\(i=\frac{210}{n\ }-L\)
By design, as shown in Figure 2a(i), a cross-sectional corrugation with amplitude d and half-wavelength w is formed at each crease, as shown in Figure 2a(ii). L becomes an arc of the cross-sectional corrugation. Furthermore, n changes the number of creases, resulting in a change in the number of half-wavelengths in the cross-sectional corrugation. In the experiment, we ensured that n equalled the number of half wavelengths by setting the line interval as i/2 between both edges. In the following paragraphs, we discuss the effect of the design parameters, L and n, on the final shape of the SCS.
First, we investigated the effect of L on the shape of the SCS. Figure 2b shows an example of an SCS designed with n = 4 and L = 5, 10, and 15 mm. As shown, the thicker the linewidth, the more the paper folds, i.e., the arc length increases, assuming that the curvature is constant for any linewidth. Consequently, the gradient of the straight-line parts joining each arc increases. Therefore, as L in the design becomes higher, the amplitude of the corrugated structure d increases, and the half-wavelength w decreases. Thus, d and w are related so that when one increases, the other decreases.
Next, we investigated the effect of n on the shape of the SCS. Figure 2c shows an example of the SCS with L = 15 mm and n = 4, 6, 8, and 10. As shown in Figure 2c, the number of w corresponds to n. In this study, all SCSs were fabricated in A4 size. If the number of half-wavelengths increases, the length of paper that can be used to form one half-wavelength decreases. Therefore, as n increases, d and w decrease. The length of the paper that can be used to form one half-wavelength wmax is determined by the following equation.
\(w_{\max}=\frac{210}{n}\)
The effects of the linewidth L and the number of lines n on the amplitude d and half-wavelength w of the cross-sectional corrugation were verified based on measured values. In an experiment, 60 SCSs were fabricated by changing L by 1 mm in the range of 1–15 mm for each structure with n = 4, 6, 8, and 10. Then, the amplitude d and the half-wavelength w of the cross-sectional corrugation were measured. Figure 2d shows the results of the experiment, with the amplitude d as a circle plot and the half-wavelength w as a cross plot. To improve legibility, the horizontal axis is set to i, which is determined by L and n from Equation (1). From the blue plot (n = 4), as i decreases, L increases. Subsequently, d increases, and w decreases. The trend was the same for all the structures with n = 6, 8, or 10. Additionally, as n increased, the amount of change in amplitude and half-wavelength with linewidth decreased due to the decrease in the length of paper that can be used to form one half-wavelength wmax.