# The Influence of Mechanical Stress Micro Fields around Pores on the Strength of Elongated Etched Membrane

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The reduction in the working section of a sample in a real single-axis, elongated cross-section, which is smaller than that of a normal cross-section because of pores.
- (2)
- The concentration of mechanical stresses on the pores, as on each defect of a solid structure, the local mechanical stress is more than a normal structure’s stress.
- (3)
- The interaction of elastic stress fields around closely spaced pores.

## 2. Materials and Methods

^{6}cm

^{−2}, their diameter d was varied. The variation of the diameter was achieved with different track-etching times (Table 1). Reductions in film thickness over the etching time were taken into account in the strength tests.

## 3. Results and Discussion

_{m}are the strength or yield point of the TM and matrix, respectively, and P is the TM porosity.

^{1/2}:

^{1/2}was only an approximation, strictly valid only for a regular pore network. Table 4 shows the relative average distances calculated using the different methods described below.

^{2}test. The studied area of the surface of the TM was divided into squares of equal area, and the frequencies of several pores (centers of pores) in each square were calculated. The values of the number of pores regulated in ascending order formed a sample. The criterion for the selection of the area of one square was that the number of frequencies below 5 was not more than 20% of the total number of frequencies. The preliminary number of squares on which the surface of the TM was partitioned was determined by the Sturges formula: m = 1 + 3322·lg (N), where N is the number of pores on the surface [9]. A Poisson distribution is characterized by the equality of its mathematical expectation a and variance D, and the probability that a random quantity X takes the value k is expressed by the formula:

^{2}was compared with the table value ${\chi}_{table}^{2}$ at a given level of significance. If ${\chi}^{2}{\chi}_{table}^{2}$, then there was no reason to reject the hypothesis regarding Poisson law for the pore distribution over the surface of the TM.

^{6}cm

^{−2}and d = 4 μm (photography of the TM sections contained more than 700 pores). The Pearson criterion value was χ

^{2}= 16.98 at f = 9, where f is the number of degrees of freedom and ${\chi}_{table}^{2}=16.9190$ at the commonly used significance level of 0.05. Thus, with a validity of 0.05, the Poisson law hypothesis regarding the pore distribution over the surface of the TM could be rejected.

^{6}cm

^{−2}and d = 4 μm, the Pearson criterion value was χ

^{2}= 18.8 at f = 7, and ${\chi}_{table}^{2}=14.0671$ at a significance level of 0.05. ${\chi}^{2}{\chi}_{table}^{2}$, so, for these TMs, one must reject the hypothesis of the pore distribution at the nearest distances of Formula (2) (Figure 6).

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Modeling of the pore with a 0.3 mm hole drilled into the polyimide film; (

**b**) distortion of the hole shape at a tensile rate of 2 mm/min to 100% deformation; (

**c**) elongation of the pore shape of the TM with d = 4 μm and n = 10

^{6}cm

^{−2}at a 10% deformation.

**Figure 3.**Dependence of strength on pore diameter for TMs with different densities: (

**a**) n = 4 × 10

^{6}cm

^{−2}; (

**b**) n = 1.18 × 10

^{7}cm

^{−2}; and (

**c**) n = 4.41 × 10

^{7}cm

^{−2}. The dotted line on all plots corresponds to the unirradiated PET sample (without pores). For (

**b**,

**c**), it was considered that the film had passed the same etching mode as the TM with this diameter value.

**Figure 4.**Fields of elastic stresses around holes in polyimide film with diameter d = 0.3 mm applied at different angles to the direction of tension; from left to right: (

**a**) 90°, (

**b**) 45°, and (

**c**) 0°. Tensile stress was 15 MPa. The direction of the tensile stress is indicated by needles.

**Figure 6.**Theoretical (Formula (2)) and experimental distance distribution functions for TM with n = 10

^{6}cm

^{−2}and d = 4 μm (Table 2).

**Figure 7.**Experimental distance distribution functions for TM with n = 4 × 10

^{6}cm

^{−2}and d = 0.54 μm.

**Figure 9.**Simulation of the plastic–elastic deformation of TMs by the net-point method at one axis extension. The direction of the extension is horizontal. Dark-red zones correspond to zones with maximum stress concentrations; green represents zones of unloading.

**Figure 10.**Development of crack under one-axis elongation, where the direction of deformation is vertical: (

**a**) ɛ = 1.7%; (

**b**) ɛ = 1.8%; (

**c**) ɛ = 1.9%; (

**d**) ɛ = 4%. Values of φ from ~0.9 to 1 (color scale) characterize a destroyed material.

**Figure 11.**TM with d = 4 μm and n = 10

^{6}cm

^{−2}in polarized light, with deformation of 10%, followed by relaxation. Microcracks are visible along the pore chains. Photo was captured with a Nikon LV100 microscope.

Sample № | Etching Time, Min | Pore Diameter, μm | Film Thickness, μm |
---|---|---|---|

1 | 30 | 0.22 | 11.4 ± 1.1 |

2 | 50 | 0.54 | 11.0 ± 1.1 |

3 | 70 | 0.84 | 10.7 ± 1.1 |

4 | 90 | 1.16 | 10.4 ± 1.0 |

5 | 120 | 1.60 | 10.0 ± 0.9 |

6 * | 0 | - | 12.0 ± 1.1 |

Radiation Density, n, cm^{−2} | Pore Diameter, μm | Film Thickness, μm |
---|---|---|

1.18 × 10^{7} | 0.067 | 11.7 ± 0.2 |

1.18 × 10^{7} | 0.105 | 11.7 ± 0.1 |

1.18 × 10^{7} | 0.302 | 11.6 ± 0.2 |

1.18 × 10^{7} | 0.487 | 11.4 ± 0.2 |

1.18 × 10^{7} | 0.674 | 11.2 ± 0.2 |

1.18 × 10^{7} | 1.221 | 10.8 ± 0.2 |

4.41 × 10^{7} | 0.060 | 11.9 ± 0.2 |

4.41 × 10^{7} | 0.158 | 11.9 ± 0.2 |

4.41 × 10^{7} | 0.301 | 11.8 ± 0.2 |

4.41 × 10^{7} | 0.772 | 11.2 ± 0.1 |

4.41 × 10^{7} | 0.951 | 11.0 ± 0.2 |

10^{6} | 4.0 | 10 ± 0.9 |

Sample № | d, μm | Tensile Strength with Consideration of Reduced Working Section, σ, MPa | Conditional Yield Point with Consideration of Reduced Working Section, σ_{T}, MPa | Breaking Deformation ε, % | β |
---|---|---|---|---|---|

1 | 0.22 | 138 ± 4 | 106 ± 3 | 19 ± 4 | 1.44 |

2 | 0.54 | 139 ± 11 | 111 ± 8 | 13 ± 1 | 1.43 |

3 | 0.84 | 140 ± 7 | 113 ± 6 | 11 ± 2 | 1.42 |

4 | 1.16 | 142 ± 5 | 114 ± 4 | 9 ± 2 | 1.40 |

5 | 1.60 | 125 ± 11 | 102 ± 7 | 5 ± 0.7 | 1.60 |

6 | 0 | 199 ± 9 | 106 ± 11 | 42 ± 2 | 1 |

**Table 4.**Relative (expressed in diameters) mean distances between pore centers for TMs with n = 4 × 10

^{6}cm

^{−2}and different diameters.

Sample № | d, μm | $\overline{\mathit{r}}=\frac{1}{\sqrt{\mathit{n}}\mathit{\xb7}\mathit{d}}$ (For Regular Network) | ${\overline{\mathit{r}}}_{\mathit{m}\mathit{i}\mathit{n}}=\frac{1}{2\sqrt{\mathit{n}}\mathit{\xb7}\mathit{d}}$ (For Poisson Distribution) | ${\overline{\mathit{r}}}_{\mathit{m}\mathit{i}\mathit{n}}=\frac{\sum {\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}}{\mathit{N}\mathit{\xb7}\mathit{d}}$ (Calc. by SEM Images up to One Nearest Pore) | ${\overline{\mathit{r}}}_{\mathit{m}\mathit{i}\mathit{n}}=\frac{\sum {\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}}{\mathit{N}\mathit{\xb7}\mathit{d}}$ (Calc. by SEM Images up to 4 Nearest Pores) | ${\overline{\mathit{r}}}_{\mathit{m}\mathit{i}\mathit{n}}=\frac{\sum {\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}}{\mathit{N}\mathit{\xb7}\mathit{d}}$ (Calc. by SEM Images up to 6 Nearest Pores) |
---|---|---|---|---|---|---|

1 | 0.22 | 23 | 11.5 ± 3.4 | 16.9 ± 9.7 | 19.8 ± 9.2 | 23.9 ± 10.8 |

2 | 0.54 | 9 | 4.5 ± 2.0 | 6.9 ± 3.9 | 8.1 ± 3.7 | 9.7 ± 4.4 |

3 | 0.84 | 6 | 3 ± 1.7 | 4.5 ± 2.5 | 5.2 ± 2.4 | 6.2 ± 2.8 |

4 | 1.16 | 4 | 2 ± 1.4 | 3.2 ± 1.8 | 3.8 ± 1.7 | 4.5 ± 2.0 |

5 | 1.60 | 3 | 1.5 ± 1.2 | 2.3 ± 1.3 | 2.7 ± 1.2 | 3.3 ± 1.5 |

Number of Holes | The Angle between Holes Relative to the Direction of Tension | Strength, σ, MPa | β |
---|---|---|---|

0 | - | 190 ± 9 | 1 |

1 | - | 161 ± 8 | 1.2 |

2 | 0° | 128 ± 9 | 1.5 |

2 | 45° | 112 ± 6 | 1.7 |

2 | 90° | 118 ± 5 | 1.6 |

**Table 6.**Fractal dimension for TM surface pore distribution and TM computer models (regular network and Poisson distribution) at n = 4·10

^{6}cm

^{−2}.

d, μm | ${\mathit{r}}_{\mathbf{Average}\mathbf{smallest}}=\frac{\sum {\mathit{r}}_{\mathbf{least}}}{\mathit{N}\mathit{\xb7}\mathit{d}}$ (calc. by SEM Images up to One Nearest Pore) | Fractal Dimension for TM | Fractal Dimension for a Model with Pores Distributed by Poisson’s Law | Fractal Dimension for a Model with Pores Distributed by Poisson’s Law |
---|---|---|---|---|

0.22 | 16.9 ± 9.7 | 1.10 | 1–1.10 | 1.10 |

0.54 | 6.9 ± 3.9 | 1.25 | 1.30 | 1.27 |

0.84 | 4.5 ± 2.5 | 1.30 | 1.35 | 1.30 |

1.16 | 3.2 ± 1.8 | 1.40 | 1.40 | 1.34 |

1.60 | 2.3 ± 1.3 | 1.55 | 1.50 | 1.50 |

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**MDPI and ACS Style**

Gumirova, V.; Razumovskaya, I.; Apel, P.; Bedin, S.; Naumov, A.
The Influence of Mechanical Stress Micro Fields around Pores on the Strength of Elongated Etched Membrane. *Membranes* **2022**, *12*, 1168.
https://doi.org/10.3390/membranes12111168

**AMA Style**

Gumirova V, Razumovskaya I, Apel P, Bedin S, Naumov A.
The Influence of Mechanical Stress Micro Fields around Pores on the Strength of Elongated Etched Membrane. *Membranes*. 2022; 12(11):1168.
https://doi.org/10.3390/membranes12111168

**Chicago/Turabian Style**

Gumirova, Venera, Irina Razumovskaya, Pavel Apel, Sergey Bedin, and Andrey Naumov.
2022. "The Influence of Mechanical Stress Micro Fields around Pores on the Strength of Elongated Etched Membrane" *Membranes* 12, no. 11: 1168.
https://doi.org/10.3390/membranes12111168