# Actuation Behavior of Hydraulically Amplified Self-Healing Electrostatic (HASEL) Actuator via Dimensional Analysis

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## Abstract

**:**

## 1. Introduction

## 2. The Analytical Equation

#### 2.1. Derivation of the Electrostatics

#### 2.2. Derivation of the Fluid Mechanics

#### 2.3. Derivation of the Solid Mechanics

#### 2.4. The Electro-Mechanical-Hydraulic Equation

## 3. The Dimensional Analysis

## 4. Experimental Testing

#### 4.1. Testing the Material Properties

#### 4.2. Relevant Results from Testing the Material Properties

#### 4.3. Relevant Results from Testing the Actuator Performance

## 5. Discussion

_{1}, Π

_{2}, Π

_{3}, Π

_{6}, and Π

_{8}. These five Π groups will be discussed in detail.

#### 5.1. Π_{1} Group: Mass Flow Rate—Fluid Displacement, Viscous, and Viscoelastic Contributions

#### 5.2. Π_{2} Group: Fluid Velocity and Film Stiffness

#### 5.3. Π_{3} Group: Output Force—Fluid Pressure and Viscoelastic Contributions

#### 5.4. Π_{6} Group [Washington–Kim Group]: The Electro-Mechanical-Hydraulic Relationship

_{6}describes the contribution of all the parameters to the actuation system. This is of particular importance from a theoretical standpoint because this group directly shows the contributions from the electrical, mechanical, and hydraulic relationship, thus this Π group is denoted as the Washington–Kim (WK) group. The analytical equation shows that there is some sort of multiphysics relationship, but the dimensional analysis clarifies the contributions of the independent and dependent parameters. However, the most important feature to note is that this Π group shows the contribution of the electrostatics and the electric field and how that relates to the viscous forces, stiffness, and density.

^{2}due to the additional factors such as the viscous forces of the fluid, stiffness of the film, and the three-dimensional edge effects. The contributions of these factors can be seen throughout the data. However, based on the results from the experimental data, the applied voltage is the focus. Future work will quantify the contributions of the other factors. To further illustrate the connection between the electrostatic force and the applied voltage, a statistical significance test was conducted on the correlation coefficient between the experimental data and the WK group (Figure 14b). A correlation coefficient (r

^{2}) of 0.6225 was determined [28]. From the statistical significance test, it was determined that there is a strong (p < 0.05) correlation with 62.3% of variance explained (derivation in the Appendix A). Similar to the other Π groups, the stiffness of the film is the next major contribution to the result after the electrostatics. This is also seen in the experimental data.

#### 5.5. Π_{8} Group: Electrode Geometry Contribution

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Electrostatic Force Derivation

#### Appendix A.2. Fluid under a Hinged Plate Derivation

#### Appendix A.3. Thin Beam in Bending with Large-Deformation Derivation

#### Appendix A.4. Development of the Π Groups

#### Appendix A.5. Film Material Thickness

Film Material | Thickness [mm] |

BOPP | 0.025 |

LDPE | 0.050 |

CPP | 0.100 |

Ecoflex 00-30 | 0.800 |

#### Appendix A.6. Correlation Coefficient Calculation and the Statistical Significance Test

_{e}is the estimated/predicted values, $\overline{\mathrm{Y}}$ is the mean of the sample Y values, and Y is the observed data. This resulted in an r

^{2}value of 0.623. From this point, the t-score was calculated using the following equation [28]:

^{2}is the correlation coefficient and n is the number of sample values. This t-score will be compared to the Student’s t-distribution with n − 2 degrees of freedom at 95% confidence. The calculated t-score is 3.63. At 95% confidence, the t-score from Student’s t-distribution is 2.306. The calculated t-score is greater than the t-score from Student’s t-distribution thus r

^{2}is statistically significant.

#### Appendix A.7. The Common Nomenclature Used throughout the Text in the Analysis Sections

Variable | Meaning | Common Units |

$\mathrm{E}$ | Electric field or electric field through a substance | $\frac{\mathrm{V}}{\mathrm{m}}$ |

$\mathrm{q}$ | Charge | $\mathrm{C}$ |

${\mathsf{\epsilon}}_{0}$ | Vacuum permittivity constant, $8.854\times \left({10}^{12}\right)$ | $\frac{\mathrm{F}}{\mathrm{m}}$ |

${\mathsf{\epsilon}}_{\mathrm{r}}$ | Dielectric constant or relative permittivity | |

$\mathrm{A}$ | Surface area of conductive medium or electrode | ${\mathrm{m}}^{2}$ |

${\mathrm{E}}_{0}$ | Electric field in a vacuum | $\frac{\mathrm{V}}{\mathrm{m}}$ |

$\mathrm{V}$ | Voltage potential | $\mathrm{V}$ |

$\mathrm{C}$ | Capacitance between the electrodes | $\mathrm{F}$ |

${\mathrm{d}}_{\mathrm{f}}$ | Thickness of the film dielectric material | $\mathrm{m}$ |

${\mathrm{d}}_{\mathrm{l}}$ | Thickness of the liquid dielectric material | $\mathrm{m}$ |

${\mathsf{\epsilon}}_{\mathrm{rf}}$ | The dielectric constant for the film dielectric material | |

${\mathsf{\epsilon}}_{\mathrm{rl}}$ | The dielectric constant for the liquid dielectric material | |

$\mathrm{U}$ | The electrostatic potential energy | $\mathrm{J}$ |

${\mathrm{F}}_{\mathrm{ele}}$ | The electrostatic force | $\mathrm{N}$ |

${\mathsf{\tau}}_{\mathrm{yx}}$ | The one-dimensional shear stress for a Newtonian fluid with laminar flow | $\frac{\mathrm{N}}{{\mathrm{m}}^{2}}$ |

$\mathrm{u}$ | The fluid velocity in the x direction | $\frac{\mathrm{m}}{\mathrm{s}}$ |

$\mathsf{\rho}$ | Density of the fluid | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

${\mathrm{A}}_{\mathrm{f}}$ | Area of the flow channel | ${\mathrm{m}}^{2}$ |

$\mathsf{\theta}$ | Angle of the plate | $\mathrm{rad}$ |

$\mathrm{t}$ | Time | s |

$\mathrm{W}$ | Width of the electrode | $\mathrm{m}$ |

$\mathrm{L}$ | Length of the electrode | $\mathrm{m}$ |

$\mathrm{p}$ | The local thermodynamic pressure | $\frac{\mathrm{N}}{{\mathrm{m}}^{2}}$ |

${\mathrm{d}}_{\mathrm{max}}$ | The maximum thickness of the HASEL actuator above the x-axis | $\mathrm{m}$ |

$\mathsf{\sigma}$ | Mechanical stress | $\frac{\mathrm{N}}{{\mathrm{m}}^{2}}$ |

$\mathrm{Y}$ | Young’s modulus | $\frac{\mathrm{N}}{{\mathrm{m}}^{2}}$ |

$\mathsf{\u03f5}$ | Strain | $\frac{\mathrm{m}}{\mathrm{m}}$ |

$\mathsf{\nu}$ | Poisson’s ratio | $\frac{\mathrm{m}}{\mathrm{m}}$ |

$\mathrm{G}$ | Shear modulus | $\frac{\mathrm{N}}{{\mathrm{m}}^{2}}$ |

$\mathrm{w}$ | The beam deflection | $\mathrm{m}$ |

$\mathrm{I}$ | Second moment of inertia | ${\mathrm{m}}^{4}$ |

$\mathrm{m}$ | The mass of the electrode and film | $\mathrm{kg}$ |

${\mathrm{w}}_{\mathrm{x}}$ | The first derivative of the beam deflection | |

${\mathrm{w}}_{\mathrm{xx}}$ | The second derivative of the beam deflection |

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**Figure 1.**Example of a HASEL actuator created in the lab. The actuator pictured has a carbon tape electrode, a low-density polyethylene shell, and is filled with mineral oil.

**Figure 2.**The HASEL actuation mechanism. (

**Top**) The inactive actuator. (

**Middle**) The actuator has been activated and the critical draw-in voltage has been reached, thus the zipping of the electrode has begun. (

**Bottom**) Full draw-in of the electrodes has occurred and a majority of the liquid dielectric has been displaced causing the expansion of the film shell. This creates an output displacement and force.

**Figure 5.**The connection of the electrostatics, fluid dynamics, and solid mechanics in the analytical equation.

**Figure 6.**The 3D render of an approximate Peano-HASEL actuator as described by the analytical equation.

**Figure 9.**The experimental viscosity data of the liquid dielectric materials at laboratory temperature 20 °C.

**Figure 10.**The experimental performance testing setup. (

**a**). The testing setup for measuring the displacement. (

**b**). Testing setup for measuring the force using the load cell. (

**c**). Testing setup for measuring the lifting force.

**Figure 11.**The compiled strain–voltage data organized by the Film Type. A second order polynomial fit (y = ax

^{2}+ bx + c) is applied to the data to illustrate the V

^{2}relationship.

**Figure 12.**The compiled pressure–voltage data organized by the Film Type. A second order polynomial fit (y = ax

^{2}+ bx + c) is applied to the data to illustrate the V

^{2}relationship.

**Figure 14.**(

**a**). The normalized output displacement against the normalized dimensionless parameters of the WK group (Π

_{6}). The figure illustrates how the displacement is affected as each parameter gets larger. (

**b**). The normalized output displacement from the computed WK group (Π

_{6}) against the normalized experimental voltage. The WK group (Π

_{6}) was computed using experimental data (LDPE and olive oil).

Variables | $\mathbf{SI}\mathbf{Units}(\mathit{M},\mathit{L},\mathit{T},\mathit{i})$ | Meaning | Dependence |
---|---|---|---|

${\mathrm{D}}_{\mathrm{h}}$ | $\mathrm{L}$ | Characteristic length | D |

$\mathrm{u}$ | $\frac{\mathrm{L}}{\mathrm{T}}$ | Fluid velocity | D |

$\mathrm{p}$ | $\frac{\mathrm{M}}{\mathrm{L}{\mathrm{T}}^{2}}$ | Fluid pressure | D |

$\mathrm{I}$ | ${\mathrm{L}}^{4}$ | Second moment of inertia | I |

$\mathrm{A}$ | ${\mathrm{L}}^{2}$ | Area of the electrode | I |

$\mathrm{m}$ | $\mathrm{M}$ | Mass of the electrode and film | I |

$\mathrm{Y}$ | $\frac{\mathrm{M}}{\mathrm{L}{\mathrm{T}}^{2}}$ | Young’s modulus | I |

$\mathrm{g}$ | $\frac{\mathrm{L}}{{\mathrm{T}}^{2}}$ | Acceleration due to gravity | I |

$\mathsf{\rho}$ | $\frac{\mathrm{M}}{{\mathrm{L}}^{3}}$ | Fluid density | I |

$\mathsf{\mu}$ | $\frac{\mathrm{M}}{\mathrm{LT}}$ | Dynamic viscosity of the fluid | I |

*ε | $\frac{{\mathrm{i}}^{2}{\mathrm{T}}^{4}}{\mathrm{M}{\mathrm{L}}^{4}}$ | The permittivity | D/I |

CoV | $\frac{\mathrm{M}{\mathrm{L}}^{2}}{{\mathrm{T}}^{3}\mathrm{i}}$ | Applied voltage | I |

Dimensionless Group | Physical Meaning |
---|---|

${\Pi}_{1}=\frac{{\mathrm{D}}_{\mathrm{h}}\sqrt{\mathsf{\rho}\mathrm{Y}}}{\mathsf{\mu}}$ | The mass flow rate contribution and the viscous force contribution. Related to the displacement of the fluid. |

${\Pi}_{2}=\frac{\mathrm{u}\sqrt{\mathsf{\rho}}}{\sqrt{\mathrm{Y}}}$ | The fluid velocity contribution as it relates to the density and film stiffness. |

${\Pi}_{3}=\frac{\mathrm{p}}{\mathrm{Y}}$ | The output force contribution. |

${\Pi}_{4}=\frac{\mathrm{I}{\mathsf{\rho}}^{2}{\mathrm{E}}^{2}}{{\mathsf{\mu}}^{4}}$ | Film/electrode geometry contribution, thus a solid–fluid interface connection. |

${\Pi}_{5}=\frac{\mathrm{g}\mathsf{\mu}\sqrt{\mathsf{\rho}}}{\sqrt{{\mathrm{Y}}^{3}}}$ | The gravity contribution. |

${\Pi}_{6}=\frac{\mathsf{\epsilon}{\mathrm{V}}^{2}\sqrt{\mathsf{\rho}}}{\mathsf{\mu}\sqrt{\mathrm{Y}}}$ WK Group | The electrostatic contribution and its relationship to the viscous forces and material stiffness, the electro-mechanical-hydraulic connection. |

${\Pi}_{7}=\frac{\mathrm{m}\sqrt{\mathsf{\rho}{\mathrm{Y}}^{3}}}{{\mathsf{\mu}}^{3}}$ | The mass contribution of the film/electrode. |

${\Pi}_{8}=\frac{\mathrm{A}\mathsf{\rho}\mathrm{Y}}{{\mathsf{\mu}}^{2}}$ | The electrode geometry contribution, this is a direct solid–fluid interface connection and an indirect electro-mechanical connection. |

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**MDPI and ACS Style**

Washington, A.; Su, J.; Kim, K.J.
Actuation Behavior of Hydraulically Amplified Self-Healing Electrostatic (HASEL) Actuator via Dimensional Analysis. *Actuators* **2023**, *12*, 208.
https://doi.org/10.3390/act12050208

**AMA Style**

Washington A, Su J, Kim KJ.
Actuation Behavior of Hydraulically Amplified Self-Healing Electrostatic (HASEL) Actuator via Dimensional Analysis. *Actuators*. 2023; 12(5):208.
https://doi.org/10.3390/act12050208

**Chicago/Turabian Style**

Washington, Alexandrea, Ji Su, and Kwang J. Kim.
2023. "Actuation Behavior of Hydraulically Amplified Self-Healing Electrostatic (HASEL) Actuator via Dimensional Analysis" *Actuators* 12, no. 5: 208.
https://doi.org/10.3390/act12050208