# Dielectric Elastomer Actuator for Soft Robotics Applications and Challenges
## 3 Modeling of DEA  
As introduced in the previous section, DEs are recognized for their large planar strain. To incorporate this expanding nature into an actuator, various actuator configurations were designed, such as spherical, stack, bending, and rolled actuators [7; 28]. However, in order to operate DEAs to their full potential under various conditions, a fundamental understanding of their working principle is essential. The first step in modeling DEAs was taken by [PERSON] and [PERSON]. They revealed that electromechanical transduction effects in DEs are due to the Maxwell stress [1; 23; 29]. While the working principle may seem simple, an accurate modeling of DEAs still proves to be a challenge. In order to overcome such a challenge, many researchers proposed DE actuation models based on nonequilibrium thermodynamics of the material. Such thermodynamics of DEs were reviewed by [PERSON] and colleagues [30; 31]. Taking the planar expansion in Figure 3a,b as an example, in the reference state, the DE film is of the dimensions L\({}_{1}\), L\({}_{2}\), and L\({}_{3}\). In the current  
and \(l_{3}\), and subject to forces \(P_{1}\), \(P_{2}\), and \(P_{3}\), and to voltage \(\Phi\). When the DE film reaches equilibrium with the forces and the voltage, the change in the free energy equals the work done on the film:  
\[\delta\mathrm{F}=P_{1}\delta l_{1}+P_{2}\delta l_{2}+P_{3}\delta l_{3}+\Phi\delta Q \tag{6}\]  
where F is the Helmholtz free energy of the film, while \(Q\) is the electric charge accumulated on the film and is related to electric displacement by \(Q=Dl_{1}l_{2}\). Defining the nominal density of the Helmholtz free energy by W = F/(L\({}_{1}\)L\({}_{2}\)L\({}_{3}\)), the stretches by \(\lambda_{1}\) = \(l_{1}\)/L\({}_{1}\), \(\lambda_{2}\) = \(l_{2}\)/L\({}_{2}\), and \(\lambda_{3}\) = \(l_{3}\)/L\({}_{3}\), and assuming the incompressibility of the material as \(\lambda_{1}\lambda_{2}\lambda_{3}\) = 1, and the electric field as E = \(D\)/\(\varepsilon\), where \(\varepsilon\) is the permittivity of the film, the following relationship is obtained:  
\[\mathrm{W}(\lambda_{1},\lambda_{2},D)=\mathrm{W_{s}}(\lambda_{1},\lambda_{2}) +\frac{D^{2}}{2\varepsilon} \tag{7}\]  
The first term on the right-hand side of the equation is the Helmholtz free energy associated with the stretching of the elastomer, and the second term represents the electrostatic energy of the film [9; 32]. Early works modeled the strain energy, W\({}_{s}\), with elastic material models. However, elastic material models alone are insufficient to model the DE's strain behavior correctly. This is because DEs have nonlinear material properties and undergo time-dependent dissipative processes, such as viscoelastic, dielectric, and conductive relaxations [28; 30]. To overcome this limitation, many researchers have proposed various methods to model the viscoelastic behavior of DEs, such as the Prony series or rheological models [28]. The viscoelastic material model that is being frequently used is the three-parameter rheological model shown in Figure 3c. The three-parameter solid model is composed of springs, each with their own shear modulus, \(\mu^{\alpha}\) and \(\mu^{\beta}\), and a damper with a coefficient of viscosity, \(\eta\). Here, the sum of two spring shear moduli is the shear modulus of the material [32; 33]. Recent works utilize hyperelastic material models, such as the Gent, the Neo-Hookean, or the Yeoh model with the rheological model to predict the DEA's behavior [28; 34; 35; 36]. As mentioned, there are various configurations for DEAs. As stated by [PERSON], these actuators can be roughly categorized into two groups depending on their morphology. One group of actuators utilizes basic in-plane expansion, such as circular (diaphrag), extender, and bending beam actuators. Another group exploits the desirable features of elastomers such as the monolithic structure, flexibility, and multifunctionality. Examples of this group are spring roll actuator, bow actuator, and spider actuation [7].