# Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures

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## Abstract

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## 1. Introduction

## 2. Geometry of the Structure

## 3. Viscoelastic Property

## 4. Nonlinear Elastic Material and Governing Equations

## 5. Solution Method

## 6. Discussion

#### 6.1. Validation

#### 6.1.1. The Solving Method (SAPM)

#### 6.1.2. Comparisons

#### 6.2. Numerical Results

## 7. Conclusions

- The new theory of hyperelastic structures can be used with appropriate confidence for viscoelastic properties.
- Structures made of nonlinear elastic material are sensitive to changes in applied transverse loads, and the changes are entirely nonlinear, even with the low-load application.
- For low loads, it has a significant impact on the deformation by the boundary conditions. However, as the load increases, these effects decrease.
- As the viscosity increases, the duration of the final deformation increases, which has a direct relationship with the viscosity of the material.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Boyce, M.C.; Arruda, E.M. Constitutive models of rubber elasticity: A review. Rubber Chem. Technol.
**2000**, 73, 504–523. [Google Scholar] [CrossRef] - Gent, A.N. A new constitutive relation for rubber. Rubber Chem. Technol.
**1996**, 69, 59–61. [Google Scholar] [CrossRef] - Anssari-Benam, A.; Bucchi, A. A generalised neo-Hookean strain energy function for application to the finite deformation of elastomers. Int. J. Non-Linear Mech.
**2021**, 128, 103626. [Google Scholar] [CrossRef] - Yeoh, O.H. Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates. Rubber Chem. Technol.
**1990**, 63, 792–805. [Google Scholar] [CrossRef] - Erchiqui, F.; Gakwaya, A.; Rachik, M. Dynamic finite element analysis of nonlinear isotropic hyperelastic and viscoelastic materials for thermoforming applications. Polym. Eng. Sci.
**2005**, 45, 125–134. [Google Scholar] [CrossRef] [Green Version] - Kocaturk, T.; Akbas, S.D. Geometrically nonlinear static analysis of a simply supported beam made of hyperelastic material. Struct. Eng. Mech.
**2010**, 35, 677–697. [Google Scholar] [CrossRef] - Li, Y.L.; Oh, I.; Chen, J.H.; Zhang, H.H.; Hu, Y.H. Nonlinear dynamic analysis and active control of visco-hyperelastic dielectric elastomer membrane. Int. J. Solids Struct.
**2018**, 152, 28–38. [Google Scholar] [CrossRef] - Alibakhshi, A.; Dastjerdi, S.; Akgoz, B.; Civalek, O. Parametric vibration of a dielectric elastomer microbeam resonator based on a hyperelastic cosserat continuum model. Compos. Struct.
**2022**, 287, 115386. [Google Scholar] [CrossRef] - Almasi, A.; Baghani, M.; Moallemi, A. Thermomechanical analysis of hyperelastic thick-walled cylindrical pressure vessels, analytical solutions and FEM. Int. J. Mech. Sci.
**2017**, 130, 426–436. [Google Scholar] [CrossRef] - Asgari, M.; Hashemi, S.S. Dynamic visco-hyperelastic behavior of elastomeric hollow cylinder by developing a constitutive equation. Struct. Eng. Mech.
**2016**, 59, 601–619. [Google Scholar] [CrossRef] - Pascon, J.P. Finite element analysis of functionally graded hyperelastic beams under plane stress. Eng. Comput.
**2020**, 36, 1265–1288. [Google Scholar] [CrossRef] - Pascon, J.P. Large deformation analysis of functionally graded visco-hyperelastic materials. Comput. Struct.
**2018**, 206, 90–108. [Google Scholar] [CrossRef] - Gharooni, H.; Ghannad, M. Nonlinear analysis of radially functionally graded hyperelastic cylindrical shells with axially-varying thickness and non-uniform pressure loads based on perturbation theory. J. Comput. Appl. Mech.
**2019**, 50, 324–340. [Google Scholar] - Hosseini, S.; Rahimi, G. Nonlinear Bending Analysis of Hyperelastic Plates Using FSDT and Meshless Collocation Method Based on Radial Basis Function. Int J Appl Mech
**2021**, 13, 2150007. [Google Scholar] [CrossRef] - Xu, Q.P.; Liu, J.Y.; Qu, L.Z. A Higher-Order Plate Element Formulation for Dynamic Analysis of Hyperelastic Silicone Plate. J. Mech.
**2019**, 35, 795–808. [Google Scholar] [CrossRef] - Dadgar-Rad, F.; Firouzi, N. Large deformation analysis of two-dimensional visco-hyperelastic beams and frames. Arch. Appl. Mech.
**2021**, 91, 4279–4301. [Google Scholar] [CrossRef] - Ansari, R.; Hassani, R.; Oskouie, M.F.; Rouhi, H. Nonlinear bending analysis of hyperelastic Mindlin plates: A numerical approach. Acta Mech.
**2021**, 232, 741–760. [Google Scholar] [CrossRef] - Tashiro, K.; Shobayashi, Y.; Ota, I.; Hotta, A. Finite element analysis of blood clots based on the nonlinear visco-hyperelastic model. Biophys. J.
**2021**, 120, 4547–4556. [Google Scholar] [CrossRef] [PubMed] - Shariyat, M.; Abedi, S. An accurate hyperelasticity-based plate theory and nonlinear energy-based micromechanics for impact and shock analyses of compliant particle-reinforced FG hyperelastic plates. Zamm-Z. Angew. Math. Mech.
**2022**, 102, e202100099. [Google Scholar] [CrossRef] - Karimi, S.; Ahmadi, H.; Foroutan, K. Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane resting on a Winkler-Pasternak elastic medium under hydrostatic pressure. J. Vib. Control
**2022**. [Google Scholar] [CrossRef] - Alibakhshi, A.; Heidari, H. Nonlinear dynamics of dielectric elastomer balloons based on the Gent-Gent hyperelastic model. Eur. J. Mech. A-Solid
**2020**, 82, 103986. [Google Scholar] [CrossRef] - Alibakhshi, A.; Rahmanian, S.; Dastjerdi, S.; Malikan, M.; Karami, B.; Akgoz, B.; Civalek, O. Hyperelastic Microcantilever AFM: Efficient Detection Mechanism Based on Principal Parametric Resonance. Nanomaterials
**2022**, 12, 2598. [Google Scholar] [CrossRef] [PubMed] - Falope, F.O.; Lanzoni, L.; Tarantino, A.M. FE Analyses of Hyperelastic Solids under Large Bending: The Role of the Searle Parameter and Eulerian Slenderness. Materials
**2020**, 13, 1597. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hosseini, S.; Rahimi, G.; Shahgholian-Ghahfarokhi, D. A meshless collocation method on nonlinear analysis of functionally graded hyperelastic plates using radial basis function. Zamm-Z. Angew. Math. Mech.
**2022**, 102, e202100216. [Google Scholar] [CrossRef] - Coda, H.B.; Bernardo, C.C.L.G.; Paccola, R.R. A FEM formulation for the analysis of laminated and functionally graded hyperelastic beams with continuous transverse shear stresses. Compos. Struct.
**2022**, 292, 115606. [Google Scholar] [CrossRef] - Dastjerdi, S.; Alibakhshi, A.; Akgoz, B.; Civalek, O. A Novel Nonlinear Elasticity Approach for Analysis of Nonlinear and Hyperelastic Structures. Eng. Anal. Bound. Elem.
**2022**, 143, 219–236. [Google Scholar] [CrossRef] - Zhao, Z.T.; Niu, D.T.; Zhang, H.W.; Yuan, X.G. Nonlinear dynamics of loaded visco-hyperelastic spherical shells. Nonlinear Dyn.
**2020**, 101, 911–933. [Google Scholar] [CrossRef] - Zhao, Z.T.; Yuan, X.G.; Zhang, W.Z.; Niu, D.T.; Zhang, H.W. Dynamical modeling and analysis of hyperelastic spherical shells under dynamic loads and structural damping. Appl. Math. Model.
**2021**, 95, 468–483. [Google Scholar] [CrossRef] - Bacciocchi, M.; Tarantino, A.M. Bending of hyperelastic beams made of transversely isotropic material in finite elasticity. Appl. Math. Model.
**2021**, 100, 55–76. [Google Scholar] [CrossRef] - Khaniki, H.B.; Ghayesh, M.H.; Chin, R.; Amabili, M. A review on the nonlinear dynamics of hyperelastic structures. Nonlinear Dyn.
**2022**, 110, 963–994. [Google Scholar] [CrossRef] - Zenkour, A.M. Nonlocal thermal vibrations of embedded nanoplates in a viscoelastic medium. Struct. Eng. Mech.
**2022**, 82, 701–711. [Google Scholar] - Yuan, Y.; Niu, Z.Q.; Smitt, J. Magneto-hygro-thermal vibration analysis of the viscoelastic nanobeams reinforcedwith carbon nanotubes resting on Kerr’s elastic foundation based on NSGT. Adv. Compos. Mater.
**2022**. [Google Scholar] [CrossRef] - Soleimani-Javid, Z.; Arshid, E.; Amir, S.; Bodaghi, M. On the higher-order thermal vibrations of FG saturated porous cylindrical micro-shells integrated with nanocomposite skins in viscoelastic medium. Def. Technol.
**2022**, 18, 1416–1434. [Google Scholar] [CrossRef] - Moayeri, M.; Darabi, B.; Sianaki, A.H.; Adamian, A. Third order nonlinear vibration of viscoelastic circular microplate based on softening and hardening nonlinear viscoelastic foundation under thermal loading. Eur. J. Mech. A-Solid
**2022**, 95, 104644. [Google Scholar] [CrossRef] - Li, Y.; Liu, B. Thermal buckling and free vibration of viscoelastic functionally graded sandwich shells with tunable auxetic honeycomb core. Appl. Math. Model.
**2022**, 108, 685–700. [Google Scholar] [CrossRef] - Dang, R.Q.; Yang, A.M.; Chen, Y.M.; Wei, Y.Q.; Yu, C.X. Vibration analysis of variable fractional viscoelastic plate based on shifted Chebyshev wavelets algorithm. Comput. Math. Appl.
**2022**, 119, 149–158. [Google Scholar] [CrossRef] - Alizadeh, A.; Shishehsaz, M.; Shahrooi, S.; Reza, A. Free vibration characteristics of viscoelastic nano-disks based on modified couple stress theory. J. Strain Anal. Eng.
**2022**. [Google Scholar] [CrossRef] - Ghobadi, E.; Shutov, A.; Steeb, H. Parameter Identification and Validation of Shape-Memory Polymers within the Framework of Finite Strain Viscoelasticity. Materials
**2021**, 14, 2049. [Google Scholar] [CrossRef] - Dacol, V.; Caetano, E.; Correia, J.R. A New Viscoelasticity Dynamic Fitting Method Applied for Polymeric and Polymer-Based Composite Materials. Materials
**2020**, 13, 5213. [Google Scholar] [CrossRef] - Chang, J.J.; Li, Y.Y.; Zeng, X.F.; Zhong, H.Y.; Wan, T.L.; Lu, C. Study on the Viscoelasticity Measurement of Materials Based on Surface Reflected Waves. Materials
**2019**, 12, 1875. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, D.Z.; de Boer, G.; Ghanbarzadeh, A. A Numerical Model for Investigating the Effect of Viscoelasticity on the Partial Slip Solution. Materials
**2022**, 15, 5182. [Google Scholar] [CrossRef] [PubMed] - Itou, H.; Kovtunenko, V.A.; Rajagopal, K.R. The Boussinesq flat-punch indentation problem within the context of linearized viscoelasticity. Int. J. Eng. Sci.
**2020**, 151, 103272. [Google Scholar] [CrossRef] - Dastjerdi, S.; Akgoz, B.; Civalek, O. On the effect of viscoelasticity on behavior of gyroscopes. Int. J. Eng. Sci.
**2020**, 149, 103236. [Google Scholar] [CrossRef] - Dastjerdi, S.; Akgoz, B.; Civalek, O. On the shell model for human eye in Glaucoma disease. Int. J. Eng. Sci.
**2021**, 158, 103414. [Google Scholar] [CrossRef] - Dastjerdi, S.; Malikan, M.; Akgoz, B.; Civalek, O.; Wiczenbach, T.; Eremeyev, V.A. On the deformation and frequency analyses of SARS-CoV-2 at nanoscale. Int. J. Eng. Sci.
**2022**, 170, 103604. [Google Scholar] [CrossRef] - Li, C.; Zhu, C.; Sui, S.; Yan, J. A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory. Appl. Sci.
**2022**, 12, 40. [Google Scholar] [CrossRef] - Hu, W.; Xu, M.; Song, J.; Gao, Q.; Deng, Z. Coupling dynamic behaviors of flexible stretching hub-beam system. Mech. Syst. Signal Process.
**2021**, 151, 107389. [Google Scholar] [CrossRef] - Yan, J.W.; Lai, S.K.; He, L.H. Nonlinear dynamic behavior of single-layer graphene under uniformly distributed loads. Compos. B Eng.
**2019**, 165, 473–490. [Google Scholar] [CrossRef] - Pascon, J.P.; Coda, H.B. Finite deformation analysis of visco-hyperelastic materials via solid tetrahedral finite elements. Finite Elem. Anal. Des.
**2017**, 133, 25–41. [Google Scholar] [CrossRef] - López-Campos, J.A.; Segade, A.; Fernández, J.R.; Casarejos, E.; Vilán, J.A. Behavior characterization of visco-hyperelastic models for rubber-like materials using genetic algorithms. Appl. Math. Model.
**2019**, 66, 241–255. [Google Scholar] [CrossRef] - Yarali, E.; Baniasadi, M.; Bodaghi, M.; Bodaghi, M. 3D constitutive modeling of electro-magneto-viscohyperelastic elastomers: A semi-analytical solution for cylinders under large torsion–extension deformation. Smart Mater. Struct.
**2020**, 29, 085031. [Google Scholar] [CrossRef] - Dastjerdi, S.; Tadi Beni, Y.; Malikan, M. A comprehensive study on nonlinear hygro-thermo-mechanical analysis of thick functionally graded porous rotating disk based on two quasi-three-dimensional theories. Mech Based Des Struct
**2022**, 50, 3596–3625. [Google Scholar] [CrossRef] - Zhang, D.G.; Zhou, H.M. Nonlinear bending analysis of FGM circular plates based on physical neutral surface and higher-order shear deformation theory. Aerosp. Sci. Technol.
**2015**, 41, 90–98. [Google Scholar] [CrossRef]

**Figure 5.**Comparison between the results of this study and [51] for the visco-hyperelastic analysis.

**Figure 6.**Deflection changes versus the applied transverse loading (${q}_{z}$) for different types of boundary conditions.

**Table 1.**Comparison between the maximum deflection (mm) results of the present study and [53] for aluminum circular plates.

Maximum Deflection (mm) | ||||||||
---|---|---|---|---|---|---|---|---|

Boundary Conditions | ${\mathit{q}}_{\mathit{z}}=5\mathit{M}\mathit{P}\mathit{a}$ | ${\mathit{q}}_{\mathit{z}}=20\mathit{M}\mathit{P}\mathit{a}$ | ${\mathit{q}}_{\mathit{z}}=50\mathit{M}\mathit{P}\mathit{a}$ | ${\mathit{q}}_{\mathit{z}}=100\mathit{M}\mathit{P}\mathit{a}$ | ||||

Present | [53] | Present | [53] | Present | [53] | Present | [53] | |

SS | 3.789 | 3.883 | 8.232 | 8.333 | 12.11 | 12.15 | 15.56 | 15.68 |

CC | 1.261 | 1.263 | 4.559 | 4.562 | 8.819 | 8.820 | 12.93 | 12.93 |

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**MDPI and ACS Style**

Dastjerdi, S.; Akgöz, B.; Civalek, Ö.
Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures. *Materials* **2022**, *15*, 8425.
https://doi.org/10.3390/ma15238425

**AMA Style**

Dastjerdi S, Akgöz B, Civalek Ö.
Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures. *Materials*. 2022; 15(23):8425.
https://doi.org/10.3390/ma15238425

**Chicago/Turabian Style**

Dastjerdi, Shahriar, Bekir Akgöz, and Ömer Civalek.
2022. "Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures" *Materials* 15, no. 23: 8425.
https://doi.org/10.3390/ma15238425