discussed. The actuation behavior of a circular actuator made of a dielectric elastomer, VHB 4910, was theoretically analyzed. The effects of the deformation dependence of permittivity on electromechanical instability, loss of tension, and dielectric breakdown of the actuator during the actuation process were studied. It was found that compared with constant permittivity, the deformation dependent behavior of permittivity affected the actuation performance of the circular actuator in a variety of ways and the impact differed at different pre-stretch levels.

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Next, the effect of material anisotropy on the stability of dielectric elastomer plate was studied. A linearized incremental theory was used and the plate thickness was considered. Results of transversely isotropic materials were compared with the isotropic case. By comparing with the isotropic case, it was found that material anisotropy had a significant effect on the bifurcation instabilities of the DE plates and the impact differed at different plate thicknesses.

Finally, finite element method was used to present a numerical model of a basic buckling actuator made of dielectric elastomer. A user element subroutine [1] in Abaqus was used to simulate the electromechanical coupling behavior of the material. Different material models were programmed into the user element subroutine. The effect of a few model parameters on the performance of the buckling actuators was discussed.

DEDICATION

Dedicated to my beloved parents and grandparents.

“There is only one true heroism in the world: to see the world as it is, and to love it” ― Roman Rolland

ACKNOWLEDGMENTS

First, I would like to thank my advisor Dr. Xiaosheng Gao. He offered me this opportunity to study toward my Ph.D. degree in his group at the University of Akron. He is extremely capable in his research field and I value each meeting and discussion with him.

His deep insights and way of thinking inspired me a lot in doing my research. He also leaves me plenty of freedom to study the topics that I’m interested in and guides me in the right direction if I go wrong. I wouldn’t have accomplished my current research achievements if without the consistent support and inspiration from my advisor.

I also would like to thank my committee members: Dr. Gregory Morscher, Dr.

Kwek-Tze Tan, Dr. Ernian Pan, and Dr. Lingxing Yao. I appreciate the time you spend in helping me improve the quality of my work. I value the productive discussions with you and the wonderful feedback from your perspectives and areas of expertise. I am grateful to my committee members for your support and encouragement during this academic journey.

I would like to thank my group members and friends at the university: Jinyuan Zhai,

Tuo Luo, Chuanshi Huang, Clayton Reakes, Guanyue Rao, Fengyu Yang, Patricia

Eaglewolf, Menglong Ding, Chong Zhong and many others. Thank you for all the academic discussions and the various activities and fun we had.

I am also grateful to the Department of Mechanical Engineering at the University of Akron for financial support throughout my 5-year doctoral program. I would like to

vi thank Ms. Shannon Skelton and Ms. Ellen Wise for their assistance and technical supports in preparing all the paperwork required for graduation.

I would like to thank my family and friends in China for their support, including my parents and grandparents (my grandparents both passed away during my Ph.D. study), my other relatives in the family, my family doctors, my best friends, and many others.

Growing up, my family provided me a carefree childhood and the best public education in

China. It was later in life when I realized that not everyone is lucky enough to grow up in an environment full of love, trust, and laughter. My family will always be the best treasure of my life.

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TABLE OF CONTENTS

Page

LIST OF TABLES ...... xii

LIST OF FIGURES ...... xiii

CHAPTER

I. INTRODUCTION …………………………………………………...…………………1

1.1 Electroactive Polymers ...... 1

1.2 Working Principle of Dielectric Elastomers ...... 1

1.3 DE Materials ...... 3

1.4 Advantages of Dielectric Elastomers ...... 3

1.5 Potential Applications of Dielectric Elastomers ...... 3

1.6 Failure Modes of Dielectric Elastomers ...... 5

1.7 The Objectives of the Present Study ...... 7

II. THEORETICAL BACKGROUND AND LITERATURE REVIEWS ...... 9

2.1 Literature Review on the Nonlinear Theory of Continuum Electromechanics ...... 9

2.2 The Nonlinear Theory of Electroelasticity ...... 11

2.2.1 Kinematics ...... 11

2.2.2 Electrostatics and the Maxwell’s Equations...... 12

viii

2.2.3 Governing Equations and Boundary Conditions ...... 14

2.2.4 Constitutive Equations for Electroelasticity ...... 15

2.2.5 The Incremental Formulation ...... 17

2.3 Literature Review on the Modeling of the Dielectric Elastomers ...... 22

III. EFFECT OF THE DEFORMATION DEPENDENT PERMITTIVITY ON THE

ACTUATION OF A PRE-STRETCHED CIRCULAR DIELECTRIC ACTUATOR .... 26

3.1 Circular Actuators ...... 26

3.2 Deformation Dependent Permittivity of the Dielectric Elastomer ...... 27

3.3 Summary of the Basic Equations ...... 28

3.3.1 Field Equations ...... 29

3.3.2 Equations of State ...... 31

3.3.3 Elastic Free Energy ...... 32

3.3.4 Permittivity Function ...... 33

3.4 Actuation of the Circular Dielectric Elastomer Actuators ...... 34

3.4.1 Circle A...... 35

3.4.2 Annulus B ...... 37

3.5 Results and Discussions...... 38

3.5.1 Stretches in Annulus B ...... 38

3.5.2 Electromechanical Instability ...... 41

3.5.3 Loss of Tension ...... 43

3.5.4 Discrepancy in Voltage-stretch Response ...... 45

3.5.5 Dielectric Breakdown ...... 46

3.5.6 Variation of c ...... 47

3.6 Summary ...... 50

IV. STABILITY OF AN ANISOTROPIC DIELECTRIC ELASTOMER PLATE ...... 52

4.1 Material Anisotropy of the Dielectric Elastomer ...... 52

4.2 The Incremental Theory ...... 54

4.3 Applying the Incremental Theory to an Anisotropic DE Plate ...... 54

4.3.1 Geometry ...... 56

4.3.2 Constitutive model ...... 58

4.3.3 Incremental Formulation and Bifurcation Criterion...... 60

4.4 Results and Discussions ...... 66

4.5 Summary ...... 72

V. FINITE ELEMENT ANALYSIS OF DIELECTRIC ELASTOMERIC MEMBRANES

AS BUCKLING ACTUATORS ...... 74

5.1 Buckling Actuators Based on Dielectric Elastomer ...... 74

5.2 The Finite Element Modeling Method ...... 76

5.2.1 Electromechanical Modeling Method ...... 76

5.2.2 Constitutive Equations ...... 76

5.2.3 Finite Element Model of the Buckling Actuator in Abaqus ...... 82

5.3 Results and Discussions ...... 84

5.3.1 The Influence of the Thickness ...... 84

5.3.2 The Influence of the Material Models ...... 87

5.4 Summary ...... 89

VI. CONCLUDING REMARKS...... 91

BIBLIOGRAPHY ...... 94

APPENDICES ...... 107

APPENDIX A. THE DERIVATIVES OF THE INVARIANTS ...... 108

LIST OF TABLES

Table Page

3.1 Values of c obtained by fitting experimental data reported by different research

groups…………………………………...... 34

5.1 The material parameters for the neo-Hookean model and the Ogden

model……………………………………………………...... 84

5.2 The normalized critical voltage for two different plate thicknesses ...... 85

5.3 The breakdown electric field for two different plate thicknesses …...…...... 87

5.4 The normalized critical voltage for two different material models ..…...... 88

5.5 The breakdown electric field for two different material models ………..….….……89

xii

LIST OF FIGURES

Figure Page

1.1 An actuator made of dielectric elastomer film [2] ……………….……....…….….... 2

1.2 Dielectric elastomer used as grippers ……………………..……………...... ………...4

1.3 Dielectric elastomer used as micropumps ………………………….……...…………5

1.4 Wrinkles form on the film during pull-in instability .....…..……………...…..…...….6

1.5 Illustration of three failure modes of dielectric elastomer film .………...... 7

3.1 The working principle of the circular actuator …..………………………………….35

3.2 Comparison of the variations of radial and hoop stretches in annulus B with constant and varying permittivities ………..………………………………………………….39

3.3 Comparison of the variations of 휆3 along the radial direction for constant and varying permittivities ……………………………..……………………………………….....40

3.4 Voltage-stretch curves of circle A for c = 0 and different pre-stretch values …….....42

3.5 Voltage-stretch curves of circle A for 휆푃 = 1.25 and different c values ………...…43

(퐴) 3.6 The variation of 휎1 with 휆1 for the case of 휆푃 = 3 and c = 0 …………….…..…..44

3.7 Comparison of the voltage-stretch curves of circle A with constant and varying permittivities over different pre-stretches ……………………………….…….…....45

3.8 Comparison of the voltage-stretch curves of circle A with different c values for the cases 휆푃 = 2 and 3 …………………………………………………………….…....48

3.9 Variations of 휆푚푎푥 and the corresponding 훷̅ over a range of c-values for the 휆푃 = 1.25, 2 and 3 cases …………………………………………………………….…....49

xiii

4.1 Dimensions of the dielectric elastomer plate in (a) undeformed state and (b) deformed state…………………………………………………………….……………….…....56

4.2 Comparisons of 휆푐푟 versus 푘퐻 for isotropic and anisotropic materials with values of 퐷̂2 = 0, 1, 2, 2.2, 2.4, 2.6, 3, 4 and 5, (a)-(i), respectively.………………………...... 67

4.3 The values of the area of the unstable region for each case measured with AutoCAD.…………………………………………………………………………....69

4.4 Comparisons of 휆푐푟 for isotropic and anisotropic materials as 푘퐻 → 0.………….....70

4.5 Comparisons of 휆푐푟 versus 푘퐻 for different degrees of anisotropy, ξ = 0, 3, 5, 10, where 퐷̂2 = 2.…………………………………………………………....71

4.6 The critical electric displacement versus the anisotropic parameter ξ.……………....72

5.1 Lateral view of a buckling actuator: (a) rest state and (b) actuated state……….…....75

5.2 Mesh of the 1 mm thickness model.………………………………………………....83

5.3 Mesh of the 0.5 mm thickness model.…………………………………………….....83

5.4 The vertical displacement of the center point on the plate surface versus voltage for two different plate thicknesses………………………………………………….…....85

5.5 The deformed shape and contours of the displacement after wrinkling occurs for the 0.5 mm thickness case.…………………………………………………………….....86

5.6 The vertical displacement of the center point on the plate surface versus voltage for two different material models.…………………………………………………….....88

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CHAPTER I

INTRODUCTION

1.1 Electroactive Polymers

Functional polymers that respond mechanically to electric, chemical, pneumatic, optical, or magnetic stimulations are highly attractive and can be used in broad applications.

Among them, the type of polymers that respond to electric stimulus is called electroactive polymers (EAPs) [3].

Based on two different actuation mechanisms, EAPs are classified into two types: electronic EAPs and ionic EAPs. The electronic EAPs are activated by Coulomb force, while the ionic EAPs are activated by the diffusion or mobility of ions [3]. One type of the electronic EAPs called dielectric elastomers (DEs) demonstrates excellent properties [4] and will be the focus of the current study.

1.2 Working Principle of Dielectric Elastomers

Transducers made from dielectric elastomers can be used either in actuator mode or generator mode. In the generator mode, DEs convert mechanical energy to electrical energy. In the actuator mode, DEs convert electrical energy to mechanical energy [4]. The actuator mode is considered in this work.

As actuators, a basic form of dielectric elastomer transducers consists of a thin elastomeric membrane with two compliant electrodes coated on both sides [5] shown in Figure 1.1. When voltage is applied, the membrane will reduce in thickness and expand in area. This is because the positive charges on one electrode attract the negative charges on the other electrode. Meanwhile, the charges on the same electrode repel each other. This resulted in in-plane tensile stress and compressive stress in the thickness direction [6]. The DE material itself is insulating.

The membrane and its two conducting surfaces with electrodes form a capacitor.

For the DE material, the charges on the two surfaces will polarize the molecules in the DE. Positive charges are attracted to one surface, while negative charges are attracted to the other surface, which resulted in an increase of capacitance. The dielectric constant or dielectric permittivity represents the specific inductive capacity of this capacitor. Greater polarizability of the molecules resulted in higher dielectric constant [7][8].

Figure 1.1 An actuator made of dielectric elastomer film [2].

1.3 DE Materials

To achieve large actuated deformation, researchers have investigated different kinds of DE materials. These DE materials include silicones, acrylates, polyurethanes (PU), rubbers, latex rubbers, acrylonitrile-butadiene rubbers, olefinic, polymer foams, fluorinated, and styrenic copolymers [9]. Silicon rubbers and acrylates are the most widely used DE materials because they have good performances and they are commercially available at low cost. Silicon rubbers and acrylates could easily achieve actuated strain greater than 100% [10]. The most widely used acrylates are the VHB acrylate adhesive film from the 3M company. The VHB acrylates could achieve area expansion by 1692% due to high dielectric constant and breakdown strength [11]. The VHB acrylates will be the material considered in the present study.

1.4 Advantages of Dielectric Elastomers

DEs have other advantages such as lightweight, high work density, good frequency responses, high degree of electromechanical coupling, high breakdown strength [12], etc., and therefore, are considered as a promising class of smart materials.

1.5 Potential Applications of Dielectric Elastomers

Consequently, over the last two decades, dielectric elastomer transducers (DETs) have potential use and have been intensively studied in broad applications, such as artificial muscles [13], soft robotics [14] and energy harvesters [15], generators [16], sensors [17], and many others.

For example, Figure 1.2 shows grippers based on dielectric elastomers, an application of the DEs in soft robotics. Stiff fibers are incorporated into DE so that the grippers can be controlled to bend vertically or horizontally in different situations [18].

Figure 1.2 Dielectric elastomer used as grippers [18].

Figure 1.3 shows buckling of the dielectric elastomers used as micropumps.

In this example, the circular DE film will buckle out of plane with the application of voltage and alter the fluid flow in the channel. In this way, the flows in different channels can be controlled by electric signals [19].

Figure 1.3 Dielectric elastomer used as micropumps [19].

1.6 Failure Modes of Dielectric Elastomers

Electromechanical instability, or the so-called pull-in instability, represents a major failure mechanism of the DE actuators [20]. When voltage is applied, the DE film in Figure

1.1 reduces in thickness and expands in area, this leads to a higher electric field at the same voltage. Higher electric field causes the material to thin down further. This is positive feedback. As voltage reaches a relatively high value, the film eventually becomes unstable leading to material or dielectric failure [21]. This phenomenon has been observed in experiments and shown in Figure 1.4. In Figure 1.4, the flat film becomes unstable and forms wrinkles before failure [22].

Figure 1.4 Wrinkles form on the film during pull-in instability [22].

Another commonly seen failure of DE is dielectric breakdown. Pre-existing defects are assumed to be randomly distributed in the material. The defects grow in the material as electric field increases. When a threshold electric field is reached, the defects will form a channel in which a disruptive and intense flow of charges travels from one electrode to the other [23]. The material suddenly losses its insulation. This phenomenon is called dielectric breakdown and this critical electric field is called breakdown strength [4].

The third type of failure mode is material failure. When the maximum stress in the material is greater than the critical value of the material strength, the material will fail. Figure 1.5 gives an illustration of the three types of failure mode.

Figure 1.5 Illustration of three failure modes of dielectric elastomer film [22].

1.7 The Objectives of the Present Study

The main objective of the present work is to do modeling of the electromechanical coupling behavior of the dielectric elastomer as actuators and to study the effect of the material properties and the configuration parameters on the performance of the dielectric elastomer actuators.

First, the deformation dependent behavior of the permittivity of dielectric elastomers was discussed. The actuation behavior of a circular actuator made of a dielectric elastomer, VHB 4910, was theoretically analyzed. The effects of the deformation dependence of permittivity on electromechanical instability, loss of tension, and dielectric breakdown of the actuator during the actuation process were studied. Next, the effect of material anisotropy on the stability of a dielectric elastomer plate was studied. A linearized incremental theory was used and the plate thickness was considered. Results of transversely isotropic materials were compared with the isotropic case. Finally, finite element method was used to present a numerical model of a basic buckling actuator made of dielectric elastomer. A user element subroutine [1] in Abaqus was used to simulate the

7 electromechanical coupling behavior of the material. Different material models were programmed into the user element subroutine. The effect of a few model parameters on the performance of the buckling actuators was discussed.

The rest of the dissertation is organized as follows. Chapter II gave a literature review on the development of the nonlinear theory of continuum electromechanics. The theory adopted in this work is described in detail. Chapter

III studied the effect of the deformation dependent permittivity on the actuation of a pre-stretched circular dielectric actuator. Stability of an anisotropic dielectric elastomer plate was analyzed in Chapter IV. In Chapter V, finite element method was used to study the inflated dielectric membranes as buckling actuators. Some concluding remarks were made in Chapter VI.

CHAPTER II

THEORETICAL BACKGROUND AND LITERATURE REVIEWS

2.1 Literature Review on the Nonlinear Theory of Continuum Electromechanics

The early developments of the nonlinear theory of continuum electromechanics were made by Toupin [24], Eringen [25], and Tiersten [26].

In 1956, Toupin developed the static elasticity theory for isotropic dielectrics [24].

Following a procedure analogous to CAUCHY’s method, a stored energy function is assumed to be the function of deformation and polarization per unit mass. The principle of virtual work was used to obtain the field equations and boundary conditions. The principle of virtual work method also led to additional result, the constitutive equations for the local stress and the effective local field. He showed that the stored energy function could also be written as a function of the scalar invariants. In his work, some simple solutions to example problems were given to validate their theories. In 1963, Toupin developed the dynamics of elastic dielectrics [27]. Moving and finitely deformed elastic dielectrics were considered.

A theory of the electromagnetic field of the material was developed.

In 1963, Eringen [25] developed the electrostatics of elastic dielectrics. The principle of virtual work was also used to derive the field equations, boundary conditions, and constitutive laws of the materials under large deformations and polarizations. He gave the specific form of the constitutive laws of isotropic dielectrics and applied the theory to

9 a hollow circular cylinder as an example. In this work, polarization per unit reference volume was used instead.

In 1971, Tiersten [26] studied the thermoelectroelasticity of elastic dielectrics. Thermal effect was considered in his study. The field equations and boundary conditions were derived for dielectrics under finite deformation, polarization, and heat conduction. The mechanical inertia was included by using the theory of quasi-electrostatics.

In 1971, Lex and Nelson [28] developed the linear and nonlinear theory of electrodynamics for anisotropic dielectrics. In 1976, Lex and Nelson [29] developed the Maxwell equations of electromagnetic field in moving and finitely deformed dielectrics. In 1978, based on their previous theories, Nelson [30] derived the dynamic equations and constitutive laws for electroacoustic dielectrics.

Many remarkable works followed in applications to DE in recent years [31–

36]. These methods apply to finite and inhomogeneous deformation of DE under electrical field and can be extended to anisotropic DEs and other types of electroactive polymers. A detailed review of the nonlinear theory of electromechanics is given in [37].

In 2005, McMeeking and Landis used the principle of virtual work to formulate a Eulerian form of the governing equations for quasi-electrostatics [35].

In 2007, McMeeking et al. [38] developed the constitutive laws for dissipative materials based on their theoretical framework.

In 2005, Dorfmann and Ogden developed a Lagrangian formulation based on a total energy function, leading to relatively simple forms of the governing equations of equilibrium and constitutive equations [31]. In 2010, based on their earlier formulation, they developed an incremental theory for small deformations and electric fields superposed on a finite deformation and electric field [32]. In 2014, Dorfmann and Ogden applied their incremental theory of nonlinear electroelasticity to study the instabilities of an elastomeric plate. Their theories are adopted in the present work and will be described in detail in the next section.

In 2008, Suo et al. [33] developed a nonlinear field theory of deformable dielectrics by defining consistent work conjugates. The definitions lead to decoupled field equations, and the electromechanical coupling enters the theory through material laws.

2.2 The Nonlinear Theory of Electroelasticity

In this section, the nonlinear theory of electroelasticity [31][32][37][39][40] adopted in this work was introduced.

2.2.1 Kinematics

Consider an electroactive material body, it is defined by ℬ푟 in the reference configuration. The material body’s boundary is defined by 휕ℬ푟 . The material body is defined by ℬ in the deformed configuration. Its boundary is defined by 휕ℬ. The material body is defined as a set of particles, where vector 푿 identifies the position of an arbitrary

11 particle in the reference configuration and 풙 is the corresponding position vector in the current configuration [31][32][37][39][40]. The general equation of motion is expressed as

풙 = 흌(푿) (2.1) and the deformation gradient is defined by

푭 = 휵흌 (2.2) where 휵 is the gradient operator with respect to 푿. The Volumetric changes are accounted for by the quantity

퐽 = 풅풆풕 (푭) (2.3)

For incompressible materials

퐽 = 1 (2.4)

The left Cauchy-Green deformation tensor is defined by

퐁 = 푭푭T (2.5)

The left Cauchy-Green deformation tensor is defined by

퐂 = 푭T푭 (2.6) where superscript T denotes the transpose of a tensor.

2.2.2 Electrostatics and the Maxwell’s Equations

Purely electrostatic fields are considered here [31][32][37][39][40]. The polarization density 퐏 of the material body is defined by

퐏 = 퐃 − 휀0퐄 (2.7) where 퐃 is the electric displacement, 퐄 is the electric field and 휀0 is the permittivity of the vacuum.

The field equations are

curl 퐄 = ퟎ (2.8)

div 퐃 = 0 (2.9) where curl is the curl operator and div is the divergence operator, both with respect to 풙.

Assume the material body is in the vacuum. Let 퐄∗ and 퐃∗ be the electric field and electric displacement in the vacuum. They are related by

∗ ∗ 퐃 = 휀0퐄 (2.10)

퐄∗ and 퐃∗ must also satisfy the field equations

curl 퐄∗ = ퟎ (2.11)

div 퐃∗ = 0 (2.12)

Let 퐄L and 퐃L be the Lagrangian counterparts of the Eulerian fields 퐄 and 퐃, they are related by

T 퐄L = 푭 퐄 (2.13)

−1 퐃L = 퐽푭 퐃 (2.14)

The field equations in the reference configuration are

Curl 퐄L = ퟎ (2.15)

Div 퐃L = 0 (2.16)

13 where Curl is the curl operator and Div is the divergence operator, both with respect to 푿.

2.2.3 Governing Equations and Boundary Conditions

Assume there is no mechanical body force [31][32][37][39][40], the electromechanical equilibrium equation written in Eulerian form is

div 훔 = ퟎ (2.17) where 훔 is the total Cauchy stress tensor consisting of the mechanical part and electrical part.

The electromechanical traction boundary condition written in Eulerian form is

∗ 훔퐧 = 퐭a + 퐭e (2.18)

∗ ∗ where 퐧 is the unit outward normal to 휕ℬ, 퐭a is the mechanical traction and 퐭e = 훕e퐧 is

∗ ∗ the electrical traction resulted from the electrostatic Maxwell stress 훕e. 훕e is defined by

ퟏ 훕∗ = 휺 퐄∗⨂퐄∗ − 휺 (퐄∗ ⋅ 퐄∗)퐈 (2.19) e 0 ퟐ 0 where 퐈 is the identity tensor.

The electric boundary conditions written in Eulerian form are

퐧 × (퐄∗ − 퐄) = ퟎ (2.20)

∗ 퐧 ∙ (퐃 − 퐃) = σf (2.21)

where σf is the surface charge per unit area on boundary 휕ℬ.

The electromechanical equilibrium equation written in Lagrangian form is

Div 퐓 = ퟎ (2.22) where 퐓 is the total nominal stress tensor and is defined by 퐓 = 퐽푭−1훔.

The electromechanical traction boundary condition written in Lagrangian form is

퐓 ∗ 퐓 퐍 = 퐭A + 퐭E (2.23)

∗ ∗T where 퐍 is the unit outward normal to 휕ℬ푟, 퐭A is the mechanical traction and 퐭E = 퐓E 퐍.

∗ 퐓E is defined by

∗ −ퟏ ∗ 퐓E = 퐽푭 훕e (2.24)

The electric boundary conditions written in Lagrangian form is

T ∗ 퐍 × (푭 퐄 − 퐄L) = ퟎ (2.25)

−ퟏ ∗ 퐍 ∙ (퐽푭 퐃 − 퐃퐋) = 훔F (2.26)

2.2.4 Constitutive Equations for Electroelasticity

Assume the material is mechanically unconstrained [31][32][37][39][40], then the total stress tensor in Lagrangian form is defined by

휕푊 퐓 = (2.27) 휕푭 where 푊(푭, 퐃L) is the total energy density function based on 푭 and 퐃L.

The electric field in Lagrangian form is defined by

휕푊 퐄L = (2.28) 휕퐃L

For incompressible material, det푭 = 1, the total stress tensor, and the electric field in Lagrangian form become

휕푊 퐓 = − 푝푭−1 (2.29) 휕푭

휕푊 퐄L = (2.30) 휕퐃L where 푝 is a Lagrange multiplier related to the incompressibility.

The total stress tensor in Eulerian form is defined by

휕푊 훔 = 퐽푭−1 (2.31) 휕푭

The electric field in Eulerian form is defined by

휕푊 퐄 = 푭−T (2.32) 휕퐃L

For incompressible material, det푭 = 1, the total stress tensor, and the electric field in Eulerian form become

휕푊 훔 = 푭 − 푝퐈 (2.33) 휕푭

휕푊 퐄 = 푭−T (2.34) 휕퐃L

The electroelastic material invariants are defined by

퐼1 = 푡푟퐂 (2.35)

1 퐼 = [(푡푟퐂)2 − 푡푟(퐂2)] (2.36) 2 2

2 퐼3 = det(퐂) = 퐽 (2.37)

퐼4 = 퐃L ∙ 퐃L (2.38)

퐼5 = 퐃L ∙ 퐂퐃L (2.39)

ퟐ 퐼6 = 퐃L ∙ 퐂 퐃L (2.40)

For incompressible material

퐼3 = 1 (2.41)

푊 can be written in terms of the invariants as 푊(퐼1, 퐼2, 퐼4, 퐼5, 퐼6), then the total stress tensor and the electric field in Eulerian form for incompressible material become

ퟐ 훔 = 2푊1 퐁 + 2푊2 (퐼1퐁 − 퐁 ) − 푝퐈 + 2푊6 (퐃 ⊗ 퐁퐃 − 퐁퐃 ⊗ 퐃) (2.42)

−1 퐄 = 2(푊4 퐁 퐃 + 푊5 퐃 + 푊6 퐁퐃) (2.43)

where 푊푖 = 휕푊⁄휕퐼푖 (i=1,2,4,5,6).

2.2.5 The Incremental Formulation

A superimposed dot is used to denote the increment of a variable in the incremental formulation [31][32][37][39][40]. The incremental governing equations are expressed as

Curl 퐄̇ L = ퟎ (2.44)

Div 퐃̇ L = 0 (2.45)

Div 퐓̇ = 0 (2.46)

퐄̇ ∗ and 퐃̇ ∗ denote the increments in electric field and electric displacement in the vacuum. They are related by

∗ ∗ 퐃̇ = 휀0퐄̇ (2.47)

퐄̇ ∗ and 퐃̇ ∗ must satisfy the field equations

curl 퐄̇ ∗ = ퟎ (2.48)

div 퐃̇ ∗ = 0 (2.49)

Consider material incompressibility, the incremental traction, and electric boundary conditions are

T ∗ −T ∗ −T T −T 퐓̇ 퐍 = 퐭̇A + 훕̇ e푭 퐍 − 훕e푭 푭̇ 푭 퐍 on 휕ℬ푟 (2.50)

T ∗ T ∗ (푭̇ 퐄 + 푭 퐄̇ − 퐄̇ L) × 퐍 = ퟎ on 휕ℬ푟 (2.51)

−1 ∗ −1 −1 ∗ (푭 퐃̇ − 푭 푭̇ 푭 퐃 − 퐃̇ L) ∙ 퐍 = σ̇ F on 휕ℬ푟 (2.52)

∗ where the incremental electrostatic Maxwell stress 훕̇ e is defined by

∗ ∗ ∗ ∗ ∗ ∗ ∗ 훕̇ e = 휀0[퐄̇ ⨂퐄 + 퐄 ⨂퐄̇ − (퐄 ⋅ 퐄̇ )퐈] (2.53)

The push-forward incremental variables are defined by

−T 퐄̇ L0 = 푭 퐄̇ L (2.54)

퐃̇ L0 = 푭퐃̇ L (2.55)

퐓̇0 = 푭퐓̇ (2.56)

The incremental equations in terms of push-forward incremental variables are

curl 퐄̇ L0 = ퟎ (2.57)

div 퐃̇ L0 = 0 (2.58)

div 퐓̇0 = 0 (2.59)

The incremental traction and electric boundary conditions in terms of push- forward incremental variables become

T ∗ ∗ T 퐓̇0 퐧 = 퐭̇A0 + 훕̇ e퐧 − 훕e퐋 퐧 on 휕ℬ (2.60)

∗ T ∗ (퐄̇ + 퐋 퐄 − 퐄̇ L0) × 퐧 = ퟎ on 휕ℬ (2.61)

∗ ∗ (퐃̇ − 퐋퐃 − 퐃̇ L0) ∙ 퐧 = σ̇ F0 on 휕ℬ (2.62) where 퐋 = 휵풙̇ .

The incremental deformation gradient 푭̇ and incremental Lagrangian electric displacement 퐃̇ L result in the incremental stress 퐓̇ and the incremental Lagrangian electric field 퐄̇ L.The linearized incremental constitutive equations are expressed as

퐓̇ = 퓐푭̇ + 픸퐃̇ L (2.63)

T 퐄̇ L = 픸 푭̇ + 퐀퐃̇ L (2.64) where 퓐, 픸, 퐀 are the electroelastic moduli tensors. 퓐 is a fourth-order tensor. 픸 is a third-order tensor. 퐀 is a second-order tensor. The component form of the electroelastic moduli tensors can be derived from the total energy 푊

휕2푊 𝒜훼푖훽푗 = (2.65) 휕퐹푖훼휕퐹푗훽

휕2푊 픸훼푖훽 = (2.66) 휕퐹푖훼휕퐷퐿훽

휕2푊 A훼훽 = (2.67) 휕퐷퐿훼휕퐷퐿훽

The component form of the electroelastic moduli tensors for isotropic material are expressed as

2 6 6 휕퐼푚 휕퐼푛 6 휕 퐼푛 𝒜훼푖훽푗 = ∑푚=1 ∑푛=1 푊푚푛 + ∑푛=1 푊푛 (2.68) 휕퐹푖훼 휕퐹푗훽 휕퐹푖훼휕퐹푗훽

2 6 6 휕퐼푚 휕퐼푛 6 휕 퐼푛 픸훼푖훽 = ∑푚=4 ∑푛=4 푊푚푛 + ∑푛=5 푊푛 (2.69) 휕퐹푖훼 휕퐷퐿훽 휕퐹푖훼휕퐷퐿훽

2 6 6 휕퐼푚 휕퐼푛 6 휕 퐼푛 A훼훽 = ∑푚=4 ∑푛=4 푊푚푛 + ∑푛=4 푊푛 (2.70) 휕퐷퐿훼 휕퐷퐿훽 휕퐷퐿훼휕퐷퐿훽

2 where 푊푛 = 휕푊⁄휕퐼푛, 푊푚푛 = 휕 푊⁄휕퐼푚휕퐼푛 and various derivatives of 퐼푛.

For incompressible materials, the incremental total stress can be written as

−1 −1 −1 퐓̇ = 퓐푭̇ + 픸퐃̇ L + 푝푭 푭̇ 푭 − 푝̇푭 (2.71)

The push-forward forms of the incremental total stress and the incremental electric field are expressed as

퐓̇0 = 퓐0퐋 + 픸0퐃̇ L0 (2.72)

T 퐄̇ L = 픸0퐋 + 퐀0퐃̇ L0 (2.73) where 퓐0, 픸0, 퐀0 are the push-forward forms of the electroelastic moduli tensors.

For incompressible materials, the push-forward form of the incremental total stress can be written as

퐓̇0 = 퓐0퐋 + 픸0퐃̇ L0 + 푝퐋 − 푝̇퐈 (2.74)

The component form of the first derivatives of the invariants with respect to 푭 and 퐃L respectively:

휕퐼1 = 2퐹푖훼 (2.75) 휕퐹푖훼

휕퐼2 = 2(푐훾훾퐹푖훼 − 푐훼훾퐹푖훾) (2.76) 휕퐹푖훼

휕퐼3 −1 = 2퐼3퐹훼푖 (2.77) 휕퐹푖훼

휕퐼5 = 2퐷퐿훼퐹푖훾퐷퐿훾 (2.78) 휕퐹푖훼

휕퐼6 = 2(푐훼훽퐷퐿훽퐹푖훾퐷퐿훾 + 퐷퐿훼퐹푖훾푐훾훽퐷퐿훽) (2.79) 휕퐹푖훼

휕퐼4 = 2퐷퐿훼 (2.80) 휕퐷퐿훼

휕퐼5 = 2푐훼훽퐷퐿훽 (2.81) 휕퐷퐿훼

휕퐼6 2 = 2푐훼훽퐷퐿훽 (2.82) 휕퐷퐿훼

The component form of the second derivatives of the invariants with respect to 푭:

2 휕 퐼1 = 2훿푖푗훿훼훽 (2.83) 휕퐹푖훼휕퐹푗훽

2 휕 퐼2 = 2(2퐹푖훼퐹푗훽 − 퐹푖훽퐹푗훼 + 푐훾훾훿푖푗훿훼훽 − 푏푖푗훿훼훽 − 푐훼훽훿푖푗) (2.84) 휕퐹푖훼휕퐹푗훽

2 휕 퐼3 −1 −1 −1 −1 = 2퐼3(2퐹훼푖 퐹훽푗 − 퐹훼푗 퐹훽푖 ) (2.85) 휕퐹푖훼휕퐹푗훽

2 휕 퐼5 = 2훿푖푗퐷퐿훼퐷퐿훽 (2.86) 휕퐹푖훼휕퐹푗훽

2 휕 퐼6 = 2[훿푖푗(푐훼훾퐷퐿훾퐷퐿훽 + 푐훽훾퐷퐿훾퐷퐿훼) + 훿훼훽퐹푖훾퐷퐿훾퐹푗훿퐷퐿훿 + 휕퐹푖훼휕퐹푗훽

퐹푖훾퐷퐿훾퐹푗훼퐷퐿훽 + 퐹푗훾퐷퐿훾퐹푖훽퐷퐿훼 + 푏푖푗퐷퐿훼퐷퐿훽] (2.87)

The component form of the second derivatives of the invariants with respect to 퐃L:

2 휕 퐼4 = 2훿훼훽 (2.88) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼5 = 2푐훼훽 (2.89) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼6 2 = 2푐훼훽 (2.90) 휕퐷퐿훼휕퐷퐿훽

The component form of the mixed derivatives of the invariants with respect to 푭 and 퐃L:

2 휕 퐼5 = 2훿훼훽퐹푖훾퐷퐿훾 + 2퐷퐿훼퐹푖훽 (2.91) 휕퐹푖훼휕퐷퐿훽

2 휕 퐼6 = 2퐹푖훽푐훼훾퐷퐿훾 + 2퐹푖훾퐷퐿훾푐훼훽 + 2퐹푖훾푐훾훽퐷퐿훼 + 2훿훼훽퐹푖훾푐훾훿퐷퐿훿 (2.92) 휕퐹푖훼휕퐷퐿훽

2.3 Literature Review on the Modeling of the Dielectric Elastomers

In this section, a brief review of some of the most important works in the modeling of the dielectric elastomers was presented.

In 1998, Pelrine et al. [5] proposed an equation describing the electrostatic pressure acting on the dielectric elastomeric film under electric field. The equation is expressed as

2 푝푒 = 휀0휀E (2.93)

where 푝푒 is the effective pressure, 휀0 is the permittivity of vacuum, 휀 is the

relative dielectric constant of the material and E is the electric field. The effective 22 pressure 푝푒 is twice the value of that in a parallel-plate capacitor. This is because with the use of compliant electrodes, as the material reduces in thickness and expand in area, compressive stress in the thickness direction is generated by the unlike charges on the opposite electrodes, at the same time, a tensile stress in the length and width direction is generated by the like charges on the same electrode. The effect of the compressive stress and the tensile stress can be expressed as the effective pressure 푝푒 . They used this equation to model the actuation of a DE film with free boundary conditions and compared with the numerical and experimental results of a model constrained on the edges.

In 2006, Plante and Dubowsky [22] did analytical, finite element and experimental study of a pre-stretched circular actuator. Three failure modes are considered. They noted that for the dielectric elastomer actuators to operate reliably, high stretch rate and short periods of time were required.

In 2005, Wissler and Mazza [41] modeled the actuation of a pre-stretched circular actuator using finite element method. In their modeling analysis, they decoupled the passive mechanical response of the material from its electromechanical behavior. Material properties were obtained from their uniaxial tensile and relaxation tests. Large deformation and material viscoelasticity were considered in their analysis. Good agreement was obtained between simulation and experiment results of the equal-biaxially pre-stretched actuators. Later they compared three different hyperelastic material models in [42]. In

2007, Wissler and Mazza [43] did experimental and numerical analysis under different pre- stretch levels of the actuators. They optimized the material parameters for three different hyperelastic material models based on the test data. Then different hyperelastic material models were used to simulate the circular actuators and compared with the experimental

23 results. Satisfactory agreement was obtained. Since agreement was not obtained for uniaxial behavior, Wissler and Mazza [44] investigated the electromechanical coupling behavior of the actuators to validate the Pelrine’s equation and gave a new interpretation of the electrostatic forces acting on the actuators. They then measured the dielectric constant for the acrylic elastomer VHB 4910 and found it to be deformation dependent.

In 2007, Zhao and Suo [45] studied the electromechanical instability using the Hessian method. Based on this method, when the Hessian of the free-energy function turns negative, the electromechanical instability will occur.

In 2011, Koh et al. [46] theoretically analyzed a pre-stretched circular actuator using the method proposed by Suo et al. [33]. In their analysis, the mechanisms of large deformation of the dielectric elastomers are illustrated with an equally biaxial dielectric elastomer film. They classified the voltage-stretch responses of the dielectric elastomer film into two types based on the characteristics of the responses. For each type of response, the actuator could encounter different failure modes.

In 2010, based on their nonlinear electroelastic formulation, Dorfmann and

Ogden [32] developed an incremental theory for small deformations and electric fields superposed on a finite deformation and electric field. They illustrated their incremental theory by considering the instability of a half-space in the presence of electric field. In 2014, they applied their incremental theory of nonlinear electroelasticity to study the instabilities of an elastomeric plate. In 2017, Melnikov

[40] adopted the incremental theory to study the bifurcation of thick-walled electroelastic cylindrical and spherical shells under large deformation.

Several research groups have made great efforts in developing the numerical tools to study the dielectric elastomers. Among them, in 2003, Henann et al. [47] developed a three-dimensional, fully coupled theory that governs the electromechanical behavior of the dielectric elastomers and implemented the theory in Abaqus by programming a user element subroutine. In their code, an additional nodal degree of freedom is added to the user element to take the electric potential into account. They then analyzed several types of dielectric elastomer actuators and energy harvesters using the numerical simulation tool.

In their analysis, the constitutive behavior was assumed to be time-independent. In 2006,

Wang et al. [48] updated the constitutive laws of the previous theory to account for viscoelasticity and time-dependent mechanical response of the material. Then the numerical simulation tool was used to study the effect of viscoelasticity on the instabilities of the dielectric elastomers. Several types of instabilities were considered in their study: the pull-in instability, electrocreasing, electrocavitation and wrinkling. They found that material viscoelasticity could delay the occurrence of the instabilities.

CHAPTER III

EFFECT OF THE DEFORMATION DEPENDENT PERMITTIVITY ON THE ACTUATION OF A PRE-STRETCHED CIRCULAR DIELECTRIC ACTUATOR

Experiments have shown that under large deformation, the permittivity of some dielectric elastomers demonstrates significant deformation dependent behavior. In this chapter, we theoretically analyzed the effect of this behavior on the actuation of a circular actuator made of a dielectric elastomer, VHB 4910. The dependency of permittivity on deformation was considered through the equations of state. The equilibrium equations of the actuator were solved by MATLAB. The electromechanical instability, loss of tension and dielectric breakdown of the actuator during the actuation process under different pre-stretch levels were discussed.

3.1 Circular Actuators

As actuators, a basic form of dielectric elastomer transducers consists of a thin elastomeric membrane with two compliant electrodes coated on both sides [5].

In particular, a pre-stretched circular actuator consisting of an active area and a passive area is a commonly used configuration since being proposed by Pelrine et al. [10]. With this configuration, the pre-strain can simply be achieved by stretching

26 and fixing the film on a circular rigid frame [49] or by a dead load [50]. It is an excellent test configuration to evaluate the biaxial actuation properties of new elastomers and electrodes [10,22,49,51–53] and to study the electromechanical instabilities [54]. This configuration has also been studied theoretically by several researchers in the purpose of disclosing the fundamental electromechanical coupling principles and guiding the test designs [22,46,49]. Among all the DE materials, the acrylic elastomer VHB 4910 from 3M company is the most widely used and studied for actuators and thus it will be the material considered in this study.

When actuated, the active region of a VHB4910 DE circular actuator reduces in thickness, expands in area, and interacts with the outer passive region through the interface.

The Maxwell stress resulted from the electrostatic forces between the electrodes is usually regarded as the stress induced by the external electric field for the active region [10,49].

This is appropriate for small deformation. For circular actuators, large deformation needs to be considered since the membrane can usually be pre-stretched up to 5 times its original radius.

3.2 Deformation Dependent Permittivity of the Dielectric Elastomer

To study the electromechanical ability of the VHB 4910, the permittivity is a key material property and has been experimentally studied by many groups [53,55–62] [19].

The permittivity was found to decrease slightly as the stretch increases [55,60]. Di Lillo et al. [60] observed the value of permittivity ranging from 4.2 (unstretched) to 4.05 (5×5 stretch level) and suggested that artifacts associated with deficiencies of the instrumentation can generate misleading measurements. In contrast, many other groups

27 observed significant drops in permittivity as deformation increases [53,56,58,61].

For example, Wissler and Mazza [56] observed that the permittivity drops by 44 % when the pre-stretch increases from 1 to 5 in a circular, equal-biaxial configuration.

The reason for the discrepancy in experimental results on permittivity by different groups remains unclear. But based on the large number of experimental findings, the effect of deformation on permittivity is worth considering especially for large- deformation scenarios, where the stress term generated from the change of permittivity with deformation can become a significant part of the voltage-induced stress [37,64–66]. For circular actuators, large deformation needs to be considered since the membrane can usually be pre-stretched up to 5 times its original radius.

Therefore, how the dependency of the permittivity on deformation affects a working circular actuator is studied analytically in this chapter.

3.3 Summary of the Basic Equations

The early developments of the nonlinear theory of continuum electromechanics were made by Toupin [24], Eringen [25] and Tiersten [26]. Many remarkable works followed in applications to DE in recent years [31–36]. These methods are applicable to finite and inhomogeneous deformation of DE under electrical field and can be extended to anisotropic DEs and other types of electro-active polymers. A detailed review of the nonlinear theory of electromechanics is given in [37]. McMeeking and Landis used the principle of virtual work to formulate an Eulerian form of the governing equations for quasi-electrostatics [35]. Dorfmann and Ogden developed a Lagrangian formulation based on a total energy function, leading to relatively simple forms of the governing equations of

28 equilibrium and constitutive equations [31]. Later they also developed an incremental theory for small deformations and electric fields superposed on a finite deformation and electric field [32]. Suo et al. [33] developed a nonlinear field theory of deformable dielectrics by defining consistent work conjugates. The definitions lead to decoupled field equations, and the electromechanical coupling enters the theory through material laws.

3.3.1 Field Equations

It is more convenient to describe the circular actuator in cylindrical coordinates, where the geometry can be described in terms of cylindrical coordinates 푅, 훩, 훧 in the reference configuration and 푟, 휃, 푧 in the current configuration. The circular membrane is defined as a set of particles, where 푿(푅, 훩, 훧) and 풙(푟, 휃, 푧) identify the position of an arbitrary particle in the reference and current configurations, respectively. The circular actuator retains axisymmetric under deformation. The general equation of motion is expressed as

풙 = 흌(푿, 푡) (3.1) and the deformation gradient is defined as

푭 = 휵흌 (3.2)

The equation of motion of a particle in the circular membrane is expressed as

푟 = 푟(푅) (3.3)

휃 = 훩 (3.4)

푧 = 휆3푍 (3.5)

Therefore,

푑푟⁄푑푅 0 0 푭 = ( 0 푟⁄푅 0 ) (3.6) 0 0 휆3 and the in-plane principal stretches are

푑푟 휆 = (3.7) 1 푑푅

푟 휆 = (3.8) 2 푅

From Eqns. (3.7)- (3.8), we obtain

푑휆 1 2 = (휆 − 휆 ) (3.9) 푑푅 푅 1 2

In the absence of body force, the equilibrium equation in terms of true stress is written as

div흈 = 0 (3.10) where 흈 represents the true stress tensor and div is the divergence operator with respect to

풙.

Let 퐓 denote the nominal stress. The true stress and the nominal stress are related by

퐓 = 퐽푭−ퟏ흈 (3.11)

where 퐽 = 푑푒푡 (푭) = 휆1휆2휆3.

3.3.2 Equations of State

It is a common practice to assume DE to be fully incompressible [5], namely

휆1휆2휆3 = 1 (3.12)

Here we adopted the quasilinear dielectric behavior [64], where the true electric field 퐄 is linear in the true electric displacement 퐃, while the permittivity tensor of an incompressible DE, 휺, is deformation dependent

퐃 = 휺(휆1, 휆2)퐄 (3.13)

For an incompressible DE membrane sandwiched between two electrodes, the electric field is only applied in the thickness direction, and the nonzero components of 퐃 and 퐄 are D3 and E3 respectively. Assuming the material is isotropic, the permittivity is the same in all directions. For simplicity, notations D, E and 휀 will be used instead of D3, E3 and 휀3 in the following text. According to the theoretical framework put forth by Dorfmann and Ogden [37], the free energy density of the DE membrane, 푊(휆1, 휆2, 퐼4), is a function

2 of three independent variables, namely 휆1, 휆2, 퐼4, where 퐼4 = EL, and EL is the electrical field in the reference configuration, i.e., EL = 휆3E. The stress differences and the true electric displacement D can be expressed as

휕 푊(휆1,휆2,퐼4) 휎1 − 휎3 = 휆1 (3.14) 휕휆1

휕 푊(휆1,휆2,퐼4) 휎2 − 휎3 = 휆2 (3.15) 휕휆2

휕 푊(휆1,휆2,퐼4) −2 −2 D = −2 휆1 휆2 E (3.16) 휕퐼4

31 where 휎푖 (푖 = 1,2,3) are the principal values of the true stress tensor. Eqns. (3.14)-(3.16) constitute the equations of state once 푊(휆1, 휆2, 퐼4) is prescribed. From Eqns. (3.6)-(3.8) and (3.11), the principal values of the nominal stress are given as

T1 = 휎1⁄휆1 (3.17)

T2 = 휎2⁄휆2 (3.18)

T3 = 휎3휆1휆2 (3.19)

A particular form of the free energy which decomposes 푊 into mechanical and electrical parts takes the following form [37]

1 푊(휆 , 휆 , 퐼 ) = 푊 (휆 , 휆 ) − 휀(휆 , 휆 )휆2휆2퐼 (3.20) 1 2 4 푠 1 2 2 1 2 1 2 4

2 2 2 where 푊푠 is the free energy due to stretching. It is noted that 휆1휆2퐼4 = E .

Substituting Eqn. (3.20) into Eqns. (3.14)-(3.16), we obtain

휕푊푠 2 휆1 휕휀(휆1,휆2) 2 휎1 − 휎3 = 휆1 − 휀(휆1, 휆2)E − E (3.21) 휕휆1 2 휕휆1

휕푊푠 2 휆2 휕휀(휆1,휆2) 2 휎2 − 휎3 = 휆2 − 휀(휆1, 휆2)E − E (3.22) 휕휆2 2 휕휆2

3.3.3 Elastic Free Energy

We choose the Gent model [67], which is capable of demonstrating the stretch- stiffening effect when the elastomer approaches the limiting stretch [34]. The material properties used in this study are to represent the widely used dielectric acrylic elastomer

VHB 4910. The strain energy density function for the Gent model takes the form

휇퐽푚 퐼1−3 푊푠 = − ln (1 − ) (3.23) 2 퐽푚

2 2 2 where 휇 = 51 kPa [68] is the shear modulus, 퐼1 = 휆1+휆2+휆3, and 퐽푚 represents the upper limit of (퐼1 − 3) related to the limiting stretch. Usually for elastomers, 20 < 퐽푚 < 200

[69]. In this study, 퐽푚 = 120 is used [70].

3.3.4 Permittivity Function

Many studies have experimentally investigated the dependence of the permittivity of acrylic VHB 4910 on deformation and found that the value of permittivity decreases as the stretch increases. In some studies, the permittivity was found to decrease drastically as the equal-biaxial prestretch increases [53,56,58,61], while other studies showed a slight decrease of the permittivity under sufficient equal-biaxial prestretch [55,57,59,62]. Based on different experimental observations, different forms of permittivity function

[58,64,65,71,72] in terms of principal stretches were proposed. Dorfmann and Ogden proposed a permittivity expression by involving two dimensionless material constants that serve as electroelastic coupling parameters [37]. Jiménez and McMeeking [71] derived a general three dimensional expression of deformation dependent permittivity based on the statistical mechanics analysis of a freely jointed polymer chain, due to Kuhn and Grün [73], that relates the force of extension and polarizability anisotropy of a polymer chain to its fractional extension through the inverse Langevin function. The permittivity model derived by Jiménez and McMeeking takes the form

2 2 2 휀1 = 휀0̅ + 푐0(2휆1 − 휆2 − 휆3) (3.24)

2 2 2 휀2 = 휀0̅ + 푐0(2휆2 − 휆1 − 휆3) (3.25)

2 2 2 휀3 = 휀0̅ + 푐0(2휆3 − 휆1 − 휆2) (3.26) where 휀0̅ and 푐0 are the isotropic part of the permittivity tensor and the stretch dependent coefficient respectively. As discussed in section 2.2, only 휀3 needs to be considered in this study. Many experimental studies found a linear function was suitable to describe the dependence of the permittivity on the principal stretches [56]. Zhao and Suo [64] proposed the following function

휀 = 휀[̅ 1 + c(휆1 + 휆2 − 2)] (3.27) where 휀̅ = 휀푟휀0, with 휀푟 being the relative permittivity and the permittivity of the vacuum

−12 휀0 = 8.85 × 10 F/m. It is worth noting that Eq. (17) is a truncated Taylor expansion of

Eq. (16c). The function proposed by Zhao and Suo [64] is adopted in this study. Table 3.1 lists different c values obtained by fitting experimental data from different research groups, and the value of c falls in the range from 0 to -0.1. The effect of different c values on the actuation behavior of the DE circular membrane will be discussed in detail in Section 4.

Table 3.1 Values of c obtained by fitting experimental data reported by different research groups. Experimental data [55] [62] [59] [57] [74] c -0.0073 -0.0107 -0.0236 -0.0241 -0.0275 Experimental data [62] [56] [61] [53] [58] c -0.0317 -0.053 -0.0777 -0.0823 -0.0892

3.4 Actuation of the Circular Dielectric Elastomer Actuators

As shown in Figure 3.1, the circular actuator consists of two regions. The relatively small circular region in the center is the active region, which will be sandwiched with electrodes after pre-stretch. The outer annular region is the passive region, which does not have any electrode coated on it. We name the active region as circle A and the passive 34 region as annulus B. In the reference configuration shown in Figure 3.1(a), let the radii be

푅퐴 and 푅퐵 for circle A and annulus B respectively. The circular membrane will first be pre-stretched equal biaxially to a stretch 휆P and fixed onto a rigid circular frame as shown in Figure 3.1(b). After sufficient relaxation time, voltage Φ will be applied on circle A,

Figure 3.1(c).

Figure 3.1 The working principle of the circular actuator. In the reference state, circle A is the central circular area with a radius 푅퐴, and in the stretched state, it is the area coated

with electrodes. Annulus B is the area outside circle A.

3.4.1 Circle A

Under the Maxwell and electrostrictive stresses, circle A expands in area to reach total radial and hoop stretches of 휆1 and 휆2 respectively with respect to the reference configuration

휆1 = 휆p휆1,a (3.28)

휆2 = 휆p휆2,a (3.29)

where 휆1,a and 휆2,a are actuated stretches.

The directions of stretches and stresses in 푟, 휃, 푧 coincide with those of the principal stresses and stretches in the circular membrane. During the actuation process, circle A is in homogenous equal biaxial state, i.e., 휆1 = 휆2 and 휎1 = 휎2. Setting 휎3 = 0 and inserting 푊푠(휆1, 휆2) and ε(휆1, 휆2) into Eqn. (3.21), we have

2 −4 휇퐽푚(휆1 −휆1 ) 2 5휆1 휎1 = 2 −4 − 휀E̅ [1 + c( − 2)] (3.30) [퐽푚−(2휆1+휆1 −3)] 2

훷 where E2 = ( )2휆4 , 훷 is the applied voltage and H is the initial thickness of the H 1 membrane.

Circle A and annulus B interact through the following boundary conditions

no slip at 푅 = 푅퐴:

(퐴) (퐵) 휆2 = 휆2 (3.31)

force balance at 푅 = 푅퐴:

(퐴) (퐵) T1 = T1 (3.32) where T1 is defined in Eqn. (3.17).

3.4.2 Annulus B

As circle A expands, annulus B reduces in area and expands in thickness inhomogeneously. The stresses in B must satisfy the equilibrium Eqn. (3.10). Specialized to the current problem, Eqn. (3.10) leads to

푑휎 휎 −휎 1 = 2 1 (3.33) 푑푟 푟

Combining Eqn. (3.9) and Eqn. (3.33), a first-order nonlinear ODE system with 푅 being the independent variable and 휆1, 휆2 being the dependent variables can be formulated as

휕휎1 휆1(휎2−휎1)+ (휆2−휆1)휆2 푑휆1 휕휆2 = 휕휎1 (3.34) 푑푅 푅휆2 휕휆1

푑휆 1 2 = (휆 − 휆 ) (3.35) 푑푅 푅 1 2 where 휎1 and 휎2 are obtained from Eqn. (3.21) and (3.22) by setting 휎3 as well as the electric related terms to 0

2 −2 −2 휕푊푠 휇퐽푚(휆1 −휆2 휆1 ) 휎1 = 휆1 = 2 2 −2 −2 (3.36) 휕휆1 [퐽푚−(휆1+휆2+휆1 휆2 −3)]

2 −2 −2 휕푊푠 휇퐽푚(휆2 −휆1 휆2 ) 휎2 = 휆2 = 2 2 −2 −2 (3.37) 휕휆2 [퐽푚−(휆1+휆2+휆1 휆2 −3)]

The boundary condition at 푅 = 푅퐵 is

(퐵) 휆2 = 휆푝 (3.38)

With boundary values prescribed at 푅 = 푅퐴 and 푅 = 푅퐵, the ODE system can be solved with MATLAB solver bvp4c, which is a finite difference code that implements the three-

(퐴) stage Lobatto IIIa formula [75]. For boundary condition Eqn. (3.31), 휆2 is prescribed. For

(퐴) each given 휆2 , the corresponding 훷 applied on circle A can be calculated.

3.5 Results and Discussions

The variations of the radial and hoop stretches along the radial direction of the circular membrane under actuation can be obtained at a given voltage. Consequently, the voltage-stretch relations can be obtained for circle A with any given 휆푝 and 푅퐵⁄푅퐴 values.

푅퐵⁄푅퐴 values of 5, 10 and 15 were considered in this study and it was found that the results show a similar trend. In the calculations presented below, radial ratio 푅퐵⁄푅퐴 = 15 was

휀̅ 훷 used as an example and the voltage is normalized as 훷̅ = √ . As shown in Table 3.1, 휇 H experimental data obtained by different researchers suggested different values of c between

0 and -0.1. For example, the experimental data by Wissler and Mazza [56] suggested that c = −0.053. This value was used in most of the calculations presented here.

3.5.1 Stretches in Annulus B

As an example, Figure 3.2 shows the total stretches 휆1 and 휆2 along the radial direction respectively of the annulus B for the case of 휆푃 = 3, 푅퐵⁄푅퐴 = 15 and 훷̅ = 0.2.

Here the normalized radius 푅⁄푅퐴 was used as the horizontal axis. The solid lines resulted from a constant permittivity (c = 0), while the dashed lines were calculated using a varying permittivity with c = −0.053. The variation of the curves showed that 휆1 increases while

휆2 decreases with 푅, and they both converge to 휆푃 as 푅 approaches 푅퐵. On the other hand,

38 as 푅 approaches 푅퐴, 휆2 increases steeply while 휆1 decreases sharply. The total stretches for other values of pre-stretch, radius ratio and applied voltage showed similar variation.

Figure 3.2 Comparison of the variations of radial and hoop stretches in annulus B

with constant and varying permittivities.

Figure 3.3 displays the variation of 휆3 , which represented the variation of the thickness, along the radial direction. Comparing to the c = 0 case, the thickness of circle A was 25.3 % higher under the influence of the varying permittivity at the same applied voltage of 훷̅ = 0.2. It is worth noting that there is a jump in thickness between circle A and annulus B at the interface. This is because an idealized model was used in the analysis, where homogeneous deformation was assumed in circle A. In real applications, the

39 deformation in circle A is actually inhomogeous near the interface and the thickness changes smoothly from circle A to annulus B.

Both Figure 3.2 and Figure 3.3 suggested that there was an obvious difference between the total stretches at the interface in annulus B with the two different permittivity assumptions. The influence of the varying permittivity resulted in 휆1 being 15.6 % higher,

휆2 being 10.7 % lower and 휆3 being 3.2 % lower. This difference gradually disappeared as

푅 approaches 푅퐵. This indicated that the dependence of permittivity on deformation in circle A has a nonnegligible effect on annulus B through the interface, and the effect decays as the distance increases from the interface.

Figure 3.3 Comparison of the variations of 휆3 along the radial direction for

constant and varying permittivities.

3.5.2 Electromechanical Instability

Electromechanical instability, or the so-called pull-in instability, represents a major failure mechanism of the DE actuators [20], and it is well-known that pre-stretch can suppress the pull-in instability [76]. The pull-in instability of the pre-stretched circular actuator discussed above (푅⁄푅퐴 = 15) was studied here. As an example, Figure 3.4 shows the voltage-stretch curves for the constant permittivity case. At low pre-stretch levels, the voltage-stretch curve did not monotonically increase and exhibits a local maximum voltage value. This point marked the onset of pull-in instability [76]. As 휆푃 increases, the local maximum on the voltage-stretch curve gradually disappears. As 휆푃 reaches greater than

2.53, the voltage-stretch curve becomes monotonic and the electromechanical instability is suppressed. In our calculations, the initial thickness of the plate was incorporated into the

휀̅ 훷 normalized voltage 훷̅ = √ . As the initial thickness increases, to have the same 휇 H normalized voltage, the actual applied voltage needs to increase proportionally. This means that for a thicker plate, a larger actual voltage is required to reach the electromechanical instability point. However, the corresponding total stretch at the electromechanical instability remains the same.

Figure 3.4 Voltage-stretch curves of circle A for c = 0 and different pre-stretch values.

However, at large deformation, the dependence of permittivity on deformation becomes significant. For permittivity varying according to Eqn. (3.27), Figure 3.5 shows the voltage-stretch curves for the case of 휆푃 = 1.25. When c = 0, i.e., the permittivity was a constant, pull-in instability occurred as the applied voltage reaches 훷̅ = 0.518, as marked by the solid diamond symbol. As c decreases, a higher voltage is needed to achieve the same total stretch and the electromechanical instability is delayed. As c drops below -

0.085, the voltage-stretch curve becomes monotonic and the electromechanical instability was suppressed by sufficient influence of the dependence of permittivity on deformation.

Figure 3.5 Voltage-stretch curves of circle A for 휆푃 = 1.25 and different c values.

3.5.3 Loss of Tension

A thin film subjected to a lateral compression is easy to buckle. Therefore, loss of tension is of interest because it is a turning point for the tension-compression behavior of the dielectric film. The loss of tension of the pre-stretched circular actuator discussed above

(푅⁄푅퐴 = 15) was studied here. As circle A expands laterally against annulus B, the tensile

(퐴) stress in circle A reduces with 휆1. Figure 3.6 shows the variation of 휎1 with 휆1 for the

(퐴) case of c = 0 (constant permittivity) and 휆푃 = 3. When 휆1 = 6.73, 휎1 = 0, which

(퐴) marked the loss of tension in the membrane. As 휆1 becomes larger than 6.73, 휎1 becomes

43 compressive, causing the membrane to wrinkle due to buckling. For the results presented hereafter, the calculations will be terminated upon the occurrence of loss of tension.

(퐴) (퐴) Figure 3.6 The variation of 휎1 with 휆1 for the case of 휆푃 = 3 and c = 0. 휎1 turns

negative when 휆1 is larger than 6.73, which marks the onset of loss of tension of the

membrane.

Figure 3.7 shows the voltage-stretch curves of circle A for constant permittivity

(c = 0), solid lines, and varying permittivity (c = −0.053), dotted lines, respectively over different pre-stretches. Pre-stretches from 1.1 to 5 are considered since it is suggested that a safe practical area expansion limit for VHB4910 is 36 [6]. In this figure, the pull-in instability was denoted by the diamond symbols and the loss of tension was denoted by the circle symbols. The curves were terminated at the onset of electromechanical instability or

44 the onset of loss of tension, whichever comes first. In the case where electromechanical instability was suppressed, the stretch can go further and reach the material’s electric breakdown strength [24].

Figure 3.7 Comparison of the voltage-stretch curves of circle A with constant and varying permittivities over different pre-stretches. The pull-in instability, loss of tension and the

electric breakdown are considered for each curve.

3.5.4 Discrepancy in Voltage-stretch Response

As shown in Figure 3.7, for 휆푃 = 1.1, the onset of loss of tension happened first and was at the same total stretch value for both constant and varying permittivity. When the pre-stretch was increased to 휆푃 = 1.25, the voltage-stretch curves for c = 0 and c = -

0.053 still coincide for most of the actuation process, only a small discrepancy showing up

45 right before failure. However, as 휆푃 further increased, the discrepancy of the voltage- stretch curves between constant and varying permittivity cases started to appear at an early stage of the actuation process. Moreover, for a given pre-stretch value, the discrepancy became larger as the total stretch increased. For example, for the case of 휆푃 = 3, to obtain a total stretch of 4.5, 훷̅푣푎푟푦 was 40.1 % higher for the c = -0.053 than c = 0.

3.5.5 Dielectric Breakdown

It has been observed that the electric breakdown strength of DE was dependent on the deformation, i.e., the breakdown electric field increases as deformation increases

[10,53,55,59,63]. Several phenomenological relations between the breakdown strength and the stretch have been suggested based on experimental data. Huang et al. [63] suggested

1.13 퐸퐵 = 51휆 based on equal-biaxial stretching experiments of VHB4910, which was adopted in this study. Consequently, the normalized voltage at electric breakdown can be

휀̅ 퐸 expressed as 훷̅ = √ 퐵, which was represented by the dashed line in Figure 3.7. Liu et 퐵 휇 휆2 al. [23] proposed a more general electric breakdown strength model which further involved the nonlinearities in permittivity and elastic modulus. In their study, the experimental data by Huang et al. [63] was used as part of the model validation and similar variation of the breakdown strength with respect to stretch was obtained.

Consider again the pre-stretched circular actuator discussed above (푅⁄푅퐴 = 15).

Figure 3.7 indicated that when 휆푃 was close to 1, loss of tension occurred first. As 휆푃 increased, for example, when 휆푃 = 1.25, the electromechanical instability occurred first because the membrane gradually gained more tension with higher pre-stretch. At 휆푃 = 2,

46 the actuator encountered electromechanical instability before loss of tension when the permittivity was constant. However, for varying permittivity with c = -0.053, the electromechanical instability was suppressed and loss of tension occurred at a larger stretch level. The dependence of permittivity on deformation enabled the actuator to achieve a total stretch 28.4 % higher than the c = 0 case. At 휆푃 = 3, the pull-in instability was suppressed for both constant and varying permittivity, and loss of tension took place at the same total stretch for both cases. When 휆푃 ≥ 4, the electric breakdown determined the actuator’s maximum total stretch and the dependence of permittivity on deformation significantly lowered the maximum total stretch achievable by the actuator. For examples, when 휆푃 = 4, the maximum total stretch for the c = -0.053 case was 19.12 % lower than the c = 0 case, and to achieve the same total stretch of 5, the applied voltage for the c = -

0.053 case was 50.16 % higher than the c = 0 case.

3.5.6 Variation of c

This section examines the effect of the electrostrictive factor c on the behavior of the actuator. Figure 3.8 depicts the voltage-stretch curves with different c values for the

휆푃 = 2 and 3 cases. Figure 3.9 shows the variations of the maximum achievable stretch and the corresponding applied voltage over a range of c values for the 휆푃 = 1.25, 2 and 3 cases respectively.

Figure 3.8 Comparison of the voltage-stretch curves of circle A with different c

values for the cases 휆푃 = 2 and 3.

When the pre-stretch was low, e.g., 휆푃 = 1.25, Figure 3.9(a) indicated that the potential failure of the actuator was due to pull-in instability, and as the value of c decreased, both the maximum stretch and the corresponding voltage increased.

When the pre-stretch was increased to 휆푃 = 2, as suggested by Figure 3.9(b), the situation became more complicated. When c was close to 0, the pull-in instability happened first; as c decreased, the pull-in instability was gradually suppressed and loss of tension happened first; as c decreased further, loss of tension was delayed, and the actuator can reach its electric breakdown strength. The maximum stretch increased as c decreased over the region where failure was due to pull-in instability; it is invariant of c over the region

48 where failure was due to loss of tension; and it decreased as c decreased over the region where failure was due to dielectric breakdown. The applied voltage at the maximum stretch always increased as c decreased.

Figure 3.9 Variations of 휆푚푎푥 and the corresponding 훷̅ over a range of c values

for the 휆푃 = 1.25, 2 and 3 cases.

When the pre-stretch was further increased to 휆푃 = 3, Figure 3.9(c) indicated that the pull-in instability had been suppressed while the loss of tension was delayed, and the actuator was able to reach its electric breakdown strength. As c decreased, the voltage at breakdown increased while the maximum stretch decreased. The results for 휆푃 = 4 and 5 had similar trends to the 휆푃 = 3 case.

3.6 Summary

As an essential test configuration, the circular actuator was used to analyze the electrostrictive effect on the actuation behavior of dielectric elastomer transducers. We have shown that, compared with constant permittivity, the deformation dependent behavior of permittivity affected the actuation performance of the circular actuator in a variety of ways and the impact differed at different pre-stretch levels.

The dependence of permittivity on deformation had little influence when the pre- stretch was close to 1. As the pre-stretch increased, the dependence of permittivity on deformation strongly affected the failure mode and the maximum achievable total stretch in the active region. Depending on the values of the electrostrictive factor c and the pre- stretch, either pull-in instability, loss of tension, or dielectric breakdown may occur. At low pre-stretch levels, the dependence of permittivity on deformation was found to suppress the electromechanical instability that enabled the actuator to achieve a higher total stretch.

At high pre-stretch levels, however, as the cause of failure was due to electric breakdown, the dependence of permittivity on deformation significantly lowered the maximum achievable total stretch.

Compared to the case of constant permittivity, the dependence of permittivity on deformation resulted in higher voltage needed to achieve the same actuated stretch, and this discrepancy in voltage increased as the pre-stretch increased. Furthermore, it is found that the maximum achievable stretch and the corresponding voltage showed a strong dependence on the electrostrictive factor c, while the stretch at the onset of loss of tension was independent of c.

Although a circular actuator was considered in this study, where the Gent model was used to describe the mechanical behavior of the material and the linear function proposed by Zhao and Suo [64] was used describe the dependency of permittivity on deformation, similar analyses can be done for other geometries and material models. The analysis results can provide useful insights in material selection and design of dielectric elastomer transducers.

CHAPTER IV

STABILITY OF AN ANISOTROPIC DIELECTRIC ELASTOMER PLATE

Based on experimental studies in the literatures, a variety of methods have been proposed to improve the performance of dielectric elastomer transducers. However, many of these methods leaded to the anisotropic electromechanical behavior of the material. This chapter theoretically analyzed the effect of material anisotropy on the diffuse modes of instability in a pre-stretched dielectric elastomer plate under electrical field with plate thickness being included in the model. The general incremental formulation provided by

Dorfmann and Ogden was adopted and extended to anisotropic application. The stability behavior of the anisotropic DE plate was compared to its isotropic counterpart.

4.1 Material Anisotropy of the Dielectric Elastomer

A typical DE transducer configuration usually involves a DE thin film coated with electrodes. When applied voltage increases, the DE film thins down in thickness and, at some point, goes through a pull-in instability (PI) [77]. PI usually leads to failure of the actuator before the maximum large deformation can be achieved, which has become a limitation to DE’s applications. Pre-stretching DE before actuation is a common way to suppress PI and to enhance dielectric strength [10]. To avoid external prestrain-supporting

52 structures, interpenetrating polymer networks were developed to retain the performance benefits from prestrain without mechanically prestraining the material [78][79]. The interpenetrating networks result in anisotropy of the elastomer films. Many other innovative methods to improve the performance also introduce anisotropy into DE materials. Actuators that are made of fiber-stiffened elastomeric films can produce large unidirectional deformations with or without prestrain [80][81][82][83]. By dispersing high dielectric constant inclusions in DE matrix, the permittivity of the composite is increased.

This leads to several times larger actuated strain under the same driving electric field

[84][85]. High permittivity can also be achieved by creating a three-component dielectric- percolative composite [86]. Large deformation under lower electric field can be achieved with heterogeneous actuators [87].

Since DE films turn out to be anisotropic under applications of the above methods, the effect of material anisotropy on the instability of the DE composite solids in such situations has become an important topic in analytical [88][89][90][91] and numerical

[92][93][87] studies. In particular, the instability of a DE plate has been theoretically studied by researchers from different perspectives and using different methods. For example, Yong et al. analyzed two different electromechanical instabilities in anisotropic

DE films [94]. Rudykh and deBotton investigated the stable domain of anisotropic electroactive layered composites [95]. Xiao et al. developed a constitutive model to analyze the PI of fiber-reinforced DE membranes without in-plane loading [96]. So far in literature, to simplify the derivation of the problem, the plate thickness has not been considered when studying the effect of anisotropy on the bifurcation instability of a DE plate. However, studies have found that plate thickness affects the stability of the DE plate [97]. Moreover,

53 pre-stretch is a frequently used technique and should also be added to the model. In the present study, the plate thickness and its effect are included in the stability analysis of a pre-stretched anisotropic DE plate. We give predictions of the onset and wavelength of the bifurcation instability, which have not been presented before for an anisotropic DE plate.

4.2 The Incremental Theory

In order to study the electromechanical instabilities of DEs, many theoretical methods have been developed [98][99][100][101][102]. Dorfmann and Ogden proposed a general method to analyze the diffuse modes of instability for electroelastic solids

[32][97][103]. The advantage of this method is that, by using the linearized incremental deformation and electric displacement, the plate thickness and non-homogeneous deformation can be considered in the stability analysis. This theory is adopted in the present study.

4.3 Applying the Incremental Theory to an Anisotropic DE Plate

Based on their nonlinear theory of electroelasticity [104][105], Dorfmann and

Ogden proposed a general incremental formulation to analyze the bifurcation instability of electroelastic solids, and then specialized the formulation to isotropic materials

[32][97][103]. In the present study, the incremental governing equations and the corresponding boundary conditions from their work are adopted and specialized to an anisotropic DE plate. The reader is referred to [97] for details of the general formulation.

In this section, the stability problem of a pre-stretched anisotropic plate is formulated. The

54 bifurcation equation is obtained from the resulted linear system, and the bifurcation instability curves are obtained from the bifurcation equation.

A rectangular plate is defined as a set of particles, where vector 푿 identifies the position of an arbitrary particle in the reference configuration and 풙 is the corresponding position vector in the current configuration. The deformation gradient tensor is defined as

푭 = 휕풙⁄휕 푿. The right Cauchy-Green deformation tensor 퐂 is defined as 퐂 = 푭푇푭. The first three invariants are given as

퐼1 = tr 퐂 (4.1)

1 퐼 = [(tr 퐂)2 − tr 퐂2] (4.2) 2 2

퐼3 = det 퐂 (4.3)

The transversely isotropic material is considered in the present study. 푨 is defined as a unit vector that represents the fiber direction in the reference configuration. The material is assumed to be transversely isotropic about the 푨 direction. In general, two pseudo-invariants are used to describe the effect of fiber reinforcement in purely mechanical case. Based on the findings in [92] (section 5.1 of [92]), where the authors pointed out that “From Fig. 3b, it is evident that the electromechanical response is independent of the parameter ξ2.”, where ξ2 is the parameter of I5 in [92]. Therefore, the electromechanical response is assumed to be only dependent on one of the pseudo- invariants [92]. Only pseudo-invariant 퐼4 is used in this study and is defined as

퐼4 = 푨 ∙ 퐂푨 (4.4)

Two invariants that are related to the electric field are defined as

퐼5 = 퐃L ∙ (퐂퐃L) (4.5)

퐼6 = 퐃L ∙ 퐃L (4.6)

where 퐃L is the Lagrangian electric displacement.

4.3.1 Geometry

Figure 4.1 Dimensions of the dielectric elastomer plate in (a) undeformed state and (b)

deformed state.

Figure 4.1 shows a rectangular DE plate in undeformed and deformed states. The dimensions in the reference configuration are defined as

−퐿1 ≤ 푋1 ≤ 퐿1 (4.7)

0 ≤ 푋2 ≤ 퐻 (4.8)

−퐿3 ≤ 푋3 ≤ 퐿3 (4.9)

56 where 퐻 is the initial thickness of the plate. Here (X1, X2, X3) represents a material point in the reference, undeformed configuration and (x1, x2, x3) represents its position in the current, deformed configuration. The plate is coated with flexible electrodes on 푥2 = 0 and 푥2 = ℎ surfaces. Charge control is used in the present case. Charges ±휎푓 per unit deformed area are assumed to be on each electrode. Another way to actuate the plate is by voltage control, which will be discussed in a future study. For the case of voltage control, the 휆푐푟 versus

푘퐻 curves will look different from those of charge control because the incremental boundary conditions are different. The external electric field is considered in voltage control. Based on the study of an isotropic electroelastic plate in [97], the 휆푐푟 versus 푘퐻 curves and the unstable regions are different for charge control and voltage control. One similarity for both cases is that the influence of the electric field (퐷̂2) in the material reverses at a critical 퐷̂2 as 퐷̂2 increases. The case of voltage control is a good point worth studying for anisotropic materials.

Assume the plate is subjected to principal stretches 휆1, 휆2, 휆3. The deformation gradient tensor and the right Cauchy-Green deformation tensor become

휆1 푭 = [ 휆2 ] (4.10) 휆3

2 휆1 2 퐂 = [ 휆2 ] (4.11) 2 휆3

In the present study, the DE plate is assumed to be transversely isotropic about the

푋2 direction, thus 푨 becomes {0,1,0}. In general, the material can be transversely isotropic about other directions as well. For example, [88] considered a special case where the

57 direction of A is perpendicular to the thickness direction. A detailed study of the effect of the direction of 푨 can be conducted in the future work. Material incompressibility is considered in this study. Then the invariants become

2 2 2 퐼1 = 휆1 + 휆2 + 휆3 (4.12)

2 2 2 2 2 2 퐼2 = 휆2휆3 + 휆3휆1 + 휆1휆2 (4.13)

퐼3 = 1 (4.14)

2 퐼4 = 퐶22 = 휆2 (4.15)

2 2 퐼5 = 휆2퐷퐿2 (4.16)

2 퐼6 = 퐷퐿2 (4.17)

4.3.2 Constitutive Model

A general form of the free energy density 푊 of the DE plate can be written as

푊(퐼1, 퐼2, 퐼4, 퐼5, 퐼6). The free energy density of an anisotropic DE consists of three parts

[94][92][88]

푖푠표 푎푛푖 푒푙 푊 = 푊 (퐼1, 퐼2) + 푊 (퐼4) + 푊 (퐼5, 퐼6) (4.18)

In this work, the material anisotropy is assumed to affect the electromechanical response through the anisotropic term in the free energy density. The anisotropic term is similar to that of the purely mechanical case, where the anisotropic term is assumed to be only associated with fiber stretch. The electric part in the free energy density is assumed to be independent of the mechanical anisotropy. This is a simplification as an early step to analytically explore material anisotropy of DEs. This assumption can also be found in

58 previous studies such as [88], [92], [94]. With more evidence from experiments in the future, the assumption can be adjusted and refined based on certain practical observations.

The base material is assumed to be a neo-Hookean dielectric, and the isotropic part

푊푖푠표 is written as

휇 푊푖푠표 = (퐼 − 3) (4.19) 2 1 where 휇 is the shear modulus.

The electrical part 푊푒푙 proposed by Dorfmann and Ogden [97] is adopted as

1 푊푒푙 = (훼퐼 + 훽퐼 ) (4.20) 2휀 6 5 where 훼 and 훽 are two dimensionless material constants, and 휀 is the vacuum permittivity.

The anisotropic part 푊푎푛푖 is written as

휇 푊푎푛푖 = 휉(퐼 − 1)2 (4.21) 2 4 where 휉 is a material constant that represents the strength of reinforcement in the fiber direction [106].

Then the free energy density of the anisotropic neo-Hookean dielectric can be expressed as

휇 휇 1 푊 = (퐼 − 3) + 휉(퐼 − 1)2 + (훼퐼 + 훽퐼 ) (4.22) 2 1 2 4 2휀 6 5

4.3.3 Incremental Formulation and Bifurcation Criterion

As in [97], equibiaxial stretch in 푋1 and 푋3 directions are considered in the

−2 following analysis. Set 휆1 = 휆3 = 휆, thus 휆2 = 휆 , and the invariants become

2 −4 퐼1 = 2휆 + 휆 (4.23)

−2 4 퐼2 = 2휆 + 휆 (4.24)

−4 퐼4 = 휆 (4.25)

−4 2 퐼5 = 휆 퐷퐿2 (4.26)

2 퐼6 = 퐷퐿2 (4.27)

The free energy density of the transversely isotropic rectangular DE plate under equibiaxial deformation become

휇 휇 (훼+훽휆−4)퐷2 푊 = (2휆2 + 휆−4 − 3) + 휉(휆−8 − 2휆−4 + 1) + 퐿2 (4.28) 2 2 2휀

According to [32], the electroelastic moduli tensors are defined by

휕2푊 𝒜훼푖훽푗 = (4.29) 휕퐹푖훼휕퐹푗훽

휕2푊 픸훼푖훽 = (4.30) 휕퐹푖훼휕퐷퐿훽

휕2푊 퐴훼훽 = (4.31) 휕퐷퐿훼휕퐷퐿훽

The electroelastic moduli tensors can be rewritten for transversely isotropic materials as

2 5 5 휕퐼푚 휕퐼푛 5 휕 퐼푛 𝒜훼푖훽푗 = ∑푚=1 ∑푛=1 푊푚푛 + ∑푛=1 푊푛 (4.32) 휕퐹푖훼 휕퐹푗훽 휕퐹푖훼휕퐹푗훽

2 6 6 휕퐼푚 휕퐼푛 5 휕 퐼푛 픸훼푖훽 = ∑푚=5 ∑푛=5 푊푚푛 + ∑푛=5 푊푛 (4.33) 휕퐹푖훼 휕퐷퐿훽 휕퐹푖훼휕퐷퐿훽

2 6 6 휕퐼푚 휕퐼푛 6 휕 퐼푛 퐴훼훽 = ∑푚=5 ∑푛=5 푊푚푛 + ∑푛=5 푊푛 (4.34) 휕퐷퐿훼 휕퐷퐿훽 휕퐷퐿훼휕퐷퐿훽

2 where 푊푛 = 휕푊⁄휕퐼푛 , 푊푚푛 = 휕 푊⁄휕퐼푚휕퐼푛 and various derivatives of 퐼푛 are listed below.

The component form of the first derivatives of the invariants with respect to 푭 and

퐃L respectively:

휕퐼1 = 2퐹푖훼 (4.35) 휕퐹푖훼

휕퐼2 = 2(푐훾훾퐹푖훼 − 푐훼훾퐹푖훾) (4.36) 휕퐹푖훼

휕퐼3 −1 = 2퐼3퐹훼푖 (4.37) 휕퐹푖훼

휕퐼4 = 푎훼(퐹푖푘푎푘 − 퐹푖푚푎푚) (4.38) 휕퐹푖훼

휕퐼5 = 2퐷퐿훼퐹푖훾퐷퐿훾 (4.39) 휕퐹푖훼

휕퐼 6 = 0 (4.40) 휕퐹푖훼

휕퐼5 = 2푐훼훽퐷퐿훽 (4.41) 휕퐷퐿훼

휕퐼6 = 2퐷퐿훼 (4.42) 휕퐷퐿훼

The component form of the second derivatives of the invariants with respect to 푭:

2 휕 퐼1 = 2훿푖푗훿훼훽 (4.43) 휕퐹푖훼휕퐹푗훽

2 휕 퐼2 = 2(2퐹푖훼퐹푗훽 − 퐹푖훽퐹푗훼 + 푐훾훾훿푖푗훿훼훽 − 푏푖푗훿훼훽 − 푐훼훽훿푖푗) (4.44) 휕퐹푖훼휕퐹푗훽

2 휕 퐼3 −1 −1 −1 −1 = 2퐼3(2퐹훼푖 퐹훽푗 − 퐹훼푗 퐹훽푖 ) (4.45) 휕퐹푖훼휕퐹푗훽

2 휕 퐼4 = 2푎훼푎훽훿푖푗 (4.46) 휕퐹푖훼휕퐹푗훽

2 휕 퐼5 = 2훿푖푗퐷퐿훼퐷퐿훽 (4.47) 휕퐹푖훼휕퐹푗훽

휕2퐼 6 = 0 (4.48) 휕퐹푖훼휕퐹푗훽

The component form of the second derivatives of the invariants with respect to 퐃L:

2 휕 퐼5 = 2푐훼훽 (4.49) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼6 = 2훿훼훽 (4.50) 휕퐷퐿훼휕퐷퐿훽

The component form of the mixed derivatives of the invariants with respect to 푭 and 퐃L:

2 휕 퐼5 = 2훿훼훽퐹푖훾퐷퐿훾 + 2퐷퐿훼퐹푖훽 (4.51) 휕퐹푖훼휕퐷퐿훽

Following [97], let 풖 = 풙̇ be the incremental displacement field and assume it is plane strain. Thus the out of plane component 푢3 is 0, and the in-plane components 푢1 and

푢2 depend only on 푥1 and 푥2 . Let 휓 = 휓(푥1, 푥2) such that 푢1 = 휓,2 and 푢2 = −휓,1 .

Similarly, let the incremental electric displacement field in the current configuration be

퐃̇ L0 and assume the out of plane component 퐷̇ 퐿03 is 0. Let 휑 = 휑(푥1, 푥2) such that 퐷̇ 퐿01 =

휑,2 and 퐷̇ 퐿02 = −휑,1.

The equilibrium equations can be expressed as

푎휓,1111 + 2푏휓,1122 + 푐휓,2222 + (푒 − 푑)휑,112 + 푑휑,222 = 0 (4.52)

푑휓,222 + (푒 − 푑)휓,112 + 푓휑,22 + 푔휑,11 = 0 (4.53) where the material coefficients for transversely isotropic neo-Hookean dielectrics are as follows

푎 = 휇휆2 (4.54)

훽퐷2 2푏 = 휇휆2 + μ휆−4 + 2 + 6휇ξ휆−8 (4.55) 휀

훽퐷2 푐 = μ휆−4 + 2 + 2μξ휆−8 (4.56) 휀

훽퐷 푑 = 2 (4.57) 휀

2훽퐷 푒 = 2 (4.58) 휀

1 푓 = (훼휆−2 + 훽) (4.59) 휀

1 푔 = (훼휆4 + 훽) (4.60) 휀

−2 with 퐷2 = 휆 퐷퐿2.

To solve the incremental boundary-value problem, let

휓 = 퐴푒−푘푠푥2푒푖푘푥1 (4.61)

휑 = 푘퐵푒−푘푠푥2푒푖푘푥1 (4.62) where 푖 = √−1, 푘 > 0 is the wavenumber of the perturbation, and 푠 is to be solved [97].

For a non-trivial solution of the equilibrium equations to exist, the following bi-cubic equation must hold

(푠2 − 1)[(휆−6 + 2휉휆−10)푠4 − (1 + 휆−6 + 6휉휆−10)푠2 + 1] = 0 (4.63)

The solutions to this equation are given as

푠1 = −푠4 = 1 (4.64)

1⁄2 (1+휆−6+6휉휆−10)+(휆−12+36휉2휆−20+1+12휉휆−16−2휆−6+4휉휆−10)1⁄2 푠 = −푠 = { } (4.65) 2 5 2(휆−6+2휉휆−10)

1⁄2 1⁄2 (1+휆−6+6휉휆−10)−(휆−12+36휉2휆−20+1+12휉휆−16−2휆−6+4휉휆−10) 푠 = −푠 = { } (4.66) 3 6 2(휆−6+2휉휆−10)

Therefore,

6 −푘푠푗푥2 푖푘푥1 휓 = ∑푗=1 퐴푗푒 푒 (4.67)

6 −푘푠푗푥2 푖푘푥1 휑 = 푘 ∑푗=1 퐵푗푒 푒 (4.68)

where 퐴푗 and 퐵푗 are unknown constants. Since there is no incremental mechanical traction on 푥2 = 0 and 푥2 = ℎ, the incremental traction and electric boundary conditions [97] are expressed as

푥2 = 0 6 2 ∑푗=1[푐(1 + 푠푗 )퐴푗 − 푑푠푗퐵푗] = 0, (4.69)

6 2 2 ∑푗=1[(2푏 + 푐 − 푐푠푗 )푠푗퐴푗 − (푒 − 푑푠푗 )퐵푗] = 0, (4.70)

6 2 ∑푗=1[푑(1 + 푠푗 )퐴푗 − 푓푠푗퐵푗] = 0, (4.71)

푥2 = ℎ

6 2 −푘ℎ푠푗 ∑푗=1[푐(1 + 푠푗 )퐴푗 − 푑푠푗퐵푗] 푒 = 0, (4.72)

6 2 2 −푘ℎ푠푗 ∑푗=1[(2푏 + 푐 − 푐푠푗 )푠푗퐴푗 − (푒 − 푑푠푗 )퐵푗] 푒 = 0, (4.73)

6 2 −푘ℎ푠푗 ∑푗=1[푑(1 + 푠푗 )퐴푗 − 푓푠푗퐵푗] 푒 = 0, (4.74)

The incremental traction and electric boundary conditions take the following forms for the transversely isotropic DE plate

On 푥2 = 0:

6 −4 ̂2 −8 2 ̂ ̂ ∑푗=1[(휆 + 퐷2 + 2ξ휆 )(1 + 푠푗 )퐴푗 − 퐷2푠푗퐵푗] = 0 (4.75)

6 2 −4 ̂2 −8 −4 ̂2 −8 2 ∑푗=1[(휆 + 2휆 + 2퐷2 + 6ξ휆 − (휆 + 퐷2 + 2ξ휆 )푠푗 )푠푗퐴푗 −

̂ 2 ̂ 퐷2(2 − 푠푗 )퐵푗] = 0 (4.76)

6 ̂ 2 ̂ ∑푗=1[퐷2(1 + 푠푗 )퐴푗 − 푠푗퐵푗] = 0 (4.77) where 퐷̂2 = 퐷2⁄√μ휀, 퐵̂푗 = 퐵푗⁄√μ휀.

On 푥2 = ℎ:

6 −4 ̂2 −8 2 ̂ ̂ −푘ℎ푠푗 ∑푗=1[(휆 + 퐷2 + 2ξ휆 )(1 + 푠푗 )퐴푗 − 퐷2푠푗퐵푗]푒 = 0 (4.78)

6 2 −4 ̂2 −8 −4 ̂2 −8 2 ∑푗=1[(휆 + 2휆 + 2퐷2 + 6ξ휆 − (휆 + 퐷2 + 2ξ휆 )푠푗 )푠푗퐴푗 −

̂ 2 ̂ −푘ℎ푠푗 퐷2(2 − 푠푗 )퐵푗] 푒 = 0 (4.79)

6 ̂ 2 ̂ −푘ℎ푠푗 ∑푗=1[퐷2(1 + 푠푗 )퐴푗 − 푠푗퐵푗] 푒 = 0 (4.80) where ℎ = 퐻휆−2.

Constants 퐴푗 and 퐵푗 are not independent and related through the equilibrium equations [97]. Specialization to the current case is obtained as

퐵̂푗 = 0, 푗 = 1, 4 (4.81)

퐵̂푗 = 푠푗퐷̂2퐴푗, 푗 = 2, 3, 5, 6 (4.82)

Substituting Equations (4.64)-(4.66) and Equation (4.81)-(4.82) into Equations (4.75)-

(4.80), a system of 6×6 linear equations is obtained. The bifurcation equation can be obtained by setting the determinant of the coefficients of the linear system to zero, which marks the onset of the unstable state of the plate. An implicit relation between the lateral critical strength, 휆푐푟 , and 푘퐻 can then be obtained, and the results are calculated and plotted using Wolfram Mathematica [107].

4.4 Results and Discussions

In this section, the effect of anisotropy on the stability of the plate was analyzed with comparison to the isotropic cases.

In the following analysis, the dependence of permittivity on deformation [108] was not considered. The electroelastic material parameters were set as 훼 = 0, 훽 = 0.5.

Figure 4.2 displays the lateral critical stretch 휆푐푟 as a function of 푘퐻 for dimensionless electric displacement 퐷̂2 = 0, 1, 2, 2.2, 2.4, 2.6, 3, 4 and 5 respectively, where the solid line represented the transversely isotropic material and the dashed line represented the isotropic material. For the transversely isotropic material, the anisotropy parameter ξ = 5 was used. It can be seen that the bifurcation curves of the two materials were quite different under each electric field. The values of 휆푐푟 and 푘퐻 in the region enclosed by the upper curve and the lower curve for each case leaded to determinant < 0, which means the plate was unstable. In general, the unstable region of the transversely isotropic material was larger than that of the isotropic material under the same 퐷̂2.

Figure 4.2 Comparisons of 휆푐푟 versus 푘퐻 for isotropic and anisotropic materials with

values of 퐷̂2 = 0, 1, 2, 2.2, 2.4, 2.6, 3, 4 and 5, (a)-(i), respectively.

We measured the area of the unstable region for each case using AutoCAD [109], the area measurement data was in Figure 4.3. For each case in Figures 4.2(a)-(c) the unstable region enclosed by the upper curve and the lower curve became larger as 퐷̂2

67 increased, indicating a destabilizing effect of the electric field. The upper curve started at a higher value from 푘퐻 close to 0 for the transversely isotropic material, which means a higher lateral stretch was needed to prevent the plate from instability. The discrepancy between the upper curves for the two materials decreased as 퐷̂2 increased. For example,

Figure 4.4 listed the critical stretch 휆푐푟 above which the plate could attain stability with

푘퐻 → 0 for cases in Figures 4.2(a)-(c). Figures 4.2(a)-(c) also showed that as 푘퐻 increased, the unstable range of 휆푐푟 reduced for both materials, indicating a thick plate was more stable.

Figure 4.3 The values of the area of the unstable region for each case measured with

AutoCAD.

Figure 4.4 Comparisons of 휆푐푟 for isotropic and anisotropic materials as 푘퐻 → 0.

In Figures 4.2(d)-(i), for the isotropic material, as 퐷̂2 increased, the bifurcation curves changed into different shapes and the unstable region became smaller, indicating a stabilizing effect of the electric field. However, for the transversely isotropic material, the change occurred at a higher 퐷̂2, which can be seen in case (f). The discrepancy between the upper curves for the two materials decreased further as 퐷̂2 increased. But the discrepancy between the lower curves remained significant.

Figure 4.5 showed 휆푐푟 as a function of 푘퐻 under different levels of anisotropy, ξ =

0, 3, 5, 10 in the thickness direction, when 퐷̂2 = 2. ξ = 0 corresponded to the isotropic case. As the anisotropy parameter increased, the same 푘퐻 corresponded to larger values of

휆푐푟 on the upper and lower curves, and the unstable region became larger and moved upward. This means that anisotropy increased the risk of bifurcation instability, especially for thin plates. Similar conclusions can be drawn under larger electric field.

Figure 4.5 Comparisons of 휆푐푟 versus 푘퐻 for different degrees of anisotropy, ξ =

0, 3, 5, 10, where 퐷̂2 = 2.

Figure 4.6 displays the values of 퐷̂2, at which the electric field started to stabilize the plate, as a function of anisotropy parameter for ξ = 0, 3, 5, 10 [92]. As the anisotropy parameter increased, the critical electric displacement, at which the influence of the electric field transformed from destabilizing to stabilizing, increased.

Figure 4.6 The critical electric displacement versus the anisotropic parameter ξ.

4.5 Summary

In this chapter we studied the effect of material anisotropy on the bifurcation instabilities of DE plates using a linearized incremental theory, where the plate thickness was considered in the stability analysis. As a first step, a transversely isotropic material was considered in the present study. By comparing with the isotropic case, we showed that material anisotropy had significant effect on the bifurcation instabilities.

For the models considered in this study, material anisotropy resulted in larger unstable regions and higher critical stretch, above which the plate was stable. Increasing from zero, the electric field first had a destabilizing effect, and it then transformed to a stabilizing effect for both isotropic and transversely isotropic materials. However, the electric displacement 퐷̂2 , at which this transformation took place, increased as the

72 anisotropy parameter ξ increased. It also showed that material anisotropy had a stronger influence for a thinner plate. As the plate thickness increased, the influence decreased.

Future research may consider more complex anisotropic material models to study the effect of material anisotropy on bifurcation instabilities of DEs.

The work in this chapter was inspired by various experimental methods, such as

[78] [81] [85], that imparted material anisotropy to improve the actuated strain of the DE materials. The focus of these experimental work was to achieve larger deformation under lower electric field. But bifurcation instability and wrinkles can be a failure mode in the actuation of anisotropic DE material that had not been addressed in detail in experimental studies. We hope our analytical study could give some predictions for this issue that has not been emphasized by experiments.

CHAPTER V

FINITE ELEMENT ANALYSIS OF DIELECTRIC ELASTOMERIC MEMBRANES AS BUCKLING ACTUATORS

Dielectric elastomer is a promising type of smart material that has potential applications in many areas. One of those applications is in the area of the buckling actuators.

Under the influence of electric field, a flat dielectric elastomer membrane will buckle out of plane, deform in the out-of-plane direction and form a concave or convex shape. This kind of model can be used as buckling actuators/sensors. In this chapter, finite element method was used to present a numerical model of a basic buckling actuator made of dielectric elastomer. A user element subroutine [1] in Abaqus was used to simulate the electromechanical coupling behavior of the material. Different material models were programmed into the user element subroutine. The effect of a few model parameters on the performance of the buckling actuators was discussed.

5.1 Buckling Actuators Based on Dielectric Elastomer

A dielectric elastomer buckling actuator is a specific type of electromechanical transducers. It consists of a flat circular elastomeric membrane with compliant electrodes coated on both surfaces and is designed to operate with out-of-plane unidirectional displacements [4]. Figure 5.1 shows the lateral view of a buckling actuator in the rest state

74 and actuated state from an experiment [4]. The working principle is that by applying electric potential difference on the electrodes, the membrane will expand in area. As voltage increases, the material will finally buckle out-of-plane and produce vertical displacement since the edge of the membrane is fixed [110].

Figure 5.1 Lateral view of a buckling actuator: (a) rest state and (b) actuated state. [4]

The advantage of the buckling actuators made of dielectric elastomer is that large out-of-plane deformation can be achieved which enables their potential applications as soft tunable lenses, loud-speakers, and pumps.

The buckling actuator investigated in the present study is prepared by the following steps. Step 1: The thin flat elastomeric membrane is fixed onto a rigid ring without pre- stretch; Step 2: Compliant electrodes are applied on both surfaces of the membrane; Step

3: Voltage is applied on the membrane to actuate the material. When voltage is applied, the membrane will expand in-plane. Since the edge of the membrane is fixed, as voltage increases, the material will buckle out-of-plane and the membrane deforms in the vertical direction. As voltage further increases to sufficient high value, instability occurs and the material fails.

5.2 The Finite Element Modeling Method

As described in Chapter II, there are several theoretical methods proposed to study the electromechanical coupling behavior of dielectric elastomers. However, numerical tools are still needed to study the materials under large three-dimensional inhomogeneous deformations.

5.2.1 Electromechanical Modeling Method

Several research groups have made great efforts in developing the numerical tools to study the dielectric elastomers. Among them, Henann et al. [47] developed a three- dimensional, fully coupled theory that governs the electromechanical behavior of the dielectric elastomers and implemented the theory in Abaqus by programming a user element subroutine. In their code, an additional nodal degree of freedom is added to the user element to take the electric potential into account. In the present study, we make use of their user element subroutine as a tool to simulate the buckling actuators. We also programmed our material models into the user element subroutine to compare the difference between the material models.

5.2.2 Constitutive Equations

The free energy density of the material is assumed to have a mechanical part and an electrical part

푚푒푐ℎ 푒푙푒 푊 = 푊 (퐼1̅ , 퐼2̅ , 퐽) + 푊 (퐼4, 퐼5, 퐼6, 퐽) (5.1)

The invariants are given as

퐼1̅ = 푡푟(퐂dis) (5.2)

̅ 2 퐼2 = 푡푟(퐂dis) (5.3)

퐽 = det (푭) (5.4)

퐼4 = 퐃L ∙ 퐃L (5.5)

퐼5 = 퐃L ∙ 퐂퐃L (5.6)

ퟐ 퐼6 = 퐃L ∙ 퐂 퐃L (5.7) where 푭 is the deformation gradient tensor, 퐂 is the right Cauchy-Green deformation tensor, 퐃L is the referential electric displacement. 퐂dis is the distortional right Cauchy-

Green deformation tensor

−2⁄3 퐂dis = 퐽 퐂 (5.8)

푒푙푒 The electrical part 푊 (퐼4, 퐼5, 퐼6, 퐽) is assumed to be the “ideal dielectric elastomer” and takes the following form

1 푊푒푙푒 = 퐽−1퐼 (5.9) 2ε 5 where ε is the dielectric permittivity of the dielectric elastomer.

Following [47], for the stresses, the second Piola stress is defined by

∂푊 푺 = 2 (5.10) ∂퐂

The first Piola stress is given by

푻L = 푭푺 (5.11)

The Cauchy stress is written as

훔 = 퐽−1 푭푺푭T (5.12) The electric field in the reference configuration is defined by

∂푊 퐄L = (5.13) ∂퐃L

The electric field in the current configuration is defined by

−T 퐄 = 푭 퐄L (5.14)

5.2.2.1 The neo-Hookean model

In this section, the mechanical part 푊푚푒푐ℎ of the free energy density is assumed to be a neo-Hookean dielectric and is written as

푚푒푐ℎ 1 2 푊 = 퐶10(퐼1̅ − 3) + (퐽 − 1) (5.15) 퐷1 where 퐶10 and 퐷1 are material parameters. 퐶10 and 퐷1 are defined by

휇 퐶 = (5.16) 10 2

2 퐷 = (5.17) 1 푘 where 휇 is the shear modulus and 푘 is the bulk modulus of the material.

For the stresses, the second Piola stress is written as

2 − 1 1 1 1 3 ̅ −1 −T −1 −1 −1 푺 = 2 (퐶10퐽 (푰 − 퐼1퐂dis) + 퐽(퐽 − 1)퐂 ) − 퐽 퐂 (퐃L ∙ 퐂퐃L) + 퐽 퐃L⨂퐃L 3 퐷1 2ε ε

(5.18)

The first Piola stress is given by

2 − 1 −T 1 −T 1 −T 1 푻L = 2 (퐶10퐽 3 (푭 − (tr퐂)푭 ) + 퐽(퐽 − 1)푭 ) − 퐽(퐃 ∙ 퐃)푭 + 퐃⨂퐃L (5.19) 3 퐷1 2ε ε

The Cauchy stress is written as

2 1 2 1 1 훔 = 퐶10 (퐁dis − (푡푟퐁dis)) + (퐽 − 1)푰 − (퐃 ∙ 퐃) + 퐃⨂퐃 (5.20) 퐽 3 퐷1 2ε ε The material tangents in indicial forms for the neo-Hookean dielectric are as follows

푚푒푐ℎ 2 ∂푇L,푖푗 − 2 −1 2 −1 −1 2 −1 1 −1 −1 = 2퐶10퐽 3(− 퐹푙푘 퐹푖푗 + 퐹푙푘 푡푟푐퐹푗푖 + δ푖푘δ푗푙 − 퐹푘푙퐹푗푖 + 푡푟푐퐹푗푘 퐹푙푖 ) (5.21) ∂퐹푘푙 3 9 3 3

푒푙푒푐 ∂푇L,푖푗 −1 −1 −1 −1 −1 −1 −1 = 휀퐽(−퐹푙푖 퐶푗푚 EL,푚E푘 − E푖퐹푗푘 퐶푙푚 EL,푚 − E푖E푘퐶푗푙 + 퐹푗푖 E푘퐶푙푚 EL,푚 + ∂퐹푘푙

1 1 E 퐶−1E 퐹−1 + E E 퐹−1퐹−1 − E E 퐹−1퐹−1) (5.22) 푖 푗푚 L,푚 푙푘 2 푚 푚 푙푖 푗푘 2 푚 푚 푗푖 푙푘

푒푙푒푐 ∂푇L,푖푗 −1 −1 −1 −1 −1 = 휀퐽(퐹푙푖 퐶푗푚 EL,푚 + 퐶푗푙 E푖 − 퐹푗푖 퐶푙푚 EL,푚) (5.23) ∂EL,푙

∂퐷퐿,푗 −1 −1 −1 −1 −1 −1 = 휀퐽(퐹푙푘 퐶푗푚 EL,푚 − 퐹푗푘 퐶푙푚 EL,푚 − 퐹푚푘퐶푗푙 EL,푚) (5.24) ∂퐹푘푙

∂퐷퐿,푗 −1 = 휀퐽퐶푗푙 (5.25) ∂EL,푙

5.2.2.2 The Ogden model

In this section, the mechanical part 푊푚푒푐ℎ of the free energy density is assumed to be an Ogden [111][112][113] dielectric and can be written as

푚푒푐ℎ 푁 μ푖 훼̅ 푖 ̅훼푖 ̅훼푖 푊 (λ1, λ2, λ3) = ∑푖=1 2 2 (휆1 + 휆2 + 휆3 − 3) (5.26) 훼푖 where μ푖 and 훼푖 are material parameters and can be measured in experiments, λ푖(푖 =

1, 2, 3) are principal stretches. The electrical part 푊푒푙푒 remains the same as the neo-

Hookean dielectric.

For the stresses, the principal Kirchhoff stress is written as

1 푁 μ푝 − 훼푝 훼푝 1 훼푝 훼푝 훼푝 τ푖 = ∑푝=1 2 퐽 3 (휆 − (휆 + 휆 + 휆 )) + 푘퐽(퐽 − 1) (5.27) 훼푝 푖 3 1 2 3 where 푖 = 1, 2, 3.

There are several methods to derive the material tangents, such as the eigenvalue- bases method, the eigenvector method, and the invariants method. In this work, the eigenvalue-bases method is used.

The eigenvalue derivatives are expressed as

1 ∂τ푖 ∂τ푖 ∂λ푗 μ푝 − 훼 1 훼푝 훼푝 훼푝 훼푝 훼푝 훼푝 푘(2퐽−1)퐽 푁 3 푝 = = ∑푝=1 2 퐽 ( (휆1 + 휆2 + 휆3 ) − 휆푖 − 휆푗 + 3휆푖 δ푖푗) + 2 ∂b푗 ∂λ푗 ∂b푗 3휆푗 3 2휆푗

(5.28) where b푗 (푗 = 1, 2, 3) are the eigenvalues of the left Cauchy-Green deformation tensor 퐁.

The eigenvalue derivatives are needed to compute the isotropic tensor function derivative

∂훕 ∂τ . The indicial form of the isotropic tensor function derivative 푖푚 is required for the ∂퐁 ∂B푝푞

∂푇푚푒푐ℎ computation of the indicial spatial elasticity tensor L,푖푗 . ∂F푘푙

The Kirchhoff stress is given by

푁 훕 = ∑푖=1 √b푖퐐푖 (5.29) where N is the number of distinct eigenvalues, 퐐푖 (푖 = 1, 2, 3) is the eigenprojections of 퐁.

The first Piola stress is given by

1 1 푻 = 훕푭−T − 퐽(퐃 ∙ 퐃)푭−T + 퐃⨂퐃 (5.30) L 2ε ε L

The Cauchy stress is computed as

−1 T 훔 = 퐽 푻L푭 (5.31) The electro-elastic material tangents in indicial forms for the Ogden dielectric are derived as follows

푚푒푐ℎ ∂푇L,푖푗 ∂τ푖푚 −1 −1 −1 = (δ푝푘F푞푙 + F푝푙δ푞푘)퐹푗푚 − τ푖푚퐹푗푘 퐹푙푚 (5.32) ∂F푘푙 ∂B푝푞

푒푙푒푐 ∂푇L,푖푗 −1 −1 −1 −1 −1 −1 −1 = 휀퐽(−퐹푙푖 퐶푗푚 EL,푚E푘 − E푖퐹푗푘 퐶푙푚 EL,푚 − E푖E푘퐶푗푙 + 퐹푗푖 E푘퐶푙푚 EL,푚 + ∂F푘푙

1 1 E 퐶−1E 퐹−1 + E E 퐹−1퐹−1 − E E 퐹−1퐹−1) (5.33) 푖 푗푚 L,푚 푙푘 2 푚 푚 푙푖 푗푘 2 푚 푚 푗푖 푙푘

푒푙푒푐 ∂푇L,푖푗 −1 −1 −1 −1 −1 = 휀퐽(퐹푙푖 퐶푗푚 EL,푚 + 퐶푗푙 E푖 − 퐹푗푖 퐶푙푚 EL,푚) (5.34) ∂E푅,푙

∂퐷퐿,푗 −1 −1 −1 −1 −1 −1 = 휀퐽(퐹푙푘 퐶푗푚 EL,푚 − 퐹푗푘 퐶푙푚 EL,푚 − 퐹푚푘퐶푗푙 EL,푚) (5.35) ∂F푘푙

∂퐷퐿,푗 −1 = 휀퐽퐶푗푙 (5.36) ∂EL,푙

∂τ where the isotropic tensor function derivative 푖푚 can be computed as follows [114] ∂B푝푞

(1) if b1 ≠ b2 ≠ b3

2 ∂훕 3 √b푖 d퐁 = ∑푎=1{ [ − (b푏 + b푐)퐈푆 − [(b푎 − b푏) + (b푎 − b푐)]퐐a⨂퐐a − ∂퐁 (b푎−b푏)(b푎−b푐) d퐁

3 3 휕√b푖 (b푏 − b푐)(퐐b⨂퐐b − 퐐c⨂퐐c)]} + ∑푖=1 ∑푗=1 퐐i⨂퐐j (5.37) 휕b푗

(2) if b1 ≠ b2 = b3

∂훕 d퐁2 = 푔 − 푔 퐈 − 푔 퐁⨂퐁 + 푔 퐁⨂푰 + 푔 푰⨂퐁 − 푔 푰⨂푰 (5.38) ∂퐁 1 d퐁 2 푆 3 4 5 6

(3) if b1 = b2 = b3

∂훕 휕√b1 휕√b1 휕√b1 = ( − ) 퐈푆 + 푰⨂푰 (5.39) ∂퐁 휕b1 휕b2 휕b2 where the subscripts (a, b, c) are cyclic permutations of (1, 2, 3), the scalars 푔1, 푔2, 푔3, 푔4,

푔5, 푔6 are defined as

λ푎−λ푐 1 휕λ푐 휕λ푐 푔1 = 2 + ( − ) (5.40) (b푎−b푐) b푎−b푐 휕b푏 휕b푐

λ푎−λ푐 b푎+b푐 휕λ푐 휕λ푐 푔2 = 2b푐 2 + ( − ) (5.41) (b푎−b푐) b푎−b푐 휕b푏 휕b푐

λ푎−λ푐 1 휕λ푎 휕λ푐 휕λ푎 휕λ푐 푔3 = 2 3 + 2 ( + − − ) (5.42) (b푎−b푐) (b푎−b푐) 휕b푐 휕b푎 휕b푎 휕b푐

λ푎−λ푐 1 휕λ푎 휕λ푐 b푐 휕λ푎 휕λ푐 휕λ푎 휕λ푐 푔4 = 2b푐 3 + ( − ) + 2 ( + − − ) (5.43) (b푎−b푐) b푎−b푐 휕b푐 휕b푏 (b푎−b푐) 휕b푐 휕b푎 휕b푎 휕b푐

λ푎−λ푐 1 휕λ푐 휕λ푐 b푐 휕λ푎 휕λ푐 휕λ푎 휕λ푐 푔5 = 2b푐 3 + ( − ) + 2 ( + − − ) (5.44) (b푎−b푐) b푎−b푐 휕b푎 휕b푏 (b푎−b푐) 휕b푐 휕b푎 휕b푎 휕b푐

2 2 λ푎−λ푐 b푎b푐 휕λ푐 휕λ푎 b푐 휕λ푎 휕λ푐 b푎+b푐 휕λ푐 푔6 = 2b푐 3 + 2 ( + ) − 2 ( + ) − (5.45) (b푎−b푐) (b푎−b푐) 휕b푎 휕b푐 (b푎−b푐) 휕b푎 휕b푐 b푎−b푐 휕b푏

퐈푆 is given by

1 (퐈 ) = (δ δ + δ δ ) (5.46) 푆 푖푗푘푙 2 푖푘 푗푙 푖푙 푗푘 81

d퐁2 The component form of is given by d퐁

d퐁2 1 [ ] = (δ B + δ B + δ B + δ B ) (5.47) d퐁 푖푗푘푙 2 푖푘 푙푗 푖푙 푘푗 푗푙 푖푘 푘푗 푖푙

The Kirchhoff stress can be computed as

(1) if b1 ≠ b2 ≠ b3

3 훕 = ∑푖=1 λ푖퐐푖 (5.48)

(2) if b1 ≠ b2 = b3

훕 = λ푎퐐푎 + λ푏(푰 − 퐐푎) (5.49)

(3) if b1 = b2 = b3

훕 = λ1푰 (5.50) The stresses, electric filed and material tangents computed in this section will be used to do constitutive update for each integration point of the user elements.

5.2.3 Finite Element Model of the Buckling Actuator in Abaqus

In the present finite element simulations, a thin elastomeric film with circular shape is considered. 10 is used for the radius to thickness ratio. The model is meshed with 8-node brick element with 5 layers of elements through the membrane thickness.

Figure 5.2 shows the mesh of the 1 mm thickness model which consists of 49250 elements. Figure 5.3 shows the mesh of the 0.5 mm thickness model which also consists of 29550 elements. The edge of the finite element model is fixed. To trigger the material

−3 to buckle out-of-plane, a geometric imperfection of order 10 푡0 (푡0 denotes the thickness) in the thickness direction is added to the finite element model. Two material models are considered, the neo-Hookean model and the Ogden model. Two different thicknesses are

82 considered, 0.5 mm and 1 mm, which represent two available thicknesses of the commercial acrylic elastomers VHB 4905 and VHB 4910.

Figure 5.2 Mesh of the 1 mm thickness model.

Figure 5.3 Mesh of the 0.5 mm thickness model.

After the model is fixed at the edge and the geometric imperfection is applied. The membrane is actuated upon the application of an electric potential difference on its surfaces.

5.3 Results and Discussions

The material was assumed to be incompressible (퐽 = 1) throughout the analysis in this section. Table 5.1 lists the values of the material parameters [115] used in the present analysis. The material parameters were adopted from the same experiment in [115]. The variations of the vertical displacement of the center point on the plate surface versus voltage can be obtained for each case.

Table 5.1 The material parameters for the neo-Hookean model and the Ogden model.

neo-Hookean model Shear modulus 휇 (푃푎) Bulk modulus 푘 (푃푎) 156000 7748003.92 Ogden model

훼1 훼2 훼3 휇1 (푃푎) 휇2 (푃푎) 휇3 (푃푎) 1.293 2.3252 2.561 8580 84300 -23300

5.3.1 The Influence of the Thickness

In Figure 5.4, the vertical displacement of the center point on the plate surface versus voltage was sketched for both plate thicknesses. It can be seen that as voltage increased, both plates buckle out of plane at their corresponding critical voltages, and the vertical deformations of the plates increased. Table 5.2 lists the normalized critical voltage for each case. The 0.5 mm thickness plate buckled at a normalized voltage of 0.012. The 1 mm thickness plate buckled at a normalized voltage of 0.050. Compared to the 1 mm thickness case, plate with thickness 0.5 mm buckled out of plane under a lower voltage. To achieve the same vertical deformation, a lower voltage was needed for the 0.5 mm thickness case.

Figure 5.4 The vertical displacement of the center point on the plate surface

versus voltage for two different plate thicknesses.

Table 5.2 The normalized critical voltage for two different plate thicknesses.

Normalized critical buckling voltage 0.5 mm thickness 1 mm thickness 0.012 0.050

In Figure 5.4, instability occurred for each case. For the 0.5 mm thickness case, the wrinkling instability occurred first and was marked by a solid diamond symbol. Figure 5.5 showed the deformed shape of the 0.5 mm thickness plate when the wrinkling instability

85 occurred. It can be seen that the wrinkling patterns started to form in the membrane. For the 0.5 mm thickness case, the pull-in instability occurred first and was marked by a solid circle symbol. It can be seen that in Figure 5.4, the deformation voltage curve increased sharply right before the pull-in instability. This indicated that further increasing the voltage resulted in positive feedback in the continuous increase of the deformation that made the material unstable and fail eventually. Based on the numerical calculations, the plate with thickness 0.5 mm encountered wrinkling instability around the vertical displacement of

3.64 mm while the 1 mm thickness case could reach around 5.98 mm before pull-in instability occurred. This indicated that a thin plate was more susceptible to wrinkling instability.

Figure 5.5 The deformed shape and contours of the displacement after wrinkling

occurs for the 0.5 mm thickness case.

Electric breakdown strength 50V/휇m [59] was used in the present study. Table 5.3 lists the breakdown electric field for the two plates under investigation. It showed that the

0.5 mm thickness plate encountered electromechanical instability at an electric field of 29.3

V/휇m and the 1 mm thickness plate encountered electromechanical instability at an electric field of 47.6 V/휇m. This indicated that electromechanical instability occurred before the material reaches its dielectric breakdown strength for the two cases considered here.

Table 5.3 The breakdown electric field for two different plate thicknesses.

Dielectric breakdown Breakdown electric field (V/휇m) strength (V/휇m) 0.5 mm thickness 1 mm thickness 50 29.3 47.6

5.3.2 The Influence of the Material Models

In this section, 1 mm thickness was used throughout the analysis. In Figure 5.6, the vertical displacement of the center point on the plate surface versus voltage was sketched for both plates with different material models. The solid line denoted the curve for the neo-

Hookean model. The dashed line denoted the curve for the Ogden model. It can be seen that, as voltage increased, both plates buckled out of plane at their corresponding critical voltages, and the vertical deformations of the plates increased. Table 5.4 lists the normalized critical voltage for each case. The plate with the neo-Hookean model buckled at a normalized voltage of 0.078. The plate with the Ogden model buckled at a normalized voltage of 0.050. Compared to the neo-Hookean model, plate with the Ogden model

87 buckled out of plane under a lower voltage. To achieve the same vertical deformation, a lower voltage was needed for the Ogden model case.

Figure 5.6 The vertical displacement of the center point on the plate surface

versus voltage for two different material models.

Table 5.4 The normalized critical voltage for two different material models.

Normalized critical buckling voltage neo-Hookean model Ogden model 0.078 0.050

In Figure 5.6, pull-in instability occurred in both cases and was marked by a solid circle symbol. It can be seen that the deformation voltage curve increased sharply right before the pull-in instability. Based on the numerical calculations, both plates encountered the pull-in instability around the vertical displacement of 5.98 mm. The corresponding normalized voltage for the plate with the neo-Hookean model was 0.268 while for the

Ogden model case the corresponding normalized voltage was 0.179 before the pull-in instability occurred.

Table 5.5 lists the breakdown electric field for the two plates under investigation in this section. It showed that the plate with the neo-Hookean model encountered the pull-in instability at an electric field of 41.6 V/휇m and the plate with the Ogden model encountered the pull-in instability at an electric field of 47.6 V/ 휇 m. This indicated that electromechanical instability occurred before the material reached its dielectric breakdown strength for the two cases considered here.

Table 5.5 The breakdown electric field for two different material models.

Dielectric breakdown Breakdown electric field (V/휇m) strength (V/휇m) neo-Hookean model Ogden model 50 41.6 47.6

5.4 Summary

In this chapter, the actuation of dielectric elastomer buckling actuators was under investigation. A dielectric elastomer buckling actuator consisted of a flat circular elastomeric membrane with compliant electrodes coated on both surfaces and was designed to operate with out-of-plane unidirectional displacements. Finite element method was used

89 to present a numerical model of a basic buckling actuator made of dielectric elastomer. A user element subroutine in Abaqus was used to simulate the electromechanical coupling behavior of the material. Different material models were programmed into the user element subroutine. The neo-Hookean model and the Ogden model were considered here. The stresses and material tangents were derived for each model and implemented into the material part of the user element subroutine. The effect of plate thickness and material models on the performance of the buckling actuators were discussed.

For the cases considered in this chapter, compared to the 1 mm thickness case, the plate with thickness 0.5 mm buckled out of plane under a lower voltage. To achieve the same vertical deformation, a lower voltage was needed for the 0.5 mm thickness case.

However, the 0.5 mm thickness plate was more susceptible to wrinkling instability and fails at a lower achievable deformation. The 1 mm thickness plate failed due to the pull-in instability. Both plates failed before the dielectric breakdown strength. For the different material models situation, compared to the neo-Hookean model, plate with the Ogden model buckled out of plane under a lower voltage. To achieve the same vertical deformation, a lower voltage was needed for the Ogden model case. Both plates failed due to the pull-in instability before reaching the dielectric breakdown strength.

CHAPTER VI

CONCLUDING REMARKS

Dielectric elastomer is a type of smart materials that can be potentially used as electromechanical transducers. Dielectric elastomer produces large reversible deformation under applied voltage [116]. It also has many other advantages, such as high work density, good frequency responses, high degree of electromechanical coupling, and commercially available with low prices [12]. Consequently, Dielectric elastomers have potential use in a variety of applications, such as artificial muscles [13], generators [16], sensors [17], and many others.

The objective of the present work is to do modeling and numerical simulations of the electromechanical coupling behavior of the dielectric elastomer as actuators and to study the effect of some key factors, such as the material properties and the configuration parameters, on the performance of the dielectric elastomer actuators.

The permittivity of some dielectric elastomers has been found to demonstrate significant deformation dependent behavior. In this work, we theoretically analyzed the effect of this behavior on the actuation of a circular actuator made of a dielectric elastomer,

VHB 4910. The dependency of permittivity on deformation was considered through the equations of state. The equilibrium equations of the actuator were solved by MATLAB.

The electromechanical instability, loss of tension, and dielectric breakdown of the actuator

91 during the actuation process under different pre-stretch levels are discussed. It was found that the dependency of permittivity on deformation could effectively suppress the electromechanical instability of the actuator at low pre-stretch levels. At high pre-stretch levels, however, it lowered the maximum stretch the actuator can achieve. Compared to the case of a constant permittivity, the deformation-dependent effect resulted in higher voltage needed to achieve the same actuated stretch, and this discrepancy in voltage increased as the pre-stretch increased.

Some methods to improve the performance of dielectric elastomer transducers have led to the anisotropic electromechanical behavior of the material. We theoretically analyzed the effect of material anisotropy on the diffuse modes of instability in dielectric elastomer plates. As a first step, a transversely isotropic material was considered in the present study. The linearized incremental theory was used and the plate thickness was considered in the stability analysis. The bifurcation equations were solved with Wolfram

Mathematica [107] to obtain the critical stretches under different electric fields. It was found that, compared to the isotropic material, material anisotropy resulted in larger unstable regions and higher critical stretches above which the plate was stable. It was also shown that material anisotropy had a stronger influence on a thinner plate. As the plate thickness increased, the influence decreased.

The buckling actuators made of dielectric elastomer could produce large out-of- plane actuated deformation and have potential applications as soft tunable lenses, loud- speakers, and pumps. Finite element method was used to present a numerical model of a basic buckling actuator made of dielectric elastomer. A user element subroutine [1] in

Abaqus was used to simulate the electromechanical coupling behavior of the material.

Different material models were programmed into the user element subroutine. The effect of a few model parameters on the performance of the buckling actuators was discussed.

BIBLIOGRAPHY

[1] Henann D L, Chester S A and Bertoldi K 2013 Modeling of dielectric elastomers:

Design of actuators and energy harvesting devices J. Mech. Phys. Solids 61 2047–

[2] Zeng C and Gao X 2020 Stability of an anisotropic dielectric elastomer plate Int.

J. Non. Linear. Mech. 124 103510

[3] Bar-Cohen, Y., Cardoso, V.F., Ribeiro, C. and Lanceros-Méndez, S., 2017.

Electroactive polymers as actuators. In Advanced piezoelectric materials (pp. 319-

352). (Woodhead Publishing).

[4] Carpi, F., De Rossi, D., Kornbluh, R., Pelrine, R.E. and Sommer-Larsen, P. eds.,

2011. Dielectric elastomers as electromechanical transducers: Fundamentals,

materials, devices, models and applications of an emerging electroactive polymer

technology. (Elsevier).

[5] Pelrine R E, Kornbluh R D and Joseph J P 1998 Electrostriction of polymer

dielectrics with compliant electrodes as a means of actuation Sensors Actuators A

Phys. 64 77–85

[6] Palsule, S. ed., 2016. Polymers and Polymeric Composites: A Reference Series.

(Springer Berlin Heidelberg).

[7] Ku, C.C. and Liepins, R., 1987. Electrical properties of polymers (pp. 25-92).

(Munich: Hanser publishers).

[8] Flory P J and Volkenstein M 1969 Statistical mechanics of chain molecules

Biopolymers 8 699–700

[9] Biggs J, Danielmeier K, Hitzbleck J, Krause J, Kridl T, Nowak S, Orselli E, Quan

X, Schapeler D, Sutherland W and Wagner J 2013 Electroactive Polymers:

Developments of and Perspectives for Dielectric Elastomers Angew. Chemie Int.

Ed. 52 9409–21

[10] Pelrine R, Kornbluh R, Pei Q and Joseph J 2000 High-Speed Electrically Actuated

Elastomers with Strain Greater Than 100 % Science. 287.5454 836–9

[11] Li T, Keplinger C, Baumgartner R, Bauer S, Yang W and Suo Z 2013 Giant

voltage-induced deformation in dielectric elastomers near the verge of snap-

through instability J. Mech. Phys. Solids 61 611–28

[12] Carpi F, Rossi D De, Kornbluh R, Pelrine R and Sommer-Larsen P 2011

Dielectric elastomers as electromechanical transducers: Fundamentals, materials,

devices, models and applications of an emerging electroactive polymer

technology. (Elsevier)

[13] Anderson I A, Gisby T A, McKay T G, O’Brien B M and Calius E P 2012 Multi-

functional dielectric elastomer artificial muscles for soft and smart machines J.

Appl. Phys. 112 041101–20

[14] Li J, Liu L, Liu Y and Leng J 2018 Dielectric Elastomer Spring-Roll Bending

Actuators: Applications in Soft Robotics and Design Soft Robot. 6 69–81

[15] Lv X, Liu L, Liu Y and Leng J 2015 Dielectric elastomer energy harvesting:

maximal converted energy, viscoelastic dissipation and a wave power generator

Smart Mater. Struct. 24 115036 (15pp)

[16] Moretti G, Santos Herran M, Forehand D, Alves M, Jeffrey H, Vertechy R and

Fontana M 2020 Advances in the development of dielectric elastomer generators

for wave energy conversion Renew. Sustain. Energy Rev. 117 109430

[17] Böse H, Stier S and Muth S 2019 Glove with versatile operation tools based on

dielectric elastomer sensors Electroactive Polymer Actuators and Devices

(EAPAD) XXI ed Y Bar-Cohen (SPIE) p 47

[18] Shian S, Bertoldi K and Clarke D R 2015 Dielectric Elastomer Based “grippers”

for Soft Robotics Adv. Mater. 27 6814–9

[19] Tavakol B and Holmes D P 2016 Voltage-induced buckling of dielectric films

using fluid electrodes Appl. Phys. Lett. 108

[20] Leng J, Liu L, Liu Y, Yu K and Sun S 2009 Electromechanical stability of

dielectric elastomer Appl. Phys. Lett. 94 211901

[21] Zhao X 2009 Mechanics of Soft Active Materials (Harvard University).

[22] Plante J S and Dubowsky S 2006 Large-scale failure modes of dielectric

elastomer actuators Int. J. Solids Struct. 43 7727–51

[23] Liu X, Jia S, Li B, Xing Y, Chen H, Ma H and Sheng J 2017 An

electromechanical model for the estimation of breakdown voltage in stretchable

dielectric elastomer IEEE Trans. Dielectr. Electr. Insul. 24 3099–112

[24] Toupin R A 1956 The Elastic Dielectric J. Ration. Mech. Anal. 5 849–915

[25] Eringen A C 1963 On the foundations of electroelastostatics Int. J. Eng. Sci. 1

127–53

[26] Tiersten H F 1971 On the nonlinear equations of thermo-electroelasticity Int. J.

Eng. Sci. 9 587–604

[27] Toupin R A 1963 A dynamical theory of elastic dielectrics 1 101–26

[28] Eringen A C 1963 On the foundations of electroelastostatics Int. J. Eng. Sci. 1

127–53

[29] Lax M and Nelson D F 1976 Maxwell equations in material form Phys. Rev. B 13

1777–84

[30] Nelson D F 1978 Theory of nonlinear electroacoustics of dielectric, piezoelectric,

and pyroelectric crystals J. Acoust. Soc. Am. 63 1738–48

[31] Dorfmann A and Ogden R W 2005 Nonlinear electroelasticity Acta Mech. 174

167–83

[32] Dorfmann A and Ogden R W 2010 Nonlinear electroelastostatics: Incremental

equations and stability Int. J. Eng. Sci. 48 1–14

[33] Suo Z, Zhao X and Greene W H 2008 A nonlinear field theory of deformable

dielectrics J. Mech. Phys. Solids 56 467–86

[34] Suo Z 2010 Theory of dielectric elastomers Acta Mech. Solida Sinica. 23 549–78

[35] McMeeking R M and Landis C M 2005 Electrostatic Forces and Stored Energy

for Deformable Dielectric Materials J. Appl. Mech. 72 581–90

[36] Goulbourne N, Mockensturm E and Frecker M 2005 A Nonlinear Model for

Dielectric Elastomer Membranes J. Appl. Mech. 72 899–906

[37] Dorfmann L and Ogden R W 2017 Nonlinear electroelasticity: Material

properties, continuum theory and applications Proc. R. Soc. A Math. Phys. Eng.

Sci. 473 20170311

[38] McMeeking R M, Landis C M and Jimenez S M A 2007 A principle of virtual

work for combined electrostatic and mechanical loading of materials Int. J. Non.

Linear. Mech. 42 831–8

[39] Dorfmann L and Ogden R W 2019 Instabilities of soft dielectrics Philos. Trans. R.

Soc. A Math. Phys. Eng. Sci. 377 20180077

[40] Melnikov A 2017 Bifurcation of Thick-walled Electroelastic Cylindrical and

Spherical Shells at Finite Deformation (University of Glasgow)

[41] Wissler M and Mazza E 2005 Modeling and simulation of dielectric elastomer

actuators Smart Mater. Struct. 14 1396

[42] Wissler M and Mazza E 2005 Modeling of a pre-strained circular actuator made

of dielectric elastomers Sensors Actuators, A Phys. 120 184–92

[43] Wissler M and Mazza E 2007 Mechanical behavior of an acrylic elastomer used in

dielectric elastomer actuators Sensors Actuators, A Phys. 134 494–504

[44] Wissler M and Mazza E 2007 Electromechanical coupling in dielectric elastomer

actuators Sensors Actuators, A Phys. 138 384–93

[45] Zhao X and Suo Z 2007 Method to analyze electromechanical stability of

dielectric elastomers J. Appl. Phys. 91 71101

[46] Koh S J A, Li T, Zhou J, Zhao X, Hong W, Zhu J and Suo Z 2011 Mechanisms of

large actuation strain in dielectric elastomers J. Polym. Sci. Part B Polym. Phys. 49

504–15

[47] Henann D L, Chester S A and Bertoldi K 2013 Modeling of dielectric elastomers:

Design of actuators and energy harvesting devices J. Mech. Phys. Solids 61 2047–

[48] Wang S, Decker M, Henann D L and Chester S A 2016 Modeling of dielectric

viscoelastomers with application to electromechanical instabilities J. Mech. Phys.

Solids 95 213–29

[49] Wissler M and Mazza E 2005 Modeling of a pre-strained circular actuator made

of dielectric elastomers Sensors Actuators, A Phys. 120 184–92

[50] Huang J, Li T, Chiang Foo C, Zhu J, Clarke D R and Suo Z 2012 Giant, voltage-

actuated deformation of a dielectric elastomer under dead load Appl. Phys. Lett.

100 041911

[51] Kumar A, Ahmad D and Patra K 2018 Dependence of Actuation Strain of

Dielectric Elastomer on Equi-biaxial, Pure Shear and Uniaxial Modes of Pre-

stretching IOP Conference Series: Materials Science and Engineering vol 310

(IOP Publishing) p 012104

[52] Liu L, Huang Y, Zhang Y and Allahyarov E 2018 Understanding reversible

Maxwellian electroactuation in a 3M VHB dielectric elastomer with prestrain

Polymer (Guildf). 144 150–8

[53] Choi H R, Jung K, Chuc N H, Jung M, Koo I, Koo J, Lee J, Lee J, Nam J, Cho M

and Lee Y 2005 Effects of prestrain on behavior of dielectric elastomer actuator

Smart Structures and Materials 2005: Electroactive Polymer Actuators and

Devices (EAPAD) vol 5759, ed Y Bar-Cohen (International Society for Optics and

Photonics) pp 283–92

[54] Bense H, Trejo M, Reyssat E, Bico J and Roman B 2017 Buckling of elastomer

sheets under non-uniform electro-actuation 13 2876–85

[55] Kofod G, Sommer-Larsen P, Kornbluh R and Pelrine R 2003 Actuation Response

of Polyacrylate Dielectric Elastomers J. Intell. Mater. Syst. Struct. 14 787–93

[56] Wissler M and Mazza E 2007 Electromechanical coupling in dielectric elastomer

actuators Sensors Actuators, A Phys. 138 384–93

[57] Jean-Mistral C, Sylvestre A, Basrour S and Chaillout J J 2010 Dielectric

properties of polyacrylate thick films used in sensors and actuators Smart Mater.

Struct. 19 075019

[58] Li B, Chen H, Qiang J, Hu S, Zhu Z and Wang Y 2011 Effect of mechanical pre-

stretch on the stabilization of dielectric elastomer actuation J. Phys. D. Appl. Phys.

44 155301

[59] Tröls A, Kogler A, Baumgartner R, Kaltseis R, Keplinger C, Schwödiauer R,

Graz I and Bauer S 2013 Stretch dependence of the electrical breakdown strength

and dielectric constant of dielectric elastomers Smart Mater. Struct. 22 104012

(5pp)

[60] Lillo L Di, Schmidt A, Carnelli D A, Ermanni P, Kovacs G, Mazza E and

Bergamini A 2012 Measurement of insulating and dielectric properties of acrylic

elastomer membranes at high electric fields J. Appl. Phys. 024904

[61] Qiang J, Chen H and Li B 2012 Experimental study on the dielectric properties of

polyacrylate dielectric elastomer Smart Mater. Struct. 21 025006

[62] McKay T G, Calius E and Anderson I A 2009 The dielectric constant of 3M

VHB: a parameter in dispute Electroactive Polymer Actuators and Devices

(EAPAD) vol 7287, ed Y Bar-Cohen (International Society for Optics and

Photonics) p 72870P

[63] Huang J, Shian S, Diebold R M, Suo Z and Clarke D R 2012 The thickness and

stretch dependence of the electrical breakdown strength of an acrylic dielectric

100

elastomer Appl. Phys. Lett. 101 122905

[64] Zhao X and Suo Z 2008 Electrostriction in elastic dielectrics undergoing large

deformation J. Appl. Phys. 104 123530

[65] Gei M, Colonnelli S and Springhetti R 2014 The role of electrostriction on the

stability of dielectric elastomer actuators Int. J. Solids Struct. 51 848–60

[66] Dorfmann L and Ogden R W 2018 The effect of deformation dependent

permittivity on the elastic response of a finitely deformed dielectric tube Mech.

Res. Commun. 93 47–57

[67] Gent A N 1996 A New Constitutive Relation for Rubber Rubber Chem. Technol.

69 59–61

[68] Wissler M and Mazza E 2007 Mechanical behavior of an acrylic elastomer used in

dielectric elastomer actuators Sensors Actuators, A Phys. 134 494–504

[69] Goriely A, Destrade M and Ben Amar M 2006 Instabilities in elastomers and in

soft tissues Q. J. Mech. Appl. Math. 59 615–30

[70] Lu T, Huang J, Jordi C, Kovacs G, Huang R, Clarke D R and Suo Z 2012

Dielectric elastomer actuators under equal-biaxial forces, uniaxial forces, and

uniaxial constraint of stiff fibers Soft Matter 8 6167–73

[71] Jiménez S M A and Mcmeeking R M 2013 Deformation dependent dielectric

permittivity and its effect on actuator performance and stability 57 183–91

[72] Jiménez S M A and McMeeking R M 2016 A constitutive law for dielectric

elastomers subject to high levels of stretch during combined electrostatic and

mechanical loading: Elastomer stiffening and deformation dependent dielectric

permittivity Int. J. Non. Linear. Mech. 87 125–36

101

[73] Kuhn W and Grün F 1942 Beziehungen zwischen elastischen Konstanten und

Dehnungsdoppelbrechung hochelastischer Stoffe Kolloid-Zeitschrift 101 248–71

[74] Cohen N, Oren S S and deBotton G 2017 The evolution of the dielectric constant

in various polymers subjected to uniaxial stretch Extrem. Mech. Lett. 16 1–5

[75] MathWorks, Inc. (R2010a). MATLAB, Version 11.3, Natick, Massachusetts.

[76] Zhao X and Suo Z 2010 Theory of dielectric elastomers capable of giant

deformation of actuation Phys. Rev. Lett. 104 178302

[77] Zhao X and Suo Z 2010 Theory of dielectric elastomers capable of giant

deformation of actuation Phys. Rev. Lett. 104 178302

[78] Ha S M, Yuan W, Pei Q, Pelrine R and Stanford S 2006 Interpenetrating polymer

networks for high-performance electroelastomer artificial muscles Adv. Mater. 18

887–91

[79] Ha S M, Yuan W, Pei Q, Pelrine R and Stanford S 2007 Interpenetrating networks

of elastomers exhibiting 300% electrically-induced area strain Smart Mater. Struct.

16 S280–7

[80] Huang J, Lu T, Zhu J, Clarke D R and Suo Z 2012 Large, uni-directional actuation

in dielectric elastomers achieved by fiber stiffening Appl. Phys. Lett. 100 211901–

[81] Lu T, Shi Z, Shi Q and Wang T J 2016 Bioinspired bicipital muscle with fiber-

constrained dielectric elastomer actuator Extrem. Mech. Lett. 6 75–81

[82] Subramani K B, Cakmak E, Spontak R J and Ghosh T K 2014 Enhanced

electroactive response of unidirectional elastomeric composites with high-

dielectric-constant fibers Adv. Mater. 26 2949–53

102

[83] Lu T, Huang J, Jordi C, Kovacs G, Huang R, Clarke D R and Suo Z 2012

Dielectric elastomer actuators under equal-biaxial forces, uniaxial forces, and

uniaxial constraint of stiff fibers Soft Matter 8 6167–73

[84] Carpi F and De Rossi D 2005 Improvement of electromechanical actuating

performances of a silicone dielectric elastomer by dispersion of titanium dioxide

powder IEEE Trans. Dielectr. Electr. Insul. 12 835–43

[85] Razzaghi Kashani M, Javadi S and Gharavi N 2010 Dielectric properties of

silicone rubber-titanium dioxide composites prepared by dielectrophoretic

assembly of filler particles Smart Mater. Struct. 19 035019(7pp)

[86] Huang C, Zhang Q M, DeBotton G and Bhattacharya K 2004 All-organic

dielectric-percolative three-component composite materials with high

electromechanical response Appl. Phys. Lett. 84 4391–3

[87] Rudykh S, Lewinstein A, Uner G and DeBotton G 2013 Analysis of

microstructural induced enhancement of electromechanical coupling in soft

dielectrics Appl. Phys. Lett. 102 151905–3

[88] Sharma A K and Joglekar M M 2019 Effect of anisotropy on the dynamic

electromechanical instability of a dielectric elastomer actuator Smart Mater.

Struct. 28 015006(12pp)

[89] Hossain M and Steinmann P 2018 Modelling electro-active polymers with a

dispersion-type anisotropy Smart Mater. Struct. 27 025010 (18pp)

[90] He L, Lou J, Du J and Wu H 2017 Voltage-induced torsion of a fiber-reinforced

tubular dielectric elastomer actuator Compos. Sci. Technol. 140 106–15

[91] Ponte Castañeda P and Siboni M H 2012 A finite-strain constitutive theory for

103

electro-active polymer composites via homogenization Int. J. Non. Linear. Mech.

47 293–306

[92] Sharma A K and Joglekar M M 2019 A numerical framework for modeling

anisotropic dielectric elastomers Comput. Methods Appl. Mech. Eng. 344 402–20

[93] Polukhov E, Vallicotti D and Keip M A 2018 Computational stability analysis of

periodic electroactive polymer composites across scales Comput. Methods Appl.

Mech. Eng. 337 165–97

[94] Yong H, He X and Zhou Y 2012 Electromechanical instability in anisotropic

dielectric elastomers Int. J. Eng. Sci. 50 144–50

[95] Rudykh S and Debotton G 2011 Stability of anisotropic electroactive polymers

with application to layered media Z. Angew. Math. Phys 62 1131–42

[96] Xiao R, Gou X and Chen W 2016 Suppression of electromechanical instability in

fiber-reinforced dielectric elastomers AIP Adv. 6 035321

[97] Dorfmann L and Ogden R W 2014 Instabilities of an electroelastic plate Int. J.

Eng. Sci. 77 79–101

[98] Zhao X, Hong W and Suo Z 2007 Electromechanical hysteresis and coexistent

states in dielectric elastomers Phys. Rev. B - Condens. Matter Mater. Phys. 76

134113–9

[99] De Tommasi D, Puglisi G, Saccomandi G and Zurlo G 2010 Pull-in and wrinkling

instabilities of electroactive dielectric actuators J. Phys. D. Appl. Phys. 43 325501

[100] De Tommasi D, Puglisi G and Zurlo G 2013 Electromechanical instability and

oscillating deformations in electroactive polymer films Appl. Phys. Lett. 102

011903–4

104

[101] Díaz-Calleja R, Sanchis M J and Riande E 2009 Effect of an electric field on the

deformation of incompressible rubbers: Bifurcation phenomena J. Electrostat. 67

158–66

[102] Leng J, Liu L, Liu Y, Yu K and Sun S 2009 Electromechanical stability of

dielectric elastomer Appl. Phys. Lett. 94 211901–3

[103] Dorfmann L and Ogden R W 2019 Instabilities of soft dielectrics Proc. R. Soc. A

Math. Phys. Eng. Sci. 377 20180077

[104] Dorfmann A and Ogden R W 2005 Nonlinear electroelasticity Acta Mech. 174

167–83

[105] Dorfmann A and Ogden R W 2006 Nonlinear electroelastic deformations J. Elast.

82 99–127

[106] Feng Y, Okamoto R J, Namani R, Genin G M and Bayly P V. 2013 Measurements

of mechanical anisotropy in brain tissue and implications for transversely isotropic

material models of white matter J. Mech. Behav. Biomed. Mater. 23 117–32

[107] Wolfram Research, Inc. (2019). Mathematica, Version 11.3, Champaign, Illinois.

[108] Zeng C and Gao X 2019 Effect of the deformation dependent permittivity on the

actuation of a pre-stretched circular dielectric actuator Mech. Res. Commun. 101

103420

[109] Autodesk, Inc. (2013). AutoCAD, Version 19.0, San Rafael, CA.

[110] Lampani L 2012 Finite Element Modeling of Dielectric Elastomer Actuators for

Space Applications (Università di Roma La Sapienza)

[111] OGDEN R W 1982 Elastic Deformations of Rubberlike Solids Mechanics of

Solids (Elsevier) pp 499–537

105

[112] OGDEN R W 1984 Non-linear Elastic Deformations (Courier Corporation).

[113] OGDEN R W 1972 Large deformation isotropic elasticity – on the correlation of

theory and experiment for incompressible rubberlike solids Proc. R. Soc. London.

A. Math. Phys. Sci. 326 565–84

[114] Eduardo A. de Souza Neto, Djordje Peric, David R. J. Owen,

2011. Computational methods for plasticity: theory and applications (John Wiley

& Sons).

[115] Wissler M T 2007 MODELING DIELECTRIC ELASTOMER ACTUATORS

(SWISS FEDERAL INSTITUTE OF TECHNOLOGY in ZURICH)

[116] Hines L, Petersen K and Sitti M 2016 Inflated Soft Actuators with Reversible

Stable Deformations Adv. Mater. 28 3690–6

106

APPENDICES

107

APPENDIX A

THE DERIVATIVES OF THE INVARIANTS

1. The derivatives of the invariants for transversely isotropic electroelastic materials are given below.

The component form of the first derivatives of the invariants with respect to 푭 and

퐃L respectively:

휕퐼1 = 2퐹푖훼 (A.1) 휕퐹푖훼

휕퐼2 = 2(푐훾훾퐹푖훼 − 푐훼훾퐹푖훾) (A.2) 휕퐹푖훼

휕퐼3 −1 = 2퐼3퐹훼푖 (A.3) 휕퐹푖훼

휕퐼4 = 푎훼(퐹푖푘푎푘 − 퐹푖푚푎푚) (A.4) 휕퐹푖훼

휕퐼5 = 2퐷퐿훼퐹푖훾퐷퐿훾 (A.5) 휕퐹푖훼

휕퐼 6 = 0 (A.6) 휕퐹푖훼

휕퐼5 = 2푐훼훽퐷퐿훽 (A.7) 휕퐷퐿훼

휕퐼6 = 2퐷퐿훼 (A.8) 휕퐷퐿훼

The component form of the second derivatives of the invariants with respect to 푭:

108

2 휕 퐼1 = 2훿푖푗훿훼훽 (A.9) 휕퐹푖훼휕퐹푗훽

2 휕 퐼2 = 2(2퐹푖훼퐹푗훽 − 퐹푖훽퐹푗훼 + 푐훾훾훿푖푗훿훼훽 − 푏푖푗훿훼훽 − 푐훼훽훿푖푗) (A.10) 휕퐹푖훼휕퐹푗훽

2 휕 퐼3 −1 −1 −1 −1 = 2퐼3(2퐹훼푖 퐹훽푗 − 퐹훼푗 퐹훽푖 ) (A.11) 휕퐹푖훼휕퐹푗훽

2 휕 퐼4 = 2푎훼푎훽훿푖푗 (A.12) 휕퐹푖훼휕퐹푗훽

2 휕 퐼5 = 2훿푖푗퐷퐿훼퐷퐿훽 (A.13) 휕퐹푖훼휕퐹푗훽

휕2퐼 6 = 0 (A.14) 휕퐹푖훼휕퐹푗훽

The component form of the second derivatives of the invariants with respect to 퐃L:

2 휕 퐼5 = 2푐훼훽 (A.15) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼6 = 2훿훼훽 (A.16) 휕퐷퐿훼휕퐷퐿훽

The component form of the mixed derivatives of the invariants with respect to 푭 and 퐃L:

2 휕 퐼5 = 2훿훼훽퐹푖훾퐷퐿훾 + 2퐷퐿훼퐹푖훽 (A.17) 휕퐹푖훼휕퐷퐿훽

2. The derivatives of the invariants for isotropic electroelastic materials are given below.

The component form of the first derivatives of the invariants with respect to

푭 and 퐃L respectively:

109

휕퐼1 = 2퐹푖훼 (A.18) 휕퐹푖훼

휕퐼2 = 2(푐훾훾퐹푖훼 − 푐훼훾퐹푖훾) (A.19) 휕퐹푖훼

휕퐼3 −1 = 2퐼3퐹훼푖 (A.20) 휕퐹푖훼

휕퐼5 = 2퐷퐿훼퐹푖훾퐷퐿훾 (A.21) 휕퐹푖훼

휕퐼6 = 2(푐훼훽퐷퐿훽퐹푖훾퐷퐿훾 + 퐷퐿훼퐹푖훾푐훾훽퐷퐿훽) (A.22) 휕퐹푖훼

휕퐼4 = 2퐷퐿훼 (A.23) 휕퐷퐿훼

휕퐼5 = 2푐훼훽퐷퐿훽 (A.24) 휕퐷퐿훼

휕퐼6 2 = 2푐훼훽퐷퐿훽 (A.25) 휕퐷퐿훼

The component form of the second derivatives of the invariants with respect to 푭:

2 휕 퐼1 = 2훿푖푗훿훼훽 (A.26) 휕퐹푖훼휕퐹푗훽

2 휕 퐼2 = 2(2퐹푖훼퐹푗훽 − 퐹푖훽퐹푗훼 + 푐훾훾훿푖푗훿훼훽 − 푏푖푗훿훼훽 − 푐훼훽훿푖푗) (A.27) 휕퐹푖훼휕퐹푗훽

2 휕 퐼3 −1 −1 −1 −1 = 2퐼3(2퐹훼푖 퐹훽푗 − 퐹훼푗 퐹훽푖 ) (A.28) 휕퐹푖훼휕퐹푗훽

2 휕 퐼5 = 2훿푖푗퐷퐿훼퐷퐿훽 (A.29) 휕퐹푖훼휕퐹푗훽

2 휕 퐼6 = 2[훿푖푗(푐훼훾퐷퐿훾퐷퐿훽 + 푐훽훾퐷퐿훾퐷퐿훼) + 훿훼훽퐹푖훾퐷퐿훾퐹푗훿퐷퐿훿 + 휕퐹푖훼휕퐹푗훽

퐹푖훾퐷퐿훾퐹푗훼퐷퐿훽 + 퐹푗훾퐷퐿훾퐹푖훽퐷퐿훼 + 푏푖푗퐷퐿훼퐷퐿훽] (A.30)

110

The component form of the second derivatives of the invariants with respect to 퐃L:

2 휕 퐼4 = 2훿훼훽 (A.31) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼5 = 2푐훼훽 (A.32) 휕퐷퐿훼휕퐷퐿훽

2 휕 퐼6 2 = 2푐훼훽 (A.33) 휕퐷퐿훼휕퐷퐿훽

The component form of the mixed derivatives of the invariants with respect to

푭 and 퐃L:

2 휕 퐼5 = 2훿훼훽퐹푖훾퐷퐿훾 + 2퐷퐿훼퐹푖훽 (A.34) 휕퐹푖훼휕퐷퐿훽

2 휕 퐼6 = 2퐹푖훽푐훼훾퐷퐿훾 + 2퐹푖훾퐷퐿훾푐훼훽 + 2퐹푖훾푐훾훽퐷퐿훼 + 2훿훼훽퐹푖훾푐훾훿퐷퐿훿 (A.35) 휕퐹푖훼휕퐷퐿훽

111