# Bias Truck Tire Deformation Analysis with Finite Element Modeling
## 2 Description of Bias Tire Model  
For analysis of pneumatic bias tire mounting on the rim and its inflation, the axisymmetric finite element model was employed. The analysis was based on the tire meridional cross-section with a plane containing the tire axis of revolution, also referred to as the tire profile (Figure 1). The tire profile is subject to discretization with axisymmetric finite elements. Each finite element node has two degrees of freedom--axial \(x\) and radial \(y\) displacement--but in each Gaussian point of the element, in the coordinate system \(x\), \(y\), \(\varphi\) (\(\varphi\)--angular coordinate, measured around the tire rotation axis), there are three normal stresses and one shear stress. When analyzing tire deformation, large displacements occur, thus a non-linear mechanics problem was considered with total Lagrangian formulation [19]. Therefore, for description of deformation and stresses, the energetically conjugate Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor were used. Their components are invariant against the rigid body motion. As an initial configuration, a tire profile in vulcanization mold was assumed immediately following the end of the vulcanization process, as, in this condition, the tire geometry is uniquely determined. The cord-rubber composite used in pneumatic tire building is a specific one owing to two main reasons. 1. The rubber ply, reinforced with cords, is characterized by a high degree of anisotropy. The composite matrix, i.e., rubber, is a hyperelastic material, and the cord has high tensile stiffness. By assuming the quotient of initial, tangent Young moduli for cord and rubber as composite anisotropy indicators, it amounts to ca. 640 for a nylon-rubber combination and ca. 6000 for a steel-rubber combination. For typical composites with a rigid matrix, this indicator amounts to 20-70 (glass/graphite-epoxy resin). 2. In the tire forming stages, bead wires are moved towards the tire symmetry plane, and the body ply takes toroidal shape. As a result, the cord density and the cord angle (with exception of radial tire body ply) with the toroid parallel lines in a tire profile location become the function of the location distance to the tire rotation axis (Figure 2). In the layer arranged on the tire building drum, the cord density \(epdm_{0}\) (ends per decimeter) and the cord angles \(\theta_{0}\) to drum surface meridians are constant. As a result of \"lifting\" of the layer from the  
Figure 1: Meridional bias truck tire profile. a—crown, b—shoulder, c—sidewall, d—bead; 1—3 x \(\pm\)\(\theta\) doubled body piles, 2—flipper strips, 3—chafer, 4—bead wires, 5—bead fillers, 6—sidewall, 7—tread. building drum to the profile in the mold, the values of these two parameters vary along the profile of the cord ply. The lowest cord density _epdm_ is in the tire crown area, and the highest is in the vicinity of bead wire. Similar variability can be observed for the cord angle \(\theta\) in the layer of formed tire (Figure 3). Equations for effective material constants of cord-rubber composite ply, considered as transversely isotropic, two-dimensional continuums, can be found in the literature, e.g., in [20]. However, due to the first characteristic feature given for such a composite, using material constants defined this way in the tire computational model results in lack of convergence of the solution that is searched for. Thus, in tire analysis, overlaying reinforcement elements onto rubber elements is commonly used. The elements have the same nodes and occupy the same geometrical space, but the materials have distinctly described stress-strain relationships. As in the paper [17], here, the determination of reinforcement material constants was adopted from the Halpin-Tsai equations, with the assumption of a very small value of Young modulus for rubber. The rubber elements were assigned the characteristics of a hyperelastic and incompressible neo-Hookean material.