# Calculation of Homogenized Mechanical Coefficients of Fiber-Reinforced Composite Using Finite Element Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{A}, the transverse Young’s modulus E

_{T}, the transverse and longitudinal Poisson’s ratio (ν

_{A}and ν

_{T}) and the shear modulus G

_{A}).

## 3. Results

_{m}= 4.14 GPa; ν

_{m}= 0.22; E

_{f}= 86.90 GPa; ν

_{f}= 0.34.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Katouzian, M. On the Effect of Tempeature on Creep Behavior of Neat and Carbon Fiber Reinforced PEEK and Epoxy—A Micromechanical Approach. Ph.D. Thesis, University of Munich, Munich, Germany, 1995. [Google Scholar]
- Findley, W.N.; Khosla, G. Application of the Superposition Principle and Theories of Mechanical Equation of State, Strain, and Time Hardening to Creep of Plastics under Changing Loads. J. Appl. Phys.
**1955**, 26, 821. [Google Scholar] [CrossRef] - Hashin, Z. On Elastic Behavior of Fibre Reinforced Materials of Arbitrary Transverse Phase Geometry. J. Mech. Phys. Solids
**1965**, 13, 119–134. [Google Scholar] [CrossRef] - Hashin, Z.; Shtrikman, S. A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials. J. Mech. Phys. Solids
**1963**, 11, 127–140. [Google Scholar] [CrossRef] - Hashin, Z.; Rosen, B.W. The Elastic Moduli of Fiber-Reinforced Materials. J. Appl. Mech.
**1964**, 31, 223–232. [Google Scholar] [CrossRef] - Bowles, D.E.; Griffin, O.H., Jr. Micromechanics Analysis of Space Simulated Thermal Stresses in Composites. Part I: Theory and Unidirectional Laminates. J. Reinf. Plast. Compos.
**1991**, 10, 504–521. [Google Scholar] [CrossRef] - Zhao, Y.H.; Weng, G.J. Effective Elastic Moduli of Ribbon-Reinforced Composites. J. Appl. Mech.
**1990**, 57, 158–167. [Google Scholar] [CrossRef] - Hill, R. Theory of Mechanical Properties of Fiber-strengthened Materials: I Elastic Behavior. J. Mech. Phys. Solids
**1964**, 12, 199–212. [Google Scholar] [CrossRef] - Hill, R. Theory of Mechanical Properties of Fiber-strengthened Materials: II Inelastic Behavior. J. Mech. Phys. Solids
**1964**, 12, 213–218. [Google Scholar] [CrossRef] - Hill, R. Theory of Mechanical Properties of Fiber-strengthened Materials: III Self-Consistent Model. J. Mech. Phys. Solids
**1965**, 13, 189–198. [Google Scholar] [CrossRef] - Hill, R. Continuum Micro-Mechanics of Elastoplastic Polycrystals. J. Mech. Phys. Solids
**1965**, 13, 89–101. [Google Scholar] [CrossRef] - Aboudi, J. Micromechanical characterization of the non-linear viscoelastic behavior of resin matrix composites. Compos. Sci. Technol.
**1990**, 38, 371–386. [Google Scholar] [CrossRef] - Aboudi, J. Mechanics of Composite Materials—A Unified Micromechanical Approach; Elsevier: Amsterdam, The Netherlands, 1991. [Google Scholar]
- Othman, M.I.A.; Fekry, M.; Marin, M. Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating. Struct. Eng. Mech.
**2020**, 73, 621–629. [Google Scholar] - Marin, M.; Hobiny, A.; Abbas, I. The Effects of Fractional Time Derivatives in Porothermoelastic Materials Using Finite Element Method. Mathematics
**2021**, 9, 1606. [Google Scholar] [CrossRef] - Vlase, S.; Teodorescu-Draghicescu, H.; Motoc, D.L.; Scutaru, M.L.; Serbina, L.; Calin, M.R. Behavior of Multiphase Fiber-Reinforced Polymers Under Short Time Cyclic Loading. Optoelectron. Adv. Mater. Rapid Commun.
**2011**, 5, 419–423. [Google Scholar] - Abbas, I.; Hobiny, A.; Marin, M. Photo-thermal interactions in a semi-conductor material with cylindrical cavities and variable thermal conductivity. J. Taibah Univ. Sci.
**2020**, 14, 1369–1376. [Google Scholar] [CrossRef] - Bratu, P.; Dobrescu, C.; Nitu, M.C. Dynamic Response Control of Linear Viscoelastic Materials as Resonant Composite Rheological Models. Rom. J. Acoust. Vib.
**2023**, 20, 73–77. [Google Scholar] - Niculiţă, C.; Vlase, S.; Bencze, A.; Mihălcică, M.; Calin, M.R.; Serbina, L. Optimum stacking in a multiply laminate used for the skin of adaptive wings. Optoelectron. Adv. Mater. Rapid Commun.
**2011**, 5, 1233–1236. [Google Scholar] - Katouzian, M.; Vlase, S.; Calin, M.R. Experimental procedures to determine the viscoelastic parameters of laminated composites. J. Optoelectron. Adv. Mater.
**2011**, 13, 1185–1188. [Google Scholar] - Abo-Dahab, S.M.; Abouelregal, A.E.; Marin, M. Generalized Thermoelastic Functionally Graded on a Thin Slim Strip Non-Gaussian Laser Beam. Symmetry
**2020**, 12, 1094. [Google Scholar] [CrossRef] - Fliegener, S.; Hohe, J. An anisotropic creep model for continuously and discontinuously fiber reinforced thermoplastics. Compos. Sci. Technol.
**2020**, 194, 108168. [Google Scholar] [CrossRef] - Xu, B.; Xu, W.; Guo, F. Creep behavior due to interface diffusion in unidirectional fiber-reinforced metal matrix composites under general loading conditions: A micromechanics analysis. Acta Mech.
**2020**, 231, 1321–1335. [Google Scholar] [CrossRef] - Lal, H.M.M.; Xian, G.-J.; Thomas, S.; Zhang, L.; Zhang, Z.; Wang, H. Experimental Study on the Flexural Creep Behaviors of Pultruded Unidirectional Carbon/Glass Fiber-Reinforced Hybrid Bars. Materials
**2020**, 13, 976. [Google Scholar] [CrossRef] - Wang, Z.; Smith, D.E. Numerical analysis on viscoelastic creep responses of aligned short fiber reinforced composites. Compos. Struct.
**2019**, 229, 111394. [Google Scholar] [CrossRef] - Fattahi, A.M.; Mondali, M. Theoretical study of stress transfer in platelet reinforced composites. J. Theor. Appl. Mech.
**2014**, 52, 3–14. [Google Scholar] - Fattahi, A.M.; Moaddab, E.; Bibishahrbanoei, N. Thermo-mechanical stress analysis in platelet reinforced composites with bonded and debonded platelet end. J. Mech. Sci. Technol.
**2015**, 29, 2067–2072. [Google Scholar] [CrossRef] - Tebeta, R.T.; Fattahi, A.M.; Ahmed, N.A. Experimental and numerical study on HDPE/SWCNT nanocomposite elastic properties considering the processing techniques effect. Microsyst. Technol.
**2021**, 26, 2423–2441. [Google Scholar] [CrossRef] - Selmi, A.; Friebel, C.; Doghri, I.; Hassis, H. Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: A comparative study of several micromechanical models. Compos. Sci. Technol.
**2007**, 67, 2071–2084. [Google Scholar] [CrossRef] - Stanciu, A.; Teodorescu-Drǎghicescu, H.; Vlase, S.; Scutaru, M.L.; Cǎlin, M.R. Mechanical behavior of CSM450 and RT800 laminates subjected to four-point bend tests. Optoelectron. Adv. Mater. Rapid Commun.
**2012**, 6, 495–497. [Google Scholar] - Tran, A.B.; Yvonnet, J.; He, Q.C.; Toulemonde, C.; Sanahuja, J. A simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials. Comput. Methods Appl. Mech. Eng.
**2011**, 200, 2956–2970. [Google Scholar] [CrossRef] - Katouzian, M.; Vlase, S.; Scutaru, M.L. Finite Element Method-Based Simulation Creep Behavior of Viscoelastic Carbon-Fiber Composite. Polymers
**2021**, 13, 1017. [Google Scholar] [CrossRef] - Fung, Y.C. Fundamentals of Solid Mechanics; Prentice-Hall: Kent, OH, USA, 1965. [Google Scholar]
- Schapery, R.A. Stress Analysis of Viscoelastic Composite Materials. J. Compos. Mater.
**1967**, 1, 228. [Google Scholar] [CrossRef] - Morris, D.H.; Yeow, Y.T.; Brinson, H.F. The Viscoelastic Behavior of the Principal Compliance Matrix of Unidirectional Graphite Epoxy Composite; VPI&SU: Blacksburg, VA, USA, 1979. [Google Scholar]
- Huang, Z.-M. Constitutive relation, deformation, failure and strength of composites reinforced with continuous/short fibers or particles. Compos. Struct.
**2021**, 262, 113279. [Google Scholar] [CrossRef] - Huang, Z.-M. A micromechanics approach to stiffness and strength of unidirectional composites. J. Reinf. Plastics Comp.
**2019**, 38, 167–196. [Google Scholar] [CrossRef] - Hinton, M.J.; Kaddour, A.S.; Soden, P.D. The world-wide failure exercise: Its origin, concept and content. In The World-Wide Failure Exercise; Elsevier: Amsterdam, The Netherlands, 2004; pp. 2–28. [Google Scholar]
- Ahmadi, M.; Ansari, R.; Hassanzadeh-Aghdam, M.K. Micromechanical finite element analysis of Young’s modulus, yield strength and thermal expansion coefficient of nano-sized ceramic particle/metal matrix nanocomposites. J. Braz. Soc. Mech. Sci. Eng.
**2023**, 45, 478. [Google Scholar] [CrossRef] - Oz, F.E. Computational examination of the effect of voids on the mechanical response of composites with emphasize on the cure hardening behavior. Mech. Adv. Mater. Struct.
**2022**, 29, 1–14. [Google Scholar] [CrossRef] - Christofi, I.; Hadjiloizi, D.A.; Georgiades, A.V. Dynamic micromechanical model for smart composite and reinforced shells. Zamm-Z. Fur Angew. Math. Und Mech.
**2022**, 102, e202100211. [Google Scholar] [CrossRef] - Gupta, M.; Ray, M.C.; Kundalwal, S.I. Dynamic modelling and analysis of smart carbon nanotube-based hybrid composite beams: Analytical and finite element study. Proc. Inst. Mech. Engineers. Part L-J. Mater. Des. Appl.
**2021**, 235, 2185–2206. [Google Scholar] [CrossRef] - Bratu, P.; Vlase, S.; Dragan, N.; Vasile, O.; Itu, C.; Nitu, C.M.; Toderita, A. Modal Analysis of the Inertial Platform of the Laser ELI-NP Facility in Magurele-Bucharest. Rom. J. Acoust. Vib.
**2022**, 19, 112–120. [Google Scholar] - Mishra, V.N.; Sarangi, S.K. A Numerical Model for the Effective Damping Properties of Unidirectional Fiber-Reinforced Composites. Mech. Compos. Mater.
**2023**, 59, 1031–1044. [Google Scholar] [CrossRef] - Qin, F.P.; Lu, F.C.; Huang, L. Numerical simulation and experimental validation of ratchetting deformation of short fiber-reinforced polymer composites. Compos. Part B Eng.
**2023**, 266, 110974. [Google Scholar] [CrossRef] - Vázquez, J.M.C.; Wu, L.; Noels, L. A micromechanical mean-field homogenization surrogate for the stochastic multiscale analysis of composite materials failure. Int. J. Numer. Methods Eng.
**2023**, 124, 5200–5262. [Google Scholar] [CrossRef] - Manchiraju, V.N.M.; Bhagat, A.R.; Dwivedi, V.K. Estimation of Elastic Constants Using Numerical Methods and Their Validation Through Experimental Results for Unidirectional Carbon/Carbon Composite Materials. Jpn. J. Metrol. Soc. India
**2023**, 38, 923–937. [Google Scholar] [CrossRef] - de Morais, A.B. Transverse moduli of continuous-fibre-reinforced polymers. Compos. Sci. Technol.
**2000**, 60, 997–1002. [Google Scholar] [CrossRef] - Fuchs, M.B.; Paley, M.; Miroshny, E. Evaluation of failure criteria for fiber composites using finite element micromechanics. J. Compos. Mater.
**1998**, 32, 766–782. [Google Scholar] - Astaraki, S.; Zamani, E.; Mohamadipoor, R. Determination of mechanical properties of nanocomposites reinforced with spherical silica nanoparticles using experiments, micromechanical model and finite elements method. J. Compos. Mater.
**2023**, 57, 2689–2702. [Google Scholar] [CrossRef] - Scutaru, M.L.; Vlase, S.; Marin, M.; Modrea, A. New analytical method based on dynamic response of planar mechanical elastic systems. Bound. Value Probl.
**2020**, 2020, 104. [Google Scholar] [CrossRef] - Li, Y.; Li, Y.Q. Evaluation of elastic properties of fiber reinforced concrete with homogenization theory and finite element simulation. Constr. Build. Mater.
**2019**, 200, 301–309. [Google Scholar] [CrossRef] - Nguyen, A.V.; Nguyen, T.C. Homogenization of Viscoelastic Composite Reinforced Woven Flax Fibers. In Proceedings of the 11th Joint Canada-Japan Workshop on Composites/1st Joint Canada-Japan-Vietnam Workshop on Composites, Ho Chi Minh, Vietnam, 8–10 August 2016; pp. 187–192. [Google Scholar]
- Matsuda, T.; Ohno, N. Predicting the elastic-viscoplastic and creep behaviour of polymer matrix composites using the homogenization theory. In Creep and Fatigue in Polymer Matrix Composites; Woodhead: Cambridge, UK, 2011; pp. 113–148. [Google Scholar]
- Tian, W.L.; Qi, L.H.; Jing, Z. Numerical simulation on elastic properties of short-fiber-reinforced metal matrix composites: Effect of fiber orientation. Compos. Struct.
**2016**, 152, 408–417. [Google Scholar] [CrossRef] - Zhu, T.L.; Wang, Z. Research and application prospect of short carbon fiber reinforced ceramic composites. J. Eur. Ceram. Soc.
**2023**, 43, 6699–6717. [Google Scholar] [CrossRef] - Griffith, W.I. The Accelerated Characterization of Viscoelatic Composite Materials; VPI&SU: Blacksburg, VA, USA, 1979. [Google Scholar]
- Teodorescu-Draghicescu, H.; Stanciu, A.; Vlase, S.; Scutaru, L.; Calin, M.R.; Serbina, L. Finite Element Method Analysis of Some Fibre-Reinforced Composite Laminates. Optoelectron. Adv. Mater. Rapid Commun.
**2011**, 5, 782–785. [Google Scholar] - Sá, M.F.; Gomes, A.; Correia, J.; Silvestre, N. Creep behavior of pultruded GFRP elements—Part 1: Literature review and experimental study. Compos. Struct.
**2011**, 93, 2450–2459. [Google Scholar] [CrossRef] - Wu, J.; Zhu, Y.; Li, C. Experimental Investigation of Fatigue Capacity of Bending-Anchored CFRP Cables. Polymers
**2023**, 15, 2483. [Google Scholar] [CrossRef] [PubMed] - Alhoubi, Y.; Mahaini, Z.; Abed, F. The Flexural Performance of BFRP-Reinforced UHPC Beams Compared to Steel and GFRP-Reinforced Beams. Sustainability
**2022**, 14, 15139. [Google Scholar] [CrossRef] - Xian, G.; Guo, R.; Li, C. Combined effects of sustained bending loading, water immersion and fiber hybrid mode on the mechanical properties of carbon/glass fiber reinforced polymer composite. Compos. Struct.
**2022**, 281, 115060. [Google Scholar] [CrossRef] - Kamiński, M.; Ostrowski, P. Homogenization of heat transfer in fibrous composite with stochastic interface defects. Compos. Struct.
**2021**, 261, 113555. [Google Scholar] [CrossRef] - Ostoja-Starzewski, M.; Sheng, P.Y.; Jasiuk, I. Damage patterns and constitutive response of random matrix-inclusion composites. Eng. Fract. Mech.
**1997**, 58, 581–606. [Google Scholar] [CrossRef]

**Figure 4.**The elementary cell made up of 16 fibers and a qualitative representation of displacements (in colors).

**Figure 9.**Sections of the first and second rows. Colored are a qualitative representation of the displacements.

**Figure 10.**Sections of the third and fourth rows. Colored are a qualitative representation of the displacements.

Loading Case | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 3.60 × 10^{−4} | −1.22 × 10^{−4} | −1.22 × 10^{−4} | −1.60 × 10^{−13} | −1.01 × 10^{−17} | −3.86 × 10^{−17} |

2 | 3.60 × 10^{−4} | −1.22 × 10^{−4} | −1.22 × 10^{−4} | −8.54 × 10^{−8} | −3.27 × 10^{−9} | −5.43 × 10^{−8} |

3 | 3.11 × 10^{−4} | −1.05 × 10^{−4} | −1.05 × 10^{−4} | −2.45 × 10^{−4} | −7.56 × 10^{−9} | 2.41 × 10^{−10} |

4 | 1.79 × 10^{−4} | −6.03 × 10^{−5} | −6.02 × 10^{−5} | −4.25 × 10^{−4} | −1.46 × 10^{−8} | 2.33 × 10^{−9} |

5 | −1.46 × 10^{−6} | 4.46 × 10^{−7} | 4.56 × 10^{−7} | −4.91 × 10^{−4} | −1.67 × 10^{−8} | 4.06 × 10^{−11} |

Loading Case | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 3.60 × 10^{−4} | −8.83 × 10^{−5} | −8.83 × 10^{−5} | −4.25 × 10^{−17} | 4.38 × 10^{−11} | −1.11 × 10^{−17} |

2 | 3.60 × 10^{−4} | −8.70 × 10^{−5} | −9.15 × 10^{−5} | −1.08 × 10^{−7} | −2.74 × 10^{−6} | −1.42 × 10^{−7} |

3 | 3.11 × 10^{−4} | −7.95 × 10^{−5} | −7.23 × 10^{−5} | −2.48 × 10^{−4} | 4.86 × 10^{−9} | −4.83 × 10^{−10} |

4 | 1.79 × 10^{−4} | −4.56 × 10^{−5} | −4.16 × 10^{−5} | −4.29 × 10^{−4} | 8.71 × 10^{−9} | −1.05 × 10^{−9} |

5 | −1.33 × 10^{−6} | 3.69 × 10^{−7} | 2.32 × 10^{−7} | −4.95 × 10^{−4} | 1.43 × 10^{−8} | −1.53 × 10^{−9} |

Loading Case | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\epsilon}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 3.60 × 10^{−4} | −1.09 × 10^{−4} | −1.09 × 10^{−4} | −2.84 × 10^{−7} | −9.09 × 10^{−7} | −2.84 × 10^{−7} |

2 | 3.60 × 10^{−4} | −1.06 × 10^{−4} | −1.08 × 10^{−4} | −9.58 × 10^{−8} | −1.25 × 10^{−6} | −9.40 × 10^{−8} |

3 | 3.11 × 10^{−4} | −9.33 × 10^{−5} | −9.01 × 10^{−5} | −2.46 × 10^{−4} | −1.92 × 10^{−9} | −8.84 × 10^{−11} |

4 | 1.79 × 10^{−4} | −5.36 × 10^{−5} | −5.18 × 10^{−5} | −4.27 × 10^{−4} | −4.01 × 10^{−9} | 7.93 × 10^{−10} |

5 | −1.40 × 10^{−6} | 4.11 × 10^{−7} | 3.54 × 10^{−7} | −4.93 × 10^{−4} | −2.58 × 10^{−9} | −6.73 × 10^{−10} |

Loading Case | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 3.14 × 10 | 4.71 × 10^{−2} | 4.71 × 10^{−2} | −4.38 × 10^{−8} | −5.65 × 10^{−13} | −2.24 × 10^{−12} |

2 | 3.60 × 10^{−4} | −1.22 × 10^{−4} | −1.22 × 10^{−4} | −8.54 × 10^{−8} | −3.27 × 10^{−9} | −5.43 × 10^{−8} |

3 | 3.11 × 10^{−4} | −1.05 × 10^{−4} | −1.05 × 10^{−4} | −2.45 × 10^{−4} | −7.56 × 10^{−9} | 2.41 × 10^{−10} |

4 | 1.79 × 10^{−4} | −6.03 × 10^{−5} | −6.02 × 10^{−5} | −4.25 × 10^{−4} | −1.46 × 10^{−8} | 2.33 × 10^{−9} |

5 | −1.46 × 10^{−6} | 4.46 × 10^{−7} | 4.56 × 10^{−7} | −4.91 × 10^{−4} | −1.67 × 10^{−8} | 4.06 × 10^{−11} |

Loading Case | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 1.46 × 10^{0} | −5.72 × 10^{−2} | −5.72 × 10^{−2} | 1.08 × 10^{−18} | −3.93 × 10^{−14} | −1.60 × 10^{−13} |

2 | 3.60 × 10^{−4} | −8.70 × 10^{−5} | −9.15 × 10^{−5} | −1.08 × 10^{−7} | −2.74 × 10^{−6} | −1.42 × 10^{−7} |

3 | 3.11 × 10^{−4} | −7.95 × 10^{−5} | −7.23 × 10^{−5} | −2.48 × 10^{−4} | 4.86 × 10^{−9} | −4.83 × 10^{−10} |

4 | 1.79 × 10^{−4} | −4.56 × 10^{−5} | −4.16 × 10^{−5} | −4.29 × 10^{−4} | 8.71 × 10^{−9} | −1.05 × 10^{−9} |

5 | −1.33 × 10^{−6} | 3.69 × 10^{−7} | 2.32 × 10^{−7} | −4.95 × 10^{−4} | 1.43 × 10^{−8} | −1.53 × 10^{−9} |

Loading Case | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{y}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{y}\mathit{z}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{z}\mathit{x}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{y}}$ |
---|---|---|---|---|---|---|

1 | 1.78 × 10 | −7.15 × 10^{−3} | −7.14 × 10^{−3} | −9.64 × 10^{−4} | −3.08 × 10^{−3} | −9.64 × 10^{−4} |

2 | 3.60 × 10^{−4} | −1.06 × 10^{−4} | −1.08 × 10^{−4} | −9.58 × 10^{−8} | −1.25 × 10^{−6} | −9.40 × 10^{−8} |

3 | 3.11 × 10^{−4} | −9.33 × 10^{−5} | −9.01 × 10^{−5} | −2.46 × 10^{−4} | −1.92 × 10^{−9} | −8.84 × 10^{−11} |

4 | 1.79 × 10^{−4} | −5.36 × 10^{−5} | −5.18 × 10^{−5} | −4.27 × 10^{−4} | −4.01 × 10^{−9} | 7.93 × 10^{−10} |

5 | −1.40 × 10^{−6} | 4.11 × 10^{−7} | 3.54 × 10^{−7} | −4.93 × 10^{−4} | −2.58 × 10^{−9} | −6.73 × 10^{−10} |

Modulus [MPa] | Matrix | Fiber | Average |
---|---|---|---|

${E}_{11}$ | 4140.0 | 86,900.0 | 56,278.0 |

${E}_{23}={E}_{13}$ | 4140.0 | 86,899.0 | 12,741.0 |

${\nu}_{1}$ | 0.34 | 0.22 | 0.28 |

${\nu}_{23}$ | 0.34 | 0.22 | 0.31 |

${G}_{23}$ | 1544.0 | 35,614.7 | 12,318.2 |

${K}_{23}$ | 4827.4 | 63,597.7 | 27,953.2 |

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**MDPI and ACS Style**

Katouzian, M.; Vlase, S.; Itu, C.; Scutaru, M.L.
Calculation of Homogenized Mechanical Coefficients of Fiber-Reinforced Composite Using Finite Element Method. *Materials* **2024**, *17*, 1334.
https://doi.org/10.3390/ma17061334

**AMA Style**

Katouzian M, Vlase S, Itu C, Scutaru ML.
Calculation of Homogenized Mechanical Coefficients of Fiber-Reinforced Composite Using Finite Element Method. *Materials*. 2024; 17(6):1334.
https://doi.org/10.3390/ma17061334

**Chicago/Turabian Style**

Katouzian, Mostafa, Sorin Vlase, Calin Itu, and Maria Luminita Scutaru.
2024. "Calculation of Homogenized Mechanical Coefficients of Fiber-Reinforced Composite Using Finite Element Method" *Materials* 17, no. 6: 1334.
https://doi.org/10.3390/ma17061334