# Analysis of Stress Intensity Factor of a Fibre Embedded in a Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Numerical Evaluation of SIF

#### 2.2. Non-Dimensional Parametric Analysis

## 3. Results and Discussion

#### 3.1. Isotropic Fibre Embedded in a Matrix

#### 3.2. Orthotropic Free Fibre

#### 3.3. Orthotropic Fibre Embedded in a Matrix

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Verification of the Finite Element Model

#### Appendix A.1. Free Fibre Model

#### Appendix A.2. Embedded Fibre Model with Very Low Stiffness Matrix

**Figure A2.**Non-dimensional SIF of an isotropic free fibre resulting from a computational model of the isolated fibre (line) and with embedding material of nearly zero stiffness (dots).

#### Appendix A.3. Comparison with an Analytical Model

**Figure A3.**Geometric illustration of a model consisting of a broken fibre embedded in a matrix as a solid cylinder with central circular crack. The fibre is broken in the middle (${x}_{1}=L/2$) and shown as the grey area.

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**Figure 1.**(

**a**) 3D geometry of a notched fibre with straight-fronted edge crack (indicated in grey) embedded in a matrix; (

**b**) fibre cross-section at crack plane.

**Figure 2.**FE model with details of the boundary conditions and finite-element meshing of the crack tip region.

**Figure 4.**(

**a**) Non-dimensional SIF for an isotropic fibre embedded in a matrix for different fibre–matrix stiffness ratios, $\alpha $. The dots refer to the FEM results and the dashed lines to the fitting curves according to Equation (10). The continuous grey curve corresponds to the SIF for a free fibre ($\alpha =1$). (

**b**) Relative reduction of the non-dimensional SIF for an isotropic fibre embedded in a matrix with respect to the free fibre (continuous grey line).

**Figure 5.**Non-dimensional CMOD for an isotropic fibre embedded in a matrix for different fibre–matrix stiffness ratios, $\alpha $. The continuous grey curve corresponds to the CMOD for a free fibre ($\alpha =1$).

**Figure 6.**(

**a**) Comparison of non-dimensional stress intensity factor of an orthotropic free fibre for different longitudinal/transverse stiffness ratios, $\beta $. The dots refer to the FEM results and the dashed lines indicate the fitting results. (

**b**) Percentage of non-dimensional stress intensity factor changes of an orthotropic free fibre for different longitudinal/transverse stiffness ratio, $\beta $. The percentage change was calculated with respect to the model with $\beta =0$ (${E}_{1}={E}_{2}$) as the reference.

**Figure 7.**Comparison of non-dimensional stress intensity factor of an orthotropic fibre embedded in a matrix, for different fibre longitudinal/transverse stiffness ratios ($\beta $) at fibre–matrix stiffness ratios: (

**a**) $\alpha =0.95$ and (

**b**) $\alpha =-0.95$.

**Table 1.**Coefficients for the fitting equation given in Equation (10).

Coefficients of Equation (10) | |||||
---|---|---|---|---|---|

${A}_{1}$ | 1.187 | ${B}_{1}$ | −0.4268 | ${C}_{1}$ | −0.4647 |

${A}_{2}$ | −0.4349 | ${B}_{2}$ | −0.1893 | ${C}_{2}$ | 0.7246 |

${A}_{3}$ | −0.3255 | ${B}_{3}$ | −0.07555 | ${C}_{3}$ | 0.4295 |

**Table 2.**Coefficients for the fitting of equation given in Equation (11).

Coefficients of Equation (11) | |||||||
---|---|---|---|---|---|---|---|

${A}_{1}$ | −0.9022 | ${B}_{1}$ | 0.932 | ${C}_{1}$ | −4.647 | ${D}_{1}$ | 17.67 |

${A}_{2}$ | 0.3677 | ${B}_{2}$ | −0.7516 | ${C}_{2}$ | 4.415 | ${D}_{2}$ | −10.7 |

${A}_{3}$ | 1.833 | ${B}_{3}$ | 1.81 | ${C}_{3}$ | −1.928 | ||

${A}_{4}$ | −1.993 | ${B}_{4}$ | −1.727 |

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**MDPI and ACS Style**

Barzegar, M.; Costa, J.; Trias, D.; Guerrero, J.M.; Lopes, C.; Gonzalez, C.
Analysis of Stress Intensity Factor of a Fibre Embedded in a Matrix. *J. Compos. Sci.* **2023**, *7*, 22.
https://doi.org/10.3390/jcs7010022

**AMA Style**

Barzegar M, Costa J, Trias D, Guerrero JM, Lopes C, Gonzalez C.
Analysis of Stress Intensity Factor of a Fibre Embedded in a Matrix. *Journal of Composites Science*. 2023; 7(1):22.
https://doi.org/10.3390/jcs7010022

**Chicago/Turabian Style**

Barzegar, Mostafa, Josep Costa, Daniel Trias, Jose M. Guerrero, Claudio Lopes, and Carlos Gonzalez.
2023. "Analysis of Stress Intensity Factor of a Fibre Embedded in a Matrix" *Journal of Composites Science* 7, no. 1: 22.
https://doi.org/10.3390/jcs7010022