# Calibration of the PA6 Short-Fiber Reinforced Material Model for 10% to 30% Carbon Mass Fraction Mechanical Characteristic Prediction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Plate Molding Experiment

^{3}/s.

#### 2.2. Investigation of Fibers under Scanning Electronic Microscope

#### 2.3. The Effective Fiber Length

#### 2.4. Calculation of Short-Reinforced Composite Tensile Curves

_{ijkl}is the fourth-order fiber orientation tensor, δ

_{ij}is the second-order unit tensor, and the coefficients B are related to the components of the stiffness matrix of the transversely isotropic unidirectional composite [34]. The fourth-order tensor in Tucker’s averaging procedure was reduced to a second-order tensor by applying the orthotropic closure approximation presented by Cintra and Tucker in [35]. The approximation parameters of the fiber orientation tensors mainly influence the calculation of the stress–strain curves. The fiber direction from 0 to 90° was divided into 20 equal parts, with a tolerance interval in the fiber orientation tensor of 0.01.

_{e}, the elastic shear modulus, is defined using a Lame parameter, based on the given Young’s modulus and the Poisson’s ratio of the matrix in the elastic range.

- Power law [39]:$$R\left({\epsilon}_{p}\right)=k{\epsilon}_{p}{}^{m};$$
- Exponential law:$$R\left({\epsilon}_{p}\right)={R}_{\infty}\left[1-{e}^{-m{\epsilon}_{p}}\right];$$
- Exponential and linear law:$$R\left({\epsilon}_{p}\right)=k{\epsilon}_{p}+{R}_{\infty}\left[1-{e}^{-m{\epsilon}_{p}}\right].$$

## 3. Results

#### 3.1. Melt Front and Microstucture Experimental Investigation

_{11}. In addition, the fiber orientation tensor predicts that the component a

_{11}has the highest probability. The fibers presented a chaotic appearance within the core layer. The skin layer comprised approximately 3% of the thickness, while the core layers constituted 19%. The start and end of every layer were difficult to determine, especially at the transition from the skin layer to the shell layer.

#### 3.2. Injection Molding and the Fiber Orientation Models Validation

#### 3.3. Determination Length and Diameter of the Fibers

#### 3.4. Experimental Stress–strain Curves

#### 3.5. Determination of the Matrix Material Models

#### 3.6. Comparison of the Effect of Considering the Distribution of Fiber Lengths and the Distribution of the Orientation Tensor on the Accuracy of Approximation of the Tensile Curves of a Short-Reinforced Composite

#### 3.7. Determination of the Parameters of the Composite Material Models, Common for Different Fiber Mass

#### 3.8. Failure Criterion Parameter Identification for Short-Fiber Reinforced Thermoplastic Composites

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Chawla, K.K. Composite Materials: Science and Engineering; Springer: New York, NY, USA, 2016. [Google Scholar]
- Rana, S.; Fangueiro, R. Advanced Composites in Aerospace Engineering. In Advanced Composite Materials for Aerospace Engineering; Elsevier: Amsterdam, The Netherlands, 2016; pp. 1–15. [Google Scholar] [CrossRef]
- Fu, S.-Y.; Lauke, B.; Mai, Y.-W. Science and Engineering of Short Fibre Reinforced Polymers Composites. In Woodhead Publishing in Materials; CRC Press: Boca Raton, FL, USA, 2009. [Google Scholar]
- Nakonieczny, D.S.; Kern, F.; Dufner, L.; Antonowicz, M.; Matus, K. Alumina and Zirconia-Reinforced Polyamide PA-12 Composites for Biomedical Additive Manufacturing. Materials
**2021**, 14, 6201. [Google Scholar] [CrossRef] [PubMed] - Díaz, J.; Rubio, L. Developments to Manufacture Structural Aeronautical Parts in Carbon Fibre Reinforced Thermoplastic Materials. J. Mater. Processing Technol.
**2003**, 143–144, 342–346. [Google Scholar] [CrossRef] - Miaris, A.; Edelmann, K.; von Hayek Boelingen, M. A350 WXB: Thousands of Thermoplastic Composite Parts in an FRP Aircraft-SAMPE. Available online: https://www.nasampe.org/store/viewproduct.aspx?id=4417947 (accessed on 25 March 2022).
- Chang, I.Y.; Pratte, J.F. LDF™ Thermoplastic Composites Technology. J. Thermoplast. Compos. Mater.
**1991**, 4, 227–252. [Google Scholar] [CrossRef] - Eguemann, N.; Giger, L.; Roux, M.; Dransfeld, C.; Thiebaud, F.; Perreux, D. Compression Moulding of Complex Parts for the Aerospace with Discontinuous Novel and Recycled Thermoplastic Composite Materials. In Proceedings of the 19th International Conference on Composite Materials, Paris, France, 28 July–2 August 2013; pp. 1–11. [Google Scholar]
- European Bioplastics. Fact Sheet: Mechanical Recycling; European Bioplastics eV: Berlin, Germany, 2010. [Google Scholar]
- Andrzejewski, J.; Aniśko, J.; Szulc, J. A Comparative Study of Biocarbon Reinforced Polyoxymethylene and Polyamide: Materials Performance and Durability. Compos. Part A Appl. Sci. Manuf.
**2022**, 152, 106715. [Google Scholar] [CrossRef] - Steffens, M.; Himmel, N.; Maier, M. Design and Analysis of Discontinuous Long Fiber Reinforced Thermoplastic Structures for Car Seat Applications. Trans. Eng. Sci.
**1998**, 21, 35–44. [Google Scholar] - Kondapalli, P.S.; Grumm, K.; Cao, Y.; Laurent, V. Application and CAE simulation of over molded short and continuous fiber thermoplastic composite: Part II. 12th Int LS-DYNA
^{®}Users Conf. FEA Inf. Eng. J.**2013**, 2, 1–10. [Google Scholar] - Fu, S. Effects of Fiber Length and Fiber Orientation Distributions on the Tensile Strength of Short-Fiber-Reinforced Polymers. Compos. Sci. Technol.
**1996**, 56, 1179–1190. [Google Scholar] [CrossRef] - Fu, S.-Y.; Hu, X.; Yue, C.-Y. Effects of Fiber Length and Orientation Distributions on the Mechanical Properties of Short-Fiber-Reinforced Polymers. J. Soc. Mater. Sci. Jpn.
**1999**, 48, 74–83. [Google Scholar] [CrossRef][Green Version] - Capela, C.; Oliveira, S.E.; Pestana, J.; Ferreira, J.A.M. Effect of Fiber Length on the Mechanical Properties of High Dosage Carbon Reinforced. Procedia Struct. Integr.
**2017**, 5, 539–546. [Google Scholar] [CrossRef] - Capela, C.; Oliveira, S.E.; Ferreira, J.A.M. Mechanical Behavior of High Dosage Short Carbon Fiber Reinforced Epoxy Composites. Fibers Polym.
**2017**, 18, 1200–1207. [Google Scholar] [CrossRef] - Karsli, N.G.; Aytac, A.; Deniz, V. Effects of Initial Fiber Length and Fiber Length Distribution on the Properties of Carbon-Fiber-Reinforced-Polypropylene Composites. J. Reinf. Plast. Compos.
**2012**, 31, 1053–1060. [Google Scholar] [CrossRef] - Kufel, A.; Para, S.; Kuciel, S. Basalt/Glass Fiber Polypropylene Hybrid Composites: Mechanical Properties at Different Temperatures and under Cyclic Loading and Micromechanical Modelling. Materials
**2021**, 14, 5574. [Google Scholar] [CrossRef] [PubMed] - Moritzer, E.; Heiderich, G.; Hirsch, A. Fiber Length Reduction during Injection Molding; AIP Publishing: Dresden, Germany, 2019; p. 070001. [Google Scholar] [CrossRef]
- Kurkin, E.I.; Lukyanov, O.E.; Chertykovtseva, V.O.; Espinosa Barcenas, O.U. Molding Gate Optimization for Weld Line Location Away from Structures Loaded Area. J. Phys. Conf. Ser.
**2021**, 1925, 012056. [Google Scholar] [CrossRef] - Eshelby, J. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proc. R. Soc. Lond. A
**1957**, 241, 376–396. [Google Scholar] [CrossRef] - Mori, T.; Tanaka, K. Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions. Acta Metall.
**1973**, 21, 571–574. [Google Scholar] [CrossRef] - Su, N.; Pierce, R.S.; Rudd, C.; Liu, X. Comprehensive Investigation of Reclaimed Carbon Fibre Reinforced Polyamide (RCF/PA) Filaments and FDM Printed Composites. Compos. Part B Eng.
**2022**, 233, 109646. [Google Scholar] [CrossRef] - Mentges, N.; Dashtbozorg, B.; Mirkhalaf, S.M. A Micromechanics-Based Artificial Neural Networks Model for Elastic Properties of Short Fiber Composites. Compos. Part B Eng.
**2021**, 213, 108736. [Google Scholar] [CrossRef] - Pietrogrande, R.; Carraro, P.A.; De Monte, M.; Quaresimin, M. A Novel Pseudo-Grain Approach for the Estimation of the Elastic Stress Distributions within the Matrix of Short Fiber-Reinforced Polymers. Compos. Part B Eng.
**2018**, 150, 115–123. [Google Scholar] [CrossRef] - Templeton, P.A. Strength Predictions of Injection Molding Compounds. J. Reinf. Plast. Compos.
**1990**, 9, 210–225. [Google Scholar] [CrossRef] - Chin, W.-K.; Liu, H.-T.; Lee, Y.-D. Effects of Fiber Length and Orientation Distribution on the Elastic Modulus of Short Fiber Reinforced Thermoplastics. Polym. Compos.
**1988**, 9, 27–35. [Google Scholar] [CrossRef] - Friedrich, K. Microstructural efficiency and fracture toughness of short fiber/thermoplastic matrix composites. Comp. Sci. Tech.
**1985**, 22, 43–74. [Google Scholar] [CrossRef] - Verdejo de Toro, E.; Coello Sobrino, J.; Martínez Martínez, A.; Miguel Eguía, V.; Ayllón Pérez, J. Investigation of a Short Carbon Fibre-Reinforced Polyamide and Comparison of Two Manufacturing Processes: Fused Deposition Modelling (FDM) and Polymer Injection Moulding (PIM). Materials
**2020**, 13, 672. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rosen, B.W.; Dow, N.F.; Hashin, Z. Mechanical Properties of Fibrous Composites; General Electric Co.: Philadelphia, PA, USA, 1964. [Google Scholar]
- Jain, A.; Abdin, Y.; Van Paepegem, W.; Verpoest, I.; Lomov, S.V. Effective Anisotropic Stiffness of Inclusions with Debonded Interface for Eshelby-Based Models. Compos. Struct.
**2015**, 131, 692–706. [Google Scholar] [CrossRef] - Tandon, G.P.; Weng, G.J. The Effect of Aspect Ratio of Inclusions on the Elastic Properties of Unidirectionally Aligned Composites. Polym. Compos.
**1984**, 5, 327–333. [Google Scholar] [CrossRef] - Li, H.; Zare, Y.; Rhee, K.Y. Mathematical Simplification of the Tandon–Weng Approach to the Mori–Tanaka Model for Estimating the Young’s Modulus of Clay/Polymer Nanocomposites. JOM
**2017**, 69, 2819–2824. [Google Scholar] [CrossRef] - Gusev, A.A. Finite Element Estimates of Viscoelastic Stiffness of Short Glass Fiber Reinforced Composites. Compos. Struct.
**2017**, 171, 53–62. [Google Scholar] [CrossRef] - Cintra, J.S.; Tucker, C.L. Orthotropic Closure Approximations for Flow-induced Fiber Orientation. J. Rheol.
**1995**, 39, 1095–1122. [Google Scholar] [CrossRef] - Borja, R.I.; Borja, R.I. Plasticity: Modeling & Computation; SpringerLink; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Castañeda, P.P. Exact Second-Order Estimates for the Effective Mechanical Properties of Nonlinear Composite Materials. J. Mech. Phys. Solids
**1996**, 44, 827–862. [Google Scholar] [CrossRef] - Digimat Documentation. MF and MX User’s Guide. Digimat 2022.1—Online Help (HTML). Available online: https://help.mscsoftware.com/bundle/digimat_2022.1/page/digimat_main.htm (accessed on 25 March 2022).
- Hollomon, J. Tensile Deformation. AIME Trans
**1945**, 32, 268–290. [Google Scholar] - Chen, J.; Zhao, L.; Zhou, K. Improvement in the Mechanical Performance of Multi Jet Fusion–Printed Aramid Fiber/Polyamide 12 Composites by Fiber Surface Modification. Addit. Manuf.
**2022**, 51, 102576. [Google Scholar] [CrossRef] - Xu, Z.; Yang, T.; Nakamura, M.; Yang, Y.; Hamada, H. Effect of Carbon Powder Surface Treatment on Carbon Fiber Reinforced PA Composites. Energy Procedia
**2016**, 89, 15–23. [Google Scholar] [CrossRef][Green Version] - Foss, P.H.; Tseng, H.-C.; Snawerdt, J.; Chang, Y.-J.; Yang, W.-H.; Hsu, C.-H. Prediction of Fiber Orientation Distribution in Injection Molded Parts Using Moldex3D Simulation. Polym. Compos.
**2014**, 35, 671–680. [Google Scholar] [CrossRef] - Rosato, D.V.; Rosato, D.V. Plastics Engineered Product Design; Elsevier Advanced Technology: Inglaterra/Estados Unidos, UK, 2003. [Google Scholar]

**Figure 3.**Short fiber extraction from composite: (

**a**) Sample preparation; (

**b**) burning samples in the oven; (

**c**) fibers after matrix burning.

**Figure 4.**Experimentally obtained plate melt front at molding composites with different fiber mass fraction: (

**a**) 0%; (

**b**) 10%; (

**c**) 15%; (

**d**) 20%; (

**e**) 30%.

**Figure 5.**Morphology of the fracture of short-fiber reinforced PA6 with 30% carbon-fiber mass fraction.

**Figure 7.**Fracture surface of short-fiber reinforced PA6 with 15% carbon-fiber mass fraction, ISO 527-2 sample, orientated at 90°. Dashed lines demark the size in micrometers of the skin and core layer.

**Figure 8.**Comparison of simulated and experimental filling: (

**a**) Melt front, blue lines—calculated isolines at each 2% of filling time, black and red lines—experimentally obtained melt fronts; (

**b**) Orientation tensor components, solid lines—calculated values, dash lines—mean experimentally obtained zones sizes.

**Figure 9.**The microstructure of short-fiber reinforced PA6 with 30% carbon-fiber mass fraction along the following orientations: (

**a**) 0°; (

**b**) 90°.

**Figure 11.**Measured fiber aspect ratio distribution for 15% (u15) and 30% (u30) fiber mass fractions.

**Figure 12.**Tensile test of ISO 527-2 PA6 matrix samples: (

**a**) Tension curves; (

**b**) Yield stress; (

**c**) Poisson’s ratio.

**Figure 13.**Experimental stress–strain curves for ISO 527-2 samples of PA6 composites with different fiber mass fractions, cut at 0°, 45° and 90° angles to flow direction: (

**a**) 10%; (

**b**) 15%; (

**c**) 20%; (

**d**) 30%.

**Figure 14.**The approximation of PA6 matrix ISO 527-2 tension curves, dashed lines—experiment, solid lines—model, with hardening laws: (

**a**) power; (

**b**) exponential; (

**c**) exponential and linear. The solid lines show the optimized curves. Dashed lines are the experimental data.

**Figure 15.**Stress–strain curves of short-fiber reinforced PA6 with 30% carbon fiber were modeled using the exponential law. Comparison between: (

**a**) Single AR value and AR distribution; (

**b**) Single layer and multilayer analysis.

**Figure 16.**Stress–strain curves for the model material for PA6 with different short carbon fiber mass fractions, orientated in different directions. From left to right: 10%, 15%, 20%, and 30% fiber mass fraction. From top to bottom: non-calibrated parameters; calibrated fiber aspect ratio; all matrix parameters and fiber aspect ratio calibrated. Material model—color lines: 0°—red lines, 45°—green lines, 90°—blue lines. Experimental data—black dashed lines.

**Figure 17.**Matrix models (solid lines) after the composite stiffness calibration with different hardening laws: (

**a**) power; (

**b**) exponential; (

**c**) exponential and linear. Dashed lines—experiment.

**Figure 18.**Stress–strain curves prepared from the mean obtained strain limits, and using Tsai–Hill 3D transversely isotropic strain-based values for 10%, 15%, 20%, and 30% fiber mass fractions. Material model—color lines: 0°—red lines, 45°—green lines, 90°—blue lines. Experimental data—black dashed lines.

**Figure 19.**Common mass fraction composite material models for all fibers. Exponent and linear hardening law case. For angles between the flow and load directions: (

**a**) 0°; (

**b**) 45°; (

**c**) 90°.

Layer | Thickness, mm | a_{11} | a_{22} | a_{33} | a_{12} | a_{13} | a_{23} |
---|---|---|---|---|---|---|---|

1 | 0.1667 | 0.603 | 0.304 | 0.0929 | 0.00104 | 0.00019 | 0.00011 |

2 | 1.5003 | 0.821 | 0.126 | 0.0533 | 0.00087 | 0.00001 | 0.00003 |

3 | 0.1667 | 0.711 | 0.244 | 0.0451 | −0.00131 | −0.00033 | 0.00008 |

4 | 0.3334 | 0.461 | 0.486 | 0.0526 | −0.01870 | −0.00113 | −0.00495 |

5 | 0.1667 | 0.711 | 0.244 | 0.0451 | −0.00131 | −0.00033 | 0.00008 |

6 | 1.5003 | 0.821 | 0.126 | 0.0533 | 0.00087 | 0.00001 | 0.00003 |

7 | 0.1667 | 0.603 | 0.304 | 0.0929 | 0.00104 | 0.00019 | 0.00011 |

Hardening Law | Power | Exponential | Exponential and Linear |
---|---|---|---|

Young’s modulus, MPa | 3341 | 3547 | 3552 |

Yield stress, MPa | 12.57 | 8.27 | 6.68 |

Hardening modulus, ${R}_{\infty}$ MPa | - | 60.59 | 55.41 |

Hardening Exponent m | 0.21891 | 433.69 | 528.67 |

Linear hardening modulus k, MPa | 158.12 | - | 623.17 |

Relative error, % | 5.32 | 4.81 | 4.77 |

Hardening Law | Fiber Mass Fraction, % | Mean AR Model | Param. CV, % | |||
---|---|---|---|---|---|---|

10 | 15 | 20 | 30 | |||

Aspect ratio | ||||||

Power | 14.00 | 16.53 | 14.47 | 14.82 | 15.00 | 7.35 |

Exponential | 14.53 | 16.94 | 15.14 | 15.47 | 15.52 | 6.60 |

Exponential and linear | 14.30 | 16.75 | 14.71 | 15.27 | 15.30 | 7.03 |

Parameter | Fiber Mass Fraction, % | Mean Parameters Model | Parameters CV, % | |||
---|---|---|---|---|---|---|

10 | 15 | 20 | 30 | |||

Power Law | ||||||

Young’s modulus, MPa | 4383 | 4437 | 4162 | 3759 | 4185 | 7.4 |

Yield stress, MPa | 12.3 | 12.6 | 10.2 | 9.9 | 11.2 | 12.3 |

Hardening modulus, MPa | 147.6 | 161.9 | 124.6 | 137.0 | 142.8 | 11.1 |

Hardening exponent | 0.1995 | 0.2238 | 0.2023 | 0.2475 | 0.2183 | 10.2 |

Fibers’ AR | 8.84 | 12.23 | 14.19 | 16.79 | 13.01 | 25.8 |

Exponential Law | ||||||

Young’s modulus, MPa | 4654 | 4615 | 4741 | 3994 | 4501 | 7.6 |

Yield stress, MPa | 13.8 | 21.2 | 12.4 | 16.1 | 15.9 | 24.3 |

Hardening modulus, MPa | 59.2 | 51.0 | 51.5 | 38.7 | 50.1 | 17.0 |

Hardening exponent | 404.8 | 370.6 | 353.6 | 377.8 | 376.7 | 5.7 |

Fibers’ AR | 9.02 | 12.25 | 13.37 | 16.62 | 12.8 | 24.4 |

Exponential and linear law | ||||||

Young’s modulus, MPa | 4672 | 4842 | 4625 | 3994 | 4533 | 8.2 |

Yield stress, MPa | 13.6 | 13.0 | 14.5 | 14.5 | 13.9 | 5.4 |

Hardening modulus, MPa | 56.6 | 54.9 | 46.1 | 37.0 | 48.6 | 18.6 |

Hardening exponent | 447.1 | 417.7 | 381.1 | 458.3 | 426.0 | 8.1 |

Hardening linear modulus, MPa | 208.8 | 216.0 | 144.1 | 188.4 | 189.3 | 17.1 |

Fibers’ AR | 8.82 | 12.36 | 13.80 | 16.54 | 12.9 | 24.9 |

**Table 5.**Mean relative error of stress–strain curve prediction of different fiber mass fraction composites with models based on fibers and matrix characteristics obtained experimentally, mean fiber aspect ratio calibrated models, and the mean parameter model.

Hardening Law | Fiber Mass Fraction, % | Mean Level of Error, % | |||
---|---|---|---|---|---|

10 | 15 | 20 | 30 | ||

Models, based on fibers and matrix characteristics obtained experimentally | |||||

Power | 29.0 | 25.7 | 35.8 | 35.9 | 31.6 |

Exponential | 27.8 | 25.2 | 33.7 | 34.7 | 30.4 |

Exponential and linear | 28.6 | 25.9 | 36.1 | 35.7 | 31.6 |

Mean fiber aspect ratio calibrated models | |||||

Power | 11.7 | 12.1 | 12.0 | 11.1 | 11.7 |

Exponential | 11.5 | 12.3 | 11.3 | 11.7 | 11.7 |

Exponential and linear | 11.7 | 12.5 | 12.6 | 12.2 | 12.3 |

Mean with the calibration of all matrix and fiber aspect ratio parameters | |||||

Power | 7.4 | 7.6 | 4.9 | 7.7 | 6.9 |

Exponential | 6.9 | 6.8 | 4.1 | 8.0 | 6.5 |

Exponential and linear | 7.0 | 6.9 | 3.9 | 7.9 | 6.4 |

Parameters | Fiber Mass Fraction, % | Mean Strain Limits | Parameters CV, % | |||
---|---|---|---|---|---|---|

10 | 15 | 20 | 30 | |||

Power law | ||||||

Axial tensile strain limit, 10^{−2} | 2.009 | 2.957 | 1.944 | 1.880 | 2.197 | 23.16 |

Inplane tensile strain limit, 10^{−2} | 3.516 | 3.855 | 3.573 | 2.151 | 3.274 | 23.30 |

Transverse shear strain limit, 10^{−2} | 5.606 | 5.140 | 5.235 | 4.364 | 5.086 | 10.26 |

Exponential law | ||||||

Axial tensile strain limit, 10^{−2} | 1.566 | 2.353 | 1.832 | 2.031 | 1.945 | 17.05 |

Inplane tensile strain limit, 10^{−2} | 2.999 | 3.524 | 3.430 | 2.278 | 3.058 | 18.57 |

Transverse shear strain limit, 10^{−2} | 5.873 | 5.249 | 5.355 | 4.076 | 5.138 | 14.77 |

Exponential and linear law | ||||||

Axial tensile strain limit, 10^{−2} | 2.141 | 2.402 | 1.908 | 1.944 | 2.099 | 10.81 |

Inplane tensile strain limit, 10^{−2} | 3.684 | 3.573 | 3.559 | 2.012 | 3.207 | 24.90 |

Transverse shear strain limit, 10^{−2} | 5.607 | 5.140 | 5.140 | 4.798 | 5.171 | 6.42 |

**Table 7.**Mean relative error (%) of ultimate strain prediction for different fiber mass fraction composites with models based on fiber and matrix characteristics obtained experimentally, mean fiber aspect ratio calibrated models, and the mean parameter model.

Hardening Law | Fiber Mass Fraction, % | Mean Level of Error, % | |||
---|---|---|---|---|---|

10 | 15 | 20 | 30 | ||

Power | 16.24 | 7.79 | 18.92 | 56.63 | 24.90 |

Exponential | 17.30 | 8.87 | 19.73 | 55.04 | 25.24 |

Exponential and linear | 16.26 | 7.79 | 19.02 | 56.36 | 24.86 |

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**MDPI and ACS Style**

Kurkin, E.; Spirina, M.; Espinosa Barcenas, O.U.; Kurkina, E.
Calibration of the PA6 Short-Fiber Reinforced Material Model for 10% to 30% Carbon Mass Fraction Mechanical Characteristic Prediction. *Polymers* **2022**, *14*, 1781.
https://doi.org/10.3390/polym14091781

**AMA Style**

Kurkin E, Spirina M, Espinosa Barcenas OU, Kurkina E.
Calibration of the PA6 Short-Fiber Reinforced Material Model for 10% to 30% Carbon Mass Fraction Mechanical Characteristic Prediction. *Polymers*. 2022; 14(9):1781.
https://doi.org/10.3390/polym14091781

**Chicago/Turabian Style**

Kurkin, Evgenii, Mariia Spirina, Oscar Ulises Espinosa Barcenas, and Ekaterina Kurkina.
2022. "Calibration of the PA6 Short-Fiber Reinforced Material Model for 10% to 30% Carbon Mass Fraction Mechanical Characteristic Prediction" *Polymers* 14, no. 9: 1781.
https://doi.org/10.3390/polym14091781