# Characterization of a Rectangular-Cut Kirigami Pattern for Soft Material Tuning
## 2 Materials and Methods  
### Rectangular Pattern Cuts  
The proposed pattern of rectangular kirigami cuts is shown in Figure 1. This pattern is defined by the geometry of the rectangular cuts and their distribution on the supporting material. In this work, it is proposed that the cuts have a uniform distribution throughout the entire supporting material, including the edges where the cuts are interrupted by the edge of the base material. The description of the rectangular cuts includes the width (w) and the height (h), with all rectangles having the same dimensions. The distribution of the cuts on the base material is defined by the distance (s) between cuts on the same line and the distance (h1) between lines of cuts. The length (z) of the rectangular cut that coincides with a section of the upper or lower cut is also described. The behavior of the kirigami cell can be analyzed by discretizing it and examining the behavior of one of the cuts, which can then be used to propagate the deformation phenomenon throughout the entire base material. To model the deformation that occurs along the kirigami base material, a section surrounded by a dotted line, as shown in Figure 2, is proposed to be considered as a representative section of any section along the cell. If it is assumed that when the application of force P begins, supports A and B are fixed, as shown on Figure 3; then, it can be considered as if the element forming the section is a small beam with fixed supports. This type of beam allows the ends to be fixed but can undergo rotation, which is ignored in this analysis. At the begin of the deformation, this structural element is subjected to a force P applied at the center of the beam. By considering the geometry of the beam, in this case, the section described is analyzed from support point A to the center of the beam with a length of L/2. The material is considered to have a thickness \(t\) and a width \(w\). The equation for the vertical displacement at any point between support A and the center of the beam is valid for the range \(0<x<L/2\). The equation for analyzing deformation is derived through an analysis of shear force and bending moment on the selected segment, as explained on [37]. The development of the analysis leads us to Equation (1), considering the moment of inertia as shown in Equation (2). Figure 1: Geometrical description of kirigami rectangular cuts. Figure 2: Examined segment of the kirigami pattern delineated by a dashed border. \[y=\frac{P}{48 EI}\Big{[}4x^{3}-3L^{2}x\Big{]}, \tag{1}\] \[I=\frac{tw^{3}}{12},\] (2) \[y=\frac{P}{Etw^{3}}\Big{[}x^{3}-\frac{3}{4}L^{2}x\Big{]}, \tag{3}\]  
Equation (3) indicates the displacement \"\(y\)\" of the central point of the beam in Figure 2, which is the point where the displacement is at its maximum. This allows us to identify the maximum deformation of the beam under the established considerations. Once the deformation of the bar-like element is determined, we will illustrate the stages through which the rectangular kirigami cut goes, involving other parameters related to the configuration of the rectangular kirigami cut. Figure 3 shows the different states through which the rectangular cut progresses as deformation advances, as obtained through an analysis of images captured during the experimentation with the specimens. The images are presented as illustrations to more effectively emphasize the points of interest, illustrating the location of various geometric parameters from the initial shape to highlight their impact. In Figure 3a, the rectangular kirigami cut is shown in an initial stage of deformation, where it maintains nearly the same shape, illustrating a single cut since, if the distribution of cuts is uniform along the base material, all cuts not on the boundary exhibit similar behavior. Figure 3b illustrates the same rectangular cut in a subsequent stage of deformation, still revealing the proportions that constitute the cut, albeit with the rectangular shape having deformed into an octagonal form.