Modeling and experimental investigation of multilayer DE transducers considering the influence of the electrode layers

$\sigma_\text{elast}, {i}$ in i-direction can be derived from the strain energy density $u_\text{elast}$ and the hydro-static pressure p depending on the stretch λ_i as presented in (3).

$$\begin{equation} \sigma_{\text{elast}, i} = \lambda_{i}\frac{\partial u_\text{elast}}{\partial \lambda_{i}} - p.\\\end{equation} \tag{ 3 }$$

$\sigma_{\text{fric}, i}$ is modeled with several Prandtl elements, consisting of a linear spring and a frictional element in series. Prandtl elements are used to describe stick-slip behavior of materials, which requires case distinction according to [18]. If the stress is lower than the stress parameter σ_f of the frictional element, the linear spring effect occurs, otherwise the stress is constant. The sign depends on the direction of the strain ε_i. This case distinction is demonstrated for one Prandtl element n in (4) using the linear spring parameter $Y_{\text{P}, n}$, the frictional stress parameter $\sigma_{\text{f}, n}$, and the strain dependent sign factor $\alpha_\mathrm{\varepsilon}$.

$$\begin{align} \sigma_{\text{fric}, {i}, {n}} = &\begin{cases}Y_{\text{P}, n}\varepsilon_{i} & \text{for}~ |\sigma_{\text{fric}, i, n}| \lt \sigma_{\text{f}, n} \\ \alpha_\mathrm{\varepsilon}\sigma_{\text{f}, n} &\text{for}~|\sigma_{\text{fric}, i, n}| \unicode{x2A7E} \sigma_{\text{f}, n}\end{cases},\,\\ & \text{with: } \alpha_\mathrm{\varepsilon} = \frac{\varepsilon_\text{i}}{|\varepsilon_\text{i}|},~ i \in \left\{x,\, y,\, z\right\}. \nonumber\end{align} \tag{ 4 }$$

$\sigma_{\text{viscela}, i}$ is simplified with one damping element and a generalized Maxwell model in parallel. The damper models a linear viscose stress depending on the velocity in i-direction with the linear viscosity η:

$$\begin{equation} \sigma_{\text{visc},i} = \eta\dot{\lambda}_{i}.\end{equation} \tag{ 5 }$$

Each Maxwell element consists of a linear spring (linear modulus of elasticity $Y_{\text{MW}, j}$ as material parameter) and damper (having a linear viscosity parameter $\eta_{\text{MW}, j}$ as material parameter) in series and is usually used to model the relaxation properties of elastomer materials, whereas the parallel damper describes the creeping properties [25].

The viscoelastic stress is determined by the sum of the generalized Maxwell model and a damper model in parallel:

$$\begin{align} & \sigma_{\text{viscela},i} = \eta\dot{\lambda}_{i}+\sum_{j = 1}^{N_\text{MW}}{\sigma_{{\text{MW}}, j,i}},\nonumber\\ & \sigma_{{\text{MW}}, j,i}+\frac{\eta_{\text{MW}, j}}{Y_{\text{MW}, j}}\dot{\sigma}_{\text{MW}, j,i} = \eta_{\text{MW}, j}\varepsilon_{i}.\nonumber\end{align} \tag{ 6 }$$

Is finally obtained with two systems of equations considering the material model in all three spatial directions under the assumption of incompressibility. For this, the stress balances in the three directions x, y, and z according to (1) and the stress expressions defined in (3), (4), and (6) are considered. Equation (7) consider the x- and z-direction, whereas the equations are transformed with the help of the hydrostatic pressure p. The equations in the y- and z-directions can be derived analogously by replacing the index x with the index y in (7).

$$\begin{align} \sigma_{x}-\sigma_{z} &= \sigma_{x-z} = \lambda_{x}\frac{\partial u_\text{elast}}{\partial \lambda_{x}} - \lambda_{z}\frac{\partial u_\text{elast}}{\partial \lambda_{z}}\nonumber\\ &\quad +\eta\left(\dot{\lambda}_{x}-\dot{\lambda}_{z}\right)+\sum_{j = 1}^{N_\text{MW}}{\sigma_{{\text{MW}}, j, x-z}} \nonumber\\ & \quad +\sum_{n = 1}^{N_\text{p}}{\left(\sigma_{\text{fric}, x,n}-\sigma_{\text{fric}, z,n}\right)},\nonumber\end{align} \tag{ 7 }$$

$$\begin{align} \sigma_{\text{MW}, {j, x-z}}& +\frac{\eta_{\text{MW}, j}}{Y_{\text{MW}, j}}\dot{\sigma}_{\text{MW}, j, x-z} = \eta_{\text{MW}, j}\left(\varepsilon_{x}-\varepsilon_{z}\right).\nonumber\end{align}$$

According to the generalized model (7) and the uniaxial deformation assumption (9), the effective elastic stress in x-direction is defined by:

$$\begin{equation} \sigma_{\text{elast}, x-z} = \sigma_{\text{elast}, x} = \lambda_{x}\frac{\partial u_\text{elast}}{\partial \lambda_{x}} - \lambda_{z}\frac{\partial u_\text{elast}}{\partial \lambda_{z}}.\end{equation} \tag{ 10 }$$

Several earlier investigations deal with different model assumptions for the strain energy density $u_\text{elast}$ for DE materials. For this reason, this study does not analyze different models. Since figure 4 shows a strongly non-linear behavior in the stress–strain diagram for the electrode material EL 3162, the Yeoh model was selected for this, while a Neo-Hookean model appears to be appropriate for the elastomer material EL 2030. Using the first invariant of the left Cauchy-Green deformation tensor $I_1 = \lambda_{x}^2+\lambda_{y}^2+\lambda_{z}^2$ the following assumptions of these two models are defined.

$$\begin{equation} u_\text{elast,Hook} = C_{10}\left(I_1-3\right),\\\end{equation} \tag{ 11 }$$

$$\begin{align} u_\text{elast,Yeoh} = C_{10}\left(I_1-3\right) +C_{20}\left(I_1-3\right)^2+C_{30}\left(I_1-3\right)^3.\nonumber\end{align} \tag{ 12 }$$

Using the uniaxial deformation assumption of (9), the two elastic models of (11), (12), and the effective elastic stress defined in (10), the following two hyperelastic equations are derived:

$$\begin{equation} \sigma_{\operatorname{elast}, x, \operatorname{Hook}} = 2C_{10}\left(\lambda_x^2-\frac{1}{\lambda_x}\right),\end{equation} \tag{ 13 }$$

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$$\begin{align} \sigma_{\operatorname{elast}, x, \operatorname{Yeoh}}& = 2\left(\lambda_x^2-\frac{1}{\lambda_x}\right)\nonumber\\ & \quad \times \left(C_{10} + 2C_{20}\lambda_\text{uni} + 3C_{30}\lambda_\text{uni}^2\right),\nonumber\\ & \quad \text{with: } \,\lambda_\text{uni} = \left(\lambda_x^2 +\frac{2}{\lambda_x}-3 \right).\nonumber\end{align} \tag{ 14 }$$

To define quasi-static conditions a speed variation was first carried out and a strain rate of $\dot{\lambda} = 0.002\,\,{\text{s}^{-1}}$ was found as suitable. The results are illustrated in figure 4.

Since the electrode material EL 3162 has a significant static hysteresis as expected, the characteristic curve averaged from the hysteresis is used for the parameterization. A characteristic property of the behavior is represented by the inflection point (see figure 4), at which the gradient changes from decreasing to increasing. If the gradient increases the local Young's modulus increases, resulting in stiffer material behavior. For EL 3162 the inflection point is determined at a strain of $\varepsilon\approx25\%$, for EL 2030 it is outside the measuring range. Furthermore, the measured stress is significantly higher for EL 3162 compared to EL 2030, resulting in larger material parameter values as shown in table 1.

Table 1 shows that the Neo-Hookean model for the elastomer material 2030 is sufficiently accurate in the identification range. The additional values given for the parameters $C_\mathrm{20}$ and $C_\mathrm{30}$ of the Yeoh model are negligibly small or zero. The linear Young's modulus is calculated with $Y = 6C_{10}$, resulting in $Y = 1.01\,\text{MPa}$ for EL 2030 and $Y_\text{e} = 1.44\,\text{MPa}$ for EL 3162. These values basically represents the greater stiffness of EL 3162 compared to EL 2030.

Since only EL 3162 has a significant static hysteresis as shown in the previous section, the friction material model is parameterized only for electrode material. According to (4), (7), and (9) the effective friction stress in x-direction for the Prandtl elements is defined:

$$\begin{align} & \kern-48pt \sigma_{\text{fric}, x,n}-\sigma_{\text{fric}, z,n} = \sigma_{\text{fric}, x-z,n} =\end{align} \tag{ 15 }$$

$$\begin{align} &\kern-48pt \begin{cases}Y_{\text{P},n}\varepsilon_{x}-Y_{\text{P},n}\varepsilon_{z} &\text{for}~|Y_{\text{P},n}\varepsilon_{z}| \lt \sigma_{\text{f},n}\nonumber \\ Y_{\text{P},n}\varepsilon_{x}-\alpha_{\varepsilon}\sigma_{\text{f},n} &\text{for}~|Y_{\text{P},n}\varepsilon_{x}| \lt \sigma_{\text{f},n}\unicode{x2A7D} |Y_{\text{P},n}\varepsilon_{z}| \nonumber\\ \alpha_\mathrm{\varepsilon}2\sigma_{\text{f},n} &\text{for}~|\sigma_{\text{fric}, {x,n}}| \unicode{x2A7E} \sigma_{\text{f},n} \end{cases},\end{align}$$

$$\begin{align} &\kern-48pt \text{with: }\,\alpha_\mathrm{\varepsilon} = \frac{\varepsilon_{x}}{|\varepsilon_{x}|},\qquad \varepsilon_{z} = \frac{1}{\sqrt{1+\varepsilon_{x}}}-1\nonumber.\end{align}$$

For the parameter identification several quasi-static sinusoidal-strain functions are applied for a prestrain of $\varepsilon_{x,0} = 25\%$ and a variation in strain amplitude $\hat{\varepsilon} = 0.1\%$–10%. The frequency is determined as a function of the strain amplitude so that the maximum velocity $v_\text{max} = 0.1\,\text{mm}\,\text{s}^{-1}$ is not exceeded to ensure quasi-static conditions. To determine the friction hysteresis separately, the elastic material properties must be considered (compare figure 2 and equation (1)). Therefore, the measured stress σ_x is subtracted by the hyperelastic stress σ_elast (see figure 5(a)) determined with the already parameterized elastic model from the paragraph before, according to:

$$\begin{align} \Delta \sigma_\mathrm{fric} = \sigma_x - \sigma_\mathrm{elast}\big\vert_{ \Delta \varepsilon = \varepsilon_x - \varepsilon_{x\mathrm{,0}}}.\end{align} \tag{ 16 }$$

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As figure 5(b) illustrates, two characteristic values can be exploited for the parameter identification, the hysteresis losses $U_\text{fric}$ and the stress slope $E_\text{fric}$, as demonstrated in [18]. Measurements are carried out with different maximum strain values and in each case $E_\text{fric}$ and $U_\text{fric}$ is calculated. The Prandtl parameters $\sigma_\text{f,i}$ and $Y_\text{P,i}$ are then parameterized in a way that the deviation between the measured and estimated values becomes minimal. The measurements, the basic procedure according to the identification method in [18], and the model results obtained with the estimation are shown in figures 5 and 6.

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The number of Prandtl elements was varied in the range of one to four. It was found that a number of three Prandtl elements provides satisfying accuracy within the investigated amplitude range, since no significant change in $U_\text{fric}$ and $E_\text{fric}$ for three elements compared to four elements was noticed.

The experimental data and analytical results are compared in figure 6. The corresponding parameter values can be found in table 2.

Table 2. Identified parameter values of the friction material model.

$Y_\text{P,1}$a	$\sigma_\text{f,1}$b	$Y_\text{P,2}$a	$\sigma_\text{f,2}$b	$Y_\text{P,3}$a	$\sigma_\text{f,3}$b
1.8	1.56	0.44	5.08	0.175	11

^aUnits: in MPa; ^bin kPa.

For the parameter identification of the Maxwell elements and the additionally considered damper in parallel sinusoidal strain excitations in a small signal range with a strain amplitude of $\varepsilon_\text{amp} = 0.5\%$, a prestrain of $\varepsilon_\text{x, 0} = 25\%$, and a frequency range of $f = 0.01-10\,\text{Hz}$ are experimentally investigated. To separate the viscoelastic stress from the measured stress, the already parameterized friction and hyperelastic model are used and the calculated static stresses are subtracted from the measured stress resulting in the viscoelastic stress. For the parameter identification the stress amplitude σ_amp, stretch amplitude λ_amp, and the phase angle ϕ are determined based on the measured viscoelastic stress–stretch trajectory enabling the calculation of the storage E' and loss modulus E'' with (17) as carried out in [25].

$$\begin{equation} E^{^{\prime}} = \text{cos}\left(\varphi\right)\frac{\sigma_\text{amp}}{\varepsilon_\text{amp}},\qquad E^{^{\prime\prime}} = \text{sin}\left(\varphi\right)\frac{\sigma_\text{amp}}{\varepsilon_\text{amp}}.\end{equation} \tag{ 17 }$$

From the parameters of the linear Maxwell elements ($Y_{\text{MW}, j}$ and $\eta_{\text{MW}, j}$) and the damper in parallel (η), the storage modulus E' and the loss modulus E'' can be determined as a function of the angular frequency:

$$\begin{align} E^{^{\prime}} &= \sum_{j = 1}^{N_\text{MW}}{Y_{\text{MW}, j}\frac{\omega^2\frac{{\eta_{\text{MW}, j}}^2}{{Y_{\text{MW}, j}}^2}}{1+\omega^2\frac{{\eta_{\text{MW}, j}}^2}{{Y_{\text{MW}, j}}^2}}}.\end{align} \tag{ 18 }$$

$$\begin{align} E^{^{\prime\prime}} &= \sum_{j = 1}^{N_\text{MW}}{Y_{\text{MW}, j}\frac{\omega\frac{\eta_{\text{MW}, j}}{Y_{\text{MW}, j}}}{1+\omega^2\frac{{\eta_{\text{MW}, j}}^2}{{_{\text{MW}, j}}^2}}}+\omega\eta.\end{align} \tag{ 19 }$$

In order to apply (18) and (19) for the identification, the measured stretch λ_x must first be converted to an equivalent stretch $\tilde{\lambda}_{x}$ considering the uniaxial deformation assumption of (9)

$$\begin{equation} \dot{\tilde{\lambda}}_{x} = \dot{\lambda}_{x}-\dot{\lambda}_{z} = \dot{\lambda}_{x}\left(1+0.5\lambda_{x}^{-3/2}\right).\end{equation} \tag{ 20 }$$

Corresponding to (7), (9), and (20), the viscose stress of the linear damper and Maxwell elements results in

$$\begin{align} &\sigma_{\text{visc}, {x-z}} = \eta\left(\dot{\lambda}_{x}-\dot{\lambda}_{z}\right) = \eta\dot{\tilde{\lambda}}_{x},\end{align} \tag{ 21 }$$

$$\begin{align} &\sigma_{\text{MW}, {j,x-z}}+\frac{\eta_{\text{MW}, j}}{Y_{\text{MW}, j}}\dot{\sigma}_{\text{MW}, {j,x-z}} = \eta_{\text{MW}, j}\left(\dot{\lambda}_{x}-\dot{\lambda}_{z}\right)\nonumber\\ & \kern138pt = \eta_{\text{MW}, j}\dot{\tilde{\lambda}}_{x}.\nonumber\end{align} \tag{ 22 }$$

For parameter identification the number of Maxwell elements was varied in the range of one to five elements. The challenge is to find a suitable parameter set so that the deviation for the storage and loss modulus is minimized over the measured frequency range. Both deviations of the modules between model and measurement are minimized with the same weighting. It turned out that the model error decreases only slightly with increasing number of Maxwell elements, consequently the material model is parameterized with only one Maxwell element and the linear damper in parallel, see figure 2. The resulting modules are compared with the measured ones in figure 7 in the frequency domain and the corresponding model parameters are provided in table 3.

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Table 3. Identified viscoelastic parameter values.

Material	ηb	$Y_\text{MW,1}$a	$\eta_\text{MW,1}$b
EL 2030	0.3	0.11	0.19
EL 3162	5.3	0.99	693

^aUnits: in MPa; ^bin kPa.

According to figure 7 the storage modulus of the electrode material EL 3162 has a strong increase in the range of low frequency $f \lt 1\,\text{Hz}$, resulting in a significant stiffening during dynamic operations. For frequencies $f \gt 2\,\text{Hz}$ the changes in the storage modulus and the viscosity parameter are small compared to the low frequency range. Both, the storage and loss modulus are quite high for EL 3162 compared to EL 2030, which indicates that in addition to stiffening and hysteresis, the electrode material causes both higher loss and relaxation. This can also be seen by the identified parameters, as they are approximately 5–12 times higher for EL 3162 compared to EL 2030. Although the model shows some deviations over the measured frequency range, the significant differences in the viscoelastic properties of the two investigated materials are modelled in a sufficient way and an impact of the electrode material properties on the behavior of DE transducers can be expected.

The electrical capacitance is measured with a variation in prestrain using an LCR meter (GW Instek LCR-817). In the LCR meter, a serial model is chosen as an equivalent circuit for the capacitance measurement. The measurement frequency is set to 100 Hz, which is within the bandwidth of the DE transducer. The definition of the capacitance, under consideration of parameter $\beta = \frac{A_\text{e}}{A_0}$ in [26], independent of the deformation assumption, is given in (38) and for assumption of planar deformation in (39),

$$\begin{align} &C = \varepsilon_0\varepsilon_r\frac{\beta w_0\lambda_{x}l_0\lambda_{y}}{d_0\lambda_{z}},\end{align} \tag{ 38 }$$

$$\begin{align} &C_\text{planar} = \varepsilon_0\varepsilon_r\frac{\beta w_0l_0}{d_0}\lambda_{x}^2, \text{with:} \nonumber\\ &\lambda_{y} = 1, \qquad \sigma_{z} = 0.\nonumber\end{align} \tag{ 39 }$$

According to figure 11, the measurements and calculated results coincide well in the undeformed state, but the change in capacitance when prestrain is applied, shows a distinct difference when assuming a planar deformation, although the aspect ratio of the samples is 1:9 in the undeformed state. Thus, the sample does not deform in a planar way as assumed, but rather a deformation in y-direction can be observed (showing a narrowing at the two free sides). As the prestrain is increased, the aspect ratio decreases and the effect of narrowing increases. Further, parts of the sample are squeezed due to clamping, that also affects the deformation.

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To improve the capacitance model, the occurring deformation is approximated by introducing the deformation exponent κ_y in y-direction and κ_z in z-direction with the following correlation:

$$\begin{equation} \lambda_{y} = \lambda_{x}^{\kappa_{y}},\qquad \lambda_{z} = \lambda_{x}^{\kappa_{z}}.\end{equation} \tag{ 40 }$$

In order to fulfill the property of incompressibility, condition (41) is required.

$$\begin{equation} \kappa_{y}+\kappa_{z} = -1.\end{equation} \tag{ 41 }$$

The equations (38), (40), and (41) lead to the improved capacitance model:

$$\begin{equation} C_\text{Fit} = \varepsilon_0\varepsilon_r\frac{A_\text{e}\lambda_{x}\lambda_{x}^{\kappa_{y}}}{d_0\lambda_{x}^{-1-\kappa_{y}}} = \varepsilon_0\varepsilon_r\frac{A_\text{e}\lambda_{x}^{2\left(1+\kappa_{y}\right)}}{d_0}.\end{equation} \tag{ 42 }$$

For planar deformation the coefficient is $\kappa_{y} = 0$ and for uniaxial deformation $\kappa_{y} = -0.5$. For the discussed measurements a factor of $\kappa_{y} = -0.225$ appears suitable, which is in between. The results are summarized in figure 11.

For the subsequent mechanical and electromechanical model validation, the determined assumption of real deformation, which is considered in (40) and (41), is used. The obtained model is presented in the appendix D, assuming in-plane actuation in the x-direction, hyperelastic neo-Hookean model for the elastomer, and Yeoh model for the electrode layers.

For the following evaluation of the overall modeling, the statistical error measure normalized root mean squared error (NRMSE) was selected:

$$\begin{align} \mathrm{NRMSE}_S = \frac{\frac{1}{N_S} \sum\limits_{l = 1}^{N_S} \sqrt{\frac{1}{N_e}\sum\limits_{k = 1}^{N_e}\left(y_{\text{meas,}S,l,k}-y_{\text{calc,}S,l,k}\right)^2}}{\left(\bar{y}_{\text{meas,}S}\right)},\end{align} \tag{ 43 }$$

where $S \in$ {S0, S1, S2, S3} represents the sample configuration, N_S is the number of samples with same properties, and N_e is the number of sample points where the individual measurements are evaluated. The corresponding NRMSE values can be found in tables 4-6 for mechanical elastic, viscoelastic, and electromechanical tests, respectively.

Table 6. NRMSE of the electromechanical testing (see figure 14).

Sample	S1	S2	S3
NRMSE($F_\text{e}$)	10.3%	8.56%	6.99%
NRMSE(εpre)	10.7%	20.5%	11.1%

Force–strain curves were measured in the strain range of $\varepsilon = 0\%{-}50\%$ at low velocity ($v = 0.01\,\text{mm}\,\text{s}^{-1}$), so that a quasi-static behavior assumption is justified. In figure 12(a) the mean measured curves of four comparable samples for each sample configuration are illustrated. To evaluate the changes with the thickness ratio, the averaged force increase $F_\text{inc}$ with respect to the averaged force measured for sample S0 according to (44) and the hysteresis losses $U_\text{fric}$ (see figure 5) are determined (see figures 12(b) and (c)).

$$\begin{equation} F_\text{inc} = \frac{F_\text{averaged}}{F_\text{averaged,S0}}.\end{equation} \tag{ 44 }$$

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As expected, the force and the hysteresis as well as their losses increase when the thickness ratio is enlarged, due to the material properties. Although the measured force over the strain shows certain deviations compared to analytic values, the calculated force increase $F_\text{inc}$ shows an accetable match for different thickness ratios. The calculated hysteresis loss appears slightly too high, however, the trend for varying δ coincides well with analytic values.

As next, the storage E' and loss modulus E'' were determined corresponding to the viscoelastic parameter identification in section 2.2, and in accordance with (17), by applying sinusoidal-strain-excitation. The stress is measured and calculated, respectively. The investigation is performed for the three exemplary frequencies $f = 1\,\text{Hz}$, $f = 3\,\text{Hz}$, and $f = 10\,\text{Hz}$, a prestrain of $\varepsilon_{x,0} = 12.5\%$ and a strain amplitude of $\varepsilon_\text{amp} = 0.5\%$. The measured and calculated results are illustrated in figure 13.

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Corresponding to larger storage and loss modules for the electrode material (see figure 7) both values increase with an increase in thickness ratio δ. Especially for the frequency f = 3 Hz, the calculated and measured values show a good agreement, whereas the difference is higher for f = 1 Hz and f = 10 Hz. This behavior can be explained by the model error of the viscoelastic material behavior of the pure materials (see figure 7). For example, the storage modulus of EL 3162 is underestimated for f = 10 Hz, which means that the storage modulus of the DE transducer is also underestimated. Nevertheless, the tendency for the examined frequencies is well modeled and thus, the impact of the electrodes on the viscoelastic DE transducer behavior is sufficiently calculated. It should be mentioned that these models are very sensitive to uncertainties and can hardly be improved by adding further Maxwell elements.

In [22] the two electromechanical values, the electrical force $F_\text{e}$ and electrical strain ε_e are introduced, both generated by the electrical field strength. Accordingly, these two characteristic values are measured for an electrical field strength of $E = {50}{\frac{V}{\mu\text{m}}}$ and a prestrain range of $\varepsilon_{x,0} = 10\%$–50%. The voltage is reduced as a function of the prestrain to maintain a constant electrical field strength. In figure 14 the measured and calculated results for the electrical force and strain are compared on the left and the value changes according to sample S1 (sample with the lowest electrode thickness) and a prestrain of $\varepsilon_{x,0} = 10\%$ on the right.

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Both, the measurement and the calculation do not indicate noticeable influence of the electrode thickness on the electrically generated force. Since the electrical field strength is kept constant at $E = {50}{\frac{V}{\mu\text{m}}}$, the electrical force decreases by increasing prestrain due to a reduction of the actuation area. The measured electrically generated force is about 15% lower than the calculated values, however the dependence of thickness ratio and prestrain are accurately modeled.

The electrically induced strain decreases as the electrode thickness increases as expected, and as the prestrain increases. Initially it rises to a maximum value and then it decreases. Although the absolute values are calculated slightly too high, model deviations when varying the electrode thickness are comparatively small. For both, the measured and calculated results, a shift of the maximum electrical strain to a lower prestrain can be observed by increasing the electrode thickness. An explanation for this behavior can be found in the nonlinear elasticity of the electrode material. As already discussed in section 2, EL 3162 is characterized with an inflection point in the stress–strain curve at about $\varepsilon\approx25\%$ (see figure 4), while the inflection point of EL 2030 is observed for $\varepsilon \gt 50\%$. Consequently, the point of maximum strain of the DE transducer decreases with the increase of the electrode thickness resulting in a shift of the maximum achievable electrical strain to lower prestrain $\varepsilon_{x,0}$. In conclusion, it can be stated that the calculated values for the electrically generated strain and force are slightly too high, but the influence of the thickness ratio caused by the electrode thickness is well represented.

It was shown that explicit material models for the elastomer and the electrode (section 2) can be used to derive holistic models for multilayer DE transducers (section 3) with sufficient validity. Although the absolute values deviate slightly, the influence of the electrode properties on the overall behavior can be determined very well with the tranducer model. Two main reasons are hypothesized for the model inaccuracies. First, the calculation assumes material parameters that are identified for pressed pure materials (see section 2.2), but may differ from the material properties of the printed electrode layers and the thin elastomer film. Secondly, the clamps can influence the mechanical behavior by squeezing the sample resulting in a thickness increase.