## 3 Numerical Simulations of an Aircraft Tire Rolling on a Steel Drum  
A 3-D numerical model of the tire previously developed for tire/runway studies [5; 6; 7; 8] is employed. The aim of this part is to test and extend the capabilities of this 3-D model to predict the thermal evolution of the tire tread during rolling at high velocity. Figure 12 illustrates the tire 3-D FE model built-up. In this part, the tire cross-section was cut out from the tire using a water-jet, as shown in Figure 12a. This method gives an accurate cross section and a detailed image of all the inner layers of the tire structure (rubberized fabric layers, steel belts, nylon cords, etc. ), Figure 12b. An image processing method was used to discretize the cross-sectional image and to obtain the 2-D FE section of the tire. The finite element analysis was carried out in several stages. In the first stage, the tire was modeled using an axisymmetric 2-D model and inflated at 1700 kPa. A 3-D model (Figure 12d) was then developed for performing the rolling of the tire (dynamic simulation). The 3-D tire model is generated by using the symmetric model generation technique from the 2-D axisymmetric model. The 3-D model is composed of 51 circumferential segments, coupled temperature-displacement elements (C3D8 RHT, eight-node tri-linear displacement and temperature, reduced integration with  
Figure 11: Experimental test: (**a**) thermal evolution of the tire tread rolling on a drum, (**b**) image of the contact zone. hourglass control, hybrid with constant pressure) were used in this simulation. The steel drum of 3 m in diameter was modeled as a 3-D deformable body. The friction coefficient implemented in the numerical simulation is based on experimental tests [15]. A classical rolling (CR) method based on the Lagrangian formulation was used with a fixed friction coefficient. The numerical simulations were performed using ABAQUS/Standard solver. The hyper-elastic behavior of rubber is based on the Mooney-Rivlin model [16; 17]. The viscoelastic parameters of rubber have been determined experimentally. The rubber viscoelastic behavior is modeled using the Prony series model, widely employed in tire modeling [18; 19; 20]. The rubber is considered as a homogenous, isotropic and incompressible material. Concerning the thermal parameters of the rubber material, the thermal conductivity was 0.20 W/m \({}^{\circ}\)C, the specific heat was 1900 J/(kg.\({}^{\circ}\)C) and the coefficients of expansion were 77e\({}^{-6}\)\(\times\) 1/\({}^{\circ}\)C for 20 \({}^{\circ}\)C and 150e\({}^{-6}\)\(\times\) 1/\({}^{\circ}\)C for 200 \({}^{\circ}\)C.  
The major numerical difficulties are related to the highly non-linear model of the tire, large deformation, incompressibility of elastomers, and specific boundary conditions of the tire/drum contact. The contact between the tire and the drum is modeled as a softened contact relationship [21], in which the contact pressure is an exponential contact pressure-overclosure relationship. The surfaces begin to transmit contact pressure once the clearance between them, measured in the contact normal direction, reduces to a sufficiently small distance C\({}_{0}\). The contact pressure transmitted between the two surfaces increases exponentially as the clearance continues to diminish. Figure 13 illustrates this behavior. The value of the clearance C\({}_{0}\) and the contact pressure P\({}_{0}\) were determined after a numerical sensitivity analysis, C\({}_{0}\) = 0.002 m and P\({}_{0}\) = 2000 kPa. The numerical simulation of tire-drum test is performed in several steps according to the experimental protocol. The tire was preloaded at 1\({}_{1}\) kN. The numerical simulations are done taking into account the high geometrical deformations of the tire and its hyper-elastic materials. The next Figure 14 shows the tire position on the drum at slip angles at 0\({}^{\circ}\) and 6\({}^{\circ}\). The skidding angle is imposed before the drum starts rotating. Figure 12: Tire 3-D finite element model built-up from the 2-D tire cross section.