Eng 2023 , 4
88
Figure7. Optimization evolution of the best value and mean value. As presented in Figure 8, the best value was almost constant after the 16th generation. On the other hand, the mean value did not converge. For this case, the stop criteria adopted was when the best value between generations was less than the MATLAB ® default tolerance. The stresses field for the optimum case, where the blank distance, d , and angle, α , were optimized, are presented in Figure 9, in the increment just before rupture. The normal stress field in the MD ( σ 11 in Y direction), Figure 9a, shows a stress concentration between the cuts, as expected. As the distance between cuts are only 0.1 mm, the stress concentration is approximately K t = 21 (regarding the stress in fiber direction, MD), justifying the low rupture force. The same behavior is detected for the other stresses (theCD( σ 22 in X direction) in Figure 9b and shear stress ( σ 12 ) in Figure 9c). Hence, cuttings affect the stress fields in the different directions of the paper plane. In this case, rupture begins at the center of the paper, moving fast towards the left and right edges (see Figure 9d), in the same way as it occurs experimentally in the laboratory. Considering the other case, the optimization regarding only the inclination of the cuts, the stress fields are show in Figure 10. The stress concentration factor is approximately K t =4.1 for the MD stress (significantly lower as in the previous case). As in the previous case, the rupture starts at the center of the paper and moves towards the left and right edges (see Figure 10d). Additionally, the simulations regarded the paper as a homogeneous media with no variations in fiber alignment or different concentrations throughout the model. This would not be the case in real paper, and such factors would have an influence on the paper rupture force. Figure 11a shows the MD stress field distribution ( σ 11 in Y direction) around the cuts with an orientation of 45 ◦ , and Figure 11b shows the same MD stress field distribution at the beginning of the paper rupture starting at the lower edge towards the center. Due to paper rupture (Figure 11b), stress flows in the non-ruptured region and hence stress is increased (darker green) in this region, while in the ruptured region stress field tends towards zero (darker blue).
Figure8. Optimization evolution of best and mean value for parameter d .
Made with FlippingBook flipbook maker