# Grousers Effect in Tracked Vehicle Multibody Dynamics with Deformable Terrain Contact Model
## 2 Soil Mechanics  
In this section, the soil mechanics equations implemented in the multibody environment are summarized. The soil behaviour when compressed in normal direction can be described with the Bekker equation (Equation (1)), also known as the pressure-sinkage relationship [6]. \[p=\left(\frac{k_{c}}{b}+k_{\phi}\right)\cdot y^{n}=k_{\phi}\cdot y^{n}, \tag{1}\]  
with \(y\) representing the sinkage coordinate while \(k_{c}\), \(k_{\phi}\) are empirical coefficients describing the cohesion and the internal friction angle of the considered type of soil. \(n\) is the sinkage exponent and \(b\) the reference dimension of a circular or rectangular plate in contact with the terrain. A different formulation of Bekker law was proposed by [PERSON] in Reference [16]. [PERSON] et al. [11] formulated a pressure-sinkage relationship describing also the soil response during an unloading cycle or when pressure removal occurs. The soil shear stress-shear displacement relationship in the longitudinal direction of the track can be described trough the Janosi-Hanamoto law [36] (Equation (2))  
\[\tau=\tau_{max}\cdot\left(1-e^{-j/K}\right), \tag{2}\]  
where \(j\) is the shear displacement of the terrain and \(K\) is the shear deformation parameter which represents the shear displacement required for the development of the maximum shear stress. The maximum shear stress for a certain type of terrain is expressed by Mohr-Coulomb equation (Equation (3)) and depends on soil cohesion \(c\) and internal friction angle \(\phi\). \[\tau_{max}=c+\sigma_{n}\cdot\tan\phi, \tag{3}\]  
with \(\sigma_{n}\) bring the normal pressure (\(p\) in Bekker law). Values of the above mentioned empirical parameters \(k_{eq}\),\(c\),\(\phi\) and \(n\) can be found in the literature [37] for several terrain types. Also numerical methods for their real time determination have been developed [38]. These iterative algorithms allow to know some variable terrain characteristics during tracked machine motion. Focusing on tracked or wheeled vehicles on deformable terrain, the \(\phi\) values for rubber-soil metal-soil contact are very similar to the soil internal friction angle. Adhesion values between rubber and soil or metal and soil differ from cohesion values [39]. If a shearing action is exerted on the soil by a track or a wheel, shear displacement of the terrain occurs. The shear displacement \(j\) is equivalent to the relative motion between track and soil (slip). As discussed in the following sections, the traction force developed by a tracked machine coincides with the integral of Equation (2) on the track area in contact with the soil. If no slip occurs no drawbar pull is developed. As shown in Figure 1 Janosi-Hanamoto law differs from experimental results in particular for compact terrains and high normal pressure [40]. Janosi-Hanamoto law expressed in Equation (2) is monotone increasing, whereas shear stress-shear displacement curves for dense frictional-cohesive soils exhibits a maximum [41]. To model a maximum in the shear stress--shear displacement behaviour other formulations are available in the literature [31]. \[\tau=\tau_{max}K_{r}\cdot\left[1+\left(\frac{1}{K_{r}\cdot(1-e^{-1})}-1\right) \cdot e^{(1-j/K_{W})}\right]\cdot\left(1-e^{-j/K_{W}}\right), \tag{4}\]  
where \(K_{r}\) represents the ratio between the residual shear stress and its maximum admissible value while \(K_{w}\) represents the shear displacement at which the maximum shear stress occurs. Equation (2) was implemented in the MTB model developed in this paper and also in other similar MTB models available in the literature [19]. As discussed, the application of this equation can be justified for loose soils and low normal pressure values. In the present MTB model Equation (2) was  
Figure 1: Janosi-Hanamoto model and experimental results, for different types of soil. Experimental data usually exhibit a peak in the traction capabilities for certain values of the shear displacement.