# Parameters Identification of Rubber-like Hyperelastic Material Based on General Regression Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. GRNN Method

#### 2.1. Theoretical Basis of GRNN

**x**and y are the input vector and output variable of the sample data, respectively, when the observed value of

**x**is set as

**x**

_{0}, i.e., the commonly known target value, then the regression value y with respect to the input vector

**x**is obtained by

**x**

_{0}, y) represents the probability density function. Using Parzen non-parametric estimation [40], the Gaussian kernel function is selected as the kernel function, and then the density function f(

**X**, Y) can be calculated based on the sample data ${\left\{{\mathit{X}}_{i},{Y}_{i}\right\}}_{i=1}^{n}$, shown as below

#### 2.2. Architecture of GRNN

- (1)
- Input layer

- (2)
- Pattern layer

- (3)
- Summation layer

_{j}in the ith learning sample, the corresponding transfer function is

- (4)
- Output layer

**X**), which is given as

## 3. Application of GRNN for Determining the Hyperelastic Model Parameters

#### 3.1. Hyperelastic Material Model

_{1}, I

_{2}and I

_{3}) or principal extension ratios (λ

_{1}, λ

_{2}and λ

_{3}), that is, W = W(I

_{1}, I

_{2}, I

_{3}) or W = W(λ

_{1}, λ

_{2}, λ

_{3}).

- The M–R model can be defined by two parameters, C
_{10}and C_{01}, shown as below,

_{10}and C

_{01}are model parameters which need to be determined.

- The formulation of polynomial model (N = 2) is given by,

_{ij}is the corresponding model parameter, including C

_{01}, C

_{10}, C

_{20}, C

_{11}and C

_{02}.

- Based on the principal extension ratio, the strain energy of the Ogden model can be defined as,

_{i}and α

_{i}are material parameters related to temperature. D

_{i}= 0 for incompressible strain energy, while J is the elastic volume ratio. Thereby, six parameters, i.e., μ

_{1}, α

_{1}, μ

_{2}, α

_{2}, μ

_{3}, and α

_{3}, are the unknown parameters for the definition of the Ogden model.

#### 3.2. The Parameter Identification Methodology for a Hyperelastic Model Based on Finite Element Analysis, Experiment and GRNN

- (a)
- Prepare the target values of the GRNN model. For this case, experiments, e.g., uniaxial tensile, are needed to be carried out for the purpose of obtaining the experimental force-displacement curve (i.e., target curve);
- (b)
- Provide the learning samples of the GRNN model. The corresponding simulation models of the experiments are required to establish the same boundary and the initial conditions are considered. Next, several sets of material parameters (i.e., C
_{10}and C_{01}for M-R model) will be predefined to produce different force-displacement curves. For the GRNN model, the sets of the material parameters can be taken as output vectors, and the corresponding force-displacement curves are input vectors. In this way, the learning samples of the GRNN model are given by FEA; - (c)
- Obtain the identified material parameters. Through the GRNN learning model, when the results of force-displacement calculated by FEA meet the requirements of accuracy, the corresponding output value at this moment is what we want.

**Figure 1.**The scheme of the topological structure of a GRNN-based approach for the prediction of model parameters of M–R model. Nomenclature: u displacement; F reaction force; n number of learning samples; exp experimental results; j the cycle number of the GRNN model; s the given accuracy requirement.

#### 3.3. An Example of GRNN-Based Approach Application

#### 3.3.1. Uniaxial Tensile Test with Hyperelastic Rubber Specimen

#### 3.3.2. FEA Calculation with Same Experimental Condition

_{3}= 20 mm over a time period of 60 s. The finite element mesh applied in this case consists of eight-node 3D stress elements with hybrid formulation (i.e., C3D8H). The mesh size is controlled by a global size of 0.5 mm. In special, a reference point is introduced to acquire the variations in force and displacement during the step time. So then, the displacement loading will be applied at the reference point, and transmitted through a reference point to the surface of the strip specimen.

_{10}= 0.0385 and C

_{01}= 0.4052 for M-R model, C

_{10}= −2.1506, C

_{01}= 2.7355, C

_{20}= 2.1308, C

_{11}= −6.7135 and C

_{02}= 6.3381 for polynomial model (N =2), μ

_{1}= −3.9450, α

_{1}= −2.3031, μ

_{2}= −0.3774, α

_{2}= −1.3540, μ

_{3}= 5.4133 and α

_{3}= −3.8436 for Ogden model (N = 3). Using these three sets of model parameters, the corresponding force-displacement curves are calculated and compared with the experimental force-displacement curve, as shown in Figure 7. It can found that there are significant differences between the least-squares fit result of the M–R model and experimental data when the deformation is larger than 10mm. For the polynomial model (N = 2), although the least-squares fit result is much better than the M–R model, the error becomes markedly more prominent as the deformation increases, and there is still room for improvement of accuracy.

## 4. Results and Discussion

- (a)
- M–R model;

- (b)
- Polynomial model (N = 2);

- (c)
- Ogden model (N = 3);

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dal, H.; Acikgoz, K.; Badienia, Y. On the performance of isotropic hyperelastic constitutive models for rubber-like materials: A state of the art review. Appl. Mech. Rev.
**2021**, 73, 020802. [Google Scholar] [CrossRef] - Feng, Z.G.; Kosawada, T.; Nakamura, T.; Sato, D.; Kitajima, T.; Umezu, M. Theoretical methods and models for mechanical properties of soft biomaterials. Aims Mater Sci.
**2017**, 4, 680–705. [Google Scholar] [CrossRef] - Mihai, L.A.; Goriely, A. How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2017**, 473, 20170607. [Google Scholar] [CrossRef] [PubMed][Green Version] - Puglisi, G.; Saccomandi, G. Multi-scale modelling of rubber-like materials and soft tissues: An appraisal. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2016**, 472, 20160060. [Google Scholar] [CrossRef] [PubMed][Green Version] - Destrade, M.; Saccomandi, G.; Sgura, I. Methodical fitting for mathematical models of rubber-like materials. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2017**, 473, 20160811. [Google Scholar] [CrossRef][Green Version] - Chaves, W.V. Notes on Continuum Mechanics; Springer: Dordrecht, The Netherlands, 2013; pp. 423–464. [Google Scholar] [CrossRef]
- Wilber, J.P.; Criscione, J.C. The Baker–Ericksen inequalities for hyperelastic models using a novel set of invariants of Hencky strain. Int. J. Solids Struct.
**2005**, 42, 1547–1559. [Google Scholar] [CrossRef] - Mooney, M. A Theory of large elastic deformation. J. Appl. Phys.
**1940**, 11, 582–592. [Google Scholar] [CrossRef] - Rivlin, R.S. Chapter 10—Large elastic deformations. In Rheology; Eirich, F.R., Ed.; Springer: New York, NY, USA, 1900; pp. 351–385. [Google Scholar] [CrossRef]
- Ogden, R.W. Non-Linear Elastic Deformation; Courier Corporation: New York, NY, USA, 1997. [Google Scholar] [CrossRef]
- Gent, A.N. A new constitutive relation for rubber. Rubber Chem. Technol.
**1996**, 69, 59–61. [Google Scholar] [CrossRef] - Pucci, E.; Saccomandi, G. A note on the Gent model for rubber-like materials. Rubber Chem. Technol.
**2002**, 75, 839–852. [Google Scholar] [CrossRef] - Gent, A.N.; Thomas, A.G. Forms for the stored (strain) energy function for vulcanized rubber. J. Polym. Sci.
**1958**, 28, 625–628. [Google Scholar] [CrossRef] - Carroll, M.M. A strain energy function for vulcanized rubbers. J. Elast.
**2011**, 103, 173–187. [Google Scholar] [CrossRef] - Nguyen, H.D.; Huang, S.C. The uniaxial stress-strain relationship of hyperelastic material models of rubber cracks in the platens of papermaking machines based on nonlinear strain and stress measurements with the finite element method. Materials
**2022**, 14, 7534. [Google Scholar] [CrossRef] - Horgan, C.O.; Murphy, J.G. Incompressible transversely isotropic hyperelastic materials and their linearized counterparts. J. Elast.
**2021**, 143, 187–194. [Google Scholar] [CrossRef] - Emminger, C.; Cakmak, U.D.; Preuer, P.; Graz, I.; Major, Z. Hyperelastic material parameter determination and numerical study of TPU and PDMS Dampers. Materials
**2022**, 14, 7639. [Google Scholar] [CrossRef] - Herrmann, H. A constitutive model for linear hyperelastic materials with orthotropic inclusions by use of quaternions. Contin. Mech.
**2021**, 33, 1375–1384. [Google Scholar] [CrossRef] - Kawabata, S.; Matsuda, M.; Tei, K.; Kawai, H. Experimental survey of the strain energy density function of isoprene rubber vulcanizate. Macromolecules
**1981**, 14, 154–162. [Google Scholar] [CrossRef] - Hartmann, S. Parameters estimation of hyperelasticity relations of generalized polynomial-type with constraint conditions. Int. J. Solids Struct.
**2001**, 38, 7999–8018. [Google Scholar] [CrossRef] - Ogden, R.W.; Saccomandi, G.; Sgura, I. Fitting hyperelastic models to experimental data. Comput. Mech.
**2004**, 34, 484–502. [Google Scholar] [CrossRef][Green Version] - Bazkiaei, A.K.; Shirazi, K.H.; Shishesaz, M. A framework for model base hyper-elastic material simulation. J. Rubber Res.
**2020**, 23, 287–299. [Google Scholar] [CrossRef] - Portillo, F.J.S.; Sempere, O.C.; Marques, E.A.S.; Lozano, M.S.; da Silva, L.F.M. Mechanical characterisation and comparison of hyperelastic adhesives: Modelling and experimental validation. J. Appl. Comput. Mech.
**2022**, 8, 359–369. [Google Scholar] [CrossRef] - Sunyoung, I.; Wonbae, K.; Hyungjun, K.; Maenghyo, C. Artificial neural network modeling for anisotropic hyperelastic materials based on computational crystal structure data. In Proceedings of the AIAA Scitech 2020 Forum, Orlando, FL, USA, 6–10 January 2020. [Google Scholar] [CrossRef]
- Li, Y.; Sang, J.B.; Wei, X.Y.; Yu, W.Y.; Tian, W.C.; Liu, G.R. Inverse identification of hyperelastic constitutive parameters of skeletal muscles via optimization of AI techniques. Comput. Method. Biomec.
**2021**, 24, 1647–1659. [Google Scholar] [CrossRef] [PubMed] - Lopez-Campos, J.A.; Ferreira, J.P.S.; Segade, A.; Fernandez, J.R.; Natal, R.M. Characterization of hyperelastic and damage behavior of tendons. Comput. Method. Biomec.
**2020**, 23, 213–223. [Google Scholar] [CrossRef] [PubMed] - Hashemi, M.S.; Baniassadi, M.; Baghani, M.; George, D.; Remond, Y.; Sheidaei, A. A novel machine learning based computational framework for homogenization of heterogeneous soft materials: Application to liver tissue. Biomec. Model. Mechan.
**2020**, 19, 1131–1142. [Google Scholar] [CrossRef] [PubMed] - Mendizabal, A.; Marquez-Neila, P.; Cotin, S. Simulation of hyperelastic materials in real-time using deep learning. Med. Image Anal.
**2020**, 59, 101569. [Google Scholar] [CrossRef] - Shahani, A.R.; Shooshtar, H.; Baghaee, M. On the determination of the critical J-integral in rubber-like materials by the single specimen test method. Eng. Fract. Mech.
**2017**, 184, 101–120. [Google Scholar] [CrossRef] - Nair, A.U.; Taggart, D.G.; Vetter, F.J. Use of a genetic algorithm for determining material parameters in ventricular myocardium. In Proceedings of the IEEE 30th Annual Northeast Bioengineering Conference, Western New England Coll, Springfield, MA, USA, 17–18 April 2004. [Google Scholar] [CrossRef]
- Li, Q.; Zhao, J.C.; Zhao, B.; Zhu, X.S. Parameter optimization of rubber mounts based on finite element analysis and genetic neural network. J. Macromol. Sci. A
**2009**, 46, 186–192. [Google Scholar] [CrossRef] - Specht, D.F. A general regression neural network. IEEE Trans. Neur. Net.
**1991**, 2, 568–576. [Google Scholar] [CrossRef][Green Version] - Ding, W.F.; Alharbi, A.; Almadhor, A.; Rahnamayiezekavat, P.; Mohammadi, M.; Rashidi, M. Evaluation of the performance of a composite profile at elevated temperatures using finite element and hybrid artificial intelligence techniques. Materials
**2022**, 15, 1402. [Google Scholar] [CrossRef] - Yi, S.X.; Yang, Z.J.; Xie, H.X. Hot deformation and constitutive modeling of TC21 titanium alloy. Materials
**2022**, 15, 1923. [Google Scholar] [CrossRef] - Liu, Y.; Song, S.Y.; Zhang, Y.D.; Li, W.; Xiao, G.J. Prediction of surface roughness of abrasive belt grinding of superalloy material based on RLSOM-RBF. Materials
**2021**, 14, 5701. [Google Scholar] [CrossRef] - Chi, X.M.; Han, S. Effects of servo tensile test parameters on mechanical properties of medium-Mn Steel. Materials
**2019**, 12, 3793. [Google Scholar] [CrossRef][Green Version] - Wang, K.J.; He, B.; Chen, R.L. Predicting parameters of nature oil reservoir using general regression neural network. In Proceedings of the IEEE International Conference on Mechatronics and Automation, Harbin, China, 5–8 August 2007. [Google Scholar] [CrossRef]
- Huang, L.N.; Nan, J.C. Researches on GRNN neural network in RF nonlinear systems modeling. In Proceedings of the 2011 International Conference on Computational Problem-Solving, Chengdu, China, 21–23 October 2011. [Google Scholar] [CrossRef]
- Ding, S.; Chang, X.H.; Wu, Q.H. A study on approximation performances of general regression neural network. In Proceedings of the 3rd International Conference on Machinery Electronics and Control Engineering (ICMECE 2013), Jinan, China, 29–30 November 2013. [Google Scholar] [CrossRef]
- Parzen, E. On estimation of probability density function and mode. Ann. Math. Stat.
**1962**, 33, 1065–1076. [Google Scholar] [CrossRef] - Haines, D.W.; Wilson, W.D. Strain-energy density function for rubberlike materials. J. Mech. Phys. Solids
**1979**, 27, 345–360. [Google Scholar] [CrossRef] - Destrade, M.; Murphy, J.G.; Saccomandi, G. Simple shear is not so simple. Int. J. Non-Linear Mech.
**2012**, 47, 210–214. [Google Scholar] [CrossRef] - AbuShanab, W.S.; Abd Elaziz, M.; Ghandourah, E.I.; Moustafa, E.B.; Elsheikh, A.H. A new fine-tuned random vector functional link model using hunger games search optimizer for modeling friction stir welding process of polymeric materials. J. Mater. Res. Technol.
**2021**, 14, 1482–1493. [Google Scholar] [CrossRef]

**Figure 2.**The specimen and its geometry employed in a uniaxial tensile test: (

**a**) The image of silicone rubber specimen with bonded aluminum sheets; (

**b**) The dimensions of silicone rubber specimen and aluminum sheets.

**Figure 7.**Results comparisons of: (

**a**) M-R model; (

**b**) polynomial model (N) = 2; and (

**c**) Ogden model (N = 3) fitted by least-squares with experimental data.

**Figure 8.**Numerical results of force-displacement curves and experimental curve: (

**a**) M–R model results for learning samples of GRNN; (

**b**) Polynomial model (N = 2) results for learning samples of GRNN; (

**c**) Ogden model (N = 3) results for learning samples of GRNN.

**Figure 10.**(

**a**) Results obtained by using polynomial model (N = 2); (

**b**) The partially enlarged views for specific deformation stages based on polynomial model (N = 2).

**Figure 11.**(

**a**) Results obtained by using the Ogden model (N = 3); (

**b**) The partially enlarged views for specific deformation stages based on Ogden model (N = 3).

Model | Parameters | Least-Squares Method | Sample-1 | Sample-2 | Sample-3 | Sample-4 | Sample-5 |
---|---|---|---|---|---|---|---|

M-R model | C_{10} | 0.0385 | 0.0510 | 0.0210 | 0.3160 | 0.2898 | 0.2898 |

C_{01} | 0.4052 | 0.5270 | 0.4052 | 0.0420 | 0.0395 | 0.0455 | |

Polynomial model (N = 2) | C_{10} | −2.1506 | −1.2904 | −2.7958 | −2.1506 | −2.1506 | −1.9506 |

C_{01} | 2.7355 | 1.6413 | 3.5562 | 2.7355 | 2.7355 | 2.2535 | |

C_{20} | 2.1308 | 1.2785 | 2.7700 | 2.1308 | 2.1308 | 2.1308 | |

C_{11} | −6.7135 | −4.0281 | −8.7276 | −6.8000 | −7.0000 | −6.7135 | |

C_{02} | 6.3381 | 3.9029 | 8.2395 | 6.5000 | 6.8000 | 6.2530 | |

Ogden model (N = 3) | μ_{1} | −3.9450 | −3.4560 | −4.7340 | −3.1560 | −4.7340 | −3.9513 |

α_{1} | −2.3031 | −2.7637 | −1.8425 | −1.8425 | −2.7637 | −2.3068 | |

μ_{2} | −0.3774 | −0.3019 | −0.4529 | −0.4529 | −0.4529 | −0.3780 | |

α_{2} | −1.3540 | −1.6248 | −1.0832 | −1.6248 | −1.6248 | −1.3520 | |

μ_{3} | 5.4133 | 4.3306 | 6.4960 | 4.3306 | 6.4960 | 5.4057 | |

α_{3} | −3.8436 | −4.6123 | −3.0749 | −3.0749 | −−4.6123 | −3.8382 |

Model | Parameters | GRNN-Based Approach | Least-Squares Method |
---|---|---|---|

M-R model | C_{10} | 0.2393 | 0.0385 |

C_{01} | 0.1134 | 0.4025 | |

Polynomial model (N = 2) | C_{10} | −2.1505 | −2.1506 |

C_{01} | 2.7354 | 2.7355 | |

C_{20} | 2.1006 | 2.1308 | |

C_{11} | −6.6185 | −6.7135 | |

C_{02} | 6.2484 | 6.3381 | |

Ogden model (N = 3) | μ_{1} | −3.9516 | −3.9450 |

α_{1} | −2.3069 | −2.3031 | |

μ_{2} | −0.3780 | −0.3774 | |

α_{2} | −1.3507 | −1.3540 | |

μ_{3} | 5.4001 | 5.4133 | |

α_{3} | −3.8342 | −3.8436 |

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**MDPI and ACS Style**

Hou, J.; Lu, X.; Zhang, K.; Jing, Y.; Zhang, Z.; You, J.; Li, Q.
Parameters Identification of Rubber-like Hyperelastic Material Based on General Regression Neural Network. *Materials* **2022**, *15*, 3776.
https://doi.org/10.3390/ma15113776

**AMA Style**

Hou J, Lu X, Zhang K, Jing Y, Zhang Z, You J, Li Q.
Parameters Identification of Rubber-like Hyperelastic Material Based on General Regression Neural Network. *Materials*. 2022; 15(11):3776.
https://doi.org/10.3390/ma15113776

**Chicago/Turabian Style**

Hou, Junling, Xuan Lu, Kaining Zhang, Yidong Jing, Zhenjie Zhang, Junfeng You, and Qun Li.
2022. "Parameters Identification of Rubber-like Hyperelastic Material Based on General Regression Neural Network" *Materials* 15, no. 11: 3776.
https://doi.org/10.3390/ma15113776