# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction
### Investigation of Tread Deformation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multicolumn{5}{c}{**Tire**} \\ \cline{2-6}  & & **A** & **B** & **C** & **D** \\ \hline \multirow{4}{*}{Breaker (with nylon)} & \(E_{l}\) & 1667\(\cdot\)10\({}^{3}\) & 1667\(\cdot\)10\({}^{3}\) & — & 1367\(\cdot\)10\({}^{3}\) \\ \cline{2-6}  & \(E_{t}\) & 26,736 & 26.736 & — & 24,568 \\ \cline{2-6}  & \(E_{r}\) & 26,736 & 26.736 & — & 24,568 \\ \hline \multirow{4}{*}{Breaker (with steel)} & \(E_{l}\) & 73,119 & 73,119 & 73,119 & 73,119 \\ \cline{2-6}  & \(E_{t}\) & 21,737 & 21,737 & 21,737 & 21,737 \\ \cline{2-6}  & \(E_{r}\) & 23,684 & 23,684 & 23,684 & 23,684 \\ \hline \multirow{4}{*}{Cord (with viscose)} & \(E_{l}\) & 24,625 & 24,625 & 26,819 & — \\ \cline{2-6}  & \(E_{t}\) & 3553\(\cdot\)10\({}^{3}\) & 3553\(\cdot\)10\({}^{3}\) & 4339\(\cdot\)10\({}^{3}\) & — \\ \cline{2-6}  & \(E_{r}\) & 24,625 & 24,625 & 26,819 & — \\ \hline Inner sealing layer \(E\) & 18 & 18 & 18 & 18 & 18 \\ \hline Part of the ring up to the cord \(E\) & 16,601 & 16,630 & 17,118 & 16,511 \\ \hline Whole ring \(E\) & 16,890 & 16,906 & 17,311 & 16,541 \\ \hline Sidewall elasticity modulus during radial tension \(E\), & 507.5 & 531.15 & 788.1 & 551.7 \\ \hline Equivalent sidewall elasticity modulus during radial bending \(E\) & flat band, & 23.32 & 24.37 & 29.77 & 67.89 \\ \cline{2-6}  & ring. & 42.69 & 43.04 & 61.51 & 113.4 \\ \hline Equivalent modulus of elasticity of the band bending \(E_{c}\) & 15.89 & 17.74 & 19.63 & 15.92 \\ \cline{2-6}  & perpendicular to the circular direction, & 15.35 & 17.37 & 16.34 & 15.90 \\ \cline{2-6}  & in the circular direction. & 23.73 & 24.21 & 22.50 & 18.50 \\ \hline Relative position of the neutral layer \(e/\sum\delta\) at bending & & & & \\ flat band in the circular direction, & 0.288 & 0.281 & 0.383 & 0.377 \\ \cline{2-6}  & perpendicular to the circular direction ring, & 0.108 & 0.11 & 0.088 & 0.124 \\ \cline{2-6}  & in the circular direction. & 0.286 & 0.277 & 0.376 & 0.372 \\ \hline \hline \end{tabular} Note: \(E\)—in all directions, \(E_{t}\)—the rolling direction of the tire, \(E_{t}\)—perpendicular to the rolling direction of the tire, and \(E_{r}\)—in the radial direction (in relation to the tire).  
\end{table} 
Table 2: Modulus of elasticity of the tire elements, MPa. Figure 6: Summary experimental data: 1—experimental data; 2—generalized line. Whereas the tread consists of the parts with and without a pattern and the deformation properties of the tread are evaluated in the model as a single generalized layer, the possibilities of assessment of the tread part with a pattern were examined. The simplest model assesses the filling of the part with a pattern (Figure 7), which can be executed using the expression of porous composites [44]:  
\[E=k_{u}E_{g} \tag{11}\]  
where \(k_{u}\)--tread filling factor, where \(k_{u}=A_{g}/A_{p}\), \(A_{i}\)--the area of the continuous pattern in the part of the tread part selected according to the repetition of the pattern, \(A_{p}\)--area of the selected part of the tread; and \(E_{g}\)--tread rubber modulus of elasticity. Since the tread pattern is different across differed parts of the tread (Figure 7), the filling factor for each tread band was calculated separately. In this case, the equivalent modulus of elasticity of the tread was equal to:  
\[E_{1}=E_{g}\frac{\Sigma k_{ui}k_{bi}}{1-k_{a}(1-\Sigma k_{ui}k_{bi})} \tag{12}\]  
where \(E_{g}\)--tread rubber modulus of elasticity; \(k_{a}=\delta_{1a}/\delta_{1}\); \(\delta_{1}=\delta_{1a}+\delta_{1b}\); \(k_{bi}\)--share of the width of the \(i\)-th band in the width of the tread, \(\delta_{1}\)--tire tread thickness, \(\delta_{1a}\)--tread part with a pattern, and \(\delta_{1b}\)--continuous part of the tread.