# Parameter Identification of Fractional Index Viscoelastic Model for Vegetable-Fiber Reinforced Composite

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Model

## 3. Numerical Methodology

## 4. Materials and Methods

#### 4.1. Testing Samples

#### 4.2. Experimental Procedure

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Shamsuri, A.A.; Abdan, K.; Jamil, S.N.A.M. Polybutylene succinate (PBS)/natural fiber green composites: Melt blending processes and tensile properties. Phys. Sci. Rev.
**2022**. [Google Scholar] [CrossRef] - Ullah-Arif, Z.; Yasir-Khalid, M.; Fahad-Sheikh, M.; Zolfagharian, A.; Bodaghi, M. Biopolymeric sustainable materials and their emerging applications. J. Environ Chem. Eng.
**2022**, 14, 1477. [Google Scholar] [CrossRef] - Sonkusare, P.; Agarwal, P.; Dhakad, S.K.; Rana-Ravindra, S. A Review Paper: Study of Various Renewable Resources Polymer and Different Types of Nanocomposite Materials. In Technology Innovation in Mechanical Engineering: Select Proceedings of TIME 2021; Springer Nature Singapore: Singapore, 2022; pp. 63–73. [Google Scholar] [CrossRef]
- Chaudhary, V.; Ahmad, F. A review on plant fiber reinforced thermoset polymers for structural and frictional composites. Polym. Test.
**2020**, 91, 106792. [Google Scholar] [CrossRef] - Malik, K.; Ahmad, F.; Azhani-Yunus, N.; Gunister, E.; Nakato, T.; Mouri, E.; Ali, S. A Review of Flax Fiber Reinforced Thermoset Polymer Composites: Thermal-Physical Properties, Improvements, and Application. J. Nat. Fibers
**2021**, 10, 1–19. [Google Scholar] [CrossRef] - Farazin, A.; Khan, A. An extensive study on strain dependence of glass fiber-reinforced polymer-based composites. J. Strain Anal. Eng. Des.
**2022**, 57, 411–432. [Google Scholar] [CrossRef] - Santha, R.D.; Gopala-Krishna, M.O. Evaluation of Mechanical and Micro Structural Properties of Natural Fiber Reinforced Polymer Composites. Mater. Sci. Forum.
**2022**, 1065, 69–77. [Google Scholar] [CrossRef] - Venkatarajan, S.; Subbu, C.; Athijayamani, A.; Muthuraja, R. Mechanical properties of natural cellulose fibers reinforced polymer composites—2015–2020: A review. Mater. Today Proc.
**2021**, 47, 1017–1024. [Google Scholar] [CrossRef] - Müller, S.; Kästner, M.; Brummund, J.; Ulbricht, V. A nonlinear fractional viscoelastic material model for polymers. Comput. Mater. Sci.
**2011**, 50, 2938–2949. [Google Scholar] [CrossRef] - Xu, H.; Jiang, X. Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput. Math. Appl.
**2017**, 73, 1377–1384. [Google Scholar] [CrossRef] - Fang, C.; Leng, J.; Sun, H.; Gu, J. A multi-branch thermoviscoelastic model based on fractional derivatives for free recovery behaviors of shape memory polymers. Mech. Mater.
**2018**, 120, 34–42. [Google Scholar] [CrossRef] - Martin, O. Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model. Appl. Math. Model.
**2017**, 43, 351–359. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Salem, M.G. The thermal vibration of small-sized rotating fractional viscoelastic beams positioned on a flexible foundation in the light of the Moore–Gibson–Thompson model. J. Ocean Eng. Sci.
**2022**. [Google Scholar] [CrossRef] - Muliana, A. A fractional model of nonlinear multiaxial viscoelastic behaviors. Time-Depend. Mater.
**2022**, 118, 1573–2738. [Google Scholar] [CrossRef] - Ruhani, B.; Toghraie, D.; Hekmatifar, M.; Hadian, M. Statistical investigation for developing a new model for rheological behavior of ZnO–Ag (50%–50%)/Water hybrid Newtonian nanofluid using experimental data. Phys. A Stat. Mech. Appl.
**2019**, 525, 741–751. [Google Scholar] [CrossRef] - Cappelli, L.; Montemurro, M.; Dau, F.; Guillaumat, L. Multi-scale identification of the viscoelastic behavior of composite materials through a non-destructive test. Mech. Mater.
**2019**, 137, 103137. [Google Scholar] [CrossRef] - Xiao, Z.; Haitian, Y.; Yiqian, H. Identification of constitutive parameters for fractional viscoelasticity. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 311–322. [Google Scholar] [CrossRef] - Fan, W.; Jiang, X.; Qi, H. Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method. Phys. A Stat. Mech. Its Appl.
**2015**, 427, 40–49. [Google Scholar] [CrossRef] - Shabani, M.; Jahani, K.; Di Paola, M.; Homayoun Sadeghi, M. Frequency domain identification of the fractional Kelvin-Voigt’s parameters for viscoelastic materials. Mech. Mater.
**2019**, 137, 103099. [Google Scholar] [CrossRef] - Viviani, L.; Di Paola, M.; Royer-Carfagni, G. A fractional viscoelastic model for laminated glass sandwich plates under blast actions. Int. J. Mech. Sci.
**2022**, 222, 107204. [Google Scholar] [CrossRef] - Kästner, M.; Obst, M.; Brummund, J.; Thielsch, K.; Ulbricht, V. Inelastic material behavior of polymers—Experimental characterization, formulation and implementation of a material model. Mech. Mater.
**2012**, 52, 40–57. [Google Scholar] [CrossRef] - Mashayekhi, S.; Stanisauskis, E.; Hassani, M.; Oates, W. Excluded volume effects and fractional viscoelasticity in polymers. Meccanica
**2022**, 57, 821–832. [Google Scholar] [CrossRef] - Zakria, A.; Abouelregal, A.E. Fractional viscoelastic model with a non-singular kernel for a rotating semiconductor circular cylinder permeated by a magnetic field and due to heat flow pulse heating. Waves Random Complex Media
**2022**, 32, 1–36. [Google Scholar] [CrossRef] - Sasso, M.; Palmieri, G.; Amodio, D. Application of fractional derivative models in linear viscoelastic problems. Mech Time-Depend Mater
**2011**, 15, 367–387. [Google Scholar] [CrossRef] - Zheng, Z.; Zhao, W.; Dai, H. A new definition of fractional derivative. Int. J. Non Linear Mech.
**2019**, 108, 1–6. [Google Scholar] [CrossRef] - Alaa, M.; Samy, Y.; Tawheed, H.; Mohammed, A.A. Microstructure and modeling of uniaxial mechanical properties of Polyethersulfone nanocomposite ultrafiltration membranes. Int. J. Mech. Sci.
**2021**, 204, 106568. [Google Scholar] [CrossRef] - Vaiana, N.; Sessa, S.; Rosati, L. A generalized class of uniaxial rate-independent models for simulating asymmetric mechanical hysteresis phenomena. Mech. Syst. Signal. Process.
**2021**, 146, 106984. [Google Scholar] [CrossRef] - Li, K.F.; Yang, C.Q.; Zhao, Y.B.; Pan, Y.; Wang, G.; Zheng, Y.Y.; Xu, F. Study on the creep behavior of PVA-ECC based on fractional-differential rheological model. Constr. Build. Mater.
**2020**, 230, 117064. [Google Scholar] [CrossRef] - Hofer, U.; Luger, M.; Traxl, R.; Lackner, R. Multiscale modeling of the viscoelastic response of braid-reinforced polymers: Model formulation and experimental assessment considering different rheological models. Compos. B Eng.
**2020**, 182, 107398. [Google Scholar] [CrossRef] - Fang, C.; Sun, H.; Gu, J. A Fractional Calculus Approach to the Prediction of Free Recovery Behaviors of Amorphous Shape Memory Polymers. J. Mech.
**2016**, 32, 11–17. [Google Scholar] [CrossRef] - Xiao, R.; Sun, H.; Chen, W. An equivalence between generalized Maxwell model and fractional Zener model. Mech. Mater.
**2016**, 100, 148–153. [Google Scholar] [CrossRef] - Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Vinagre Jara, B.M. Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys.
**2009**, 228, 3137–3153. [Google Scholar] [CrossRef] [Green Version] - Barile, C.; Casavola, C.; De Cillis, F. Mechanical comparison of new composite materials for aerospace applications. Compos. B. Eng.
**2019**, 162, 122–128. [Google Scholar] [CrossRef] - Podlubny, I.; Skovranek, T.; Vinagre Jara, B.M. Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox. Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C. In Proceedings of the NASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC-CIE2009, San Diego, California, USA, 30 August–2 September 2009; pp. 1173–1180. [Google Scholar] [CrossRef]
- Sessa, S.; Vaiana, N.; Paradiso, M.; Rosati, L. An inverse identification strategy for the mechanical parameters of a phenomenological hysteretic constitutive model. Mech. Syst. Signal Process.
**2020**, 139, 106622. [Google Scholar] [CrossRef] - Jin, Z.; Chen, G.; Yang, Z. Rolling Bearing Fault Diagnosis Based on WOA-VMD-MPE and MPSO-LSSVM. Entropy
**2022**, 24, 927. [Google Scholar] [CrossRef] [PubMed] - Salih, S.Q.; Alsewari, A.A.; Al-Khateeb, B.; Zolkipli, M.F. Novel Multi-swarm Approach for Balancing Exploration and Exploitation in Particle Swarm Optimization. Recent Trends Data Sci. Soft Comput. Irict 2018. Adv. Intell. Syst. Comput.
**2018**, 843, 196–206. [Google Scholar] [CrossRef] - Rodríguez Soto, A.A.; Valín Rivera, J.L.; Alves Borges, L.M.S.; Palomares Ruiz, J.E. Tensile, Impact, and Thermal Properties of an Epoxynovolac Matrix Composites with Cuban Henequen Fibers. Mech. Compos. Mater.
**2018**, 54, 341–348. [Google Scholar] [CrossRef] - Liu, D.; Song, J.; Anderson, D.P.; Chang, P.R.; Hua, Y. Bamboo fiber and its reinforced composites: Structure and properties. Cellulose
**2012**, 19, 1449–1480. [Google Scholar] [CrossRef] - Wang, X.; Petrů, M.; Xia, L. Modeling the dynamics behavior of flax fiber reinforced composite after water aging using a modified Huet-Sayegh viscoelastic model with fractional derivatives. Constr. Build Mater.
**2021**, 290, 122879. [Google Scholar] [CrossRef] - Faal, R.T.; Sourki, R.; Crawford, B.; Vaziri, R.; Milani, A.S. Using fractional derivatives for improved viscoelastic modeling of textile composites. Part I: Fabric yarns. J. Compos. Mater.
**2020**, 54, 3245–3260. [Google Scholar] [CrossRef] - Bisanda, E.T.N.; Ansell, M.P. Properties of sisal-CNSL composites. J. Mater. Sci.
**1992**, 27, 1690–1700. [Google Scholar] [CrossRef] - Van-Quy, H.O.; Thao Nguyen, S.T. Experimental Analysis of Coir Fiber Sheet Reinforced Epoxy Resin Composite. IOP Conf. Ser. Mater. Sci. Eng.
**2019**, 642, 012007. [Google Scholar] [CrossRef] - Zhongya, L.; Yueguang, W. A strain gradient linear viscoelasticity theory. Int. J. Solids Struct
**2020**, 203, 197–209. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Standard model or Linear Viscoelastic Solid. (

**b**) The fractional version or fractional Zener Model.

**Figure 3.**(

**a**) Dry henequen fibers. (

**b**) Composite plate. (

**c**) Typical traction samples of fiber-reinforced composites.

**Figure 5.**Loading–unloading tensile tests with strain as entrance signal, five specimens, (

**a**) with 22 wt%, and (

**b**) with 28 wt%.

**Figure 6.**Fluency staggered test, five specimens, (

**a**) with 9 wt%, (

**b**) with 14 wt%, and (

**c**) with 28 wt%.

**Figure 7.**Tensile strain for loading–unloading test, one specimen with 0 wt%, experimental data in dots and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 8.**Tensile strain for loading–unloading test, one specimen with 9 wt%, experimental data in dots and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 9.**Tensile strain for loading–unloading test, one specimen with 14 wt%, experimental data in dots and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 10.**Percentage relative error between the maximum values of each cycle, experimental data and response of both models.

**Figure 11.**Tensile strain for staggered fluency test, five specimens with 9 wt%, experimental data in dotted lines and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 12.**Tensile strain for staggered fluency test, five specimens with 14 wt%, experimental data in dotted lines and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 13.**Tensile strain for staggered fluency test, five specimens with 28 wt%, experimental data in dotted lines and model response in continuous line, (

**a**) integer index model, (

**b**) fractional index model.

**Figure 14.**Percentage relative error between the means of the experimental data and the response of both models, integer index model in a continuous line and gray fill, and fractional index model in dashed line and white fill.

Fiber Percent in Weight | $\mathit{\eta}$ (GPa· s) | ${\mathit{E}}_{1}$ (GPa) | E (GPa) | $\mathit{\alpha}$ |
---|---|---|---|---|

0 wt% | 10.4429 | 1.8643 | 0.3007 | 1 |

9 wt% | 15.0150 | 2.3142 | 0.6222 | 1 |

14 wt% | 15.6175 | 2.3624 | 0.6427 | 1 |

22 wt% | 16.3209 | 2.5003 | 0.6512 | 1 |

28 wt% | 14.9841 | 2.2026 | 0.5986 | 1 |

Fiber Percent in Weight | $\mathit{\eta}$ (GPa· s) | ${\mathit{E}}_{1}$ (GPa) | E (GPa) | $\mathit{\alpha}$ |
---|---|---|---|---|

0 wt% | 9.9695 | 0.5874 | 0.9964 | 0.2155 |

9 wt% | 8.1984 | 1.6223 | 0.5998 | 0.4993 |

14 wt% | 10.0255 | 2.1003 | 0.6069 | 0.7996 |

22 wt% | 12.0033 | 2.3232 | 1.1035 | 0.8101 |

28 wt% | 16.0550 | 2.1599 | 1.8237 | 0.6713 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rodríguez Soto, A.A.; Valín Rivera, J.L.; Sanabio Alves Borges, L.M.; Palomares Ruiz, J.E.
Parameter Identification of Fractional Index Viscoelastic Model for Vegetable-Fiber Reinforced Composite. *Polymers* **2022**, *14*, 4634.
https://doi.org/10.3390/polym14214634

**AMA Style**

Rodríguez Soto AA, Valín Rivera JL, Sanabio Alves Borges LM, Palomares Ruiz JE.
Parameter Identification of Fractional Index Viscoelastic Model for Vegetable-Fiber Reinforced Composite. *Polymers*. 2022; 14(21):4634.
https://doi.org/10.3390/polym14214634

**Chicago/Turabian Style**

Rodríguez Soto, Angel Alexander, José Luís Valín Rivera, Lavinia María Sanabio Alves Borges, and Juan Enrique Palomares Ruiz.
2022. "Parameter Identification of Fractional Index Viscoelastic Model for Vegetable-Fiber Reinforced Composite" *Polymers* 14, no. 21: 4634.
https://doi.org/10.3390/polym14214634