Modeling and Verification of a New Hyperelastic Model for Rubber-Like Materials

The essential evaluation criterion for the hyperelasticmodel is its ability to describe themechanical behavior of rubber-likematerials under different deformation modes over a large deformation range accurately. Based on the Seth strain tensor invariant, a new hyperelastic model for isotropic and incompressible rubber-like materials is proposed. In order to investigate the prediction ability of the new model, the parameters of the new model, the Yeoh model, and the Carroll model are identified by test data of 8% vulcanized rubber and two different types of carbon black filled rubber, respectively. To this end, the data of uniaxial tension and equibiaxial tension are used simultaneously. Then, the same set of model parameters is used for prediction of pure shear (plane tension) deformation. The results show that the new model not only can predict the test data of pure shear (or plane tension) accurately, but also can be reliable to describe the response of various rubber materials over a large deformation range. Finally, the finite element simulation and experiment on static stiffness of rubber bushing are carried out based on the new model. By comparison of the experimental data with the simulation data, the new model can accurately reflect the mechanical behavior of rubber bushing. The new model can be used for performance analysis of rubber products and has better application value.


Introduction
Rubber materials are used in engineering fields widely, such as tires, rubber tracks, vehicle seals, and vibration-isolation devices [1][2][3].Without considering the time effect, rubberlike materials are generally considered to be isotropic and incompressible hyperelastic materials.It is of great significance to study the performance of rubber products to establish a reliable hyperelastic model which can accurately reflect mechanical behavior of rubber-like materials.The essential evaluation criterion for the hyperelastic model is its ability to reproduce the mechanical behavior of rubberlike materials in different deformation states over a large range accurately.The researches of Hossain et al. [4,5] on representative hyperelastic models show that their model cannot reliably predict mechanical behavior of rubber-like materials in all deformation modes by using only a certain type of test data.In addition, these models cannot also perfectly reproduce the stress-strain relationship over a large range In recent years, many scholars have analyzed the disadvantages of existing hyperelastic models and developed many new hyperelastic models based on the work of predecessors.Hossain [6] proposed a three-dimensional electroelastic constitutive framework that can model the stiffness gaining during the curing process undergoing finite deformations.By improving Mooney-Rivlin model, Yaya and Bechir [7] proposed a new compressible hyperelastic model with four parameters, and the response quality of this model is equivalent to that of the Ogden six-parameter model.In contrast to the traditional model, Crespo et al. [8] proposed a new hyperelastic model by What-You-Prescribe-Is-What-You-Get formulations.The form of Crespo model is not assumed beforehand and there are no material parameters.Vu et al. [9] proposed a micromechanical model for rubber elasticity on the basis of analytical networking-averaging of the tube model and by applying a closed-form of the Rayleigh exact non-Gaussian chains.Based on Lie group methods, Zhao [10] postulated a partial differential equation for isotropic hyperelastic constitutive models.Nkenfack et al. [11] proposed a new approach named Hybrid Integral Approach to model the incompressible isotropic hyperelastic behavior of rubber-like materials.Based on the Arruda and Boyce model, this model includes an original part made of an integral density and an interleaving constraint part represented by a logarithmic function.However, this model contains six material parameters and the application is limited.Aiming at the deficiency of the Arruda-Boyce model in predicting the equibiaxial data, Hossain et al. [12] compared five modified versions of the Arruda-Boyce model and two modified versions of full-network model with the Arruda-Boyce model.Bahreman and Darijani [13] proposed a new polynomial hyperelastic model which is a function of the principal invariants of the left Cauchy-Green strain tensor.Based on the neo-Hookean model, Bechir et al. [14] proposed a new strain energy function, which can predict the test data in multiple deformation modes by only using uniaxial tensile test data.However, this model only can be reliable to describe the response of rubbers in the small deformation range.Horgan et al. [15] compared the Fung-Demiray model and the Vito model to the stress response quality in various deformation modes.They believe that the hyperelastic model with the second invariant of the right Cauchy-Green deformation tensor can more accurately reflect the mechanical behavior and some physical effects of rubber-like materials.Under the incompressible assumption, Lin [16] derived the mathematical relationship between the right Cauchy-Green deformation tensor principal invariants and used the interpolation algorithm to obtain a new hyperelastic model that can predict stressstrain relationships by only using data of uniaxial tension and equibiaxial tension.Based on the logarithmic strain tensor invariants, Xiao et al. [17] proposed an explicit method for constructing multiaxial elastic potential only through test data in uniaxial deformation mode.Based on the research of Xiao, Yu et al. [18] proposed a new hyperelastic model, which is suitable for general compressible deformation of rubber-like materials, and the parameters have direct physical meaning.
Compared with the classical model, the hyperelastic model proposed in recent years can generally better reflect the mechanical behavior of rubber-like materials in various deformation modes; while the constitutive equations are more complicated, parameters are more and derivation methods are more cumbersome.These defects limit the application of the models.In view of the above problems, a new hyperelastic model is proposed based on the Seth strain tensor invariant.The basic test data of 8% vulcanized rubber and NR55 carbon black filled rubber are used for parameters identification.It seems that the data of uniaxial tension and equibiaxial tension are sufficient to obtain prefect model parameters.Finally, the finite element model of rubber bushing is established based on the new model by ABAQUS software.By comparing experimental data with simulation data, the static stiffness characteristics of bushing are analyzed to examine the applicability of the new model furthermore.

The General Forms of the Hyperelastic Constitutive Model
Based on the hyperelastic theory [19], the stress-strain relationship of rubber-like materials can be derived from the strain energy function .The relationship between the first Piola-Kirchhoff stress tensor  and the deformation gradient tensor  is as follows.
Under the assumption of isotropic and isothermal materials, (1) can be written as where   = /  ( = 1, 2, 3) and   ( = 1, 2, 3) are the principal invariants of the right Cauchy-Green deformation tensor  =   .In general, rubber-like materials are considered to be incompressible; then (2) can be written as where  is the Lagrangian multiplier which can be eliminated by the boundary conditions in different deformation modes.After eliminating , the corresponding stress-strain relationship can be determined by (3).
Nowadays, three kinds of tests, namely, uniaxial tension, equibiaxial tension, and pure shear (or plane tension), are usually used to determine the mechanical behavior of rubberliked materials.Under the assumption that the base vector of the Cartesian coordinate system is ( 1 ,  2 ,  3 ), the deformation gradient  is as follows for uniaxial tension, equibiaxial tension, and pure shear (or plane tension), respectively: where  is the stretch ratio.After eliminating  by the boundary conditions under different deformation modes, the relationship between nominal stress and stretch ratio  can be written as follows.

Hyperelastic Constitutive Model Based on Seth Strain Tensor
The stress-strain relationship of rubber-like materials is significantly nonlinear over the large deformation range.
The hyperelastic models which do not contain correlation term of the second invariant I 2 cannot accurately reproduce the test data in multiaxial deformation state [15,20].Based on the invariants of Seth strain tensor, a highly nonlinear hyperelastic model is established in this paper, which can accurately reflect the mechanical behavior of rubber-like materials in various deformation modes over the large range.
When  ̸ = 0, the expression of the Seth strain tensor  () is defined as follows.
According to the polar decomposition theorem, the deformation gradient tensor  can be written as where  and  are right and left stretch tensors, respectively, and  is the rotation tensor.The eigenvalues of  and  are the principal stretches   ( = 1, 2, 3).According to spectral decomposition,  can be written as where   ( = 1, 2, 3) are orthogonal eigenvectors of .Therefore, (6) can be expressed as follows.
The first invariant of Seth strain tensor  () 1 is as follows.
Following the approach of Bechir et al. [14], the first invariant of the Seth strain tensor is chosen to generalize the strain energy function.Expanding  in powers of  ()  1 ( () ), the following function can be obtained: where  takes an arbitrary integer that is not 0. Obviously, (11) satisfies the Valanis-Landel hypothesis [21].The relationships between the principal invariants of the right Cauchy-Green deformation tensor  and the principal stretches are as follows.
When  takes positive and negative values simultaneously, the strain energy function  can more accurately reflect the influence of each invariant on the mechanical behavior of rubber-like materials [22].Higher order terms help to improve the goodness of fit under large deformation conditions [23].Considering the above factors, the strain energy function  is defined as follows.
Based on ( 12), the following expressions can be obtained.
Inserting ( 16) into ( 15), the strain energy function  can be written as a function of the invariants of the right Cauchy-Green strain tensor.

Model Parameters Identification
The parameters of the hyperelastic model are usually determined by basic tests such as uniaxial tension, equibiaxial tension, and pure shear (or plane tension).When the parameters are identified by data of uniaxial tension and equibiaxial tension, the model can more accurately reflect the stressstrain relationship for all deformations [24].Considering that the new strain energy is the function of invariants, in order to examine its performance, Yeoh model [25] as (18) and Carroll model [24] as (19) are selected as the comparison object.
Firstly, the test data of 8% vulcanized rubber reported by Treloar [26] are used for parameters identification.It should be emphasized that only the data of uniaxial tension and the equibiaxial tension are used simultaneously in the process of parameters identification.Then, the same values of model parameters are used to predict the stress-strain relationship in pure shear state.Curve fitting is performed by Universal   Global Optimization in the 1stOpt software.The parameters values of each model are shown in Table 1.The fitting and prediction results are shown in Figure 1.
In order to evaluate the goodness of fit of each model, the coefficients of determination R 2 are calculated [27]: where  err = ∑  =1 ( P −  ) 2 ,  tot = ∑  =1 (  − ) 2 ,   is the test values, P is the model fit values,  is the average values of   , and N is the number of test data.In order to save space, the goodness of fit of each model is shown in Figures 1-3.
It is obvious that both of the new model and Carroll model are better than the Yeoh model for reproducing and predicting the test data of 8% vulcanized rubber.In particular, the Yeoh model exhibits a "softer" property when fitting the equibiaxial tension data.This is maybe due to the lack of the second invariant I 2 in the Yeoh model [28].According to analysis of Carroll [24], Treloar's data for uniaxial tension and equibiaxial tension are suitable for developing strain energy function, because the stretch values are large, =7.6 in uniaxial tension and =4.45 in equibiaxial tension.However, not all hyperelastic materials have this characteristic, for example, some carbon black filled rubber materials.
Based on the above analysis, test data of two different carbon black filled rubber materials are selected to further investigate the application scope of the new model.The formula of the first carbon black filled rubber is shown in Table 2 and the test data are shown in Figure 2 [29].The other is NR55 carbon black filled rubber and the test data are shown in Figure 3 [30].Likewise, the data of uniaxial tension and equibiaxial tension are used simultaneously in the process of parameters identification.Then, the same values of model parameters are used to predict the stress-strain relationship in pure shear state.The parameters values of each model are shown in Table 1.The fitting and prediction results are shown in Figures 2 and 3.
It is obvious that the stretch values of two carbon black filled rubber materials are less than that of 8% vulcanized rubber.Although the Yeoh model can keep the curves "S" shaped in all deformation states, its accuracy is worse than that of the Carroll model and the new model.Comparing the Carroll model and the new model, one finds that the curves shapes of both models are similar.By analyzing the two models, it can be found that both models contain highorder terms of invariant I 1 , and the orders of invariant I 2 are different.The order of the I 2 terms in the Carroll model is 1/2, and the highest order of the I 2 terms in the new model is 2.Although results of the Carroll model and the new model are all satisfactory, the new model can be reliable to reproduce and predict the behavior of two carbon black filled rubber materials with a better approximation.The reason may be that the Carroll model is developed based on Treloar's data, so it is more suitable for rubber materials with large stretch values.Therefore, the new model can accurately reflect the mechanical behavior of rubber-like materials for large and small strain and has a wider range of applications.

Model Verification by Static Stiffness Test of Rubber Bushing
In order to investigate the prediction power for mechanical behavior of rubber-like materials, the finite element model of rubber bushing is established based on the new model by the subroutine UHYPER of ABAQUS software.The static  stiffness of rubber bushing is studied along with comparison between simulation and experimental results.
The object is the McPherson suspension comfort bushing, in which the rubber material is NR55 carbon black filled rubber.The bushing consists of an inner steel ring, an outer steel ring, and rubber which is bonded to steel rings by vulcanization.The finite element model of the rubber bushing is shown in Figure 4.
The hyperelastic material is defined by the subroutine UHYPER of ABAQUS based on the new model whose parameters are the same as in Table 1 The inner and outer steel rings are set as discrete rigid, using R3D4 units.The rubber is set as deformable body, using C3D8H hybrid units.In order to simulate the actual working condition, the contact surfaces between steel rings and rubber are set to tie constraint.
The model is applied with an axial load of 1000N, a radial load of 3000N, and a rotational load of 16 ∘ around Y-axis, respectively.The simulation results are shown in Figure 5.
The static stiffness of rubber bushing is tested by MST 831 stiffness testing instrument with a loading rate of 0.01mm/s and 0.1 ∘ /s, respectively, as shown in Figure 6.In order to avoid the influence of the Mullins effect, the loading-unloading process is repeated four times, and then the fifth loading curve is taken as the final results.
It can be seen from Figure 7 that the radial stiffness curve and the axial stiffness curve are in good agreement with the experimental data.The maximum error is 7.10% and 4.75%, respectively.The torsional stiffness curve is slightly larger than the experimental data, and the maximum error is 19.40%.The reason for the errors is mainly that the vulcanization viscosity effect between the steel rings and rubber is neglected in order to reduce simulation difficulty.Conclusively, the new model can be applied to the static stiffness analysis of rubber bushing and is reliable to reflect the mechanical behavior of bushing.

Conclusion
In this paper, a hyperelastic model for an isotropic and incompressible rubber-like materials is proposed based on the Seth strain tensor invariants.The new model is a function of the invariants of the right Cauchy-Green deformation tensor, satisfying the Valanis-Landel hypothesis.In order to investigate the prediction ability and applicability, the Yeoh model and the Carroll model which are based on invariants are selected as comparison objects, and the test data of 8% vulcanized rubber and two different types of carbon black filled rubber are used for parameters identification.The new model has a better prediction ability for different rubber materials than the Yeoh model and the Carroll model.The reason may be that the new model contains high-order terms of invariants I 1 and I 2 , while the Yeoh model does not contain the terms of I 2 and the Carroll model does not contain the high-order terms of I 2 .In addition, one advantage of the new model is that the satisfactory parameters can be identified by using uniaxial tension and equibiaxial tension data simultaneously.
Furthermore, the finite element simulation and experiment for the static stiffness of rubber bushing are studied based on the new model.The maximum error of radial stiffness, axial stiffness, and torsional stiffness is 7.10%, 4.75%, and 19.40%, respectively.The efficiency of the new model in finite element simulations is verified.The new model can be applied to the static stiffness analysis of rubber bushing and is reliable to reflect the mechanical behavior of bushing.

Figure 1 :
Figure 1: Comparison of test data of 8% vulcanized rubber with results of each model: (a) uniaxial tension data and fitting results; (b) equibiaxial tension data and fitting results; (c) pure shear data and predicting results.

Figure 2 :
Figure 2: Comparison of test data of the first carbon black filled rubber with results of each model: (a) uniaxial tension data and fitting results; (b) equibiaxial tension data and fitting results; (c) pure shear data and predicting results.

Figure 3 :
Figure 3: Comparison of test data of NR55 carbon black filled rubber with results of each model: (a) uniaxial tension data and fitting results; (b) equibiaxial tension data and fitting results; (c) pure shear data and predicting results.

Figure 4 :
Figure 4: The finite element model of the rubber bushing.

Figure 5 :
Figure 5: (a) The simulation result of radial stiffness; (b) the simulation result of axial stiffness; (c) the simulation result of torsional stiffness.

Figure 6 :
Figure 6: The radial stiffness and torsional stiffness test.

Table 1 :
The parameters values of each model for different rubber materials.

Table 2 :
Material formula of the first carbon black filled rubber.