Thermo-electro-mechanical behavior of dielectric elastomer actuators: experimental investigations, modeling and simulation

The mechanical behavior of the DEA primarily results from the mechanical properties of the elastomer. The silicone used as elastic dielectric shows an elastic and viscoelastic behavior. It behaves isotropic and is incompressible with a Poisson's ratio of ν ≈ 0.5.

The deformation behavior of an elastomer can be represented by a stress-strain curve. To obtain this curve, a uniaxial compression test with the measurement setup shown in figure 3 is carried out. The compression tests were performed to reach deformations which resemble the ones during operation. For testing, samples of ELASTOSIL$^{\circledR}$ P7670 with 2.5 mm thickness and 5.8 mm diameter are prepared. The test samples are compressed between the lubricated adapters with a velocity of 0.01 mm s⁻¹ and the resulting force is measured. The engineering stress is calculated by dividing the measured force by the size of the initial area on which the force is acting. The engineering strain is calculated by dividing the length change by the initial length of the sample.

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Although the mechanical behavior of the DEA primarily results from the mechanical properties of the elastomer, the influence of the graphite powder electrodes' should be known. Because it is not possible to do the compression tests with loose graphite powder, test samples with the same dimensions as the silicone test samples are cut out of the active area of a DEA and again the uniaxial compression test is carried out. Figure 4 shows the obtained stress-strains curves of the pure ELASTOSIL$^{\circledR}$ P7670 and the actuator test samples.

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It can be seen that the graphite electrodes' in the DEA sample lead to a steeper stress-strain curve compared to the pure silicone sample. In addition, both curves show a hysteresis which is due to the viscoelastic properties of the silicone.

The time-dependent viscoelastic behavior is described by creep and relaxation processes. Creep means that if the elastomer is subjected to a constant mechanical stress step it responds with an initial strain step followed by an exponential increase of the strain over time. Relaxation is the asymptotic decrease in stress over time after an initial stress step due to an applied strain step (see Zhao et al [21]).

For determining the viscoelastic behavior of the ELASTOSIL$^{\circledR}$ P7670 and the DEA a creep test and a relaxation test were performed. For the creep test, a test sample was subjected to a constant force which results in an initial compression strain of about 20 %. During the relaxation test a constant compression strain of 25 % was applied to the test sample. Displacement respectively force were measured for 20 minutes. The determined creep and relaxation curves are shown in figures 5 and 6. The forces needed for a compression of about 20 % differ for the pure silicone sample and the actuator sample. In case of the actuator, the force needed is higher because as seen in the stress-strain curves the actuator behaves stiffer than the pure silicone due to the electrodes'. For this reason, the resulting stress in the actuator is higher for a constant compression of 25 % as in the silicone sample. However, the silicone and the actuator samples show similar time-dependent creep and relaxation behavior.

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The DEA setup can be transformed into an electrical equivalent circuit, which is shown in figure 7. It consists of the capacitance $C_\text{a}$, the parallel resistance of the dielectric polymers layers $R_\text{p}$ which causes the leakage current and the series resistance $R_\text{s}$ which includes the electrode resistances and the contact resistances of the electrical connections [22].

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The value of the capacitance $C_\text{a}$ can be calculated by

$$\begin{equation} C_\text{a} = \epsilon_0 \epsilon_\text{r} \frac{A}{h_\text{d}} \,\text{.} \end{equation} \tag{ 1 }$$

It is defined by the material properties of the elastic dielectric (vacuum permittivity ₀, relative permittivity $\epsilon_\text{r}$) and the actuator design (active region A, thickness of the dielectric layer $h_\text{d}$).

A direct measurement of the relative permittivity $\epsilon_\text{r}$ of the dielectric layer is impossible. Therefore, a sample structure is fabricated. Copper electrodes' with a defined size are sputtered on a silicone membrane with 50 µm thickness. With an LCR meter the capacitance of the sample structure is measured and the relative permittivity is calculated after rearranging equation (1):

$$\begin{equation} \epsilon_\text{r} = \frac{1}{\epsilon_0} \frac{C h_\text{d}}{A} \end{equation} \tag{ 2 }$$

The value of the calculated relative permittivity $\epsilon_\text{r}$ of ELASTOSIL$^{\circledR}$ P7670 is 3.

In general, the value of the parallel resistance $R_\text{p}$ of unmodified silicone material is of the order of gigaohms up to teraohms. For ELASTOSIL$^{\circledR}$ P7670 $R_\text{p}$ is about 40 GΩ [23]. With $\rho_\text{d} = R_\text{p} A/h_\text{d}$ the specific resistance results to around 1 TΩ cm.

As already mentioned, the electrodes' consist of graphite powder particles. For this type of electrode a constant specific resistance, as it is valid for homogeneous materials, does not exist. But it is possible to measure the sheet resistance $R_\text{e}$ which is defined as

$$\begin{equation} {R_\text{e}} = \frac{\rho_\text{e}}{h_\text{e}} \end{equation} \tag{ 3 }$$

with $\rho_\text{e}$ as specific resistance and $h_\text{e}$ as electrode thickness. For the sheet resistance measurement a sheet resistance meter is used. The measured value for the graphite powder electrodes' is 10 kΩ. By rearranging equation (3), the value of $\rho_\text{e}$ is calculated. With the given electrode thickness of 5 µm the specific electrode resistance $\rho_\text{e}$ results to 5 Ω cm.

In this work the contact resistances are neglected. But previous measurements have shown that the used type of electrical connection results in contact resistances at the transition from compliant electrodes' to the stiff copper wires between 10 kΩ and 80 kΩ [24].

The application of an electric voltage to the DEA leads to an electrostatic attraction between the electrodes'. The effective electrostatic pressure $\tilde{p}_\text{el}$ can be calculated by [7]:

$$\begin{equation} \tilde{p}_\text{el} = \epsilon_0 \epsilon_\text{r} \frac{U^2}{{h_\text{d}}^2} = \epsilon_0 \epsilon_\text{r} E^2 \end{equation} \tag{ 4 }$$

The electrostatic pressure depends on the relative permittivity $\epsilon_\text{r}$ of the elastomer, the applied voltage U and the thickness $h_\text{d}$ of the dielectric elastomer layers. The voltage applied to the DEA is restricted by the dielectric breakdown field strength of the elastomer. Taking into account the thickness $h_\text{d}$ of the elastic layers and the breakdown field strength of the material a maximum voltage can be calculated. However, it is important to take into account the varying thickness of the layers.

After the application of the electric voltage the DEA deforms until the electrostatic pressure is balanced by the elastic pressure of the elastomer. The resulting deformation of the DEA depends on the mechanical material properties of the elastomer. The stiffer the material, the smaller the deformation.

Figure 8 shows the measured actuator strain depending on the applied electric voltage and field strength for the sample DEA introduced before. In figure 8(b), the straight line represents the electric field, i.e. the voltage over the actual thickness, and the dashed line the voltage over the initial thickness. This means that for a given strain the actual electric field is larger than the prescribed nominal one. Due to the breakdown strength $E_\text{EBS}$ of ELASTOSIL$^{\circledR}$ P7670 with at least 30 V µm⁻¹ the maximum applied voltage U during testing is 1200 V. This leads to an electric field strength E of around 26 V µm⁻¹ taking into account the thickness change of the elastomer layers. At the maximum applied voltage the deformation of the DEA is about 9 %. For the thickness measurement two laser displacement sensors (Keyence LK-G32) are used. During actuation, the deformation of the DEA is measured from both sides [25].

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Dielectric elastomer actuators heat up during operation. This heating up due to dynamic excitation is a thermo-electro-mechanical process. On the one hand this is due to the non-ideal conductivity of the electrodes' and the contact resistances and on the other hand this is due to the visco-elasticity of the elastomer. During operation of the DEA, charging and discharging current is flowing through the electrodes' and a leakage current is flowing through the polymer. Because of the resistances, the electric current leads to a heating along the electrodes' and at the electrical connection of the DEA. In addition, heating in the elastomer material occurs due to the deformation of the material and the appearing internal friction of the molecules and due to joule heating. The increase of temperature in the DEA depends strongly on the dynamic excitation. In general, DEAs are used at frequencies f of the electric voltage from DC to around 1 kHz.

For measuring the heating of the sample actuator, a voltage U of 1 kV at frequencies f of 10 Hz, 100 Hz and 1 kHz is applied for 30 minutes to the electrodes'. During this time the temperature of the whole DEA is measured with a thermal imaging camera (FLIR SC655). Figure 9 shows the thermal images of the tested DEA after 30 seconds and 30 minutes operation with 1 kHz and 1 kV. A symmetric temperature distribution was obtained after 30 seconds, as expected. The temperature distribution after 30 minutes was not symmetric. This behavior may be caused by an uneven degradation of the left and the right electrical connections of the DEA. The temperatures at the two points marked in figure 9 are analyzed and the results are shown in figure 10. The diagrams show the influence of the frequency to the heating of the DEA. With increasing frequency the temperature in the actuator increases as well. In addition it can be seen, that during operation the DEA heats up slowly in the middle of the active area and much faster next to the electrical connections. For higher frequencies the area next to the connections reaches a temperature of almost 125 °C and the heating process seems not to be finished. The region in the middle of the actuator reaches temperatures between 33 °C and 45 °C.

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To evaluate the mechanical behavior of the DE structure, a model is needed, which is capable of describing visco-hyperelastic behavior. The aim of this section is, to (i) present the mechanical field equations and (ii) introduce the applied mechanical material model.

In the following equations the index notation is used, as well as the Einstein summation convention for the coordinate indices k, l, m = 1, 2, 3. The deformation gradient tensor $\boldsymbol{\mathbf{F}}$ is given by:

$$\begin{equation} {F_{kl}=\dfrac{\mathop{}\partial x_k(X_m,t)}{\mathop{}\partial X_l}\,\text{.}} \end{equation} \tag{ 5 }$$

Here, $\vec{x}$ represents the current position vector and $\vec{X}$ the referential position vector. The deformation gradient can be decomposed polarly into a rotational and a stretch component

$$\begin{equation} {F_{kl}=R_{km}U_{ml}=V_{km}R_{ml}\,\text{,}} \end{equation} \tag{ 6 }$$

where $\boldsymbol{\mathbf{R}}$ is the rotation tensor, $\boldsymbol{\mathbf{U}}$ is the (material) right stretch tensor and $\boldsymbol{\mathbf{V}}$ is the (spatial) left stretch tensor. The eigenvalues of $\boldsymbol{\mathbf{U}}$ and $\boldsymbol{\mathbf{V}}$ are called the principal stretches λ_p . In this work, the material is assumed to be incompressible, which leads to

$$\begin{equation} {\lambda_1 \, \lambda_2\,\lambda_3=1\,\text{.}} \end{equation} \tag{ 7 }$$

With the mass density ϱ, the Cauchy stress tensor $\boldsymbol{\mathbf{\sigma}}$ and the volume force density vector $\vec{f}$ the balance of momentum reads

$$\begin{equation} \sigma_{lk,l}+\varrho f_k=\varrho \ddot{x}_k\,\text{.} \end{equation} \tag{ 8 }$$

$\ddot{\vec{x}}$ is the second order time derivative of the current position vector. The symmetry of the stress tensor

$$\begin{equation} \sigma_{kl}=\sigma_{lk}\,\text{,} \end{equation} \tag{ 9 }$$

follows from the balance of angular momentum. Regarding the assumption of incompressibility, the hyperelastic material behavior can be described according to Ogden [26], for which the strain energy density W is defined as

$$\begin{equation} W\left(\lambda_1, \lambda_2, \lambda_3 \right) = \displaystyle \sum_{i=1}^{N} \dfrac{\mu_i}{\alpha_i}\left[\lambda_1^{\alpha_i} +\lambda_2^{\alpha_i} + \lambda_3^{\alpha_i} -3\right]\,\text{,} \end{equation} \tag{ 10 }$$

with the material constants µ_i and α_i . By including time-dependence of the material parameters, represented in terms of Prony series, a viscoelastic material behavior can be modeled [27].

$$\begin{equation} \mu_{i} (t)=\mu_{i,\infty} \left[\beta_\infty + \displaystyle \sum_{j=1}^{M} \beta_j \exp\left(-\frac{t}{\tau_j}\right)\right] \end{equation} \tag{ 11 }$$

The parameters for the Prony series are $\beta_\infty$, β_j and τ_j . The exponential decay in stiffness has been proved appropriate to model the mechanical behavior of silicone subjected to cyclic loads [28]. Thus, the principal values of the Cauchy stress σ_p can be calculated as

$$\begin{equation} {\sigma_p=\lambda_{(p)}\,\dfrac{\mathop{}\partial W}{\mathop{}\partial \lambda_{(p)}}-\,\tilde{p}\,\text{,}} \end{equation} \tag{ 12 }$$

where $\tilde{p}$ is an arbitrary hydrostatic pressure introduced due to the incompressibility constraint equation (7). Please note, that there is no summation over indices given in brackets.

3.1.1. Dissipation in the mechanical field.

To investigate the energy which is converted into the thermal energy during the actuation, the mechanical energy 'loss' has to be considered [29]. Therefore, we define the mechanical power density $p^\text{mech}$ as

$$\begin{equation} p^\text{mech} = \sigma \, \dot{\varepsilon}\,\text{.} \end{equation} \tag{ 13 }$$

For further investigation we additively decompose the mechanical power into an elastic $p^\text{ela}$ and a viscous part $p^\text{vis}$

$$\begin{equation} p^\text{mech} = p^\text{ela} + p^\text{vis}\,\text{.} \end{equation} \tag{ 14 }$$

In the case of cyclic loading, the sum of the elastic mechanical power vanishes over each cycle due to the equilibrium of the mechanical energy of the loading and unloading phase. The mean value of the viscous mechanical power density $\overline{p}^\text{vis}$ can be calculated from the potential energy density u lost in each cycle and the frequency of the mechanical loading function $f^{{}\text{mech}}$

$$\begin{equation} \overline{p}^\text{vis}=u \, f^{{}\text{mech}}\,\text{.} \end{equation} \tag{ 15 }$$

As shown in figure 11, the 'lost' potential mechanical energy density in each cycle, which is transferred into thermal energy, is equal to the area of the hysteresis in the stress-strain-diagram.

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The electrical field formulation has to be considered in the two components of a DEA, (i) the polymer, where the electric potential leads to a polarization of the dielectric material and (ii) the electrode, where the electric charge is mobile. The balance laws of the electrical field are derived from Maxwell's equations. If we neglect magnetic effects, the Maxwell-Faraday equation is defined as

$$\begin{equation} {e_{mlk}\, E_{l,m}= 0} \end{equation} \tag{ 16 }$$

for which $\boldsymbol{\mathbf{e}}$ represents the third order Levi-Civita-tensor and $\vec{E}$ is the electric field vector. Gauss's law

$$\begin{equation} {D_{k,k}=\mathfrak{q}^\text{f}} \end{equation} \tag{ 17 }$$

provides a relation between the electric displacement field $\vec{D}$ and the density of the free electrical charge $\mathfrak{q}^\text{f}$. From Ampère's circuital law

$$\begin{equation} {j^\text{f}_k=-\dot{D}_{k}} \end{equation} \tag{ 18 }$$

we obtain the third balance equation, where ${\vec{j^\text{f}}}$ denotes the free current density. The continuity of charge provides the formal relation of charge and current

$$\begin{equation} {j_{k,k}= \left(j^\text{f}_k + j^\text{b}_k \right)_{,k} =-\dfrac{\mathop{}\partial \mathfrak{q}}{\mathop{}\partial t} = -\dfrac{\mathop{}\partial}{\mathop{}\partial t} \left(\mathfrak{q}^\text{f} + \mathfrak{q}^\text{b} \right)\,\text{.}} \end{equation} \tag{ 19 }$$

Here, the superscript $(\cdot)^\text{f}$ and $(\cdot)^\text{b}$ stand for the free and bound parts of charge and its current. For the electric field in the dielectric material we assume a linear, isotropic material behavior

$$\begin{equation} D_k= \epsilon_0 \epsilon_\text{r} E_k\,\text{.} \end{equation} \tag{ 20 }$$

Here, ₀ stands for the free space permittivity and $\epsilon_\text{r}$ for the relative permittivity. While charging and discharging, the constitutive equation for the current is given by

$$\begin{equation} j^\text{f}_k = \dfrac{1}{\rho} E_k \,\text{,} \end{equation} \tag{ 21 }$$

where ρ represents the respective specific resistance.

3.2.1. Dissipation in the electrical field.

For the dissipation in the electrical field two effects have to be considered, the dielectric loss and the resistive loss.

The dielectric loss occurs, due to the fact of dielectric relaxation. The loss tangent (or the dissipation factor) is considered to be small for silicone rubbers within the investigated frequency range (with the maximum of $1\ \mathrm{kHz}$) [30]. Thus the dielectric power loss is assumed to have a minor influence on the heating of the structure and will be neglected in this work.

The specific resistance of the electrode leads to a loss while charging and discharging the electrode layers. Due to the scales of frequency and the strength of the electric field investigated, for this work only the resistive loss is considered. The power density of an electric current within an electric field $p^\text{elec}$ is given by

$$\begin{equation} p^\text{elec} = j^\text{f}_k {E}_k\,\text{.} \end{equation} \tag{ 22 }$$

Considering the constitutive equation (21) the density of the resistive power loss $p^\text{res}$ can be calculated by

$$\begin{equation} p^\text{res} = \rho_\text{e} j^\text{f}_k j^\text{f}_k\, \text{,} \end{equation} \tag{ 23 }$$

if the current on the electrodes' is known. For the dissipation, resulting from the leakage current flowing through the polymer [31], we can determine the power loss density depending on the electric field in the dielectric material by

$$\begin{equation} p^\text{leak} = \dfrac{1}{\rho_\text{d}} {E}_k {E}_k\, \text{.} \end{equation} \tag{ 24 }$$

For the electro-mechanical coupling, we use the additive decomposition of the total stress tensor $\boldsymbol{\mathbf{\sigma}}^\text{tot}$ into a mechanical part and the Maxwell stress tensor [32]

$$\begin{equation} \boldsymbol{\mathbf{\sigma}}^\text{tot} = \boldsymbol{\mathbf{\sigma}}^\text{maxw} + \boldsymbol{\mathbf{\sigma}}^\text{mech}\,\text{.} \end{equation} \tag{ 25 }$$

The mechanical stress tensor $\boldsymbol{\mathbf{\sigma}}^\text{mech}$ is equal to the previously defined Cauchy stress. The Maxwell stress tensor $\boldsymbol{\mathbf{\sigma}}^\text{maxw}$ is defined as

$$\begin{equation} \sigma^\text{maxw}_{kl} = \epsilon E_k E_l - \dfrac{1}{2} \delta_{kl} \epsilon E_m E_m\,\text{,} \end{equation} \tag{ 26 }$$

where $\epsilon = \epsilon_0 \epsilon_\text{r}$. Considering compliant electrodes, this formulation is consistent with the previously equivalent electrostatic pressure in equation (4).

Since DEAs are subjected to a heating and therefore a temperature change, the thermal field has to be incorporated. The thermal field is described by the heat equation

$$\begin{equation} \varrho \, \chi \, \dfrac{\mathop{}\partial T}{\mathop{}\partial t} + \left(-\kappa \, T_{,k} + q^\text{c}_k \right)_{,k} = q^\text{tot}\,\text{.} \end{equation} \tag{ 27 }$$

Here χ denotes the specific heat capacity, κ the thermal conductivity, T the temperature, $\vec{q^\text{c}}$ the convective heat flux vector and $q^\text{tot}$ the total volumetric heat source. At the top and lateral surfaces of the DEA a convective cooling is assumed as boundary condition, which is dependent on the difference between surface ($T^\text{s}$) and ambient ($T^\text{a}$) temperature and the heat transfer coefficient α.

$$\begin{equation} q^\text{c}_k=\alpha \left(T^{{}\text{s}} - T^{{}\text{a}}\right) n_k\,\text{.} \end{equation} \tag{ 28 }$$

The heat transfer coefficient itself depends on the difference between surface and ambient temperature as well [33].

$$\begin{align} \alpha &= \beta_T \left(T^\text{s} - T^\text{a}\right)^{0.25} \, &\text{for top surface} \end{align} \tag{ 29 }$$

$$\begin{align} \alpha &= \alpha_{L} + \beta_{L} \left(T^\text{s} - T^\text{a}\right)^{0.25} \, &\text{for lateral surface} \end{align} \tag{ 30 }$$

Please note the different dimensions of α and β. The used values for the variables are given in table 3. The surface normal vector $\vec{n}$ is defined by the boundary of the thermal field. The former defined power losses in equations (15), (23) and (24) are supposed to completely transfer into heat

$$\begin{equation} q^\text{tot} = p^\text{vis} {+ p^\text{leak}} + p^\text{res} \text{.} \end{equation} \tag{ 31 }$$

With these assumptions the thermal field is fully defined.

It is sufficient to only distinguish between the active and the passive domain, since the mechanical field is assumed to be homogeneous within the excited layers. The intersection of these domains with an inhomogeneous mechanical behavior is small in comparison to the rest, so that by neglecting this intermediate domain the occurring inaccuracy will be insignificant.

The inhomogeneity of the electric field at the edges of the electrodes' will be neglected as well. The effective error is vanishingly small due to the ratio of electrodes' area to the distance between electrodes' [34, 35].

The parameters for the mechanical model are obtained from a curve fitting procedure between the mechanical material model and the experimental data of the relaxation test. The mass density of ELASTOSIL$^{\circledR}$ P7670 is taken from the manufacturer data sheet [36]. The mass density of the electrode is assumed to be equal to the one of the polymer. Since we use the experimental data of the actuator, the mechanical model assumes a homogenized structure of the actuator in thickness direction. The electrical material properties were measured as described in section 2.2. The parameters for density, thermal conductivity and specific heat of silicone rubber were taken from results available in literature [37, 38]. Since the structure of the electrode is not exactly known, there exist no really reliable data. Empirical values have been used, resulting from previous numerical investigations. A list of the used material parameters is shown in table 3.

Table 3. Material parameters of the numerical simulation.

Description	Symbol	Value
Mechanical material properties
Ogden-model stiffness	µ1	$6.3 \times 10^5\ \mathrm{N\ m^{-2}}$
	µ2	$1.2 \times 10^3\ \mathrm{N\ m^{-2}}$
	µ3	$-1 \times 10^4\ \mathrm{N\ m^{-2}}$
Ogden-model exponent	α1	1.3
	α2	5
	α3	−2
Prony-model decay factor	β1	1.3 × 10−1
	β2	7.5 × 10−2
	β3	2.3 × 10−2
Prony-model decay time	τ1	2.2 × 10−2 s
	τ2	4.0 × 10−1 s
	τ3	3.2 × 101 s
mass density	ϱ	$1 \times 10^{3}\ \mathrm{kg\ m^{-3}}$
Electrodynamical material properties
relative permittivity of polymer	$\epsilon_\text{r}$	3
specific resistance of electrode	$\rho_\text{e}$	$5 \times 10^{-2}\ \Omega\mathrm{m}$
specific resistance of polymer	$\rho_\text{d}$	$1 \times 10^{-10}\ \Omega\mathrm{m}$
Thermodynamical material properties
thermal conductivity of polymer	$\kappa_\text{d}$	$2 \times 10^{-1}\ \mathrm{W\ (Km)^{-1}}$
thermal conductivity of electrode	$\kappa_\text{e}$	$2.5\ \mathrm{W\ (Km)^{-1}}$
specific heat capacity of polymer	$\chi_\text{d}$	$1.4 \times 10^{3}\ \mathrm{J\ (kgK)^{-1}}$
specific heat capacity of electrode	$\chi_\text{e}$	$7 \times 10^{2}\ \mathrm{J\ (kgK)^{-1}}$
heat transfer constant top surface	βT	$4.19\ \mathrm{W\ (K^{1.25}m^{2})^{-1}}$
heat transfer constant lateral surface	αL	$1.41\ \mathrm{W\ (Km^{2})}^{-1}$
heat transfer constant lateral surface	βL	$4.00\ \mathrm{W}\ (\mathrm{K}^{1.25}\mathrm{m}^{2})^{-1}$

The results of the electro-mechanical simulation are presented in this section. The electrostatic attraction of the electrodes', due to the excitation with a sinusoidal voltage, leads to cyclic stress in thickness direction. The resulting strain becomes stationary after a certain number of cycles, as consequence of the viscous material behavior. The envelopes of the strain are displayed in figure 13, for the frequencies of $f = \{10\ \mathrm{Hz}, 100\ \mathrm{Hz\ and}\ 1000\ \mathrm{Hz}\}$. Since the difference of energy loss per cycle within the transient part is small with respect to the duration of the complete process, we only consider the stationary behavior for the subsequent analysis.

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Figure 14 shows a hysteresis plot, resulting from the numerical simulation. Here, the difference of the loading and unloading phase in one cycle at the frequency of 1 kHz is given, for when this difference becomes stationary. The abscissa is given by the curve of the loading phase. The area within the curve is equal to the viscous energy lost per cycle, as described in section 3.1. Whereas the viscous power loss and the power loss due to leakage current are evenly distributed in the active domain, the electrodes' resistive power loss is not. The charging and discharging related current density on the electrode layer is distributed as shown in figure 15, with peaks at the connections of the wires. At these locations the electrodes' current density has a maximum, thereby also the power loss owns its maximum there, compare equation (23). In table 4, the numerically obtained values of the viscous power loss density $p^{\text{vis}}$, the polymers resistive power loss density $p^{\text{leak}}$ and the maximum of the electrodes' resistive power loss density $\hat{p}^{\text{res}}$ are given for the investigated frequencies. Comparing the sources of power losses by the proportion of the cumulated amount, one finds out, that the total viscous power loss $P^{\text{vis}}$ is quite small and does not contribute significantly to the heating of the DEA. For frequencies up to $f = 190\ \mathrm{Hz}$, the leakage current in the polymer is the dominant cause for the total power loss. At higher frequencies, the electrodes' resistive power loss becomes the major contributor of the total power loss.

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Table 4. Calculated power loss density for examined frequencies.

Frequency	$p^{\text{vis}}$	$p^{\text{leak}}$	$\hat{p}^{\text{res}}$
$10\ \mathrm{Hz}$	$5.00 \times 10^2\ \mathrm{W\ m^{-3}}$	$2.70 \times 10^4\ \mathrm{W\ m^{-3}}$	$2.11 \times 10^4\ \mathrm{W\ m^{-3}}$
$100\ \mathrm{Hz}$	$1.27 \times 10^3\ \mathrm{W\ m^{-3}}$	$2.69 \times 10^4\ \mathrm{W\ m^{-3}}$	$2.11 \times 10^6\ \mathrm{W\ m^{-3}}$
$1000\ \mathrm{Hz}$	$1.32 \times 10^3\ \mathrm{W\ m^{-3}}$	$2.69 \times 10^4\ \mathrm{W\ m^{-3}}$	$2.12 \times 10^6\ \mathrm{W\ m^{-3}}$

Nevertheless, the local heating of the structure is primarily defined by the local values of the power loss density p. For all investigated frequencies, the hot spots always occurred at the wires, which can be explained (i) by the superposition of all three regarded contributions for power loss and (ii) by the maximum of the electrodes" resistive power loss density $p^{\text{res}}$ in this area.

In this section the results of the thermal field analysis are given. For this, the input load for the heat generation due to mechanical and electrical power losses were determined in section 5.2. In figure 16 the thermography of the DE test-structure is shown, gained by the numerical simulation with a cyclic excitation of $U = 1\ \mathrm{kV}$ and $f = 1\ \mathrm{kHz}$. As expected, the highest temperatures occur at the connection wires. As can be seen, the whole structure heats up with increasing time. For locations at point No. 1 (in the middle) and point No. 2 (near the hot spots) the computed temperatures versus time are shown figure 17. One may notice, that the temperatures are higher for higher frequencies.

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