Electromechanically Coupled Finite Element Model of a Circular Electro-active Polymer Actuator. by Aseem Pralhad Deodhar

Transcription

1 ABSTRACT DEODHAR, ASEEM PRALHAD. Electromechanically Coupled Finite Element (FE) Model of a Circular Electro-active Polymer Actuator. (Under the direction of Dr. Stefan Seelecke). This thesis presents a Finite Element model of a circular Dielectric Electro-active Polymer (DEAP) actuator using two different hyperelastic material models. This model is a first approximation of the material behavior of the DEAP and lays out a basic framework of modeling an electromechanically coupled DEAP as a system by neglecting the hysteric and rate-dependent material behavior. This includes the modeling of the compliant electrodes along with the polymeric dielectric. The compliant electrodes, pasted on both sides of the elastomer, have a large effect on the mechanical behavior of the DEAP which needs to be taken into consideration while modeling the DEAP system. The model is developed using the commercial Finite Element Modeling software, COMSOL Multiphysics. The material parameters are found by systematically varying them and comparing the simulation results to known experimental results. The results from the mechanical simulations are presented in the form of force-displacement curves and are validated by comparing them to experimental results. Electromechanical simulations are carried out and the stroke of the actuator for different electrode stiffness values is compared with experimental values when the DEAP is biased with a constant force. The comparisons with experimental values are good qualitatively. The capacitance variation of the DEAP as a function of deformation is also predicted by the model and is in good agreement with the experimental values. This work lays out a platform for further enhancements by including the viscoelastic material properties into a finite element model.

2 Electromechanically Coupled Finite Element Model of a Circular Electro-active Polymer Actuator by Aseem Pralhad Deodhar A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering Raleigh, North Carolina 2011 APPROVED BY: Dr. Gregory Buckner Dr. Mohammed Zikry Dr. Stefan Seelecke Chair of Advisory Committee

3 DEDICATION To Aai, Baba, Dada, Bhakti & Saee ii

4 BIOGRAPHY Aseem Deodhar was born in Pune, India on August 10, He graduated as a Bachelor of Technology in Mechanical Engineering in May 2008 from College of Engineering, Pune (CoEP) iii

5 ACKNOWLEDGMENTS I would like to thank my adviser Dr. Stefan Seelecke for all his patience with me and all the guidance, motivation and encouragement. I also thank him for giving me an opportunity to travel to Germany. It has been a great experience working with him. I also thank Dr. Gregory Buckner and Dr. Mohammed Zikry for serving on my graduate committee. I would also like to thank Dr. Alexander York for all his help with the experimental data and extremely useful tips about writing and presenting technical data. Special thanks to Micah Hodgins for his help with experiments and simulations. I thank the other members (past and present) of the Adaptive Structures Lab Stephen Furst, Gheorghe Bunget, Rohan Hangekar and Nicole Lewis for all the good times at and away from work. I specially thank my cousin sister Mrunal and her husband Anant Natu for all their help and support. Finally, I m most thankful to my parents and my elder brother for all their love, support and understanding. iv

6 TABLE OF CONTENTS LIST OF FIGURES... vii LIST OF TABLES... ix Chapter 1. INTRODUCTION Introduction and Background Literature Review Motivation Thesis Overview... 5 Chapter 2. MECHANICS & ELECTROSTATICS Problem Characterization Hyperelastic Material Models Neo-Hookean material model Mooney-Rivlin material model Determination of material parameters for Mooney-Rivlin material Chapter 3. GEOMETRY & FE MODEL Geometry of the DEAP actuator Boundary Conditions Mechanical Boundary Conditions Electrical Boundary Conditions Electromechanical Boundary condition Finite Element Mesh Chapter 4. MECHANICAL SIMULATIONS Experiments Simulations Pre-strain Solver Results v

7 Chapter 5. ELECTRO-MECHANICAL COUPLING Experiments Simulation Procedure Results Time Resolved Data Local Results Deformation and Stress Electric Field Charge Distribution DEAP Stroke Global Results Force-Voltage Characteristics Capacitance Variation (Sensing Applications) Simulations with a.75 inch 2 actuator Further Experiments Electrode Thickness variation Actuator diameter variation CONCLUSIONS REFERENCES APPENDICES APPENDIX 1 Basic Mechanics and Electrostatics A1.1 Deformation and Strain A1.2 Stresses A1.3 Electrostatics A1.3.1 Maxwell Stress Tensor vi

8 LIST OF FIGURES Figure 1: DEAP construction and Operating principle... 1 Figure 2: Sketch of a circular DEAP actuator Figure 3: Mechanical response of the elastomer without electrodes and with pasted electrodes. [33]... 4 Figure 4: Stress - Strain diagram for Hookean, Neo-Hookean and Mooney-Rivlin material models Figure 5: DEAP actuator Figure 6: Electroactive Polymer Actuator Geometry - Solidworks model [33] Figure 7: Pre-loading the DEAP in out-of-plane direction schematic. [33] Figure 8: Simplified loading & boundary conditions Figure 9: Mechanical Boundary Conditions Figure 10: Electrical Boundary conditions Figure 11: Vertical displacement at the edge marked red and radial stresses at points 1,2,3 for global and local mesh convergence respectively Figure 12: Global mesh convergence with uniform mesh (left) and non-uniform mesh (right) Figure 13: Local mesh convergence with uniform mesh (left) and non-uniform mesh (right) Figure 14: Finite Element mesh and a zoom in at the left end. [34] Figure 15: Setup for tensile experiment on elastomer specimen Figure 16: Stress vs Strain plot for simple tensile experiment on the elastomer Figure 17: Schematic for out-of-plane deformation experiments [33] Figure 18: Displacement Input in displacement control simulation Figure 19: Force vs Displacement curves for different values of elastomer Young's modulus. [34] Figure 20: Force vs Displacement for different values of electrode Young's modulus. [34]. 30 vii

9 Figure 21: Schematic for electro-mechanical experiments on the DEAP [33] Figure 22: Schematic of test setup (left) and the actual set-up (right). [34] Figure 23: Time resolved data. Input parameters (top left), Displacement output (top right). Experimental displacement output for one voltage cycle (bottom). [34] Figure 24: Time resolved data for Displacement control (Simulation Top, Experimental Bottom) Figure 25: Deformation shapes of the elastomer during pre-deflection and actuation. [34].. 36 Figure 26: Radial Stress Contour plots Figure 27: Thickness Stresses Contour Plot Figure 28: Electric Field Contour Plots Figure 29: Surface Charge densities on the internal surface of the top electrode (left) and bottom electrode (right) Figure 30: Test setup schematic (left) and actual picture of the test (right). [34] Figure 31: Suspended Mass vs Stroke Figure 32: Blocking Force vs Voltage curve for fixed displacement of 2mm Figure 33: Capacitance variation with deformation Figure 34: Force v Displacement for DEAP75 actuator with electrodes Figure 35: Force v Voltage plot for DEAP75 actuator Figure 36: Variation of pre-deflection and radial stresses in electrodes with thickness Figure 37: Variation of Stresses with change in actuator diameter DEAP75 (Top Left), DEAP100 (Top right), DEAP200 (Bottom) Figure 38: Variation in capacitance with variation in diameter. Experimentally validated plot (top) and prediction for other geometries (bottom) Figure 39: Surface Charge densities for Top electrode (top) and bottom electrode (bottom). 50 viii

10 LIST OF TABLES Table 1: Material parameters for the two material models Table 2: Material parameters for different values of young's modulus and material model.. 29 ix

11 Chapter 1. INTRODUCTION 1.1 Introduction and Background The main focus of this thesis is to develop an electromechanically coupled, multi-field finite element model of a circular Dielectric Electro-Active Polymer (DEAP). A DEAP is essentially a capacitor with a compliant elastomer as the dielectric and compliant conducting electrodes pasted on both sides. The basic construction is as shown in Figure 1. When a potential difference is applied across the electrodes the electrostatic charges build up on the electrodes and they attract each other. In the process they exert mechanical pressure on the thin elastomeric dielectric and squeeze it between them. Due to the mechanical pressure, the elastomer film contracts in the thickness direction and is enlarged in the in-plane directions. DEAPs are actuated at high voltages (typically 2kV 5kV) but draw very low current and hence have low power consumption. Electrodes High Voltage Elastomer High Voltage Figure 1: EAP construction and Operating principle The DEAPs are also pre-deflected in order to use the strain caused by the electrostatic (Maxwell s) force to produce displacement stroke. There are various DEAP configurations being studied for different applications and there are a number of ways in which the DEAP can be pre-deflected. 1

12 Figure 2: Sketch of a circular DEAP actuator. Electroactive polymers (DEAP) are a novel variety of materials which can potentially be used in active structures as actuators and sensors. With a large number of miniature applications envisioned for DEAPs [1-7, 10], they are quickly becoming a primary focus of research among industry and academia with various applications such as prosthetic devices, braille display systems and flat panel loudspeakers, to name a few. There are several properties of DEAP materials including low power consumption, high elastic energy density [1], fast response [8], light-weight [9] and silent operation which make them an attractive choice for use smart material systems and devices. 1.2 Literature Review This section presents a review of some of the previous modeling and simulation work done on Electroactive Polymer devices. It includes both the analytical models and finite element models developed for various actuator geometries and configurations. Pelrine et al [2] presented the mechanism of electrostriction in Electroactive Polymers due to the electrostatic attraction of free charges on the electrodes. They derive relationships for the effective actuation pressure of an DEAP. They found that this pressure is twice that in a parallel plate capacitor and they attribute the excessive pressure to the compliance of the electrodes. They also experimentally investigated the in-plane stresses for actuators made of different polymer materials and found a good co-relation between the analytical and 2

13 experimental values. Kofod [11] presented an electromechanically coupled model for DEAPs using elastic and hyperelastic material models. The voltage-strain response was plotted for each of the models with an applied weight to the DEAP. The Ogden [12] model predicted a response which lied in between those of Hooke and Neo-Hooke models. Zhao [13] et. al proposed an analytical model for a membrane of a dielectric elastomer deformed into an outof-plane axi-symmetric shape. Their model studies the kinematics of deformation and charging, together with thermodynamics, leads to equations that govern the state of equilibrium. Goulbourne [14] et al presented a method for modeling dielectric elastomer membrane accounting for material non-linearity and large deformations. Their model is tailored to axi-symmetric inflatable membranes, used in artificial blood pump systems. Wissler [15, 23-24] et. al presented a considerable amount of work on modeling and simulating the DEAP actuators. They simulated a pre-stretched circular DEAP actuator, activated with a predefined voltage, using the commercially available FE software ABAQUS. They propose a novel approach for FE analysis of elastomer actuators in which they use kinematic boundary conditions to apply the electromechanical pressure. They found discrepancies in the uni-axial test data from experiment and simulations due to the time dependent material response. Rosenblatt [16] et. al present a technique to accurately model DEAPs taking into account non-linearities and large deformations. They proposed an algorithm to model the electromechanical coupling that is inherent to dielectric DEAP systems. Their algorithm was validated by using finite element simulations with the commercially available software ANSYS. Wissler [17] et. al investigated the electromechanical coupling in dielectric EAPs. They analyzed the analytical equation of electrostatic forces in dielectric elastomer systems with energy considerations and numerical calculations. Their system was modeled using COMSOL, to plot charge and electrostatic force distributions. Lochmater [18] et. al modeled the viscoelastic behavior of a dielectric film using spring damper framework. The model predicted stable deformation state under activation with constant charge and an unstable equilibrium state for activation under constant voltage. Jung [19] et. al presented a computational system to describe electromechanical behavior of DEAPs. Hyperelastic material models like Mooney-Rivlin 3

14 Force (N) [20] and Yeoh [21] model are used for simulations in the commercial FE program ANSYS. The predictions of the Mooney-Rivlin model showed good agreement with the experimental data. Gao [22] et. al combined the hyperelastic Ogden material model with total Maxwell s stress methods to describe the material. They also formulated large deformation Finite Element models and compared simulation results to experimental results to validate the computational model. While all of these models provide beneficial insight into the elastomer material behavior itself, they lack the ability to accurately predict a complete DEAP actuator structure which includes compliant electrodes. Including these other elements into a coupled model is necessary for behavior predictions of various DEAP actuators as they would be used in commercial applications. The electrodes have a non-negligible effect on the mechanical response of the DEAP which can be seen from the Figure 3 below. 0 Force vs. Displacement Elastomer Elastomer with Electrodes Displacement (mm) Figure 3: Mechanical response of the elastomer without electrodes and with pasted electrodes. [33] 1.3 Motivation As it is seen from the above figure, the force-displacement graphs plotted for the elastomer with and without electrodes show that there is an appreciable difference in the mechanical 4

15 response of the actuator for the two cases. It is clear from Figure 3 that the electrodes make the actuator stiffer and add a considerable amount of hysteresis in the actuator response. The focus of this work is to develop an electromechanically coupled finite element model which takes into account the effect of electrodes in the mechanical response of the DEAP actuator. There are challenges modeling an elastomeric polymer actuators resulting from their inherent electromechanical coupling and non-linear and hysteretic material behavior. A finite element model will also help in better design of the DEAP system geometry for specific applications. The Finite Element (FE) also has to account for the electro-mechanical coupling that results due to the Maxwell s forces from the applied electric field inducing a mechanical deformation as explained in section 1.1. The finite element model presented in this work is developed using the commercial finite element code COMSOL Multiphysics and is a first step towards a DEAP actuator system model. 1.4 Thesis Overview This thesis presents a Finite Element model of a DEAP actuator as a system taking into account the influence of the electrodes on the mechanical performance of the actuator. Chapter 2 lays out the basic partial differential equations governing the electromechanically coupled problem. A brief introduction to the basic hyperelastic material models Neo Hookean and Mooney-Rivlin is also included in this chapter. Chapter 3 introduces the geometry and the finite element model for the Electroactive Polymer actuator. It describes the boundary conditions, and the finite element mesh used for the simulations. Chapter 4 and 5 describe the process of material parameter calibration for the material models used. They present the simulation results and comparison with experimental values. Chapter 5 also presents the validation of the model by simulating a different DEAP geometry and comparing it with experimental results. The sensing capabilities of the DEAP from the 5

16 model predictions are also presented. Finally, the conclusion and scope for future work are presented. 6

17 Chapter 2. MECHANICS & ELECTROSTATICS This chapter describes the electromechanical problem formulation starting from the basic momentum balance equations. A brief description of the hyperelastic materials is also included. 2.1 Problem Characterization The modeling of the DEAP system in a Finite Element program involves solving of the partial differential equations that define each of the physics mechanics, electrostatics and electro-mechanical coupling. The characterization of a structural mechanics problem involves specifying the balance principles, the Neumann and the Dirichlet boundary conditions and the constitutive laws or equations which presume the existence of a relation between forces and displacements with stresses and strains respectively, which is exclusively local i.e. at the considered material point. The momentum balance principle describes the equilibrium of internal forces and stresses. The forces on a body such as body loads, inertial forces and forces resulting from the stresses. In tensor form the local balance of momentum can be stated as [29]: u b (1) i ij, j i here, ρ is the density, u is the acceleration vector, ij, j is the divergence of the Cauchy i Stress tensor and b i is the matrix of volume specific body loads. In case of a quasi-static model, the acceleration term is zero. So the momentum balance equation becomes [29]; 0 b (2) ij, j i 7

18 The Dirichlet boundary conditions are prescribed as displacements along the boundaries: where, u i is the displacement and b u ( X, t) u ( X, t) X (3) i j i j u i is the prescribed displacement. If the prescribed displacements are identical to zero, they are referred to as homogenous boundary conditions, like those at supports. ui ( X, t) 0, X. The Neumann boundary conditions prescribe the equilibrium of the force, and can be stated in tensor notation as [29]: ijnj Tij nj ti (4) where T ij is the Maxwell s stress tensor and n j is the normal vector. It is defined as: 1 2 Tij 0Ei E j 0E ij 2 (5) The Maxwell s stress tensor is applied as a force boundary condition thus coupling the mechanical and the electrostatic fields. As seen from equation 5, the Maxwell s stress is a function of the electric field in the material. The, electric field has to be calculated on the entire domain and it is governed by the charge balance equation for electrostatics [27]: ( E P) (6) 0 i i, i f where, E is the electric field, P is the polarization and ρ f is the charge density. The Dirichlet boundary condition specifies the voltage in case of an DEAP modeling problem. The voltage can be specified as a linear function time as in equation 6. For the DEAP the maximum input voltage considered is 2500 volt. V ( t) V t (7) 8

19 In the case of the DEAP actuator, the mechanical behavior of the elastomer and electrode materials is non-linear and hence the problem is a large deformation non-linear one. Hyperelastic material models can be used to approximate the non-linear material behavior. 2.2 Hyperelastic Material Models This section provides a short introduction to hyperelastic materials and the material models Neo Hookean and Mooney-Rivlin that are used for modeling the DEAP material in this work. A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress-strain relationship derives from a strain energy density function, W. The most common example of this kind of material is rubber, whose stressstrain relationship can be defined as non-linearly elastic, isotropic, incompressible and generally independent of strain rate. The earliest and the basic models to define hyperelastic materials were proposed by Ronald Rivlin and Melvin Mooney. These models are called as Neo-Hookean and Mooney-Rivlin material models. The stresses in Hyperelastic materials can be calculated from the following relations: If W(F) is the strain energy density function, the first Piola-Kirchoff stress tensor (P) can be calculated as, W W W P F 2F F E C (8) where, F is the deformation gradient, E is the Green-Lagrangian strain tensor, and C is the right Cauchy-Green deformation tensor. The second Piola-Kirchoff stress tensor can be calculated as 1 W W W S F 2 F E C (9) 9

20 and the Cauchy Stress tensor can be calculated as 1 W 1 W 2 W F F F F F J F J E J C T T T (10) Neo-Hookean material model The energy density function for a Neo-Hookean solid is given as: W C ( I 3) D ( J 1) (11) where C 1 and D 1 are material constants and I 1 is the first invariant of the left Cauchy-Green deformation tensor. I (12) λ i are stretch ratios in the principal directions. For incompressible materials J =1 and the strain energy function is reduced to W C1( I1 3) Mooney-Rivlin material model The material model for a Mooney-Rivlin solid is a linear combination of two invariants of the left Cauchy-Green deformation tensor B. The strain energy density function for a Mooney-Rivlin material is given as W C ( I 3) C ( I 3) D ( J 1) (13) where C 10, C 01 and D 1 are material constants. For an incompressible Mooney-Rivlin solid J =1 and the strain energy function is reduced to W C10 ( I1 3) C01( I2 3) I and I (14) 10

21 The typical stress-strain curves for Neo-Hookean and Mooney-Rivlin materials are shown in Figure 4. Figure 4: Stress - Strain diagram for Hookean, Neo-Hookean and Mooney-Rivlin material models Determination of material parameters for Mooney-Rivlin material As can be seen from Figure 4, the stress-strain curves for the hyperelastic materials closely follow that of the Hookean material for small strain limits after which they start to deviate away non-linearly. The Mooney-Rivlin material parameters C 10 and C 01 can be expressed in terms of the Young s modulus E and the Poisson s ratio v by linearizing the stress-strain relations of the Mooney-Rivlin material model and comparing them to those of a Hookean solid. This would give an approximation for the Mooney-Rivlin material parameters. The relation can be found by starting with the strain energy function for the Mooney-Rivlin model as described below. The analytical stress-strain relation for hyperelastic materials is obtained by partial differentiation of the strain energy function with respect to the stretch ratios. For an 11

22 incompressible Mooney-Rivlin material the strain energy function is: W C ( I 3) C ( I 3) s (15) For a biaxial case, the stress distribution in a Mooney-Rivlin material in terms of the principal stretch ratios [30] is given as: 1 ( )( C C ) (16) ( )( C C ) (17) We substitute the stretch ratios in terms of strains as λ = 1+ ε and simplify equations 29 & 30. We also assume that the strain measures are small and, hence, all terms of ε with powers greater than 1 can be ignored without any appreciable error. Also, for a purely biaxial state of stress, the principal stresses and strains are aligned with the axial stresses and strains i.e. ζ 1 = ζ x and ζ 2 = ζ y. On simplification yields: 4C 4 C 2C 2 C x x y (18) 2C 2 C 4C 4 C y x y (19) The stress state in a Hookean material is given by: E E x 2 x 2 y 1 1 (20) E E y x y 1 1 (21) And can when compared to Eq 24 and 26 yields C C E 4(1 ) (22) 12

23 Also, for consistency with linear elasticity in the limit of small strains, it is essential that: µ 2 C C (23) where µ is the shear modulus for simple shear [31]. Writing the shear modulus in terms of Young s modulus and Poisson s ratio, we get another equation for the material parameters. The equations 28 & 29 are solved simultaneously to get the two material parameters C 10 and C

24 Chapter 3. GEOMETRY & FE MODEL This chapter describes the construction and geometry of the DEAP actuator. The boundary conditions during mechanical loading and those introduced due to actuator geometry are also detailed in the second section. It also describes the procedure for applying the mechanical and electrical boundary conditions using the commercial finite element software COMSOL Multiphysics [28]. The finite element mesh details are discussed in the third section. 3.1 Geometry of the DEAP actuator The Dielectric Electro-Active Polymer actuator considered in this study has a circular geometry. It is constructed with a film of silicone material and pasted with carbon based electrodes held together in place by a plastic frame. Figure 5 shows the schematic for the DEAP construction process. Figure 5: DEAP actuator The DEAP actuator has a circular geometry as can be seen in Figure 6. This particular actuator geometry has been designed for out-of-place actuation. The actuator is made of a 14

25 solid frame adhered to a proprietary pre-strained silicone based elastomer film. They are approx. 1 inch 2 in size, and the elastomer layer is a few 10 s of µm thick in the absence of any load. Figure 6: Electroactive Polymer Actuator Geometry. Solidworks model [33]. Two black carbon-based electrodes make up the effective DEAP area. The thickness of the carbon electrodes is assumed to be 10 µm. This makes the DEAP actuator a layered composite structure with electrode-elastomer-electrode. The circular mounting section made from the frame material is also adhered at the center of the actuator. When the actuator is loaded at its center mechanically or a potential difference is applied across its electrodes, it exhibits out-of-plane motion by displacing the center mount, as shown in Figure 6 (right). The pasted electrodes have an effect on the mechanical response of the DEAP actuator and hence they have to be accounted for in the model in order to capture their effect. 3.2 Boundary Conditions Generally, the DEAP actuator is mechanically pre-loaded and then an electric potential applied across its electrodes. This helps to use the stroke caused by the electrostriction as a displacement stroke. The actuator geometry considered in this study has a center mount, as discussed in the earlier section, which can be used to apply a mechanical force to pre-load the DEAP in the out-of plane direction. A schematic of pre-deflecting a DEAP with an actuator 15

26 attached to a load cell (LC) can be used to illustrate this further as shown in Figure 7. Figure 7: Pre-loading the DEAP in out-of-plane direction schematic. [33] The above situation can be shown in a simplified diagram as sketched in Figure 8. Figure 8: Simplified loading & boundary conditions 16

27 The circular geometry of the DEAP actuator and uniform thickness all over also makes it symmetric about an axis perpendicular to the DEAP surface through the center. Thus, axial symmetry can be used while modeling the geometry using COMSOL. This helps in converting a three dimensional problem into a simplified two dimensional axially symmetric one. Another advantage of this simplification is that the problem becomes computationally more efficient. The commercially available finite element software COMSOL Multiphysics [28] has been used to develop a finite element model. COMSOL Multiphysics is a powerful interactive environment for modeling and solving a variety of engineering and physics problems based on partial differential equations (PDEs). There are several application modes, based upon basic physics (structural, electrical, fluids etc) which can be used to model problems involving single physics or more than one of the application modes can be used in conjunction with others to model coupled field problems. For modeling the electro-mechanically coupled DEAP actuator, three COMSOL Multiphysics environments are used Structural mechanics, Electrostatics and Deformed Mesh. The Structural Mechanics module is used to model the response of the DEAP undergoing mechanical loading. The Electrostatics module is used to apply input voltage and it also generates the Maxwell s Stress Tensor along the boundaries of the electrodes. The deformed mesh module allows the electrostatic boundary conditions to be applied even though the boundaries of the computational domain move due to the applied structural boundary conditions. The technique for mesh movement is called an arbitrary Lagrangian- Eulerian (ALE) method. In the special case of a Lagrangian method, the mesh movement follows the movement of the physical material. Lagrangian method is often used in solid mechanics, where the displacements often are relatively small. When the material motion is more complicated, like in a fluid flow model, the Lagrangian method is not appropriate. For such models, an Eulerian method, where the mesh is fixed, is often used except that this method cannot account for moving boundaries. 17

28 The ALE method is an intermediate between the Lagrangian and Eulerian methods, and it combines the best features of both it allows moving boundaries without the need for the mesh movement to follow the material Mechanical Boundary Conditions The simplification of geometry for analysis enables to model the geometry as a simple strip with the width equal to the thickness of the elastomer and similar strips for the electrodes above and below the elastomer strip. The modeled geometry is thus only a three strip composite. Since, the outer plastic frame is rigid as compared to the elastomer, the outer end of the elastomer can be considered fixed to a rigid frame. Similarly, the circular part of the frame at the center of the DEAP (the center mount) is rigid as compared with the elastomer material and hence no radial displacement of the polymer occurs at the inner end. All these mechanical boundary conditions are illustrated in Figure 9 Figure 9: Mechanical Boundary Conditions The right end of the DEAP geometry is specified as the fixed end. The left end has a roller boundary condition which restricts its movement in the radial direction. The displacement or force boundary condition is also applied on the left boundary of the elastomer 2D geometry, thus allowing simulations in both force control mode and displacement control mode. The force boundary condition is used when simulating a pre-deflected DEAP to produce stroke on actuation with a voltage. 18

29 3.2.2 Electrical Boundary Conditions The electrical boundary conditions are basically the voltage boundary conditions on the electrodes. Voltage is specified as a boundary condition on the top electrode. The boundaries of the upper electrode are specified as port and the port can be used to specify input. Ground is used as a boundary condition on the boundaries of the bottom electrode. The Figure 10 below shows the electrical boundary conditions. Figure 10: Electrical Boundary conditions Electromechanical Boundary condition The Electroactive Polymer actuator system is a coupled electro-mechanical system. Hence, there should be a way to couple the electrostatics with the structural deformation in the finite element model. The Electrostatics physics environment in COMSOL Multiphysics allows the generation of a Maxwell Stress tensor. The Maxwell force can be extracted from this stress tensor is available with units of Newton per unit area to be applied as a boundary condition. A force variable is created on the electrode sub-domains to generate a Maxwell stress tensor. The Maxwell force is then applied as a force boundary condition on the upper and the lower electrodes using the structural mechanics subdomain, thus coupling the two physics. Another way to couple the two physics is by using the Pelrine s equation as described in the Section The equation for pressure as derived by Pelrine et al is applied as a boundary condition on the upper and lower electrodes using the structural mechanics module in COMSOL. This also couples the respective physics. 19

30 3.3 Finite Element Mesh The elastomer and the electrode geometry is regular and it is easier to mesh them using a mapped mesh and quad elements. The elements have a quadratic shape function. The elements on the boundary are specified before meshing the geometry. The boundary discretization can be done in a couple of ways. The elements along the horizontal can be of uniform sizes which will make a uniform mesh in the sub-domain, or their size can vary depending on the size of the elements on the vertical boundary. So the elements near the vertical boundaries get smaller and their sizes are the same as that on the vertical edges. This creates high quality elements (with an aspect ratio close to 1) near the edges which help in dealing with singularities occurring at the edges due to mechanical boundary conditions. Both of the above mentioned meshing options were evaluated by running mesh convergence tests by varying the number of elements and deciding the type of mesh and the optimum number of elements in that mesh. Local quantity stress at important locations as shown in Figure 11 and global quantity maximum displacement were used as convergence criteria. Figure 11: Vertical displacement at the edge marked red and radial stresses at points 1,2,3 for global and local mesh convergence respectively. 20

31 Sigma_rr [Pa] Solution Time (sec) Sigma_rr [Pa] Solution Time [sec] Z-displacement [m] Solution time (sec) Z-Displacement [m] Solution Time [sec] -1.22E-3 Global Mesh Convergence No. of Elements E-3 Global Mesh Convergence No. of Elements E E E E E E E-3 Z-displacement [m] Time E E-3 Z-Displacement [m] Time E E E E-3 0 Figure 12: Global mesh convergence with uniform mesh (left) and non-uniform mesh (right). 6E+5 Local Convergence E+5 Local convergence E E E+5 3E+5 2E+5 Sigma_rr (Pt.1) Sigma_rr (Pt.2) Sigma_rr (Pt.3) Solution Time E E E+5 Sigma_rr (Pt.1) Sigma_rr (Pt.2) Sigma_rr (Pt.3) Solution time E E E+0-1E No. of Elements 200 0E+0-1E No. 600 of Elements E+5 0-2E+5 0 Figure 13: Local mesh convergence with uniform mesh (left) and non-uniform mesh (right). From Figure 12 and Figure 13 it is quite clear that the rate of convergence when using a nonuniform mesh (smaller elements near the ends and coarse mesh towards the center of the elastomer strip) is much higher than that with uniform size elements throughout the sub- 21

32 domain. Hence, a non-uniform mapped mesh is chosen for the elastomer sub-domain. The elastomer has four divisions vertically and sixty divisions horizontally giving a total of 240 elements. The elastomer has a single layer of mapped meshed quad elements with a quadratic shape function. The electrostatics module also requires air to be modeled as a surrounding medium around the DEAP. The air is meshed with a free triangular mesh. The mesh for the entire geometry is as seen in Figure 14. Figure 14: Finite Element mesh and a zoom in at the left end. [34] 22

33 Chapter 4. MECHANICAL SIMULATIONS The commercial Finite Element software COMSOL is used to simulate the DEAP actuator. This chapter describes the simulation procedure for the mechanical characterization of the actuator material in COMSOL. This includes not only the elastomer but also the electrode material. It also describes the experiments used to compare the simulation results with the experimental data. A systematic approach to vary the material parameters is also discussed. Global results in the form of force-displacement plots are presented. Contour plots for stresses at specific local points on the elastomer and electrodes are also shown 4.1 Experiments This section presents the experimental procedure and tests that are compared to the simulation results for validation. Out of plane deformation experiments have been performed on actuators. A simple tensile test on a small strip of elastomer material is done. The elastomer is clamped at both the ends. A picture of the set-up is shown in Figure 15 where one of the ends is held fixed and the other end is pulled on with the help of a linear actuator. A load cell is used with the actuator to note the reaction force and a force-displacement graph is plotted. Using the geometry of the elastomer sample a stress-strain curve is plotted as shown in Figure 16. A linear approximation of the stress-strain curve gives the approximate Young s modulus of the elastomer material. This Young s modulus serves as the initial approximation for simulations of the actuator. Figure 15: Setup for tensile experiment on elastomer specimen 23

34 Stress [Pa] 1E+06 Stress Vs Strain 1E+06 1E+06 8E+05 6E+05 y = 1E+06x E+05 2E+05 0E+00 Stress V Strain Linear Approximation Strain Figure 16: Stress vs Strain plot for simple tensile experiment on the elastomer. In the experiments on the actuator, the actuator is deformed to a fixed out-of-plane displacement using a mechanical actuator and the force is measured with a load cell. A schematic of the out-of-plane deformation experiments is shown in Figure 17. The displacement is measured from the rest position of the elastomer. The experiments are conducted on actuator samples without coated electrodes and samples with electrodes coated on them. This helps to characterize the elastomer separately. The electrodes can then be included into the model and a systematic parameter variation can be done to arrive at the correct material parameters. The details of the experimental procedure can be found in [32]. Figure 17: Schematic for out-of-plane deformation experiments [33]. 24

35 4.2 Simulations As mentioned in the earlier section, the out-of-plane deformation experiments are done on actuator samples without coated electrodes and experimental conditions are simulated in COMSOL. The structural mechanics module is chosen in order to simulate response of the elastomer to purely mechanical loading. The experiments for mechanical characterization are as mentioned in the section 4.1. The structural mechanics offers choice of several material models to choose from to apply to specify the material type. Hyperelastic material models are chosen to model the elastomer. Hyperelastic materials further allow the use of three types of material models - The Neo-Hookean, Mooney-Rivlin or the Murnaghan material model. In this work, the Neo-Hookean and Mooney-Rivlin material models have been used to model the elastomer and the electrodes. The Murnaghan model is generally used for non-linear acoustoelasticity and not considered in this study. COMSOL Multiphysics also supports geometric non-linearity (or non-linear geometry as is sometimes called) through the option of Large deformation ON/OFF. This option is kept for ON all the simulations Pre-strain COMSOL Multiphysics does not support the inclusion of a pre-strain or a pre-stress when using hyperelastic material models. This warrants the use of a stationary analysis prior to the actual analysis in-order to include the required pre-strain in the material. A stationary analysis would require modeling the geometry prior to any pre-strain. It would include the following steps: 1. Modeling the un-strained geometry 2. Running a stationary analysis which would include a required pre-strain in appropriate directions (in-plane and the thickness directions). Saving this solution. 3. Running the actual analysis starting from the saved solution is step 2 with experimental boundary conditions. This method however cannot be used when the electrodes are modeled because they are not 25

36 pre-strained. So, the equations defined for the elastomer sub-domain are modified to include the appropriate amount of pre-strain. In COMSOL, for hyperelastic materials, the stresses and strains are calculated from the Strain energy function which is defined using the components of the deformation tensor F. The deformation tensor is made up of displacement components and their derivatives. A stationary analysis is used to extract the values of the derivatives of the displacements for their values in the pre-strained state. These values are added to the equations defining the elastomer in the physics equations of the structural analysis module Solver In COMSOL Multiphysics a number of solvers can be used to solve the PDEs. Their memory requirements and computation times vary over a wide range and so does their capability of solving problems having range of degrees of freedom. For the problem of the electromechanical coupling of DEAP the number of degrees of freedom to solve for are in the range of The SPOOLS solver is a good option for solving such a problem. It also uses a lot less memory (~ 1GB) than some of the other solvers and gives the solution in an acceptable amount of time (~ sec). A parametric variation is specified for the input by using a parameter which uniformly varies the input from zero to the maximum specified value. This helps with gradual loading of the actuator and helps in convergence whereas applying the input (displacement or force) in one step sometimes leads to convergence problems Results The mechanical simulations help to arrive at the correct material parameters, both, for the elastomer and the electrodes. The elastomer is characterized by using fixed displacement simulations and comparing the results in the form of force vs. displacement plots to those obtained from experiments. The variation of material parameters is done as mentioned below: 26

37 1. The initial parameters used are based upon the approximate Young s modulus obtained from the stress-strain diagram of the tensile experiment. This modulus, as shown in Figure 16, is 1 MPa. 2. A force displacement plot obtained from the simulation is compared with the experimental force displacement plot. 3. The material parameters for the elastomer are varied by varying the Young s modulus. 4. The material parameters for each of the hyperelastic model Neo-Hookean & Mooney- Rivlin are finalized based on the fit between the experimental and simulation curves. The Neo-Hookean material model in COMSOL requires the material parameters shear modulus G and bulk modulus κ and the Mooney-Rivlin material model requires the material parameters C 10 and C 01. These parameters are derived using the equations presented in Section The values of the parameters derived for the two material models are as shown in Table 1. Table 1: Material parameters for the two material models Young's Modulus Neo-Hookean Material Parameters Mooney Rivlin Material Parameters G [Pa] κ [Pa] C 10 [Pa] C 01 [Pa] 1 MPa 3.34 x x x x MPa 4.16 x x x x MPa 5 x x x x 10 5 The simulations are run in displacement control mode. The displacement is the input and is slowly ramped from zero to 3 mm in the out-of-plane direction. The plot of input displacement vs the parameter is as shown in Figure

38 Displacement (mm) 3 Input Displacement Displacement Paramater Figure 18: Displacement Input in displacement control simulation. The reaction force is plotted against the displacement as shown in Figure 19. As seen from the plots, the stiffness of the elastomer increases for a small change in the Young s modulus value. It can be seen that the simulation results for elastomer Young s modulus values of 1.25 MPa with Mooney-Rivlin material model and 1 MPa in case of Neo-Hookean material model fit the experimental data well and represent the average experimental response. The other curves are either too stiff of less stiff as compared to the experimental result. So, for further simulations we use the data corresponding to these two values of Young s modulus and the respective material models. 28

39 Force (N) Force (N) 1E-15 Force vs. Displacement (Neo-Hookean) -1E-15 Force vs. Displacement (Mooney-Rivlin) Experimental 1 MPa 1.25 MPa 1.5 MPa Displacement (mm) Experimental 1 MPa 1.25 MPa 1.5 MPa Displacement (mm) Figure 19: Force vs Displacement curves for different values of elastomer Young's modulus. [34] A similar modeling approach is used to model the electrode material. Each electrode is assumed to be a homogenous and continuous body and they are also modeled using Hyperelastic material models. The electrodes are modeled connected to the elastomer geometry and exactly similar procedure is followed as mentioned for the elastomer parameter variation. The values of material constants for the various Young s moduli values used for electrodes are as shown in Table 2. The materials constants for Mooney-Rivlin material are derived exactly as derived for the elastomer model. Table 2: Material parameters for different values of young's modulus and material model Young's Modulus Neo-Hookean Material Parameters Mooney Rivlin Material Parameters G [Pa] κ [Pa] C 10 [Pa] C 01 [Pa] 2 MPa 6.67 x x x x MPa 8.33 x x x x MPa 1 x x x x 10 5 Figure 20 shows the force vs. displacement curves for elastomer with electrodes. From the 29

40 Force (N) Force (N) experimental data, it can be seen that the DEAP exhibits a highly hysteretic behavior. The plot for Mooney-Rivlin material with electrode Young s modulus equal to 2 MPa and Neo- Hookean model with electrode modulus equal to 3 MPa are approximately in the center of the hysteresis loop. Depending upon the frequency with which the experiments are conducted, the force can vary between the two branches of the hysteresis loop for a given displacement of the actuator. The material parameters used for the simulation are chosen to represent the average values. The force-displacement curve for 2 MPa Mooney-Rivlin model and 3 MPa Neo-Hookean model (as seen in Figure 20) pass approximately through the center of the hysteresis loop and the corresponding material parameters are chosen for simulating the electrodes. There is a clear difference between the force-displacement curves for the elastomer without electrodes and the elastomer with electrodes. Hence, in order to simulate the DEAP system, the effects of electrodes on the material behavior need to be taken into account. 0 Force vs. Displacement (with Electrodes) Neo-Hookean model 0 Force vs. Displacement (with Electrodes) Mooney-Rivlin model Experimental 1.5 Mpa 2 MPa 3 MPa Displacement (mm) Experimental 1.5 MPa 2 MPa 3 MPa Displacement (mm) Figure 20: Force vs Displacement for different values of electrode Young's modulus. [34] The hyper-elastic material models used are able to account for the non-linear material behavior of the elastomer and the electrodes. Also, modeling the DEAP actuator as a composite geometry with elastomer and electrode layers enables to account for the difference 30

41 in mechanical response from that of the elastomer. An approximation for the material properties of the elastomer and the electrodes is also made possible by variation of the material parameters in a systematic way and comparing the simulation results to the experimental results. This also sets up the model to be used with electro-mechanical boundary conditions to simulate the electro-mechanical coupling which is discussed in the next chapter. 31

42 Chapter 5. ELECTRO-MECHANICAL COUPLING The actuation of the DEAP actuator is a result of an electro-mechanical phenomenon where mechanical displacement of the DEAP is caused due to electrostriction when a voltage is applied across the electrodes. This chapter describes the simulations which account for the electromechanical coupling. The simulations are performed for both displacement control and force control and the results are presented. 5.1 Experiments There are modes of experiments that are performed on the DEAP actuators The displacement control mode and the force control mode. In the displacement control mode the DEAP actuator is displaced to certain known displacement value and then the voltage is applied across the electrodes by keeping the displacement fixed. The experimental setup for this experiment is similar to that as mentioned for the out-of-plane deformation experiments. A linear actuator is used to displace the DEAP to the specified displacement and voltage is cycled from zero to 2500 V and back to zero. The schematic for the experiments is as shown in Figure 21. Figure 21: Schematic for electro-mechanical experiments on the DEAP [33]. 32

43 For force control experiments different masses are suspended from the center of the DEAP and the voltage is again cycled from zero to 2500V and back to zero. The displacement of the center of the DEAP is recorded using a laser sensor. The stroke of the DEAP is calculated using the deflection values at the start of the voltage actuation and the maximum deflection of the DEAP. The schematic and the actual setup for the suspended weight experiments are as shown in Figure 22. Figure 22: Schematic of test setup (left) and the actual set-up (right). [34] 5.2 Simulation Procedure The electromechanical simulation is run according to the following procedure: 1. The mechanical force/displacement is first applied from zero to its maximum value without applying any input voltage. 2. When the force reaches its maximum value it is held constant at that value. 3. The input voltage is cycled from zero to 2.5 kv and back to zero keeping the applied force constant at its maximum value. 33

44 Force (N) Voltage (V) Displacement (mm) 5.3 Results Time Resolved Data The procedure for the electromechanical simulations has been mentioned in Section 5.2. A graphical representation of the inputs is as shown in Figure 23 (left). The time resolved displacement data for the above input is as shown in Figure 23 (right). As it can be seen from the output plot, the displacement increases as the applied force is increased to the maximum pre-deflection value. The vertical displacement further increases as the input voltage are applied holding the force constant at its maximum value. This further increase in displacement is the stroke of the actuator Force Voltage Displacement vs. Time Time / Parameter Time / Parameter Figure 23: Time resolved data. Input parameters (top left), Displacement output (top right). Experimental displacement output for one voltage cycle (bottom). [34] 34

45 Force (N) Force (N) Displacement [mm] Voltage [V] Force [N] A similar plot can be made when the simulations are run in displacement control mode. In this case displacement is the input and the reaction force becomes the output. The reaction force is measured at the inner vertical boundary of the elastomer. The time resolved data for simulations in displacement control mode is as shown in Figure Input Parameters Displacement 500 Voltage Time / Parameter Force vs Time (simulation) Force Time / Parameter 3000 Voltage vs. Time (Experiment) 0 Force vs. Time (Experiment) Voltage 20 (V) Time Figure 24: Time resolved data for Displacement control (Simulation Top, Experimental Bottom) Local Results Deformation and Stress The Figure 25 shows the series of deformed shapes for the elastomer during its mechanical deflection and electrical actuation. The deformation is similar to a beam bending deflection. The curved bending shows that the deformation is representative of a three dimensional deformation. 35

46 Figure 25: Deformation shapes of the elastomer during pre-deflection and actuation. [34] A closer view at the stresses and the deformation at the ends of the elastomer and the electrodes are shown in Figure 26 and Figure 27. Figure 26 shows the Second-Piola Kirchoff stress in the radial direction while Figure 27 shows the Second-Piola Kirchoff stresses in the thickness directions. The radial stresses are higher at both the ends of the elastomer because of the slight bending occurring due to the boundary conditions. The stresses in the elastomer away from the ends are fairly uniform and they reduce slightly as we move from the inner end to the outer end due to the slightly increasing radial area of the elastomer. The stresses in the electrodes show similar variation as those in the elastomer. They are lower because the electrodes are not pre-strained initially. In the case of the stresses in the thickness directions the plots show stresses during maximum actuation voltage. The pressure on the elastomer in the thickness direction is the maximum at the max actuation voltage and so are the compressive stresses. At the ends of the elastomer the stresses are tensile because of the boundary conditions at the ends. The stresses along the 36

47 thickness in the electrodes are constant and equal to zero. Figure 26: Radial Stress Contour plots 37

48 Figure 27: Thickness Stresses Contour Plot 38

49 Electric Field The contour plots for the electric field are as shown in Figure 28. The electric field in the elastomer is very high and is of the order of 10 8 V/m. The upper electrode has positive charge and the lower electrode has a negative charge. Hence, the direction of the electric field is from the upper electrode to the bottom electrode. Also the value of the electric field in the electrodes is zero as expected. Figure 28: Electric Field Contour Plots 39

50 Surface Charge Density [C/m2] Surface Charge Density [C/m2] Charge Distribution The charge distributions on the internal surfaces of the electrodes are plotted against the length of the electrode and shown in Figure 29. The charges on the upper and the lower electrode are have same magnitude but opposite in sign. There are singularities at the ends of the electrode which can be seen in the plots. The surface charge density on the rest of the electrode is almost constant at C/m Surface Charge Density (Top Electrode) Surface Charge Density (Bottom Electrode) Electrode Length [m] Electrode Length [m] Figure 29: Surface Charge densities on the internal surface of the top electrode (left) and bottom electrode (right) DEAP Stroke The design of DEAP actuator in a pumping system would depend upon the requirements of the flow rate and pressure head. An appropriate amount of DEAP stroke would be required to achieve these and hence the stroke prediction is necessary for and effective design. Experiments are done for stroke estimation as per the following procedure. Different masses are suspended from the center of the elastomer and then the voltage is cycled from zero to 2.5 kv and back to zero. The pre-deflection due to the suspended weights as well as the maximum displacement after cycling the voltage are noted. The difference between maximum displacement and the pre-deflection gives the stroke of the actuator for the weight 40

51 suspended and the corresponding pre-deflection. A schematic of the experimental setup is as shown in Figure 30. Figure 30: Test setup schematic (left) and actual picture of the test (right). [34] The results presented are using the Mooney-Rivlin and the Neo-Hookean material models with material parameters corresponding to the electrode Young s modulus of 2 MPa and 3 MPa respectively. As shown in Figure 31, the value of the suspended mass is plotted against the corresponding stroke of the DEAP. As the weight suspended is increased, the elastomer gets stiffer, due to larger pre-deflections and the stroke decreases. The simulation results also show this trend of decreasing stroke similar to the experimental results. The experimental curve will shift towards the right if the DEAP is loaded quasi-statically thus giving it time to relax. 41

52 Mass (grams) 30 Mass - Stroke Experimental Mooney-Rivlin Neo-Hookean Stroke (mm) Figure 31: Suspended Mass vs Stroke Global Results Force-Voltage Characteristics A graph of Force vs. Voltage has been plotted (Figure 32) for simulations run with fixed displacement of the elastomer and with Mooney Rivlin and Neo Hookean material models. The displacement is held constant at 2 mm. The force has been normalized prior to plotting. As seen from the experimental force displacement plots of Figure 3, the DEAP shows a hysteretic behavior. So, for a displacement of 2mm, the corresponding force can vary anywhere between the two branches of the hysteresis loop, depending on the creep of the material. Also, the model currently does not include the viscoelastic behavior. Hence, for comparison, we normalize the force from the experiments and the simulations and plot it against the voltage. It can be seen that the amplitude of the force from simulations shows a good match to the experimental value. 42

53 Force (N) 0.14 Force vs. Voltage Experimental Mooney-Rivlin Neo-Hookean Voltage (V) Figure 32: Blocking Force vs Voltage curve for fixed displacement of 2mm Capacitance Variation (Sensing Applications) The capacitance of the DEAP varies as the actuator get deformed out-of-plane. The capacitance is given by the equation: A C 0 r d (37) where C is the capacitance, ε 0 and ε r are the permittivity of free space and the dielectric constant of the elastomer, A is the surface area of the elastomer and d is the elastomer thickness. As the elastomer gets deformed the area will increase and the thickness decreases. Hence, the capacitance of the DEAP is a function of the deformation field. This can be potentially utilized to use the DEAP in a dual actuation and sensing mode. 43

54 Capacitance [pf] From the COMSOL model, for a force control simulation, the capacitance variation has been plotted against z-displacement (Figure 33). The capacitance is calculated by applying a small test voltage across the electrodes of the DEAP and noting the capacitance change with deformation of the actuator. 340 Capacitance vs Displacement Experimental Simulation Displacement [mm] Figure 33: Capacitance variation with deformation From the graph it can be seen that the capacitance value is in a close range as compared to that measured experimentally with a capacitance meter. Also, the trend that the simulation results show is similar to that shown by experimental observations. The simulation result is seen to deviate away from the experimental curve as the displacement. This may be due to the dielectric constant assumed constant for simulations and in reality it is a function of displacement. 5.4 Simulations with a.75 inch 2 DEAP actuator The parameters arrived at after the systematic variation and comparison with experimental data are used to simulate an actuator of a different size but same material. The 44

55 Force [N] simulations are done with the same two hyperelastic material models which are used to simulate the larger 1 inch 2 actuator. The results in the form of the force-displacement and force voltage plots are shown in Figure 34 and Figure Force V Displacement Experimental Neo-Hookean Mooney-Rivlin Displacement [mm] Figure 34: Force v Displacement for a.75 inch 2 DEAP actuator with electrodes 45

56 Force [N] 0.06 Force vs Voltage Experimental Neo-Hookean Mooney-Rivlin Voltage [V] Figure 35: Force v Voltage plot for DEAP75 actuator The force-voltage plots show that the model predicts the correct trend. The Neo-Hookean and the Mooney-Rivlin models show very similar force-displacement and force-voltage response. The model thus has been verified for two different actuators. 5.5 Further Experiments In order to identify the right actuator geometry, electrode thickness, elastomer thickness i.e. to design a right actuator-sensor for the right application, the finite element model should be able to predict the trends in actuator response under different conditions. This section presents some trend prediction using the Finite Element model developed in terms of maximum stress values in the elastomer and the electrodes, the maximum displacement for same load, capacitance variation for equal displacement. The effect of variation of electrode stiffness was shown in Chapter 4 with force-displacement plots. Here, the effects of electrode thickness variation and elastomer diameter on the above mentioned quantities are shown Electrode Thickness variation Simulations are done for two more values of electrode thickness 15 um and 20 um. The variation in maximum stresses in the electrodes and the variation in deflection for same pre- 46

57 Pre-Deflection [mm] load applied on the actuator are shown in Figure Pre-deflection Radial Stress [Pa] x Max Stresses Electrode Thickness [µm] Electrode Thickness [µm] Figure 36: Variation of pre-deflection and radial stresses in electrodes with thickness Actuator diameter variation The variation in stresses and capacitance for same displacement are shown in the figures below. Three actuator sizes of.75, 1.0 and 2.0 inch 2 were chosen. The actuators are called DEAP75, DEAP100 and DEAP200 respectively. The capacitance variation with displacement shows a similar trend for all the three actuators. The stresses plotted are Second Piola-Kirchoff Stresses in radial direction and they show an increasing trend. 47

58 Figure 37: Variation of Stresses with change in actuator diameter DEAP75 (Top Left), DEAP100 (Top right), DEAP200 (Bottom). 48

59 Capacitance [pf] Capacitance [pf] Capacitance [pf] DEAP DEAP Displacement [mm] DEAP75 DEAP Displacement [mm] DEAP200 DEAP Displacement [mm] Figure 38: Variation in capacitance with variation in diameter. Experimentally validated plot (top) and prediction for other geometries (bottom). The Figure 39 shows the charge distribution for the three actuators. It can be seen that the charge distribution remains the same for all the three actuators. This is because of the same material and the dielectric. 49

60 Surface Charge Density [C/m2] Surface Charge Density [C/m2] DEAP75 DEAP DEAP E E E E E E-03 Electrode Length [m] DEAP75 DEAP100 DEAP E E E E E E-03 Electrode Length [m] Figure 39: Surface Charge densities for Top electrode (top) and bottom electrode (bottom). 50

61 CONCLUSIONS In this work, an electromechanically coupled Finite Element model of a Dielectric Electroactive Polymer (DEAP) actuator has been developed as a first step toward developing a fully-fledged time dependent hysteretic model. Hyperelastic material models are used for modeling the elastomer and electrode materials. A systematic procedure is followed to determine the hyperelastic material model parameters. The mechanical and the electromechanical response of the material are successfully modeled. The comparison of the simulation results with the experimental results validates the FE model. The model predicts the mechanical and the electromechanical response fairly accurately and this can be seen from the force-displacement and the force-voltage plots. The capacitive sensing capabilities of the DEAP are also successfully implemented in the model and verified by comparing with experimental results. The model can be used for predicting trends in response for changes in actuator geometry and material parameters. Although the viscoelastic and the time dependent properties of the material are ignored for this case, the model can accurately predict the trends for a good proportion of the expected operating range of the actuator. Further improvement in the finite element model can be made by implementation of the time dependent and hysteretic material properties of the elastomeric materials. This will enable prediction of the DEAP response to various loading rates which might be experienced for potential applications. 51

62 REFERENCES [1] Ron Pelrine, Roy Kornbluh, Qibing Pei, and Jose Joseph, "High-Speed Electrically Actated Elastomer with Strain Greater than 100%," Science, vol. 287, pp , February [2] Ron Pelrine, Roy Kornbluh, and Jose Joseph, "Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation," Sensors and Actuators A: Physical, vol. 64, no. 1, pp , January [3] R Heydt, R Kornbluh, R Pelrine, and V Mason, "Design and Performance of an Electrostrictive-Polymer-FIlm Acoustic Actuator," Journal of Sound and Vibration, vol. 215, no. 2, pp , August [4] R Trujillo, J Mou, P.E. Phelan, and D.S. Chau, "Investigation of electrostrictive polymers as actuators for mesoscale devices," International Journal of Advanced Manufacturing Technology, vol. 23, pp , [5] Stephanie P Lacour, Harsha Prahlad, Ron Pelrine, and Sigurd Wagner, "Mechatronic system of dielectric elastomer actuators addressed by thin film photoconductors on plastic," Sensors and Actuators A: Physical, vol. 111, no. 2-3, pp , March [6] Federico Capri, Piero Chiarelli, Alberto Mazzoldi, and Danilo De Rossi, "Electromechanical characterisation of dielectric elastomer planar actuators: comparative evaluation of different electrode materials and different counterloads," Sensors and Actuators A: Physical, vol. 107, no. 1, pp , October [7] Qibing Pei, Ron Pelrine, Scott Stanford, Roy Kornbluh, and Marcus Rosenthal, "Electroelastomer rolls and their application for biomimetic walking robots," Synthetic Metals, vol. 135, pp , April

63 [8] Ron D Kornbluh, Ron Pelrine, Qibing Pei, Seajin Oh, and Jose Joseph, "Ultrahigh strain response of field-actuated elastomeric polymers," in Proceedings of SPIE 3987, 51, Newport Beach, CA, USA, [9] Xuequn Zhang, Christiane Löwe, Michael Wissler, Bernd Jähne, and Gabor Kovacs, "Dielectric Elastomers in Actuator Technology," Advanced Engineering Materials, vol. 7, no. 5, pp , [10] Ron Pelrine et al., "Applications of dielectric elastomer actuators," in Proceedings of SPIE 4329, 335, Newport Beach, CA, USA, [11] Guggi Kofod, "Dielectric Elastomer Actuators," The Technical University of Denmark, PhD Thesis ISSN: , [12] R.W. Ogden, "Large deformation isotropic elasticity on the correlation of theory and experiment for incompressible rubberlike solids," Proceedings of the Royal Society of London, A, vol. 326, pp , [13] Tianhu He, Xuanhe Zhao, and Zhigang Suo, "Dielectric elastomer membranes undergoing inhomogeneous deformation," Journal of Applied Physics, vol. 106, [14] Nakhiah Goulbourne, Eric Mockenstrum, and Mary Frecker, "A Nonlinear Model for Dielectric Elastomer Membranes," Journal of Applied Mechanics, vol. 72, pp , November [15] M Wissler and E Mazza, "Mechanical Behavior of an Acrylic Elastomer Used in Dielectric Elastomer Actuators," Sensors and Actuators A, vol. 134, pp , [16] L Rosenblatt, J Morrison, and L Iannuci, "Modelling Electroactive Polymer (EAP) Actuators: Electro-Mechanical Coupling using Finite Element Software," in Proceedings of SPIE 6927, San Diego, CA,USA,

64 [17] M Wissler and E Mazza, "Electromechanical Coupling in Dielectric Elastomer Actuators," Sensors and Actuators A: Physical, vol. 138, no. 2, August [18] Patrick Lochmatter, Silvain Michel, and Gabor Kovacs, "Electromechanical model for static and dynamic activation of elementary dielectric elastomer actuators," in Proceedings of SPIE Vol. 6168, [19] Woosang Jung and Yutaka Toi, "Computational Modeling of Electromechanical Behaviors of Dielectric Elastomer Actuators," in Proceedings of the International MultiConference of Engineers and Computer Scientists (IMECS), [20] M Mooney, "A theory of large elastic deformation," Journal of Applied Physics, vol. 11, pp , [21] O.H. Yeoh, "Some forms of the strain energy function for rubber," Rubber Chemistry and Technology, vol. 66, no. 5, pp , November [22] Zhan Gao, Alisar Tuncer, and Alberto Cuitino, "Modeling and Simulation of the Coupled Mechanical-Electrical Response of Soft Solids," International Journal of Plasticity, [23] M Wissler and E Mazza, "Modeling of a pre-strained circular actuator made of dielectric elastomers," Sensors and Actuators A, vol. 120, pp , [24] M Wissler and E Mazza, "Modeling and Simulation of dielectric elastomer actuators," Smart Materials and Structures, vol. 14, pp ,

65 [25] G Holzapfel, Nonlinear Solid Mechanics - A Continuum Approach for Engineering.: John Wiley & Sons Ltd, [26] M Wissler, "Modeling Dielectric Elastomer Actuators," PhD Thesis, Swiss Federal Institute of Technology, Zurich, [27] David J Griffiths, Introduction to Electrodynamics, 3rd ed.: Prentice Hall, [28] COMSOL Multiphysics, User's Guide, Version 3.5a, 2008, COMSOL Inc. [29] D. Kuhl and G. Meschke, Finite Element Methods in Structural Mechanics Lecture Notes, Ruhr University Bochum, May 2005 [30] I Müller and P Strehlow, Rubber & Rubber Balloons: Paradigms of Thermodynamics, Lect. Notes Phys Berlin Heidelberg: Springer-Verlag, [31] L.R.G Treloar, The physics of rubber elasticity. London: Oxford University Press, [32] Alexander York, Jason Dunn, and Stefan Seelecke, "Experimental Characterization of the hysteretic and rate-dependent electromechanical behavior of dielectric electroactive polymer actuators," Smart Materials and Structures, vol. 19, p. 9, August [33] Micah Hodgins, Experimental Analysis of Negative-rate Bias Spring Coupled to a Circular Dielectric Electro-active Polymer, Master s Thesis, NCSU,

66 [34] Micah Hodgins and Stefan Seelecke, "Experimental Analysis of Biasing Elements for Dielectric Electro-Active Polymers," in Proc. of SPIE, Vol 7976, , San Diego, CA, [35] Aseem Deodhar, Alexander York, Micah Hodgins and Stefan Seelecke, Finite Element Modeling of Electromechanical Coupling of Electroactive Polymer Actuators, in Proc. of SPIE 7978, 79780D, San Diego, CA,

67 APPENDICES 57

68 APPENDIX 1 Basic Mechanics and Electrostatics The appendix provides the theoretical background for non-linear continuum mechanics [25-26] and the forces on dielectrics with respect to the Maxwell s stresses [27-28]. A1.1 Deformation and Strain The deformation of a body can be specified by using two body configurations, the current configuration and the reference configuration. Consider a particle at a position X in the reference configuration and moves to a point x in the current configuration. The position of the particle in the reference configuration is independent of time and the current configuration gives the position of the particle at time t. Figure A1: Deformation of a body The relation between the current configuration and the reference configuration can be mapped through space and time by the relation x f ( X, t) (A1) where, the function f gives the mapping relation. The deformation of a material line element dx (in the reference configuration) to material 58