## 3 Materials and Methods
### Complex Finite Element Model of the Rubber Tyre
#### 3.2.1 The 2D Axial Symmetric Model  
The finite element analysis (at the fulfilment of certain conditions) provides a possibility for a planar analysis of 3D bodies, resulting in significant reductions in the required computational time, space, and hardware capacity. For the modelling of the inflation of  
Figure 2: The analysed domain of the road pavement structure and its finite element mesh. the tyre and its stretching to the body of the wheel disk, it is sufficient to solve a planar axial symmetric problem, meaning that only the radial cross-section of the tyre has to be modelled. The geometry of the structure of the tyre was reconstructed based on the cross-section drawing of the DTA 275/80R22.5-type tyre published by [5] (p. 21), with the overall width of the tyres being 288 mm and the dual spacing of the tyres being 315 mm. The composite structure of the tyre is illustrated in Figure 3, the geometric values are presented in Table 3, and the material parameters are collected in Table 4, based on a literature review. The material parameters, as [PERSON]'s hyper elastic, viscoelastic, and rebar, were collected from the works of [17; 18]. The experimental determination of the material parameters of the tyre has been well summarised by [19]. The ABAQUS software prescribes the position of the cylindrical co-ordinate system, required to the definition of the axial symmetrical problem, where the x-axis is the radial axis, and the y-axis coincides with the z-axis. According to this, the cross-sectional model of the tyre is defined on the xy-plane (Figure 3b). In ABAQUS software, there is no need for special commands to define the axial symmetric problem, but, necessarily, the mesh can be constructed only from axial symmetric elements. The following element types were used for the tyre model:  
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* CAX4 (4-node bilinear continuum element) for linear elastic or elastic-plastic materials;
* CAX4H (4-node bilinear hybrid continuum element) for hyper elastic materials;
* SFMAX1 (2-node linear membrane-like surface element) for rebar. Surface elements are, by their appearance, membrane elements with no thickness. They are usually applied for defining surplus weight or distributed loading on the surface, but they are also useful for the simplified consideration of the rebar of the tyre. A great advantage of the SFMAX1 surface element is that the rebar has to be modelled only in the radial section, and the cross-section of the rebar can be given by parameters that are independent of the finite element mesh. A further advantage is that the surface elements do not need to join with nodes to the other elements of the finite element mesh; these can be defined as an embedded element. In the tyre model, the material of the rubber is the \"host element\" assigned to the rebar. The clamping of the axial symmetric model is at the intersection of the axis of the rotation of the wheel and the axis of symmetry of the tyre (Figure 3b). Some degrees of  
Figure 3: Cross-section view of radial tyre published by [5] (**a**); finite element mesh of the axial symmetric tyre model (**b**). freedom (DOFs) of the clamping point have to be constrained, namely, the 2 (U2) and the 6 (UR3) degrees of freedom, in order to obtain no displacement along the axis of rotation; moreover, the radial cross-section cannot turn around its plane. The cross-section of the tyre model in Figure 3a reflects the state before the inflation and the nesting onto the body of the wheel. The interaction between the body of wheel and the tyre was worked out based on the tear analysis of the rebar published by [20]. At the beginning of the simulation, a tensile stress value of 30 MPa was given in the strands at a longitudinal direction, which provided a fine nesting of the tyre rim onto the body of the wheel. The tyre rim on the body of the wheel may slip during inflation (here, a friction coefficient of 0.3 was chosen); moreover, in case of a radical surplus pressure, the rim may even detach from the body of the wheel.