A specific rubber bushing, with only radial and axial stiffness data having been acquired, is studied. In terms of the hyperelastic material of this bushing, threeterm Ogden law is utilized as the material constitutive model which requires to be characterized. Without the material mechanical tests provided, a parameter identification method is proposed for searching a group of acceptable parameters which are able to model rubberlike material of this rubber bushing. In this case, based on the nonlinear finite element analysis method and optimization technique, the parameters of material law are determined, and the rotational stiffness of this bushing is also evaluated. The complete stiffness information has been established.
Rubber is recognized as an engineering material since the vulcanization of rubber being pioneered by Charles Goodyear in 1839 [
Several methods are proposed at solving problems of parameter identification. Baker and Shrot [
In this paper, an inverse finite element method for parameters identification is introduced, which involves the stiffness information of a specific component. More specifically, we focus on the parameters identification for one of the material constitutive laws and the evaluation for rotational stiffness of one certain rubber bushing.
Generally, vulcanized rubber is assumed as an incompressible hyperelastic material [
Li and Yang [
The Ogden strain energy potential is expressed in terms of the principal stretches (
The Ogden model, (
Ogden constitutive law is considered one of the most successful functions in describing the large deformation range of rubberlike materials. In this paper, therefore, the constitutive equation of threeterm Ogden, with six coefficients, is utilized to define the property of this rubber material.
In this paper, based on the combination of the nonlinear FE method and optimization technique, the parameters of hyperelastic material will be defined to describe the mechanical characteristics of rubber bushing. Meanwhile, the complete stiffness information about rubber bushing can be obtained.
During the identification process, two FE simulations are performed to replicate the radial and axial stiffness tests. The parameters of the material constitutive law are determined to fit the experimental result. Firstly, a set of initial values are assigned to the parameters of constitutive law. Simulate the translational stiffness of rubber bushing described by the foregoing parameters. If this result fails to fit the experimental data, an optimization procedure needs to be conducted to find the appropriate parameters which are able to model an effective hyperelastic material achieving to describe the mechanical characteristics of rubber bushing. Figure
Typical steps in parameters identification cycle.
The axial and radial stiffness tests were conducted due to the axisymmetrical structure of rubber bushing and limited test equipment. As shown in Figure
CAD model of rubber bushing.
The translational stiffness tests provide axial and radial stiffness information which will be utilized as the standard data in the next parameter identification process.
In this work, a CAD model of rubber bushing is imported to the FE software. This model is meshed to finite elements, material properties and appropriate boundary conditions are set subsequently to acquire a completely rubber bushing FE model. To simulate the radial and axial stiffness data, the software ABAQUS is used. This process is the preparation for the next stage—parameter identification.
More specifically, two load steps are set as the boundary conditions to fully assess the models fit. In the first step, a forced displacement was applied on the outer surface of bushing simultaneously intermediate shaft fixed. This is a preloaded step for fixing the bushing and establishing nonlinear contact steadily. In the second step, the loads were applied on the pipe around the center hole, incrementally from 0.0 N to 1000 N in
Figure
FE model of rubber bushing.
The simulations were carried out for the two test described earlier: a radial stiffness test and an axial stiffness test. The simulated versus experimental displacementstress curvilinear relationship is shown in Figure
The initial loaddisplacement curves.
Parameters of the constitutive equation of rubber are determined based on the combination of translational stiffness test data and finite element optimization techniques. When the results of simulation present apparent deviations from test data, the coefficients of the constitutive model should be redefined. This process can be completed by optimization software.
The six coefficients (
The values of design variables.
Design variables  Lower bound  Upper bound 


0.5  2.0 







0.6  1.6 

10  13 

−5  −15 
Two FE simulations are performed to replicate the radial and axial stiffness tests. If the simulated curves overlap the experimental counterpart, it means that the parameters of the constitutive equation having been selected could model this rubberlike material of rubber bushing accurately. To minimize the gap between simulated loaddisplacement curves and the experimental data, four subobjectives are proposed as follows.
Since the experimental and simulated curves are linear ranging from the minimum to the maximum value of the loads applied in this research, the above four subobjectives are able to make the simulated curves overlap the experimental counterpart and then replicate the radial and axial stiffness tests.
In a usual optimization problem, only one design object should be defined, while in this specific case four subobjectives are involved in the identification process. A weighted equation is employed, therefore, to integrate the four goals mentioned above into one general objective:
Genetic algorithm [
Histories of design variable.
Having experienced 10 iterations, the result converges. Then it is extracted from the simulated model with parameters of constitutive model characterized simultaneously. The simulated versus experimental displacementstress curvilinear relationship is shown in Figure
The coefficient of material law (after optimized evaluation).
Constitutive equation 



Ogden 







The loaddisplacement curves (after optimization).
Figure
Due to the limited amount of experimental appliances, the rotational stiffness tests are not conducted. Instead, they will be obtained through simulated software based on FE theory. The rotational stiffness information can be obtained by applying torques incrementally from 0 N·mm to 5000 N·mm around
Rotational stiffness curve (around
Rotational stiffness curve (around
In this study, the parameters of the threeterm Ogden law used for modeling hyperelastic material have been identified. A nonlinear simulation is performed to replicate experimental radial and axial stiffness test for a specific rubber bushing, while the parameters of the material law are determined to fit the experimental results. Comparing the experimental and simulated data, only petty errors are found which indicates that the parameters of the constitutive equation are valid for modeling the material of this rubber bushing.
Subsequently, an accurate model is established relying on the determinate parameter of the material law in order to evaluate two rotational stiffness curves which have not been obtained due to the lack of rotational test data. In this procedure, the complete information about stiffness of rubber bushing can be acquired.
This parameter identification method is proved to model the appropriate hyperelastic material for rubber bushing validly when material tensile tests data are not provided. Additionally, the whole information about rubber bushing stiffness can be established practically.
This work was funded by the 2012 openended foundation of Key Laboratory of Advanced Manufacturing Technology for Automobile Parts, Ministry of Education by Project of cstc2013yykfB0198.