# Micro-Slit Cutting in an Aluminum Foil Using an Un-Traveling Tungsten Wire
## 2 Principle  
Figure 1 shows the process of slit cutting with a tungsten micro-wire. Briefly, the process can be divided into four steps. In the first step, the tungsten micro-wire contacts with the aluminum foil and gets bent, generating tension, as shown in Figure 1a. In second, the tension reacts to the aluminum foil and causes stress concentration at the foil edge, as shown in Figure 1b. Due to the sensitive plasticity of aluminum, the plastic strain will occur around the tungsten wire, resulting in a small \"breach\" [18]. The goal of this step is to generate enough power for cutting aluminum foil as well as to produce the breach for stress concentration [19]. The wire feeding rate in this step namely initial velocity is relatively slow, usually ranging from 10 \(\upmu\)m\(\cdot\)s\({}^{-1}\) to 50 \(\upmu\)m\(\cdot\)s\({}^{-1}\). In third, the cutting process gets easier and faster because the breach maintains along the cutting trajectory. The wire feeding rate of this step namely secondary velocity can reach 500 \(\upmu\)m\(\cdot\)s\({}^{-1}\), as shown in Figure 1c. At last, the tungsten wire returns along the cutting path to the initial position, preparing for the next slit, as shown in Figure 1d.  
Essentially, this cutting method is not much different from traditional mechanical cutting. The diameter of the tungsten microwite used in this process is dozens of micrometers. Keeping the feeding motion of tungsten wire, plastic deformation will take place at the foil edge and stress will gradually generate. Once the stress exceeds the tensile strength of aluminum, a breach, and then a slit will generate.  
A statics simulation is performed via Abaqus software to analyze the second step of generating the cutting force and producing the breach [20]. First, the initial contact force (\(F\)) is calculated by the elongation of the tungsten wire. Figure 2 shows the force model and the analysis of the simulation. Let the cross-sectional area of tungsten wire be \(S\), the original length of tungsten wire be \(L\), and its elongation be \(\Delta L\). Besides, \(d\) is the opening width of the wire fixture and \(\theta\) is the angle between tungsten wire and aluminum foil in the process. According to the definition of elastic modulus and the preload force (\(F_{d}\)) gained when fixing tungsten wire, \(F_{p}\) is derived as follows:  
\[F_{P}=E_{1}\frac{S}{L}\Delta L+F_{d}=E_{1}\frac{S}{L}d\left(\frac{1}{sin\theta }\right)+F_{d}. \tag{1}\]  
The initial contact force (\(F\)) can be obtained by employing force synthesis principle and written as  
\[F=2 Fcos\theta=2\left[E_{1}\frac{S}{L}d\left(\frac{1}{sin\theta}\right)+F_{d} \right]cos\theta. \tag{2}\]In present work, the tungsten wire used is 50 \(\upmu\)m in diameter with original length (\(L\)) of 115 \(\upmu\)m. The preload force is set at 5 N when fixing this kind of tungsten wires and the opening width of the wire fixture (\(d\)) is 2 mm. During experiments, the angle between tungsten wire and aluminum foil (\(\theta\)) is measured to be 75\({}^{\circ}\). As a result, the initial contact force (\(F\)) can be calculated referring to Equation (1) and the value is 2.01 N.  
Second, a series of parameters necessary for simulation are set, as summarized in Table 1. In this simulation, the back of aluminum foil is fixed and all its freedom degrees are limited as boundary conditions. Hexahedron mesh is employed for the finite element calculation. The stress and deformation distributions are shown in Figure 3. Obviously, the von Mises stress near the breach is much higher than in other areas and exceeds the rupture strength of aluminum. As a result, a straight slit with great width consistency can be machined.  
Figure 1: Process of cutting aluminum foil with a tungsten microwire: (**a**) the first step; (**b**) the second step; (**c**) the third step; (**d**) the fourth step.  
Figure 2: The force model and analysis of simulation.