# The FEM Model of the Pump Made of Dielectric Electroactive Polymer Membrane
## 2 Principle of Operation of DEAP Actuator  
A DEAP membrane is made of a silicone membrane that is pre-stretched during the production process. Then, the carbon grease electrodes are printed. In principle, DEAPs are similar to flexible capacitors. As can be seen in the Figure 1, they consist of an elastomeric film sandwiched between two electrodes. Applying voltage to them leads to compression of the elastomer, which causes a change in capacity [9; 10; 11; 12; 13]. For this circular geometry, the deflection of the DEA membrane is described by the radial stretch \(\lambda_{r}\), circumferential stretch \(\lambda_{c}\) and thickness stretch \(\lambda_{z}\)[14]. \[\lambda_{r}\lambda_{c}\lambda_{z}=1\quad\lambda_{r}=\frac{l}{l_{0}}=\frac{ \sqrt{l_{0}^{2}+d^{2}}}{l_{0}}\quad\lambda_{z}=\frac{z}{z_{0}}\quad\lambda_{c }=const \tag{1}\]  
\[z=z_{0}\frac{l_{0}}{l}=z_{0}\frac{l_{0}}{\sqrt{l_{0}^{2}+d^{2}}}\quad\sin( \theta)=\frac{d}{\sqrt{l_{0}^{2}+d^{2}}} \tag{2}\]  
I propose a dynamic model composed of a set of nonlinear, time-invariant differential equations describing the dynamic relationship between the input voltage \(u\) and the output actuator displacement  
Figure 1: Principle of operation of dielectric electroactive polymer actuator  
\(d\). I present a model which accurately describes the behavior of the actuator, both in transient and steady state, and for different mass loads. The vertical force equilibrium causing membrane deformation can be defined as:  
\[m\ddot{d}=mg+F_{L}-sin(\theta)\left(F_{M}+F_{h}+F_{f}\right) \tag{3}\]  
where \(d\) is a distance, \(\theta\) is an angle, \(m\) is the mass of the initial load, \(g\) means standard gravity and \(F_{L}\) defines the external load. The electromechanical coupling is produced by Maxwell Stress which compresses the membrane when a voltage \(u\) is applied to the DEAP. The actuation is in the thickness direction and due to constant volume produces the change of radial stretch \(y_{r}\). The value of \(F_{M}\) is equal to:  
\[F_{M}sin(\theta)=\overline{c_{1}}c_{2}du^{2} \tag{4}\]  
\[\overline{c_{1}}=\frac{2\pi rz_{0}}{l_{0}},\ \overline{c_{1}}=-\frac{\varepsilon_{0} \varepsilon_{r}}{z_{0}^{2}} \tag{5}\]  
where \(\varepsilon_{0}\) is vacuum permittivity and \(\varepsilon_{r}\) is the relative permittivity of the polymer material. The parameters \(z_{0}\), \(l_{0}\) and \(r\) describe the geometry of DEAP actuator and are defined in Table 1. Force \(F_{u}\) considers dynamic viscoelastic process according to:  
\[F_{h}sin(\theta)=\overline{c_{1}}d\sigma_{e}\left(\frac{l_{0}^{2}}{l_{0}^{2} +d^{2}}\right) \tag{6}\]  
\[\sigma_{e}=-k_{e}\varepsilon_{e}+k_{e}(\lambda_{r}-1)+\frac{\sigma_{e}}{ \eta_{e}} \tag{7}\]  
\[\dot{\varepsilon_{e}}=-\frac{k_{e}}{\eta_{e}}\varepsilon_{e}+\frac{k_{e}}{ \eta_{e}}(\lambda_{r}-1) \tag{8}\]  
There are many hyperelastic strain energy density functions, such as Yeoh, Ogden, Arruda-Boyce, etc. Similar to [11], I use the Ogden model and define two parameters \(\beta_{i}\) and \(\gamma_{i}\) to obtain to the expression for the hyperelastic stress \(\sigma_{e}\):  
\[z=z_{0}\frac{l_{0}}{l}=z_{0}\frac{l_{0}}{\sqrt{l_{0}^{2}+d^{2}}}\quad\sin( \theta)=\frac{d}{\sqrt{l_{0}^{2}+d^{2}}} \tag{2}\]0  
where \(\alpha_{i}\) is 2, 4 and 6. The main difficulty encountered with the identification of the DEAP parameters in the dynamic case is to obtain the damping coefficients \(b\) and the coefficients of the viscoelastic \(k_{e}\) and