#### 4.1. Force Measurement Results

The force measurement results for the conventional DEA without any rods are shown in

Figure 6. The large force outputs and the quicker rising trend were found in direction 1 compared with direction 2. The maximum force outputs were

${f}_{2}$= 0.17 N (±0.01 N) in direction 2 and

${f}_{1}$= 0.35 N (±0.01 N) in direction 1 at the maximum pre-strain condition. The results indicate that large pre-strains in direction 2 increased the force outputs in both directions. Pre-strains in direction 2 were observed to be more effective in amplifying the force output in direction 1. An approximate linear relationship is seen between the force outputs and pre-strains in direction 2. The force output from the RP-DEA, which has pre-strain

${\lambda}_{2,pre}=3$ only, is also shown in

Figure 6 for comparison.

It is recalled that Equation (7) was derived from the equation of state for the DE material to describe the effect of bi-axial pre-strain on the actuation force outputs.

Figure 6 indicates a good correlation between Equation (7) and the measured actuation forces in both directions. Offsets between the theory and experimental data may be due to the migration of the electrodes on the DEA. Although the electrode material was carefully repainted to both surfaces at each step before the activation, the quality of the re-application could be ensured only on the upper surface of the DEA. It was difficult to access and check the electrode conditions on the lower surface since the mounting condition was maintained for consistency of measurement. Furthermore, in Equation (7), the permittivity of the material is assumed to be constant. Large deformation of the elastomer changes the geometries and arrangements of the polymeric molecule chains, which may potentially vary the electromechanical coupling behavior of the material. In general, the derived correlation shows the potential for it to predict the actuation force output of the DEA structure in a bi-axial loading condition. It can also be used to guide the pre-strain configurations in a DEA design.

With the same pre-strains, ${\lambda}_{1,pre}=1.2$ and ${\lambda}_{2,pre}=3$, the force output of the RP-DEA with the rod configuration was measured to be ${f}_{1}$= 0.45 N (±0.01 N) in direction 1. It is a higher force output compared with that of the conventional DEA with the same pre-strain, 0.32 N (±0.01 N). The extra force output is due to the motion constraint, as the rods play the role of both maintaining pre-strain and constraining electrode movement in the RP-DEA.

In the durability study of the RP-DEA, the static force responses of the structure as in

Figure 1b were also measured to compare between Case 1 and Case 3. Five samples were measured for each case and the results are shown in

Figure 7. The force output of the RP-DEA with multi-ARs was found to be 6% less than the force output of the RP-DEA with a single AR. The reason is due to the decrease in the total area of the AR. However, the reduction in force output is considered to be small compared with the 20% reduction in the area of the AR. The narrow gaps between ARs may also slow down the spreading of charge over the AR during the charging phase, hence lead to the change in the dynamics of the RP-DEA. It would require further study to fully identify this effect.

#### 4.2. Lifetime Assessment Results

The lifetime results for all samples are summarized in

Figure 8. For Case 1, the operating lifetimes varied significantly from 5 s to 120 s. Overall, 80% of the samples failed within 60 s and the average lifetime was 39 s. The standard deviation was evaluated as 33 s. For Case 2, the average operating lifetime of the structures was increased significantly to 202 s. However, the measured lifetime had broad range of variation, between 103 s and 240 s. The standard deviation was calculated as 51 s. For RP-DEAs in Case 3, all samples survived to the full test duration of 240 s. The applied voltage was then increased gradually until the structures failed. The failing input voltages were found to be consistently around 9.5 kV.

The failures in all three cases were recorded as shown in

Figure 9. In Case 1, 95% of failures (19 out of 20) occurred close to the rods (

Figure 9a), while 5% of the failures (1 out of 20) occurred around the lead contact point. This indicates the regions close to rods primarily cause the premature failure of the RP-DEA. The electrode deposition should avoid these “weak” regions for the sake of improving the durability. For the RP-DEAs in Case 2), the failures were found to be associated with the center of the AR (

Figure 9b), where the tip of the cable contacts the DE in order to connect to the power supply. In this case, because the “weak” regions were no longer activated, the average lifetime increased significantly compared with those in Case 1. The in-AR lead contact was found to be the secondary cause that also causes the premature failure. For the RP-DEAs in Case 3, as both sources of failure were removed from the structure, all samples survived through the full duration of the life-assessment (4 min).

#### 4.3. Finite Element (FE) Simulation Configuration

In order to better understand the primary source of failure, the “weak” regions in the structure, the deformation of the RP-DEA was evaluated by 2D FE simulation. This was implemented using the four-noded rectangular element, Q4, in MATLAB. The elastomer is hyperelastic and if subjected to large deformations

$\left({\lambda}_{1}=3,{\lambda}_{2}=1.2\right)$, the FE analysis is nonlinear due to:

- C1.
The solid is anisotropic due to bi-axial loading

- C2.
The elastic moduli, ${E}_{i}\left({\lambda}_{1},{\lambda}_{2}\right)$ and ${E}_{2}\left({\lambda}_{1},{\lambda}_{2}\right)$, of the DE in the in-plane directions become functions of strains instead of remaining at the constant, $Y$

The nonlinear analysis was achieved by setting

${\lambda}_{2}=1$ as the initial boundary condition, and splitting the large deformation,

${\lambda}_{1}=3$, into 10 small deformation steps,

$\Delta {\lambda}_{1}=0.2$, from

${\lambda}_{1}=1$ to

${\lambda}_{1}=3$. The 11th step is applied to strain the DE with

${\lambda}_{2}=1.2$. Hence, the FE analysis for the

${j}^{th}$$\left(j=1,2,\dots .,11\right)$ step can be considered to be linear with respect to the elastic moduli,

${E}_{1-j}$ and

${E}_{2-j}$, as:

where

${d}_{j}$ is the vector of nodal displacements after current step,

$j$,

${\mathsf{\sigma}}_{j},{\mathsf{\u03f5}}_{j}$ are the corresponding vectors of the nodal stress and strain components, and

${p}_{j}$ is the vector of nodal coordinates that describes the shape of object before the current step. It is defined as:

where

${p}_{0}$ is the vector of nodal coordinates for the object in the undeformed state, and

${{\displaystyle \sum}}_{n=1}^{i-1}{d}_{n}$ is the summation of all nodal displacements from previously simulated steps. The final stress and strain distributions of the object,

${\mathsf{\sigma}}_{final}$ and

${\mathsf{\u03f5}}_{final}$, are evaluated in the same way as for total deformation in Equation (11):

For the large deformation up to

$\lambda =3$, the predictive stress–strain models of elastomer that is proposed by Carpi and Gei [

37] was used to describe such nonlinear material behavior:

The model is based on the one-term Ogden energy density function, depending uniquely on the Young’s modulus,

$Y$, which is appropriate for analysis up to the inflection (flex) point of the elastomer in nominal stress–strain correlation. For the VHB 4910 DE acrylic film,

${\lambda}_{flex}=3.3$. The nominal stresses can be obtained by differentiating Equation (13) to yield the expressions:

The elastic moduli can be solved by evaluating

$\frac{\partial {\sigma}_{i}}{\partial {\lambda}_{i}}$ to give:

This then gives the complete stress-strain relation. The incompressibility gives the shear modulus as

$G=Y/3$, the modified Hooke’s law for this 2D finite element simulation can be expressed in matrix form as:

where

$v$ is the Poisson ratio.

Figure 10 presents the pre-inflection stress-strain correlation of the elastomer when subjected to large deformation, as appropriate to Equation (15). The elastomer becomes softer as it is stretched further in both directions. The relaxations in both elastic moduli are taken over the discrete steps in the simulation.

Because the simulation is in 2D, it does not generate the compressive strain distribution in the direction of thickness directly. It is reasonable to represent the thickness strain

${\epsilon}_{33}$ as:

The applied boundary conditions were configured as shown in

Figure 11. The simulated DE was set in a square, which corresponds to the AR as in

Figure 2a. The sub-strain

$\Delta {\lambda}_{1}$ was applied on the three nodal sets in the left hand side edge of the square from the 1st to 10th steps of the simulation. The three corresponding nodal sets in the right hand side edge were fully constrained. This simulates the applied pre-strain that is held by the rod in direction 2, which was 200% in the experiment. The sub-strain

$\Delta {\lambda}_{2}$ was applied to the upper edge of the square, and the lower edge was fully constrained for the 11th step of the simulation. This simulates the 20% pre-strain from the experiment in direction 1.

The quasi-static linear FE simulation used here is to only demonstrate the potential non-uniform thickness distribution of the DE in the RP-DEA. The employed material model does not include the full viscoelasticity of hyperelastic elastomers, and the result only indicates the relative thick and thin regions across the structure. It is insufficient to provide the exact estimation on the deformation.

#### 4.4. FE Simulation Results

The simulated results for RP-DEA are presented in

Figure 12. It shows the DE to be deformed non-uniformly in the RP-DEA. In general, the thickness of the film decreases as it moves from the edge to the center in direction 2. The three thinnest regions with

${\epsilon}_{33}=-17\%$ are found around the rods, which correlate to the “weak” region that causes the premature failure shown in

Figure 8. When the RP-DEA has the single AR as in

Figure 12a, the AR covers all these thinnest regions. When the RP-DEA is configured with multi-ARs, the ARs only cover the relatively thin regions with

${\epsilon}_{33}>-15\%$. It hence avoids the resultant premature failure. In

Figure 13, the AR of the RP-DEA, as in

Figure 9a, were evenly divided into the left, center and right regions in direction 2. This evaluation suggests that in Case 1, 35% of the failures occurred in the left region; 50% of the failures occurred in the center; 15% of the failures occurred in the right region. It is in agreement with the simulation, where the “weak” regions around the rods cover a larger area in the center region compared with the side regions. More failures would therefore be expected to occur in the center region of the AR.

Considering the three potential sources of failure of the DEA: (i) dielectric strength; (ii) mechanical strength; and (iii) pull-in instability, the most likely failure mode in the RP-DEA is the dielectric strength as explained below:

- (i)
Dielectric strength failure: The failure occurs because the resultant local electrical field exceeds the dielectric strength of the film. A 7.5 kV actuation voltage was found to be close to the breakdown voltage of VHB 4910 in agreement with Plante’s work at the same pre-strain condition (

${\lambda}_{1,pre}=1.2$,

${\lambda}_{2,pre}=3)$ [

27]. Because the DE in the RP-DEA is deformed non-uniformly, the thinnest regions in the film or “weak” regions are close to the dielectric limit. All other relatively thicker regions have the smaller values of local pre-strain and higher breakdown voltages.

- (ii)
Mechanical strength failure: For VHB 4910, it has been shown that the film can be stretched up to the pre-strain of 600% [

27], which is well beyond the pre-strain configuration in this work. Therefore the mechanical strength is unlikely to be a source of failure.

- (iii)
Pull-in instability failure: This failure mode of the RP-DEA is less likely because: (1) no wrinkling was observed prior to the failure; and (2) the RP-DEAs were pre-strained with high stretch rate (

$\ge $0.01 s

^{−1}). In this case, the viscosity “stiffens” the elastomer and makes it resistant against the pull-in instability [

27].