# A Simulation of Tire Hydroplaning Based on Laser Scanning of Road Surfaces
## 3 Test Methods
### Simulation Model
#### 3.2.1 Tire Modeling
Therefore, this paper adopts the Neo-Hookean model to describe the superelastic characteristics of tire rubber materials, with its strain energy relationship given by:  
\[\text{U}=\text{C}_{10}(\text{I}_{1}-3)+\frac{1}{\text{D}_{1}(\text{J}-1)} \tag{4}\]  
Figure 8: Tire structure composition. In the formula, U is the strain potential energy, C10 is the material constant, I1 is the first invariant, J is volume change, and D1 is the material parameter, which represents the incompressibility of the material. The Neo-Hookean model parameters and material density [37], obtained from the relevant test results of rubber materials, are shown in Table 4.  
At the same time, because rubber materials have the characteristics of super elasticity and viscoelasticity, it is necessary to choose an appropriate viscoelastic constitutive model to simulate tire rubber materials. The constitutive model selected in this paper is the genetic gene model [35], and its constitutive model relationship is shown in Equation (5) [37]. \[\sigma=\int_{0}^{1}2\mathrm{G}(t-\tau)\frac{\mathrm{d}e}{\mathrm{d}\tau} \mathrm{d}\tau+\mathrm{I}\int_{0}^{1}\mathrm{K}(t-\tau)\frac{\mathrm{d} \Delta}{\mathrm{d}\tau}\mathrm{d}\tau \tag{5}\]  
In the equation, \(\sigma\) is the Cauchy stress, G (t) is the shear relaxation kernel function, K (t) is the volume relaxation kernel function, e is the shear strain, \(\Delta\) is the volume deformation strain, t is a certain time, and \(\tau\) is the past time. The viscoelastic behavior of rubber materials is described using a generalized Maxwell model, in which the time-dependent shear and bulk relaxation moduli, G(t) and K(t), are expressed as Prony series expansions (Equations (6) and (7)). This formulation is widely used in commercial finite element software such as Abaqus (Version 2021(r)) to represent linear viscoelasticity in the time domain [37]. \[\mathrm{G}(t)=\mathrm{G}_{\infty}+\sum_{i=1}^{n_{\mathrm{C}}}\mathrm{G}_{i} \exp(-\frac{t}{\tau_{i}^{\mathrm{G}}}) \tag{6}\]  
\[\mathrm{K}(t)=\mathrm{K}_{\infty}+\sum_{i=1}^{n_{\mathrm{K}}}\mathrm{K}_{i} \exp(-\frac{t}{\tau_{i}^{\mathrm{K}}}) \tag{7}\]  
In the formula, \(\mathrm{G}_{\infty},\mathrm{G}_{i}\), are the shear \(\mathrm{K}_{\infty}\) and modulus \(\mathrm{K}_{i}\), and \(\tau_{i}^{\mathrm{G}}\) and \(\tau_{i}^{\mathrm{K}}\) are the volume modulus and the relaxation time of each component. The viscoelastic constitutive model parameters [35] are obtained by referring to the relevant literature, as shown in Table 5.  
To simulate rubber-tire cord composite materials, this paper adopts the reinforcement (Rebar) element from Abaqus, where the material orientation rotates with deformation, better reflecting actual working conditions. Compared to anisotropic membrane elements, this approach can avoid computational errors under large strains. The Rebar element \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material Parameter**} & \multicolumn{3}{c}{**Neo–Hookean Model**} \\ \cline{2-4}  & **C10 (MPa)** & **D1 (MPa-1)** & \(\rho\) (g/cm\({}^{3}\)) \\ \hline green tread & 0.835 & 0.024 & 1.12 \\ Cap gum & 0.869 & 0.04 & 1.12 \\ Body glue & 1.0 & 0.02 & 1.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of rubber materials for meridian tires. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Rubber Material**} & \multicolumn{3}{c}{**Prony Series Parameters**} \\ \cline{2-4}  & \(-\)p & \(-\)p \\  & \(\mathbf{g_{1}}\) & \(\mathbf{k_{1}}\) & \(\mathbf{T_{1}}\) \\ \hline Tread, crown, body glue & 0.3 & 0 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Prony series parameters. requires defining the cross-sectional area, spacing, angle, modulus, Poisson's ratio, and density [36], as shown in Table 6.  
The tire pattern and model are shown in Figure 9.