# Angle of the Perforation Line to Optimize Partitioning Efficiency on Toilet Papers

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## Abstract

**:**

## 1. Introduction

## 2. Simulation–Materials and Methods

#### 2.1. Optimization

- a Python script to modify the FE model regarding the GA design parameters;
- a Python script to perform the FE results analyses (post-processing);
- a Fortran subroutine for the material model (more details in the section below);
- a MATLAB
^{®}script to control the FE analysis and GA.

^{®}GA algorithm. The analysis started when MATLAB

^{®}GA generated the first generation of design parameters. Then, a Python script was called to modify the FE model regarding the design parameters. After that, the MATLAB

^{®}ran the FE analysis with the material model.

^{®}code and a Python script that accesses the ABAQUS

^{TM}results several times until it detected a reduction of 20% in terms of the maximum force.

#### 2.2. Material Model

^{TM}does not have a native constitutive law to model plasticity for orthotropic materials. Hence, a user material subroutine for explicit simulations (VUMAT) was implemented to simulate the orthotropic elastic–plastic behavior for the paper sheet. The material model, proposed by Mäkelä and Östlund [13], allows the paper anisotropic behavior to be accounted for, since the paper response is highly dependent on the fiber orientation. The model assumes the decomposition of the strain tensor into an elastic strain tensor and a plastic strain tensor (Equation (2)) while conserving the volume.

#### 2.3. Finite Element Model

_{11}= 13.89 MPa, E

_{22}= E

_{33}= 4.23 MPa, μ = 0.33 and G

_{12}= 2.1 MPa. The parameters for the IPE model consider K

_{22}= K

_{33}since the mechanical behavior in the CD (direction 2) is similar to that in the thickness direction (direction 3). Thus, A = 1, B = 2.40, C = 2.40 and D = 1.38.

## 3. Experimental Tests–Materials and Methods

#### 3.1. Materials

#### 3.2. Methods

^{2}). A Mettler Toledo PB303 Delta range analytical balance (Mettler Toledo, Columbus, OH, USA) was used to determine the paper sample weight. To determine the thickness, where a stack of sheets of paper or a sheet of paper were/was compressed at a given pressure between two parallel plates, a FRANK-PTI

^{®}Micrometer (FRANK-PTI GMBH, Birkenau, Germany) was used, in accordance to the standard ISO 12625-3:2014 [17]. According to this standard [17], the bulk, which is the inverse of density, can be determined by using the grammage and thickness previously determined.

^{®}VantageNX Universal Testing Machine, tensile tests were performed in the MD for all samples. For each paper, samples were prepared with the perforation in the center (0°) and with the line of perforation at different angles (20°, 30°, 37.5°, 41° and 45°). Other samples were also prepared, of each paper, with the length of a single “sheet” without perforation but with the orientation of the corresponding angle to annulate the fiber orientation contribution (see Figure 4)

## 4. Results and Discussion

^{2}. Evaluating the outcomes for the thickness and bulk, values vary between 51% and 60%, respectively, due to the embossing process type.

^{®}default tolerance.

_{t}= 21 (regarding the stress in fiber direction, MD), justifying the low rupture force. The same behavior is detected for the other stresses (the CD (${\sigma}_{22}$ in X direction) in Figure 9b and shear stress (${\sigma}_{12}$) 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.

_{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).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Experimental set-up to test non-perforated and perforated toilet papers. (F shows the force direction applied in the tensile test).

**Figure 6.**Experimental and theoretical perforation efficiency results as function of perforation line angle.

**Figure 10.**Stress field for the optimum orientation: (

**a**) in fiber direction MD; (

**b**) normal to fiber direction CD; (

**c**) shear stress; (

**d**) rupture for half of the model.

**Figure 11.**(

**a**) Fiber direction stress field in MD for cuts at 45°; (

**b**) rupture starting in the lower edge running to the center.

**Table 1.**Physical characterization of the toilet papers: number of plies, grammage, thickness, bulk, cut and blank distance.

Toilet Paper ID | N° Plies | Grammage (g/m^{2}) | Thickness (µm) | Bulk (cm ^{3}/g) | Cut Distance (mm) | Blank Distance (mm) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\overline{x}$ | σ | $\overline{x}$ | σ | $\overline{x}$ | σ | $\overline{x}$ | σ | $\overline{x}$ | σ | ||

A | 2 | 36.6 | ±0.64 | 374 | ±10.4 | 10.2 | ±0.36 | 1.5 | ±0.05 | 1.0 | ±0.05 |

B | 2 | 35.4 | ±0.26 | 305 | ±12.4 | 8.6 | ±0.37 | 1.9 | ±0.05 | 1.2 | ±0.10 |

C | 2 | 32.4 | ±0.42 | 611 | ±4.4 | 19.1 | ±0.41 | 4.0 | ±0.05 | 1.0 | ±0.05 |

D | 2 | 44.9 | ±0.71 | 345 | ±8.7 | 7.7 | ±0.27 | 2.3 | ±0.05 | 1.0 | ±0.05 |

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**MDPI and ACS Style**

Vieira, J.C.; Vieira, A.C.; Ribeiro, M.L.; Fiadeiro, P.T.; Costa, A.P.
Angle of the Perforation Line to Optimize Partitioning Efficiency on Toilet Papers. *Eng* **2023**, *4*, 80-91.
https://doi.org/10.3390/eng4010005

**AMA Style**

Vieira JC, Vieira AC, Ribeiro ML, Fiadeiro PT, Costa AP.
Angle of the Perforation Line to Optimize Partitioning Efficiency on Toilet Papers. *Eng*. 2023; 4(1):80-91.
https://doi.org/10.3390/eng4010005

**Chicago/Turabian Style**

Vieira, Joana Costa, André Costa Vieira, Marcelo L. Ribeiro, Paulo T. Fiadeiro, and Ana Paula Costa.
2023. "Angle of the Perforation Line to Optimize Partitioning Efficiency on Toilet Papers" *Eng* 4, no. 1: 80-91.
https://doi.org/10.3390/eng4010005