# Fatigue Life Prediction of Radial Tire Bead Using a Maximum Strain Energy Density Range Method
## 4 Finite Element Analysis  
### 4.1. Tire Modelling Details  
In this section, the developed tire model is described. A 175/75R14 tire was taken as the subject for this research. This research obtained the tire cross-section from the real tire of the 175/75R14 tire, as shown in Figure 8. The cross-sectional raster image was acquired by a CNC432 tire cross-section scanner, and then imported into AutoCAD software to divide each part. The scanner is shown in Figure 9. Hypermesh was utilized to establish a two-dimensional finite element mesh model, and refine the mesh for the  
Figure 6: \(\Delta SED\)–\(N_{f}\) curve of apex.  
Figure 7: \(\Delta SED\)–\(N_{f}\) curve of abrasion.  
bead and other vulnerable areas. The size of the elements in the bead region was 2 mm. The rubber components were defined by Hyperelastic elements (CGAX3H and CGAX4H). The cord-rubber composite materials were represented by the element Rebar. The steel cord components in the bead region were meshed using a SFMGAX1 element. The two-dimensional model was in a plane strain state. The 2D finite element model is shown in Figure 10.  
Tires are mainly composed of rubber and cord-rubber materials, with typical nonlinear stress-strain behavior. Hence, the uniaxial tension tests were performed to determine the rubber hyperelastic material model [16]. The constitutive relationship curve for each rubber compound is shown in Figure 11.  
Figure 8: Cross-section of 175/75R14 tire.  
Figure 9: Tire cross-section scanner.  
Abaqus provides several commonly used constitutive models, including the Neo-Hookean model, Yeoh model, Mooney-Rivlin model, Ogden model, Arruda-Boyce model, etc. Although the Neo-Hookean model [17] includes only one material constant, under large strain, the strain energy value predicted by the model is inconsistent with the measured value. The Mooney-Rivlin model [18] is very convenient to use; however, people find that it cannot predict multi-axial data. [PERSON] [19] indicated that the Ogden model and the Mooney-Rivlin model are essentially equivalent. The Arruda-Boyce model [20] is complex and not widely used. Due to the large deformation of the grounding part during the tire driving process, the Yeoh model has the advantage of being able to cover a large range of deformation, and its accuracy and stability are still relatively reliable under large deformations. Therefore, the Yeoh constitutive model was used for modelling the hyperelastic property of rubber compounds. The Yeoh strain energy function is expressed as Equation (5).  
\[U=C_{10}(I_{1}-3)+C_{20}(I_{1}-3)^{2}+C_{30}(I_{1}-3)^{3} \tag{5}\]  
where \(U\) is the Yeoh strain energy function, \(I_{1}\) is the first strain invariant, and \(C_{10}\), \(C_{20}\), \(C_{30}\) are modulus parameters of rubber compounds, which can be obtained by fitting the experimental data [21]. The fitting curve of the abrasion compound simulated by Yeoh model is shown in Figure 12. As seen from the figure, the Yeoh model has some errors in the small strain interval, but it is in good agreement with the experimental results in the medium and large strain intervals.  
Figure 11: Stress–strain response curve of different rubber compounds.  
Figure 10: 2-dimensional finite element model.