# Evaluation of Critical Parameters in Tensile Strength Measurement of Single Fibres

^{*}

*Journal of Composites Science*in 2019)

## Abstract

**:**

## 1. Introduction

## 2. Fibre Tensile Strength

#### 2.1. Single Fibre Testing

- aligning the fibre with the loading direction as perfectly as possible; and
- controlling the gauge length as accurately as possible, since fibre strength is length dependent.

#### 2.2. Determination of Tensile Strength

#### 2.3. Measurement Uncertainty

#### 2.4. Experimentation

#### 2.5. Evaluation of Measurement Uncertainty in Input Quantities

#### 2.6. Propagation of Uncertainty to Fibre Tensile Strength

## 3. Sensitivity Analysis

#### 3.1. Sensitivity Analysis for Fibre Tensile Strength

#### 3.2. Estimation of Sensitivity for Tensile Strength at Other Gauge Lengths

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of a single fibre mounted on a card frame; (

**b**) major issues with sample preparation leading to inaccurate measurement of gauge length; and (

**c**) angular misalignment in the fibre specimen.

**Figure 3.**Schematic diagram showing the effect of fibre misalignment on the effective load on the fibre. (

**Top**) The effect of fibre elongation and system compliance on the instantaneous gauge length of the fibre, as measured by the displacement transducer. (

**Bottom**) Fibre misalignment can occur in any direction in three dimensions, i.e., the fibre can be positioned anywhere inside a cone of angle $\alpha $.

**Figure 5.**Sensitivity indices of different input quantities in tensile strength measurement of fibres at different gauge lengths.

S.N. | Quantity | Value | Unit |
---|---|---|---|

1 | ${F}_{\mathrm{R}}$ | 0.16 | N |

2 | h | 1 | mm |

3 | ${L}_{0}$ | 30 | mm |

4 | ${L}_{\mathrm{S}}$ | 0.69 | mm |

5 | D | 6.7 × ${10}^{-3}$ | mm |

**Table 2.**Measurement uncertainty values of all input quantities for the given fibre tensile test example, along with expanded uncertainty for a confidence level of 95% and 99%.

S.N. | Measurement Uncertainty | Value | Expanded Uncertainty | Value | Expanded Uncertainty | Value | Unit |
---|---|---|---|---|---|---|---|

1 | $u\left({F}_{\mathrm{R}}\right)$ | $1.27\times {10}^{-3}$ | ${U}_{95}\left({F}_{\mathrm{R}}\right)$ | $2.48\times {10}^{-3}$ | ${U}_{99}\left({F}_{\mathrm{R}}\right)$ | $3.28\times {10}^{-3}$ | N |

2 | $u\left(h\right)$ | $0.28$ | ${U}_{95}\left(h\right)$ | $0.55$ | ${U}_{99}\left(h\right)$ | $0.72$ | mm |

3 | $u\left({L}_{0}\right)$ | $0.34$ | ${U}_{95}\left({L}_{0}\right)$ | $0.66$ | ${U}_{99}\left({L}_{0}\right)$ | $0.88$ | mm |

4 | $u\left({L}_{\mathrm{S}}\right)$ | $0.71$ | ${U}_{95}\left({L}_{\mathrm{S}}\right)$ | $1.38$ | ${U}_{99}\left({L}_{\mathrm{S}}\right)$ | $1.86$ | mm |

5 | $u\left(D\right)$ | $5.5\times {10}^{-5}$ | ${U}_{95}\left(D\right)$ | $1.8\times {10}^{-4}$ | ${U}_{99}\left(D\right)$ | $3.2\times {10}^{-4}$ | mm |

S.N. | Quantity | Value | Unit |
---|---|---|---|

1 | $\frac{\partial \sigma}{\partial {F}_{\mathrm{R}}}$ | $2.81\times {10}^{10}$ | m${}^{-2}$ |

2 | $\frac{\partial \sigma}{\partial h}$ | $4.63\times {10}^{9}$ | N·m${}^{-3}$ |

3 | $\frac{\partial \sigma}{\partial {L}_{0}}$ | $1.51\times {10}^{8}$ | N·m${}^{-3}$ |

4 | $\frac{\partial \sigma}{\partial {L}_{\mathrm{S}}}$ | $1.51\times {10}^{8}$ | N·m${}^{-3}$ |

5 | $\frac{\partial \sigma}{\partial D}$ | $-1.30\times {10}^{15}$ | N·m${}^{-3}$ |

S.N. | $\mathit{\sigma}$ (GPa) | ${\mathit{u}}_{\mathit{c}}\left(\mathit{\sigma}\right)$ (GPa) | S.N. | $\mathit{\sigma}$ (GPa) | ${\mathit{u}}_{\mathit{c}}\left(\mathit{\sigma}\right)$ (GPa) | S.N. | $\mathit{\sigma}$ (GPa) | ${\mathit{u}}_{\mathit{c}}\left(\mathit{\sigma}\right)$ (GPa) |
---|---|---|---|---|---|---|---|---|

1 | 1.69 | 0.07 | 11 | 3.23 | 0.10 | 21 | 4.36 | 0.08 |

2 | 1.78 | 0.07 | 12 | 3.28 | 0.11 | 22 | 4.45 | 0.12 |

3 | 1.97 | 0.06 | 13 | 3.48 | 0.08 | 23 | 4.49 | 0.05 |

4 | 2.23 | 0.03 | 14 | 3.50 | 0.18 | 24 | 4.72 | 0.09 |

5 | 2.56 | 0.06 | 15 | 3.52 | 0.17 | 25 | 4.73 | 0.12 |

6 | 2.72 | 0.03 | 16 | 3.68 | 0.14 | 26 | 4.85 | 0.06 |

7 | 2.73 | 0.05 | 17 | 3.75 | 0.09 | 27 | 5.44 | 0.31 |

8 | 2.82 | 0.05 | 18 | 3.82 | 0.07 | 28 | 5.68 | 0.08 |

9 | 2.93 | 0.06 | 19 | 4.04 | 0.06 | 29 | 5.99 | 0.17 |

10 | 3.03 | 0.09 | 20 | 4.15 | 0.12 | 30 | 6.92 | 0.10 |

**Table 5.**Sensitivity measures of each input quantity for the given single fibre tensile test example.

Rank | Input | ∣Sensitivity∣ | Sensitivity${}^{2}$ |
---|---|---|---|

1 | D | $0.89$ | 0.79 |

2 | ${F}_{\mathrm{R}}$ | $0.45$ | 0.20 |

3 | h | 1.63 × ${10}^{-2}$ | 2.67 × ${10}^{-4}$ |

4 | ${L}_{\mathrm{S}}$ | 1.33 × ${10}^{-3}$ | 1.76 × ${10}^{-6}$ |

5 | ${L}_{0}$ | 6.35 × ${10}^{-4}$ | 4.03 × ${10}^{-7}$ |

**Table 6.**Average sensitivity measures for each input quantity for the set of 30 single fibre tensile test.

Rank | Input | ∣Sensitivity∣ | Sensitivity${}^{2}$ |
---|---|---|---|

1 | D | $0.81$ | 0.66 |

2 | ${F}_{\mathrm{R}}$ | $0.48$ | 0.23 |

3 | h | 0.02 | 2.33 × ${10}^{-4}$ |

4 | ${L}_{\mathrm{S}}$ | 1.26 × ${10}^{-3}$ | 1.58 × ${10}^{-6}$ |

5 | ${L}_{0}$ | 6.03 × ${10}^{-4}$ | 3.64 × ${10}^{-7}$ |

**Table 7.**Sensitivity indices (S) of different input quantities in tensile strength measurement of fibres at different gauge lengths.

Input ↓ | 30 mm S | 20 mm S | 10 mm S | 5 mm S | 2.5 mm S |
---|---|---|---|---|---|

D | 0.81 | 0.83 | 0.85 | 0.74 | 0.33 |

${F}_{\mathrm{R}}$ | 0.48 | 0.45 | 0.38 | 0.25 | 0.08 |

h | 0.02 | 0.04 | 0.15 | 0.47 | 0.62 |

${L}_{\mathrm{S}}$ | 1.26 × ${10}^{-3}$ | 4.36 × ${10}^{-3}$ | 0.04 | 0.23 | 0.61 |

${L}_{0}$ | 6.03 × ${10}^{-4}$ | 2.09 × ${10}^{-3}$ | 0.02 | 0.11 | 0.29 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Islam, F.; Joannès, S.; Laiarinandrasana, L. Evaluation of Critical Parameters in Tensile Strength Measurement of Single Fibres. *J. Compos. Sci.* **2019**, *3*, 69.
https://doi.org/10.3390/jcs3030069

**AMA Style**

Islam F, Joannès S, Laiarinandrasana L. Evaluation of Critical Parameters in Tensile Strength Measurement of Single Fibres. *Journal of Composites Science*. 2019; 3(3):69.
https://doi.org/10.3390/jcs3030069

**Chicago/Turabian Style**

Islam, Faisal, Sébastien Joannès, and Lucien Laiarinandrasana. 2019. "Evaluation of Critical Parameters in Tensile Strength Measurement of Single Fibres" *Journal of Composites Science* 3, no. 3: 69.
https://doi.org/10.3390/jcs3030069