Effect of the deformation dependent permittivity on the actuation of a pre-stretched circular dielectric actuator

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Effect of the deformation dependent permittivity on the actuation of a pre-stretched circular dielectric actuator Chuan Zeng , Xiaosheng Gao PII: DOI: Reference:

S0093-6413(19)30457-4 https://doi.org/10.1016/j.mechrescom.2019.103420 MRC 103420

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Mechanics Research Communications

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16 May 2019 26 June 2019 28 September 2019

Please cite this article as: Chuan Zeng , Xiaosheng Gao , Effect of the deformation dependent permittivity on the actuation of a pre-stretched circular dielectric actuator, Mechanics Research Communications (2019), doi: https://doi.org/10.1016/j.mechrescom.2019.103420

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Highlights   

The deformation dependent behavior of the permittivity of dielectric elastomers is discussed The actuation behavior of a circular actuator made of a dielectric elastomer, VHB 4910, is 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 are studied

Effect of the deformation dependent permittivity on the actuation of a pre-stretched circular dielectric actuator Chuan Zeng, Xiaosheng Gao* Department of Mechanical Engineering, University of Akron, Akron, OH, 44325, USA *Corresponding author: [email protected] (Xiaosheng Gao) Tel.:+1-330-972-2415

Abstract Experiments have shown that under large deformation, the permittivity of some dielectric elastomers demonstrates significant deformation dependent behavior. In this paper, we theoretically analyze 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 is considered through the equations of state. The equilibrium equations of the actuator are solved by MATLAB. The electromechanical instability, loss of tension and dielectric breakdown of the actuator during the actuation process under different pre-stretch levels are discussed. It is found that the dependency of permittivity on deformation can effectively suppress the electromechanical instability of the actuator at low pre-stretch levels. At high pre-stretch levels, however, it lowers the maximum stretch the actuator can achieve. Compared to the case of a constant permittivity, the deformation-dependent effect results in higher voltage needed to achieve the same actuated stretch, and this discrepancy in voltage increases as the pre-stretch increases. Furthermore, it is found that the maximum achievable stretch and the corresponding voltage show a strong dependence on factor c in the permittivity-stretch relation, while the stretch at the onset of loss of tension is independent of c. © 2015 The Authors. Published by Elsevier Ltd. Keywords: Dielectric elastomer, deformation dependent permittivity, electromechanical instability, loss of tension, dielectric breakdown

1. Introduction Electrical actuators made from dielectric elastomers (DEs) can produce large actuated strain under applied voltage [1]. DEs also have other advantages such as lightweight, fast response, low cost, etc., and therefore, are considered as a promising class of smart materials. Over the last two decades, dielectric elastomer transducers (DETs) have been intensively studied in broad applications, such as artificial muscles, soft robotics [2] and energy harvesters [3]. As actuators, a basic form of DETs consists of a thin elastomeric membrane with two compliant electrodes coated on both sides [4]. 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. [1]. With this configuration, the pre-strain can simply be achieved by stretching and fixing the film on a circular rigid frame [5] or by a dead load [6]. It is an excellent test configuration to evaluate the biaxial actuation properties of new elastomers and electrodes [1,5,7] and to study the electromechanical instabilities [8]. 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 [5,7,9]. 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 the 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 [1,5]. To study the electromechanical ability of the VHB 4910, the permittivity is a key material property and has been experimentally studied by many groups [10–18]. The permittivity was found to decrease slightly as the stretch increases [10,15]. Di Lillo et al. [15] 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 observed significant drops in permittivity as deformation increases [11,13,16,17]. For example, Wissler and Mazza [11] observed that the permittivity drops by 44 % when the pre-stretch increases from 1 to 5 in a circular, equalbiaxial 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 largedeformation scenarios, where the stress term generated from the change of permittivity with deformation can become a significant part of the voltage-induced stress [19–22]. 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 paper. The early developments of the nonlinear theory of continuum electromechanics were made by Toupin [23], Eringen [24] and Tiersten [25]. Many remarkable works followed in applications to DE in recent years [26–28]. 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.

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McMeeking and Landis formulate an Eulerian form of the governing equations for quasi-electrostatics [26]. Dorfmann and Ogden developed a Lagrangian formulation based on a total energy function [27]. Suo et al. [28] 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. The rest of the paper is organized as follows. In Section 2 we summarize the basic equations of the nonlinear electroelasticity, where the dependency of permittivity on deformation is incorporated into the constitutive model. In Section 3 we specialize the equilibrium equations to the geometry of a circular membrane subject to the pre-stretch and electric field. We consider the material to be a Gent [29] dielectric. A nonlinear formulation of a pre-stretched circular actuator that includes both geometrical and material nonlinearities is presented. The resulted differential equations along with the boundary conditions are solved to obtain the voltage-stretch relations of the active region over a variety of pre-stretch values. In Section 4 the effect of deformation dependent permittivity on the actuation performance of the circular actuator is analyzed based on the obtained numerical results. The results suggest that the deformation dependent permittivity has a remarkable effect on the applied voltage to achieve a certain actuated strain, the electromechanical instability, the maximum stretch, and the dielectric breakdown of the actuator. We close with several concluding remarks in Section 5. 2. Summary of the basic equations 2.1. 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 in the reference configuration and in the current configuration. The circular membrane is defined as a set of particles, where and 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 (1) and the deformation gradient is defined as (2) The equation of motion of a particle in the circular membrane is expressed as (3) Therefore, ⁄ ⁄ ( ) (4) and the in-plane principal stretches are (5a) (5b) From Eqns. (5a)-(5b), we obtain

(6) In the absence of body force, the equilibrium equation in terms of true stress is written as (7) where represents the true stress tensor and is the divergence operator with respect to . Let denote the nominal stress. The true stress and the nominal stress are related by (8) where . 2.2. Equations of state It is a common practice to assume DE to be fully incompressible [4], namely (9) Here we adopted the quasilinear dielectric behavior [19], where the true electric field is linear with the true electric displacement , while the permittivity tensor of an incompressible DE, , is deformation dependent (10) For an incompressible DE membrane sandwiched between two electrodes, the electric field is only applied in the thickness direction, and the nonzero components of and are and respectively. Assuming the material is isotropic, the permittivity is the same in all directions. For simplicity, notations , and will be used instead of , and in the following text. According to the theoretical framework put forth by Dorfmann and Ogden [22], the free energy density of the DE membrane, , is a function of three independent variables, namely , , , where and is the electrical field in the reference configuration, i.e., . The stress differences and the true electric displacement can be expressed as (11a) (11b) (11c) where are the principal values of the true stress tensor. Eqns. (11a)-(11c) constitute the equations of state once is prescribed. From Eqns. (4), (5) and (8), the principal values of the nominal stress are given as ⁄ (12a) ⁄ (12b) (12c) A particular form of the free energy which decomposes into mechanical and electrical parts takes the following form [22] (13) where is the free energy due to stretching. It is noted that . Substituting Eqn. (13) into Eqns. (11a)-(11c), we obtain (14a)

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(14b)

c Experimental data c

-0.0107 [11] -0.053

-0.0073 [18] -0.0317

-0.0236 [17] -0.0777

-0.0241 [16] -0.0823

-0.0275 [13] -0.0892

2.3. Elastic free energy 3. Actuation of the circular DEA We choose the Gent model [29], which is capable of demonstrating the stretch-stiffening effect when the elastomer approaches the limiting stretch [30]. 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 (15) where [31] is the shear modulus, , and represents the upper limit of related to the limiting stretch. In this study, is used [32]. 2.4. 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 [11,13,16,17], while other studies showed a slight decrease of the permittivity under sufficient equal-biaxial prestretch [10,12,14,18]. Based on different experimental observations, different forms of permittivity function [13,16,17,33,34] 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 [22]. Jiménez and McMeeking [33] 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 [35], that relates the force of extension and polarizability anisotropy of a polymer chain to its fractional extension through the inverse Langevin function. Many experimental studies found a linear function was suitable to describe the dependence of the permittivity on the principal stretches [11]. Zhao and Suo [19] proposed the following function ̅ (16) where ̅ , with being the relative permittivity and the permittivity of the vacuum F/m. It is worth noting that Eq. (16) is a truncated Taylor expansion of the expression derived by Jiménez and McMeeking [33]. The function proposed by Zhao and Suo [19] is adopted in this study. Table 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 1. Values of c obtained by fitting experimental data reported by different research groups Experimental data

[10]

[18]

[14]

[12]

[36]

As shown in Fig. 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 region as annulus B. In the reference configuration shown in Fig. 1(a), let the radii be and for circle A and annulus B respectively. The circular membrane will first be pre-stretched equal biaxially to a stretch and fixed onto a rigid circular frame as shown in Fig. 1(b). After sufficient relaxation time, voltage will be applied on circle A, Fig. 1(c).

Fig. 1 The working principle of the circular actuator. In the reference state, circle A is the central circular area with a radius , and in the stretched state, it is the area coated with electrodes. Annulus B is the area outside circle A.

3.1. Circle A Under the Maxwell and electrostrictive stresses, circle A expands in area to reach total radial and hoop stretches of respectively with respect to the reference configuration (17a) (17b) where and are actuated stretches. The directions of stretches and stresses in 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., and . Setting and inserting and into Eqn. (14a), we have ̅ where

,

(18)

is the applied voltage and

is the

initial thickness of the membrane. Circle A and annulus B interact through the following boundary conditions no slip at : (19) force balance at

: (20)

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where

is defined in Eqn. (12a).

3.2. 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. (7). Specialized to the current problem, Eqn. (7) leads to (21) Combining Eqn. (6) and Eqn. (21), a first-order nonlinear ODE system with being the independent variable and being the dependent variables can be formulated as [

]⁄[

]

(22a) (22b)

where and are obtained from Eqn. (14) by setting well as the electric related terms to 0

as

the dashed lines are calculated using a varying permittivity with . The variation of the curves shows that increases while decreases with , and they both converge to as approaches . On the other hand, as approaches , increases steeply while decreases sharply. The total stretches for other values of pre-stretch, radius ratio and applied voltage show similar variation. Fig. 2 suggests that there is 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 results in being 15.6 % higher and being 10.7 % lower. This difference gradually disappears as approaches . This indicates 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.

(23a) (23b) The boundary condition at

is (24)

With boundary values prescribed at and , the ODE system can be solved with MATLAB solver bvp4c, which is a finite difference code that implements the threestage Lobatto IIIa formula [37]. For boundary condition Eqn. (19), is prescribed. For each given , the corresponding applied on circle A can be calculated. 4. 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 voltagestretch relations can be obtained for circle A with any given ⁄ values. ⁄ values of 5, 10 and 15 were and considered in this study and it was found that the results show a similar trend. In the calculations presented below, radial ratio ⁄ is used as an example and the voltage is normalized as ̅

√

̅

. As shown in Table 1, experimental

data obtained by different researchers suggest different values of c between 0 and -0.1. For example, the experimental data by Wissler and Mazza [11] suggest that . This value is used in most of the calculations presented here. 4.1. Stretches in annulus B As an example, Fig. 2 shows the total stretches and along the radial direction respectively of the annulus B for the ⁄ case of , and ̅ . Here the normalized radius ⁄ is used as the horizontal axis. The solid lines result from a constant permittivity ( ), while

Fig. 2 Comparison of the variations of radial and hoop stretches in annulus B with constant and varying permittivities.

4.2. Electromechanical instability Electromechanical instability, or the so-called pull-in instability, represents a major failure mechanism of the DE actuators [38], and it is well-known that pre-stretch can suppress the pull-in instability [39]. The pull-in instability of the pre-stretched circular actuator discussed above ( ⁄ ) is studied here. As an example, Fig. 3 shows the voltage-stretch curves for the constant permittivity case. At low pre-stretch levels, the voltage-stretch curve does not monotonically increase and exhibits a local maximum voltage value. This point marks the onset of pull-in instability [39]. As increases, the local maximum on the voltage-stretch curve gradually disappears. As 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 is incorporated into the normalized voltage ̅

√

̅

. As the initial thickness

increases, to have the same 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.

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Fig. 3 Voltage-stretch curves of circle A for c = 0 and different prestretch values.

However, at large deformation, the dependence of permittivity on deformation becomes significant. For permittivity varying according to Eqn. (16), Fig. 4 shows the voltage-stretch curves for the case of When i.e., the permittivity is a constant, pull-in instability will occur as the applied voltage reaches ̅ as marked by the solid diamond symbol. As decreases, a higher voltage is needed to achieve the same total stretch and the electromechanical instability is delayed. As drops below 0.085, the voltage-stretch curve becomes monotonic and the electromechanical instability is suppressed by sufficient influence of the dependence of permittivity on deformation.

Fig. 4 Voltage-stretch curves of circle A for c values.

Fig. 5 The variation of with for the case of and c = 0. turns negative when is larger than 6.73, which marks the onset of loss of tension of the membrane.

Fig. 6 shows the voltage-stretch curves of circle A for constant permittivity ( ), solid lines, and varying permittivity ( ), 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 is denoted by the diamond symbols and the loss of tension is denoted by the circle symbols. The curves are terminated at the onset of electromechanical instability or the onset of loss of tension, whichever comes first. In the case where electromechanical instability is suppressed, the stretch can go further and reach the material’s electric breakdown strength [24].

and different

4.3. 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 ( ⁄ ) is studied here. As circle A expands laterally against annulus B, the tensile stress in circle A reduces with . Fig. 5 shows the variation of

with

for the case of c = 0 (constant

permittivity) and . When , marks the loss of tension in the membrane. As

, which becomes

larger than 6.73, becomes 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.

Fig. 6 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.

4.4. Discrepancy in voltage-stretch response As shown in Fig. 6, for , the onset of loss of tension happens first and is at the same total stretch value for both constant and varying permittivity. When the pre-stretch is increased to , 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 right before failure. However, as further increases, the discrepancy of the voltage-stretch curves between constant and varying permittivity cases starts to appear at an early stage of the actuation process. Moreover, for a given pre-stretch value, the discrepancy gets larger as the total stretch increases. For

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example, for the case of , to obtain a total stretch of 4.5, ̅ is 40.1 % higher for the c = -0.053 than c = 0. 4.5. Dielectric breakdown It has been observed that the electric breakdown strength of DE is dependent on the deformation, i.e., the breakdown electric field increases as deformation increases [1,10,14,16,40]. Several phenomenological relations between the breakdown strength and the stretch have been suggested based on experimental data. Huang et al. [40] suggested based on equal-biaxial stretching experiments of VHB4910, which is adopted in this study. Consequently, the normalized voltage at electric breakdown can be expressed as ̅

√

̅

, which is represented by the dashed

line in Fig. 6. Liu et al. [41] proposed a more general electric breakdown strength model which further involves the nonlinearities in permittivity and elastic modulus. In their study, the experimental data by Huang et al. [40] 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 ( ⁄ ). Fig. 6 indicates that when is close to 1, loss of tension occurs first. As increases, for example, when , the electromechanical instability occurs first because the membrane gradually gains more tension with higher pre-stretch. At , the actuator encounters electromechanical instability before loss of tension when the permittivity is constant. However, for varying permittivity with c = -0.053, the electromechanical instability is suppressed and loss of tension occurs at a larger stretch level. The dependence of permittivity on deformation enables the actuator to achieve a total stretch 28.4 % higher than the c = 0 case. At , the pull-in instability is suppressed for both constant and varying permittivity, and loss of tension takes place at the same total stretch for both cases. When , the electric breakdown determines the actuator’s maximum total stretch and the dependence of permittivity on deformation significantly lowers the maximum total stretch achievable by the actuator. For examples, when , the maximum total stretch for the c = -0.053 case is 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 is 50.16 % higher than the c = 0 case.

4.6. Variation of c This section examines the effect of the electrostrictive factor c on the behavior of the actuator. Fig. 7 depicts the voltage-stretch curves with different c values for the cases. Fig. 8 shows the variations of the maximum achievable stretch and the corresponding applied voltage over a range of c values for the cases respectively.

Fig. 7 Comparison of the voltage-stretch curves of circle A with different c values for the cases .

When the pre-stretch is low, e.g., , Fig. 8(a) indicates that the potential failure of the actuator is due to pull-in instability, and as the value of c decreases, both the maximum stretch and the corresponding voltage increase. When the pre-stretch is increased to , as suggested by Fig. 8(b), the situation becomes more complicated. When c is close to 0, the pull-in instability happens first; as c decreases, the pull-in instability is gradually suppressed and loss of tension happens first; as c decreases further, loss of tension is delayed, and the actuator can reach its electric breakdown strength. The maximum stretch increases as c decreases over the region where failure is due to pull-in instability; it is invariant of c over the region where failure is due to loss of tension; and it decreases as c decreases over the region where failure is due to electric breakdown. The applied voltage at the maximum stretch always increases as c decreases.

Fig. 8 Variations of values for the

and the corresponding ̅ over a range of ccases.

When the pre-stretch is further increased to , Fig. 8(c) indicates that the pull-in instability has been suppressed while the loss of tension is delayed, and the actuator is able to reach its electric breakdown strength. As c decreases, the voltage at breakdown increases while the maximum stretch decreases. The results for have similar trends to the case. 5. Concluding remarks As an essential test configuration, the circular actuator is used to analyze the electrostrictive effect on the actuation

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behavior of DETs. We have shown that, compared with constant permittivity, the deformation dependent behavior of permittivity affects the actuation performance of the circular actuator in a variety of ways and the impact differs at different pre-stretch levels. The dependence of permittivity on deformation has little influence when the pre-stretch is close to 1. As the pre-stretch increases, the dependence of permittivity on deformation strongly affects 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 is found to suppress the electromechanical instability that enables the actuator to achieve a higher total stretch. At high pre-stretch levels, however, as the cause of failure is due to electric breakdown, the dependence of permittivity on deformation significantly lowers the maximum achievable total stretch. Compared to the case of constant permittivity, the dependence of permittivity on deformation results in higher voltage needed to achieve the same actuated stretch, and this discrepancy in voltage increases as the pre-stretch increases. Furthermore, it is found that the maximum achievable stretch and the corresponding voltage show a strong dependence on the electrostrictive factor c, while the stretch at the onset of loss of tension is independent of c. Although a circular actuator is considered in this study, where the Gent model is used to describe the mechanical behavior of the material and the linear function proposed by Zhao and Suo [19] is used to describe the dependency of permittivity on deformation, similar analyses can be done for other geometries and material models.

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