Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators

Abstract

Soft materials exhibiting large deformation under external stimuli have gained an increasing attention in the recent past because of their potential applications in soft transducers aimed at achieving biomimetic actuation. This paper theoretically analyzes the effect of internal properties, including entanglements, crosslinks and finite extensibility of polymer chains along with inherent viscoelastic properties of polymers on the performance of Dielectric Elastomer actuator (DEA) in dynamic modes of actuation. A physics based nonaffine material model proposed by Davidson and Goulbourne is used to model the polymer chains entanglements, crosslinks and finite extensibility. To incorprate the viscoelastic properties, a rheological material model based on the additive decomposition of the isotropic strain energy density into equilibrium and viscous parts is implemented. A computationally efficient method, which relies on the principle of least action is used for extracting the governing equation representing the dynamic motion of the DE actuator. The results demonstrate that DEAs with strong entanglements and crosslinks along with small finite extensibility of polymer chains exhibits lower deformation level in DC dynamic modes of actuation. It is inferred that the strong entanglements and crosslinks in polymer chains enhances the resonant frequency, but debilitate the intensity of vibration of viscoelastic DEAs. Further, the periodicity and stability of the nonlinear oscillations exhibited by the viscoelastic DEAs are assessed by employing the Poincare maps and phase portraits. The results of the present investigation can be applied effectively in bridging the mechanism between the microcosmic polymer chains and macroscopic dynamic behavior of viscoelastic DE actuators.

Appendix: Governing equations of DE membrane subjected to equal biaxial prestress

In order to validate the efficacy of the developed framework with the experimental observations reported by [30], it is assumed that the DE membrane is also subjected to initial equal biaxial prestress S. The work done due to applied equal biaxial prestress S is expressed [43] as

$$\begin{aligned} \displaystyle W_{pre} = - 8HL^2 \left( {2S\lambda } \right) \end{aligned}$$

(A.1)

Hence, the total potential energy U (Eq. 9) of the considered DE actuator subjected to electromechanical loading become

$$\begin{aligned}&U = 8HL^2 \nonumber \\&\quad \times \left[ \displaystyle \begin{array}{l} \underbrace{\left[ {\begin{array}{*{20}c} {\displaystyle \frac{1}{6}\mu _c \left( {2\lambda ^2 + \lambda ^{ - 4} } \right) } \\ \\ {\displaystyle - \mu _c \lambda _{\max }^2 \ln \left( {3\lambda _{\max }^2 - 2\lambda ^2 - \lambda ^{ - 4} } \right) } \\ \\ {\,\displaystyle + \mu _e \left( {2\lambda + 2\lambda ^{ - 1} + \lambda ^{ - 2} + \lambda ^2 } \right) } \\ \end{array}} \right] }_{isotropic\,\,equilibrium} \\ \\ \displaystyle + \underbrace{\left[ {\begin{array}{*{20}c} {\displaystyle \frac{1}{6}\beta \mu _c \left( {2\lambda ^2 \lambda _v^{ - 2} + \lambda ^{ - 4} \lambda _v^4 } \right) } \\ \\ { \displaystyle - \beta \mu _c \lambda _{\max }^2 \ln \left( {3\lambda _{\max }^2 - 2\lambda ^2 \lambda _v^{ - 2} - \lambda ^{ - 4} \lambda _v^4 } \right) } \\ \\ { \displaystyle + \beta \mu _e \left( {2\lambda \lambda _v^{ - 1} + 2\lambda ^{ - 1} \lambda _v + \lambda ^{ - 2} \lambda _v^2 + \lambda ^2 \lambda _v^{ - 2} } \right) } \\ \end{array}} \right] }_{isotropic\,\,viscous} \\ \\ \displaystyle - \underbrace{\left[ {\displaystyle \frac{1}{2}\varepsilon E^2 \lambda ^4 } \right] }_{electrical} - \underbrace{\displaystyle \left[ {2S\lambda } \right] }_{prestress} \\ \end{array} \right] \nonumber \\ \end{aligned}$$

(A.2)

Since the relaxation due to applied prestress is assumed to be completed before voltage excitation starts, one can determine the equilibrium stretch $\lambda _p$, induced by the prestress. At equilibrium state, the viscous strain is equal to the total strain, i.e., $\displaystyle \left. {\lambda _v } \right| _{t = 0} = \left. \lambda \right| _{t = 0}$ [12]. For this particular stated state, the associated elastic energy in the hyperelastic element of the Maxwell element is released, i.e.,work done associated with viscous part $W_v$ = 0. Upon invoking this condition and on setting the first variation of the total potential energy U with respect to stretch $\lambda$ equal to zero, i.e., $\displaystyle \frac{{\mathrm{d}U}}{{\mathrm{d}\lambda }} = 0$, the equilibrium equation is expressed as

$$\begin{aligned}&\displaystyle \mu _c \left( {\lambda - \lambda ^{ - 5} } \right) \displaystyle \left( {\frac{1}{3} + \frac{{2\lambda _{\max }^2 }}{{3\lambda _{\max }^2 - 2\lambda ^2 - \lambda ^{ - 4} }}} \right) \nonumber \\&\qquad - \displaystyle \mu _e \left( {1 + \lambda - \lambda ^{ - 2} - \lambda ^{ - 3} } \right) - S \nonumber \\&\quad \displaystyle = \varepsilon \left( {\frac{\phi }{{2H}}} \right) ^2 \lambda ^3 \end{aligned}$$

(A.3)

The value of prestress S required to obtained the prestretch $\lambda _p$ in the DE membrane can be evaluated by implementing $\lambda = \lambda _p$ and $\phi = 0$ in Eq. (A.3) for any given value of material parameters as

$$\begin{aligned} \displaystyle S= & {} \mu _c \left( {\lambda _p - \lambda _p^{ - 5} } \right) \left( {\frac{1}{3} + \frac{{2\lambda _{\max }^2 }}{{3\lambda _{\max }^2 - 2\lambda _p^2 - \lambda _p^{ - 4} }}} \right) \nonumber \\&- \mu _e \left( {1 + \lambda _p - \lambda _p^{ - 2} - \lambda _p^{ - 3} } \right) \end{aligned}$$

(A.4)

The resulting nonlinear governing differential equation of motion Eq. (13) of the DE actuator including the expressions of initial equal biaxial pre stress is expressed as

$$\begin{aligned}&\displaystyle \left( {\rho L^2 + 2\rho H^2 \lambda ^{ - 6} } \right) \ddot{\lambda }= 6\rho H^2 \lambda ^{ - 7} {\dot{\lambda }} ^2 \nonumber \\&\quad { \displaystyle - 3\left[ {\begin{array}{*{20}c} \begin{array}{l} \displaystyle \mu _c \left( {\lambda - \lambda ^{ - 5} } \right) \left\{ {\frac{1}{3} + \frac{{2\lambda _{\max ^2 } }}{{\left( {3\lambda _{\max ^2 } - 2\lambda ^2 - \lambda ^{ - 4} } \right) }}} \right\} \\ \\ \displaystyle + \beta \mu _c \left( {\lambda \lambda _v^{ - 2} - \lambda ^{ - 5} \lambda _v^4 } \right) \left\{ {\frac{1}{3} + \frac{{2\lambda _{\max ^2 } }}{{\left( {3\lambda _{\max ^2 } - 2\lambda ^2 \lambda _v^{ - 2} - \lambda ^{ - 4} \lambda _v^4 } \right) }}} \right\} \\ \end{array} \\ {} \\ \\ \begin{array}{l} \displaystyle + \mu _e \left( {1 + \lambda - \lambda ^{ - 2} - \lambda ^{ - 3} } \right) + \beta \mu _e \left( {\lambda _v^{ - 1} + \lambda \lambda _v^{ - 2} - \lambda ^{ - 2} \lambda _v - \lambda ^{ - 3} \lambda _v^2 } \right) \\ \\ \\ \displaystyle - \varepsilon E^2 \lambda ^3 \displaystyle - 2S \\ \end{array} \\ \end{array}} \right] } \end{aligned}$$

(A.5)

The evolution Eq. (15) of the considered DE membrane is written as

$$\begin{aligned}&\displaystyle \frac{{\mathrm{d}\lambda _v }}{{\mathrm{d}t}} + \displaystyle \frac{{2\beta \mu _c }}{\eta }\nonumber \\&\quad \left[ \begin{array}{l} \displaystyle \left( {\lambda ^{ - 4} \lambda _v^3 - \lambda ^2 \lambda _v^{ - 3} } \right) \left\{ {\frac{1}{3} \displaystyle + \frac{{\lambda _{\max ^2 } }}{{3\left( {3\lambda _{\max }^2 - 2\lambda ^2 \lambda _v^{ - 2} - \lambda ^{ - 4} \lambda _v^4 } \right) }}} \right\} \\ \\ + \displaystyle \frac{{\mu _e }}{{\mu _c }} \left( {\lambda ^{ - 1} \displaystyle + \lambda ^{ - 2} \lambda _v - \lambda \lambda _v^{ - 2} - \lambda ^2 \lambda _v^{ - 3} } \right) \\ \end{array} \right] \nonumber \\&\qquad = 0. \end{aligned}$$

(A.6)

Hence, modified three initial conditions of the DE membrane due to initially applied pre stress (Eq. 16) are expressed as

$$\begin{aligned} \left. {\frac{{\mathrm{d}\lambda }}{{\mathrm{d}t }}} \right| _{t = 0} = 0; \,\,\,\,\,\,{\left. {{\lambda }} \right| _{t = 0}} = \lambda _p;\,\,\,\,\,\,{\left. {{\lambda _v}} \right| _{t = 0}} = 1. \end{aligned}$$

(A.7)

In order to fit the experimental data, Lu et al. [30] implemented the Gent model with the following material parameters of the DE membrane (VHB4905), shear modulus $\mu = 45$ kPa, permittivity $\varepsilon = 3.98\times 10^{ - 11}$ F m$^{ - 1}$, thicknes $H = 0.5$ mm and limiting stretch material constant $J_{\lim } = 120$ respectively.

To validate the present model, we also use the same group of material parameters. we have $\mu = \mu _e + \mu _c = 45$ kPa, permittivity $\varepsilon = 3.98\times 10^{ - 11} Fm^{ - 1}$, thicknes $2H = 0.5$ mm, $\displaystyle \lambda _{\max } = \sqrt{\frac{{J_{\lim } + 3}}{3}} = 6.4$ and modulus ratio $\displaystyle \frac{{\mu _e }}{{\mu _c }} = 0.18$. This modulus ratio can be tuned to get the best fitted curves with the experimental results. The other important required material parameters considered are the density of the DE membrane (VHB4905) $\rho = 960$ kg m$^{ - 3}$ [33] and viscoelastic relaxation time, which ranges from a few seconds to hundreds of seconds at room temperature, $\displaystyle \tau _v = \frac{\eta }{{\mu _c \beta }}=10^2$ s [7, 20].

Finally for the stated parameters, the nonlinear differential Eq. (A.5) together with evolution Eq. (A.6) and initial conditions (Eq. A.7) are solved using MATLAB for evaluating the lateral stretch $\lambda$ at applied voltage $\phi$. The voltage $\phi$ was applied using a programmable voltage source with a ramping rate of $0.5kVs^{ - 1}$.