# Modeling Flexural Failure in Carbon-Fiber-Reinforced Polymer Composites

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## Abstract

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## 1. Introduction

_{23}), which represent Young’s modulus in the fiber direction, Young’s modulus perpendicular to the fiber direction, the in-plane shear modulus, the in-plane Poisson’s ratio, the out-of-plane shear modulus, and the out-of-plane Poisson’s ratio, respectively [1]. However, there are few significant mathematical relations for material strength resulting in numerous resource-intensive destructive tests, often providing single failure mode results. Accordingly, it is advantageous to study composite behavior under combined loading, and their dominating failure modes.

## 2. Materials and Methods

#### 2.1. ASTM D7264

#### 2.2. CFRP Systems

#### 2.3. Failure Criteria

#### 2.4. Prediction of Flexural Strength

#### 2.5. Finite-Element Model

#### 2.5.1. Geometry

#### 2.5.2. Domain Discretization (Meshing)

#### 2.5.3. Boundary Conditions and Contact Settings

#### 2.5.4. Solver and Postprocessing

#### 2.6. Model Verification with Shear Stresses in Beam Bending

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**FEA mesh of (

**a**) support roller, (

**b**) sample, and (

**c**) loading roller; (

**d**) complete mesh with sample and rollers.

Parameter | Span Length L (mm) | Thickness t (mm) | Width—Into the Page (mm) | Additional Overhang (%) | Loading Nose Radius (mm) |
---|---|---|---|---|---|

Value | 128 | 4 | 13 | 20 | 5 |

Composite System (Fiber/Resin) | Tensile Strength (MPa) | Compressive Strength (MPa) | Flexural Strength (MPa) | Ultimate Tensile Strain (%) | Ultimate Compressive Strain (%) |
---|---|---|---|---|---|

T700/Toray LM PAEK | 2322 | 1226 | 1455 | 1.86 | 0.98 |

APC2/PEEK (Solvay) | 2070 | 1360 | 2000 | 1.45 | 1.10 |

AS4/Hexcel 8552 | 2205 | 1530 | 1889 | 1.56 | 1.09 |

Theory | Category | Failure Criterion |
---|---|---|

Maximal stress | Noninteractive | ${\sigma}_{ij}<{F}_{ij}$ |

Maximal strain | Noninteractive | ${\epsilon}_{ij}<{\epsilon}_{ij}^{u}$ |

Hashin–Rotem | Partially interactive | $\begin{array}{l}\frac{\left|{\sigma}_{1}\right|}{{F}_{1}}=1\\ {\left(\frac{{\sigma}_{2}}{{F}_{2}}\right)}^{2}+{\left(\frac{{\tau}_{4}}{{F}_{4}}\right)}^{2}+{\left(\frac{{\tau}_{6}}{{F}_{6}}\right)}^{2}=1\\ {\left(\frac{{\sigma}_{3}}{{F}_{3}}\right)}^{2}+{\left(\frac{{\tau}_{4}}{{F}_{4}}\right)}^{2}+{\left(\frac{{\tau}_{5}}{{F}_{5}}\right)}^{2}=1\end{array}$ |

Tsai–Wu | Fully interactive | $\begin{array}{l}{f}_{1}{\sigma}_{1}+{f}_{2}\left({\sigma}_{2}+{\sigma}_{3}\right)+{f}_{11}{\sigma}_{1}^{2}+{f}_{22}\left({\sigma}_{2}^{2}+{\sigma}_{3}^{2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{f}_{44}{\tau}_{4}^{2}\\ +{f}_{66}\left({\tau}_{5}^{2}+{\tau}_{6}^{2}\right)+2{f}_{12}\left({\sigma}_{1}{\sigma}_{2}+{\sigma}_{1}{\sigma}_{3}\right)+2{f}_{23}{\sigma}_{2}{\sigma}_{3}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=1\end{array}$ |

Hoffman | Fully interactive | Same as Tsai—Wu, coefficient definitions differ. |

Tsai–Hill | Fully interactive | $\begin{array}{l}\frac{{\sigma}_{1}^{2}-{\sigma}_{1}{\sigma}_{2}-{\sigma}_{1}{\sigma}_{3}}{{F}_{1}^{2}}+\frac{{\sigma}_{2}^{2}+{\sigma}_{3}^{2}-{\sigma}_{2}{\sigma}_{3}}{{F}_{2}^{2}}+\frac{{\tau}_{4}^{2}}{{F}_{4}^{2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\tau}_{5}^{2}+{\tau}_{6}^{2}}{{F}_{6}^{2}}=1\end{array}$ |

Composite System (Fiber/Resin) | Tensile Strength (MPa) | Compressive Strength (MPa) | Flexural Strength (MPa) | FEA-Flexural Strength (MPa) | % Error between FEA and Experiment |
---|---|---|---|---|---|

T700/Toray LM PAEK | 2322 | 1226 | 1455 | 1367 | −6.1 |

AS4/Hexcel 8552 Epoxy | 2205 | 1530 | 1889 | 1717 | −9.1 |

APC2/Solvay PEEK | 2070 | 1360 | 2000 | 1725 | −13.8 |

Compressive Strength (MPa) | FEA Maximum Compressive Stress (MPa) | Relative Difference (%) | |
---|---|---|---|

Toray T700/LM PAEK | 1226 | 1338 | 9.2 |

Solvay APC-2/PEEK | 1360 | 1700 | 25 |

Hexcel AS4/8552 epoxy | 1530 | 1690 | 10.5 |

**Table 6.**Tsai–Wu index components for three analyzed systems. Bold terms contribute to the enhanced load-bearing capacity of the systems.

Toray T700/PAEK | Hexcel AS4/8552 | Solvay APC2/PEEK | |
---|---|---|---|

σ_{1} (MPa) | −1338 | 1706 | −1700 |

σ_{2} (MPa) | −41 | 35 | −50 |

τ_{12} (MPa) | 0.6 | −0.9 | 0.9 |

F_{1}σ_{1} | 0.603 | −0.325 | 0.440 |

F_{2}σ_{2} | −0.225 | 0.452 | −0.285 |

F_{11}σ_{1}^{2} | 0.687 | 0.947 | 1.061 |

F_{22}σ_{2}^{2} | 0.135 | 0.156 | 0.145 |

F_{66}τ_{12}^{2} | 0.000 | 0.000 | 0.000 |

$\mathbf{-}\sqrt{{\mathbf{F}}_{\mathbf{11}}{\mathbf{F}}_{\mathbf{22}}}$σ_{1}σ_{2} | −0.197 | −0.225 | −0.347 |

Total | 1.003 | 1.005 | 1.014 |

Failure Criteria | FEA-Predicted Strength (MPa) | Error (%) |
---|---|---|

Tsai–Wu | 1366.6 | −6.06 |

Max strain | 1257.3 | −13.58 |

Max stress | 1243.6 | −14.52 |

Tsai–Hill | 1243.6 | −14.52 |

Hoffman | 1352.9 | −7.00 |

Hashin | 1243.6 | −14.52 |

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**MDPI and ACS Style**

Burgani, T.d.S.; Alaie, S.; Tehrani, M.
Modeling Flexural Failure in Carbon-Fiber-Reinforced Polymer Composites. *J. Compos. Sci.* **2022**, *6*, 33.
https://doi.org/10.3390/jcs6020033

**AMA Style**

Burgani TdS, Alaie S, Tehrani M.
Modeling Flexural Failure in Carbon-Fiber-Reinforced Polymer Composites. *Journal of Composites Science*. 2022; 6(2):33.
https://doi.org/10.3390/jcs6020033

**Chicago/Turabian Style**

Burgani, Thiago de Sousa, Seyedhamidreza Alaie, and Mehran Tehrani.
2022. "Modeling Flexural Failure in Carbon-Fiber-Reinforced Polymer Composites" *Journal of Composites Science* 6, no. 2: 33.
https://doi.org/10.3390/jcs6020033