The strain response of silicone dielectric elastomer actuators

The strain response of silicone dielectric elastomer actuators G. Yang a, G. Yao b, W. Ren a, G. Akhras b, J.P. Szabo c and B.K. Mukherjee a* a Department of Physics, Royal Military College of Canada, Kingston, Ontario, K7K 7B4, Canada b Department of Civil Engineering, Royal Military College of Canada, Kingston, Ontario, K7K 7B4, Canada c DRDC Atlantic, PO Box 112, Dartmouth, Nova Scotia, B2Y 3Z7, Canada ABSTRACT Dielectric elastomers are known to produce large transverse strains in response to electrically induced Maxwell stresses and thus provide a useful form of electromechanical actuation. The transverse strain response of silicone (Dow Corning HS III RTV) based Maxwell stress actuators have been measured earlier as a function of driving electric field, frequency and pre-load. Experimental results show that a pre-load initially causes an increase in the strain. However, this increase appears to be a function of the relative geometries of the electroded area and of the specimen itself. The transverse strains in these materials decrease when larger values of pre-load are applied. Models of hyperelasticity that are capable of describing the large deformation of polymer materials have been used to interpret our results. Numerical finite element simulations of the material s behavior using a hyperelastic model provides good agreement with most of our observations on the electric field and pre-strain dependencies of the transverse strain. Keywords: dielectric elastomer actuator, Maxwell stress, transverse strain, hyperelastic model, finite element analysis 1. INTRODUCTION Dielectric elastomers have unique characteristics that make them promising materials for many applications in electromechanical transduction and active vibration damping. Unlike the more commonly used piezoelectric ceramics and single crystals, dielectric elastomers usually show very large electric-field-induced strains. Classical elastic theory usually does not describe their electromechanical properties very well, and a more accurate model is necessary for better understanding the response of a dielectric elastomer. When an electric field E is applied (shown vertically in Figure 1) to a dielectric elastomer film electroded on both sides, the film is subjected to a stress T due to the electrostatic force (Maxwell stress) between the electrodes and this causes the film to deform in the plane perpendicular to the applied field and stress 1. The Maxwell stress due to the applied electric field is 2 T = εεe, (1) where ε r is the relative dielectric permittivity and ε is the vacuum dielectric permittivity. r Only when the deformation of the film is very small, the transverse strain of the polymer film, S, can be approximately described by 2 2 S = (1 2 σσεε ) r E / 2Y. (2) However, most of the materials used as dielectric elastomers until now are rubber like polymers and have a non-linear elastic behavior. Equation (2), which is derived from classical linear elastic theory based on Hooke s law, does not describe their mechanical properties well; rather, the materials are hyperelastic 3,4. The non-linearity is clearly illustrated * mukherjee@rmc.ca

Figure 1. Illustration of the actuation due to the Maxwell stress. 3. 2.5 HSIII Silicone Polymer 2. Nominal Stress (MPa) 1.5 1..5. -.5.5 1. 1.5 2. 2.5 3. 3.5 Stretch Ratio Figure 2. Comparison of stress-strain measurements (dots, with the strain expressed as a stretch ratio) with a curve representing a hyperelastic model, for an HSIII silicone polymer sample. in Figure 2 where we show the strains measured as a function of stress for an HSIII silicone polymer sample; our experimental measurements, shown as dots, clearly lie along a line representing the predictions based on a hyperelastic model. For the model, we have assumed that the sample used for the data shown in Figure 2 is experiencing a uniaxial tensile load in its length direction when a load is applied to the sample. This assumption is fairly good if the film actuator is narrow enough (the ratio of the length to the width is 1:1). Our detailed experimental investigations of the transverse strain response of silicone and polyurethane elastomers 5,6,7,8 suggested that the geometry as well as the material properties determined the measured strains. High transverse strain required not only good material properties, but also optimized actuator geometry.

2. HYPERELASTIC MODEL In hyperelastic theory the material deformation is represented as a stretch ratio, λ i, instead of the strain, S i, that is commonly used in linear elastic theory. λ i is the ratio of final length to initial length in the direction of the i-strain axis, and has the following relationship with the strain: λ i = 1+S i (3) where λ i equals 1 for the un-deformed state. The strain energy density of a hyperelastic material depends on the stretch ratio via one or more of the three invariants, I i, of the stretch ratio tensor 4 : I 2 2 2 1 1 2 3 I2 = λ1λ2 + λ2λ3 + λ3λ1) 2 I = λ + λ + λ = 2 2 ( ) ( ) ( 2 ( λλλ ) 3 1 2 3 (4) For an incompressible material: λ 1 λ 2 λ 3 = 1 (5) so that I 3 equals 1 and does not contribute to the strain energy. When subjected to an external excitation, the response mode of the film under investigation will depend on its boundary condition. Figure 3 illustrates three special modes of an incompressible isotropic hyperelastic polymer film for which the deformation is relatively simple and easy to analyze. In the case of uniaxial stretching in the length direction, λ 1 = λ, and λ 2 = λ 3 = λ 1/2. Here, the suffixes 1, 2, and 3 denote the length, width, and thickness directions, respectively. Another simple situation is the planar (pure shear) stretch. In this mode we have λ 1 = λ, λ 2 = 1, and λ 3 = λ 1. The third mode is equi-biaxial stretch (inflation), which is equivalent to uniaxial compression where λ 1 = λ 2 = λ, and λ 3 = λ 2. For the same stretch ratio in the length direction, the thickness change depends significantly on the stretch mode (Figure 4). Correspondingly, the mechanical stiffness in the thickness direction will also be different. Uniaxial λ 1 =λ, λ 2 =λ 3 =λ -1/2 Planar (pure shear) λ 1 =λ, λ 2 =1, λ 3 =λ -1 Equi-biaxial (inflation) λ 1 =λ 2 =λ, λ 3 =λ -2 Figure 3. Three particular modes of a polymer film under stretching.

When an electric field is applied in the thickness direction, the mechanical response is not the same between the different stretch modes although the pre-stretch in length direction of a film caused by a preload is the same. The material is highly nonlinear (hyperelastic). The transverse strain caused by the Maxwell effect will also vary depending on the stretch mode. If the material is not fully covered by the electrode, the situation will become more complicated. Therefore the actuation produced is not only dependent on the material properties, but is also a function of the actuator structure and the test condition. 1. Stretch ratio in thickness.9.8.7.6.5 Uniaxial Planar Equi-biaxial.4 1. 1.1 1.2 1.3 1.4 1.5 Stretch ratio in length Figure 4. The thickness direction stretch ratios for three modes when the sample has the same stretch ratio in its length direction. Many models have been developed to describe hyperelastic materials 9,1,11,12. The most commonly used constitutive model for rubber was developed by Rivlin and its simplified form, the Mooney-Rivlin model, is often used for describing finite uniaxial deformation of rubber but it is inadequate for predicting strains associated with other modes of deformations 9. The Gent model 13 has a very simple form and, in the case of uniaxial stretching of an isotropic material, it can be used to express the nominal stress T 1 as T 1 = (C/3)(λ λ -2 )/(1-J/J m ), (7) where J 2 = λ+2λ 1-3. Using Equation (3) and assuming the strain S 1 to be very small, Equation (7) can be approximately expressed as T 1 = C S 1, (8) which has the same form as Hooke s law with C being the Young s modulus. This shows that the stress strain relation of a hyperelastic material can be represented by Hooke s law when the strain is very small, and it explains why the linear elastic model describes the behavior of the load-free actuator fairly well 2,5 given that the deformation due to applied electric field is normally not very large (the highest deformation reported was 3.25%). Figure 2 shows the agreement between our experimental data and Equation (7), which is represented by the continuous line in the figure. The two parameters used for generating the curve representing Equation (7) are.6 MPa for the small strain tensile modulus, C, and 25 for the maximum value J m of the J parameter of the HSIII silicone material. The J m value of 25, corresponds to a maximum stretch ratio of λ m = 5 13. Both parameters represent the material being

investigated reasonably well. Linear elastic theory, even with added non-linear terms, is unable to provide a good fit to the experimental observations. 3. FINITE ELEMENT ANALYSIS (FEA) MODEL The recent development of finite element methods has resulted in many hyperelastic models being available in commercial software packages such as ANSYS 14, ALGOR 15, and COSMOS 16. Due to its availability, we have chosen the Ogden model from the ANSYS package as our numerical tool to simulate the actuator functions. The sample is a HSIII silicone polymer film with layers of graphite powder applied to either side to act as electrodes. As our earlier observations had suggested that both material properties and actuator/electrode geometry influence the strain, several specimen geometries were considered in the finite element analysis. Figure 5 shows a narrow electrode actuator with a regular inactive edge; the width of the electrode is 3mm while the width of the in-active edges are 2mm. The thickness of the film is 197 µm while the thickness of the electrode is 5 µm. A twenty node hyperelastic element was chosen for the model and a three term Ogden model was used to simulate the silicone rubber. Here the actuator is assumed to be clamped at the upper edge and a platform is attached to the bottom edge to suspend any pre-load. The electric field induced stress was calculated using the electric field strength, the dielectric constant of the film and the vacuum dielectric permittivity. The resulting force between the electrodes (Maxwell stress) was applied to the surfaces of the electrodes. Figure 5. Finite element model for a narrow electrode polymer actuator 4. RESULTS AND DISCUSSION The results of the FEA were compared with experimental observations for two types of HSIII silicone specimens: (i) narrow electrode actuators with a regular inactive edge of 2 mm width, and (ii) narrow electrode actuators with a narrower inactive edge of 6 mm width. Figures 6 and 7 show a comparison between our experimental results and our FEA model for the elongation of the electrode and of the whole film due to pre-load. These figures show good agreement between the FEA model and observations of strain under pre-load. The strain produced by the applied electric field is the most important property for a polymer actuator. Figure 8 shows that there is a reasonably good agreement between the FEA results and our experimental measurements for the

elongation of an HSIII silicone specimen with a narrow inactive edge as a function of the applied electric field. Since the elongation of a Maxwell stress actuator is expected to be proportional to the square of the applied electric field when the electric field is not very large, the data of Figure 8 has been re-plotted in Figure 9 to show this quadratic relationship. 35 3 25 Electrode (experiment) Whole film (experiment) Electrode (FEA) Whole film (FEA) Elongation(mm) 2 15 1 5..2.4.6.8 1. 1.2 1.4 1.6 1.8 Force(N) Figure 6. Elongation as a function of pre-load for an HSIII silicone specimen with a regular inactive edge (2mm) Elongation(mm) 35 3 25 2 15 1 Electrode (Experiment) Whole film (Experiment) Electrode (FEA) Whole film (FEA) 5..2.4.6.8 1. Force(N) Figure 7. Elongation as a function of pre-load for an HSIII silicone specimen with a narrow inactive edge (6mm).

4 Electric Field Induced Elongation(µm) 35 3 25 2 15 1 5 Experimental results FEA results 2 4 6 8 1 12 14 16 Electric field(mv/m) Figure 8. Elongation of an HSIII silicone actuator with a narrow inactive edge as a function of the applied electric field. 4 35 3 Experimental results FEA results Elongation(µm) 25 2 15 1 5 5 1 15 2 25 E 2 (MV 2 /m 2 ) Figure 9. Elongation of an HSIII silicone actuator with a narrow inactive edge as a function of the square of the electric field

Figures 1 and 11 show the electric-field-induced elongation of the HSIII silicone film as a function of pre-strain for both electrode geometries. It is interesting to find that the elongation of the actuator with the narrow inactive edge is much higher than that with the regular inactive edge. This happens because the electrostatic attraction that causes the Maxwell stress only occurs over the electroded area whereas the elongation is measured over the entire film including the non-electroded inactive areas and so, when the inactive areas are relatively smaller the elongation is larger. 12 1 Elongation(µm) 8 6 4 5MV/m 7.5MV/m 1MV/m 2 5 1 15 2 25 3 Pre-strain (%) Figure 1. Elongation as a function of pre-strain for the HSIII silicone specimen with the regular inactive edge. 2 16 5MV/m 7.5MV/m 1MV/m Elongation(µm) 12 8 4 5 1 15 2 25 3 Pre-strain(%) Figure 11. Elongation as a function of pre-strain for the HSIII silicone specimen with the narrow inactive edge.

Figures 1 and 11 also show that the elongation first increases with pre-strain, reaches a maximum at a pre-strain of around 15% and then decreases as the pre-strain is further increased, which is similar to earlier observations 5,6,7,8. In Figure 12 the FEA model predictions for the electric-field-induced elongation as a function of pre-strain for the HSIII silicone specimen with the narrow inactive edge are compared with our observed values for various applied electric fields. The FEA model does not predict any decrease in strain at the higher values of pre-strain as has been observed, although the predictions agree well with observations for pre-strains below about 15%. The reasons for the divergence between the FEA results and the observations at larger pre-strains are not yet clear and are being currently investigated. 25 2 5MV/m(experiment) 7.5MV/m(experiment) 1MV/m(experiment) 5MV/m(FEA) 7.5MV/m(FEA) 1MV/m(FEA) Elongation(µm) 15 1 5 5 1 15 2 25 3 Pre-strain(%) Figure 12. Elongation of the HSIII silicone specimen with the narrow inactive edge as a function of pre-strain: comparison of FEA model predictions with experimental results Similar tests have been performed on a set of HSIII silicone actuators with different sample sizes and electrode geometries. The FEA model gives consistently good agreement with observations at low pre-strain values as detailed in the case discussed above. Analytic modeling becomes difficult when the polymer elongation does not occur according to the simple modes showed in Figure 3. Indeed FEA appears to be a good tool for modeling dielectric elastomer actuator behavior and it is likely to be particularly useful in understanding the performance of multi-layer actuators 8. 5. CONCLUSIONS Our experimental observations of the transverse strain response of silicone (Dow Corning HS III RTV) based Maxwell stress actuators as a function of driving electric field and pre-load cannot be understood on the basis of linear elastic theories but models of hyperelasticity that are capable of describing the large deformation of polymer materials can be used to interpret the results. Hyperelastic effects can be taken into account in numerical FEA simulations of the material s behaviour in order to provide an accurate prediction of actuator performance. The FEA models can be used to predict the elongation under pre-load and electric field and to find the optimized geometry of the actuator/electrode system. Our experimental and FEA studies on single-sheet silicone polymer film actuators have helped us to understand that both the geometry and the material properties of the actuator influence the observed actuation capability. Thus obtaining a high transverse strain requires not only good material properties, but also optimized actuator/electrode geometry.

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