# Modeling and Design Enhancement of Electothermal Actuators for Microgripping Applications
## 2 Materials and Methods  
### Electothermal Actuation: Theoretical Background  
The working principle of electrothermal actuation is based on the heating of a microstructure by passing a current \(l\) through the microstructure or resistors placed on it. Due to the resistance of resistors or the structure itself, Joule heating is generated according to [10]:  
\[P=l^{2}R \tag{1}\]  
where \(P\) is the generated power and \(R\) represents either the resistance of deposited resistors or is the resistance of the structure itself which can be calculated as follows [10]:  
\[R=\frac{r^{E}L}{A} \tag{2}\]  
here, \(L\) denotes the total length of the structure, \(A\) is the cross-sectional area, and \(r^{E}\) represents the resistivity of the structure. In the U-shaped actuator, due to geometry differences between the two arms, one arm is more resistive than the other, inducing different Joule heating generation in each arm. This results in a temperature difference between arms and hence leads to having one arm with more thermal expansion than the other one has. This thermal expansion is the main driving force of the tip deflection of the electrothermal actuator [13]. We display, in Figure 1, the main modeling steps used to simulate the mechanical and thermal response of the U-shaped actuator. After introducing the geometry and material properties of the U-shaped actuator, the structure mesh is generated. Next, The Joule heating and thermal expansion interface is used. It constitutes a Multiphysics module which accounts for thermal expansion, the fundamental mechanism by which motion is generated in thermal actuators, and electromagnetic heating physics. Thermal expansion is treated as a continuum behavior. The thermal strain is computed as follows:  
\[\dot{\mu}_{th}=\alpha(T)(T-T_{ref}) \tag{3}\]  
where \(\dot{\mu}_{th}\) is the thermal strain, \(\alpha(T)\) is the thermal expansion coefficient, and \(T_{ref}\) is the reference temperature. COMSOL uses a secant coefficient of thermal expansion by default. Hence, the thermal expansion coefficient can be treated as a temperature independent constant value which is obtained from the defined material properties [44]. Figure 1: Modeling approach used to simulate the mechanical and thermal response of the U-shaped actuator. Electromagnetic heating uses electric currents and heat transfer in solids interfaces to solve the Joule heating problems. The electric current interface includes the implementation of the following equations to perform time-invariant (stationary) analysis [44]:  
\[\
abla\cdot J=0 \tag{4}\]  
\[J=sE \tag{5}\]  
\[E=-\
abla V \tag{6}\]  
where \(s\) is the electrical conductivity, \(J\) is the current density, \(E\) is the electric field intensity, and \(V\) is the electric potential. The temperature variation in the thermal actuator is dictated by the heat generation and heat transfer throughout its structure. The thermal behavior is governed by the following equations which couple the electric currents and heat transfer in solids interfaces:  
\[rc_{p}u\cdot\
abla\,T=\
abla\cdot(k\
abla\,T)+Q_{\theta} \tag{7}\]  
\[Q_{\theta}=J\cdot E \tag{8}\]  
where \(r\) is the density, \(c_{p}\) is the specific heat capacity at constant stress, \(u\) is the velocity vector, \(T\) is the absolute temperature, \(k\) is the thermal conductivity, and \(Q_{\theta}\) is the Joule heating-introduced heat load [44]. Different boundary conditions have been imposed on the model in order to closely replicate the real model. In the solid mechanics interface, a fixed constraint boundary condition is imposed on the appropriate edges which sets the displacement to zero in all the three (x, y, z) directions. The two main important boundary conditions used in the electric currents interface are the terminal and ground boundary conditions. For the terminal boundary condition, the electric potential is set to be equal to the input voltage at the given boundary, whereas for the ground condition, the potential is set to zero.