# Prediction of Fracture Toughness of Pultruded Composites Based on Supervised Machine Learning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Pultruded Composite Material

#### 2.2. Mechanical Testing

#### 2.2.1. Standard Mechanical Tests

#### 2.2.2. Fracture Toughness Test

#### 2.3. Machine-Learning Methods

#### 2.3.1. Artificial Neural Network

#### 2.3.2. Random Forest

#### 2.3.3. Gradient Boosting

#### 2.3.4. Evaluation Criteria

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wang, J.; GangaRao, H.; Liang, R.; Liu, W. Durability and prediction models of fiber-reinforced polymer composites under various environmental conditions: A critical review. J. Reinf. Plast. Compos.
**2015**, 35, 179–211. [Google Scholar] [CrossRef] - Ma, Z.; Zhang, P.; Zhu, J. Review on the fatigue properties of 3D woven fiber/epoxy composites: Testing and modelling strategies. J. Ind. Text.
**2020**, 51 (Suppl. S5), 7755S–7795S. [Google Scholar] [CrossRef] - Baran, I.; Cinar, K.; Ersoy, N.; Akkerman, R.; Hattel, J.H. A Review on the Mechanical Modeling of Composite Manufacturing Processes. Arch. Comput. Methods Eng.
**2016**, 24, 365–395. [Google Scholar] [CrossRef] [PubMed] - Bostanabad, R.; Zhang, Y.; Li, X.; Kearney, T.; Brinson, L.C.; Apley, D.W.; Liu, W.K.; Chen, W. Computational microstructure characterization and reconstruction: Review of the state-of-the-art techniques. Prog. Mater. Sci.
**2018**, 95, 1–41. [Google Scholar] [CrossRef] - Mohan, N.; Senthil, P.; Vinodh, S.; Jayanth, N. A review on composite materials and process parameters optimisation for the fused deposition modelling process. Virtual Phys. Prototyp.
**2017**, 12, 47–59. [Google Scholar] [CrossRef] - Zhou, X.-Y.; Gosling, P.; Pearce, C.; Kaczmarczyk, L.; Ullah, Z. Perturbation-based stochastic multi-scale computational homogenization method for the determination of the effective properties of composite materials with random properties. Comput. Methods Appl. Mech. Eng.
**2015**, 300, 84–105. [Google Scholar] [CrossRef] - Karabasov, S.; Nerukh, D.; Hoekstra, A.; Chopard, B.; Coveney, P.V. Multiscale modelling: Approaches and challenges. Philos. Trans. R. Soc. London. Ser. A Math. Phys. Eng. Sci.
**2014**, 372, 20130390. [Google Scholar] [CrossRef] [PubMed] - Hoekstra, A.; Chopard, B.; Coveney, P. Multiscale modelling and simulation: A position paper. Philos. Trans. R. Soc. London. Ser. A Math. Phys. Eng. Sci.
**2014**, 372, 20130377. [Google Scholar] [CrossRef] - Chen, C.-T.; Gu, G.X. Machine learning for composite materials. MRS Commun.
**2019**, 9, 556–566. [Google Scholar] [CrossRef] - Ramprasad, R.; Batra, R.; Pilania, G.; Mannodi-Kanakkithodi, A.; Kim, C. Machine learning in materials informatics: Recent applications and prospects. NPJ Comput. Mater.
**2017**, 3, 1. [Google Scholar] [CrossRef] [Green Version] - Mukherjee, A.; Schmauder, S.; Rugle, M. Artificial neural networks for the prediction of mechanical begavior of metal matrix composites. Acta Met.
**1995**, 43, 4083–4091. [Google Scholar] [CrossRef] - Koker, R.; Altinkok, N.; Demir, A. Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms. Mater. Des.
**2007**, 28, 616–627. [Google Scholar] [CrossRef] - Altinkok, N.; Koker, R. Neural network approach to prediction of bending strength and hardening behaviour of particulate reinforced (Al–Si–Mg)-aluminium matrix composites. Mater. Des.
**2004**, 25, 595–602. [Google Scholar] [CrossRef] - Sha, W.; Edwards, K. The use of artificial neural networks in materials science based research. Mater. Des.
**2007**, 28, 1747–1752. [Google Scholar] [CrossRef] - Yu, W.; Li, M.; Luo, J.; Su, S.; Li, C. Prediction of the mechanical properties of the post-forged Ti–6Al–4V alloy using fuzzy neural network. Mater. Des.
**2010**, 31, 3282–3288. [Google Scholar] [CrossRef] - Ye, S.; Li, B.; Li, Q.; Zhao, H.-P.; Feng, X.-Q. Deep neural network method for predicting the mechanical properties of composites. Appl. Phys. Lett.
**2019**, 115, 161901. [Google Scholar] [CrossRef] - Yang, C.; Kim, Y.; Ryu, S.; Gu, G.X. Prediction of composite microstructure stress-strain curves using convolutional neural networks. Mater. Des.
**2020**, 189, 108509. [Google Scholar] [CrossRef] - Qi, Z.; Zhang, N.; Liu, Y.; Chen, W. Prediction of mechanical properties of carbon fiber based on cross-scale FEM and machine learning. Compos. Struct.
**2019**, 212, 199–206. [Google Scholar] [CrossRef] - Jiang, Z.; Zhang, Z.; Friedrich, K. Prediction on wear properties of polymer composites with artificial neural networks. Compos. Sci. Technol.
**2007**, 67, 168–176. [Google Scholar] [CrossRef] - Jiang, Z.; Gyurova, L.; Zhang, Z.; Friedrich, K.; Schlarb, A.K. Neural network based prediction on mechanical and wear properties of short fibers reinforced polyamide composites. Mater. Des.
**2008**, 29, 628–637. [Google Scholar] [CrossRef] - Shirvanimoghaddam, K.; Khayyam, H.; Abdizadeh, H.; Akbari, M.K.; Pakseresht, A.; Ghasali, E.; Naebe, M. Boron carbide reinforced aluminium matrix composite: Physical, mechanical characterization and mathematical modelling. Mater. Sci. Eng. A
**2016**, 658, 135–149. [Google Scholar] [CrossRef] - Li, X.; Liu, Z.; Cui, S.; Luo, C.; Li, C.; Zhuang, Z. Predicting the effective mechanical property of heterogeneous materials by image based modeling and deep learning. Comput. Methods Appl. Mech. Eng.
**2019**, 347, 735–753. [Google Scholar] [CrossRef] - Gu, G.X.; Chen, C.-T.; Buehler, M.J. De novo composite design based on machine learning algorithm. Extrem. Mech. Lett.
**2018**, 18, 19–28. [Google Scholar] [CrossRef] - Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M.; et al. TensorFlow: A system for large-scale machine learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), Savannah, GA, USA, 2–4 November 2016. [Google Scholar]
- Yang, Z.; Yabansu, Y.C.; Jha, D.; Liao, W.-K.; Choudhary, A.N.; Kalidindi, S.R.; Agrawal, A. Establishing structure-property localization linkages for elastic deformation of three-dimensional high contrast composites using deep learning approaches. Acta Mater.
**2018**, 166, 335–345. [Google Scholar] [CrossRef] - Tiryaki, S.; Aydın, A. An artificial neural network model for predicting compression strength of heat treated woods and comparison with a multiple linear regression model. Constr. Build. Mater.
**2014**, 62, 102–108. [Google Scholar] [CrossRef] - Lu, L.; Dao, M.; Kumar, P.; Ramamurty, U.; Karniadakis, G.E.; Suresh, S. Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proc. Natl. Acad. Sci. USA
**2020**, 117, 7052–7062. [Google Scholar] [CrossRef] - Mottram, J.T. Compression Strength of Pultruded Flat Sheet Material. J. Mater. Civ. Eng.
**1994**, 6, 185–200. [Google Scholar] [CrossRef] - Alqam, M.; Bennett, R.M.; Zureick, A.-H. Three-parameter vs. two-parameter Weibull distribution for pultruded composite material properties. Compos. Struct.
**2002**, 58, 497–503. [Google Scholar] [CrossRef] - Vedernikov, A.; Safonov, A.; Tucci, F.; Carlone, P.; Akhatov, I. Pultruded materials and structures: A review. J. Compos. Mater.
**2020**, 54, 4081–4117. [Google Scholar] [CrossRef] - El-Hajjar, R.; Haj-Ali, R. Mode-I fracture toughness testing of thick section FRP composites using the ESE(T) specimen. Eng. Fract. Mech.
**2005**, 72, 631–643. [Google Scholar] [CrossRef] - Fernandes, L.A.; Silvestre, N.; Correia, J.R. Characterization of transverse fracture properties of pultruded GFRP material in tension. Compos. Part B Eng.
**2019**, 175, 107095. [Google Scholar] [CrossRef] - Almeida-Fernandes, L.; Silvestre, N.; Correia, J.R.; Arruda, M. Fracture toughness-based models for damage simulation of pultruded GFRP materials. Compos. Part B Eng.
**2020**, 186, 107818. [Google Scholar] [CrossRef] - ASTM International E399-12; Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness K Ic of Metallic Materials. ASTM: West Conshohocken, PA, USA, 2012; pp. 1–33. [CrossRef]
- Nwankpa, C.; Ijomah, W.; Gachagan, A.; Marshall, S. Activation Functions: Comparison of trends in Practice and Research for Deep Learning. arXiv
**2018**, arXiv:1811.03378. [Google Scholar] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Nature
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. In Proceedings of the 3rd International Conference on Learning Representations, San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
- Segal, M.R. Machine Learning Benchmarks and Random Forest Regression, Biostatistics; UCSF: San Francisco, CA, USA, 2004; pp. 1–14. Available online: http://escholarship.org/uc/item/35x3v9t4.pdf (accessed on 24 January 2022).
- Sexton, J.; Laake, P. Standard errors for bagged and random forest estimators. Comput. Stat. Data Anal.
**2009**, 53, 801–811. [Google Scholar] [CrossRef] - Strobl, C.; Boulesteix, A.-L.; Zeileis, A.; Hothorn, T. Bias in random forest variable importance measures: Illustrations, sources and a solution. BMC Bioinform.
**2007**, 8, 25. [Google Scholar] [CrossRef] - Archer, K.J.; Kimes, R.V. Empirical characterization of random forest variable importance measures. Comput. Stat. Data Anal.
**2008**, 52, 2249–2260. [Google Scholar] [CrossRef] - Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Friedman, J.H. Stochastic gradient boosting. Comput. Stat. Data Anal.
**2002**, 38, 367–378. [Google Scholar] [CrossRef] - Chen, T.; Guestrin, C. XGBoost: A Scalable Tree Boosting System. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13–17 August 2016; Association for Computing Machinery: New York, NY, USA, 2016; pp. 785–794. [Google Scholar] [CrossRef]
- Miyajima, T.; Sakai, M. The fracture toughness for first matrix cracking of a unidirectionally reinforced carbon/carbon composite material. J. Mater. Res.
**1991**, 6, 2312–2317. [Google Scholar] [CrossRef] - Tsouvalis, N.G.; Anyfantis, K.N. Determination of the fracture process zone under Mode I fracture in glass fiber composites. J. Compos. Mater.
**2011**, 46, 27–41. [Google Scholar] [CrossRef] - Song, L.; Meng, S.; Xu, C.; Fang, G.; Yang, Q. Finite element-based phase-field simulation of interfacial damage in unidirectional composite under transverse tension. Model. Simul. Mater. Sci. Eng.
**2019**, 27, 55011. [Google Scholar] [CrossRef]

**Figure 2.**Cutting scheme for specimens from one meter of the pultruded profile (the specimen numbers are indicated for internal use).

**Figure 4.**Example of fully connected neural network with two input neurons (input layer), nine hidden neurons (two hidden layers), and one output neuron (output layer).

**Figure 6.**Heatmap of correlations in the acquired dataset. The last row (or column) represents the correlation of the fracture toughness with other properties. Property numeration is the same as in Table 2.

**Figure 7.**Plots for predicted vs. true values with dots indicating the mean prediction for one data point and vertical lines indicating error bars across different training sessions during cross-validation: (

**a**) neural network (note the different scale for the presentation of error bars), (

**b**) random forest, (

**c**) XGBoost, (

**d**) cross-validation scheme.

**Figure 8.**Variable importance vs. normalized correlations: (

**a**) random forest, (

**b**) XGBoost, (

**c**) normalized correlations. Properties are assigned similarly as in Table 2.

**Figure 9.**Predicted vs. true plots for selected features, with dots indicating the mean predictions for individual data points and vertical lines indicating error bars across different training sessions during cross-validation: (

**a**) Neural network (note the different scale for the presentation of the error bars), (

**b**) random forest, (

**c**) XGBoost.

Mechanical Test | Standard | Machine | Strain Measurements | Output Properties |
---|---|---|---|---|

Tension | ASTM D3039 | Instron 5969 | DIC | Tensile modulus, strength |

Compression | ASTM D6641 CLC | Compression modulus, strength | ||

Flexure | ASTM D7264 | Flexure modulus, strength | ||

In-plane shear | ASTM D7078 | Shear strength, shear modulus | ||

Charpy impact | ASTM D256 | Instron CEAST 9340 | Strain gauge 64k | Failure energy |

# | Property | Min | Max | Mean | STD |
---|---|---|---|---|---|

0 | Coordinates, m | 0 | 50 | - | - |

1 | 0° bending failure stress, MPa | 239 | 676 | 559.2 | 71.3 |

2 | 0° bending modulus, GPa | 25.3 | 34.6 | 30.3 | 3.2 |

3 | 90° bending failure stress, MPa | 14.7 | 172.0 | 132.5 | 25.9 |

4 | 90° bending modulus, GPa | 5.1 | 13.8 | 11.8 | 1.3 |

5 | 0° shear strength, MPa | 24.2 | 58.1 | 45.6 | 7.9 |

6 | 0° shear modulus, GPa | 4.3 | 9.3 | 6.9 | 1.2 |

7 | 90° shear strength, MPa | 45.2 | 69.2 | 54.4 | 4.9 |

8 | 90° shear modulus, GPa | 4.9 | 9.9 | 7.8 | 0.7 |

9 | 0° compression strength, MPa | 366 | 660 | 494.4 | 58.0 |

10 | 0° compression modulus, GPa | 34.3 | 54.2 | 47.4 | 3.9 |

11 | 90° compression strength, MPa | 71.6 | 117.0 | 91.9 | 10.1 |

12 | 90° compression modulus, GPa | 5.2 | 9.1 | 6.7 | 0.8 |

13 | 0° impact failure energy, kJ/m^{2} | 30 | 142 | 70 | 27 |

14 | 90° impact failure energy, kJ/m^{2} | 9.5 | 22.5 | 14.5 | 3 |

15 | 0° tensile strength, MPa | 434 | 839 | 657.5 | 85.6 |

16 | 0° tensile modulus, GPa | 41.9 | 49.3 | 45.1 | 1.5 |

17 | 90° tensile strength, MPa | 36.4 | 57.4 | 45.6 | 4.5 |

18 | 90° tensile modulus, GPa | 4.2 | 6.6 | 5.4 | 0.5 |

19 | 0° KIC for 40 mm | 173.0 | 305.8 | 232.4 | 29.6 |

**Table 3.**Prediction results for fracture toughness; the error in relation to the mean experimental results is in brackets.

Model | Mean Prediction RMSE | Mean Prediction MAE | $\mathbf{Mean}\text{}{\mathit{R}}^{2}$ |
---|---|---|---|

Neural network | 49.1 (21%) | 35.0 (15%) | −1.92 |

Random forest | 22.7 (9.8%) | 17.0 (7.3%) | 0.58 |

XGBoost | 21.8 (9.4%) | 16.0 (6.9%) | 0.61 |

**Table 4.**Prediction results for selected features of fracture toughness; the error in relation to the mean experimental results is in brackets.

Model | Mean Prediction RMSE | Mean Prediction MAE | $\mathbf{Mean}\text{}{\mathit{R}}^{2}$ |
---|---|---|---|

Neural network | 28.4 (12.2%) | 24.0 (10.3%) | −1.15 |

Random forest | 23.4 (10.1%) | 17.5 (7.5%) | 0.52 |

XGBoost | 23.8 (10.2%) | 18.8 (8.1%) | 0.54 |

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**MDPI and ACS Style**

Karamov, R.; Akhatov, I.; Sergeichev, I.V.
Prediction of Fracture Toughness of Pultruded Composites Based on Supervised Machine Learning. *Polymers* **2022**, *14*, 3619.
https://doi.org/10.3390/polym14173619

**AMA Style**

Karamov R, Akhatov I, Sergeichev IV.
Prediction of Fracture Toughness of Pultruded Composites Based on Supervised Machine Learning. *Polymers*. 2022; 14(17):3619.
https://doi.org/10.3390/polym14173619

**Chicago/Turabian Style**

Karamov, Radmir, Iskander Akhatov, and Ivan V. Sergeichev.
2022. "Prediction of Fracture Toughness of Pultruded Composites Based on Supervised Machine Learning" *Polymers* 14, no. 17: 3619.
https://doi.org/10.3390/polym14173619