# Assessment of Tire Features for Modeling Vehicle Stability in Case of Vertical Road Excitation
## 2 Investigation the Tire-Road Interaction
### Deformation Properties of the Ring  
In order to determine the deformation characteristics of the tire, the tire is divided into elements in the band consisting of the tread, breaker, cord, and protective layers attached by a flexible joint to the sides by the cord-reinforced rubber. The deformation characteristics of the individual tire elements were determined by breaking them down into monolayers made of rubber or rubber reinforced with unidirectional fiber. The elastic characteristics of the monolayer were determined according to the composite mechanic's equations [39; 40; 41; 42]:  
\[E_{\text{C1}}=E_{f}\phi_{f}+E_{m}(1-\phi_{f}),\ E_{\text{C2}}=E_{m}\left( \left(1-\phi_{f}\right)+\frac{E_{m}}{E_{f2}}\phi_{f}\right)^{-1} \tag{7}\]  
\[G_{\text{C12}}=G_{m}\left(\left(1-\phi_{f}\right)+\frac{G_{m}}{G_{f12}}\phi_{ f}\right)^{-1},\ \
u_{\text{C12}}=\varphi_{f}\
u_{f12}+\varphi_{m}\
u_{m} \tag{8}\]  
where \(E_{c}\) and \(G_{c}\)--modulus of elasticity and shear of the composite; \(E_{m}\) and \(G_{m}\)--modulus of elasticity and shear of the matrix; \(E_{f}\) and \(G_{f}\)--the modulus of elasticity and shear of the filler; \(\varphi_{f}\) and \(\varphi_{m}\)--volumetric parts of the filler and matrix; \(\
u_{C}\), \(\
u_{f}\), and \(\
u_{m}\)--[PERSON]'s coefficients of the composite, filler and matrix; indices 1, 2--the direction along the fiber and the direction transverse to the fiber, respectively.  
The properties of the cord and breaker layers in the longitudinal and transverse directions were determined using expressions of the plane stress state for the anisotropic monolayer [39; 40; 41; 42]:  
\[\left\{\begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array}\right\}=\left\{\begin{array}{ccc}\frac{1}{E_{x}}&- \frac{\
u_{xy}}{E_{y}}&\frac{\eta_{xy}}{\alpha x_{0}}\\ -\frac{\
u}{E_{x}}&\frac{1}{E_{y}}&\frac{\mu_{xy}}{\alpha x_{0}}\\ \frac{\eta_{xy}}{E_{x}}&\frac{\mu_{xy}}{E_{y}}&\frac{1}{\alpha x_{0}}\\ \end{array}\right\}\left\{\begin{array}{c}\sigma_{x}\\ \sigma_{y}\\ \tau_{xy}\end{array}\right\} \tag{9}\]  
The components of the elasticity matrix were determined according to [39; 40; 41; 42] using the axes provided in Figure 5:\[\begin{array}{c}E_{x}(\theta)=(\frac{c^{4}}{E_{1}}+\frac{s^{4}}{E_{2}}+c^{2}s^{2}( \frac{1}{c_{12}}-2\frac{c_{11}}{E_{2}}))^{-1},\\ E_{y}(\theta)=(\frac{s^{1}}{E_{1}}+\frac{c^{2}}{E_{2}}+c^{2}s^{2}(\frac{1}{c_{ 12}}-2\frac{c_{12}}{E_{2}}))^{-1},\\ G_{xy}(\theta)=(4c^{2}s^{2}(\frac{1}{E_{1}}+\frac{1}{E_{2}}+2\frac{c_{12}}{E_{ 2}})+\frac{(c-s^{2})^{2}}{G_{12}})^{-1},\\ \frac{v_{xy}}{E_{y}}(\theta)=\frac{v_{xy}}{E_{2}}(c^{4}-s^{4})-c^{2}s^{2}( \frac{1}{E_{1}}+\frac{1}{E_{2}}-\frac{1}{G_{12}}),\\ \frac{v_{xy}}{E_{y}}(\theta)=2 cs\{\frac{s^{2}}{E_{2}}+\frac{c^{2}}{E_{1}}-(c^{ 2}-s^{2})(\frac{1}{E_{2}}-\frac{1}{G_{12}})\},\\ \frac{v_{xy}}{E_{xy}}(\theta)=2 cs\{\frac{s^{2}}{E_{2}}+\frac{c^{2}}{E_{1}}+(c^ {2}-s^{2})(\frac{1}{E_{2}}-\frac{1}{G_{12}})\},\\ \frac{v_{xy}}{E_{xy}}(\theta)=2 cs\{\frac{s^{2}}{E_{2}}+\frac{c^{2}}{E_{1}}+(c^ {2}-s^{2})(\frac{1}{E_{2}}-\frac{1}{G_{12}})\},\\ \eta_{x}=\frac{\eta_{xy}E_{x}}{G_{xy}},\;\mu_{y}=\frac{\mu_{yy}E_{y}}{G_{xy}} \end{array} \tag{10}\]  
where \(c=\cos\theta\) and \(s=\sin\theta;x\)-direction forming an angle \(\theta\) with the reinforcement direction (Figure 5).  
The modulus of elasticity of the components and the Poisson's coefficients were taken from the works [36; 43]. The methodology used for this purpose is presented in [39; 40; 41; 42]. Direct modulus of elasticity calculations does not provide an answer in relation to the characteristics of the tread and unreinforced ring section that are required for the model provided in Figure 4. To determine the stiffness of the tread \(C_{z}\), tread deformations are analyzed additionally. The obtained data are presented in Table 2.